text
stringlengths 6
128k
|
|---|
# Entity-aware Transformers for Entity Search
Emma J. Gerritse Radboud University<EMAIL_ADDRESS>, Faegheh Hasibi
Radboud University<EMAIL_ADDRESS>and Arjen P. de Vries Radboud
University<EMAIL_ADDRESS>
(2022)
###### Abstract.
Pre-trained language models such as BERT have been a key ingredient to achieve
state-of-the-art results on a variety of tasks in natural language processing
and, more recently, also in information retrieval. Recent research even claims
that BERT is able to capture factual knowledge about entity relations and
properties, the information that is commonly obtained from knowledge graphs.
This paper investigates the following question: Do BERT-based entity retrieval
models benefit from additional entity information stored in knowledge graphs?
To address this research question, we map entity embeddings into the same
input space as a pre-trained BERT model and inject these entity embeddings
into the BERT model. This entity-enriched language model is then employed on
the entity retrieval task. We show that the entity-enriched BERT model
improves effectiveness on entity-oriented queries over a regular BERT model,
establishing a new state-of-the-art result for the entity retrieval task, with
substantial improvements for complex natural language queries and queries
requesting a list of entities with a certain property. Additionally, we show
that the entity information provided by our entity-enriched model particularly
helps queries related to less popular entities. Last, we observe empirically
that the entity-enriched BERT models enable fine-tuning on limited training
data, which otherwise would not be feasible due to the known instabilities of
BERT in few-sample fine-tuning, thereby contributing to data-efficient
training of BERT for entity search.
Entity retrieval, transformers, BERT, entity embeddings
††journalyear: 2022††copyright: rightsretained††conference: Proceedings of the
45th International ACM SIGIR Conference on Research and Development in
Information Retrieval; July 11–15, 2022; Madrid, Spain††booktitle: Proceedings
of the 45th International ACM SIGIR Conference on Research and Development in
Information Retrieval (SIGIR ’22), July 11–15, 2022, Madrid, Spain††doi:
10.1145/3477495.3531971††isbn: 978-1-4503-8732-3/22/07††ccs: Information
systems Language models
## 1\. introduction
Pre-trained language models (LMs) such as BERT (Devlin et al., 2019) and its
successors (Raffel et al., 2020; Yang et al., 2019; Liu et al., 2019) learn
rich contextual information about words from large-scale unstructured corpora
and have achieved intriguing results on a variety of downstream tasks in
natural language processing (NLP) (Devlin et al., 2019; Joshi et al., 2019;
Joshi et al., 2020) and information retrieval (IR) (Nogueira and Cho, 2019;
Wang et al., 2019b). It is even shown that these language models have the
capability of capturing a tremendous amount of world knowledge, including
information about real-world entities otherwise found in knowledge graphs
(KGs) (Petroni et al., 2019; Wang et al., 2020). For example, language models
can predict masked objects in cloze sentences such as “The native language of
Mammootty is _____” and “The Sharon Cuneta Show was created in _____”, where
each of them demonstrates a $\langle$subject, relation, object$\rangle$ triple
in a knowledge graph.
Language models, however, fail to perform complex reasoning about entities, as
they cannot capture sparse world facts (Talmor et al., 2020; Petroni et al.,
2019; Jiang et al., 2020). In information retrieval systems, users ask complex
queries about entities, such as “What is the second highest mountain in the
world?” or “Most famous civic-military airports.” Answering such queries
requires leveraging rich human curated information about entities in the
knowledge graphs. To bridge the gap between LMs and KGs, recent works enhance
language models with rich structured entity information from knowledge graphs,
showing superior performance for knowledge-driven NLP tasks such as relation
classification, entity typing, and cloze-style question answering (Zhang et
al., 2019; Yamada et al., 2020b; Poerner et al., 2020; Jiang et al., 2020;
Wang et al., 2019a; Wang et al., 2020; Peters et al., 2019). Despite this
success, no previous study has examined the effect of entity-enriched language
models for answering entity-oriented queries in IR.
This work explores the benefit of enriching BERT-based retrieval models with
auxiliary entity information from KGs for the _entity retrieval_ task, where
users’ queries are better answered with entities rather than passages or
documents (Balog, 2018). It is shown that entity information improves
performance of document and entity retrieval tasks (Xiong et al., 2017b, a;
Dalton et al., 2014; Hasibi et al., 2016; Garigliotti et al., 2019). Yet,
these studies were performed using traditional retrieval models, without
utilizing LMs such as BERT.
The development of BERT-based models for entity retrieval faces a major
challenge: there is limited training data for the entity retrieval task, and
fine-tuning of BERT in data-constraint regimes leads to instabilities in model
performance (Dodge et al., 2020; Phang et al., 2019; Zhang et al., 2021). More
specifically, the official entity retrieval dataset, DBpedia-Entity v2 (Hasibi
et al., 2017c), contains 467 queries; considered a small dataset for fine-
tuning of a large neural language model. When fine-tuning BERT (especially its
large variant) on a target task, the training data should be large enough that
the model parameters get close to the ideal setting of the target task, or
otherwise it causes forgetting of what the model has already learned.
Against this background, our first and primary research question is _RQ1: Can
an entity-enriched BERT-based retrieval model improve the performance of
entity retrieval?_ To address this question, we propose an entity-enriched
BERT-based retrieval model, EM-BERT, where factual entity information is
injected into the monoBERT model (Nogueira and Cho, 2019) in the form of
Wikipedia2Vec (Yamada et al., 2020a) entity embeddings that are transformed
into the BERT’s word piece vector space (Poerner et al., 2020); see Figure 1.
The EM-BERT model is first trained with the MS MARCO passage dataset and
further fine-tuned on the DBpedia-Entity v2 collection (in a cross validation
setup). Our experiments indicate that our EM-BERT model improves state-of-the-
art entity retrieval results by 11% with respect to NDCG@10. We also make the
intriguing finding that while fine-tuning of the plain BERT-based retrieval
model (monoBERT) on DBpedia-Entity v2 is brittle and prone to degenerate
performance, the same fine-tuning process using the entity-enriched BERT model
(EM-BERT) results in a stable model and brings consistent and meaningful
improvements. We posit that this is attributed to direct injection of entity
information into the BERT model, which brings the input distribution of the
model close to the output label space, thereby contributing to data-efficient
training of BERT-based models for entity-oriented tasks.
After observing the effectiveness of our EM-BERT model, we focus on
understanding the whens and whys of entity-enriched BERT models for
information retrieval tasks. Our next research question is: _RQ2: When and
which queries are helped by the EM-BERT model?_ We observe that EM-BERT mainly
helps queries that are annotated with at least one entity; plain and entity-
enriched BERT models perform on par with each other for queries without linked
entities. We further examine queries with linked entities and find that EM-
BERT is most helpful when entity mentions are broken into multiple word pieces
by the BERT tokenizer (e.g. mention “Tuvalu” in Figure 1). This indicates that
directly injecting entity information into the BERT-based models is
particularly important for less popular entities, which BERT does not
recognize as a single word (or phrase) because they are less observed during
pre-training.
In our third question, we investigate _RQ3: Why does the EM-BERT model work
and what does it learn during the fine-tuning stage?_ We approach this by
comparing the embeddings produced at the final layer of the EM-BERT network
for both entities and their mention tokens. We find that the learned entity
representations, unlike their corresponding word piece representations, form
clear clusters in the embedding space, capturing the relation between
entities. We also study some examples, and show that attention weights of
entity tokens are larger than other tokens for queries requesting lists of
entities or queries related to less known entities.
Finally, we question _RQ4: How does our entity-enriched BERT-based model
perform on other ranking tasks?_ To address this research question, we apply
the EM-BERT model to the passage ranking task, where entity information is
known to be less important and a large amount of training data can be used. We
observe that the entity information that is embedded in the plain BERT-based
retrieval model is enough for addressing this task and auxiliary entity
information from knowledge graphs does not bring significant improvements.
In summary, this work makes the following contributions:
* •
We study the effect of enriching BERT with entity information for information
retrieval tasks and propose an entity-enhanced BERT-based re-ranker EM-BERT.
* •
We establish new state-of-the-art results for the entity retrieval task on
DBpedia-Entity v2 collection.
* •
We show that our entity-enriched model, unlike its equivalent model based on
plain BERT, is robust against instabilities of BERT in data-constraint regimes
and introduce EM-BERT as a data-efficient BERT-based model for entity
retrieval.
* •
We perform thorough analysis of our EM-BERT model and add to our understanding
of the model. We unfold when and for which queries this model work, what it
learns, and how it works.
The resources developed within the course of this paper are available at
https://github.com/informagi/EMBERT.
## 2\. Related work
Figure 1. Illustration of the EM-BERT model. Entity annotated query and
documents are tokenized and mapped to their corresponding vector
representations using $F_{BERT}$ and $F_{E-BERT}$ functions.
[]
#### Graph Embeddings
After the introduction of word embeddings methods like Word2Vec (Mikolov et
al., 2013), neural embedding methods became increasingly popular. Word2Vec
uses a single-layer neural network to either predict a word depending on its
context (Continuous Bag of Word) or the context depending on a word (Skip-
Gram). The resulting hidden layer of this network captures a representation of
words and can be used for other downstream tasks. After the success of these,
embedding methods were expanded to capture other types of knowledge as well,
for example knowledge graphs. These so-called graph embeddings aim to embed
the nodes of a knowledge graph in a way that entities and their relations are
mapped accordingly into the vector space. One of the earliest introduced
methods was TransE (Bordes et al., 2013), which optimizes embeddings for head-
relation-tail triples.
The graph embedding method utilized in this paper is Wikipedia2Vec (Yamada et
al., 2020a). This method is similar to Word2Vec, but is extended to be used on
text-heavy knowledge bases like Wikipedia. Instead of solely relying on words
or edges between entities, it uses a combination of words, entities, and
anchor texts. Using either Skip-Gram or Continuous Bag of Words method like in
Word2Vec, it employs a single layer neural network to predict neighboring
words to capture word similarities, entities relations (based on neighboring
entities), and word-entity relations (based on surrounding anchor-text).
Through this combination, words and entities get embedded into the same vector
space.
#### Entity Retrieval
Graph embeddings have been used to assist in _entity-oriented information
retrieval_. Gerritse et al. (2020) uses Wikipedia2Vec embeddings for entity
retrieval on the DBpedia-Entity v2 collection (Hasibi et al., 2017c). This is
done by first tagging entities in the queries by using the TAGME entity linker
(Ferragina and Scaiella, 2010). Retrieved entities are then re-ranked by
computing the cosine similarity score between the retrieved entities and the
tagged entities, improving upon the state of the art for entity retrieval.
KEWER (Nikolaev and Kotov, 2020) is another method for entity retrieval based
on graph embeddings. It learns joint embeddings for both words and entities,
combining this with entities annotated queries using SMAPH (Cornolti et al.,
2016). KEWER also improved previous state of the art for entity retrieval. The
most recent graph-based entity ranking method, ENT Rank (Dietz, 2019),
utilizes entity-neighbor-text relations in a learning-to-rank model. ENT Rank
incorporates a variety of entity features based on text, co-occurring
entities, and entity neighbor relations, and performs competitively on entity
ranking tasks. The paper, however, uses a different version of the DBpedia-
Entity v2 collection and calculates the evaluation metrics only over judged
entities, making it incomparable to our and the aforementioned entity
retrieval methods.
The BEIR (Thakur et al., 2021) benchmark provides a number of zero-shot
retrieval models for several IR databases, including the DBpedia-Entity v2
entity retrieval collection. The best performing entity retrieval model in
BIER, BM25 + CE, re-ranks the top 100 entities retrieved by BM25, and then
uses the MiniLM cross-encoder model (Wang et al., 2020), trained with the
knowledge distillation setup provided by Hofstätter et al. (2020).
#### Entity Linking
Utilizing entity information for retrieval models requires identifying entity
information from documents and queries (Hasibi et al., 2017a; Blanco et al.,
2015; Hasibi et al., 2015). This is performed by annotating text with entity
linking toolkits (Ferragina and Scaiella, 2010; Cornolti et al., 2016; van
Hulst et al., 2020; Hasibi et al., 2017b; Cao et al., 2021). In this paper, we
use the REL entity linker (van Hulst et al., 2020), which is the state-of-the-
art open source entity linking toolkit. REL detects mentions using Flair
(Akbik et al., 2018) embeddings. REL performs candidate selection based on
Wikipedia2Vec embeddings, and entity disambiguation based on latent relations
between entity mentions in the text.
#### Transformer-based Rankers
Bidirectional Encoder Representations from Transformers (BERT) (Devlin et al.,
2019) is a language representation model, pre-trained to capture bi-
directional relations between words. The pre-trained BERT model can be fine-
tuned for various NLP tasks. BERT can be used for binary relevance
classification by feeding it a sentence A and sentence B, and using the BERT
classifier token (further denoted as [CLS]) to predict relevance. This setup
has been shown to be effective for question answering.
BERT-based models are effective for _information retrieval_ as well, and due
to their relatively low number of input tokens became especially popular for
_passage retrieval_. Several BERT-based methods have made enormous
improvements on the previous state of the art (Wang et al., 2019b; Nogueira
and Cho, 2019). MonoBERT (Nogueira and Cho, 2019) is a point-wise re-ranker,
in which the BERT model is used as a binary relevance classifier. MonoBERT, at
the time of release, obtained state-of-the-art results on the MS MARCO and
TREC CAR datasets. After pre-training, the ranking is done by first making an
initial ranking with a baseline method like BM25, and then re-ranking a query
and passage by feeding a pair of sentences A (query) and B (passage) to BERT.
#### Entities and Transformers
During pre-training, BERT does not explicitly get structured data about
entities. Some researchers have claimed that adding this may not be necessary
since BERT captures this information implicitly (Wang et al., 2020; Petroni et
al., 2019). Others set out to enrich BERT with entity information. One of the
first works combining entities with transformers, ERNIE (Zhang et al., 2019),
is an enhanced language representation model, where the combination of
entities and text is used to fine-tune the model. In KnowBert (Peters et al.,
2019), contextual word embeddings are also enhanced by integrating information
about entities. This is done by introducing a Knowledge Attention and
Recontextualization component, which takes information in one layer of a BERT
network, computes the enhanced entity information of that layer, and feeds it
to the next layer. This method improves on BERT for entity-related tasks like
entity linking.
E-BERT (Poerner et al., 2020) is another method to enhance BERT with entity
information. The main point which distinguishes E-BERT from the previously
mentioned work is that it does not require any additional training of the
network; it only requires the computation of one large transformation matrix.
This allows the method to be applied to any fine-tuned transformer model.
Since Wikipedia2Vec embeds both words and entities in the same vector space,
the embeddings of words in Wikipedia2Vec can be used to align a mapping.
E-BERT has been shown to improve performance on unsupervised QA tasks like
LAMA, as well as downstream tasks like relation classification.
Other works combining language models and knowledge graphs include (Broscheit,
2019), which enhances BERT with entities by using BERT itself for entity
linking and seeing how much information of knowledge graphs is already
contained in BERT. Next, BERT classifies texts to predict whether certain
tokens belong to an entity. The authors show that specifically fine-tuning
BERT on entity information greatly improves their scores on entity linking,
thus showing that pre-trained BERT does not embed all information about
entities yet, claiming that additional entity information does not help if
either entities are too scarce in the data or if the task does not require
entity knowledge.
## 3\. Method
In this section, we first provide a brief overview of knowledge graph
embeddings and Wikipedia2Vec (Section 3.1), and then describe the EM-BERT
model, which is an entity-enhanced version of a BERT-based retrieval model
(Section 3.2), followed by a description of EM-BERT query-document input
representation.
### 3.1. Background
Knowledge graph embeddings provide vector representations of entities in a
knowledge graph, projecting entity properties and relations into a continuous
vector space. These embeddings have proven effective not only for entity-
related tasks such as entity linking (Yamada et al., 2016), but also for
general tasks such as question answering (Nikolaev and Kotov, 2020). Knowledge
graph embeddings can be constructed purely based on graph topology (i.e.,
entities and their relations) (Bordes et al., 2013), or combining graph
topology with additional textual information such as the entity descriptions
(Yamada et al., 2016). In this paper, we use Wikipedia2Vec embeddings (Yamada
et al., 2020a), which extends the skip-gram model (Mikolov et al., 2013) with
entity relations and descriptions from Wikipedia and maps words and entities
into a shared vector space. Following (Poerner et al., 2020), we transform
Wikipedia2Vec entity embeddings into the BERT word piece embedding space and
use them as input to the BERT model, similar to a BERT word piece vector.
Table 1. Statistics about document length and number of linked entities in queries and documents of the used datasets, with mean and standard deviation. Document and query length information is computed without adding entity information. Dataset | size | avg length | avg entities | total linked
---|---|---|---|---
MS MARCO dev queries | 6980 | $6.95\pm 2.54$ | $0.34\pm 0.57$ | 2392
MS MARCO passages | 8841823 | $58.27\pm 22.94$ | $2.55\pm 3.22$ | 22532200
DBpedia-Entity v2 queries | 467 | $6.55\pm 2.88$ | $0.82\pm 0.64$ | 381
DBpedia-Entity v2 short abstracts | 389622 | $55.93\pm 25.95$ | $6.88\pm 3.93$ | 2678843
Formally, Wikipedia2Vec embeds a list of words $\mathbb{L}_{word}$ and a list
of entities $\mathbb{L}_{ent}$ into a vector space
$\mathbb{R}^{d_{\mathit{Wikipedia}}}$, where $d_{\mathit{Wikipedia}}$ is
dimensions of the Wikipedia embeddings. The embeddings are trained by fitting
a one layer neural network with the loss:
$\mathcal{L}=\mathcal{L}_{w}+\mathcal{L}_{e}+\mathcal{L}_{a},$
which consist of three cross-entropy loss functions:
* $\mathcal{L}_{w}$:
predicting context words of a given word.
* $\mathcal{L}_{e}$:
predicting neighboring entities of a given entity.
* $\mathcal{L}_{a}$:
predicting anchor text of an entity to combine words and entities in a single
embedding space.
The weight layer of this neural network results in the function
$F_{\mathit{Wikipedia}}\
:\mathbb{L}_{word}\cup\mathbb{L}_{ent}\rightarrow\mathbb{R}^{d_{\mathit{Wikipedia}}}$,
which embeds both words and entities in the Wikipedia vector space. BERT, on
the other hand, tokenizes text into the word piece dictionary
$\mathbb{L}_{wordpiece}$. We use the lookup function
$F_{BERT}:\mathbb{L}_{wordpiece}\rightarrow\mathbb{R}^{d_{BERT}}$ to transform
word pieces into the $d_{BERT}$-dimensional native BERT word piece vector
space, the resulting vector being the input for the BERT model.
### 3.2. EM-BERT
Our proposed model, referred to as EM-BERT, incorporates entity embeddings
into a point-wise document ranking approach. In its essence, EM-BERT combines
E-BERT (Poerner et al., 2020) with monoBERT (Nogueira and Cho, 2019) and
predicts a retrieval score for a query-document pair, each represented by a
sequence of words and entities. The model takes as input the concatenation of
the query tokens $t_{q_{1}},...,t_{q_{n}}$ and the document tokens
$t_{d_{1}},...,t_{d_{m}}$, where each token $t_{i}$ is either a BERT native
word piece token or an entity token.
#### Entity-enriched BERT
The off-the-shelf BERT model only accepts its native word pieces. To
incorporate entity embeddings into BERT, we need to align entity vectors with
word piece vectors. Following (Poerner et al., 2020), this is done by a linear
transformation of entity vectors to BERT-like vectors. Since
$\mathbb{L}_{wordpiece}$ does not contain any entities, the Wikipedia2Vec word
dictionary $\mathbb{L}_{Word}$ is used to obtain the linear transformation
$\mathbf{W}\in\mathbb{R}^{d_{BERT}\times d_{\mathit{Wikipedia}}}$, learned
from the Wikipedia2Vec word vector space
$F_{\mathit{Wikipedia}}[\mathbb{L}_{Word}]$ and BERT word piece space
$F_{BERT}[\mathbb{L}_{wordpiece}]$:
(1)
$\mathbf{W}=argmin_{W}\sum_{\mathclap{x\in\mathbb{L}_{word}\cap\mathbb{L}_{wordpiece}}}||W\cdot
F_{\mathit{Wikipedia}}(x)-F_{BERT}(x)||^{2}_{2}.$
Here, the intersection between $\mathbb{L}_{word}$ and
$\mathbb{L}_{wordpiece}$ is taken to ensure that the selected words have
embeddings in both Wikipedia and BERT vector spaces. The equation computes
matrix $\mathbf{W}$, in which the distance between $\mathbf{W}\cdot
F_{\mathit{Wikipedia}}(x)$ and $F_{BERT}(x)$ is minimal for all $x$. Using
$\mathbf{W}$, we can then construct the function $F_{E-BERT}$, which maps both
entities and word tokens to the BERT input vector space:
(2) $F_{E-BERT}(a)=\begin{cases}\mathbf{W}\cdot
F_{\mathit{Wikipedia}}(a),&a\in\mathbb{L}_{ent}\\\
F_{\mathit{wordpiece}}(a),&\text{otherwise}.\end{cases}$
The aligned entity vectors are then fed into BERT when an entity is mentioned
in the input text. The entities are obtained by annotating the text with an
entity; e.g., given the text “Who produced films starring Natalie Portman”,
the annotated sentence becomes “Who produced films starring Natalie Portman
ENTITY/NataliePortman”. This text is then tokenized as “who produced films
starring natalie port ##man /ENTITY/Natalie_Portman,” where the entity ID (in
bold face) is a single token, embedded by $F_{E-BERT}$ (Eq. 2). We note that
the transformation matrix $\mathbf{W}$ can be fit into any pre-trained or
fine-tuned BERT model, thus making E-BERT embeddings usable for every
available BERT model.
#### Retrieval
The retrieval phase is based on monoBERT (Nogueira et al., 2019), which is a
multi-stage ranking method with BERT and has been shown to achieve competitive
results on MS MARCO passage retrieval (Nguyen et al., 2016) and TREC-CAsT
(Dalton et al., 2020). The simplicity of this model, coupled with its high
effectiveness enables us to discern the effect of entity information on BERT-
based retrieval models and obtain an understanding of the whys and whens of
entity-enriched BERT models for information retrieval tasks.
In the multi-stage ranking method, for a query $q$, documents are ranked with
an initial ranking method; e.g., BM25. The top-$k$ documents, denoted as
$D_{K}=\\{d_{1},d_{2},\dots,d_{k}\\}$, are then passed to a second ranker for
re-ranking; here a BERT-based retrieval model. Every query-document pair
$\langle q,d_{i}\in D_{K}\rangle$ is passed to BERT as two sentences $A$ and
$B$, with the separation token [SEP] in between. The BERT [CLS] classification
vector is used as input to a single layer neural network to obtain the
probability $s_{i}$ of the document $d_{i}$ being relevant to the query $q$.
To train this model, the cross-entropy loss is used:
(3) $L=-\sum\limits_{j\in J_{pos}}log(s_{j})-\sum\limits_{j\in
J_{neg}}log(1-s_{j}),$
where $J_{pos}$ and $J_{neg}$ are indexes of relevant and non-relevant
documents for all queries.
Putting all the pieces together, the training process of the EM-BERT model is
summarized as follows. First, the transformation matrix $\mathbf{W}$ is
trained using Wikipedia2Vec embeddings and BERT word pieces. All queries and
documents are annotated with an entity linker, and the tagged queries and
documents are tokenized and mapped to the corresponding vector representations
using function $F_{E-BERT}$ . This input is fed into the EM-BERT model and the
model is first fine-tuned on MS MARCO passages and then on the entity
retrieval collection.
#### Query-Document Representation
In the EM-BERT model (similar to monoBERT), queries are truncated to have a
maximum token length of 64, and the combination of the query, passage, and
separation tokens are truncated to have a maximum length of 512 tokens.
Following (Poerner et al., 2020), we concatenate mentions and entities,
separated by the token ‘/’; i.e., “mention + ‘/’ + EntityID.” The final
representation of query-document pairs is “[CLS] + Query + [SEP] + Document +
[SEP];” see Figure 1.
## 4\. Experimental setup
Table 2. Results on DBpedia-Entity v2 collection. Superscripts 1/2/3/4 denote statistically significant differences (better or worse) compared to base method/monoBERT (1st)/monoBERT/EM-BERT (1st), respectively. The highest value per column is marked with underline and in bold. The highest value per column and per block is marked with underline. NDCG | SemSearch | INEX-LD | ListSearch | QALD-2 | Total
---|---|---|---|---|---
@10 | @100 | @10 | @100 | @10 | @100 | @10 | @100 | @10 | @100
ESIMcg (Gerritse et al., 2020) | 0.417 | 0.478 | 0.217 | 0.286 | 0.211 | 0.302 | 0.212 | 0.282 | 0.262 | 0.335
KEWER (Nikolaev and Kotov, 2020) | - | - | - | - | - | - | - | - | 0.270 | 0.310
BLP-TransE (Daza et al., 2021) | 0.631 | 0.723 | 0.446 | 0.546 | 0.442 | 0.540 | 0.401 | 0.482 | 0.472 | 0.562
BM25F+KEWER (Nikolaev and Kotov, 2020) | 0.661 | 0.733 | 0.468 | 0.530 | 0.440 | 0.521 | 0.386 | 0.474 | 0.483 | 0.560
BM25 | 0.425 | 0.523 | 0.298 | 0.330 | 0.274 | 0.322 | 0.192 | 0.243 | 0.291 | 0.349
+monoBERT (1st) | $0.588^{1}$ | $0.655^{1}$ | $0.406^{1}$ | $\underline{0.459}^{1}$ | $0.422^{1}$ | $0.458^{1}$ | $0.350^{1}$ | $0.406^{1}$ | $0.437^{1}$ | $0.490^{1}$
+monoBERT | $0.577^{1}$ | $0.640^{12}$ | $\underline{0.409}^{1}$ | $0.452^{1}$ | $0.423^{1}$ | $0.449^{12}$ | $0.347^{1}$ | $0.392^{12}$ | $0.435^{1}$ | $0.479^{12}$
+EM-BERT (1st) | $0.593^{1}$ | $0.667^{13}$ | $0.395^{1}$ | $0.448^{1}$ | $0.436^{1}$ | $0.465^{1}$ | $0.351^{1}$ | $0.403^{1}$ | $0.440^{1}$ | $0.492^{13}$
+EM-BERT | $\underline{0.612}^{1}$ | $\underline{0.672}^{1}$ | $0.392^{1}$ | $0.434^{12}$ | $\underline{0.478}^{1234}$ | $\underline{0.469}^{13}$ | $\underline{0.375}^{1234}$ | $\underline{0.418}^{134}$ | $\underline{0.461}^{1234}$ | $\underline{0.495}^{13}$
BM25F-CA (Hasibi et al., 2017c) | $\underline{0.628}$ | $\underline{0.720}$ | $\underline{0.439}$ | $\underline{0.530}$ | $0.425$ | $0.511$ | $0.369$ | $0.461$ | $0.461$ | $0.551$
+monoBERT (1st) | $0.605$ | $0.682^{1}$ | $0.427$ | $0.501$ | $0.440$ | $0.510$ | $0.405$ | $0.490$ | $0.467$ | $0.544$
+monoBERT | $0.591^{2}$ | $0.663^{12}$ | $0.434$ | $0.489^{12}$ | $0.438$ | $0.496^{2}$ | $0.401$ | $0.467^{2}$ | $0.463$ | $0.526^{12}$
+EM-BERT (1st) | $0.609$ | $0.695^{3}$ | $0.418$ | $0.490^{1}$ | $0.452$ | $0.518^{3}$ | $0.398$ | $0.483$ | $0.466$ | $0.544^{3}$
+EM-BERT | $0.621$ | $0.695$ | $0.399$ | $0.469^{12}$ | $\underline{0.492}^{1234}$ | $\underline{0.535}^{23}$ | $\underline{0.435}^{1234}$ | $\underline{0.512}^{1234}$ | $\underline{0.486}^{1234}$ | $\underline{0.553}^{3}$
GEEER (Gerritse et al., 2020) | $0.660$ | $0.736$ | $0.466$ | $0.552$ | $0.452$ | $0.535$ | $0.390$ | $0.483$ | $0.487$ | $0.572$
+monoBERT (1st) | $0.643$ | $0.733$ | $\mathbf{\underline{0.486}}$ | $\mathbf{\underline{0.564}}$ | $0.493^{1}$ | $0.554^{1}$ | $0.449^{1}$ | $0.525^{1}$ | $0.515^{1}$ | $0.591^{1}$
+monoBERT | $0.633^{2}$ | $0.725^{2}$ | $0.481$ | $0.563$ | $0.492^{1}$ | $0.553^{1}$ | $0.439^{12}$ | $0.519^{12}$ | $0.508^{12}$ | $0.586^{12}$
+EM-BERT (1st) | ${{0.650}}$ | $0.743$ | $0.483$ | $0.561$ | $0.504^{1}$ | $0.562^{1}$ | $0.444^{1}$ | $0.522^{1}$ | $0.517^{13}$ | $0.594^{13}$
+EM-BERT | $\mathbf{\underline{{0.664}}}$ | $\mathbf{\underline{0.744}}$ | $0.479$ | $0.561$ | $\mathbf{\underline{0.544}}^{1234}$ | $\mathbf{\underline{0.579}}^{1234}$ | $\mathbf{\underline{0.483}}^{1234}$ | $\mathbf{\underline{0.543}}^{1234}$ | $\mathbf{\underline{0.541}}^{1234}$ | $\mathbf{\underline{0.604}}^{1234}$
To address our research question, we compare the EM-BERT model with several
baselines on the entity retrieval task and analyze the results. Due to limited
training data for entity retrieval, we perform two-stage fine-tuning: we first
fine-tune our models on the annotated MS MARCO passage dataset, and then
continue fine-tuning on DBpedia-Entity v2 collection. In the following, we
describe our experimental setup.
### 4.1. Entity Linking and Embeddings
We use the REL entity linking toolkit (van Hulst et al., 2020) to annotate
texts with entities. REL is known to have high precision compared to other
entity linking toolkits such as GENRE (Cao et al., 2021) and TAGME (Ferragina
and Scaiella, 2010), ensuring that limited noise (incorrectly annotated
entities) is added to the model. REL incorporates Wikipedia2Vec embeddings
trained on the Wikipedia link graph. To avoid a mismatch between annotated
entities and Wikipedia2Vec entity embeddings (Gerritse et al., 2020; Dietz,
2019), we follow the REL Wikipedia2Vec training procedure, using the Wikipedia
dump of 2019-07-01, setting min-entity-count parameter to zero, with a
dimensionality of 500.
### 4.2. First Stage Fine-tuning
#### Collection
For the first stage fine-tuning of EM-BERT, we use the MS MARCO passage
ranking collection, consisting of 8,841,823 passages, extracted from web
documents retrieved by Bing. MS MARCO provides a training set with
approximately 800k queries, a development set with 6980 queries with public
relevance judgments, and an evaluation set with 6837 queries and private
relevance judgments.
Although MS MARCO passage ranking is not an entity-related task, REL annotates
29% of queries in the development sets with at least one entity; e.g., “what
were the _brothers grimm_ names.” The statistics of this collection (with REL
annotations) are reported in Table 1. It shows that on average every query and
document is annotated with 0.34 and 2.55 entities respectively, indicating
that a fair amount of entities is present in this dataset.
#### Training
We fine-tune our EM-BERT model on the officially provided MS MARCO training
set. This set provides triples of a query, a positive example, and a negative
example, making the ratio of positive-negative documents equal during
training. We refer to these triples as “sample.” Starting from the monoBERT
model provided by HuggingFace,111https://huggingface.co/castorini/monobert-
large-msmarco we fine-tune EM-BERT with randomly selected 300k samples (cf.
§3.2); fine-tuning with more samples, up until 900k, did not give any further
improvements on the development set. We use an accumulated batch size of 64, a
learning rate of $10^{-6}$, 40k warm-up steps, and AdamW as optimizer. Fine-
tuning for 300k samples took around 34 hours on a single GeForce RTX 3090 with
24 GB memory.
### 4.3. Second Stage Fine-tuning
#### Collection
For the second stage fine-tuning, we use the standard entity retrieval
collection, DBpedia-Entity v2 (Hasibi et al., 2017c). The collection is based
on DBpedia 2015-10 and consists of four different categories of queries: (i)
_SemSearch_ : named entity queries looking for particular entities; e.g.,
“mario bros”, (ii) _INEX-LD_ : keyword-style entity-oriented queries; e.g.,
“bicycle benefits health”, (iii) _ListSearch_ : queries asking for a list of
entities; e.g., “Hybrid cars sold in Europe”, and (iv) _QALD-2_ : natural
language questions about entities; e.g., “Who produces Orangina?” Relevance
judgements in this dataset are graded: Highly relevant (2), Relevant (1), and
irrelevant (0).
Documents in the DBpedia-Entity v2 collection are entities, each represented
by an abstract and other metadata (stored in RDF format). These entity
abstracts provide concise descriptions about entities and are comparable in
length to MS MARCO passages. Both queries and entity abstracts are highly
entity-oriented, reflected by the average number of entity annotations for
queries and abstracts in Table 1.
#### Training
We consider entity abstracts as entity descriptions and use them as documents
in monoBERT and EM-BERT. Since the DBpedia-Entity v2 collection provides a
limited amount of training data (only 467 queries), we take the fine-tuned
model on the MS MARCO passage dataset (Section 4.2) and continue fine-tuning
for both monoBERT and EM-BERT models using the provided folds for 5-fold cross
validation, each fold containing 48k query-entity pairs with approximately 34%
relevant and 66% non-relevant entities. In this process, both models are fine-
tuned for one epoch using all training data of each fold, resulting in 5
different models each. The fine-tuned models are then used for ranking the
test queries in the corresponding folds. Note that in this fine-tuning
process, no hyper-parameter optimization is performed. In this process, both
models are fine-tuned for one epoch using the training data of each fold,
resulting in 5 different models each. The fine-tuned models are used for
ranking the test queries in the corresponding folds. Note that in this fine-
tuning process, no hyper-parameter optimization is performed. For both models
an aggregated batch size of 64, learning-rate of $10^{-6}$, and AdamW
optimizer is used. Warmup step is changed to 4000, following the 10% warmup
step rule.
### 4.4. Baselines
We compare EM-BERT with the state-of-the-art entity retrieval methods using
term-matching, neural, and BERT-based approaches. The baselines are:
Figure 2. Comparison of re-ranked BM25F-CA run with monoBERT and EM-BERT for
different mention tokenization categories.
[]
Figure 3. Mention tokenization categories for the for types of queries in
DBPedia-Entity v2 collection.
[]
* BM25: The BM25 run, based on the short abstracts of entities, with parameters $k=0.9$ and $b=0.4$. Similar to monoBERT and EM-BERT, this run uses only abstract information of entities (unlike other methods that use reach representation of entities).
* BM25F-CA (Hasibi et al., 2017c): The fielded variant of BM25, obtaining scores from 5-field representation of entities. Field weights are computed using Coordinate Ascent on each train fold, and then scored on the corresponding test fold. This is the best non-neural run and also the best baseline reported in (Hasibi et al., 2017c).
* ESIMcg (Gerritse et al., 2020): ESIMcg ranks entities based on the aggregated cosine similarity scores between the Wikipedia2Vec embedding of each linked entity in the query and the target entity. Entities in this method are linked using TAGME (Ferragina and Scaiella, 2010). The method re-ranks the BM25F-CA run.
* GEEER (Gerritse et al., 2020): GEEER linearly combines the BM25F-CA score and entity-query similarities based on ESIMcg. The weights of this linear combination are computed on each train fold using Coordinate Ascent, and scored accordingly on each test fold. Similar to ESIMcg, GEEER re-ranks the BM25F-CA run.
* KEWER (Nikolaev and Kotov, 2020): KEWER uses an embedding method with joint word and entity embeddings. A query-entity score is computed on all query terms by the weighted sum of the similarity between the target entity embedding and the term, re-ranking the BM25F-CA run.
* BM25F+KEWER (Nikolaev and Kotov, 2020): The model linearly combines the KEWER score as listed above with BM25F-CA and re-ranks the BM25F-CA run.
* BLP-TransE (Daza et al., 2021): BLP-TransE is a BERT-based method, combining a BERT cross-encoder architecture with TransE graph embeddings. It encodes queries and entities, uses the similarity between these encodings as query score, and then re-ranks and linearly combines the BM25F-CA scores.
* monoBERT: Our Pygaggle implementation of the monoBERT model. This implementation achieves slightly better performance compared to results reported in (Nogueira et al., 2019): MRR@10 of 0.379 vs 0.372 on the MS MARCO development set. Following the multi-stage re-ranking method (Nogueira et al., 2019), we re-rank three entity retrieval runs using monoBERT: BM25, BM25F-CA, and GEEER (the best baseline run). The re-ranking is performed on the top-1000 entities of the BM25 and BM25F-CA, and top-100 entities of the GEEER run (as GEEER re-ranks the BM25F-CA run).
Table 3. Comparison of results for queries without and with at least one linked entity. Superscripts 1/2/3 show statistically significant differences compared to BM25F-CA/monoBERT (1st)/monoBERT, respectively. DBpedia-Entity v2 | NDCG@10 (¿1 en) | NDCG@10 (no-en)
---|---|---
BM25F-CA | 0.481 | 0.414
+monoBERT (1st) | 0.487 | 0.422
+monoBERT | 0.482 | 0.420
+EM-BERT | $0.516^{123}$ | 0.420
### 4.5. Evaluation Metrics
For evaluation of entity retrieval, we use the commonly reported metric in
(Hasibi et al., 2017c); the Normalized Discounted Cumulative Gain (NDCG) at
ranks 10 and 100. Statistical significant differences of NDCG@10 and NDCG@100
values are determined using the two-tailed paired t-test with p-value $<0.05$.
## 5\. Results and Analysis
In the following, we evaluate our entity-enriched BERT-based retrieval model
and answer our four research questions (Sections 5.1-5.4 ) listed in Section
1.
### 5.1. Entity Retrieval Results
In this section, we answer our first research question: _RQ1: Can an entity-
enriched BERT-based retrieval model improve the performance of entity
retrieval?_
We compare retrieval performance of EM-BERT with a variety of models on the
DBpedia-Entity v2 collection in Table 2. The baseline (cf. Section 4.4)
results are presented in the top part of the table. The next three
compartments in the Table summarise results of re-ranking BM25, BM25F-CA, and
GEEER runs with the following models:
* •
monoBERT (1st): The monoBERT model after first stage fine-tuning; cf. Section
4.4.
* •
monoBERT: The monoBERT model after second stage fine-tuning on DBpedia-Entity
v2.
* •
EM-BERT (1st): The EM-BERT model fine-tuned on the MS MARCO passage
collection. Re-ranking setup is similar to the monoBERT runs; cf. Section 4.4.
* •
EM-BERT: The EM-BERT model after second stage fine-tuning. Setup is similar to
the previous run.
As shown in Table 2, re-ranking of the GEEER run with EM-BERT markedly
outperforms all baselines and establishes a new state-of-the-art result on the
DBpedia-Entity v2 collection, with 11% improvements over the best run. When
re-ranking the BM25 and BM25F-CA runs, we observe the same trend that EM-BERT
outperforms all the corresponding BERT-based baselines. Comparing different
query categories, EM-BERT improves over all runs by a large margin for
ListSearch and most importantly for QALD-2 queries; improvements over GEEER
are 20% and 24% with respect to NDCG@10 for ListSearch and QALD-2 queries,
respectively. QALD queries are complex natural language queries that are hard
to answer for most neural and non-neural models, reflected by the lowest NDCG
scores on all baseline models.
Table 2 also shows that monoBERT performance is decreased when fine-tuned with
a limited amount of training data. The drop in the performance of monoBERT is
expected due to the known instabilities of BERT with few-sample training
(Zhang et al., 2021), which can cause forgetting of what the model has already
learned. The striking observation, however, is that EM-BERT results are
improved when trained on the same data and using the same procedure. This
indicates that entity-enriched BERT models enable fine-tuning on limited
training data, thereby contributing to data-efficient training of BERT for
entity-oriented tasks.
Based on these results, we can positively answer our first research question:
_our entity-enriched BERT-based model significantly improves state-of-the-art
results on entity retrieval, and more interestingly, it is robust against
instabilities of BERT when fine-tuned with limited training data._
(a) Embeddings of entity tokens.
(b) Embeddings of entity mention tokens.
Figure 4. UMAP plots of final layer EM-BERT embeddings of entity tokens and
their corresponding mention tokens. Queries were randomly selected from
queries with between 100 and 200 relevant entities.
### 5.2. Query Analysis
In the following we answer our research question _RQ2: When and which queries
are helped by the EM-BERT model?_
#### Entity annotations of queries
Knowing that not all queries in DBpedia-Entity v2 have a linked entity, we
expect EM-BERT to perform better on queries with linked entities than on
queries without them. To study this hypothesis, we compare NDCG@10 for two
subsets of DBpedia-Entity v2 queries: with and without at least one linked
entity. Table 3 shows the results. We see substantial improvements of EM-BERT
over monoBERT and monoBERT (1st) for queries with linked entities, confirming
our hypothesis.
#### Tokenization of entity mentions
To further our understanding of helped queries, we investigate the difference
in EM-BERT performance for various types of linked queries by looking into
their mentions tokenization. The BERT tokenizer splits words into word pieces,
which can be categorized into _beginning_ word pieces, indicating either the
start of a word or a full word, and _middle/end_ word pieces starting with a
‘##’. BERT employs around 30k tokens, consisting of around 24k beginning
tokens and 6k middle/end tokens. Given that Wikipedia has around 12 million
distinct entities, we expect only the most common entities to be present as a
single word piece. We hypothesize that there is a performance difference
between these very common entities, which are often seen in their natural form
during the BERT pre-training process, and the rarer entities, where their
mentions are broken into several word pieces. We, therefore, divide entity
mentions into four categories:
* •
One token: Mentions that are considered as a single token by the BERT
tokenizer; e.g., “France” is tokenized to ‘France’.
* •
Multiple tokens, no ##: Mentions that are tokenized to more than one token,
where each token represents a single unbroken word; e.g., “Yoko Ono” is
tokenized to ‘Yoko’, ‘Ono’.
* •
One ##: Mentions that are split into multiple word pieces, of which exactly
one is a middle/end word piece, e.g., “Weser” is be tokenized to ‘wes’, ‘##er’
* •
Multiple ##: Mentions which gets split into multiple word pieces, of which
more than one is a middle/end word piece, for example “Frisian” is tokenized
to ‘fr’, ‘##isi’, ‘##an’.
We categorize each query with a linked entity in one of the previously
mentioned categories, prioritizing the most complicated entity; i.e., if a
query has two mentions belonging to “One token” and “Multiple ##” categories,
the query is categorized as “Multiple ##”. Figure 3 compares the performance
of monoBERT and EM-BERT models for these four mention tokenization categories.
It shows that gains of EM-BERT are more pronounced for queries in “Multiple
##” and “One ##” categories than other categories. This supports our claim
that queries with uncommon entity mentions are most helped by including their
entity embeddings in the EM-BERT model. Additionally, we illustrate the
relation between different tokenization categories and the query types mention
(cf. Section 4.3) in Figure 3. The plot shows that 51% of INEX-LD queries have
no linked entities, explaining marginal improvements of EM-BERT for these
queries. This plot also describes why EM-BERT performs best on ListSearch and
QALD-2 queries: most of the less known entities fall in “Multiple ##” and “One
##” categories.
These results provide an answer to RQ2: _EM-BERT helps queries that are linked
to at least one entity. Additionally, entity information provided by the EM-
BERT model is most useful for less popular entities, which BERT cannot treat
their mentions as a single token._
Table 4. Comparison between EM-BERT and monoBERT for two example queries. Each query is listed twice, first with the normal BERT tokenization, then with the EM-BERT tokenization, both with the attentions of the first layer and first attention head. The Rel. column indicates the relevance judgement for that entity. The last two columns show the corresponding ranks obtained by the monoBERT/EM-BERT models for the entity. Query | Entity abstract | Rel. | Baseline | Comparison
---|---|---|---|---
give me all movies directed by francis ford cop ##pol ##a | christopher cop ##pol ##a ( born january 25 , 1962 ) is a film director and producer | 0 | monoBERT: 2 | EM-BERT: 551
give me all movies directed by francis ford cop ##pol ##a / ENTITY/Francis_Ford_Coppola | rumble fish / ENTITY/Rumble_Fish is an american / ENTITY/United_States 1983 drama film directed by francis ford cop ##pol ##a / ENTITY/Francis_Ford_Coppola . | 2 | monoBERT: 119 | EM-BERT: 1
give me the capitals of all countries in africa | list of african depend ##encies — including the respective capitals . | 0 | monoBERT: 1 | EM-BERT: 105
give me the capitals of all countries in africa / ENTITY/Africa | da ##kar / ENTITY/Dakar is the capital and largest city of senegal . it is located on the cap - ve ##rt peninsula / ENTITY/Cap-Vert on the atlantic coast / ENTITY/East_Coast_of_the_United_States and is the western ##most city in the old world / ENTITY/Europe and on the african / ENTITY/Africa mainland . | 1 | monoBERT: 82 | EM-BERT: 1
### 5.3. Model Analysis
To investigate _RQ3: Why does the EM-BERT model work and what does it learn
during the fine-tuning stage?_, we examine the proximity of the embedding
tokens compared to the mention tokens in the final layer of the BERT network,
and also discuss several examples of helped queries.
#### Entities vs. mention embeddings
To get an understanding of what BERT learns from entity information, we
compare entity and mention embeddings in the final layer of EM-BERT. We
randomly choose eight queries with 100-200 relevant entities. For each of
those documents, we take the first four linked entities and extract their
entity and mention embeddings from the final layer of EM-BERT. The mention
embeddings are taken from the first starting token before that entity. We use
UMAP (McInnes et al., 2018) with the default settings to plot these embeddings
in 2-dimensional space; see Figure 4. As is evident from the plots, the entity
embeddings form more pronounced clusters than the entity mention embeddings.
For example, in the entity embedding plot, American bridges and rivers build a
cluster together with other entities based in the US or Canada, while being
close to the corresponding queries. This also indicates that the entity
information injected through entity embeddings prevails into the final layers
of the BERT model, keeping a better representation of entity information than
provided with the default BERT word piece tokens.
#### Insightful Examples
We discuss in detail a few (hand-picked) example queries that are especially
helped or hurt by re-ranking with the EM-BERT model. Table 4 visualizes the
attention weights of the [CLS] token for the first attention head on the first
layer, for sample query-entity pairs. In the query “Give me all movies
directed by Francis Ford Coppola”, monoBERT incorrectly ranks other filmmaker
members of the Coppola family, matching the last name only. EM-BERT, in
contrast, finds the correct entity, as movies directed by Francis Ford Coppola
are situated close to him in the embedding space (as opposed to the movies by
his daughter Sofia Coppola). Table 4 shows that EM-BERT gives high attention
to the entity tokens relevant to the entity and query. It also shows high
attention to the words “movies”, which is essential here.
We can also see that through fine-tuning on DBpedia-entity, EM-BERT learns to
retrieve specific entities for queries requesting a list. For example for the
query “Give me the capitals of all countries in Africa”, while monoBERT’s
highest ranked entities are general overview pages like ‘List of African
dependencies’, all of EM-BERT’s highest-ranked entities are capitals of
African countries like ‘Dakar’ and ‘Porto-Novo’.
### 5.4. Performance on Passage Retrieval
Finally, we turn to our last research question, _RQ4: How does our entity-
enriched BERT-based model perform on other ranking tasks?_ To further our
understanding on the effect of entity-enriched BERT for IR tasks, we compare
the monoBERT and EM-BERT models after the first fine tuning stage on the MS
MARCO collection. We observe no significant differences between the two models
for both queries with and without linked entities. A similar observation is
made for the TREC Conversation Assistant Track (TREC-CAsT) (Joko et al.,
2022). We posit that the added value of enhancing BERT-based retrieval models
with explicit entity representations is mainly pronounced for entity-oriented
tasks.
## 6\. Conclusion and future work
In this research, we investigated the value of adding entity embeddings to
BERT for entity search. We proposed an entity-enhanced BERT-based retrieval
model, EM-BERT, and compared it with a regular BERT model. We found that EM-
BERT improves substantially over the state-of-the-art results, showing that
entity-oriented tasks benefit from entity embeddings in BERT, as is
hypothesized in (Broscheit, 2019). We found three categories where entities
help the most: complex natural language queries, queries requesting a list of
entities with a certain property, and queries with less known entities, in
which their mentions get split in multiple tokens by the BERT tokenizer. We
also showed that EM-BERT, unlike BERT-based retrieval models, can be fine-
tuned with limited training data, thus being effective for lesser-resourced
entity-related tasks.
For further research, the effect of the entity linking method can be
investigated; here we used REL, which is an entity linker with high precision.
It would be also interesting to evaluate the performance of other dense and
sparse retrieval methods on entity-oriented queries and tasks. This, however,
may require additional training data for fine-tuning on entity-oriented
queries, which would create a supplementary MS Marco subset to train and
evaluate ranking systems.
## References
* (1)
* Akbik et al. (2018) Alan Akbik, Duncan Blythe, and Roland Vollgraf. 2018\. Contextual String Embeddings for Sequence Labeling. In _Proc. of 27th International Conference on Computational Linguistics_ _(COLING ’18)_. 1638–1649.
* Balog (2018) Krisztian Balog. 2018\. _Entity-Oriented Search_. The Information Retrieval Series, Vol. 39. Springer.
* Blanco et al. (2015) Roi Blanco, Giuseppe Ottaviano, and Edgar Meij. 2015\. Fast and Space-Efficient Entity Linking in Queries. _Proc. of the Eighth ACM International Conference on Web Search and Data Mining_ (2015), 179–188.
* Bordes et al. (2013) Antoine Bordes, Nicolas Usunier, Alberto Garcia-Duran, Jason Weston, and Oksana Yakhnenko. 2013\. Translating embeddings for modeling multi-relational data. In _Proc. of the 26th International Conference on Neural Information Processing Systems_ _(NeurIPS ’13)_. 2787–2795.
* Broscheit (2019) Samuel Broscheit. 2019\. Investigating Entity Knowledge in BERT with Simple Neural End-To-End Entity Linking. In _Proc. of the 23rd Conference on Computational Natural Language Learning_ _(CoNLL ’19)_. 677–685.
* Cao et al. (2021) Nicola De Cao, Gautier Izacard, Sebastian Riedel, and Fabio Petroni. 2021. Autoregressive Entity Retrieval. In _Proc. of the International Conference on Learning Representations_ _(ICLR ’21)_.
* Cornolti et al. (2016) Marco Cornolti, Paolo Ferragina, Massimiliano Ciaramita, Stefan Rüd, and Hinrich Schütze. 2016\. A Piggyback System for Joint Entity Mention Detection and Linking in Web Queries. In _Proc. of the 25th International Conference on World Wide Web_ _(WWW ’16)_. 567–578.
* Dalton et al. (2020) Jeffrey Dalton, Chenyan , and Jamie Callan. 2020. TREC CAsT 2019: The conversational assistance track overview. _arXiv preprint_ (2020). arXiv:2003.13624
* Dalton et al. (2014) Jeffrey Dalton, Laura Dietz, and James Allan. 2014\. Entity Query Feature Expansion Using Knowledge Base Links. In _Proc. of the 37th International ACM SIGIR Conference on Research and Development in Information Retrieval_ _(SIGIR ’14)_. 365–374.
* Daza et al. (2021) Daniel Daza, Michael Cochez, and Paul Groth. 2021\. Inductive Entity Representations from Text via Link Prediction. In _Proc. of the Web Conference 2021_ _(WWW ’21)_. 798–808.
* Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019. BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding. In _Proc. of The North American Chapter of the Association for Computational Linguistics ’19_ _(NAACL ’19)_. 4171–4186.
* Dietz (2019) Laura Dietz. 2019\. ENT Rank: Retrieving Entities for Topical Information Needs through Entity-Neighbor-Text Relations. In _Proc. of the 42nd International ACM SIGIR Conference on Research and Development in Information Retrieval_ _(SIGIR’19)_. 215–224.
* Dodge et al. (2020) Jesse Dodge, Gabriel Ilharco, Roy Schwartz, Ali Farhadi, Hannaneh Hajishirzi, and Noah Smith. 2020\. Fine-tuning pretrained language models: Weight initializations, data orders, and early stopping. _arXiv preprint arXiv:2002.06305_ (2020).
* Ferragina and Scaiella (2010) Paolo Ferragina and Ugo Scaiella. 2010. TAGME: On-the-Fly Annotation of Short Text Fragments (by Wikipedia Entities). In _Proc. of the International Conference on Information and Knowledge Management_ _(CIKM ’10)_. 1625–1628.
* Garigliotti et al. (2019) Darío Garigliotti, Faegheh Hasibi, and Krisztian Balog. 2019\. Identifying and Exploiting Target Entity Type Information for Ad hoc Entity Retrieval. _Information Retrieval Journal_ 22, 3 (2019), 285–323.
* Gerritse et al. (2020) Emma Gerritse, Faegheh Hasibi, and Arjen De Vries. 2020\. Graph-Embedding Empowered Entity Retrieval. In _Proc. of European Conference on Information Retrieval_ _(ECIR ’20)_.
* Hasibi et al. (2015) Faegheh Hasibi, Krisztian Balog, and Svein Erik Bratsberg. 2015\. Entity Linking in Queries: Tasks and Evaluation. In _Proc. of the 2015 International Conference on The Theory of Information Retrieval_ _(ICTIR ’15)_. 171–180.
* Hasibi et al. (2016) Faegheh Hasibi, Krisztian Balog, and Svein Erik Bratsberg. 2016\. Exploiting Entity Linking in Queries for Entity Retrieval. In _Proc. of the 2016 ACM International Conference on the Theory of Information Retrieval_ _(ICTIR ’16)_. 209–218.
* Hasibi et al. (2017a) Faegheh Hasibi, Krisztian Balog, and Svein Erik Bratsberg. 2017a. Entity Linking in Queries: Efficiency vs. Effectiveness. In _Proc. of the 39th European Conference on Information Retrieval_ _(ECIR ’17)_. 40–53.
* Hasibi et al. (2017b) Faegheh Hasibi, Krisztian Balog, Daŕio Garigliotti, and Shuo Zhang. 2017b. Nordlys: A toolkit for entity-oriented and semantic search. In _Proc. of the 40th International ACM SIGIR Conference on Research and Development in Information Retrieval_ _(SIGIR ’17)_. 1289–1292.
* Hasibi et al. (2017c) Faegheh Hasibi, Fedor Nikolaev, Chenyan Xiong, Krisztian Balog, Svein Erik Bratsberg, Alexander Kotov, and Jamie Callan. 2017c. DBpedia-Entity V2: A Test Collection for Entity Search. In _Proc. of the 40th International ACM SIGIR Conference on Research and Development in Information Retrieval_ _(SIGIR ’19)_. 1265–1268.
* Hofstätter et al. (2020) Sebastian Hofstätter, Sophia Althammer, Michael Schröder, Mete Sertkan, and Allan Hanbury. 2020\. Improving Efficient Neural Ranking Models with Cross-Architecture Knowledge Distillation. _arXiv preprint_ (2020). arXiv:2010.02666
* Jiang et al. (2020) Zhengbao Jiang, Antonios Anastasopoulos, Jun Araki, Haibo Ding, and Graham Neubig. 2020. X-FACTR: Multilingual Factual Knowledge Retrieval from Pretrained Language Models. In _Proc. of the 2020 Conference on Empirical Methods in Natural Language Processing_ _(EMNLP ’20)_. 5943–5959.
* Joko et al. (2022) Hideaki Joko, Emma J Gerritse, Faegheh Hasibi, and Arjen P de Vries. 2022. Radboud University at TREC CAsT 2021. In _TREC_.
* Joshi et al. (2020) Mandar Joshi, Danqi Chen, Yinhan Liu, Daniel S. Weld, Luke Zettlemoyer, and Omer Levy. 2020\. SpanBERT: Improving Pre-training by Representing and Predicting Spans. _Transactions of the Association for Computational Linguistics_ (2020), 64–77.
* Joshi et al. (2019) Mandar Joshi, Omer Levy, Luke Zettlemoyer, and Daniel Weld. 2019\. BERT for Coreference Resolution: Baselines and Analysis. In _Proc. of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing_ _(EMNLP-IJCNLP ’19)_. 5803–5808.
* Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019. RoBERTa: A Robustly Optimized BERT Pretraining Approach. _arXiv preprint_ (2019). arXiv:1907.11692
* McInnes et al. (2018) Leland McInnes, John Healy, Nathaniel Saul, and Lukas Grossberger. 2018. UMAP: Uniform Manifold Approximation and Projection. _The Journal of Open Source Software_ 3, 29 (2018), 861\.
* Mikolov et al. (2013) Tomas Mikolov, Ilya Sutskever, Kai Chen, Gregory S. Corrado, and Jeffrey Dean. 2013. Distributed Representations of Words and Phrases and their Compositionality. In _Proc. of Advances in Neural Information Processing Systems_ _(NeurIPS ’13)_. 3111–3119.
* Nguyen et al. (2016) Tri Nguyen, Mir Rosenberg, Xia Song, Jianfeng Gao, Saurabh Tiwary, Rangan Majumder, and Li Deng. 2016. MS MARCO: A Human Generated MAchine Reading COmprehension Dataset. In _Proc. of the Workshop on Cognitive Computation: Integrating neural and symbolic approaches 2016_.
* Nikolaev and Kotov (2020) Fedor Nikolaev and Alexander Kotov. 2020. Joint Word and Entity Embeddings for Entity Retrieval from a Knowledge Graph. In _Proc. of the European Conference on Information Retrieval_ _(ECIR ’20)_. 141–155.
* Nogueira and Cho (2019) Rodrigo Nogueira and Kyunghyun Cho. 2019. Passage Re-ranking with BERT. _arXiv preprint_ (2019). arXiv:1901.04085
* Nogueira et al. (2019) Rodrigo Nogueira, Wei Yang, Kyunghyun Cho, and Jimmy Lin. 2019\. Multi-stage document ranking with BERT. _arXiv preprint_ (2019). arXiv:1910.14424
* Peters et al. (2019) Matthew E. Peters, Mark Neumann, Robert Logan, Roy Schwartz, Vidur Joshi, Sameer Singh, and Noah A. Smith. 2019. Knowledge Enhanced Contextual Word Representations. In _Proc. of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing_ _(EMNLP-IJCNLP ’19)_. 43–54.
* Petroni et al. (2019) Fabio Petroni, Tim Rocktäschel, Patrick S. H. Lewis, Anton Bakhtin, Yuxiang Wu, Alexander H. Miller, and Sebastian Riedel. 2019. Language Models as Knowledge Bases?. In _Proc. of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing_ _(EMNLP-IJCNLP ’19)_. 2463–2473.
* Phang et al. (2019) Jason Phang, Thibault Fevry, and Samuel R. Bowman. 2019\. Sentence Encoders on STILTs: Supplementary Training on Intermediate Labeled-data Tasks. _arXiv preprint arXiv:1811.01088_ (2019).
* Poerner et al. (2020) Nina Poerner, Ulli Waltinger, and Hinrich Schütze. 2020\. E-BERT: Efficient-Yet-Effective Entity Embeddings for BERT. In _Findings of the Association for Computational Linguistics_ _(ELMNLP ’20)_. 803–818.
* Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2020. Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer. _Journal of Machine Learning Research_ 21, 140 (2020), 1–67.
* Talmor et al. (2020) Alon Talmor, Yanai Elazar, Yoav Goldberg, and Jonathan Berant. 2020. oLMpics - On what Language Model Pre-training Captures. _Transactions of the Association for Computational Linguistics_ 8 (2020), 743–758.
* Thakur et al. (2021) Nandan Thakur, Nils Reimers, Andreas Rücklé, Abhishek Srivastava, and Iryna Gurevych. 2021\. BEIR: A Heterogeneous Benchmark for Zero-shot Evaluation of Information Retrieval Models. In _Proc. of 35th Conference on Neural Information Processing Systems Datasets and Benchmarks Track (Round 2)_ _(NeurIPS ’21)_.
* van Hulst et al. (2020) Johannes M. van Hulst, Faegheh Hasibi, Koen Dercksen, Krisztian Balog, and Arjen P. de Vries. 2020. REL: An Entity Linker Standing on the Shoulders of Giants. In _Proc. of the 43rd International ACM SIGIR Conference on Research and Development in Information Retrieval_ _(SIGIR ’20)_.
* Wang et al. (2020) Chenguang Wang, Xiao Liu, and Dawn Song. 2020. Language Models are Open Knowledge Graphs. _arXiv preprint_ (2020). arXiv:2010.11967
* Wang et al. (2020) Ruize Wang, Duyu Tang, Nan Duan, Zhongyu Wei, Xuanjing Huang, Jianshu ji, Guihong Cao, Daxin Jiang, and Ming Zhou. 2020. K-Adapter: Infusing Knowledge into Pre-Trained Models with Adapters. _arXiv preprint_ (2020). arXiv:2002.01808
* Wang et al. (2020) Wenhui Wang, Furu Wei, Li Dong, Hangbo Bao, Nan Yang, and Ming Zhou. 2020. MiniLM: Deep Self-Attention Distillation for Task-Agnostic Compression of Pre-Trained Transformers. In _Proc. of Advances in Neural Information Processing Systems_ _(NeurIPS ’21)_ , H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin (Eds.). 5776–5788.
* Wang et al. (2019a) Xiaozhi Wang, Tianyu Gao, Zhaocheng Zhu, Zhiyuan Liu, Juanzi Li, and Jian Tang. 2019a. KEPLER: A unified model for knowledge embedding and pre-trained language representation. _arXiv preprint_ (2019). arXiv:1911.06136
* Wang et al. (2019b) Zhiguo Wang, Patrick Ng, Xiaofei Ma, Ramesh Nallapati, and Bing Xiang. 2019b. Multi-passage BERT: A Globally Normalized BERT Model for Open-domain Question Answering. In _Proc. of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing_ _(EMNLP-IJCNLP ’19)_. 5878–5882.
* Xiong et al. (2017a) Chenyan Xiong, Jamie Callan, and Tie-Yan Liu. 2017a. Word-Entity Duet Representations for Document Ranking. In _Proc. of the 40th International ACM SIGIR Conference on Research and Development in Information Retrieval_ _(SIGIR ’17)_. 763–772.
* Xiong et al. (2017b) Chenyan Xiong, Russell Power, and Jamie Callan. 2017b. Explicit Semantic Ranking for Academic Search via Knowledge Graph Embedding. In _Proc. of the 26th International Conference on World Wide Web_ _(WWW ’17)_. 1271–1279.
* Yamada et al. (2020a) Ikuya Yamada, Akari Asai, Jin Sakuma, Hiroyuki Shindo, Hideaki Takeda, Yoshiyasu Takefuji, and Yuji Matsumoto. 2020a. Wikipedia2Vec: An Efficient Toolkit for Learning and Visualizing the Embeddings of Words and Entities from Wikipedia. In _Proc. of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ _(EMNLP ’20)_. 23–30.
* Yamada et al. (2020b) Ikuya Yamada, Akari Asai, Hiroyuki Shindo, Hideaki Takeda, and Yuji Matsumoto. 2020b. LUKE: Deep Contextualized Entity Representations with Entity-aware Self-attention. In _Proc. of the 2020 Conference on Empirical Methods in Natural Language Processing_ _(EMNLP ’20)_. 6442–6454.
* Yamada et al. (2016) Ikuya Yamada, Hiroyuki Shindo, Hideaki Takeda, and Yoshiyasu Takefuji. 2016. Joint learning of the embedding of words and entities for named entity disambiguation. In _Proc. of The SIGNLL Conference on Computational Natural Language Learning_ _(SIGNLL ’16)_. 250–259.
* Yang et al. (2019) Zhilin Yang, Zihang Dai, Yiming Yang, Jaime Carbonell, Russ R Salakhutdinov, and Quoc V Le. 2019\. XLNet: Generalized Autoregressive Pretraining for Language Understanding. In _Advances in Neural Information Processing Systems_ _(NeurIPS ’19)_. 5753–5763.
* Zhang et al. (2021) Tianyi Zhang, Felix Wu, Arzoo Katiyar, Kilian Q Weinberger, and Yoav Artzi. 2021. Revisiting Few-sample BERT Fine-tuning. In _Proc. of 2021 International Conference on Learning Representations_ _(ICLR ’20)_.
* Zhang et al. (2019) Zhengyan Zhang, Xu Han, Zhiyuan Liu, Xin Jiang, Maosong Sun, and Qun Liu. 2019\. ERNIE: Enhanced Language Representation with Informative Entities. In _Proc. of the 57th Annual Meeting of the Association for Computational Linguistics_ _(ACL ’19)_. 1441–1451.
|
# Multiply Robust Estimation of Causal Effect Curves for Difference-in-
Differences Designs
Gary Hettinger
Department of Biostatistics, Epidemiology, and Informatics
University of Pennsylvania
Philadelphia, PA, U.S.A.
<EMAIL_ADDRESS>
&Youjin Lee
Department of Biostatistics
Brown University
Providence, RI, U.S.A.
&Nandita Mitra
Department of Biostatistics, Epidemiology, and Informatics
University of Pennsylvania
Philadelphia, PA U.S.A.
###### Abstract
Researchers commonly use difference-in-differences (DiD) designs to evaluate
public policy interventions. While established methodologies exist for
estimating effects in the context of binary interventions, policies often
result in varied exposures across regions implementing the policy. Yet,
existing approaches for incorporating continuous exposures face substantial
limitations in addressing confounding variables associated with intervention
status, exposure levels, and outcome trends. These limitations significantly
constrain policymakers’ ability to fully comprehend policy impacts and design
future interventions. In this study, we propose innovative estimators for
causal effect curves within the DiD framework, accounting for multiple sources
of confounding. Our approach accommodates misspecification of a subset of
treatment, exposure, and outcome models while avoiding any parametric
assumptions on the effect curve. We present the statistical properties of the
proposed methods and illustrate their application through simulations and a
study investigating the diverse effects of a nutritional excise tax.
_Keywords_ Dose Response $\cdot$ Doubly Robust $\cdot$ Health Policy $\cdot$
Semiparametric
## 1 Introduction
Public policies wield significant influence on a broad spectrum of population
outcomes, including those related to public health and economics.(Pollack
Porter et al., 2018) Accordingly, decision-makers require rigorous evaluation
of such policies to comprehend their effectiveness and inform the design of
subsequent policy initiatives. Complicating the comprehensive assessment of
policy effects, units within a study may be differentially exposed to a policy
for various reasons. For example, exposure to an excise tax policy may depend
on the ease in which a buyer can evade the tax (e.g., geographic proximity to
cross-border shopping) or responsive price adjustments made by the seller
which may diminish the actual tax rate seen by consumers.(Chaloupka et al.,
2019) These differential exposures can lead to effect heterogeneity that, if
understood, can help identify affected populations and inspire future
interventions that may generate a different exposure distribution.(Cintron et
al., 2022)
When evaluating effects of binary policy interventions, researchers commonly
use difference-in-differences (DiD) approaches, which exploit the so-called
counterfactual parallel trends assumption to utilize outcomes from a control
group, measured before and after the intervention, to impute what would have
happened in the treated group in the absence of intervention.(Heckman et al.,
1997; Abadie, 2005) This assumption requires that there are no confounding
differences between treated and control groups that also affect the outcome
trend between the before- and after-intervention windows. Recent work has
demonstrated that incorporating a continuous measure of exposure requires even
stricter parallel trends assumptions.(Callaway et al., 2021) As DiD methods to
incorporate continuous exposure measures are limited, many researchers
alternatively conduct subgroup analyses to explore effect heterogeneity, but
these cannot identify causal differences between subgroups as they adjust for
confounding within, rather than between, subgroups.
In the DiD setting with a binary exposure variable, substantial methodology
has been developed to relax the counterfactual parallel trends assumption by
adjusting for observed confounders between the treated and control
groups.(Abadie, 2005; Li and Li, 2019; Sant’Anna and Zhao, 2020) However, few
works have targeted confounding between different exposure levels among the
treated group when estimating effect curves in DiD studies. Han and Yu (2019)
considered DiD extensions of the approach developed by Hirano and Imbens
(2004) to estimate causal effect curves by utilizing an estimate for the
generalized propensity score with continuous exposures as a bias-adjusting
covariate in a regression model. However, this approach hinges entirely on
accurately specifying the conditional density function for the continuous
exposure given covariates and may yield unstable estimates even under correct
specification.(Austin, 2018) Beyond DiD studies, there have been notable
advancements in estimating dose-response curves in the presence of
confounding. These include doubly robust estimators that utilize both
propensity scores and outcome regression models to estimate effect curves and
generally improve upon approaches solely reliant on the propensity score in
terms of both efficiency and robustness.(Díaz and van der Laan, 2013; Kennedy
et al., 2017; Bonvini and Kennedy, 2022) Still, methods to harness these
properties have yet to be developed to account for the multiple levels of
confounding – between treated and control groups and between different
exposure levels among the treated group – and longitudinal design present in
policy evaluations that employ a control group.
In this work, we develop novel estimators for causal effect curves that
account for multiple sources of confounding under a DiD framework. Our
methodology incorporates both outcome and treatment models to rigorously
adjust for confounding effects and exploits the efficiency of semi-parametric
influence-function based estimators. Our estimators are multiply robust in the
sense that only a subset of the four required nuisance function models need to
be well-specified for consistent effect estimation. To further mitigate
potential model misspecification, our approach allows for flexible modeling
techniques to capture complex confounding relationships and non-parametric
effect curves.
The remainder of the paper is organized as follows: in Section 2, we introduce
the setting of interest, identification assumptions, and proposed estimators.
In Section 3, we demonstrate finite sample properties of our estimators under
different nuisance function model specifications through simulation studies.
We then apply our methodology to study the effects of the Philadelphia
beverage tax in Section 4 and conclude with a discussion in Section 5.
## 2 Methods
### 2.1 Setting
We assume an intervention is introduced between two time-periods, $t=0,1$.
Units, denoted with subscript $i$, are either in the intervention group,
$A_{i}=1$, or not, $A_{i}=0$, and the intervention assignment status at time
$t$ is denoted by $Z_{it}=tA_{i}\in\\{0,1\\}$. The actual exposure to the
intervention received by a unit is measured by $D_{i}\in\mathbbm{R}$, with the
exposure status given by $\zeta_{it}=tD_{i}$. Depending on our choice of
exposure, a “zero" exposure may not represent true protection from the
intervention and therefore we denote this by $D_{i}=\emptyset$ or
$\zeta_{it}=\emptyset$. We denote population-level vectors for intervention
and exposure variables as $\mathbf{A}$, $\mathbf{Z_{t}}$, $\mathbf{D}$, and
$\boldsymbol{\zeta_{t}}$, respectively. We observe outcomes of interest at
each time point, $Y_{it}$. Finally, units and their outcomes may differ
according to a vector of observed pre-intervention covariates,
$\mathbf{X_{i}}$, although we could also include time-varying covariates
assuming time-invariant effects.(Zeldow and Hatfield, 2021)
The following assumptions allow us to define potential outcomes in terms of
intervention and exposure status as $Y_{it}^{(Z_{it},\zeta_{it})}$:
###### Assumption 1 (Arrow of Time)
$Y_{i0}^{(\mathbf{Z_{0}},\boldsymbol{\zeta_{0}},\mathbf{Z_{1}},\boldsymbol{\zeta_{1}})}=Y_{i0}^{(\mathbf{Z_{0}},\boldsymbol{\zeta_{0}})}$
###### Assumption 2 (Stable Unit Treatment Value Assumption (SUTVA))
$Y_{it}^{(\mathbf{Z_{t}},\boldsymbol{\zeta_{t}})}=Y_{it}^{(Z_{it},\zeta_{it})}$
Assumption 1 states that potential outcomes do not depend on future treatment
or exposure, which would be violated if for example units adapted their
behavior in anticipation of an upcoming effect. After connecting potential
outcomes to treatment at a given time, we further assume that potential
outcomes of a given unit only depend on population intervention and exposure
status through a unit’s own intervention and exposure status (Assumption 2).
This formulation can be considered a relaxation of the traditional SUTVA from
the setting with only a binary intervention by allowing potential outcomes to
vary according to $\zeta_{it}$ in addition to $Z_{it}$.
We are primarily interested in estimating a curve for the causal effect of the
intervention at different exposure levels, $\delta$, as represented by what we
denote the Average Dose Effect on the Treated:
$ADT(\delta):=\Psi(\delta)=E[Y_{1}^{(1,\delta)}-Y_{1}^{(0,\emptyset)}|A=1]$
(1)
This estimand answers the question, what would be the average effect of a
policy on the treated group where all treated units received the exposure of
$\delta$? The Average Treatment Effect on the Treated (ATT) commonly estimated
in DiD studies under a binary treatment study has one unobserved potential
outcome for treated units – $Y_{1}^{(0,\emptyset)}$, which is unobserved for
all treated units. Additionally, standard dose response curves focus on
identifying analogies to $Y_{1}^{(1,\delta)}$, which is unobserved for treated
units $i$, where $D_{i1}\neq\delta$. To estimate (1) in our setting, we must
identify both sets of unobserved potential outcomes.
A natural extension of the $ADT(\delta)$ would consider the expected value of
(1) over a relevant distribution of doses to quantify an aggregate effect,
which we refer to as the Stochastic Average Dose Effect on the Treated:
$\begin{split}SADT&:=\Psi=E_{\mathbbm{D}}[E[Y_{1}^{(1,\delta)}-Y_{1}^{(0,\emptyset)}|A=1]|A=1]\\\
&=\int\limits_{\mathbbm{D}}ADT(\delta)df(\delta|A=1)\end{split}$ (2)
This estimand represents the average effect of an intervention that randomly
assigns treatment to the treated group based on the density of exposures, $f$,
and thus carries a different interpretation than other proposed aggregate DiD
estimands like the ATT or average slope effects from Callaway et al. (2021)
and de Chaisemartin and D’Haultfoeuille (2022). While stochastic estimands
like the $SADT$ have many potential uses (e.g., (Muñoz and van der Laan,
2012)), we primarily will use the $SADT$ for its relevance in deriving an
estimator for the $ADT(\delta)$.
### 2.2 Identification
To identify the $ADT(\delta)$ and $SADT$, we require additional assumptions to
link the unobservable potential outcomes to observable data. The strong
assumption of binary DiD approaches is that of counterfactual parallel trends:
###### Assumption 3 (Conditional Counterfactual Parallel Trends Between
Treated and Control)
$E[Y_{1}^{(0,\emptyset)}-Y_{0}^{(0,\emptyset)}|A=1,\mathbf{X}]=E[Y_{1}^{(0,\emptyset)}-Y_{0}^{(0,\emptyset)}|A=0,\mathbf{X}]$
Assumption 3 allows us to connect $E[Y_{1}^{(0,\emptyset)}|A=1]$ in (1) to
observable potential outcomes in the control group, $A=0$. While the DiD
design innately adjusts for baseline differences between the treated and
control groups by within-group differencing, it does not address confounding
between treatment group and outcome trends. Here, we have adopted the approach
several have taken to relax this assumption by assuming parallel trends holds
after conditioning on a set of observed baseline confounders.(Abadie, 2005; Li
and Li, 2019; Sant’Anna and Zhao, 2020)
To connect $E[Y_{1}^{(1,\delta)}|A=1]$ to observable potential outcomes for
all treated units, we require an additional, parallel trends assumption:
###### Assumption 4 (Conditional Counterfactual Parallel Trends Among Treated
Between Doses)
$E[Y_{1}^{(1,\delta)}-Y_{0}^{(0,\emptyset)}|A=1,D=\delta,\mathbf{X}]=E[Y_{1}^{(1,\delta)}-Y_{0}^{(0,\emptyset)}|A=1,\mathbf{X}]\text{
for all }\delta\in\mathbbm{D}$
For intuition, consider similar units, as defined by $\mathbf{X}$, between (i)
the subset of the treated group that receives a particular exposure level,
$\delta$, and (ii) the entire treated group. In principle, we then require
that the observed trends in group (i) are representative of the counterfactual
trends in group (ii) were all of group (ii) given that exposure level when
intervened on. A visual example of this assumption is presented in Figure 1.
Previous works have noted the necessity of similar assumptions in the
continuous exposure setting either unconditionally or with regards to
conditional distributions rather than expectations.(Callaway et al., 2021; Han
and Yu, 2019)
Figure 1: An example of Assumption 4 where exposure level is discretized into
high and low dose for visualization.
While assumptions regarding parallel trends are by definition untestable using
observed data, it is common for practitioners to test for counterfactual
parallel trends between the treated and control groups leading up to the
intervention when multiple pre-treatment observations are available (e.g.,
$t=-1$) under a binary treatment.(Bilinski and Hatfield, 2018) The test
derives its usefulness by assuming the fully observable relationship between
$E[Y_{0}^{(0,\emptyset)}-Y_{-1}^{(0,\emptyset)}|A=1]$ and
$E[Y_{0}^{(0,\emptyset)}-Y_{-1}^{(0,\emptyset)}|A=0]$ is a reasonable proxy
for the unobservable relationship between
$E[Y_{1}^{(0,\emptyset)}-Y_{0}^{(0,\emptyset)}|A=1]$ and
$E[Y_{1}^{(0,\emptyset)}-Y_{0}^{(0,\emptyset)}|A=0]$. Similar placebo tests
are possible for Assumption 4, however part of this assumption,
$E[Y_{t}^{(1,\delta)}|A=1]$, is unobservable even in the pre-treatment period.
Therefore, the usefulness of comparable tests assessing
$E[Y_{0}^{(0,\emptyset)}-Y_{-1}^{(0,\emptyset)}|A=1,D=\delta]=E[Y_{0}^{(0,\emptyset)}-Y_{-1}^{(0,\emptyset)}|A=1]$
also depends on how well these trends represent counterfactual trends at
different exposure levels.
The final two assumptions are extensions of standard causal inference
assumptions to the continuous exposure setting:
###### Assumption 5 (Consistency)
$Y_{it}^{(z_{it},d_{it})}=Y_{it}$ when $(Z_{it},D_{it})=(z_{it},d_{it})$
###### Assumption 6 (Positivity)
There is $\epsilon>0$ such that
(i) $\epsilon\leq\pi_{A}(\mathbf{x})<1$ for all $\mathbf{x}\in\mathbbm{X}$
and
(ii) $\epsilon\leq\pi_{D}(\delta|\mathbf{x},A=1)$ for all
$\mathbf{x}\in\mathbbm{X}|A=1,\text{ }\delta\in\mathbbm{D}$,
where $\pi_{A}(\mathbf{x})=P(A=1|\mathbf{x})$ and
$\pi_{D}(\delta|\mathbf{x},A=1)=f(\delta|A=1,\mathbf{X})$.
Here, we require a stricter positivity assumption (Assumption 6) than the
binary setting, which only requires Assumption 6(i). Assumption 6(ii) relates
to covariate overlap between the different dose levels and would violated if
certain subpopulations, defined by confounders $\mathbf{X}$, were restricted
from certain exposure levels.
After these assumptions, we then can frame (1) and (2) fully with observable
data as:
###### Theorem 1 (Identifiability)
Under Assumptions 1-6, (1) is identifiable through
$\Psi(\delta)=E[\mu_{1,\Delta}(\delta,\mathbf{X})-\mu_{0,\Delta}(\mathbf{X})|A=1]$,
where
$\mu_{1,\Delta}(\delta,\mathbf{X})=E[Y_{1}-Y_{0}|A=1,D=\delta,\mathbf{X}]$ and
$\mu_{0,\Delta}(\mathbf{X})=E[Y_{1}-Y_{0}|A=0,\mathbf{X}]$.
Accordingly, (2) is then identifiable through $\Psi=E[\Psi(D)|A=1]$.
### 2.3 Multiply-Robust Estimator
An ideal estimator for (1) would then be based on the efficient influence
function (EIF) for this estimand, as EIF-based estimators come with optimal
convergence properties and often, though not necessarily, certain robustness
to model misspecification. However, the EIF is not tractable for (1) without
requiring parametric restrictions on the form of the otherwise infinite-
dimensional curve as such a curve does not satisfy the requirement for
pathwise differentiability.(Kennedy, 2022) Parametric restrictions generally
limit method applicability and data fit, so we do not consider this a viable
path forward. However, the EIF for (2) is tractable and given by:
###### Theorem 2 (Efficient Influence Function for $\Psi$)
The efficient influence function for $\Psi$, is given by:
$\phi(\mathbf{O})=\frac{A}{P(A=1)}\xi(Y_{1},Y_{0},A,D,\mathbf{X})+\tau(Y_{1},Y_{0},A,\mathbf{X})+J(A,\mathbf{X})-\Psi$
Where
$\displaystyle\xi(Y_{1},Y_{0},A,D,\mathbf{X})$
$\displaystyle=m(D)+\frac{(Y_{1}-Y_{0})-\mu_{1,\Delta}(D,\mathbf{X})}{\pi_{D}(D|A=1,\mathbf{X})}f(D|A=1)$
$\displaystyle\tau(Y_{1},Y_{0},A,\mathbf{X})$
$\displaystyle=\frac{A}{P(A=1)}\mu_{0,\Delta}(\mathbf{X})+\frac{1-A}{P(A=1)}\frac{\pi_{A}(\mathbf{X})}{1-\pi_{A}(\mathbf{X})}[(Y_{1}-Y_{0})-\mu_{0,\Delta}(\mathbf{X})]$
$\displaystyle J(A,\mathbf{X})$
$\displaystyle=\frac{A}{P(A=1)}[\int\limits_{\mathbbm{D}}\\{\mu_{1,\Delta}(d,\mathbf{X})-m(d)\\}df(d|A=1)]$
And
$m(D)=\int\limits_{\mathbbm{X}}\mu_{1,\Delta}(D,\mathbf{x})dP(\mathbf{x}|A=1)$,
$f(D|A=1)=\int\limits_{\mathbbm{X}}\pi_{D}(D|A=1,\mathbf{x}))dP(\mathbf{x}|A=1)$
While notationally complex, the resulting influence function is largely
intuitive. Note that $\xi(\mathbf{Y_{1},Y_{0},A,D,\mathbf{X}})$ strongly
corresponds to $E[Y_{1}-Y_{0}|A=1,D=D]=E[Y_{1}(1,D)-Y_{0}(0,\emptyset)|A=1]$.
Here, $m(D)$ averages the conditional expectation,
$\mu_{1,\Delta}(D,\mathbf{X})=E[Y_{1}-Y_{0}|A=1,D,\mathbf{X}]$ across all
covariates in the treated group. The second term of $\xi$ re-weights observed
deviations from expectation by the balance-inducing generalized propensity
score weight, $w^{(1)}=f(D)/\pi_{D}(D|A=1,\mathbf{X})$, as a sort of bias
correction term when $\mu_{1,\Delta}$ is misspecified. Similarly,
$\tau(Y_{1},Y_{0},A,\mathbf{X})$ correseponds to
$E[Y_{1}-Y_{0}|A=1]=E[Y_{1}(0,\emptyset)-Y_{0}(0,\emptyset)|A=1]$, where now
we plug in treated unit covariates to
$\mu_{0,\Delta}(\mathbf{X})=E[Y_{1}-Y_{0}|A=0,\mathbf{X}]$. The second term
re-weights observed deviations from expectation for the control group by ATT
propensity score weights,
$w^{(0)}=\pi_{A}(\mathbf{X})/(1-\pi_{A}(\mathbf{X}))$ as a sort of bias
correction term when $\mu_{0}$ is misspecified. Finally, we note that
$J(\mathbf{A,\mathbf{X}})$ is independent of exposure dose and
$E[J(\mathbf{A,\mathbf{X}})|A=1]=0$, so will not factor into point estimation
for either (1) or (2). A similar decomposition for $\xi$ and $J$ was noted by
Kennedy et al. (2017) to generate their so-called pseudo-outcomes when
estimating a dose response curve for $E[Y^{(\delta)}]$ under a cross-sectional
study design.
Utilizing this influence function and subsequent decomposition, we then can
estimate (1) and (2) with an multi-step, plug-in procedure:
1. 1.
Estimate nuisance functions as
$\hat{\pi}_{D},\hat{\pi}_{A},\hat{\mu}_{1,\Delta},\hat{\mu}_{0,\Delta}$.
2. 2.
Plug in data and estimated nuisance functions to construct
$\hat{\tau}(\mathbf{O};\hat{\pi}_{A},\hat{\mu}_{0,\Delta})$ and take the
empirical mean of $\hat{\tau}$ to obtain $\hat{\theta}_{0}$.
3. 3.
Plug in data and estimated nuisance functions to construct
$\hat{\xi}(\mathbf{O};\hat{\pi}_{D},\hat{\mu}_{1,\Delta})$ and regress
$\hat{\xi}$ on dose variable $D$ using a flexible regression function
$r_{n}(\hat{\xi},D)$ to obtain $\hat{\theta}(\delta)$.
4. 4.
For (1), take the difference $\hat{\theta}(\delta)-\hat{\theta}_{0}$ as
$\hat{\Psi}(\delta)$.
5. 5.
For (2), take the empirical mean of $\frac{A}{P(A=1)}\hat{\xi}-\hat{\tau}$ as
$\hat{\Psi}$.
Practically speaking, $\hat{\pi}_{D}$ and $\hat{\mu}_{1,\Delta}$ must be
estimated only using data from the $A=1$ group whereas $\hat{\pi}_{A}$
requires data from both $A=1$ and $A=0$ groups. Estimation of
$\hat{\mu}_{0,\Delta}$ could incorporate data from both $A=1$ and $A=0$
groups, but the efficiency gained from utilizing $A=1$ data comes at the cost
of robustness in that correctly specifying $\hat{\mu}_{0,\Delta}$ would
require properly modeling treatment effect dynamics, linking its specification
to that of $\hat{\mu}_{1,\Delta}$. Another generally beneficial step in
practice is to normalize $w^{(1)}$ and $w^{(0)}$ to both have mean one as in
Hajek weighted estimators, which has previously been noted as important for
improving the stability of doubly robust DiD estimators.(Sant’Anna and Zhao,
2020)
The decomposition of the influence function $\phi$ also suggests components
related to doubly-robust estimators for both
$E[Y_{1}(1,D)-Y_{0}(0,\emptyset)|A=1]$ and
$E[Y_{1}(0,\emptyset)-Y_{0}(0,\emptyset)|A=1]$, foreshadowing the multiple
robustness properties of our proposed estimator:
###### Theorem 3 (Multiple Robustness)
Suppose that Assumptions 1-6 hold. Additionally assume that:
1. i.
At least one of the dose-specific nuisance functions {$\pi_{D}$,
$\mu_{1,\Delta}$} are correctly specified.
2. ii.
At least one of the intervention-specific nuisance functions {$\pi_{A}$,
$\mu_{0,\Delta}$} are correctly specified.
Then, $\hat{\Psi}$ is a consistent estimator for (2). Further, if
$r_{n}(\xi,d)$ is a regression estimator such that
$r_{n}(\xi,d)\overset{p}{\to}E[\xi|D=d]$, then $\hat{\Psi}(\delta)$ is a
consistent estimator for (1) when $\delta$ is an exposure value in the
interior of the compact support of $D$.
Thereby, we only require one correct model specification in each specified
pair of nuisance functions corresponding to
$E[Y_{1}(1,D)-Y_{0}(0,\emptyset)|A=1]$ and
$E[Y_{1}(0,\emptyset)-Y_{0}(0,\emptyset)|A=1]$, respectively. Accordingly, our
estimator will result in consistent effect estimation in 9 of the 16 ($56\%$)
permutations of correct/incorrect nuisance function specifications (i.e.,
{$\pi_{D}\in\\{\text{Correct,Incorrect}\\},\mu_{1,\Delta}\in\\{\text{Correct,Incorrect}\\},\pi_{A}\in\\{\text{Correct,Incorrect}\\},\mu_{0,\Delta}\in\\{\text{Correct,Incorrect}\\}$}).
Compared to standard doubly robust estimators which only require two nuisance
functions and thus achieve consistent effect estimation in 3 of the 4 ($75\%$)
specification permutations, we see the required cost of addressing multiple
levels of confounding in terms of robustness. However, if we consider the
specification of both outcome models, $\\{\mu_{1,\Delta},\mu_{0,\Delta}\\}$,
together and both propensity score models, $\\{\pi_{D},\pi_{A}\\}$, together,
then we have consistent estimates in 3 of the 4 specification permutations and
further achieve consistency under additional “partial” misspecifications.
The requirements regarding $r_{n}$ suggest the use of a flexible non-
parametric regression function to avoid misspecifying the conditional density
of $\xi$ given $D$. Following Kennedy et al. (2017), we specifically consider
a local linear kernel regression estimator for $r_{n}$, which is a non-
parametric estimator imposing relatively few regularity conditions regarding
effect curves and nuisance functions for consistency in this framework
(Appendix).
### 2.4 Uncertainty Quantification
Calculating confidence intervals for these multi-step estimators is non-
trivial, particularly for $\Psi(D)$ which does not absorb the semi-parametric
efficiency guarantees of influence-function based estimators for binary
treatments when all nuisance functions are well-specified. Since standard non-
parametric regressors are consistent but centered at a smoothed version of the
effect curve, rather than the true curve, we adapt previous works in
expressing uncertainty about this smoothed curve (Wasserman, 2006; Kennedy et
al., 2017). While our estimator is asymptotically normal under mild
restrictions on nuisance function convergence rates (Appendix), we note that
such results ignore finite-sample uncertainty in the estimation of nuisance
functions and may lead to underestimated variances (Lunceford and Davidian,
2004). Thereby, we consider two alternative approaches.
First, we follow previous approaches that represent plug-in causal inference
estimators as a series of unbiased estimating equations (i.e., score equations
for specified parametric nuisance function models, sample mean equation for
$\theta_{0}$ using $\hat{\tau}_{i}$, and kernel regression equation for
$\hat{\theta}(D)$ using $\hat{\xi}_{i}$) and apply M-estimation theory to
engineer sandwich estimators for the variance of estimated
parameters.(Stefanski and Boos, 2002; Cole et al., 2023) Concretely, this
approach first defines the estimating equation $\Gamma_{i}(\eta,\delta)$ by:
$\sum\limits_{i=1}^{n}\Gamma_{i}(\eta,\delta)=\sum\limits_{i=1}^{n}\begin{bmatrix}\theta(\delta)-\ell_{i}(\delta)\hat{\xi_{i}}\\\
\theta_{0}-\hat{\tau_{i}}\\\
A_{i}S_{\pi_{D}}(Y_{i1},Y_{i0},D_{i},\mathbf{X}_{i};\alpha^{(D)},\sigma^{2})\\\
A_{i}S_{\mu_{1}}(Y_{i1},Y_{i0},D_{i},\mathbf{X}_{i};\beta^{(D)})\\\
S_{\pi_{A}}(Y_{i1},Y_{i0},\mathbf{X}_{i};\alpha^{(A)})\\\
(1-A_{i})S_{\mu_{0}}(Y_{i1},Y_{i0},\mathbf{X}_{i};\beta^{(0)})\end{bmatrix}=0$
where
$\eta=(\theta(\delta),\tau,\alpha^{(D)},\sigma^{2},\beta^{(D)},\alpha^{(A)},\beta^{(0)})$
are estimable parameters; $S_{\pi_{D}}$, $S_{\mu_{1}}$, $S_{\pi_{A}}$, and
$S_{\mu_{0}}$ are score equations for parametric nuisance models; and
$\ell_{i}(\delta)$ are the local linear kernel regression estimation
weights.(Wasserman, 2006) We then use the sandwich variance formula as:
$\hat{V}(\hat{\eta})=\\{\sum\limits_{i=1}^{n}\frac{\partial}{\partial\eta^{\prime}}\Gamma_{i}(\hat{\eta})\\}^{-1}\\{\sum\limits_{i=1}^{n}\Gamma_{i}^{\bigotimes
2}(\hat{\eta})\\{\sum\limits_{i=1}^{n}\frac{\partial}{\partial\eta}\Gamma_{i}^{\prime}(\hat{\eta})\\}^{-1}$
Finally, we can calculate
$\hat{V}(\hat{\Psi}(\delta))=V(\hat{\theta}(\delta)-\hat{\theta}_{0})=\hat{V}_{1,1}(\hat{\eta})+\hat{V}_{2,2}(\hat{\eta})-2\hat{V}_{1,2}(\hat{\eta})$
and derive $95\%$ confidence intervals under a normal approximation for each
$\delta$. The calculated sandwich estimator under linear
($\pi_{D},\mu_{1,\Delta},\mu_{0,\Delta}$) and logistic ($\pi_{A}$) models is
provided in the Appendix.
Still, this sandwich estimator only applies for a restricted set of nuisance
function models, which may be inadequate for complex dynamics, and rely on
correct specification of all nuisance functions. Alternatively, bootstrapping
approaches can be used to express pointwise uncertainty about the smoothed
kernel estimate under certain regularity conditions like those required for
consistency of $r_{n}$ described in the Appendix.(Wasserman, 2006) Further,
bootstrap approaches have been shown to better account for finite sample
variability and model misspecification than parametric formulations of
asymptotic approximations for previous doubly robust estimators.(Funk et al.,
2011) Throughout this paper, we implement a non-parametric weighted bootstrap
where we sample unit-specific, rather than observation-specific, weights to
account for correlation between multiple observations of the same unit. After
sampling these weights from an $exponential(1)$ distribution, we then scale
the weights such that the sum of weights in each treatment group are equal to
their observed sample sizes and utilize these weights for each fitted model
and population mean required in the estimation process. Allowing for small but
non-zero weights, unlike a discrete bootstrap sampling approach, is
particularly helpful for stabilizing bootstrap sample estimates which
otherwise may only have data from a few exposure levels. We then estimate the
$2.5$ and $97.5$ percentiles among 500 bootstrap replicates as our interval
bounds.
### 2.5 Multiple Observation Times
For simplicity, we introduced methods with one observation before the
intervention ($t=0$) and one observation after ($t=1$). In practice, we often
have multiple observation times within these periods that we would like to
incorporate additional information from. Following the design of our data
example, we consider multiple post-intervention observation times,
$m=1,...,M$, where each of these observation times has a corresponding pre-
intervention observation time (e.g., corresponding months of the year before
and after a policy implementation).(Hettinger et al., 2023) Further, to
distinguish from studies involving related to staggered adoption and dynamic
treatment regimes, we consider a single intervention time for all treated
units where the exposure is constant for all times after the intervention.
Presuming our parallel trends assumptions now hold between each of the $m$
pairs of corresponding observation times, we can calculate $m$-time specific
effects as defined in the $M=1$ setting and average them over all $m$.
Alternative approaches for multiple observation times will depend on the study
and we refer the reader to Callaway and Sant’Anna (2021) for frameworks under
more complex designs. When estimating outcome-based nuisance functions under
multiple observation times, practitioners must consider efficiency and
robustness trade-offs when deciding between modeling each observation time
separately or incorporating time into a single model. For propensity score
models, we utilize a single model to mimic that interventions and exposures
are only assigned once, but time-specific weights and their interpretation
have been further explored by Stuart et al. (2014). Finally, both mentioned
sandwich and bootstrapping uncertainty approaches are readily extensible to
this setting. Sandwich estimators require stacking additional estimating
equations for each observation time into the formulation of
$\Gamma_{i}(\eta,\delta)$, while described bootstrap approaches, which re-
sample units, already account for within-unit correlation induced by multiple
observations.(Li and Li, 2019)
## 3 Simulation Studies
We conducted numerical studies to examine the finite-sample properties of our
proposed methods relative to alternative approaches under different forms of
model misspecification. Specifically, our data generating mechanism, which
roughly matches the structure of our data example in the following section,
simulated normally distributed covariates,
$\mathbf{X_{i}}=(X_{i1},X_{i2},X_{i3},X_{i4})^{T}\sim
N(\mathbf{0},\mathbf{I}_{4})$
a binary intervention,
$\displaystyle A_{i}\mid\mathbf{X_{i}}\sim
Bern(expit(-0.3+0.15X_{i1}+0.15X_{i2}-0.15X_{i3}+0.25X_{i4}))$ $\displaystyle
expit(z)=exp(z)/[1+exp(z)]$
a normally distributed exposure among treated units,
$D_{i}\mid\mathbf{X_{i}},A=1\sim
N(\mu=6+0.45X_{i1}+0.4X_{i2}-0.45X_{i3}+0.9X_{i4},\sigma^{2}=4)$
a normally distributed outcome at time $t=0$,
$Y_{i0}\mid\mathbf{X_{i}},A_{i}\sim
N(\mu=10+0.4X_{i1}-X_{i2}+0.4X_{i3}+0.3X_{i4}+2A_{i},\sigma^{2}=0.3^{2})$
and a normally distributed outcome at time $t=1$ where the expected trend is
determined by confounders, intervention group, and exposure level,
$\displaystyle Y_{i1}\mid\mathbf{X_{i}},A_{i},D_{i}\sim
N(\mu=Y_{i0}+A_{i}\lambda_{1}(\mathbf{X_{i}},D_{i})+(1-A_{i})\lambda_{0}(\mathbf{X_{i}}),\sigma^{2}=0.1^{2})$
$\displaystyle\lambda_{1}(\mathbf{X_{i}},D_{i})=1+0.2X_{i1}+0.2X_{i2}+0.3X_{i3}-0.1X_{i4}+2A_{i}+$
$\displaystyle\hskip 113.81102ptD(0.1-0.1X_{i1}+0.1X_{i3}-0.01/10D^{2})$
$\displaystyle\lambda_{0}(\mathbf{X_{i}})=-3-X_{i1}+0.7X_{i2}+0.6X_{i3}-0.6X_{i4}$
To analyze the simulated data, we compared our proposed, multiply robust
method (MR) to four alternative estimators. First, we estimated a standard
linear two-way fixed effects (TWFE) model,
$Y_{it}=\beta_{0}+\boldsymbol{\beta}^{\prime}\mathbf{X_{i}}+\gamma_{T}t+\gamma_{A}A_{i}+\gamma_{D}A_{i}D_{i}+\tau_{A}tA_{i}+\tau_{D}tA_{i}D_{i}+\epsilon_{it}$,
and calculated an effect curve as
$\hat{\Psi}^{(twfe)}(\delta)=\hat{\tau}_{A}+\hat{\tau}_{D}\delta$. This is an
analogous approach to the model studied by Callaway et al. (2021) that does
not adjust for confounding between intervention or exposure levels and outcome
trends. Second, we estimated a flexible, confounding-naive difference-in-
differences estimator as
$\hat{\Psi}^{(naive)}(\delta)=\hat{E}[Y_{1}-Y_{0}|D=\delta,A=1]-\hat{E}[Y_{1}-Y_{0}|A=0]$,
where the first expectation is estimated with a local linear kernel regression
and the second with a sample mean. Then, we considered outcome regression and
IPW analogies to our proposed estimator. The outcome regression approach
relies entirely on specification of $\mu_{1,\Delta}$ and $\mu_{0,\Delta}$ and
estimates
$\hat{\Psi}^{(or)}(\delta)=\hat{E}[\hat{\mu}_{1,\Delta}(\mathbf{X})-\hat{\mu}_{0,\Delta}(\mathbf{X})|A=1]$
with a sample mean over treated units. The IPW approach relies entirely on
specification of $\pi_{D}$ and $\pi_{A}$, estimating
$\hat{\theta}^{(ipw)}(\delta)=\hat{E}[\hat{w}^{(1)}(Y_{1}-Y_{0})|A=1,D=\delta]$
with a local linear kernel regression and
$\hat{\theta}^{(ipw)}_{0}=\hat{E}[\hat{w}^{(0)}(Y_{1}-Y_{0})|A=0]$ with a
sample mean over control units, to get
$\hat{\Psi}^{(ipw)}(\delta)=\hat{\theta}^{(ipw)}(\delta)-\hat{\theta}^{(ipw)}_{0}$.
This is a similar approach to that of Han and Yu (2019) but using the
estimated propensity scores for weights rather than covariate adjustment.
We consider nuisance function misspecification by developing models under the
specified mean functions, but using $\mathbf{X}$ for covariates in our model
under a correct specification and $\mathbf{W}$ for covariates under an
incorrect specification, where $\mathbf{W}$ is the same covariate
transformation as given in Kang and Schafer (2007). For each of the 16
permuations of nuisance model specification, we then generate 1000 simulated
datasets for each of three sample sizes: $n=200$, $1000$, and $10000$.
Potential estimators are compared using integrated absolute mean bias and root
mean squared error (RMSE) where summations are taken over the density of $D$
as estimated from a super-population with $n=1,000,000$. The $10\%$ of mass at
each boundary are excluded from evaluation as these values may not frequently
be observed in simulations. Finally, for $n=200$ datasets, we assess finite
sample inference properties by calculating the integrated coverage
probabilities and interval width of confidence intervals derived from the
described sandwich and bootstrap variance approaches for our proposed,
multiply robust estimator.
### 3.1 Results
Figure 2: An example fit for one simulated dataset ($n=1000$) showing the true effect curve and estimated effect curves using the multiply robust (MR), confounding-naive (Naive), and two-way fixed effects (TWFE) approaches. A constant difference between the MR and Naive approaches represents confounding between treated and control groups while the increasing difference over values of $\delta$ represents confounding between units at different exposure levels. Table 1: Comparison of Bias (RMSE) of effect curve estimates under multiply robust (MR), outcome regression (OR), and inverse probability weighting (IPW) approaches in simulation scenarios varying by nuisance function specification ($n=200$). Green cells represent scenarios where both sources of confounding are correctly adjusted for asymptotically, whereas orange and red cells represent scenarios where one and both sources are not correctly adjusted for, respectively. Confounding-naive and two-way fixed effects estimators have Bias (RMSE) of 0.329 (0.169) and 0.384 (0.184) in all scenarios. Incorrect Models | MR | OR | IPW
---|---|---|---
None | 0.008 (0.019) | 0.002 (0.019) | 0.015 (0.064)
$\pi_{A}$ | 0.008 (0.019) | 0.002 (0.019) | 0.117 (0.09)
$\mu_{0,\Delta}$ | 0.034 (0.03) | 0.089 (0.046) | 0.015 (0.064)
$\pi_{D}$ | 0.008 (0.019) | 0.002 (0.019) | 0.112 (0.08)
$\mu_{1,\Delta}$ | 0.028 (0.054) | 0.138 (0.056) | 0.015 (0.064)
$\pi_{A},\pi_{D}$ | 0.008 (0.019) | 0.002 (0.019) | 0.149 (0.112)
$\mu_{0,\Delta},\mu_{1,\Delta}$ | 0.033 (0.063) | 0.186 (0.095) | 0.015 (0.064)
$\mu_{0,\Delta},\pi_{D}$ | 0.034 (0.03) | 0.089 (0.046) | 0.112 (0.08)
$\pi_{A},\mu_{1,\Delta}$ | 0.028 (0.054) | 0.138 (0.056) | 0.117 (0.09)
$\pi_{A},\mu_{0,\Delta}$ | 0.132 (0.053) | 0.089 (0.046) | 0.117 (0.09)
$\pi_{D},\mu_{1,\Delta}$ | 0.119 (0.068) | 0.138 (0.056) | 0.112 (0.08)
$\pi_{A},\mu_{0,\Delta},\pi_{D}$ | 0.131 (0.053) | 0.089 (0.046) | 0.149 (0.112)
$\pi_{A},\mu_{0,\Delta},\mu_{1,\Delta}$ | 0.140 (0.088) | 0.186 (0.095) | 0.117 (0.09)
$\pi_{A},\pi_{D},\mu_{1,\Delta}$ | 0.119 (0.068) | 0.138 (0.056) | 0.149 (0.112)
$\mu_{0,\Delta},\pi_{D},\mu_{1,\Delta}$ | 0.116 (0.075) | 0.186 (0.095) | 0.112 (0.08)
$\pi_{A},\mu_{0,\Delta},\pi_{D},\mu_{1,\Delta}$ | 0.179 (0.11) | 0.186 (0.095) | 0.149 (0.112)
Table 2: A comparison of Coverage Probabilities (Interval Widths) for confidence intervals derived from sandwich and bootstrap variances for the proposed, multiply robust estimator in simulation scenarios varying by nuisance function specification ($n=200$). Incorrect Models | Sandwich | Bootstrap
---|---|---
None | 99.1 (0.77) | 94.4 (0.52)
$\pi_{A},\pi_{D}$ | 99.0 (0.77) | 94.5 (0.52)
$\mu_{0,\Delta},\mu_{1,\Delta}$ | 96.2 (0.93) | 94.3 (0.85)
$\pi_{A},\mu_{0,\Delta},\pi_{D},\mu_{1,\Delta}$ | 85.2 (0.90) | 84.4 (0.87)
Results summarizing the performance of causal effect curve point estimates are
given in Table 1. In all analyzed scenarios, $\hat{\Psi}^{(twfe)}(\delta)$ and
$\hat{\Psi}^{(naive)}(\delta)$ exhibit significant bias as they fail to
account for confounding between the treated and control groups and between the
different exposure levels within the treated group. When $\hat{\pi}_{A}$ or
$\hat{\pi}_{D}$ are misspecified ($75\%$ of scenarios),
$\hat{\Psi}^{(ipw)}(\delta)$ demonstrates bias, the severity of which depends
upon the specific models that are misspecified. Similarly,
$\hat{\Psi}^{(or)}(\delta)$ is biased whenever $\hat{\mu}_{0,\Delta}$ or
$\hat{\mu}_{1,\Delta}$ are misspecified ($75\%$ of scenarios). On the
contrary, our proposed estimator, $\hat{\Psi}^{(mr)}(\delta)$, exhibits bias
only when both $\hat{\pi}_{A}$ and $\hat{\mu}_{0,\Delta}$ or both
$\hat{\pi}_{D}$ and $\hat{\mu}_{1,\Delta}$ are misspecified ($44\%$ of
scenarios). Further, $\hat{\Psi}^{(mr)}(\delta)$ correctly accounts for at
least one of the two sources of confounding as long as one of the four
nuisance functions are well-specified.
Under correct nuisance function specification, $\hat{\Psi}^{(mr)}(\delta)$ and
$\hat{\Psi}^{(or)}(\delta)$ demonstrate the lowest RMSE and thus most
efficient estimates. The RMSE of $\hat{\Psi}^{(ipw)}(\delta)$, roughly 5x that
of $\hat{\Psi}^{(mr)}(\delta)$ and $\hat{\Psi}^{(or)}(\delta)$ in well-
specified scenarios, demonstrates the instability of approaches that rely
solely on weighting even under correct model specification. Still,
$\hat{\Psi}^{(mr)}(\delta)$ performance may be subject to similar instability
when outcome models are misspecified.
Bootstrapped confidence intervals for $\hat{\Psi}^{(mr)}$ generally achieve
the specified coverage probabilities in scenarios with consistent effect
estimates, suggesting multiply robust inference in these simulations.
Robustness of inference has not been proven theoretically for bootstrap
approaches but has been observed frequently in practice (Funk et al., 2011).
The sandwich variance estimator is conservative under correct nuisance
function specification, likely due to the sample size considered ($n=200$) and
the asymptotic reliance of similar approaches noted previously (Lunceford and
Davidian, 2004). Additionally, the sandwich variance estimator is not robust
to any nuisance function misspecification due to its fully parametric
specification and only achieves the specified coverage bounds when all models
are correctly specified.
## 4 Application to Nutritional Excise Taxes
### 4.1 Philadelphia Beverage Tax Study
We applied the proposed methodology to estimate the effects of an excise tax
on sugary and sweetened beverages implemented in January 2017 in Philadelphia,
Pennsylvania. Previous studies have largely used difference-in-differences
methods to estimate the average effect of the tax on volume sales of taxed
beverages in Philadelphia and non-taxed neighboring county stores (Roberto et
al., 2019; Hettinger et al., 2023; Cawley et al., 2019). Although these
studies assessed the impact of the tax as a binary intervention, they
identified two significant mechanisms contributing to heterogeneous exposure
to the policy. First, as the tax was imposed on the distributor, each
individual store had the discretion to determine the extent of corresponding
price adjustments. While prior commentaries have speculated that consumers
might be more responsive to the awareness of the tax than actual price
fluctuations, this hypothesis has yet to be rigorously investigated. Second,
compelling evidence suggests a pattern of cross-border shopping in that
neighboring non-taxed counties demonstrated a concurrent increase in sales
during the tax implementation period. Prior studies have established
associations between distance to the border and magnitude of tax effects, but
lacked a robust causal framework to effectively control for confounding
factors between border proximity and beverage sales (Cawley et al., 2019;
Hettinger et al., 2023). A comprehensive understanding of the heterogeneity in
tax effects related to both of these exposures is crucial for informing the
design of future policies.
In the analysis, we considered 140 pharmacies from Philadelphia as our treated
group and 123 pharmacies from Baltimore and non-neighboring Pennsylvania
counties as our control group. Data provided by Information Resources Inc.
(IRI) and described previously includes volume sales of taxed beverages
aggregated in each 4-week period in the year prior to (2016) and after (2017)
tax implementation ($m=1,...,13$) (Muth et al., 2016; Roberto et al., 2019).
To account for variations in store size, we scaled volume sales for each store
by their average 4-week sales in 2016. This scaling not only mitigates many
differences in outcome trends between stores in our study, but also
facilitates attempts to generalize effects to regions where stores may have a
different magnitude of beverage sales.
To quantify the first source of exposure heterogeneity, we assessed the
weighted price of taxed beverages per ounce. The measure for price change was
determined by calculating the difference between the average metric in the
first 4-week period of 2017 (post-tax price) and the average metric across all
of 2016 (pre-tax price). Here, we excluded prices after January 2017 from our
calculation as these may be influenced by post-tax sales and could incorrectly
associate our exposure measure with effect heterogeneity. For the second
source of exposure heterogeneity, we calculated, for each store, the
percentage of adjacent zip codes also subject to the tax as a measure of
protection from cross-border shopping. Unfortunately, the de-identification of
stores in our data besides zip code prevented the calculation of a more
detailed distance or protection metric. When evaluating effects according to
each exposure, we considered several confounding variables including the
alternative exposure measure (for models on the treated group), the pre-tax
price and sales at a given store, and both the percentage of white residents
and average income per household in the zip-code.
We used the proposed kernel smoothing approach to estimate the full effect
curve flexibly and estimated our nuisance functions using SuperLearner to
limit the potential for model misspecification (van der Laan et al., 2007).
Specifically, we included a comprehensive ensemble of generalized linear
models, generalized additive models, Lasso, and boosting algorithms as
candidate learners for the conditional mean functions of our binomial
($\hat{\pi}_{A}$) and gaussian
($\hat{\pi}_{D},\hat{\mu}_{0,\Delta},\hat{\mu}_{1,\Delta}$) exposures and
outcomes. To model the conditional density function of the continuous
exposure, we followed previous methodologies to model the squared residuals
from the mean model, $\hat{\epsilon}_{i}^{2}$, using SuperLearner under the
same set of learners and then estimate the density of
$(D_{i}-\hat{D}_{i})/\sqrt{\hat{\epsilon}_{i}^{2}}$ (Hirano and Imbens, 2004;
Kreif et al., 2015). To account for the point mass at $D=100\%$ in the second
exposure, we estimated the density at this point separately by treating the
point as a binomial dependent variable.
### 4.2 Analysis of Effect Curves
Our exposure measures exhibited substantial heterogeneity. $10\%$ of
Philadelphia stores actually decreased their price of taxed beverages
following the tax, whereas $59\%$ increased their price by more than the tax
amount ($\$0.015$/oz.). $40\%$ of Philadelphia stores were $100\%$ surrounded
by other Philadelphia zip codes and the IQR of the remaining $60\%$ of stores
was $55-92\%$ surrounded. All confounding variables were significantly
associated with the intervention, exposure, and outcome trends in generalized
linear models where all confounding variables were included in an additive
fashion.
We first used our methodology to assess for parallel trends in the pre-
treatment period as described in a previous section by testing known placebo
effects for the trends between $m=1$ and $m=2,...,13$ in 2016. In spirit, we
treat January 2016 as the “pre-intervention”, $t=0$, time period and the
subsequent 2016 periods individually as “post-intervention”, $t=1$, time
periods. In addition to only serving as a proxy for the untestable
counterfactual parallel trends assumptions, these placebo tests do not
directly assess conditional parallel trends in the pre-treatment period as
such tests would require estimation of particular parameters in parametric
models (limiting confounder adjustment) or conditional effects by $\mathbf{X}$
(an active and challenging area of research in its own right). Instead, these
placebo tests assess whether any confounding exists on the population level
after adjusting for potential confounders using our methodology. Previous work
has used this approach to assess conditional parallel trends in the binary
setting (Hettinger et al., 2023). In our setting, our methods generally
estimated placebo effects with magnitudes between $10-30\%$ those of non-
placebo effects presented in the subsequent section, suggesting observed
deviations from parallel trends would explain only a portion of our estimated
effects. The only trend exhibiting high risk for violations of parallel trends
in the pre-treatment period came between periods $m=1$ and $m=13$, which may
indicate consumers adjusting shopping behavior prior to the tax, a violation
of Assumption 1 rather than Assumption 4.
Figure 3: Causal effect curve estimates representing the effect of the tax
where all Philadelphia stores change their price by $x$ cents per ounce.
Estimates are calculated using the multiply robust (MR), inverse probability
weighted (IPW), outcome regression (OR), and confounder-naive (Naive)
approaches with pointwise confidence intervals derived for the multiply robust
estimator from bootstrapped variance estimates (shaded).
Figure 4: Causal effect curve estimates representing the effect of the tax
where all Philadelphia stores are $x\%$ surrounded by taxed zip codes.
Estimates are calculated using the multiply robust (MR), inverse probability
weighted (IPW), outcome regression (OR), and confounder-naive (Naive)
approaches with pointwise confidence intervals derived for the multiply robust
estimator from bootstrapped variance estimates (shaded).
Effect curves for both exposure measures under the described multiply robust,
IPW, outcome regression, and confounder-naive approaches are visualized with
bootstrapped confidence intervals for the multiply robust estimator in Figures
3 and 4. When assessing effect curves related to specific price changes, we
saw strong evidence that consumers are responding to specific price changes
even after adjusting for observed confounders, regardless of estimator. If all
stores were to actually decrease their price following the tax, we estimated
that the tax would actually increase sales, whereas if all stores increased
their price by $\$0.02$/oz., we estimated that the tax would decrease sales by
about $25\%$. While these estimates demonstrate substantial influences of
economic competition, we likely would require incorporation of some sort of
economic network to truly interpret such effects under counterfactual
scenarios like where all pharmacies had decreased their prices since many of
these additional sales come from Philadelphia consumers unlikely to change
shopping habits under such a scenario. Still, policymakers could utilize this
curve to determine the relative elasticity of consumer shopping habits
conditional on the observed alternative price points when designing future
policies.
Estimated curves for effects of the tax under different exposures to cross-
border shopping may assist policy evaluators in understanding effect
heterogeneity resulting from cross-border shopping in Philadelphia. Further,
policymakers may use the estimated curve to assess what would have happened
had all neighboring counties also been taxed,
$\hat{\Psi}^{(dr)}(\delta=100\%)$, compared to if only some had been taxed,
e.g., , $\hat{\Psi}^{(dr)}(\delta=80\%)$. This counterfactual scenario was
previously explored by Lee et al. (2023), wherein they estimated that the tax
would still have had a substantial impact under this implementation. Rather
than utilizing a cross-border shopping exposure measure among Philadelphia
stores, their analytical approach exploited assumptions about which sale
increases in non-taxed neighboring counties would be returned to Philadelphia
had neighboring counties also been taxed. In our analysis, we similarly
estimated a significant tax effect in this counterfactual scenario, suggesting
that the tax would have had a $13\%$ effect on store-level sales had all
Philadelphia stores been fully surrounded by taxed zip codes, which falls
between the estimated effects ignoring confounding ($9\%$) and the estimated
aggregate effect over the observed distribution of doses ($17\%$). Notably,
$\hat{\Psi}^{(ipw)}(\delta)$ is quite volatile due to large weights accruing
near exposure boundaries, supporting the need for outcome models to improve
estimate stability.
## 5 Discussion
In this paper, we introduced a novel approach for estimating causal effect
curves within the framework of difference-in-differences designs. These effect
curves enable researchers to explore questions such as, “What would be the
average effect of a policy where all units were exposed to the policy at the
$\delta$-level?”, while accounting for multiple sources of observed
confounding commonly encountered in policy evaluations. Crucially, our
approach offers a flexible and multiply robust mechanism for confounder
adjustment without requiring any parametric assumptions about the form of the
effect curve. Further, our approach can incorporate general machine learning
and non-parametric modeling techniques by either imposing a limited set of
restrictions or utilizing sample splitting techniques (Bonvini and Kennedy,
2022; Chang, 2020). This flexibility and robustness are particularly vital
given the additional modeling complexities in this setting with a continuous
exposure and two sources of confounding compared to the standard binary
exposure setting. Moreover, by harnessing information from both
treatment/exposure and outcome models through an influence function-based
estimator, our approach enhances the efficiency of existing approaches that
rely solely on the propensity score.
While designed for difference-in-differences studies like the application
presented, the fundamental concepts of our approach have broader applications.
For example, one can readily extend our methodology to accommodate alternative
study designs, such as the controlled interrupted time series, by modifying
the presented parallel trends assumptions to the new design and altering the
derived influence function according to the newly identified conditional
expectations (Lopez Bernal et al., 2018). Further, our specification of
potential outcomes under a two-dimensional exposure vector resembles the
exposure mapping framework introduced by Aronow and Samii (2017). In this
framework, our continuous exposure, $D$, is analogous to their exposure
mapping function, $h_{it}(\mathbf{A})$, which represents the amount of
exposure received through one’s neighbors. As a result, our approach holds
promise for estimation of effects specified under interference, where current
approaches are highly sensitive to confounding (Clarke, 2017; Butts, 2021). To
fully realize these applications, however, future work will need to address
uncertainty estimation under such dependencies between units.
## Acknowledgements
This material is based upon work supported by the National Science Foundation
under Grant No. 2149716.
## Available Code
Code and an example simulated dataset are provided on GitHub at
https://github.com/garyhettinger/DiD-interference.
## References
* Pollack Porter et al. [2018] Keshia M. Pollack Porter, Lainie Rutkow, and Emma E. McGinty. The Importance of Policy Change for Addressing Public Health Problems. _Public Health Reports_ , 133(1_suppl):9S–14S, 11 2018. ISSN 0033-3549. doi:10.1177/0033354918788880.
* Chaloupka et al. [2019] Frank J. Chaloupka, Lisa M. Powell, and Kenneth E. Warner. The Use of Excise Taxes to Reduce Tobacco, Alcohol, and Sugary Beverage Consumption. _Annual Review of Public Health_ , 40(1):187–201, 4 2019. ISSN 0163-7525. doi:10.1146/annurev-publhealth-040218-043816.
* Cintron et al. [2022] Dakota W. Cintron, Nancy E. Adler, Laura M. Gottlieb, Erin Hagan, May Lynn Tan, David Vlahov, Madellena Maria Glymour, and Ellicott C. Matthay. Heterogeneous treatment effects in social policy studies: An assessment of contemporary articles in the health and social sciences. _Annals of Epidemiology_ , 70:79–88, 6 2022. ISSN 10472797. doi:10.1016/j.annepidem.2022.04.009.
* Heckman et al. [1997] J. J. Heckman, H. Ichimura, and P. E. Todd. Matching As An Econometric Evaluation Estimator: Evidence from Evaluating a Job Training Programme. _The Review of Economic Studies_ , 64(4):605–654, 10 1997. ISSN 0034-6527. doi:10.2307/2971733.
* Abadie [2005] Alberto Abadie. Semiparametric Difference-in-Differences Estimators. _The Review of Economic Studies_ , 72(1):1–19, 1 2005. ISSN 1467-937X. doi:10.1111/0034-6527.00321.
* Callaway et al. [2021] Brantly Callaway, Andrew Goodman-Bacon, and Pedro H. C. Sant’Anna. Difference-in-Differences with a Continuous Treatment. _arXiv_ , 7 2021.
* Li and Li [2019] Fan Li and Fan Li. Double-Robust Estimation in Difference-in-Differences with an Application to Traffic Safety Evaluation. _Observational Studies_ , 5(1):1–23, 2019. ISSN 2767-3324. doi:10.1353/obs.2019.0009.
* Sant’Anna and Zhao [2020] Pedro H.C. Sant’Anna and Jun Zhao. Doubly robust difference-in-differences estimators. _Journal of Econometrics_ , 219(1):101–122, 11 2020. ISSN 03044076. doi:10.1016/j.jeconom.2020.06.003.
* Han and Yu [2019] Bing Han and Hao Yu. Causal difference-in-differences estimation for evaluating the impact of semi-continuous medical home scores on health care for children. _Health Services and Outcomes Research Methodology_ , 19(1):61–78, 3 2019. ISSN 1387-3741. doi:10.1007/s10742-018-00195-9.
* Hirano and Imbens [2004] Keisuke Hirano and Guido W. Imbens. The Propensity Score with Continuous Treatments. pages 73–84. 7 2004. doi:10.1002/0470090456.ch7.
* Austin [2018] Peter C. Austin. Assessing the performance of the generalized propensity score for estimating the effect of quantitative or continuous exposures on binary outcomes. _Statistics in Medicine_ , 37(11):1874–1894, 5 2018. ISSN 0277-6715. doi:10.1002/sim.7615.
* Díaz and van der Laan [2013] Iván Díaz and Mark J. van der Laan. Targeted Data Adaptive Estimation of the Causal Dose–Response Curve. _Journal of Causal Inference_ , 1(2):171–192, 12 2013. ISSN 2193-3677. doi:10.1515/jci-2012-0005.
* Kennedy et al. [2017] Edward H. Kennedy, Zongming Ma, Matthew D. McHugh, and Dylan S. Small. Non-parametric methods for doubly robust estimation of continuous treatment effects. _Journal of the Royal Statistical Society: Series B (Statistical Methodology)_ , 79(4):1229–1245, 9 2017. ISSN 1369-7412. doi:10.1111/rssb.12212.
* Bonvini and Kennedy [2022] Matteo Bonvini and Edward H. Kennedy. Fast convergence rates for dose-response estimation. _arXiv_ , 7 2022.
* Zeldow and Hatfield [2021] Bret Zeldow and Laura A. Hatfield. Confounding and regression adjustment in <scp>difference-in-differences</scp> studies. _Health Services Research_ , 56(5):932–941, 10 2021. ISSN 0017-9124. doi:10.1111/1475-6773.13666.
* de Chaisemartin and D’Haultfoeuille [2022] Clément de Chaisemartin and Xavier D’Haultfoeuille. Two-Way Fixed Effects and Differences-in-Differences with Heterogeneous Treatment Effects: A Survey. Technical report, National Bureau of Economic Research, Cambridge, MA, 1 2022.
* Muñoz and van der Laan [2012] Iván Díaz Muñoz and Mark van der Laan. Population Intervention Causal Effects Based on Stochastic Interventions. _Biometrics_ , 68(2):541–549, 6 2012. ISSN 0006-341X. doi:10.1111/j.1541-0420.2011.01685.x.
* Bilinski and Hatfield [2018] Alyssa Bilinski and Laura A. Hatfield. Nothing to see here? Non-inferiority approaches to parallel trends and other model assumptions. _arXiv_ , 5 2018.
* Kennedy [2022] Edward H. Kennedy. Semiparametric doubly robust targeted double machine learning: a review. _arXiv_ , 3 2022.
* Wasserman [2006] Larry Wasserman. _All of Nonparametric Statistics_. Springer New York, New York, NY, 2006. ISBN 978-0-387-25145-5. doi:10.1007/0-387-30623-4.
* Lunceford and Davidian [2004] Jared K. Lunceford and Marie Davidian. Stratification and weighting via the propensity score in estimation of causal treatment effects: a comparative study. _Statistics in Medicine_ , 23(19):2937–2960, 10 2004. ISSN 0277-6715. doi:10.1002/sim.1903.
* Stefanski and Boos [2002] Leonard A Stefanski and Dennis D Boos. The Calculus of M-Estimation. _The American Statistician_ , 56(1):29–38, 2 2002.
* Cole et al. [2023] Stephen R Cole, Jessie K Edwards, Alexander Breskin, Samuel Rosin, Paul N Zivich, Bonnie E Shook-Sa, and Michael G Hudgens. Illustration of 2 Fusion Designs and Estimators. _American Journal of Epidemiology_ , 192(3):467–474, 2 2023. ISSN 0002-9262. doi:10.1093/aje/kwac067.
* Funk et al. [2011] Michele Jonsson Funk, Daniel Westreich, Chris Wiesen, Til Stürmer, M. Alan Brookhart, and Marie Davidian. Doubly Robust Estimation of Causal Effects. _American Journal of Epidemiology_ , 173(7):761–767, 4 2011. ISSN 1476-6256. doi:10.1093/aje/kwq439.
* Hettinger et al. [2023] Gary Hettinger, Christina Roberto, Youjin Lee, and Nandita Mitra. Estimation of Policy-Relevant Causal Effects in the Presence of Interference with an Application to the Philadelphia Beverage Tax. _arXiv_ , 1 2023.
* Callaway and Sant’Anna [2021] Brantly Callaway and Pedro H.C. Sant’Anna. Difference-in-Differences with multiple time periods. _Journal of Econometrics_ , 225(2):200–230, 12 2021. ISSN 03044076. doi:10.1016/j.jeconom.2020.12.001.
* Stuart et al. [2014] Elizabeth A. Stuart, Haiden A. Huskamp, Kenneth Duckworth, Jeffrey Simmons, Zirui Song, Michael E. Chernew, and Colleen L. Barry. Using propensity scores in difference-in-differences models to estimate the effects of a policy change. _Health Services and Outcomes Research Methodology_ , 14(4):166–182, 12 2014. ISSN 1387-3741. doi:10.1007/s10742-014-0123-z.
* Kang and Schafer [2007] Joseph D. Y. Kang and Joseph L. Schafer. Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data. _Statistical Science_ , 22(4), 11 2007. ISSN 0883-4237. doi:10.1214/07-STS227.
* Roberto et al. [2019] Christina A. Roberto, Hannah G. Lawman, Michael T. LeVasseur, Nandita Mitra, Ana Peterhans, Bradley Herring, and Sara N. Bleich. Association of a Beverage Tax on Sugar-Sweetened and Artificially Sweetened Beverages With Changes in Beverage Prices and Sales at Chain Retailers in a Large Urban Setting. _JAMA_ , 321(18):1799, 5 2019. ISSN 0098-7484. doi:10.1001/jama.2019.4249.
* Cawley et al. [2019] John Cawley, David Frisvold, Anna Hill, and David Jones. The impact of the Philadelphia beverage tax on purchases and consumption by adults and children. _Journal of Health Economics_ , 67:102225, 9 2019. ISSN 01676296. doi:10.1016/j.jhealeco.2019.102225.
* Muth et al. [2016] MK Muth, M Sweitzer, D Brown, K Capogrossi, S Karns, D Levin, A Okrent, P Siegel, and C Zhen. Understanding IRI household-based and store-based scanner data, 4 2016.
* van der Laan et al. [2007] Mark J. van der Laan, Eric C Polley, and Alan E. Hubbard. Super Learner. _Statistical Applications in Genetics and Molecular Biology_ , 6(1), 1 2007. ISSN 1544-6115. doi:10.2202/1544-6115.1309.
* Kreif et al. [2015] Noémi Kreif, Richard Grieve, Iván Díaz, and David Harrison. Evaluation of the Effect of a Continuous Treatment: A Machine Learning Approach with an Application to Treatment for Traumatic Brain Injury. _Health Economics_ , 24(9):1213–1228, 9 2015. ISSN 1057-9230. doi:10.1002/hec.3189.
* Lee et al. [2023] Youjin Lee, Gary Hettinger, and Nandita Mitra. Policy effect evaluation under counterfactual neighborhood interventions in the presence of spillover. _arXiv_ , 3 2023.
* Chang [2020] Neng-Chieh Chang. Double/debiased machine learning for difference-in-differences models. _The Econometrics Journal_ , 23(2):177–191, 5 2020. ISSN 1368-4221. doi:10.1093/ectj/utaa001.
* Lopez Bernal et al. [2018] James Lopez Bernal, Steven Cummins, and Antonio Gasparrini. The use of controls in interrupted time series studies of public health interventions. _International Journal of Epidemiology_ , 47(6):2082–2093, 12 2018. ISSN 0300-5771. doi:10.1093/ije/dyy135.
* Aronow and Samii [2017] Peter M. Aronow and Cyrus Samii. Estimating average causal effects under general interference, with application to a social network experiment. _The Annals of Applied Statistics_ , 11(4), 12 2017. ISSN 1932-6157. doi:10.1214/16-AOAS1005.
* Clarke [2017] Damian Clarke. Estimating Difference-in-Differences in the Presence of Spillovers. _Munich Personal RePEc Archive_ , 2017.
* Butts [2021] Kyle Butts. Difference-in-Differences Estimation with Spatial Spillovers. _arXiv_ , 5 2021.
|
# A State-Distribution Matching Approach to Non-Episodic Reinforcement
Learning
Archit Sharma Rehaan Ahmad Chelsea Finn
###### Abstract
While reinforcement learning (RL) provides a framework for learning through
trial and error, translating RL algorithms into the real world has remained
challenging. A major hurdle to real-world application arises from the
development of algorithms in an episodic setting where the environment is
reset after every trial, in contrast with the continual and non-episodic
nature of the real-world encountered by embodied agents such as humans and
robots. Enabling agents to learn behaviors autonomously in such non-episodic
environments requires that the agent to be able to conduct its own trials.
Prior works have considered an alternating approach where a forward policy
learns to solve the task and the backward policy learns to reset the
environment, but what initial state distribution should the backward policy
reset the agent to? Assuming access to a few demonstrations, we propose a new
method, MEDAL, that trains the backward policy to match the state distribution
in the provided demonstrations. This keeps the agent close to the task-
relevant states, allowing for a mix of easy and difficult starting states for
the forward policy. Our experiments show that MEDAL matches or outperforms
prior methods on three sparse-reward continuous control tasks from the EARL
benchmark, with 40% gains on the hardest task, while making fewer assumptions
than prior works. Code and videos are at: https://sites.google.com/view/medal-
arl/home
reinforcement learning, autonomous, discriminator
## 1 Introduction
A cornerstone of human and animal intelligence is the ability to learn
autonomously through trial and error. To that extent, reinforcement learning
(RL) presents a natural framework to develop learning algorithms for embodied
agents. Unfortunately, the predominant emphasis on episodic learning
represents a departure from the continual non-episodic nature of the real-
world, which presents multiple technical challenges. First, episodic training
undermines the autonomy of the learning agent by requiring repeated extrinsic
interventions to reset the environment after every trial, which can be both
time-consuming and expensive as these interventions may have to be conducted
by a human. Second, episodic training from narrow initial state distributions
can lead to less robust policies that are reliant on environment resets to
recover; e.g. Sharma et al. (2022) show that policies learned in episodic
settings with narrow initial state distributions are more sensitive to
perturbations than those trained in non-episodic settings.
Figure 1: An overview of our proposed method MEDAL (right) contrasting it with
forward-backward RL (Han et al., 2015; Eysenbach et al., 2017) (left). MEDAL
trains a backward policy $\pi_{b}$ to pull the agent back to the state
distribution defined by the demonstrations, enabling the forward policy
$\pi_{f}$ to the learn the task efficiently in contrast to FBRL that retrieves
the agent to the initial state distribution before every trial of $\pi_{f}$.
Prior works have found that conventional RL algorithms substantially
depreciate in performance when applied in non-episodic settings (Co-Reyes et
al., 2020; Zhu et al., 2020a; Sharma et al., 2022). Why do such algorithms
struggle to learn in non-episodic, autonomous RL (ARL) settings? Resetting the
environment after every single episode allows for natural repetition: the
agent can repeatedly practice the task under a narrow set of initial
conditions to incrementally improve the policy. Critically, algorithms
developed for episodic learning do not have to learn how to reach these
initial conditions in the first place. Thus, the main additional challenge in
non-episodic, autonomous RL settings is to enable the repetitive practice that
is necessary to learn an adept policy. For example, an autonomous robot that
is practicing how to close a door will also need to learn how to open a door.
Several recent works learn a backward policy to enable the main forward policy
to practice the task: for example, Han et al. (2015); Eysenbach et al. (2017)
propose a backward policy that learns to match the initial state distribution.
However, unlike the episodic setting, the agent can practice the task from any
initial state, and not just the narrow initial state distribution that is
usually provided by resets. Can the backward policy create starting conditions
that enable the forward policy to improve efficiently? It could be useful for
the agent to try the task both from “easy” states that are close to the goal
and harder states that are representative of the starting conditions at
evaluation. Easier and harder initial conditions can be seen as a curriculum
that simplifies exploration. Kakade & Langford (2002) provide a theoretical
discussion on how the initial state distribution affects the performance of
the learned policy. One of the results show that the closer the starting state
distribution is to the state distribution of the optimal policy $\rho^{*}$,
the faster the policy moves toward the optimal policy $\pi^{*}$. While an
oracle access to $\rho^{*}$ is rarely available, we often have access to a
modest set of demonstrations. In this work, we aim to improve autonomous RL by
learning a backward policy that matches the starting state distribution to the
state distribution observed in the demonstrations. This enables the agent to
practice the task from a variety of initial states, including some that are
possibly easier to explore from. An intuitive representation of the algorithm
is shown in Figure 1.
The primary contribution of our work is an autonomous RL algorithm Matching
Expert Distributions for Autonomous Learning (MEDAL), which learns a backward
policy that matches the state distribution of a small set of demonstrations,
in conjunction with a forward policy that optimizes the task reward. We use a
classification based approach that implicitly minimizes the distance between
the state distribution of the backward policy and the state distribution in
the demonstrations without requiring the density under either distribution. In
Section 5, we empirically analyze the performance of MEDAL on the Environments
for Autonomous RL (EARL) benchmark (Sharma et al., 2022). We find that MEDAL
matches or outperforms competitive baselines in all of the sparse-reward
environments, with a more than a 40% gain in success rate on the hardest task
where all other comparisons fail completely. Our ablations additionally
indicate the importance of matching the state distribution in the
demonstrations, providing additional empirical support for the hypothesis that
the expert state distribution constitutes a good starting state distribution
for learning a task.
## 2 Related Work
Autonomous RL. Using additional policies to enable autonomous learning goes
back to the works of (Rivest & Schapire, 1993) in context of finite state
automaton, also referred to as “homing strategies” in (Even-Dar et al., 2005)
in context of POMDPs. More recently, in context of continuous control, several
works propose autonomous RL methods targeting different starting distributions
to learn from: Han et al. (2015); Eysenbach et al. (2017) match the initial
state distribution, Zhu et al. (2020a) leverage state-novelty (Burda et al.,
2018) to create new starting conditions for every trial, and Sharma et al.
(2021) create a curriculum of starting states based on the performance of the
forward policy to accelerate the learning. In addition, (Xu et al., 2020; Lu
et al., 2020) leverage ideas from unsupervised skill discovery (Gregor et al.,
2016; Eysenbach et al., 2018; Sharma et al., 2019; Hazan et al., 2019; Campos
et al., 2020), with the former using it to create an adversarial initial state
distribution and the latter to tackle non-episodic lifelong learning with a
non-stationary task-distribution. Our work proposes a novel algorithm MEDAL
that, unlike these prior works, opts to match the starting distribution to the
state distribution of demonstrations. Value-accelerated Persistent RL (VaPRL)
(Sharma et al., 2021) also considers the problem of autonomous RL with a few
initial demonstrations. Unlike VaPRL, our algorithm does not rely on
relabeling transitions with new goals (Andrychowicz et al., 2017), and thus
does not require access to the functional form of the reward function,
eliminating the need for additional hyperparameters that require task-specific
tuning. A simple and task-agnostic ARL method would accelerate the development
of autonomous robotic systems, the benefits of such autonomy being
demonstrated by several recent works (Chatzilygeroudis et al., 2018; Gupta et
al., 2021; Smith et al., 2021; Ha et al., 2020; Bloesch et al., 2022).
Distribution Matching in RL. Critical to our method is matching the state
distribution of the demonstrations. Such a distribution matching perspective
is often employed in inverse RL (Ng et al., 2000; Ziebart et al., 2008, 2010;
Finn et al., 2016) and imitation learning (Ghasemipour et al., 2020; Argall et
al., 2009) or to encourage efficient exploration (Lee et al., 2019). More
recently, several works have leveraged implicit distribution matching by
posing a classification problem, pioneered in Goodfellow et al. (2014), to
imitate demonstrations (Ho & Ermon, 2016; Baram et al., 2017; Kostrikov et
al., 2018; Rafailov et al., 2021), to imitate sequences of observations
(Torabi et al., 2019; Zhu et al., 2020b), or to learn reward functions for
goal-reaching (Fu et al., 2018; Singh et al., 2019). Our work employs a
similar discriminator-based approach to encourage the state distribution
induced by the policy to match that of the demonstrations. Importantly, our
work focuses on creating an initial state distribution that the forward policy
can learn efficiently from, as opposed to these prior works that are designed
for the episodic RL setting. As the experiments in Section 5.2 and Section 5.3
show, naïve extensions of these methods to non-episodic settings don’t fare
well.
Accelerating RL using Demonstrations. There is rich literature on using
demonstrations to speed up reinforcement learning, especially for sparse
reward problems. Prior works have considering shaping rewards using
demonstrations (Brys et al., 2015), pre-training the policy (Rajeswaran et
al., 2017), using behavior cloning loss as a regularizer for policy gradients
(Rajeswaran et al., 2017) and $Q$-learning (Nair et al., 2018), and
initializing the replay buffer (Nair et al., 2018; Vecerik et al., 2017;
Hester et al., 2018). MEDAL leverages demonstrations to accelerate non-
episodic reinforcement learning by utilizing demo distribution to create
initial conditions for the forward policy. The techniques proposed in these
prior works are complimentary to our proposal, and can be leveraged for non-
episodic RL in general as well. Indeed, for all methods in our experiments,
the replay buffer is initialized with demonstrations.
## 3 Preliminaries
Autonomous Reinforcement Learning. We use the ARL framework for non-episodic
learning defined in Sharma et al. (2022), which we briefly summarize here.
Consider a Markov decision process
${\mathcal{M}\equiv(\mathcal{S},\mathcal{A},p,r,\rho_{0})}$, where
$\mathcal{S}$ denotes the state space, $\mathcal{A}$ denotes the action space,
${p:\mathcal{S}\times\mathcal{A}\times\mathcal{S}\mapsto\mathbb{R}_{\geq 0}}$
denotes the transition dynamics,
${r:\mathcal{S}\times\mathcal{A}\mapsto\mathbb{R}}$ denotes the reward
function and $\rho_{0}$ denotes the initial state distribution. The learning
algorithm $\mathbb{A}$ is defined as
${\mathbb{A}:\\{s_{i},a_{i},s_{i+1},r_{i}\\}_{i=0}^{t}\mapsto\\{a_{t},\pi_{t}\\}}$,
which maps the transitions collected in the environment until time $t$ to an
action $a_{t}$ and its best guess at the optimal policy
${\pi_{t}:\mathcal{S}\times\mathcal{A}\mapsto\mathbb{R}_{\geq 0}}$. First, the
initial state is sampled exactly once ($s_{0}\sim\rho_{0}$) at the beginning
of the interaction and the learning algorithm interacts with the environment
through the actions $a_{t}$ till $t\to\infty$. This is the key distinction
from an episodic RL setting where the environment resets to a state from the
initial state distribution after a few steps. Second, the action taken in the
environment does not necessarily come from $\pi_{t}$, for example, a backward
policy $\pi_{b}$ may generate the action taken in the environment.
ARL defines two metrics: Continuing Policy Evaluation measures the reward
accumulated by $\mathbb{A}$ over the course of training, defined as
${\mathbb{C}(\mathbb{A})=\lim_{h\to\infty}\frac{1}{h}\mathbb{E}\left[\sum_{t=0}^{h}r(s_{t},a_{t})\right]}$
and Deployed Policy Evaluation metric measures how quickly an algorithm
improves the output policy $\pi_{t}$ at the task defined by the reward
function $r$, defined as:
$\mathbb{D}(\mathbb{A})=\sum_{t=0}^{\infty}J(\pi^{*})-J(\pi_{t}),$ (1)
where
${J(\pi)=\mathbb{E}\left[\sum_{j=0}^{\infty}\gamma^{j}r(s_{j},a_{j})\right],s_{0}\sim\rho_{0},a_{t}\sim\pi(\cdot\mid
s_{t})}$, ${s_{t+1}\sim p(\cdot\mid s_{t},a_{t})}$ and
${\pi^{*}\in\operatorname*{arg\,max}_{\pi}J(\pi)}$. The goal for an algorithm
$\mathbb{A}$ is to minimize $\mathbb{D}(\mathbb{A})$, that is to bring
$J(\pi_{t})$ close to $J(\pi^{*})$ in the least number of samples possible.
Intuitively, minimizing $\mathbb{D}(\mathbb{A})$ corresponds to maximizing the
area under the curve for $J(\pi_{t})$ versus $t$.
$\mathbb{C}(\mathbb{A})$ corresponds to the more conventional average-reward
reinforcement learning. While algorithms are able to accumulate large rewards
during training, they do not necessarily recover the optimal policy in non-
episodic settings (Zhu et al., 2020a; Co-Reyes et al., 2020; Sharma et al.,
2022). In response, Sharma et al. (2022) introduce $\mathbb{D}(\mathbb{A})$ to
explicitly encourage algorithms to learn task-solving behaviors and not just
accumulate reward through training.
Imitation Learning via Distribution Matching. Generative Adversarial Networks
(Goodfellow, 2016) pioneered implicit distribution matching for distributions
where likelihood cannot be computed explicitly. Given a dataset of samples
$\\{x_{i}\\}_{i=1}^{N}$, where $x_{i}\sim p^{*}(\cdot)$ for some target
distribution $p^{*}$ over the data space $\mathcal{X}$, generative
distribution $p_{\theta}(\cdot)$ can be learned through the following minimax
optimization:
$\min_{p_{\theta}}\max_{D}\mathbb{E}_{x\sim p^{*}}\left[\log
D(x)\right]+\mathbb{E}_{x\sim p_{\theta}}\left[\log(1-D(x))\right]$ (2)
where $D:\mathcal{X}\mapsto[0,1]$ is discriminator solving a binary
classification problem. This can be shown to minimize the Jensen-Shannon
divergence, that is $\mathcal{D}_{\textrm{JS}}(p_{\theta}\mid\mid p^{*})$
(Goodfellow et al., 2014; Nowozin et al., 2016) by observing that the Bayes-
optimal classifier satisfies
$D^{*}(x)=\frac{p^{*}(x)}{p^{*}(x)+p^{\theta}(x)}$ (assuming that prior
probability of true data and fake data is balanced). Because we do not require
an explicit density under the generative distribution and only require the
ability to sample the distribution, this allows construction of imitation
learning methods such as GAIL (Ho & Ermon, 2016) which minimizes
$\mathcal{D}_{\textrm{JS}}(\rho^{\pi}(s,a)\mid\mid\rho^{*}(s,a))$, where the
policy $\pi$ is rolled out in the environment starting from initial state
distribution $\rho_{0}$ to generate samples from the state-action distribution
$\rho^{\pi}(s,a)$ and $\rho^{*}(s,a)$ is the target state-action distribution
of the demonstrations.
## 4 Matching Expert Distributions for Autonomous Learning (MEDAL)
Several prior works demonstrate the ineffectiveness of standard RL methods in
non-episodic settings (Co-Reyes et al., 2020; Zhu et al., 2020a; Sharma et
al., 2022). Adding noise to actions, for example $\epsilon$-greedy in DQN
(Mnih et al., 2015) or Gaussian noise in SAC (Haarnoja et al., 2018)), can be
sufficient for exploration in episodic setting where every trial starts from a
narrow initial state distribution. However, such an approach becomes
ineffective in non-episodic settings because the same policy is expected to
both solve the task and be sufficiently exploratory. As a result, a common
solution in non-episodic autonomous RL settings is to learn another policy in
addition to the forward policy $\pi_{f}$ that solves the task (Han et al.,
2015; Eysenbach et al., 2017; Zhu et al., 2020a): a backward policy $\pi_{b}$
that targets a set of states to explore solving the task from. More precisely,
the forward policy $\pi_{f}$ learns to solve the task from a state sampled
from $\rho^{b}$, the marginal state distribution of $\pi^{b}$. An appropriate
$\rho^{b}$ can improve the efficiency of learning $\pi_{f}$ by creating an
effective initial state distribution for it. What should the $\pi_{b}$
optimize? We discuss this question in Section 4.1 and a practical way to
optimize the suggested objective in Section 4.2. An overview of our proposed
algorithm is given in Section 4.3.
Figure 2: Comparison of sampling initial states $s_{0}$ from the state
distribution of the optimal policy $\rho^{*}$, with sampling the initial state
from the default distribution $\rho_{0}$ in the episodic setting. The episodic
return is computed by initializing the agent at $s_{0}\sim\rho_{0}$ in both
the cases. The former improves both the sample efficiency and the performance
of the final policy.
### 4.1 Finding Better Starting States
In episodic settings, $\pi_{f}$ always starts exploring from $\rho_{0}$, which
is the same distribution from which it will be evaluated. A natural objective
for $\pi_{b}$ then is to minimize the distance between $\rho^{b}$ and
$\rho_{0}$. And indeed, prior works have proposed this approach (Han et al.,
2015; Eysenbach et al., 2017) by learning a backward controller to retrieve
the initial state distribution $\rho_{0}$. While the initial state
distribution cannot be changed in the episodic setting, $\pi_{b}$ does not
have any restriction to match $\rho_{0}$ in the autonomous RL setting. Is
there a better initial state distribution to efficiently learn $\pi_{f}$ from?
Interestingly, Kakade & Langford (2002) provide a theoretical discussion on
how the initial state distribution affects the performance. The main idea is
that learning an optimal policy often requires policy improvement at states
that are unlikely to be visited. Creating a more uniform starting state
distribution can accelerate policy improvement by encouraging policy
improvement at those unlikely states. The formal statement can be found in
(Kakade & Langford, 2002, Corollary 4.5). Informally, the result states that
the upper bound on the difference between the optimal performance and that of
policy $\pi$ is proportional to
$\lVert\frac{\rho^{*}(s)}{\mu}\rVert_{\infty}$, where $\rho^{*}$ is the state
distribution of the optimal policy and $\mu$ is the initial state
distribution. This suggests that an initial state distribution $\mu$ that is
close to the optimal state distribution $\rho^{*}$ would enable efficient
learning. Intuitively, some initial states in the optimal state distribution
would simplify the exploration by being closer to high reward states, which
can be bootstrapped upon to learn faster from the harder initial states. To
empirically verify the theoretical results, we compare the learning speed of
RL algorithm in the episodic setting on tabletop organization (environment
details in Section 5) when starting from (a) the standard initial state
distribution, that is $s_{0}\sim\rho_{0}$, versus (b) states sampled from the
stationary distribution of the optimal policy, that is $s_{0}\sim\rho^{*}(s)$.
We find in Figure 2 that the latter not only improves the learning speed, but
also improves the performance by nearly 18%.
### 4.2 Resetting to Match the Expert State Distribution
The theoretical and empirical results in the previous section suggest that
$\pi_{f}$ should attempt to solve the task from an initial state distribution
that is close to $\rho^{*}(s)$, thus implying that $\pi_{b}$ should try to
match $\rho^{*}(s)$. How do we match $\rho^{b}$ to $\rho^{*}$? We will assume
access to a small set of samples from $\rho^{*}(s)$ in the form of
demonstrations $\mathcal{D}_{f}$. Because we are limited to sampling
$\rho^{b}$ and only have a fixed set of samples from $\rho^{*}$, we consider
the following optimization problem:
$\min_{\pi_{b}}\max_{C}\mathbb{E}_{s\sim\rho^{*}}\big{[}\log
C(s)\big{]}+\mathbb{E}_{s\sim\rho^{b}}\big{[}\log(1-C(s))\big{]}$ (3)
where $C:\mathcal{S}\mapsto[0,1]$ is a state-space classifier. This
optimization is very much reminiscent of implicit distribution matching
techniques used in (Goodfellow et al., 2014; Nowozin et al., 2016; Ho & Ermon,
2016; Ghasemipour et al., 2020) when only the samples are available and
densities cannot be explicitly measured. This can be interpreted as minimizing
the Jensen-Shannon divergence
$\mathcal{D}_{\textrm{JS}}(\rho^{b}\mid\mid\rho^{*})$. Following these prior
works, $C(s)$ solves a binary classification where $s\sim\rho^{*}$ has a label
$1$ and $s\sim\rho^{b}$ has a label $0$. Further, $\pi_{b}$ solves a RL
problem to maximize ${\mathbb{E}_{s\sim\rho^{b}}[r(s,a)]}$, where the reward
function $r(s,a)=-\log(1-C(s))$. Assuming sufficiently expressive non-
parametric function classes, $(\rho^{*},0.5)$ is a saddle point for the
optimization in Equation 3.
Relationship to Prior Imitation Learning Methods. GAIL (Ho & Ermon, 2016)
proposes to match the state-action distribution $\rho^{\pi}(s,a)$ to that of
the expert $\rho^{*}(s,a)$, that is minimize
$\mathcal{D}_{\textrm{JS}}(\rho^{\pi}(s,a)\mid\mid\rho^{*}(s,a))$. Prior works
have considered the problem of imitation learning when state-only observations
are available (Torabi et al., 2019; Zhu et al., 2020b) by minimizing
$\mathcal{D}_{f}(\rho^{\pi}(s,s^{\prime})\mid\mid\rho^{*}(s,s^{\prime}))$,
where $f$-divergence enables generalized treatment of different discrepancy
measures such KL-divergence of JS-divergence used in prior work (Nowozin et
al., 2016). In contrast to these works, our work proposes to minimize
$\mathcal{D}_{\textrm{JS}}(\rho^{\pi}(s)\mid\mid\rho^{*}(s))$. Furthermore,
state distribution matching is only used for the backward policy in our
algorithm, whereas the forward policy is maximizing return, as we summarize in
the next section. Finally, unlike prior works, the motivation for matching the
state distributions is to create an effective initial state distribution for
the forward policy $\pi_{f}$. Our experimental results in Section 5.2 suggest
that naively extending GAIL to non-episodic settings is not effective,
validating the importance of these key differences.
Algorithm 1 Matching Expert Distributions for Autonomous Learning (MEDAL)
require: forward demos $\mathcal{D}_{f}$;
optional: backward demos $\mathcal{D}_{b}$;
initialize: $\mathcal{R}_{f},\pi_{f}(a\mid s),Q^{\pi_{f}}(s,a)$; // forward
policy
initialize: $\mathcal{R}_{b},\pi_{b}(a\mid s),Q^{\pi_{b}}(s,a)$; // backward
policy
initialize: $C(s)$; // state-space discriminator
$\mathcal{R}_{f}\leftarrow\mathcal{R}_{f}\cup\mathcal{D}_{f},\mathcal{R}_{b}\leftarrow\mathcal{R}_{b}\cup\mathcal{D}_{b}$;
$s\sim\rho_{0}$; // sample initial state
while not done do
// run forward policy for a fixed number of steps or until goal is reached,
otherwise run backward policy
if forward then
$a\sim\pi_{f}(\cdot\mid s)$;
$s^{\prime}\sim p(\cdot\mid s,a),r\leftarrow r(s,a)$;
$\mathcal{R}_{f}\leftarrow\mathcal{R}_{f}\cup\\{(s,a,s^{\prime},r)\\}$;
update $\pi_{f},Q^{\pi_{f}}$;
else
$a\sim\pi_{b}(\cdot\mid s)$;
$s^{\prime}\sim p(\cdot\mid s,a),r\leftarrow-\log(1-C(s^{\prime}))$;
$\mathcal{R}_{b}\leftarrow\mathcal{R}_{b}\cup\\{(s,a,s^{\prime},r)\\}$;
update $\pi_{b},Q^{\pi_{b}}$;
end if
// train disriminator every $K$ steps
if train-discriminator then
// sample a batch of positives $S_{p}$ from the forward demos
$\mathcal{D}_{f}$, and a batch of negatives $S_{n}$ from backward replay
buffer $\mathcal{R}_{b}$
$S_{p}\sim\mathcal{D}_{f},S_{n}\sim\mathcal{R}_{b}$;
update $C$ on $S_{p}\cup S_{n}$;
end if
$s\leftarrow s^{\prime}$;
end while
### 4.3 MEDAL Overview
With these components in place, we now summarize our proposed algorithm,
_Matching Expert Distributions for Autonomous Learning_ (MEDAL). We
simultaneously learn the following components: a _forward policy_ that learns
to solve the task and will also be used for evaluation, a backward policy that
learns creates the initial state distribution for the forward policy by
matching the state distribution in the demonstrations, and finally a state-
space discriminator that learns to distinguish between the states visited by
the backward policy and the states visited in the demonstrations. MEDAL
assumes access to a set of forward demonstrations $\mathcal{D}_{f}$,
completing the task from the initial state distribution, and optionally, a set
of backward demonstrations $\mathcal{D}_{b}$ undoing the task back to the
initial state distribution. The forward policy $\pi_{f}$ is trained to
maximize $\mathbb{E}[{\sum_{t=0}^{\infty}\gamma^{t}r(s_{t},a_{t})}]$ and the
replay buffer for the forward policy is initialized using $\mathcal{D}_{f}$.
The backward policy $\pi_{b}$ trains to minimize
${\mathcal{D}_{\textrm{JS}}(\rho^{b}(s)\mid\mid\rho^{*}(s))}$ which translates
into maximizing
${-\mathbb{E}[\sum_{t=0}^{\infty}\gamma^{t}\log(1-C(s_{t+1}))]}$ and the
replay buffer for the backward policy is initialized using the backward
demonstrations $\mathcal{D}_{b}$, if available. Finally, the state-space
discriminator $C(s)$ trains to classify states sampled from the forward
demonstrations $\mathcal{D}_{f}$ with label $1$ and states visited by
$\pi_{b}$ as label $0$. Note, we are trying to match the state marginal of
policy $\pi_{b}$ (i.e. $\rho^{b}(s)$) to the optimal state distribution
$\rho^{*}(s)$ (approximated via forward demonstrations $\mathcal{D}_{f}$, not
backward demonstrations), thereby motivating the classification problem for
$C(s)$.
When interacting with the environment during training, we alternate between
collecting samples using $\pi_{f}$ for a fixed number of steps and collecting
samples using $\pi_{b}$ for a fixed number of steps. The policies can be
updated using any RL algorithm. The state-space discriminator $C(s)$ is
updated every $K$ steps collected in the environment, with the states visited
by $\pi_{b}$ being labeled as $0$ and states in $\mathcal{D}_{f}$ labeled as
$1$. The minibatch for updating the parameters of $C(s)$ is balanced to ensure
equal samples from $\rho^{*}(s)$ and $\rho^{b}(s)$. The pseudocode for MEDAL
is provided in Algorithm 1, and further implementation details can be found in
Appendix A.
Figure 3: Continuous-control environments from the EARL benchmark: (_left_)
Table-top organization (TO) where a gripper is tasked with moving a mug to one
of the four goal locations, (_center_) sawyer door closing (SD) where the
sawyer robot is tasked with closing the door, (_right_) sawyer peg insertion
(SP) where the robot is tasked with picking up the peg and inserting into the
goal location.
## 5 Experiments
In this section, we empirically analyze the performance of MEDAL to answer to
following questions: (1) How does MEDAL compare to other non-episodic,
autonomous RL methods? (2) Given the demonstrations, can existing imitation
learning methods suffice? (3) How important is it for the backward controller
to match the entire state distribution, instead of just the initial state
distribution?
Environments. To analyze these questions, we consider three sparse-reward
continuous-control environments from the EARL benchmark (Sharma et al., 2022).
The _table-top organization_ is a simplified manipulation environment where a
gripper is tasked to move the mug to one of four coasters. The _sawyer door
closing_ environment requires a sawyer robot arm to learn how to close a door
starting from various initial positions. The challenge in the ARL setting
arises from the fact that the agent has to open the door to practice closing
it again. Finally, the _sawyer peg insertion_ environment requires the sawyer
robot arm to pick up a peg and insert it into a designated goal location. This
is a particularly challenging environment as the autonomously operating robot
can push the peg into places where it can be hard to retrieve it back, a
problem that is not encountered in the episodic setting as the environment is
reset to the initial state distribution every few hundred steps.
Evaluation. We follow the evaluation protocol laid down in the EARL benchmark.
All algorithms are reset to a state $s_{0}\sim\rho_{0}$ and interact with
their environments almost fully autonomously thereon, only being reset to an
initial state intermittently after several hundreds of thousands of steps of
interaction. Since our objective is to acquire task policies in a sample
efficient way, we will focus on deployed policy evaluation. Specifically, we
approximate
$J(\pi_{t})=\mathbb{E}[\sum_{t=0}^{\infty}\gamma^{t}r(s_{t},a_{t})]$ by
averaging the return of the policy over $10$ episodes starting from
$s_{0}\sim\rho_{0}$, every $10,000$ steps collected in the training
environment. Note, the trajectories collected for evaluation are not provided
to the learning algorithm $\mathbb{A}$. For all considered environments, the
reward functions are sparse in nature and correspondingly, EARL provides a
small set of demonstrations to the algorithms, that correspond to doing and
undoing the task (a total of $10$-$30$ demonstrations depending on the
environment). Environment specific details such as reward functions and
intermittent resets can be found in Appendix B.
### 5.1 Benchmarking MEDAL on EARL
Method | Tabletop | Sawyer | Sawyer
---|---|---|---
| Organization | Door | Peg
_naïve RL_ | 0.32 (0.17) | 0.00 (0.00) | 0.00 (0.00)
_FBRL_ | 0.94 (0.04) | 1.00 (0.00) | 0.00 (0.00)
_R3L_ | 0.96 (0.04) | 0.54 (0.18) | 0.00 (0.00)
_VaPRL_ | 0.98 (0.02) | 0.94 (0.05) | 0.00 (0.00)
_MEDAL_ | 0.98 (0.02) | 1.00 (0.00) | 0.40 (0.16)
_oracle RL_ | 0.80 (0.11) | 1.00 (0.00) | 1.00 (0.00)
Table 1: Average return of the final learned policy. Performance is averaged
over 5 random seeds. The mean and and the standard error are reported, with
the best performing entry in bold. For all domains, $1.0$ indicates the
maximum performance and $0.0$ indicates minimum performance.
First, we benchmark our proposed method MEDAL on the aforementioned EARL
environments against state-of-the-art non-episodic ARL methods.
Comparisons. We briefly review the methods benchmarked on EARL, which MEDAL
will be compared against: (1) forward-backward RL (FBRL) (Han et al., 2015;
Eysenbach et al., 2017), where the backward policy recovers the initial state
distribution; (2) value-accelerated persistent RL (VaPRL) (Sharma et al.,
2021), where the backward policy creates a curriculum based on the forward
policy’s performance; (3) R3L (Zhu et al., 2020a) has a backward policy that
optimizes a state-novelty reward (Burda et al., 2018) to encourage the forward
policy to solve the tasks from new states in every trial; (4) naïve RL
represents the episodic RL approach where only a forward policy optimizes the
task-reward throughout training; and finally (5) oracle RL is the same
episodic RL baseline but operating in the episodic setting. For a fair
comparison, the forward policy for all baselines use SAC (Haarnoja et al.,
2018), and the replay buffer is always initialized with the forward
demonstrations. Further, the replay buffers for backward policies in FBRL,
VaPRL is also initialized with the backward demos. The replay buffer of the
backward policy in R3L is not initialized with backward demos as it will
reduce the novelty of the states in the backward demos for the RND reward
without the backward policy ever visiting those states.
Figure 4: Performance of each method on (left) the table-top organization
environment, (center) the sawyer door closing environment, and (right) the
sawyer peg environment. Plots show learning curves with mean and standard
error over 5 random seeds.
It’s important to note that some of these comparisons make additional
assumptions compared to MEDAL:
* •
oracle RL operates in the episodic setting, that is the environment is reset
to a state from the initial state distribution every few hundred steps. The
baseline is included as a reference to compare performance of baselines in ARL
versus the conventional episodic setting. It also enables us to compare the
performance of conventional RL algorithms when moving from the episodic
setting to the ARL setting, by comparing the performance of oracle RL and
naïve RL.
* •
VaPRL relies on relabeling goals which requires the ability to query the
reward function for any arbitrary state and goal (as the VaPRL curriculum can
task the agent to reach arbitrary goals from the demonstrations).
Additionally, VaPRL has a task specific hyperparameter that controls how
quickly the curriculum moves towards the initial state distribution.
In a real-world settings, where the reward function often needs to be learned
as well (for example from images), these assumptions can be detrimental to
their practical application. While FBRL also requires an additional reward
function to reach the initial state distribution, the requirement is not as
steep. Additionally, we consider a version of FBRL that learns this reward
function in Section 5.3. However, the ability to query the reward function for
arbitrary states and goals, as is required by VaPRL, can be infeasible in
practice. The impact of these additional assumptions cannot be overstated, as
the primary motivation for the autonomous RL framework is to be representative
of real-world RL training.
Results. Table 1 shows the performance of the final forward policy, and Figure
4 shows the deployed performance of the forward policy versus the training
time for different methods. MEDAL consistently outputs the best performing
final policy, as can be seen in Table 1. Particularly notable is the
performance on sawyer peg insertion, where the final policy learned by MEDAL
gets 40% success rate on average, while all other methods fail completely.
With the exception of VaPRL on tabletop organization, MEDAL also learns more
efficiently compared to any of the prior methods. Notably, MEDAL substantially
reduces the sample efficiency gap between ARL methods and episodic methods on
sawyer door closing.
We posit two reasons for the success of MEDAL: (a) Learning a backward policy
that retrieves the agent close to the task distribution enables efficient
exploration, producing the speedup in performance. (b) Bringing the agent
closer to the state distribution implicit in the demonstrations may be easier
to maximize compared to other objectives, for example, retrieving the agent to
the initial state distribution.
Figure 5: MEDAL in comparison to imitation learning methods on tabletop
organization and sawyer door closing. Behavior cloning (BC) does not fare
well, suggesting the importance of online data collection. The success of
online imitation learning methods such as GAIL in episodic settings does not
translate to the non-episodic ARL setting, as indicated by the substantial
drop in performance of naïve GAIL compared to oracle GAIL.
### 5.2 Imitation Learning
Given that MEDAL assumes access to a set of demonstrations, a natural
alternative to consider is imitation learning. In this section, we focus our
experiments on the tabletop organization environment. We first test how a
behavior cloning fares (BC). Results in Figure 5 suggest that behavior cloning
does not do well on tabletop organization, completely failing to solve the
task and leaves substantial room for improvement on sawyer door. This is to be
expected as EARL provides only a small number of demonstrations, and errors
compounding over time from imperfect policies generally leads to poor
performance. How do imitation learning methods with online data collection
fare? We consider an off-policy version of GAIL, Discriminator Actor-Critic
(DAC) (Kostrikov et al., 2018), which matches $\rho^{\pi}(s,a)$ to
$\rho^{*}(s,a)$ with an implicit distribution matching approach similar to
ours. Assuming that $\rho^{\pi}(s,a)$ can match $\rho^{*}(s,a)$, the method
should in principle recover the optimal policy – there is nothing specific
about GAIL that restricts it to the episodic setting. However, as the results
in Figure 5 suggest, there is a substantial drop in performance when running
GAIL in episodic setting (oracle GAIL) versus the non-episodic ARL setting
(naïve GAIL). While such a distribution matching could succeed, naïvely
extending the methods to the ARL setting is not as successful, suggesting that
it may require an additional policy (similar to the backward policy) to be
more effective.
### 5.3 The Choice of State Distribution
The key element in MEDAL is matching the state distribution of the backward
policy to the states in the demonstrations. To isolate the role of our
proposed scheme of minimizing
$\mathcal{D}_{\textrm{JS}}(\rho^{b}\mid\mid\rho^{*})$, we compare it to an
alternate method that minimizes
$\mathcal{D}_{\textrm{JS}}(\rho^{b}\mid\mid\rho_{0})$, i.e., matching the
initial state distribution $\rho_{0}$ instead of $\rho^{*}$. This makes
exactly one change to MEDAL: instead of sampling positives for the
discriminator $C(s)$ from forward demonstrations $\mathcal{D}_{f}$, the
positives are sampled from $\rho_{0}$. Interestingly, this also provides a
practically realizable implementation of FBRL, as it removes the requirement
of the additional reward function required for the learning a backward policy
to reach the initial state distribution. We call this method FBRL + VICE as
VICE (Singh et al., 2019) enables learning a goal reaching reward function
using a few samples of the goal distribution, in this case the goal
distribution for $\pi_{b}$ being $\rho_{0}$. As can be seen in Figure 6, the
FBRL + VICE learns slower than MEDAL, highlighting the importance of matching
the entire state distribution as done in MEDAL.
Figure 6: Isolating the effect of matching demonstration data. The speed up of
MEDAL compared to FBRL + VICE, which matches the initial state distribution,
suggests that the performance gains of MEDAL can be attributed to the better
initial state distribution created by the backward controller.
## 6 Conclusion
We propose MEDAL, an autonomous RL algorithm that learns a backward policy to
match the expert state distribution using an implicit distribution matching
approach. Our empirical analysis indicates that this approach creates an
effective initial state distribution for the forward policy, improving both
the performance and the efficiency. The simplicity of MEDAL also makes it more
amenable for the real-world, not requiring access to additional reward
functions.
MEDAL assumes access to a (small) set of demonstrations, which may not be
feasible in several real-world scenarios. Identifying good initial state
distributions without relying on a set of demonstrations would increase the
applicability of MEDAL. Similarly, in applications where safe exploration is a
requirement, MEDAL can be adapted to constrain the forward policy such that it
stays close to the task-distribution defined by the demonstrations. While
MEDAL pushes further the improvements in ARL, as exemplified by the reduction
of sample efficiency gap on sawyer door closing results, there is still a
substantial gap in performance between ARL methods and oracle RL on sawyer
peg, motivating the search for better methods.
## References
* Andrychowicz et al. (2017) Andrychowicz, M., Wolski, F., Ray, A., Schneider, J., Fong, R., Welinder, P., McGrew, B., Tobin, J., Abbeel, P., and Zaremba, W. Hindsight experience replay. _arXiv preprint arXiv:1707.01495_ , 2017.
* Argall et al. (2009) Argall, B. D., Chernova, S., Veloso, M., and Browning, B. A survey of robot learning from demonstration. _Robotics and autonomous systems_ , 57(5):469–483, 2009.
* Baram et al. (2017) Baram, N., Anschel, O., Caspi, I., and Mannor, S. End-to-end differentiable adversarial imitation learning. In _International Conference on Machine Learning_ , pp. 390–399. PMLR, 2017.
* Bloesch et al. (2022) Bloesch, M., Humplik, J., Patraucean, V., Hafner, R., Haarnoja, T., Byravan, A., Siegel, N. Y., Tunyasuvunakool, S., Casarini, F., Batchelor, N., et al. Towards real robot learning in the wild: A case study in bipedal locomotion. In _Conference on Robot Learning_ , pp. 1502–1511. PMLR, 2022.
* Brys et al. (2015) Brys, T., Harutyunyan, A., Suay, H. B., Chernova, S., Taylor, M. E., and Nowé, A. Reinforcement learning from demonstration through shaping. In _Twenty-fourth international joint conference on artificial intelligence_ , 2015.
* Burda et al. (2018) Burda, Y., Edwards, H., Storkey, A., and Klimov, O. Exploration by random network distillation. _arXiv preprint arXiv:1810.12894_ , 2018.
* Campos et al. (2020) Campos, V., Trott, A., Xiong, C., Socher, R., Giró-i Nieto, X., and Torres, J. Explore, discover and learn: Unsupervised discovery of state-covering skills. In _International Conference on Machine Learning_ , pp. 1317–1327. PMLR, 2020.
* Chatzilygeroudis et al. (2018) Chatzilygeroudis, K., Vassiliades, V., and Mouret, J.-B. Reset-free trial-and-error learning for robot damage recovery. _Robotics and Autonomous Systems_ , 100:236–250, 2018.
* Co-Reyes et al. (2020) Co-Reyes, J. D., Sanjeev, S., Berseth, G., Gupta, A., and Levine, S. Ecological reinforcement learning. _arXiv preprint arXiv:2006.12478_ , 2020.
* Even-Dar et al. (2005) Even-Dar, E., Kakade, S. M., and Mansour, Y. Reinforcement learning in pomdps without resets. 2005\.
* Eysenbach et al. (2017) Eysenbach, B., Gu, S., Ibarz, J., and Levine, S. Leave no trace: Learning to reset for safe and autonomous reinforcement learning. _arXiv preprint arXiv:1711.06782_ , 2017.
* Eysenbach et al. (2018) Eysenbach, B., Gupta, A., Ibarz, J., and Levine, S. Diversity is all you need: Learning skills without a reward function. _arXiv preprint arXiv:1802.06070_ , 2018.
* Finn et al. (2016) Finn, C., Levine, S., and Abbeel, P. Guided cost learning: Deep inverse optimal control via policy optimization. In _International conference on machine learning_ , pp. 49–58. PMLR, 2016.
* Fu et al. (2018) Fu, J., Singh, A., Ghosh, D., Yang, L., and Levine, S. Variational inverse control with events: A general framework for data-driven reward definition. _arXiv preprint arXiv:1805.11686_ , 2018.
* Ghasemipour et al. (2020) Ghasemipour, S. K. S., Zemel, R., and Gu, S. A divergence minimization perspective on imitation learning methods. In _Conference on Robot Learning_ , pp. 1259–1277. PMLR, 2020.
* Goodfellow (2016) Goodfellow, I. Nips 2016 tutorial: Generative adversarial networks. _arXiv preprint arXiv:1701.00160_ , 2016.
* Goodfellow et al. (2014) Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., and Bengio, Y. Generative adversarial nets. _Advances in neural information processing systems_ , 27, 2014.
* Gregor et al. (2016) Gregor, K., Rezende, D. J., and Wierstra, D. Variational intrinsic control. _arXiv preprint arXiv:1611.07507_ , 2016.
* Gupta et al. (2021) Gupta, A., Yu, J., Zhao, T. Z., Kumar, V., Rovinsky, A., Xu, K., Devlin, T., and Levine, S. Reset-free reinforcement learning via multi-task learning: Learning dexterous manipulation behaviors without human intervention. _arXiv preprint arXiv:2104.11203_ , 2021.
* Ha et al. (2020) Ha, S., Xu, P., Tan, Z., Levine, S., and Tan, J. Learning to walk in the real world with minimal human effort. _arXiv preprint arXiv:2002.08550_ , 2020.
* Haarnoja et al. (2018) Haarnoja, T., Zhou, A., Abbeel, P., and Levine, S. Soft actor-critic: Off-policy maximum entropy deep reinforcement learning with a stochastic actor. In _International conference on machine learning_ , pp. 1861–1870. PMLR, 2018.
* Han et al. (2015) Han, W., Levine, S., and Abbeel, P. Learning compound multi-step controllers under unknown dynamics. In _2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_ , pp. 6435–6442. IEEE, 2015.
* Hazan et al. (2019) Hazan, E., Kakade, S., Singh, K., and Van Soest, A. Provably efficient maximum entropy exploration. In _International Conference on Machine Learning_ , pp. 2681–2691. PMLR, 2019.
* Hester et al. (2018) Hester, T., Vecerik, M., Pietquin, O., Lanctot, M., Schaul, T., Piot, B., Horgan, D., Quan, J., Sendonaris, A., Osband, I., et al. Deep q-learning from demonstrations. In _Proceedings of the AAAI Conference on Artificial Intelligence_ , volume 32, 2018.
* Ho & Ermon (2016) Ho, J. and Ermon, S. Generative adversarial imitation learning. _Advances in neural information processing systems_ , 29:4565–4573, 2016.
* Kakade & Langford (2002) Kakade, S. and Langford, J. Approximately optimal approximate reinforcement learning. In _In Proc. 19th International Conference on Machine Learning_. Citeseer, 2002.
* Kostrikov et al. (2018) Kostrikov, I., Agrawal, K. K., Dwibedi, D., Levine, S., and Tompson, J. Discriminator-actor-critic: Addressing sample inefficiency and reward bias in adversarial imitation learning. _arXiv preprint arXiv:1809.02925_ , 2018.
* Lee et al. (2019) Lee, L., Eysenbach, B., Parisotto, E., Xing, E., Levine, S., and Salakhutdinov, R. Efficient exploration via state marginal matching. _arXiv preprint arXiv:1906.05274_ , 2019.
* Lu et al. (2020) Lu, K., Grover, A., Abbeel, P., and Mordatch, I. Reset-free lifelong learning with skill-space planning. _arXiv preprint arXiv:2012.03548_ , 2020.
* Mnih et al. (2015) Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. A., Veness, J., Bellemare, M. G., Graves, A., Riedmiller, M., Fidjeland, A. K., Ostrovski, G., et al. Human-level control through deep reinforcement learning. _nature_ , 518(7540):529–533, 2015.
* Nair et al. (2018) Nair, A., McGrew, B., Andrychowicz, M., Zaremba, W., and Abbeel, P. Overcoming exploration in reinforcement learning with demonstrations. In _2018 IEEE international conference on robotics and automation (ICRA)_ , pp. 6292–6299. IEEE, 2018.
* Ng et al. (2000) Ng, A. Y., Russell, S. J., et al. Algorithms for inverse reinforcement learning. In _Icml_ , volume 1, pp. 2, 2000.
* Nowozin et al. (2016) Nowozin, S., Cseke, B., and Tomioka, R. f-gan: Training generative neural samplers using variational divergence minimization. In _Proceedings of the 30th International Conference on Neural Information Processing Systems_ , pp. 271–279, 2016.
* Rafailov et al. (2021) Rafailov, R., Yu, T., Rajeswaran, A., and Finn, C. Visual adversarial imitation learning using variational models. _Advances in Neural Information Processing Systems_ , 34, 2021.
* Rajeswaran et al. (2017) Rajeswaran, A., Kumar, V., Gupta, A., Vezzani, G., Schulman, J., Todorov, E., and Levine, S. Learning complex dexterous manipulation with deep reinforcement learning and demonstrations. _arXiv preprint arXiv:1709.10087_ , 2017.
* Rivest & Schapire (1993) Rivest, R. L. and Schapire, R. E. Inference of finite automata using homing sequences. _Information and Computation_ , 103(2):299–347, 1993.
* Sharma et al. (2019) Sharma, A., Gu, S., Levine, S., Kumar, V., and Hausman, K. Dynamics-aware unsupervised discovery of skills. _arXiv preprint arXiv:1907.01657_ , 2019.
* Sharma et al. (2021) Sharma, A., Gupta, A., Levine, S., Hausman, K., and Finn, C. Autonomous reinforcement learning via subgoal curricula. _Advances in Neural Information Processing Systems_ , 34, 2021.
* Sharma et al. (2022) Sharma, A., Xu, K., Sardana, N., Gupta, A., Hausman, K., Levine, S., and Finn, C. Autonomous reinforcement learning: Formalism and benchmarking. _International Conference on Learning Representations_ , 2022.
* Singh et al. (2019) Singh, A., Yang, L., Hartikainen, K., Finn, C., and Levine, S. End-to-end robotic reinforcement learning without reward engineering. _arXiv preprint arXiv:1904.07854_ , 2019.
* Smith et al. (2021) Smith, L., Kew, J. C., Peng, X. B., Ha, S., Tan, J., and Levine, S. Legged robots that keep on learning: Fine-tuning locomotion policies in the real world. _arXiv preprint arXiv:2110.05457_ , 2021.
* Torabi et al. (2019) Torabi, F., Warnell, G., and Stone, P. Adversarial imitation learning from state-only demonstrations. In _Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems_ , pp. 2229–2231, 2019.
* Vecerik et al. (2017) Vecerik, M., Hester, T., Scholz, J., Wang, F., Pietquin, O., Piot, B., Heess, N., Rothörl, T., Lampe, T., and Riedmiller, M. Leveraging demonstrations for deep reinforcement learning on robotics problems with sparse rewards. _arXiv preprint arXiv:1707.08817_ , 2017.
* Xu et al. (2020) Xu, K., Verma, S., Finn, C., and Levine, S. Continual learning of control primitives: Skill discovery via reset-games. _arXiv preprint arXiv:2011.05286_ , 2020.
* Zhang et al. (2017) Zhang, H., Cisse, M., Dauphin, Y. N., and Lopez-Paz, D. mixup: Beyond empirical risk minimization. _arXiv preprint arXiv:1710.09412_ , 2017.
* Zhu et al. (2020a) Zhu, H., Yu, J., Gupta, A., Shah, D., Hartikainen, K., Singh, A., Kumar, V., and Levine, S. The ingredients of real world robotic reinforcement learning. In _International Conference on Learning Representations_ , 2020a.
* Zhu et al. (2020b) Zhu, Z., Lin, K., Dai, B., and Zhou, J. Off-policy imitation learning from observations. In _the Thirty-fourth Annual Conference on Neural Information Processing Systems (NeurIPS 2020)_ , 2020b.
* Ziebart et al. (2008) Ziebart, B. D., Maas, A. L., Bagnell, J. A., Dey, A. K., et al. Maximum entropy inverse reinforcement learning. In _Aaai_ , volume 8, pp. 1433–1438. Chicago, IL, USA, 2008.
* Ziebart et al. (2010) Ziebart, B. D., Bagnell, J. A., and Dey, A. K. Modeling interaction via the principle of maximum causal entropy. In _ICML_ , 2010.
## Appendix A MEDAL Implementation
MEDAL is implemented with TF-Agents, built on SAC as the base RL algorithm.
Hyperparameters follow the default values: initial collect steps: 10,000,
batch size sampled from replay buffer for updating policy and critic: 256,
steps collected per iteration: 1, trained per iteration: 1, discount factor:
0.99, learning rate: $3e-4$ (for critics, actors, and discriminator). The
actor and critic network were parameterized as neural networks with two hidden
layers each of size $256$. For the discriminator, it was parameterized as a
neural network with one hidden layer of size $128$. This discriminator is
updated once every 10 collection steps for all environments. Due to a small
positive dataset, mixup (Zhang et al., 2017) is used as a regularization
technique on the discriminator for all environments. Additionally, the batch
size for the discriminator is set to $800$ for all environments as this
significantly larger value was found to stabilize training.
Another choice that improved the stability was the choice of reward function
for the backward controller: both $r(s,a)=-\log(1-C(s))$ and
$r(s,a)=\log(C(s))$ preserve the saddle point $(\rho^{*},0.5)$ for the
optimization in Equation 3. However, as can be seen in Figure 7,
$r(s,a)=-\log(1-C(s))$ leads to both better and stable performance. We
hypothesize that this is due to smaller gradients of the $-\log(1-C(s))$ when
$C(s)\leq 0.5$, which is where the discriminator is expected to be for most of
the training as the discriminator can easily distinguish between expert states
and those of the backward policy to begin with.
Figure 7: Performance comparison of $r(s,a)=\log(C(s))$ and
$r(s,a)=-\log(1-C(s))$ on (left) the table-top organization environment,
(center) the sawyer door closing environment, and (right) the sawyer peg
environment. Plots show learning curves with mean and standard error over 5
random seeds.
## Appendix B Environments
The environment details can be found in (Sharma et al., 2022). We briefly
describe environments for completeness. For every environment, $H_{T}$ defines
the number of steps after which the environment is reset to a state
$s_{0}\sim\rho_{0}$, and $H_{E}$ defines the evaluation horizon over which the
return is computed for deployed policy evaluation:
table-top organization: Table-top organization is run with a training horizon
of $H_{T}=200,000$ and $H_{E}=200$. The sparse reward function is given by:
$r(s,g)=\mathbb{I}(\lVert s-g\rVert_{2}\leq 0.2),$
where $\mathbb{I}$ denotes the indicator function. The environment has $4$
possible goal locations for the mug, and goal location for the gripper is in
the center. EARL provides a total of $12$ forward demonstrations and $12$
backward demonstrations ($3$ per goal).
sawyer door closing: Sawyer door closing is run with a training horizon of
$H_{T}=200,000$ and an episode horizon of $H_{E}=300$. The sparse reward
function is:
$r(s,g)=\mathbb{I}(\lVert s-g\rVert_{2}\leq 0.02),$
where $\mathbb{I}$ again denotes the indicator function. The goal for the door
and the robot arm is the closed door position. EARL provides $5$ forward
demonstrations and $5$ backward demonstrations.
sawyer peg: Sawyer peg is run with a training horizon of $H_{T}=100,000$ and
an episode horizon of $H_{E}=200$. The sparse reward function is:
$r(s,g)=\mathbb{I}(\lVert s-g\rVert_{2}\leq 0.05),$
where $\mathbb{I}$ again denotes the indicator function. The goal for the peg
is to be placed in the goal slot. EARL provides $10$ forward demonstrations
and $20$ backward demonstrations.
|
# Cross-Entropy Method Variants for Optimization ††thanks: Code available at
https://github.com/mossr/CrossEntropyVariants.jl
Robert J. Moss Stanford University, Computer Science
Stanford, CA, 94305
<EMAIL_ADDRESS>
###### Abstract
The cross-entropy (CE) method is a popular stochastic method for optimization
due to its simplicity and effectiveness. Designed for rare-event simulations
where the probability of a target event occurring is relatively small, the CE-
method relies on enough objective function calls to accurately estimate the
optimal parameters of the underlying distribution. Certain objective functions
may be computationally expensive to evaluate, and the CE-method could
potentially get stuck in local minima. This is compounded with the need to
have an initial covariance wide enough to cover the design space of interest.
We introduce novel variants of the CE-method to address these concerns. To
mitigate expensive function calls, during optimization we use every sample to
build a surrogate model to approximate the objective function. The surrogate
model augments the belief of the objective function with less expensive
evaluations. We use a Gaussian process for our surrogate model to incorporate
uncertainty in the predictions which is especially helpful when dealing with
sparse data. To address local minima convergence, we use Gaussian mixture
models to encourage exploration of the design space. We experiment with
evaluation scheduling techniques to reallocate true objective function calls
earlier in the optimization when the covariance is the largest. To test our
approach, we created a parameterized test objective function with many local
minima and a single global minimum. Our test function can be adjusted to
control the spread and distinction of the minima. Experiments were run to
stress the cross-entropy method variants and results indicate that the
surrogate model-based approach reduces local minima convergence using the same
number of function evaluations.
## I Introduction
The cross-entropy (CE) method is a probabilistic optimization approach that
attempts to iteratively fit a distribution to elite samples from an initial
input distribution [1, 2]. The goal is to estimate a rare-event probability by
minimizing the cross-entropy between the two distributions [3]. The CE-method
has gained popularity in part due to its simplicity in implementation and
straightforward derivation. The technique uses importance sampling which
introduces a proposal distribution over the rare-events to sample from then
re-weights the posterior likelihood by the likelihood ratio of the true
distribution over the proposal distribution.
There are a few key assumptions that make the CE-method work effectively.
Through random sampling, the CE-method assumes that there are enough objective
function evaluations to accurately represent the objective. This may not be a
problem for simple applications, but can be an issue for computationally
expensive objective functions. Another assumption is that the initial
parameters of the input distribution are wide enough to cover the design space
of interest. For the case with a multivariate Gaussian distribution, this
corresponds to an appropriate mean and wide covariance. In rare-event
simulations with many local minima, the CE-method can fail to find a global
minima especially with sparse objective function evaluations.
This work aims to address the key assumptions of the CE-method. We introduce
variants of the CE-method that use surrogate modeling to approximate the
objective function, thus updating the belief of the underlying objective
through estimation. As part of this approach, we introduce evaluation
scheduling techniques to reallocate true objective function calls earlier in
the optimization when we know the covariance will be large. The evaluation
schedules can be based on a distribution (e.g., the Geometric distribution) or
can be prescribed manually depending on the problem. We also use a Gaussian
mixture model representation of the prior distribution as a method to explore
competing local optima. While the use of Gaussian mixture models in the CE-
method is not novel, we connect the use of mixture models and surrogate
modeling in the CE-method. This connection uses each elite sample as the mean
of a component distribution in the mixture, optimized through a subroutine
call to the standard CE-method using the learned surrogate model. To test our
approach, we introduce a parameterized test objective function called sierra.
The sierra function is built from a multivariate Gaussian mixture model with
many local minima and a single global minimum. Parameters for the sierra
function allow control over both the spread and distinction of the minima.
Lastly, we provide an analysis of the weak areas of the CE-method compared to
our proposed variants.
## II Related Work
The cross-entropy method is popular in the fields of operations research,
machine learning, and optimization [4, 5]. The combination of the cross-
entropy method, surrogate modeling, and mixture models has been explored in
other work [6]. The work in [6] proposed an adaptive grid approach to
accelerate Gaussian-process-based surrogate modeling using mixture models as
the prior in the cross-entropy method. They showed that a mixture model
performs better than a single Gaussian when the objective function is
multimodal. Our work differs in that we augment the “elite” samples both by an
approximate surrogate model and by a subroutine call to the CE-method using
the learned surrogate model. Other related work use Gaussian processes and a
modified cross-entropy method for receding-horizon trajectory optimization
[7]. Their cross-entropy method variant also incorporates the notion of
exploration in the context of path finding applications. An approach based on
relative entropy, described in section III-A, proposed a model-based
stochastic search that seeks to minimize the relative entropy [8]. They also
explore the use of a simple quadratic surrogate model to approximate the
objective function. Prior work that relate cross-entropy-based adaptive
importance sampling with Gaussian mixture models show that a mixture model
require less objective function calls than a naïve Monte Carlo or standard
unimodal cross-entropy-based importance sampling method [9, 10].
## III Background
This section provides necessary background on techniques used in this work. We
provide introductions to cross-entropy and the cross-entropy method, surrogate
modeling using Gaussian processes, and multivariate Gaussian mixture models.
### III-A Cross-Entropy
Before understanding the cross-entropy method, we first must understand the
notion of cross-entropy. Cross-entropy is a metric used to measure the
distance between two probability distributions, where the distance may not be
symmetric [3]. The distance used to define cross-entropy is called the
Kullback-Leibler (KL) distance or KL divergence. The KL distance is also
called the relative entropy, and we can use this to derive the cross-entropy.
Formally, for a random variable $\mathbf{X}=(X_{1},\ldots,X_{n})$ with a
support of $\mathcal{X}$, the KL distance between two continuous probability
density functions $f$ and $g$ is defined to be:
$\displaystyle\mathcal{D}(f,g)$
$\displaystyle=\mathbb{E}_{f}\left[\log\frac{f(\mathbf{X})}{g(\mathbf{X})}\right]$
$\displaystyle=\int\limits_{\mathbf{x}\in\mathcal{X}}f(\mathbf{x})\log
f(\mathbf{x})d\mathbf{x}-\int\limits_{\mathbf{x}\in\mathcal{X}}f(\mathbf{x})\log
g(\mathbf{x})d\mathbf{x}$
We denote the expectation of some function with respect to a distribution $f$
as $\mathbb{E}_{f}$. Minimizing the KL distance $\mathcal{D}$ between our true
distribution $f$ and our proposal distribution $g$ parameterized by
$\bm{\uptheta}$, is equivalent to choosing $\bm{\uptheta}$ that minimizes the
following, called the cross-entropy:
$\displaystyle H(f,g)$ $\displaystyle=H(f)+\mathcal{D}(f,g)$
$\displaystyle=-\mathbb{E}_{f}[\log g(\mathbf{X})]$ (using KL distance)
$\displaystyle=-\int\limits_{\mathbf{x}\in\mathcal{X}}f(\mathbf{x})\log
g(\mathbf{x}\mid\bm{\uptheta})d\mathbf{x}$
where $H(f)$ denotes the entropy of the distribution $f$ (where we conflate
entropy and continuous entropy for convenience). This assumes that $f$ and $g$
share the support $\mathcal{X}$ and are continuous with respect to
$\mathbf{x}$. The minimization problem then becomes:
$\displaystyle\operatornamewithlimits{minimize}_{\bm{\uptheta}}$
$\displaystyle-\int\limits_{\mathbf{x}\in\mathcal{X}}f(\mathbf{x})\log
g(\mathbf{x}\mid\bm{\uptheta})d\mathbf{x}$ (1)
Efficiently finding this minimum is the goal of the cross-entropy method
algorithm.
### III-B Cross-Entropy Method
Using the definition of cross-entropy, intuitively the cross-entropy method
(CEM or CE-method) aims to minimize the cross-entropy between the unknown true
distribution $f$ and a proposal distribution $g$ parameterized by
$\bm{\uptheta}$. This technique reformulates the minimization problem as a
probability estimation problem, and uses adaptive importance sampling to
estimate the unknown expectation [3]. The cross-entropy method has been
applied in the context of both discrete and continuous optimization problems
[2, 11].
The initial goal is to estimate the probability
$\displaystyle\ell=P_{\bm{\uptheta}}(S(\mathbf{x})\geq\gamma)$
where $S$ can the thought of as an objective function of $\mathbf{x}$, and
$\mathbf{x}$ follows a distribution defined by
$g(\mathbf{x}\mid\bm{\uptheta})$. We want to find events where our objective
function $S$ is above some threshold $\gamma$. We can express this unknown
probability as the expectation
$\displaystyle\ell=\mathbb{E}_{\bm{\uptheta}}[\mathbbm{1}_{(S(\mathbf{x})\geq\gamma)}]$
(2)
where $\mathbbm{1}$ denotes the indicator function. A straightforward way to
estimate eq. 2 can be done through Monte Carlo sampling. But for rare-event
simulations where the probability of a target event occurring is relatively
small, this estimate becomes inadequate. The challenge of the minimization in
eq. 1 then becomes choosing the density function for the true distribution
$f(\mathbf{x})$. Importance sampling tells us that the optimal importance
sampling density can be reduced to
$\displaystyle
f^{*}(\mathbf{x})=\frac{\mathbbm{1}_{(S(\mathbf{x})\geq\gamma)}g(\mathbf{x}\mid\bm{\uptheta})}{\ell}$
thus resulting in the optimization problem:
$\displaystyle\bm{\uptheta}_{g}^{*}$
$\displaystyle=\operatornamewithlimits{arg\,min}_{\bm{\uptheta}_{g}}-\int\limits_{\mathbf{x}\in\mathcal{X}}f^{*}(\mathbf{x})\log
g(\mathbf{x}\mid\bm{\uptheta}_{g})d\mathbf{x}$
$\displaystyle=\operatornamewithlimits{arg\,min}_{\bm{\uptheta}_{g}}-\int\limits_{\mathbf{x}\in\mathcal{X}}\frac{\mathbbm{1}_{(S(\mathbf{x})\geq\gamma)}g(\mathbf{x}\mid\bm{\uptheta})}{\ell}\log
g(\mathbf{x}\mid\bm{\uptheta}_{g})d\mathbf{x}$
Note that since we assume $f$ and $g$ belong to the same family of
distributions, we get that $f(\mathbf{x})=g(\mathbf{x}\mid\bm{\uptheta}_{g})$.
Now notice that $\ell$ is independent of $\bm{\uptheta}_{g}$, thus we can drop
$\ell$ and get the final optimization problem of:
$\displaystyle\bm{\uptheta}_{g}^{*}$
$\displaystyle=\operatornamewithlimits{arg\,min}_{\bm{\uptheta}_{g}}-\int\limits_{\mathbf{x}\in\mathcal{X}}\mathbbm{1}_{(S(\mathbf{x})\geq\gamma)}g(\mathbf{x}\mid\bm{\uptheta})\log
g(\mathbf{x}\mid\bm{\uptheta}_{g})d\mathbf{x}$ (3)
$\displaystyle=\operatornamewithlimits{arg\,min}_{\bm{\uptheta}_{g}}-\mathbb{E}_{\bm{\uptheta}}[\mathbbm{1}_{(S(\mathbf{x})\geq\gamma)}\log
g(\mathbf{x}\mid\bm{\uptheta}_{g})]$
The CE-method uses a multi-level algorithm to estimate $\bm{\uptheta}_{g}^{*}$
iteratively. The parameter $\bm{\uptheta}_{k}$ at iteration $k$ is used to
find new parameters $\bm{\uptheta}_{k^{\prime}}$ at the next iteration
$k^{\prime}$. The threshold $\gamma_{k}$ becomes smaller that its initial
value, thus artificially making events less rare under $\mathbf{X}\sim
g(\mathbf{x}\mid\bm{\uptheta}_{k})$.
In practice, the CE-method algorithm requires the user to specify a number of
elite samples $m_{\text{elite}}$ which are used when fitting the new
parameters for iteration $k^{\prime}$. Conveniently, if our distribution $g$
belongs to the natural exponential family then the optimal parameters can be
found analytically [5]. For a multivariate Gaussian distribution parameterized
by $\bm{\upmu}$ and $\bm{\Sigma}$, the optimal parameters for the next
iteration $k^{\prime}$ correspond to the maximum likelihood estimate (MLE):
$\displaystyle\bm{\upmu}_{k^{\prime}}$
$\displaystyle=\frac{1}{m_{\text{elite}}}\sum_{i=1}^{m_{\text{elite}}}\mathbf{x}_{i}$
$\displaystyle\bm{\Sigma}_{k^{\prime}}$
$\displaystyle=\frac{1}{m_{\text{elite}}}\sum_{i=1}^{m_{\text{elite}}}(\mathbf{x}_{i}-\bm{\upmu}_{k^{\prime}})(\mathbf{x}_{i}-\bm{\upmu}_{k^{\prime}})^{\top}$
The cross-entropy method algorithm is shown in algorithm 1. For an objective
function $S$ and input distribution $g$, the CE-method algorithm will run for
$k_{\text{max}}$ iterations. At each iteration, $m$ inputs are sampled from
$g$ and evaluated using the objective function $S$. The sampled inputs are
denoted by $\mathbf{X}$ and the evaluated values are denoted by $\mathbf{Y}$.
Next, the top $m_{\text{elite}}$ samples are stored in the elite set
$\mathbf{e}$, and the distribution $g$ is fit to the elites. This process is
repeated for $k_{\text{max}}$ iterations and the resulting parameters
$\bm{\uptheta}_{k_{\text{max}}}$ are returned. Note that a variety of input
distributions for $g$ are supported, but we focus on the multivariate Gaussian
distribution and the Gaussian mixture model in this work.
function CrossEntropyMethod($S,g,m,m_{\text{elite}},k_{\text{max}}$)
for $k\in[1,\ldots,k_{\text{max}}]$ do
$\mathbf{X}\sim g(\;\cdot\mid\bm{\uptheta}_{k})$ where
$\mathbf{X}\in\mathbb{R}^{m}$
$\mathbf{Y}\leftarrow S(\mathbf{x})$ for $\mathbf{x}\in\mathbf{X}$
$\mathbf{e}\leftarrow$ store top $m_{\text{elite}}$ from $\mathbf{Y}$
$\bm{\uptheta}_{k^{\prime}}\leftarrow\textsc{Fit}(g(\;\cdot\mid\bm{\uptheta}_{k}),\mathbf{e})$
return $g(\;\cdot\mid\bm{\uptheta}_{k_{\text{max}}})$
Algorithm 1 Cross-entropy method.
### III-C Mixture Models
A standard Gaussian distribution is unimodal and can have trouble generalizing
over data that is multimodal. A mixture model is a weighted mixture of
component distributions used to represent continuous multimodal distributions
[4]. Formally, a Gaussian mixture model (GMM) is defined by its parameters
$\bm{\upmu}$ and $\bm{\Sigma}$ and associated weights $\mathbf{w}$ where
$\sum_{i=1}^{n}w_{i}=1$. We denote that a random variable $\mathbf{X}$ is
distributed according to a mixture model as
$\mathbf{X}\sim\operatorname{Mixture}(\bm{\upmu},\bm{\Sigma},\mathbf{w})$. The
probability density of the GMM then becomes:
$\displaystyle
P(\mathbf{X}=\mathbf{x}\mid\bm{\upmu},\bm{\Sigma},\mathbf{w})=\sum_{i=1}^{n}w_{i}\operatorname{\mathcal{N}}(\mathbf{x}\mid\bm{\upmu}_{i},\bm{\Sigma}_{i})$
To fit the parameters of a Gaussian mixture model, it is well known that the
expectation-maximization (EM) algorithm can be used [12, 13]. The EM algorithm
seeks to find the maximum likelihood estimate of the hidden variable $H$ using
the observed data defined by $E$. Intuitively, the algorithm alternates
between an expectation step (E-step) and a maximization step (M-step) to
guarantee convergence to a local minima. A simplified EM algorithm is provide
in algorithm 2 for reference and we refer to [12, 13] for further reading.
function ExpectationMaximization($H,E,\bm{\uptheta}$)
for E-step do
Compute $Q(h)=P(H=h\mid E=e,\bm{\uptheta})$ for each $h$
Create weighted points: $(h,e)$ with weight $Q(h)$
for M-step do
Compute $\mathbf{\hat{\bm{\uptheta}}}_{\text{MLE}}$
Repeat until convergence.
return $\mathbf{\hat{\bm{\uptheta}}}_{\text{MLE}}$
Algorithm 2 Expectation-maximization.
### III-D Surrogate Models
In the context of optimization, a surrogate model $\hat{S}$ is used to
estimate the true objective function and provide less expensive evaluations.
Surrogate models are a popular approach and have been used to evaluate rare-
event probabilities in computationally expensive systems [14, 15]. The
simplest example of a surrogate model is linear regression. In this work, we
focus on the Gaussian process surrogate model. A Gaussian process (GP) is a
distribution over functions that predicts the underlying objective function
$S$ and captures the uncertainty of the prediction using a probability
distribution [5]. This means a GP can be sampled to generate random functions,
which can then be fit to our given data $\mathbf{X}$. A Gaussian process is
parameterized by a mean function $\mathbf{m}(\mathbf{X})$ and kernel function
$\mathbf{K}(\mathbf{X},\mathbf{X})$, which captures the relationship between
data points as covariance values. We denote a Gaussian process that produces
estimates $\hat{\mathbf{y}}$ as:
$\displaystyle\hat{\mathbf{y}}$
$\displaystyle\sim\mathcal{N}\left(\mathbf{m}(\mathbf{X}),\mathbf{K}(\mathbf{X},\mathbf{X})\right)$
$\displaystyle=\begin{bmatrix}\hat{S}(\mathbf{x}_{1}),\ldots,\hat{S}(\mathbf{x}_{n})\end{bmatrix}$
where
$\displaystyle\mathbf{m}(\mathbf{X})=\begin{bmatrix}m(\mathbf{x}_{1}),\ldots,m(\mathbf{x}_{n})\end{bmatrix}$
$\displaystyle\mathbf{K}(\mathbf{X},\mathbf{X})=\begin{bmatrix}k(\mathbf{x}_{1},\mathbf{x}_{1})&\cdots&k(\mathbf{x}_{1},\mathbf{x}_{n})\\\
\vdots&\ddots&\vdots\\\
k(\mathbf{x}_{n},\mathbf{x}_{1})&\cdots&k(\mathbf{x}_{n},\mathbf{x}_{n})\end{bmatrix}$
We use the commonly used zero-mean function $m(\mathbf{x}_{i})=\mathbf{0}$.
For the kernel function $k(\mathbf{x}_{i},\mathbf{x}_{i})$, we use the squared
exponential kernel with variance $\sigma^{2}$ and characteristic scale-length
$\ell$, where larger $\ell$ values increase the correlation between successive
data points, thus smoothing out the generated functions. The squared
exponential kernel is defined as:
$\displaystyle
k(\mathbf{x},\mathbf{x}^{\prime})=\sigma^{2}\exp\left(-\frac{(\mathbf{x}-\mathbf{x}^{\prime})^{\top}(\mathbf{x}-\mathbf{x}^{\prime})}{2\ell^{2}}\right)$
We refer to [5] for a detailed overview of Gaussian processes and different
kernel functions.
## IV Algorithms
We can now describe the cross-entropy method variants introduced in this work.
This section will first cover the main algorithm introduced, the cross-entropy
surrogate method (CE-surrogate). Then we introduce a modification to the CE-
surrogate method, namely the cross-entropy mixture method (CE-mixture).
Lastly, we describe various evaluation schedules for redistributing objective
function calls over the iterations.
### IV-A Cross-Entropy Surrogate Method
The main CE-method variant we introduce is the cross-entropy surrogate method
(CE-surrogate). The CE-surrogate method is a superset of the CE-method, where
the differences lie in the evaluation scheduling and modeling of the elite set
using a surrogate model. The goal of the CE-surrogate algorithm is to address
the shortcomings of the CE-method when the number of objective function calls
is sparse and the underlying objective function $S$ has multiple local minima.
The CE-surrogate algorithm is shown in algorithm 3. It takes as input the
objective function $S$, the distribution $\mathbf{M}$ parameterized by
$\bm{\uptheta}$, the number of samples $m$, the number of elite samples
$m_{\text{elite}}$, and the maximum iterations $k_{\text{max}}$. For each
iteration $k$, the number of samples $m$ are redistributed through a call to
EvaluationSchedule, where $m$ controls the number of true objective function
evaluations of $S$. Then, the algorithm samples from $\mathbf{M}$
parameterized by the current $\bm{\uptheta}_{k}$ given the adjusted number of
samples $m$. For each sample in $\mathbf{X}$, the objective function $S$ is
evaluated and the results are stored in $\mathbf{Y}$. The top
$m_{\text{elite}}$ evaluations from $\mathbf{Y}$ are stored in $\mathbf{e}$.
Using all of the current function evaluations $\mathbf{Y}$ from sampled inputs
$\mathbf{X}$, a modeled elite set $\mathbf{E}$ is created to augment the
sparse information provided by a low number of true objective function
evaluations. Finally, the distribution $\mathbf{M}$ is fit to the elite set
$\mathbf{E}$ and the distribution with the final parameters
$\bm{\uptheta}_{k_{\text{max}}}$ is returned.
function CE-Surrogate($S$, $\mathbf{M}$, $m$, $m_{\text{elite}}$,
$k_{\text{max}}$)
for $k\in[1,\ldots,k_{\text{max}}]$ do
$m,m_{\text{elite}}\leftarrow\textsc{EvaluationSchedule}(k,k_{\text{max}})$
$\mathbf{X}\sim\mathbf{M}(\;\cdot\mid\bm{\uptheta}_{k})$ where
$\mathbf{X}\in\mathbb{R}^{m}$
$\mathbf{Y}\leftarrow S(\mathbf{x})$ for $\mathbf{x}\in\mathbf{X}$
$\mathbf{e}\leftarrow$ store top $m_{\text{elite}}$ from $\mathbf{Y}$
$\mathbf{E}\leftarrow\textsc{ModelEliteSet}(\mathbf{X},\mathbf{Y},\mathbf{M},\mathbf{e},m,m_{\text{elite}})$
$\bm{\uptheta}_{k^{\prime}}\leftarrow\textsc{Fit}(\mathbf{M}(\;\cdot\mid\bm{\uptheta}_{k}),\mathbf{E})$
return $\mathbf{M}(\;\cdot\mid\bm{\uptheta}_{k_{\text{max}}})$
Algorithm 3 Cross-entropy surrogate method.
The main difference between the standard CE-method and the CE-surrogate
variant lies in the call to ModelEliteSet. The motivation is to use all of the
already evaluated objective function values $\mathbf{Y}$ from a set of sampled
inputs $\mathbf{X}$. This way the expensive function evaluations—otherwise
discarded—can be used to build a surrogate model of the underlying objective
function. First, a surrogate model $\hat{S}$ is constructed from the samples
$\mathbf{X}$ and true objective function values $\mathbf{Y}$. We used a
Gaussian process with a specified kernel and optimizer, but other surrogate
modeling techniques such as regression with basis functions can be used. We
chose a Gaussian process because it incorporates probabilistic uncertainty in
the predictions, which may more accurately represent our objective function,
or at least be sensitive to over-fitting to sparse data. Now we have an
approximated objective function $\hat{S}$ that we can inexpensively call. We
sample $10m$ values from the distribution $\mathbf{M}$ and evaluate them using
the surrogate model. We then store the top $10m_{\text{elite}}$ values from
the estimates $\mathbf{\hat{\mathbf{Y}}}_{\text{m}}$. We call these estimated
elite values $\mathbf{e}_{\text{model}}$ the model-elites. The surrogate model
is then passed to SubEliteSet, which returns more estimates for elite values.
Finally, the elite set $\mathbf{E}$ is built from the true-elites
$\mathbf{e}$, the model-elites $\mathbf{e}_{\text{model}}$, and the
subcomponent-elites $\mathbf{e}_{\text{sub}}$. The resulting concatenated
elite set $\mathbf{E}$ is returned.
function
ModelEliteSet($\mathbf{X},\mathbf{Y},\mathbf{M},\mathbf{e},m,m_{\text{elite}}$)
$\hat{S}\leftarrow\textsc{GaussianProcess}(\mathbf{X},\mathbf{Y},\text{kernel},\text{optimizer})$
$\mathbf{X}_{\text{m}}\sim\mathbf{M}(\;\cdot\mid\bm{\uptheta}_{k})$ where
$\mathbf{X}_{\text{m}}\in\mathbb{R}^{10m}$
$\mathbf{\hat{\mathbf{Y}}}_{\text{m}}\leftarrow\hat{S}(\mathbf{x}_{\text{m}})$
for $\mathbf{x}_{\text{m}}\in\mathbf{X}_{\text{m}}$
$\mathbf{e}_{\text{model}}\leftarrow$ store top $10m_{\text{elite}}$ from
$\mathbf{\hat{\mathbf{Y}}}_{\text{m}}$
$\mathbf{e}_{\text{sub}}\leftarrow\textsc{SubEliteSet}(\hat{S},\mathbf{M},\mathbf{e})$
$\mathbf{E}\leftarrow\\{\mathbf{e}\\}\cup\\{\mathbf{e}_{\text{model}}\\}\cup\\{\mathbf{e}_{\text{sub}}\\}$
$\triangleright$ elite set
return $\mathbf{E}$
Algorithm 4 Modeling elite set.
To encourage exploration of promising areas of the design space, the algorithm
SubEliteSet focuses on the already marked true-elites $\mathbf{e}$. Each elite
$e_{x}\in\mathbf{e}$ is used as the mean of a new multivariate Gaussian
distribution with covariance inherited from the distribution $\mathbf{M}$. The
collection of subcomponent distributions is stored in $\mathbf{m}$. The idea
is to use the information given to us by the true-elites to emphasize areas of
the design space that look promising. For each distribution
$\mathbf{m}_{i}\in\mathbf{m}$ we run a subroutine call to the standard CE-
method to fit the distribution $\mathbf{m}_{i}$ using the surrogate model
$\hat{S}$. Then the best objective function value is added to the
subcomponent-elite set $\mathbf{e}_{\text{sub}}$, and after iterating the full
set is returned. Note that we use $\theta_{\text{CE}}$ to denote the
parameters for the CE-method algorithm. In our case, we recommend using a
small $k_{\text{max}}$ of around $2$ so the subcomponent-elites do not over-
fit to the surrogate model but have enough CE-method iterations to tend
towards optimal.
function SubEliteSet($\hat{S},\mathbf{M},\mathbf{e}$)
$\mathbf{e}_{\text{sub}}\leftarrow\emptyset$
$\mathbf{m}\leftarrow\\{e_{x}\in\mathbf{e}\mid\operatorname{\mathcal{N}}(e_{x},\mathbf{M}.\Sigma)\\}$
for $\mathbf{m}_{i}\in\mathbf{m}$ do
$\mathbf{m}_{i}\leftarrow\textsc{CrossEntropyMethod}(\hat{S},\mathbf{m}_{i}\mid\theta_{\text{CE}})$
$\mathbf{e}_{\text{sub}}\leftarrow\\{\mathbf{e}_{\text{sub}}\\}\cup\\{\textsc{Best}(\mathbf{m}_{i})\\}$
return $\mathbf{e}_{\text{sub}}$
Algorithm 5 Subcomponent elite set.
### IV-B Cross-Entropy Mixture Method
We refer to the variant of our CE-surrogate method that takes an input mixture
model $\mathbf{M}$ as the cross-entropy mixture method (CE-mixture). The CE-
mixture algorithm is identical to the CE-surrogate algorithm, but calls a
custom Fit function to fit a mixture model to the elite set $\mathbf{E}$. The
input distribution $\mathbf{M}$ is cast to a mixture model using the
subcomponent distributions $\mathbf{m}$ as the components of the mixture. We
use the default uniform weighting for each mixture component. The mixture
model $\mathbf{M}$ is then fit using the expectation-maximization algorithm
shown in algorithm 2, and the resulting distribution is returned. The idea is
to use the distributions in $\mathbf{m}$ that are centered around each true-
elite as the components of the casted mixture model. Therefore, we would
expect better performance of the CE-mixture method when the objective function
has many competing local minima. Results in section V-C aim to show this
behavior.
function Fit($\mathbf{M},\mathbf{m},\mathbf{E}$)
$\mathbf{M}\leftarrow\operatorname{Mixture}(\mathbf{m})$
$\mathbf{\hat{\bm{\uptheta}}}\leftarrow\textsc{ExpectationMaximization}(\mathbf{M},\mathbf{E})$
return $\mathbf{M}(\;\cdot\mid\mathbf{\hat{\bm{\uptheta}}})$
Algorithm 6 Fitting mixture models (used by CE-mixture).
### IV-C Evaluation Scheduling
Given the nature of the CE-method, we expect the covariance to shrink over
time, thus resulting in a solution with higher confidence. Yet if each
iteration is given the same number of objective function evaluations $m$,
there is the potential for elite samples from early iterations dominating the
convergence. Therefore, we would like to redistribute the objective function
evaluations throughout the iterations to use more truth information early in
the process. We call these heuristics evaluation schedules. One way to achieve
this is to reallocate the evaluations according to a Geometric distribution.
Evaluation schedules can also be ad-hoc and manually prescribed based on the
current iteration.
We provide the evaluation schedule we use that follows a Geometric
distribution with parameter $p$ in algorithm 7. We denote
$G\sim\operatorname{Geo}(p)$ to be a random variable that follows a truncated
Geometric distribution with the probability mass function
$p_{G}(k)=p(1-p)^{k}$ for $k\in\\{0,1,2,\ldots,k_{\text{max}}\\}$. Note the
use of the integer rounding function (e.g., $\lfloor x\rceil$), which we later
have to compensate for towards the final iterations. Results in section V-C
compare values of $p$ that control the redistribution of evaluations.
function EvaluationSchedule($k,k_{\text{max}}$)
$G\sim\operatorname{Geo}(p)$
$N_{\text{max}}\leftarrow k_{\text{max}}\cdot m$
$m\leftarrow\lfloor N_{\text{max}}\cdot p_{G}(k)\rceil$
if $k=k_{\text{max}}$ then
$s\leftarrow\displaystyle\sum_{i=1}^{k_{\text{max}}-1}\lfloor
N_{\text{max}}\cdot p_{G}(i)\rceil$
$m\leftarrow\min(N_{\text{max}}-s,N_{\text{max}}-m)$
$m_{\text{elite}}\leftarrow\min(m_{\text{elite}},m)$
return ($m,m_{\text{elite}}$)
Algorithm 7 Evaluation schedule using a Geometric distr.
## V Experiments
In this section, we detail the experiments we ran to compare the CE-method
variants and evaluation schedules. We first introduce a test objective
function we created to stress the issue of converging to local minima. We then
describe the experimental setup for each of our experiments and provide an
analysis and results.
### V-A Test Objective Function Generation
$-15$$-10$$-5$$0$$5$$10$$15$$\eta=0.5$, decay=1$\eta=2.0$, decay=1$\eta=6.0$,
decay=1$-2$$-1.5$$-1$$-0.5$$0$$\cdot
10^{-2}$$-15$$-10$$-5$$0$$5$$10$$15$$-15$$-10$$-5$$0$$5$$10$$15$$\eta=0.5$,
decay=0$-15$$-10$$-5$$0$$5$$10$$15$$\eta=2.0$,
decay=0$-15$$-10$$-5$$0$$5$$10$$15$$\eta=6.0$,
decay=0$-2$$-1.5$$-1$$-0.5$$0$$\cdot 10^{-2}$
Figure 1: Example test objective functions generated using the sierra
function.
To stress the cross-entropy method and its variants, we created a test
objective function called sierra that is generated from a mixture model
comprised of $49$ multivariate Gaussian distributions. We chose this
construction so that we can use the negative peeks of the component
distributions as local minima and can force a global minimum centered at our
desired $\mathbf{\tilde{\bm{\upmu}}}$. The construction of the sierra test
function can be controlled by parameters that define the spread of the local
minima. We first start with the center defined by a mean vector
$\mathbf{\tilde{\bm{\upmu}}}$ and we use a common covariance
$\mathbf{\tilde{\bm{\Sigma}}}$:
$\displaystyle\mathbf{\tilde{\bm{\upmu}}}$
$\displaystyle=[\mu_{1},\mu_{2}],\quad\mathbf{\tilde{\bm{\Sigma}}}=\begin{bmatrix}\sigma&0\\\
0&\sigma\end{bmatrix}$
Next, we use the parameter $\delta$ that controls the clustered distance
between symmetric points:
$\displaystyle\mathbf{G}$
$\displaystyle=\left\\{[+\delta,+\delta],[+\delta,-\delta],[-\delta,+\delta],[-\delta,-\delta]\right\\}$
We chose points $\mathbf{P}$ to fan out the clustered minima relative to the
center defined by $\mathbf{\tilde{\bm{\upmu}}}$:
$\displaystyle\mathbf{P}$
$\displaystyle=\left\\{[0,0],[1,1],[2,0],[3,1],[0,2],[1,3]\right\\}$
The vector $\mathbf{s}$ is used to control the $\pm$ distance to create an ‘s’
shape comprised of minima, using the standard deviation $\sigma$:
$\mathbf{s}=\begin{bmatrix}+\sigma,-\sigma\end{bmatrix}$. We set the following
default parameters: standard deviation $\sigma=3$, spread rate $\eta=6$, and
cluster distance $\delta=2$. We can also control if the local minima clusters
“decay”, thus making those local minima less distinct (where
$\text{decay}\in\\{0,1\\})$. The parameters that define the sierra function
are collected into
$\bm{\uptheta}=\langle\mathbf{\tilde{\bm{\upmu}}},\mathbf{\tilde{\bm{\Sigma}}},\mathbf{G},\mathbf{P},\mathbf{s}\rangle$.
Using these parameters, we can define the mixture model used by the sierra
function as:
$\displaystyle\mathbf{M}_{\mathcal{S}}\sim\operatorname{Mixture}\left(\left\\{\bm{\uptheta}~{}\Big{|}~{}\operatorname{\mathcal{N}}\left(\mathbf{g}+s\mathbf{p}_{i}+\mathbf{\tilde{\bm{\upmu}}},\;\mathbf{\tilde{\bm{\Sigma}}}\cdot
i^{\text{decay}}/\eta\right)\right\\}\right)$ $\displaystyle\text{for
}(\mathbf{g},\mathbf{p}_{i},s)\in(\mathbf{G},\mathbf{P},\mathbf{s})$
We add a final component to be our global minimum centered at
$\mathbf{\tilde{\bm{\upmu}}}$ and with a covariance scaled by $\sigma\eta$.
Namely, the global minimum is
$\mathbf{x}^{*}=\mathbb{E}[\operatorname{\mathcal{N}}(\mathbf{\tilde{\bm{\upmu}}},\mathbf{\tilde{\bm{\Sigma}}}/(\sigma\eta))]=\mathbf{\tilde{\bm{\upmu}}}$.
We can now use this constant mixture model with $49$ components and define the
sierra objective function $\mathcal{S}(\mathbf{x})$ to be the negative
probability density of the mixture at input $\mathbf{x}$ with uniform weights:
$\displaystyle\mathcal{S}(\mathbf{x})$
$\displaystyle=-P(\mathbf{M}_{\mathcal{S}}=\mathbf{x})=-\frac{1}{|\mathbf{M}_{\mathcal{S}}|}\sum_{j=1}^{n}\operatorname{\mathcal{N}}(\mathbf{x}\mid\bm{\upmu}_{j},\bm{\Sigma}_{j})$
An example of six different objective functions generated using the sierra
function are shown in fig. 1, sweeping over the spread rate $\eta$, with and
without decay.
### V-B Experimental Setup
Experiments were run to stress a variety of behaviors of each CE-method
variant. The experiments are split into two categories: algorithmic and
scheduling. The algorithmic category aims to compare features of each CE-
method variant while holding common parameters constant (for a better
comparison). While the scheduling category experiments with evaluation
scheduling heuristics.
$\displaystyle-15$$\displaystyle-10$$\displaystyle-5$$\displaystyle
0$$\displaystyle 5$$\displaystyle 10$$\displaystyle
15$$\displaystyle-15$$\displaystyle-10$$\displaystyle-5$$\displaystyle
0$$\displaystyle 5$$\displaystyle 10$$\displaystyle 15$Experiment 1A: CE-
method ($\displaystyle k=5$)
(a) The cross-entropy method.
$\displaystyle-15$$\displaystyle-10$$\displaystyle-5$$\displaystyle
0$$\displaystyle 5$$\displaystyle 10$$\displaystyle
15$$\displaystyle-15$$\displaystyle-10$$\displaystyle-5$$\displaystyle
0$$\displaystyle 5$$\displaystyle 10$$\displaystyle 15$Experiment 1A: CE-
surrogate ($\displaystyle k=5$)
(b) The cross-entropy surrogate method.
$\displaystyle-15$$\displaystyle-10$$\displaystyle-5$$\displaystyle
0$$\displaystyle 5$$\displaystyle 10$$\displaystyle
15$$\displaystyle-15$$\displaystyle-10$$\displaystyle-5$$\displaystyle
0$$\displaystyle 5$$\displaystyle 10$$\displaystyle 15$Experiment 1A: CE-
mixture ($\displaystyle k=5$)
(c) The cross-entropy mixture method.
Figure 2: Iteration $k=5$ illustrated for each algorithm. The covariance is
shown by the contours.
Because the algorithms are stochastic, we run each experiment with 50
different random number generator seed values. To evaluate the performance of
the algorithms in their respective experiments, we define three metrics.
First, we define the average “optimal” value $\bar{b}_{v}$ to be the average
of the best so-far objective function value (termed “optimal” in the context
of each algorithm). Again, we emphasize that we average over the 50 seed
values to gather meaningful statistics. Another metric we monitor is the
average distance to the true global optimal
$\bar{b}_{d}=\mathinner{\\!\left\lVert\mathbf{b}_{\mathbf{x}}-\mathbf{x}^{*}\right\rVert}$,
where $\mathbf{b}_{\mathbf{x}}$ denotes the $\mathbf{x}$-value associated with
the “optimal”. We make the distinction between these metrics to show both
“closeness” in value to the global minimum and “closeness” in the design space
to the global minimum. Our final metric looks at the average runtime of each
algorithm, noting that our goal is to off-load computationally expensive
objective function calls to the surrogate model.
For all of the experiments, we use a common setting of the following
parameters for the sierra test function (shown in the top-right plot in fig.
1):
$(\mathbf{\tilde{\bm{\upmu}}}=[0,0],\;\sigma=3,\;\delta=2,\;\eta=6,\;\text{decay}=1)$
#### V-B1 Algorithmic Experiments
We run three separate algorithmic experiments, each to test a specific
feature. For our first algorithmic experiment (1A), we want to test each
algorithm when the user-defined mean is centered at the global minimum and the
covariance is arbitrarily wide enough to cover the design space. Let
$\mathbf{M}$ be a distribution parameterized by
$\bm{\uptheta}=(\bm{\upmu},\bm{\Sigma})$, and for experiment (1A) we set the
following:
$\bm{\upmu}^{(\text{1A})}=[0,0]\qquad\bm{\Sigma}^{(\text{1A})}=\begin{bmatrix}200&0\\\
0&200\end{bmatrix}$
For our second algorithmic experiment (1B), we test a mean that is far off-
centered with a wider covariance:
$\bm{\upmu}^{(\text{1B})}=[-50,-50]\qquad\bm{\Sigma}^{(\text{1B})}=\begin{bmatrix}2000&0\\\
0&2000\end{bmatrix}$
This experiment is used to test the “exploration” of the CE-method variants
introduced in this work. In experiments (1A) and (1B), we set the following
common parameters across each CE-method variant:
$(k_{\text{max}}=10,\;m=10,\;m_{\text{elite}}=5)^{(\text{1A,1B})}$
This results in $m\cdot k_{\text{max}}=100$ objective function evaluations,
which we define to be relatively low.
For our third algorithmic experiment (1C), we want to test how each variant
responds to an extremely low number of function evaluations. This sparse
experiment sets the common CE-method parameters to:
$(k_{\text{max}}=10,\;m=5,\;m_{\text{elite}}=3)^{(\text{1C})}$
This results in $m\cdot k_{\text{max}}=50$ objective function evaluations,
which we defined to be extremely low. We use the same mean and covariance
defined for experiment (1A):
$\bm{\upmu}^{(\text{1C})}=[0,0]\qquad\bm{\Sigma}^{(\text{1C})}=\begin{bmatrix}200&0\\\
0&200\end{bmatrix}$
#### V-B2 Scheduling Experiments
In our final experiment (2), we test the evaluation scheduling heuristics
which are based on the Geometric distribution. We sweep over the parameter $p$
that determines the Geometric distribution which controls the redistribution
of objective function evaluations. In this experiment, we compare the CE-
surrogate methods using the same setup as experiment (1B), namely the far off-
centered mean. We chose this setup to analyze exploration schemes when given
very little information about the true objective function.
### V-C Results and Analysis
$2$$4$$6$$8$$10$$-2$$-1$$0$$\cdot 10^{-2}$Iteration$\bar{b}_{v}$Experiment
1ACE-methodCE-surrogateCE-mixture
Figure 3: Average optimal value for experiment (1A) when the initial mean is
centered at the global minimum and the covariance sufficiently covers the
design space.
Figure 3 shows the average value of the current optimal $\bar{b}_{v}$ for the
three algorithms for experiment (1A). One standard deviation is plotted in the
shaded region. Notice that the standard CE-method converges to a local minima
before $k_{\text{max}}$ is reached. Both CE-surrogate method and CE-mixture
stay below the standard CE-method curve, highlighting the mitigation of
convergence to local minima. Minor differences can be seen between CE-
surrogate and CE-mixture, differing slightly towards the tail in favor of CE-
surrogate. The average runtime of the algorithms along with the performance
metrics are shown together for each experiment in table I.
TABLE I: Experimental results. Exper. | Algorithm | Runtime | $\bar{b}_{v}$ | $\bar{b}_{d}$
---|---|---|---|---
1A | CE-method | 0.029 $\operatorname{s}$ | $-$0.0134 | 23.48
CE-surrogate | 1.47 $\operatorname{s}$ | $-$0.0179 | 12.23
CE-mixture | 9.17 $\operatorname{s}$ | $-$0.0169 | 16.87
1B | CE-method | 0.046 $\operatorname{s}$ | $-$0.0032 | 138.87
CE-surrogate | 11.82 $\operatorname{s}$ | $-$0.0156 | 18.24
CE-mixture | 28.10 $\operatorname{s}$ | $-$0.0146 | 33.30
1C | CE-method | 0.052 $\operatorname{s}$ | $-$0.0065 | 43.14
CE-surrogate | 0.474 $\operatorname{s}$ | $-$0.0156 | 17.23
CE-mixture | 2.57 $\operatorname{s}$ | $-$0.0146 | 22.17
2 | CE-surrogate, $\operatorname{Uniform}$ | — | $-$0.0193 | 8.53
CE-surrogate, $\operatorname{Geo}(0.1)$ | — | $-$0.0115 | 25.35
CE-surrogate, $\operatorname{Geo}(0.2)$ | — | $-$0.0099 | 27.59
| CE-surrogate, $\operatorname{Geo}(0.3)$ | — | $-$0.0089 | 30.88
| | | $-\text{0.0220}\approx\mathbf{x}^{*}$
An apparent benefit of the standard CE-method is in its simplicity and speed.
As shown in table I, the CE-method is the fastest approach by about 2-3 orders
of magnitude compared to CE-surrogate and CE-mixture. The CE-mixture method is
notably the slowest approach. Although the runtime is also based on the
objective function being tested, recall that we are using the same number of
true objective function calls in each algorithm, and the metrics we are
concerned with in optimization are to minimize $\bar{b}_{v}$ and
$\bar{b}_{d}$. We can see that the CE-surrogate method consistently out
performs the other methods. Surprisingly, a uniform evaluation schedule
performs the best even in the sparse scenario where the initial mean is far
away from the global optimal.
$2$$4$$6$$8$$10$$-1.5$$-1$$-0.5$$0$$\cdot
10^{-2}$Iteration$\bar{b}_{v}$Experiment 1BCE-methodCE-surrogateCE-mixture
Figure 4: Average optimal value for experiment (1B) when the initial mean is
far from the global minimum with a wide covariance.
When the initial mean of the input distribution is placed far away from the
global optimal, the CE-method tends to converge prematurely as shown in fig.
4. This scenario is illustrated in fig. 5. We can see that both CE-surrogate
and CE-mixture perform well in this case.
$\displaystyle-100$$\displaystyle-80$$\displaystyle-60$$\displaystyle-40$$\displaystyle-20$$\displaystyle
0$$\displaystyle-100$$\displaystyle-80$$\displaystyle-60$$\displaystyle-40$$\displaystyle-20$$\displaystyle
0$Experiment 1B: $\displaystyle k=1$
Figure 5: First iteration of the scenario in experiment (1B) where the
initial distribution is far away form the global optimal. The red dots
indicate the true-elites, the black dots with white outlines indicate the
“non-elites” evaluated from the true objective function, and the white dots
with black outlines indicate the samples evaluated using the surrogate model.
$2$$4$$6$$8$$10$$-2$$-1$$0$$\cdot 10^{-2}$Iteration$\bar{b}_{v}$Experiment
1CCE-methodCE-surrogateCE-mixture
Figure 6: Average optimal value for experiment (1C) when we restrict the
number of objective function calls.
Given the same centered mean as before, when we restrict the number of
objective function calls even further to just 50 we see interesting behavior.
Notice that the results of experiment (1C) shown in fig. 6 follow a curve
closer to the far away mean from experiment (1B) than from the same setup as
experiment (1A). Also notice that the CE-surrogate results cap out at
iteration 9 due to the evaluation schedule front-loading the objective
function calls, thus leaving none for the final iteration (while still
maintaining the same total number of evaluations of 50).
## VI Conclusion
We presented variants of the popular cross-entropy method for optimization of
objective functions with multiple local minima. Using a Gaussian processes-
based surrogate model, we can use the same number of true objective function
evaluations and achieve better performance than the standard CE-method on
average. We also explored the use of a Gaussian mixture model to help find
global minimum in multimodal objective functions. We introduce a parameterized
test objective function with a controllable global minimum and spread of local
minima. Using this test function, we showed that the CE-surrogate algorithm
achieves the best performance relative to the standard CE-method, each using
the same number of true objective function evaluations.
## References
* [1] RY Rubinstein and DP Kroese “The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning” Springer, 2004
* [2] Reuven Rubinstein “The Cross-Entropy Method for Combinatorial and Continuous Optimization” In _Methodology and Computing in Applied Probability_ 1.2 Springer, 1999, pp. 127–190
* [3] Pieter-Tjerk De Boer, Dirk P Kroese, Shie Mannor and Reuven Y Rubinstein “A Tutorial on the Cross-Entropy Method” In _Annals of Operations Research_ 134.1 Springer, 2005, pp. 19–67
* [4] Mykel J. Kochenderfer “Decision Making Under Uncertainty: Theory and Application” MIT Press, 2015
* [5] Mykel J Kochenderfer and Tim A Wheeler “Algorithms for Optimization” MIT Press, 2019
* [6] Rémi Bardenet and Balázs Kégl “Surrogating the Surrogate: Accelerating Gaussian-Process-Based Global Optimization with a Mixture Cross-Entropy Algorithm” In _International Conference on Machine Learning (ICML)_ , 2010, pp. 55–62
* [7] Y.. Tan, A. Kunapareddy and M. Kobilarov “Gaussian Process Adaptive Sampling Using the Cross-Entropy Method for Environmental Sensing and Monitoring” In _International Conference on Robotics and Automation (ICRA)_ , 2018, pp. 6220–6227
* [8] Abbas Abdolmaleki et al. “Model-Based Relative Entropy Stochastic Search” In _Advances in Neural Information Processing Systems (NIPS)_ , 2015, pp. 3537–3545
* [9] Nolan Kurtz and Junho Song “Cross-entropy-based adaptive importance sampling using Gaussian mixture” In _Structural Safety_ 42 Elsevier, 2013, pp. 35–44
* [10] Ziqi Wang and Junho Song “Cross-entropy-based adaptive importance sampling using von Mises-Fisher mixture for high dimensional reliability analysis” In _Structural Safety_ 59 Elsevier, 2016, pp. 42–52
* [11] Dirk P Kroese, Sergey Porotsky and Reuven Y Rubinstein “The cross-entropy method for continuous multi-extremal optimization” In _Methodology and Computing in Applied Probability_ 8.3 Springer, 2006, pp. 383–407
* [12] Arthur P Dempster, Nan M Laird and Donald B Rubin “Maximum likelihood from incomplete data via the EM algorithm” In _Journal of the Royal Statistical Society: Series B (Methodological)_ 39.1 Wiley Online Library, 1977, pp. 1–22
* [13] Murray Aitkin and Granville Tunnicliffe Wilson “Mixture models, outliers, and the EM algorithm” In _Technometrics_ 22.3 Taylor & Francis Group, 1980, pp. 325–331
* [14] Jing Li and Dongbin Xiu “Evaluation of failure probability via surrogate models” In _Journal of Computational Physics_ 229.23 Elsevier, 2010, pp. 8966–8980
* [15] Jing Li, Jinglai Li and Dongbin Xiu “An efficient surrogate-based method for computing rare failure probability” In _Journal of Computational Physics_ 230.24 Elsevier, 2011, pp. 8683–8697
|
that the full integrand localises on the maximal non-separating boundary of
the moduli space, corresponding to a bi-nodal sphere, parametrised by four
‘loop marked points’, one pair per node,
$\displaystyle\mathfrak{I}^{(2)}_{4}$
$\displaystyle=\frac{\mathcal{K}\tilde{\mathcal{K}}}{\prod_{I}(\ell^{I})^{2}}\int_{{\mathfrak{M}}_{0,4+2g}}c^{(g)}\big{(}{\mathcal{J}}^{(g)}\mathcal{Y}^{(g)}\big{)}^{2}\prod_{{\scalebox{0.6}{$A$}}=1}^{4+4}{}^{\prime}\bar{\delta}(\mathcal{E}_{\scalebox{0.6}{$A$}})\
\,.$ (5.16)
The resulting integrand formula, presented here for $n=4$ particles for
simplicity, takes the form of an integral over the moduli space
${\mathfrak{M}}_{0,n+2g}$, fully localised on solutions to the nodal
scattering equations $\mathcal{E}_{\scalebox{0.6}{$A$}}=0$. While the
structure is reminiscent of the one-loop case, new features appear as well; in
particular the factors $c^{(g)}$ and ${\mathcal{J}}^{(g)}$ arise from the
degeneration of ${\mathfrak{M}}_{g,n}$ to ${\mathfrak{M}}_{0,n+2g}$. We
briefly discuss these ingredients below, more detailed expositions, as well as
$n$-point formulæ for supergravity and super Yang-Mills theory (constructed
again using the double copy) can be found in the original paper [108].
* •
_Moduli:_ The residue theorem localises the integrand on the non-separating
boundary, where $q_{II}=e^{i\pi\Omega_{II}}=0$. In this limit, the remaining
moduli, given by the off-diagonal components of the period matrix, become
cross-ratios of the nodal marked points $\sigma_{I^{\pm}}$,
$q_{IJ}=e^{2i\pi\Omega_{IJ}}=\frac{\sigma_{I^{+}J^{+}}\sigma_{I^{-}J^{-}}}{\sigma_{I^{+}J^{-}}\sigma_{I^{-}J^{+}}}\,.$
(5.17)
The measure is then naturally expressed in terms of all marked points
(including the nodes) modulo Möbius transformation, leading to a Jacobian
${\mathcal{J}}^{(g)}$ with
$\prod_{I<J}\frac{dq_{IJ}}{q_{IJ}}=\frac{{\mathcal{J}}^{(g)}}{\mathrm{vol\;SL}(2,\mathbb{C})}\,,\quad{\mathcal{J}}^{(g)}=\frac{1}{\sigma_{1^{+}2^{+}}\sigma_{1^{+}2^{-}}\sigma_{1^{-}2^{+}}\sigma_{1^{-}2^{-}}}\prod_{I^{\pm}}d\sigma_{I^{\pm}}\,.$
(5.18)
The analogous change of variables in the scattering equations results in
another copy of the same Jacobian factor.
The cross-ratio factor $\,c^{(2)}=1/(1-q_{12})$ originates in the degeneration
of the moduli space ${\mathfrak{M}}_{2,n}$ to ${\mathfrak{M}}_{0,n+4}$; to be
precise, from mapping the last modular parameter $q_{12}$ to the nodal sphere
[108]. It can be given a concrete physical interpretation in projecting out
unphysical poles from the integrand $\mathfrak{I}_{n}^{(2)}$.
* •
_Nodal scattering equations:_ On the nodal sphere, $P_{\mu}$ takes the
following form;
$P_{\mu}(\sigma)=\ell^{I}_{\mu}\,\omega_{I^{+}I^{-}}(\sigma)+\sum_{i}\frac{k_{i\mu}}{\sigma-\sigma_{i}}\,d\sigma\,,$
(5.19)
where $\omega_{I^{+}I^{-}}(\sigma)$ are the genus-two holomorphic Abelian
differentials in the maximal non-separating degeneration. In this limit, these
differentials acquire simple poles at the corresponding nodes:
$\omega_{I}=\frac{\omega_{I^{+}I^{-}}}{2\pi
i}\,,\quad\omega_{I^{+}I^{-}}(\sigma)=\frac{(\sigma_{I^{+}}-\sigma_{I^{-}})\,d\sigma}{(\sigma-\sigma_{I^{+}})(\sigma-\sigma_{I^{-}})}\,,$
(5.20)
The nodal scattering equations can then be compactly expressed as the
vanishing of a meromorphic quadratic differential $\mathfrak{P}^{(g)}$ with
only simple poles,
$\mathcal{E}_{\scalebox{0.6}{$A$}}=\mathrm{Res}_{\sigma_{\\!A}}\mathfrak{P}^{(g)}\,,\qquad\mathfrak{P}^{(2)}=P^{2}-(\ell^{I}\\!\omega_{I^{+}I^{-}})^{2}+(\ell_{1}^{2}+\ell_{2}^{2})\,\omega_{1^{+}1^{-}}\omega_{2^{+}2^{-}}\,.$
(5.21)
Note in particular the last term, a novel feature at genus two that plays a
crucial role in obtaining the correct loop propagators. 191919and is closely
related to the absence of a straightforward Feynman tree-theorem at two loops.
* •
_Integrand:_ The chiral integrand is defined straightforwardly by the nodal
sphere-limit of the genus-two expression,
$\mathcal{Y}=\mathcal{Y}^{(2)}\big{|}_{\mathrm{nodal}}$, and can be calculated
using (5.20).
### 5.6 Further topics
The nodal sphere formulation of loop integrands proved to be a starting point
for many further exciting avenues of research.
* •
At the level of the worldsheet model, the simple structure of one-loop
correlators, supported on a nodal sphere is reflected by the presence of a so-
called ‘gluing operator’ in the ambitwistor string [159]. This gluing operator
$\Delta$ encodes the propagator of the target-space field theory, and is thus
a BRST-invariant but non-local worldsheet operator. Genus zero correlators
with an insertion of $\Delta$ directly give the one-loop integrand formulæ
(5.2) localized on the nodal sphere, without need for further simplifications.
Extensions to higher loops are currently not known.
* •
Focusing on the integrand expressions, there has been tremendous progress on
extending many of the tree-level evaluation techniques to loop level, [160,
161, 162, 163].
* •
The nodal sphere, and in particular the forward limit structure of the
integrand, also inspired loop formulæ in the twistorial models [100]. These
were obtained from a forward limit of the 6d spinorial tree-level amplitudes
mentioned in section 3.
* •
Very recently, a proposal has also appeared for one-loop correlators in
massive $\phi^{4}$ theory on de Sitter spacetime, based on the nodal sphere
[164]. This builds on earlier work expressing (tree-level) de Sitter
‘cosmological correlators’ as worldsheet integrals, supported on so-called
cosmological scattering equations, which are now differential operators
expressed as functions of the conformal generators [165].
* •
Finally, the ambitwistor string progress at loop level has also inspired
calculations and proposals in the full superstring. In ref. [166], the authors
used an ambitwistor-string-inspired method, based on forward limits of the
moduli space integrals, to construct one-loop matrix elements with insertions
of operators $D^{2k}F^{n}$ and $D^{2k}R^{n}$ in the tree-level effective
action. Progress has also been made at higher loop orders, where the double
copy and the close relation between the ambitwistor string and the superstring
chiral integrands has been used to propose a formula for the three-loop four-
particle superstring integrand [91]. This is achieved by constructing first an
expression on the three-nodal sphere using BCJ numerators, and then lifting
this to a fully modular invariant proposal for the superstring chiral
integrand on a $g=3$ Riemann surface.
## 6 Frontiers
As we have seen, ambitwistor-strings give one of the most direct routes from a
physical theory to compact formulæ for tree amplitudes and loop integrands.
Despite these successes, many open questions remain and it is questionable as
to whether it will one day be possible to understand these worldsheet models
as providing secure basic formulations of physical theories. To consolidate
them, more work needs to be done to relate them to more standard fully
nonlinear formulations of physical theories, either via field theory, string
theory or holography. We briefly expand on these connections.
### 6.1 Curved backgrounds
Amplitudes on curved backgrounds are a relatively new subject with the
frontier being, until recently, at three points at tree level. They provide a
stepping stone to connect with conventional nonlinear field theory. Spaces of
complexified null geodesics make good sense on an analytic curved space with
metric $g(x)_{\mu\nu}$ [33] and one can ask whether ambitwistor strings can be
defined on such curved ambitwistor spaces. In particular in the RNS models of
§2 we can replace $P^{2}\rightarrow H=g^{\mu\nu}(x)P_{\mu}P_{\nu}+\ldots$. It
was shown in [167] that, in the type II case, curved analogues of the
constraints $H$, $G=\Psi\cdot P$ and $\tilde{G}=\tilde{\Psi}\cdot P$ can be
constructed so that they satisfy the flat space OPEs iff they are obtained
from a solution to the NS-NS-sector of 10d type II supergravity. This can be
used to construct vertex operators and amplitudes at three points on a plane
wave [168, 169], and one can similarly encode the Yang-Mills equations in the
heterotic model [170]. This provides a completely different perspective to
that pursued in the 1970’s and 1980’s when space-time field equations in 4d
were shown [24, 34, 171] to correspond to the existence of formal
neighbourhoods of $\mathbb{A}$ inside
$\mathbb{P}\mathbb{T}\times\mathbb{P}\mathbb{T}^{*}$ or supersymmetric
extensions [23, 172]. In the ambitwistor string, the field equations are
encoded in the quantum consistency of the worldsheet model on a curved
ambitwistor space. In all these cases and unlike the conventional string, one
can use field redefinitions to make the gauge-fixed action linear, so that
correlators are relatively easy to compute. However, the complexity of the
curved background constraints makes it problematic to identify the generic
integrated vertex operators.
The 4d twistor models and twistor-strings can also be defined on curved
backgrounds built from curved twistor spaces. These are equivalent to curved
but self-dual Yang-Mills or Einstein backgrounds, which are integrable.
Exploiting this integrability, one can construct amplitudes of arbitrary
multiplicity, [83, 173, 174] with explicit formulæ for certain classes of
backgrounds.
### 6.2 Relationship with conventional, null and chiral strings
All the amplitude formulæ we have discussed localize on the solutions to the
scattering equations. These equations were first obtained by Fairlie [175,
176] in a semiclassical study of string solutions, and made famous in the work
of Gross and Mende [177, 178], where they were shown to govern the string path
integral in the limit of high energy scattering at fixed angle ($s\gg
1/\alpha^{\prime}$). The scattering equations thus play a prominent role both
in the $\alpha^{\prime}\rightarrow\infty$ limit and in the field theory limit
$\alpha^{\prime}\rightarrow 0$ , albeit indirectly via the ambitwistor string.
This clash, with the scattering equations appearing in both the low tension
and high tension field theory limit, has so far impeded attempts to connect
the ambitwistor string directly with conventional strings. In this context, it
is interesting to note that the $\alpha^{\prime}\rightarrow\infty$ limit can
also be understood as a null limit in which the worldsheet becomes null ruled
by null geodesics [124].
More generally Siegel proposed that so-called chiral string theories could be
obtained by flipping certain worldsheet boundary conditions [123], so that
both left-moving and right-moving modes in the conventional string become
holomorphic on the worldsheet in the sectorized chiral string. A bosonic such
model had already been introduced in [179], and pure spinor versions discussed
in [180]. In these models both the left moving and right-moving Virasoro
constraints become holomorphic but commute with each other and each satisfies
the holomorphic Virasoro algebra. Ambitwistor have a similar character in the
sense that $X^{\mu}$ and $\int^{\sigma}P^{\mu}$ can be thought of as being
independent holomorphic functions on the worldsheet playing roles as different
combinations of left and right movers. However, the ambitwistor-string only
contains one holomorphic Virasoro generator, $P\cdot\partial X$. The gauged
constraint $P^{2}$ is analogous to difference between the two Virasoro
generators, but has trivial OPE with itself, even on a curved background. This
reflects the degeneration of the two copies of the Virasoro algebra to a
Galilean Conformal Algebra [124, 125]. On the other hand, in the model of
[179], the gauged constraint $P^{2}$ is replaced by a more general quadratic
expression in $P$ and $\partial X$
$P^{2}\rightarrow\mathcal{H}:=A_{\mu\nu}(X)P^{\mu}P^{\nu}+B_{\mu\nu}P^{\mu}\partial
X^{\nu}+C(x)_{\mu\nu}\partial X^{\mu}\partial X^{\nu}\,,$ (6.1)
that is constrained to obey a nontrivial OPE. The imposition of these OPEs
yields field equations for the background with a finite number of
$\alpha^{\prime}$ corrections. However, since the OPE of $\mathcal{H}$ with
itself is nontrivial, there no longer appears to be a reduction to ambitwistor
space nor localization on the scattering equations. The connections between
sectorized strings, null strings and ambitwistor strings are now well studied
in different models [124, 125, 180, 181], see also [182, 183]. For example,
versions of T-duality become possible in sectorized strings whereas they are
not in the ambitwistor-string [184].
The amplitude formulæ to which such sectorized chiral strings give rise appear
to be problematic. The Koba-Nielsen factor of the conventional string,
consisting of a product $\prod_{i<j}|\sigma_{ij}|^{\alpha^{\prime}s_{ij}}$, is
replaced by a product
$\prod_{i<j}(\sigma_{ij}/\bar{\sigma}_{ij})^{\alpha^{\prime}s_{ij}}$ so that
the branching would seem to make the contour prescription problematic. This is
resolved in the work of Mizera who uses twisted cohomology and residues to
define the amplitudes of these theories [185]. As
$\alpha^{\prime}\rightarrow\infty$ he shows localization on the scattering
equations.
### 6.3 Celestial holography and soft theorems
Celestial holography [186] seeks to understand the S-matrices of massless
theories by formulating these theories on the conformal boundary of
asymptotically flat space-times, $\mathscr{I}$, the _light-cone at infinity_.
This approach emerged from the study of connections between soft theorems for
amplitudes and asymptotic BMS symmetries of space-time [187, 188], and aims to
establish a holographic dictionary for the S-matrix from $\mathscr{I}$ or the
celestial sphere [189, 190]. In an asymptotically simple space-time, all light
rays reach $\mathscr{I}$ and we can represent ambitwistor space
$\mathbb{A}=T^{*}\mathscr{I}_{\mathbb{C}}$ as the cotangent bundle of the
complexification of null infinity. Both the RNS and 4d twistor ambitwistor
models can be expressed in this representation [191, 192] and from this
perspective, the connection between soft theorems and asymptotic symmetries
can be understood directly at the level of the vertex operators as generators
of supertranslations and super-rotations in their soft limits. The ambitwistor
strings then provide an underpinning theory for the generation of the S-matrix
now seen as being based at $\mathscr{I}$. The recently discovered
gravitational ‘$w_{1+\infty}$-symmetry’ [193] can be seen in this formulation
as arising directly from the geometry of the asymptotic twistor space at
$\mathscr{I}$ [194].
## Acknowledgements
This work was supported by the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant agreement No.
764850 “SAGEX”. YG is grateful for the support from the CUniverse research
promotion project “Toward World-class Fundamental Physics” of Chulalongkorn
University (grant reference CUAASC). The research of LJM is supported by the
STFC grant ST/T000864/1.
## Appendix A Penrose transform
The Penrose transform for a general ambitwistor space $\mathbb{P}\mathbb{A}$
maps spin $s$ fields on space-time to elements of
$H^{1}(\mathbb{P}\mathbb{A},\mathcal{O}(s-1))$, with the classes corresponding
to spin $s$ plane-waves of the form
$\epsilon_{\mu_{1}}\ldots\epsilon_{\mu_{s}}\rme^{ik\cdot X}$ mapping to
$(\epsilon\cdot P)^{s}\,\bar{\delta}(k\cdot P)\,\rme^{ik\cdot x}$.
Briefly, this follows by taking a cohomology class $\phi\in
H^{1}(\mathbb{P}\mathbb{A},\mathcal{O}(s-1))$ on $\mathbb{P}\mathbb{A}$,
pulling it back to $p^{*}\phi$ on $PT^{*}M|_{P^{2}=0}$, the lightcone inside
the projective cotangent bundle, where it becomes necessarily trivial as there
is no first cohomology on this space for the given homogeneity weights. Thus
we can find $g$ of weight $s-1$ on $T^{*}M|_{P^{2}}=0$ such that
$p^{*}\phi=\bar{\partial}g\,,\qquad\mbox{e.g.\ for $\phi$ in \eqref{drep}
}\qquad g=(\epsilon\cdot P)^{s}\frac{\rme^{ik\cdot X}}{k\cdot P}\,.$ (A.1)
Then we can obtain the field via
$(\epsilon\cdot P)^{s}\,\rme^{ik\cdot x}=P\cdot\partial_{X}g\,.$ (A.2)
Since $g$ is defined up to the gauge freedom, $\delta g=(\epsilon\cdot
P)^{s-1}\rme^{ik\cdot X}$ (or indeed some a general global holomorphic
function of $P$), we have the usual gauge freedom
$\delta(\epsilon_{\mu_{1}}\ldots\epsilon_{\mu_{s}}\rme^{ik\cdot
X})=k_{(\mu_{1}}\epsilon_{\mu_{2}}\ldots\epsilon_{\mu_{s})}\,\rme^{ik\cdot
X}.$ (A.3)
This can be proven more abstractly as follows. The Penrose transform is the
connecting map
$H^{0}(PT^{*}M_{P^{2}=0},\mathcal{O}(s))/P\cdot\partial_{X}\left(H^{0}(PT^{*}M_{P^{2}=0},\mathcal{O}(s-1))\right)\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}H^{1}(\mathbb{P}\mathbb{A},\mathcal{O}(s-1)),,$
(A.4)
from the long exact sequence in cohomology arising from the short exact
sequence:
$0\rightarrow\mathcal{O}(s-1)_{\mathbb{P}\mathbb{A}}\rightarrow\mathcal{O}(s-1)_{T^{*}M_{P^{2}=0}}\xrightarrow{P\cdot\partial_{X}}\mathcal{O}(s)_{T^{*}M_{P^{2}=0}}\rightarrow
0\,,.$ (A.5)
that defines the functions on $\mathbb{P}\mathbb{A}$, see [34, 22] for a full
discussion.
## Appendix B Conformal Field Theory basics
In this appendix, we present a lightning review of some aspects of chiral two-
dimensional conformal field theories (CFTs) particularly relevant for the
ambitwistor string. For a more extensive introduction, we refer the interested
reader to the multitude of excellent textbooks, for example [28, 29, 118, 195,
196, 197].
All ambitwistor string models are two-dimensional CFTs with a local action in
terms of a set of fields defined over a closed Riemann surface $\Sigma$,
referred to as the worldsheet. These fields can be characterized by their
statistics (bosonic vs fermionic) and their conformal weight, as well as
additional quantum numbers such as the spin structure at higher genus. The
conformal weight of a field $\Phi$ is a pair of half-integers
$(h,\bar{h})\in\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$ labeling
the transformation properties of $\Phi$ under two-dimensional conformal
transformations. 202020A field is called _primary_ if its conformal weights
are well-defined. Since 2d local conformal transformations are equivalent to
holomorphic coordinate transformations, we can identify the conformal weight
with the form degree. This means that $\Phi$ is section of $(h,\bar{h})$
powers fo the holomorphic and antiholomorphic canonical bundles
$K_{\Sigma}^{h}\otimes\overline{K}_{\Sigma}^{\bar{h}}$, i.e.
$\Phi\in\Omega^{0}(\Sigma,K_{\Sigma}^{h}\otimes\overline{K}_{\Sigma}^{\bar{h}})\,.$
(B.1)
Here, negative weights are to be interpreted as sections of the respective
tangent bundles, using the isomorphism $K_{\Sigma}^{-1}\cong T_{\Sigma}$.
Equivalently, $\Phi$ may be expressed locally in affine worldsheet coordinates
$z$ as
$\Phi(z,\bar{z})=\phi_{h,\bar{h}}\,(dz)^{h}\,(d\bar{z})^{\bar{h}}\,.$ (B.2)
In the ambitwistor string, we will only encounter _chiral_ CFTs, known as
$\beta\gamma$-systems (for bosons) or $bc$-systems (for fermions). We will
describe these jointly below, keeping track of the statistics via a variable
$\epsilon$, with $\epsilon=-1$ for fermionic statistics, and $\epsilon=1$ for
bosons. A chiral CFT is then defined by the action
$S=\frac{1}{2\pi}\int b\bar{\partial}c\,,$ (B.3)
in conformal gauge. The fields $b$ has conformal weight $(h,0)$, giving
conformal weight $(1-h,0)$ to the conjugate field $c$. Fields such as these
with conformal weight $\bar{h}=0$ are often referred to as ‘left-moving’. The
OPE between these conjugate fields is
$c(z)\,b(w)\sim\frac{1}{z-w}\,,\qquad\qquad
b(z)\,c(w)\sim-\frac{\epsilon}{z-w}\,.$ (B.4)
A standard calculation gives the holomorphic stress-energy tensor
$T_{bc}=-h\,b\partial c+(1-h)\,(\partial b)c\,.$ (B.5)
From this expression, we find the central charge anomaly as (twice) the
coefficient of the fourth order pole in the $T(z)T(w)$ OPE,
$\mathfrak{c}=2\epsilon\left(6h^{2}-6h+1\right)\,.$ (B.6)
Finally we note the following useful formula for the number of zero modes
$n_{b}$ and $n_{c}$ of the two fields on a Riemann surface of genus $g$,
derived via the Riemann-Roch theorem,
$n_{c}-n_{b}=\frac{1}{2}\left(2h-1\right)\chi\,,\qquad\mathrm{with}\;\;\chi=2(1-g)\,.$
(B.7)
## Bibliography
## References
* [1] E. Witten, _Perturbative gauge theory as a string theory in twistor space_ , _Commun.Math.Phys._ 252 (2004) 189–258, [hep-th/0312171].
* [2] N. Berkovits, _An Alternative string theory in twistor space for N=4 superYang-Mills_ , _Phys.Rev.Lett._ 93 (2004) 011601, [hep-th/0402045].
* [3] N. Berkovits and E. Witten, _Conformal supergravity in twistor-string theory_ , _JHEP_ 0408 (2004) 009, [hep-th/0406051].
* [4] V. P. Nair, _A Current Algebra for Some Gauge Theory Amplitudes_ , _Phys. Lett._ B214 (1988) 215–218.
* [5] R. Roiban, M. Spradlin and A. Volovich, _On the tree level S matrix of Yang-Mills theory_ , _Phys.Rev._ D70 (2004) 026009, [hep-th/0403190].
* [6] E. Witten, _Parity invariance for strings in twistor space_ , _Adv.Theor.Math.Phys._ 8 (2004) 779–796, [hep-th/0403199].
* [7] L. Dolan and P. Goddard, _Tree and Loop Amplitudes in Open Twistor String Theory_ , _JHEP_ 0706 (2007) 005, [hep-th/0703054].
* [8] Z. Bern, J. J. M. Carrasco and H. Johansson, _Perturbative Quantum Gravity as a Double Copy of Gauge Theory_ , _Phys.Rev.Lett._ 105 (2010) 061602, [1004.0476].
* [9] Z. Bern, J. J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, _The SAGEX Review on Scattering Amplitudes, Chapter 2: An Invitation to Color-Kinematics Duality and the Double Copy_ , 2203.13013.
* [10] H. Kawai, D. C. Lewellen and S. H. H. Tye, _A Relation Between Tree Amplitudes of Closed and Open Strings_ , _Nucl. Phys._ B269 (1986) 1.
* [11] F. Cachazo, _Fundamental BCJ Relation in N=4 SYM From The Connected Formulation_ , 1206.5970.
* [12] F. Cachazo and Y. Geyer, _A ‘Twistor String’ Inspired Formula For Tree-Level Scattering Amplitudes in N=8 SUGRA_ , 1206.6511.
* [13] A. Hodges, _A simple formula for gravitational MHV amplitudes_ , 1204.1930.
* [14] F. Cachazo and D. Skinner, _Gravity from Rational Curves in Twistor Space_ , _Phys.Rev.Lett._ 110 (2013) 161301, [1207.0741].
* [15] F. Cachazo, L. Mason and D. Skinner, _Gravity in Twistor Space and its Grassmannian Formulation_ , _SIGMA_ 10 (2014) 051, [1207.4712].
* [16] D. Skinner, _Twistor strings for $\mathcal{N}$ = 8 supergravity_, _JHEP_ 04 (2020) 047, [1301.0868].
* [17] F. Cachazo, S. He and E. Y. Yuan, _Scattering equations and Kawai-Lewellen-Tye orthogonality_ , _Phys. Rev._ D90 (2014) 065001, [1306.6575].
* [18] F. Cachazo, S. He and E. Y. Yuan, _Scattering of Massless Particles in Arbitrary Dimensions_ , _Phys.Rev.Lett._ 113 (2014) 171601, [1307.2199].
* [19] F. Cachazo, S. He and E. Y. Yuan, _Scattering of Massless Particles: Scalars, Gluons and Gravitons_ , _JHEP_ 1407 (2014) 033, [1309.0885].
* [20] F. Cachazo, S. He and E. Y. Yuan, _Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations_ , _JHEP_ 01 (2015) 121, [1409.8256].
* [21] F. Cachazo, S. He and E. Y. Yuan, _Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM_ , _JHEP_ 07 (2015) 149, [1412.3479].
* [22] L. Mason and D. Skinner, _Ambitwistor strings and the scattering equations_ , _JHEP_ 1407 (2014) 048, [1311.2564].
* [23] E. Witten, _An Interpretation of Classical Yang-Mills Theory_ , _Phys.Lett._ B77 (1978) 394.
* [24] J. Isenberg, P. Yasskin and P. Green, _Non selfdual Gauge Fields_ , _Phys.Lett._ B78 (1978) 462–464.
* [25] R. S. Ward, _On Selfdual gauge fields_ , _Phys. Lett. A_ 61 (1977) 81–82.
* [26] Y. Geyer, A. E. Lipstein and L. J. Mason, _Ambitwistor Strings in Four Dimensions_ , _Phys. Rev. Lett._ 113 (2014) 081602, [1404.6219].
* [27] E. Casali, Y. Geyer, L. Mason, R. Monteiro and K. A. Roehrig, _New Ambitwistor String Theories_ , _JHEP_ 11 (2015) 038, [1506.08771].
* [28] M. B. Green, J. H. Schwarz and E. Witten, _Superstring Theory_ , vol. 1. CUP, 1987.
* [29] M. B. Green, J. H. Schwarz and E. Witten, _Superstring Theory_ , vol. 2. CUP, 1987.
* [30] E. Witten, _Superstring Perturbation Theory Revisited_ , 1209.5461.
* [31] T. Adamo and L. Mason, _Conformal and Einstein gravity from twistor actions_ , _Class.Quant.Grav._ 31 (2014) 045014, [1307.5043].
* [32] K. Ohmori, _Worldsheet Geometries of Ambitwistor String_ , _JHEP_ 06 (2015) 075, [1504.02675].
* [33] C. LeBrun, _Spaces of complex null geodesics in Complex Riemmannian Geometry_ , _Transactions of the AMS._ 278 (1983) 209–31.
* [34] R. Baston and L. Mason, _Conformal gravity, the Einstein equations and spaces of complex null geodesics_ , _Class.Quant.Grav._ 4 (1987) 815–826.
* [35] E. Witten, _Two-dimensional models with (0,2) supersymmetry: Perturbative aspects_ , _Adv. Theor. Math. Phys._ 11 (2007) 1–63, [hep-th/0504078].
* [36] R. Britto, F. Cachazo and B. Feng, _New recursion relations for tree amplitudes of gluons_ , _Nucl.Phys._ B715 (2005) 499–522, [hep-th/0412308].
* [37] R. Britto, F. Cachazo, B. Feng and E. Witten, _Direct proof of tree-level recursion relation in Yang-Mills theory_ , _Phys.Rev.Lett._ 94 (2005) 181602, [hep-th/0501052].
* [38] F. Cachazo and P. Svrcek, _Tree level recursion relations in general relativity_ , hep-th/0502160.
* [39] J. Bedford, A. Brandhuber, B. J. Spence and G. Travaglini, _A Recursion relation for gravity amplitudes_ , _Nucl.Phys._ B721 (2005) 98–110, [hep-th/0502146].
* [40] A. Brandhuber, J. Plefka and G. Travaglini, _The SAGEX Review on Scattering Amplitudes, Chapter 1: Modern Fundamentals of Amplitudes_ , 2203.13012.
* [41] N. Arkani-Hamed and J. Kaplan, _On Tree Amplitudes in Gauge Theory and Gravity_ , _JHEP_ 04 (2008) 076, [0801.2385].
* [42] N. Arkani-Hamed and J. Kaplan, _On Tree Amplitudes in Gauge Theory and Gravity_ , _JHEP_ 0804 (2008) 076, [0801.2385].
* [43] T. Cohen, H. Elvang and M. Kiermaier, _On-shell constructibility of tree amplitudes in general field theories_ , _JHEP_ 04 (2011) 053, [1010.0257].
* [44] H. Elvang and Y.-t. Huang, _Scattering Amplitudes_ , 1308.1697.
* [45] H. Elvang and Y.-t. Huang, _Scattering Amplitudes in Gauge Theory and Gravity_. Cambridge University Press, 4, 2015.
* [46] C. Cheung, _TASI Lectures on Scattering Amplitudes_ , pp. 571–623. 2018\. 1708.03872. 10.1142/9789813233348-0008.
* [47] B. Feng and M. Luo, _An Introduction to On-shell Recursion Relations_ , _Front. Phys. (Beijing)_ 7 (2012) 533–575, [1111.5759].
* [48] L. Dolan and P. Goddard, _Proof of the Formula of Cachazo, He and Yuan for Yang-Mills Tree Amplitudes in Arbitrary Dimension_ , _JHEP_ 1405 (2014) 010, [1311.5200].
* [49] P. Deligne and D. Mumford, _The irreducibility of the space of curves of given genus_ , _Publ. Math. IHES_ (1969) 75–110.
* [50] H. Johansson and J. Nohle, _Conformal Gravity from Gauge Theory_ , 1707.02965.
* [51] N. Berkovits and M. Lize, _Field theory actions for ambitwistor string and superstring_ , _JHEP_ 09 (2018) 097, [1807.07661].
* [52] Y. Geyer, L. Mason and D. Skinner, _Ambitwistor strings in six and five dimensions_ , _JHEP_ 08 (2021) 153, [2012.15172].
* [53] T. Azevedo and O. T. Engelund, _Ambitwistor formulations of R 2 gravity and (DF)2 gauge theories_, _JHEP_ 11 (2017) 052, [1707.02192].
* [54] C. Cheung, K. Kampf, J. Novotny, C.-H. Shen and J. Trnka, _On-Shell Recursion Relations for Effective Field Theories_ , _Phys. Rev. Lett._ 116 (2016) 041601, [1509.03309].
* [55] H. Luo and C. Wen, _Recursion relations from soft theorems_ , _JHEP_ 03 (2016) 088, [1512.06801].
* [56] N. Berkovits, _Infinite Tension Limit of the Pure Spinor Superstring_ , _JHEP_ 1403 (2014) 017, [1311.4156].
* [57] H. Gomez and E. Y. Yuan, _N-point tree-level scattering amplitude in the new Berkovits‘ string_ , _JHEP_ 04 (2014) 046, [1312.5485].
* [58] O. Chandia and B. C. Vallilo, _Ambitwistor superstring in the Green–Schwarz formulation_ , _Eur. Phys. J. C_ 77 (2017) 473, [1612.01806].
* [59] N. Berkovits, E. Casali, M. Guillen and L. Mason, _Notes on the $D=11$ pure spinor superparticle_, _JHEP_ 08 (2019) 178, [1905.03737].
* [60] M. Guillen, _Notes on the 11D pure spinor wordline vertex operators_ , _JHEP_ 08 (2020) 122, [2006.06022].
* [61] Z. Bern, J. Carrasco and H. Johansson, _New Relations for Gauge-Theory Amplitudes_ , _Phys.Rev._ D78 (2008) 085011, [0805.3993].
* [62] C.-H. Fu, Y.-J. Du, R. Huang and B. Feng, _Expansion of Einstein-Yang-Mills Amplitude_ , _JHEP_ 09 (2017) 021, [1702.08158].
* [63] S. Mizera, _Kinematic Jacobi Identity is a Residue Theorem: Geometry of Color-Kinematics Duality for Gauge and Gravity Amplitudes_ , _Phys. Rev. Lett._ 124 (2020) 141601, [1912.03397].
* [64] A. Edison and F. Teng, _Efficient Calculation of Crossing Symmetric BCJ Tree Numerators_ , _JHEP_ 12 (2020) 138, [2005.03638].
* [65] S. He, L. Hou, J. Tian and Y. Zhang, _Kinematic numerators from the worldsheet: cubic trees from labelled trees_ , _JHEP_ 08 (2021) 118, [2103.15810].
* [66] H. Frost and L. Mason, _Lie Polynomials and a Twistorial Correspondence for Amplitudes_ , 1912.04198.
* [67] C. R. Mafra, _Planar binary trees in scattering amplitudes_ , 11, 2020, 2011.14413, DOI.
* [68] H. Frost, C. R. Mafra and L. Mason, _A Lie bracket for the momentum kernel_ , 2012.00519.
* [69] N. Arkani-Hamed, T.-C. Huang and Y.-t. Huang, _Scattering amplitudes for all masses and spins_ , _JHEP_ 11 (2021) 070, [1709.04891].
* [70] H. Frost, _Universal aspects of perturbative gauge theory_ , Ph.D. thesis, Oxford U., 2020.
* [71] H. Frost, _The Algebraic Structure of the KLT Relations for Gauge and Gravity Tree Amplitudes_ , _SIGMA_ 17 (2021) 101, [2111.07257].
* [72] T. Adamo, _Lectures on twistor theory_ , _PoS_ Modave2017 (2018) 003, [1712.02196].
* [73] T. Adamo, M. Bullimore, L. Mason and D. Skinner, _Scattering Amplitudes and Wilson Loops in Twistor Space_ , _J.Phys.A_ A44 (2011) 454008, [1104.2890].
* [74] S. A. Huggett and K. P. Tod, _An Introduction to Twistor Theory_. Cambridge University Press, 1986, 10.1017/CBO9780511624018.
* [75] N. Woodhouse, _Real methods in twistor theory_ , _Class.Quant.Grav._ 2 (1985) 257–291.
* [76] Y. Geyer, _Ambitwistor Strings: Worldsheet Approaches to perturbative Quantum Field Theories_ , Ph.D. thesis, Oxford U., Inst. Math., 2016. 1610.04525.
* [77] D. Skinner, _A Direct Proof of BCFW Recursion for Twistor-Strings_ , _JHEP_ 1101 (2011) 072, [1007.0195].
* [78] N. Berkovits and L. Motl, _Cubic twistorial string field theory_ , _JHEP_ 0404 (2004) 056, [hep-th/0403187].
* [79] T. Adamo, _Worldsheet factorization for twistor-strings_ , _JHEP_ 04 (2014) 080, [1310.8602].
* [80] R. Penrose and W. Rindler, _Spinors and Space-Time: Spinor And Twistor Methods In Space-time Geometry_ , vol. 2. Cambridge University Press, 1986.
* [81] M. Wolf, _Self-Dual Supergravity and Twistor Theory_ , _Class.Quant.Grav._ 24 (2007) 6287–6328, [0705.1422].
* [82] T. Adamo, _Gravity with a cosmological constant from rational curves_ , _JHEP_ 11 (2015) 098, [1508.02554].
* [83] T. Adamo, L. Mason and A. Sharma, _Twistor sigma models for quaternionic geometry and graviton scattering_ , 2103.16984.
* [84] K. A. Roehrig, _Chiral splitting and $\mathcal{N}=4$ Einstein-Yang-Mills tree amplitudes in 4d_, _JHEP_ 08 (2017) 033, [1705.09315].
* [85] T. Adamo, E. Casali, K. A. Roehrig and D. Skinner, _On tree amplitudes of supersymmetric Einstein-Yang-Mills theory_ , _JHEP_ 12 (2015) 177, [1507.02207].
* [86] G. Albonico, Y. Geyer and L. Mason, _Recursion and worldsheet formulae for 6d superamplitudes_ , _JHEP_ 08 (2020) 066, [2001.05928].
* [87] S. He and Y. Zhang, _Connected formulas for amplitudes in standard model_ , _JHEP_ 03 (2017) 093, [1607.02843].
* [88] Y. Zhang, _CHY formulae in 4d_ , _JHEP_ 07 (2017) 069, [1610.05205].
* [89] Y. Geyer and L. Mason, _Polarized Scattering Equations for 6D Superamplitudes_ , _Phys. Rev. Lett._ 122 (2019) 101601, [1812.05548].
* [90] Y. Geyer and L. Mason, _Supersymmetric S-matrices from the worldsheet in 10 \ & 11d_, _Phys. Lett. B_ 804 (2020) 135361, [1901.00134].
* [91] Y. Geyer, R. Monteiro and R. Stark-Muchão, _Superstring Loop Amplitudes from the Field Theory Limit_ , _Phys. Rev. Lett._ 127 (2021) 211603, [2106.03968].
* [92] M. Heydeman, J. H. Schwarz and C. Wen, _M5-Brane and D-Brane Scattering Amplitudes_ , _JHEP_ 12 (2017) 003, [1710.02170].
* [93] M. Heydeman, J. H. Schwarz, C. Wen and S.-Q. Zhang, _All Tree Amplitudes of 6D $(2,0)$ Supergravity: Interacting Tensor Multiplets and the $K3$ Moduli Space_, _Phys. Rev. Lett._ 122 (2019) 111604, [1812.06111].
* [94] F. Cachazo, A. Guevara, M. Heydeman, S. Mizera, J. H. Schwarz and C. Wen, _The S Matrix of 6D Super Yang-Mills and Maximal Supergravity from Rational Maps_ , _JHEP_ 09 (2018) 125, [1805.11111].
* [95] N. Berkovits, M. Guillen and L. Mason, _Supertwistor description of ambitwistor strings_ , _JHEP_ 01 (2020) 020, [1908.06899].
* [96] R. A. Reid-Edwards and D. A. Riccombeni, _A Superstring Field Theory for Supergravity_ , _JHEP_ 09 (2017) 103, [1701.05495].
* [97] D. G. Sepúlveda and M. Guillen, _A Pure Spinor Twistor Description of Ambitwistor Strings_ , 2006.06025.
* [98] R. Penrose, _Twistors and Particles: An Outline_ , in _Feldafing Conference of the Max-Planck Inst. on Quantum Theory and the Structure of Space-time, 1974_ , pp. 129–145, Carl Hanser Verlag, Munich, 1975.
* [99] G. Albonico, Y. Geyer and L. Mason, _From twistor-particle models to massive amplitudes_ , 2203.08087.
* [100] C. Wen and S.-Q. Zhang, _D3-Brane Loop Amplitudes from M5-Brane Tree Amplitudes_ , _JHEP_ 07 (2020) 098, [2004.02735].
* [101] M. Bullimore, L. Mason and D. Skinner, _Twistor-Strings, Grassmannians and Leading Singularities_ , _JHEP_ 1003 (2010) 070, [0912.0539].
* [102] N. Arkani-Hamed, J. Bourjaily, F. Cachazo and J. Trnka, _Unification of Residues and Grassmannian Dualities_ , _JHEP_ 01 (2011) 049, [0912.4912].
* [103] N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, _A Duality For The S Matrix_ , _JHEP_ 1003 (2010) 020, [0907.5418].
* [104] J. H. Schwarz and C. Wen, _Unified Formalism for 6D Superamplitudes Based on a Symplectic Grassmannian_ , _JHEP_ 08 (2019) 125, [1907.03485].
* [105] T. Adamo, E. Casali and D. Skinner, _Ambitwistor strings and the scattering equations at one loop_ , _JHEP_ 1404 (2014) 104, [1312.3828].
* [106] E. Casali and P. Tourkine, _Infrared behaviour of the one-loop scattering equations and supergravity integrands_ , _JHEP_ 1504 (2015) 013, [1412.3787].
* [107] T. Adamo and E. Casali, _Scattering equations, supergravity integrands, and pure spinors_ , _JHEP_ 1505 (2015) 120, [1502.06826].
* [108] Y. Geyer and R. Monteiro, _Two-Loop Scattering Amplitudes from Ambitwistor Strings: from Genus Two to the Nodal Riemann Sphere_ , _JHEP_ 11 (2018) 008, [1805.05344].
* [109] J. A. Farrow and A. E. Lipstein, _From 4d Ambitwistor Strings to On Shell Diagrams and Back_ , _JHEP_ 07 (2017) 114, [1705.07087].
* [110] E. D’Hoker and D. H. Phong, _Two loop superstrings. 1. Main formulas_ , _Phys. Lett._ B529 (2002) 241–255, [hep-th/0110247].
* [111] E. D’Hoker and D. H. Phong, _Two loop superstrings. 2. The Chiral measure on moduli space_ , _Nucl. Phys._ B636 (2002) 3–60, [hep-th/0110283].
* [112] E. D’Hoker and D. H. Phong, _Two loop superstrings. 3. Slice independence and absence of ambiguities_ , _Nucl. Phys._ B636 (2002) 61–79, [hep-th/0111016].
* [113] E. D’Hoker and D. H. Phong, _Two loop superstrings 4: The Cosmological constant and modular forms_ , _Nucl. Phys._ B639 (2002) 129–181, [hep-th/0111040].
* [114] E. D’Hoker and D. H. Phong, _Two-loop superstrings. V. Gauge slice independence of the N-point function_ , _Nucl. Phys._ B715 (2005) 91–119, [hep-th/0501196].
* [115] E. D’Hoker and D. H. Phong, _Two-loop superstrings VI: Non-renormalization theorems and the 4-point function_ , _Nucl. Phys._ B715 (2005) 3–90, [hep-th/0501197].
* [116] E. D’Hoker and D. H. Phong, _Lectures on two loop superstrings_ , _Conf. Proc._ C0208124 (2002) 85–123, [hep-th/0211111].
* [117] E. D’Hoker and D. H. Phong, _The Geometry of String Perturbation Theory_ , _Rev. Mod. Phys._ 60 (1988) 917.
* [118] J. Polchinski, _String theory. Vol. 1: An introduction to the bosonic string_. Cambridge University Press, 2007.
* [119] N. Seiberg and E. Witten, _Spin Structures in String Theory_ , _Nucl. Phys. B_ 276 (1986) 272.
* [120] E. D’Hoker and D. H. Phong, _Conformal Scalar Fields and Chiral Splitting on Superriemann Surfaces_ , _Commun. Math. Phys._ 125 (1989) 469.
* [121] P. Tourkine, _Integrands and loop momentum in string and field theory_ , _Phys. Rev. D_ 102 (2020) 026006, [1901.02432].
* [122] N. Kalyanapuram, _On Chiral Splitting and the Ambitwistor String_ , _Phys. Rev. D_ 104 (2021) 086027, [2103.08584].
* [123] W. Siegel, _Amplitudes for Left-Handed Strings_ , 1512.02569.
* [124] E. Casali and P. Tourkine, _On the null origin of the ambitwistor string_ , _JHEP_ 11 (2016) 036, [1606.05636].
* [125] E. Casali, Y. Herfray and P. Tourkine, _The complex null string, Galilean conformal algebra and scattering equations_ , _JHEP_ 10 (2017) 164, [1707.09900].
* [126] M. Yu, C. Zhang and Y.-Z. Zhang, _One loop amplitude from null string_ , _JHEP_ 06 (2017) 051, [1704.01290].
* [127] Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, _Loop Integrands for Scattering Amplitudes from the Riemann Sphere_ , _Phys. Rev. Lett._ 115 (2015) 121603, [1507.00321].
* [128] Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, _One-loop amplitudes on the Riemann sphere_ , _JHEP_ 03 (2016) 114, [1511.06315].
* [129] Y. Geyer, L. Mason, R. Monteiro and P. Tourkine, _Two-Loop Scattering Amplitudes from the Riemann Sphere_ , _Phys. Rev._ D94 (2016) 125029, [1607.08887].
* [130] P. Tourkine and P. Vanhove, _One-loop four-graviton amplitudes in ${\cal N}=4$ supergravity models_, _Phys. Rev._ D87 (2013) 045001, [1208.1255].
* [131] S. He and E. Y. Yuan, _One-loop Scattering Equations and Amplitudes from Forward Limit_ , _Phys. Rev._ D92 (2015) 105004, [1508.06027].
* [132] F. Cachazo, S. He and E. Y. Yuan, _One-Loop Corrections from Higher Dimensional Tree Amplitudes_ , _JHEP_ 08 (2016) 008, [1512.05001].
* [133] B. Feng, _CHY-construction of Planar Loop Integrands of Cubic Scalar Theory_ , _JHEP_ 05 (2016) 061, [1601.05864].
* [134] B. Feng and C. Hu, _One-loop CHY-Integrand of Bi-adjoint Scalar Theory_ , _JHEP_ 02 (2020) 187, [1912.12960].
* [135] F. Porkert and O. Schlotterer, _One-loop amplitudes in Einstein-Yang-Mills from forward limits_ , 2201.12072.
* [136] R. P. Feynman, _Quantum theory of gravitation_ , _Acta Phys. Polon._ 24 (1963) 697–722.
* [137] S. Caron-Huot, _Loops and trees_ , _JHEP_ 05 (2011) 080, [1007.3224].
* [138] C. Baadsgaard, N. E. J. Bjerrum-Bohr, J. L. Bourjaily, S. Caron-Huot, P. H. Damgaard and B. Feng, _New Representations of the Perturbative S-Matrix_ , _Phys. Rev. Lett._ 116 (2016) 061601, [1509.02169].
* [139] S. Abreu, R. Britto and C. Duhr, _The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals_ , 2203.13014.
* [140] H. Gomez, _Quadratic Feynman Loop Integrands From Massless Scattering Equations_ , _Phys. Rev. D_ 95 (2017) 106006, [1703.04714].
* [141] H. Gomez, C. Lopez-Arcos and P. Talavera, _One-loop Parke-Taylor factors for quadratic propagators from massless scattering equations_ , _JHEP_ 10 (2017) 175, [1707.08584].
* [142] N. Ahmadiniaz, H. Gomez and C. Lopez-Arcos, _Non-planar one-loop Parke-Taylor factors in the CHY approach for quadratic propagators_ , _JHEP_ 05 (2018) 055, [1802.00015].
* [143] J. A. Farrow, Y. Geyer, A. E. Lipstein, R. Monteiro and R. Stark-Muchão, _Propagators, BCFW recursion and new scattering equations at one loop_ , _JHEP_ 10 (2020) 074, [2007.00623].
* [144] J. J. Carrasco and H. Johansson, _Five-Point Amplitudes in N=4 Super-Yang-Mills Theory and N=8 Supergravity_ , _Phys.Rev._ D85 (2012) 025006, [1106.4711].
* [145] Z. Bern, J. J. M. Carrasco, L. J. Dixon, H. Johansson and R. Roiban, _Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes_ , _Phys. Rev._ D85 (2012) 105014, [1201.5366].
* [146] Z. Bern, S. Davies, T. Dennen and Y.-t. Huang, _Absence of Three-Loop Four-Point Divergences in N=4 Supergravity_ , _Phys. Rev. Lett._ 108 (2012) 201301, [1202.3423].
* [147] Z. Bern, S. Davies and T. Dennen, _The Ultraviolet Structure of Half-Maximal Supergravity with Matter Multiplets at Two and Three Loops_ , _Phys. Rev._ D88 (2013) 065007, [1305.4876].
* [148] Z. Bern, S. Davies, T. Dennen, A. V. Smirnov and V. A. Smirnov, _Ultraviolet Properties of N=4 Supergravity at Four Loops_ , _Phys. Rev. Lett._ 111 (2013) 231302, [1309.2498].
* [149] Z. Bern, S. Davies and T. Dennen, _Enhanced ultraviolet cancellations in $\mathcal{N}=5$ supergravity at four loops_, _Phys. Rev._ D90 (2014) 105011, [1409.3089].
* [150] C. R. Mafra and O. Schlotterer, _Two-loop five-point amplitudes of super Yang-Mills and supergravity in pure spinor superspace_ , _JHEP_ 10 (2015) 124, [1505.02746].
* [151] H. Johansson, G. Kalin and G. Mogull, _Two-loop supersymmetric QCD and half-maximal supergravity amplitudes_ , _JHEP_ 09 (2017) 019, [1706.09381].
* [152] S. He and O. Schlotterer, _New Relations for Gauge-Theory and Gravity Amplitudes at Loop Level_ , _Phys. Rev. Lett._ 118 (2017) 161601, [1612.00417].
* [153] S. He, O. Schlotterer and Y. Zhang, _New BCJ representations for one-loop amplitudes in gauge theories and gravity_ , _Nucl. Phys._ B930 (2018) 328–383, [1706.00640].
* [154] Y. Geyer and R. Monteiro, _Gluons and gravitons at one loop from ambitwistor strings_ , _JHEP_ 03 (2018) 068, [1711.09923].
* [155] A. Edison, S. He, O. Schlotterer and F. Teng, _One-loop Correlators and BCJ Numerators from Forward Limits_ , _JHEP_ 09 (2020) 079, [2005.03639].
* [156] Y. Geyer, R. Monteiro and R. Stark-Muchão, _Two-Loop Scattering Amplitudes: Double-Forward Limit and Colour-Kinematics Duality_ , _JHEP_ 12 (2019) 049, [1908.05221].
* [157] C. R. Mafra and O. Schlotterer, _Towards one-loop SYM amplitudes from the pure spinor BRST cohomology_ , _Fortsch. Phys._ 63 (2015) 105–131, [1410.0668].
* [158] M. Berg, I. Buchberger and O. Schlotterer, _String-motivated one-loop amplitudes in gauge theories with half-maximal supersymmetry_ , _JHEP_ 07 (2017) 138, [1611.03459].
* [159] K. A. Roehrig and D. Skinner, _A Gluing Operator for the Ambitwistor String_ , _JHEP_ 01 (2018) 069, [1709.03262].
* [160] C. Baadsgaard, N. E. J. Bjerrum-Bohr, J. L. Bourjaily, P. H. Damgaard and B. Feng, _Integration Rules for Loop Scattering Equations_ , _JHEP_ 11 (2015) 080, [1508.03627].
* [161] M. Zlotnikov, _Polynomial reduction and evaluation of tree- and loop-level CHY amplitudes_ , _JHEP_ 08 (2016) 143, [1605.08758].
* [162] H. Gomez, S. Mizera and G. Zhang, _CHY Loop Integrands from Holomorphic Forms_ , _JHEP_ 03 (2017) 092, [1612.06854].
* [163] G. Chen, Y.-K. E. Cheung, T. Wang and F. Xu, _A differential operator for integrating one-loop scattering equations_ , _JHEP_ 01 (2017) 028, [1609.07621].
* [164] H. Gomez, R. L. Jusinskas and A. Lipstein, _Cosmological Scattering Equations at Tree-level and One-loop_ , 2112.12695.
* [165] H. Gomez, R. L. Jusinskas and A. Lipstein, _Cosmological Scattering Equations_ , _Phys. Rev. Lett._ 127 (2021) 251604, [2106.11903].
* [166] A. Edison, M. Guillen, H. Johansson, O. Schlotterer and F. Teng, _One-loop matrix elements of effective superstring interactions: $\alpha$’-expanding loop integrands_, _JHEP_ 12 (2021) 007, [2107.08009].
* [167] T. Adamo, E. Casali and D. Skinner, _A Worldsheet Theory for Supergravity_ , _JHEP_ 02 (2015) 116, [1409.5656].
* [168] T. Adamo, E. Casali, L. Mason and S. Nekovar, _Amplitudes on plane waves from ambitwistor strings_ , _JHEP_ 11 (2017) 160, [1708.09249].
* [169] T. Adamo, E. Casali and S. Nekovar, _Ambitwistor string vertex operators on curved backgrounds_ , _JHEP_ 01 (2019) 213, [1809.04489].
* [170] T. Adamo, E. Casali and S. Nekovar, _Yang-Mills theory from the worldsheet_ , _Phys. Rev._ D98 (2018) 086022, [1807.09171].
* [171] C. LeBrun, _Thickenings and conformal gravity_ , _Commun.Math.Phys._ 139 (1991) 1–43.
* [172] E. Witten, _Twistor - Like Transform in Ten-Dimensions_ , _Nucl.Phys._ B266 (1986) 245.
* [173] T. Adamo, L. Mason and A. Sharma, _Gluon scattering on self-dual radiative gauge fields_ , 2010.14996.
* [174] T. Adamo, L. Mason and A. Sharma, _Graviton scattering in self-dual radiative space-times_ , 2203.02238.
* [175] D. B. Fairlie and D. E. Roberts, _Dual models without tachyons - a new approach_ , .
* [176] D. B. Fairlie, _A Coding of Real Null Four-Momenta into World-Sheet Co-ordinates_ , _Adv. Math. Phys._ 2009 (2009) 284689, [0805.2263].
* [177] D. J. Gross and P. F. Mende, _String Theory Beyond the Planck Scale_ , _Nucl.Phys._ B303 (1988) 407.
* [178] D. J. Gross and P. F. Mende, _The High-Energy Behavior of String Scattering Amplitudes_ , _Phys.Lett._ B197 (1987) 129.
* [179] O. Hohm, W. Siegel and B. Zwiebach, _Doubled $\alpha^{\prime}$-geometry_, _JHEP_ 02 (2014) 065, [1306.2970].
* [180] T. Azevedo and R. L. Jusinskas, _Connecting the ambitwistor and the sectorized heterotic strings_ , _JHEP_ 10 (2017) 216, [1707.08840].
* [181] T. Azevedo, R. L. Jusinskas and M. Lize, _Bosonic sectorized strings and the $(DF)^{2}$ theory_, _JHEP_ 01 (2020) 082, [1908.11371].
* [182] R. L. Jusinskas, _Asymmetrically twisted strings_ , 2108.13426.
* [183] M. L. Lize, B. Lyu, W. Siegel and Y.-P. Wang, _Chiral string theories as an interpolation between strings and particles_ , 2109.10401.
* [184] E. Casali and P. Tourkine, _Windings of twisted strings_ , _Phys. Rev._ D97 (2018) 061902, [1710.01241].
* [185] S. Mizera, _Aspects of Scattering Amplitudes and Moduli Space Localization_ , Ph.D. thesis, Princeton, Inst. Advanced Study, 2020. 1906.02099. 10.1007/978-3-030-53010-5.
* [186] T. McLoughlin, A. Puhm and A.-M. Raclariu, _The SAGEX Review on Scattering Amplitudes, Chapter 11: Soft Theorems and Celestial Amplitudes_ , 2203.13022.
* [187] A. Strominger, _On BMS Invariance of Gravitational Scattering_ , _JHEP_ 07 (2014) 152, [1312.2229].
* [188] A. Strominger, _Lectures on the Infrared Structure of Gravity and Gauge Theory_ , 1703.05448.
* [189] S. Pasterski and S.-H. Shao, _Conformal basis for flat space amplitudes_ , _Phys. Rev. D_ 96 (2017) 065022, [1705.01027].
* [190] S. Pasterski, _Lectures on celestial amplitudes_ , _Eur. Phys. J. C_ 81 (2021) 1062, [2108.04801].
* [191] T. Adamo, E. Casali and D. Skinner, _Perturbative gravity at null infinity_ , _Class.Quant.Grav._ 31 (2014) 225008, [1405.5122].
* [192] Y. Geyer, A. E. Lipstein and L. Mason, _Ambitwistor strings at null infinity and (subleading) soft limits_ , _Class.Quant.Grav._ 32 (2015) 055003, [1406.1462].
* [193] A. Strominger, _w(1+infinity) and the Celestial Sphere_ , 2105.14346.
* [194] T. Adamo, L. Mason and A. Sharma, _Celestial $w_{1+\infty}$ Symmetries from Twistor Space_, _SIGMA_ 18 (2022) 016, [2110.06066].
* [195] S. V. Ketov, _Conformal Field Theory_. World Scientific, Singapore, 1995.
* [196] P. Di Francesco, P. Mathieu and D. Senechal, _Conformal Field Theory_. Graduate Texts in Contemporary Physics. Springer-Verlag, New York, 1997, 10.1007/978-1-4612-2256-9.
* [197] J. Polchinski, _String theory. Vol. 2: Superstring theory and beyond_ , .
|
pnasresearcharticle Panahi Human activities are having increasingly negative
impacts on natural systems and it is of interest to understand how the “pace”
of parameter change may lead to catastrophic consequences. This work studies
the phenomenon of rate-induced tipping (R-tipping) in high-dimensional
ecological networks, where the rate of parameter change can cause the system
to undergo a tipping point from healthy survival to extinction. A quantitative
scaling law between the probability of R-tipping and the rate was uncovered,
with a striking and devastating consequence: in order to reduce the
probability, parameter changes must be slowed down to such an extent that the
rate practically reaches zero. This may pose an extremely significant
challenge in our efforts to protect and preserve the natural environment. YCL
and AH conceived the idea. SP performed computations. All analyzed data. YCL
and SP wrote the paper. YCL and AH edited the paper. The authors declare no
competing interests. 1To whom correspondence should be addressed. E-mail:
<EMAIL_ADDRESS>
# Rate-induced tipping in complex high-dimensional ecological networks
Shirin Panahi School of Electrical, Computer and Energy Engineering, Arizona
State University, Tempe, Arizona 85287, USA Younghae Do Department of
Mathematics, Nonlinear Dynamics $\&$ Mathematical Application Center,
Kyungpook National University, Daegu 41566, Republic of Korea Alan Hastings
Department of Environmental Science and Policy, University of California, One
Shields Avenue, Davis, CA 95616, USA and Santa Fe Institute, 1399 Hyde Park
Road, Santa Fe, NM 87501, USA Ying-Cheng Lai School of Electrical, Computer
and Energy Engineering, Arizona State University, Tempe, Arizona 85287, USA
Department of Physics, Arizona State University, Tempe, Arizona 85287, USA
###### Abstract
In an ecosystem, environmental changes as a result of natural and human
processes can cause some key parameters of the system to change with time.
Depending on how fast such a parameter changes, a tipping point can occur.
Existing works on rate-induced tipping, or R-tipping, offered a theoretical
way to study this phenomenon but from a local dynamical point of view,
revealing, e.g., the existence of a critical rate for some specific initial
condition above which a tipping point will occur. As ecosystems are subject to
constant disturbances and can drift away from their equilibrium point, it is
necessary to study R-tipping from a global perspective in terms of the initial
conditions in the entire relevant phase space region. In particular, we
introduce the notion of the probability of R-tipping defined for initial
conditions taken from the whole relevant phase space. Using a number of real-
world, complex mutualistic networks as a paradigm, we discover a scaling law
between this probability and the rate of parameter change and provide a
geometric theory to explain the law. The real-world implication is that even a
slow parameter change can lead to a system collapse with catastrophic
consequences. In fact, to mitigate the environmental changes by merely slowing
down the parameter drift may not always be effective: only when the rate of
parameter change is reduced to practically zero would the tipping be avoided.
Our global dynamics approach offers a more complete and physically meaningful
way to understand the important phenomenon of R-tipping.
###### keywords:
Rate-induced tipping $|$ nonlinear dynamics $|$ scaling law
###### doi:
www.pnas.org/cgi/doi/10.1073/pnas.XXXXXXXXXX
This manuscript was compiled on
In complex dynamical systems, the phenomenon of tipping point, characterized
by an abrupt transition from one type of behavior (typically normal, healthy,
or survival) to another type (e.g., catastrophic), has received growing
attention (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25). A tipping point is often of significant concern
because it is a point of “no return” in the parameter space, manifested by the
collapse of the system as a parameter passes through a critical value. In
ecological systems, sudden extinction of species on a large scale can be the
result of a tipping point (1, 2, 3, 4). Tipping points can arise in diverse
contexts such as the outbreak of epidemics (26), global climate changes (27),
and the sudden transition from normal to depressed mood in bipolar patients
(28). In nonlinear dynamics, a common type of bifurcation responsible for a
tipping point is saddle-node bifurcations (forward or backward). Consider the
situation where, in the parameter regime of interest, there are two coexisting
stable steady states: a “high” state corresponding to normal or “survival”
functioning and a “low” or “extinction” state, each with its own basin of
attractor. Suppose external factors such as climate change cause a bifurcation
parameter of the system to increase. A tipping point is a backward saddle-node
bifurcation, after which the “survival” fixed point disappears, leaving the
“extinction” state as the only destination of the system, where the original
basin of the “survival” state is absorbed into the basin of the “extinction”
state.
In real-world dynamical systems, parameters are not stationary but constantly
drift in time. A known example is the slow increase in the average global
temperature with time due to human activities. In ecological systems, some key
parameters such as the carrying capacity or the species decay rate can change
with time, and there is a global tendency for such parameter changes to “speed
up.” In fact, the rate of environmental change is an important driver across
different scales in ecology (29). The behavior of the parameter variations
with time introduces another “meta” parameter into the dynamical process: the
rate of change of the parameters. About ten years ago, it was found that the
rate can induce a tipping point - the phenomenon of rate-induced tipping or
R-tipping (9), which is relevant to fields such as climate science (30, 31),
neuroscience (32, 33), vibration engineering (34), and even competitive
economy (35). Existing studies of R-tipping were for low-dimensional dynamical
systems and the analysis approaches were “local” in the sense that they
focused on the behaviors of some specific initial condition and trajectory,
addressing issues such as the critical rate for tipping (36, 37). In
particular, with respect to a specific initial condition, R-tipping in these
studies was defined as an abrupt change in the behavior of the system (or a
critical transition) taking place at a specific rate of change of a
bifurcation parameter (9).
The state that an ecological system is in depends on a combination of
deterministic dynamics, small scale stochastic influences (e.g., demographic
stochasticity (38)), and large stochastic disturbances such as drought or
other significant climatic event (39). So when considering future dynamics of
ecological systems it makes sense to consider systems that may be far from
equilibrium, but still within the basin of attraction of an equilibrium,
rather than starting at the equilibrium. High-dimensional ecological systems
are particularly likely to be found far from equilibrium. In fact, it has been
suggested that it can be common for ecosystems to be in some transient state
(40, 41).
In this paper, we focus on high-dimensional, empirical ecological networks and
investigate R-tipping with two key time-varying parameters by presenting a
computationally feasible, “global” approach to exploring the entire relevant
phase space region with analytic insights. We focus on a representative class
of such systems: mutualistic networks (42, 43, 44, 14, 45, 46, 47, 17, 21, 19,
48) that are fundamental to ecological systems, which are formed through
mutually beneficial interactions between two groups of species. In a
mutualistic network, a species in one group derives some form of benefit from
the relationship with some species in the other group. Examples include the
corals and the single-celled zooxanthellae that form the large-scale coral
reefs, and the various networks of pollinators and plants in different
geographical regions of the world. These networks influence biodiversity,
ecosystem stability, nutrient cycling, community resilience, and evolutionary
dynamics (49), and they are a key aspect of ecosystem functioning with
implications for conservation and ecosystem management. Understanding the
ecological significance of mutualistic networks is crucial for unraveling the
complexities of ecological communities and for implementing effective
strategies to safeguard biodiversity and ecosystem health (50).
Mathematically, because of the typically large number of species involved in
the mutualistic interactions, the underlying networked systems are high-
dimensional nonlinear dynamical systems.
We first ask if R-tipping can arise in such high-dimensional systems through
simulating a number of empirical pollinator-plant mutualistic networks (see
Tab. 1) and obtain an affirmative answer. Our computations reveal that the
critical rate above which a tipping point occurs depends strongly on the
initial condition. Rather than studying the critical rate for any specific
initial condition, we go beyond the existing local analysis approaches by
investigating the probability of R-tipping for a large ensemble of initial
conditions taken from the whole relevant phase space and asking how this
probability, denoted as $\Phi(r)$, depends on the rate $r$ of parameter
change. We discover a scaling law between $\Phi(r)$ and $r$: as the rate
increases, the probability first increases rapidly, then slowly, and finally
saturates. Using a universal two-dimensional (2D) effective model that was
validated to be particularly suitable for predicting and analyzing tipping
points in high-dimensional mutualistic networks (17), we analytically derive
the scaling law. Specifically, let $\kappa(t)$ be the parameter that changes
with time linearly at the rate $r$ in a finite range, denoted as $[\kappa_{\rm
min},\kappa_{\rm max}]$. The scaling law is given by
$\displaystyle\Phi(r)\sim\exp{[-C(\kappa_{\rm max}-\kappa_{\rm min})/r]},$ (1)
where $C>0$ is a constant. Our theoretical analysis indicates that, for 2D
systems, $C$ is nothing but the maximum possible unstable eigenvalue of the
mediating unstable fixed point on the boundary separating the basins of the
survival and extinction attractors when the parameter $\kappa$ varies in the
range $[\kappa_{\rm min},\kappa_{\rm max}]$. However, for high-dimensional
systems, such correspondence does not hold, but $C$ can be determined through
a numerical fitting.
The scaling law (1) has the following features. First, the probability
$\Phi(r)$ is an increasing function of $r$ for $r>0$ [$\Phi^{\prime}(r)>0$].
Second, $\Phi^{\prime}(r)$ is a decreasing function of $r$, i.e., the increase
of $\Phi(r)$ with $r$ slows down with $r$ and the rate of increase becomes
zero for $r\rightarrow\infty$. Third, the rate of increase in $\Phi(r)$ with
$r$ is the maximum for $r\gtrsim 0$, and $\Phi(r)$ becomes approximately
constant for $r>r^{*}\sim\sqrt{(\kappa_{\rm max}-\kappa_{\rm min})}$. This
third feature has a striking implication because the probability of R-tipping
will grow dramatically as soon as the rate of parameter change increases from
zero. The real-world implication is alarming because it means that even a slow
parameter change can lead to a system collapse with catastrophic consequences.
To control or mitigate the environmental changes by merely slowing down the
parameter drift may not always be effective: only when the rate of parameter
change is reduced to practically zero would the tipping be avoided!
Alternatively, the scaling law (1) can be expressed as the following explicit
formula:
$\displaystyle\Phi(r)=B\exp{[-C(\kappa_{\rm max}-\kappa_{\rm min})/r]}$ (2)
with an additional positive constant $B$. For 2D systems, $B$ is the
difference between the basin areas of the extinction state for
$\kappa=\kappa_{\rm max}$ and $\kappa=\kappa_{\rm min}$. For high-dimensional
systems, $B$ can be determined numerically.
While a comprehensive understanding of the entire parameter space as well as
the rates and directions of change associated with these parameters are worth
investigating, the picture that motivated our work was the potential impacts
of ongoing climate change on ecological systems. Under climate change, various
parameters can vary in different directions, and some changes might offset the
effects of others. Our focus is on the scenarios where the changes in
parameters align with the observed detrimental environmental impacts. For
example, consistent with environmental deterioration, in a mutualistic network
the species decay rate can increase, and/or the mutualistic interaction
strength can decrease. A simultaneous increase in the species decay rate and
mutualistic strength does not seem physically reasonable in this context. We
thus study scenarios of multiple parameter variations that align with the
realistic climate change impacts by carrying out computations with a
systematic analysis of the tipping-point transitions in the two-dimensional
parameter plane of the species decay rate and the mutualistic interaction
strength.
## Results
We consider the empirical mutualistic pollinator–plant networks from the Web
of Life database (www.Web-of-Life.es). The needs to search for the parameter
regions exhibiting R-tipping and to simulate a large number of initial
conditions in the phase space as required by our global analysis approach, as
well as the high dimensionality of the empirical mutualistic networks demand
extremely intensive computation (51). To make the computation feasible, we
select ten networks to represent a diverse range of mutualistic interactions
from different regions of the world and highlight the generality of our
approach to R-tipping and the scaling law. The basic information about these
networks such as the name of the networks, the countries where the empirical
data were collected, and the number of species in each network, is listed in
Tab. 1. The dynamics of a mutualistic network of $N_{A}$ pollinator and
$N_{P}$ plant species, taking into account the generic plant-pollinator
interactions (42), can be described by a total of $N=N_{A}+N_{P}$ nonlinear
differential equations of the Holling type (52) in terms of the species
abundances as
$\displaystyle\dot{P}_{i}=$ $\displaystyle
P_{i}\Big{(}\alpha_{i}^{P}-\sum_{l=1}^{N_{p}}\beta_{il}^{P}P_{l}+\frac{\sum_{j=1}^{N_{A}}\gamma_{ij}^{P}A_{j}}{1+h\sum_{j=1}^{N_{A}}\gamma_{ij}^{P}A_{j}}\Big{)},$
(3a) $\displaystyle\dot{A}_{j}=$ $\displaystyle
A_{j}\Big{(}\alpha_{j}^{A}-\kappa_{j}-\sum_{l=1}^{N_{A}}\beta_{jl}^{A}A_{l}+\frac{\sum_{i=1}^{N_{P}}\gamma_{ji}^{A}P_{i}}{1+h\sum_{i=1}^{N_{P}}\gamma_{ji}^{A}P_{i}}\Big{)},$
(3b)
where $P_{i}$ and $A_{j}$ are the abundances of the $i^{th}$ and $j^{th}$
plant and pollinator species, respectively, $i=1,\ldots,N_{P}$,
$j=1,\ldots,N_{A}$, $\kappa$ is the pollinator decay rate, $\alpha^{P(A)}$ is
the intrinsic growth rate in the absence of intraspecific competition and any
mutualistic effect, $h$ is the half-saturation constant. Intraspecific and
interspecific competition of the plants (pollinators) is characterized by the
parameters $\beta_{ii}^{P}$ ($\beta_{jj}^{A}$) and $\beta_{il}^{P}$
($\beta_{jl}^{A}$), respectively. The mutualistic interactions as
characterized by the parameter $\gamma_{ij}^{P}$ can be written as
$\gamma_{ij}^{P}=\xi_{ij}\gamma_{0}/K_{i}^{\tau}$, where $\xi_{ij}=1$ if there
is a mutualistic interaction between the $i^{th}$ plant and $j^{th}$
pollinator (zero otherwise), $\gamma_{0}$ is a general interaction parameter,
$K_{i}$ is the degree of the $i^{th}$ plant species in the network, and $\tau$
determines the strength of the trade-off between the interaction strength and
the number of interactions. (The expression for $\gamma_{ji}^{A}$ is similar.)
Table 1: Empirical mutualistic networks studied in this work and their corresponding range of parameter change and fitting constants in Eq. (1) Network | Country | $N_{P}$ | $N_{A}$ | $\kappa$-interval | $B$ | $C$
---|---|---|---|---|---|---
M_PL_008 | Canary Islands | $11$ | $38$ | ${[0.90,0.93]}$ | $0.36$ | $0.19$
M_PL_013 | South Africa | $9$ | $56$ | ${[0.70,0.99]}$ | $0.07$ | $0.11$
M_PL_022 | Argentina | $21$ | $45$ | ${[0.75,0.93]}$ | $0.88$ | $0.12$
M_PL_023 | Argentina | $23$ | $72$ | $[0.87,0.96]$ | $0.59$ | $0.09$
M_PL_027 | New Zealand | $18$ | $60$ | $[0.90,0.95]$ | $0.49$ | $0.09$
M_PL_032 | USA | $7$ | $33$ | $[0.82,0.99]$ | $0.55$ | $0.19$
M_PL_036 | Açores | $10$ | $12$ | $[0.74,0.88]$ | $0.39$ | $0.12$
M_PL_037 | Denmark | $10$ | $40$ | $[0.87,0.93]$ | $0.23$ | $0.15$
M_PL_038 | Denmark | $8$ | $42$ | $[0.85,0.95]$ | $0.30$ | $0.15$
M_PL_045 | Greenland | $17$ | $26$ | $[0.96,0.98]$ | $0.09$ | $0.05$
The computational setting of our study is as follows. In the network system
described by Eqs. 3a and 3b, the number of the equations determines the phase-
space dimension of the underlying nonlinear dynamical system. There are a
large number of basic parameters in the model, such as $\gamma$ that
quantifies the strength of the mutualistic interactions and $\kappa$
characterizing the species decay rate. In the context of R-tipping, while all
the parameters should be time-varying in principle, to make our study
computationally feasible, we assume that the two key parameters ($\kappa$ and
$\gamma$) are time-dependent while keeping the other parameters fixed. Since
the defining characteristic of a system exhibiting a tipping point is the
coexistence of two stable steady states: survival and extinction, we focus on
the range of parameter variations in which the network system under study
exhibits the two stable equilibria. When presenting our results (Fig. 1 below
and Figs. S4-S14 in Supplementary Information), in each case the parameter
variation is along a specific direction in the two-dimensional parameter
plane: $\kappa$ and $\gamma$ with the goal to understand how changes in these
two parameters impact the R-tipping probability.
To introduce the rate change of a parameter, we consider the scenario where
negative environmental impacts cause the species decay rate to increase
linearly with time at the rate $r$ from a minimal value $\kappa_{\rm min}$ to
a maximal value $\kappa_{\rm max}$:
$\displaystyle\dot{\kappa}_{j}=\begin{cases}r\quad{\rm if}\quad\kappa_{\rm
min}<\kappa_{j}<\kappa_{\rm max}\\\ 0\quad\quad{\rm otherwise.}\end{cases}$
(4)
(For mutualistic networks, another relevant parameter that is vulnerable to
environmental change is the mutualistic interaction strength. The pertinent
results are presented in Supplementary Information.) To calculate the
probability of R-tipping, $\Phi(r)$, we set $r=0$ so that $\kappa=\kappa_{\rm
min}$, solve Eqs. (3a) and (3b) numerically for a large number of random
initial conditions chosen uniformly from the whole high-dimensional phase
space, and determine $10^{5}$ initial conditions that approach the high stable
steady state in which no species becomes extinct. We then increase the rate
$r$ from zero. For each fixed value of $r$, we calculate, for each of the
selected $10^{5}$ initial conditions, whether or not the final state is the
high stable state. If yes, then there is no R-tipping for the particular
initial condition. However, if the final state becomes the extinction state,
R-tipping has occurred for this value of $r$. The probability $\Phi(r)$ can be
approximated by the fraction of the number of initial conditions leading to
R-tipping out of the $10^{5}$ initial conditions.
Figure 1: Probability $\Phi(r)$ of R-tipping versus the time rate $r$ of
parameter change for ten real-world mutualistic networks, together with the
corresponding bipartite structure for each network. The dots are the
probability calculated by simulating Eqs. (3a), (3b), and (4), with an
ensemble of random initial conditions. The solid curves are the analytic
results from Eq. (1), with the two fitting parameters $B$ and $C$ given in
Tab. 1. Other parameter values are $\alpha=0.3$, $\beta=1$, $h=0.4$,
$\gamma^{P}=1.93$, and $\gamma^{A}=1.77$.
Figure 1 shows $\Phi(r)$ versus $r$ for the ten empirical mutualistic networks
specified in Tab. 1, together with the bipartite network structure, which are
distinguished by different colors. For each network, the interval
$[\kappa_{\rm min},\kappa_{\rm max}]$ of the bifurcation parameter (listed in
the fifth column of Tab. 1) is chosen such that the dynamical network for
static parameter values exhibits two stable steady states and a tipping point
in this interval, as determined by a computational bifurcation analysis of the
species abundances versus $\kappa$. Despite the differences in the topology
and the specific parameter values among the empirical networks, it is
remarkable that the probability of R-tipping exhibits a characteristic
behavior common among all the networks: as the time-varying rate of the
bifurcation parameter increases from zero, the probability increases rapidly
and then saturates at an approximately finite constant value, as quantified by
the analytic scaling law (1). The final or asymptotic value of the R-tipping
probability attained in the regime of large time rate of change of the
bifurcation parameter depends on the specifics of the underlying mutualistic
network in terms of its topological structure and basic system parameters. For
example, the total number of species, or the phase-space dimension of the
underlying dynamical system, and the relative numbers of the pollinator and
plant species vary dramatically across the ten networks, as indicated by the
ten surrounding network-structure diagrams in Fig. 1. The structural and
parametric disparities among the networks lead to different relative basin
volumes of the survival and extinction stable steady states, giving rise to
distinct asymptotic probabilities of R-tipping.
The ecological interpretation of the observed different asymptotic values of
the R-tipping probability is, as follows. Previous studies revealed the
complex interplay between the structural properties of the network and
environmental factors in shaping extinction probabilities (53, 54, 55, 56, 57,
58, 59, 60). For example, it was discovered (53) that the plant species tend
to play a more significant role in the health of the network system as
compared to pollinators (53) in the sense that plant extinction due to climate
change is more likely to trigger pollinator coextinction than the other way
around. In another example, distinct topological features were found to be
associated with the networks with a higher probability of extinction (54, 55).
It was also found that robust plant-pollinator mutualistic networks tend to
exhibit a combination of compartmentalized and nested patterns (55). More
generally, mutualistic networks in the real world are quite diverse in their
structure and parameters, each possessing one or two or all the features
including lower interaction density, heightened specialization, fewer
pollinator visitors per plant species, lower nestedness, and lower modularity,
etc. (54, 55, 56, 57, 58, 59, 60). It is thus reasonable to hypothesize that
the asymptotic value of the R-tipping probability can be attributed to the
sensitivity of the underlying network to environmental changes. For example, a
higher (lower) saturation value of the extinction probability can be
associated with the networks with a large (small) number of plants and lower
(higher) nestedness, network $M\\_PL\\_22$ (network $M\\_PL\\_13$). In other
examples, network $M\\_PL\\_32$ can be categorized as “vulnerable” due to its
low modularity, despite having a small number of plants, and network
$M\\_PL\\_45$ is resilient against extinction due to its high nestedness and
high modularity.
The remarkable phenomenon is that, in spite of these differences, the rapid
initial increase in the R-tipping probability is shared by all ten networks!
That is when a parameter begins to change with time from zero, even slowly,
the probability of R-tipping increases dramatically. The practical implication
is that ecosystems with time-dependent parameter drift are highly susceptible
to R-tipping. Parameter drifting, even at a slow pace, will be detrimental.
This poses a daunting challenge to preserving ecosystems against negative
environmental changes. In particular, according to conventional wisdom,
ecosystems can be effectively protected by slowing down the environmental
changes, but our results suggest that catastrophic tipping can occur with a
finite probability unless the rate of the environmental changes is reduced to
a near zero value.
To further explore the effects of parameter changes, we study a scenario in
which negative environmental impacts cause the species decay rate ($\kappa$)
and the mutualistic interaction strength ($\gamma_{0}$) to linearly increase
and decrease, respectively. By considering the rate of changing of $\kappa$
($\gamma_{0}$) as $r_{\kappa}$ ($r_{\gamma_{0}}$), we set
$r_{\kappa}=\mathcal{C}r_{\gamma_{0}}$, which allows us to investigate three
different scenarios by varying the parameter $\mathcal{C}$: $\mathcal{C}<1$,
$\mathcal{C}=1$, and $\mathcal{C}>1$. Figure 2 shows a systematic analysis of
the tipping-point transitions in the 2D parameter plane along with the
probability of R-tipping for the three different scenarios for the network
${\rm M}\\_{\rm PL}\\_032$. (The pertinent results for the other nine networks
are presented in Sec. 4 in Supporting Information.)
Figure 2: Scaling law of probability of R-tipping in two-dimensional
parameter space. (a) Two-dimensional parameter plane of the species decay rate
($\kappa$) and the mutualistic interaction strength ($\gamma_{0}$) for the
network ${\rm M}\\_{\rm PL}\\_032$. The solid lines correspond to the three
different scenarios of the parameter $\mathcal{C}$ in calculating the
probability of the R-tipping point from $10^{7}$ initial conditions for (b)
$\mathcal{C}<1$, (c) $\mathcal{C}=1$, and (d) $\mathcal{C}>1$. Other
parameters are $t=0.5$, $\beta=1$, $\alpha=0.3$, $\mu=0$, and $h=0.4$.
To understand the behavior of the R-tipping probability in Fig. 1, we resort
to the approach of dimension reduction (61). In particular, for tipping-point
dynamics, a high-dimensional mutualistic network can be approximated by an
effective model of two dynamical variables: the respective mean abundances of
all pollinator and plant species. The effective 2D model can be written as
(17)
$\displaystyle\dot{P}$ $\displaystyle=P\Big{(}\alpha-\beta
P+\frac{\gamma^{P}A}{1+h\gamma^{P}A}\Big{)},$ (5a) $\displaystyle\dot{A}$
$\displaystyle=A\Big{(}\alpha-\kappa-\beta
A+\frac{\gamma^{A}P}{1+h\gamma^{A}P}\Big{)},$ (5b)
where $P$ and $A$ are the average abundances of all the plants and
pollinators, respectively, $\alpha$ is the effective growth rate and parameter
$\beta$ characterizes the combined effects of intraspecific and interspecific
competition. The parameters $\gamma^{P}$ and $\gamma^{A}$ are the effective
mutualistic interaction strengths that can be determined by the method of
eigenvector weighting (17) (Supplementary Note 1). Equations (5a) and (5b)
possess five possible equilibria: $\mathbf{f}_{1}\equiv(0,0)^{T}$,
$\mathbf{f}_{2}\equiv(\alpha/\beta,0)^{T}$,
$\mathbf{f}_{3}\equiv(0,(\alpha-\kappa)/\beta)^{T}$, and
$\mathbf{f}_{4,5}\equiv(g_{1},g_{2})^{T}$, where $g_{1}$ and $g_{2}$ are two
possible equilibria that depend on the values of the parameters of the model
and can be calculated by setting zero the factor in the parentheses of Eqs.
(5a) and (5b).
Figure 3: Dynamics of the 2D reduced model Eqs. (5a) and (5b). (a,b) A tipping
point occurs as the species decay rate $\kappa$ increases towards one, for the
mean plant and pollinator abundances, respectively. The green and yellow
curves correspond to the stable and unstable steady states, respectively,
where the yellow curve with a red segment is the unstable fixed point
$\mathbf{f}_{5}$. The region of bistability is indicated by the blue dot-
dashed ellipses. (c,d) Examples of basins of attraction of the 2D model whose
parameters are determined as the corresponding averages from the empirical
network ${\rm M}\\_{\rm PL}\\_036$ in Tab. 1 for $\kappa_{min}=0.74$ and
$\kappa_{max}=0.88$, respectively. Other parameters are $\alpha=0.3$,
$\beta=1$, $h=0.4$, $\gamma^{P}=1.93$, and $\gamma^{A}=1.77$. The pink and
gray regions correspond to the basins of the stable steady states $f_{2}$
(extinction) and $f_{4}$ (survival), respectively.
The first equilibrium $\mathbf{f}_{1}$, an extinction state, is at the origin
$(P^{*},\,A^{*})=(0,0)$ and is unstable. The second equilibrium
$\mathbf{f}_{2}$ is located at the $(P^{*},\,A^{*})=(\alpha/\beta,0)$ and it
can be stable or unstable depending on the parameters $\kappa$. The locations
of the remaining equilibria depend on the value of $\kappa$. In particular,
the third equilibrium $\mathbf{f}_{3}$ can be unstable or nonexistent and the
fourth equilibrium $\mathbf{f}_{4}$ is stable and coexists with stable
equilibrium $\mathbf{f}_{2}$ in some interval of $\kappa$. The fifth
equilibrium $\mathbf{f}_{5}$ is an unstable saddle fixed point in some
relevant interval of $\kappa$. Figures 3(a) and 3(b) exemplify the behaviors
of the equilibria as $\kappa$ increases from zero to one for $\alpha=0.3$,
$\beta=1$, $h=0.4$, $\gamma^{P}=1.93$, and $\gamma^{A}=1.77$, for the average
plant and pollinator abundances, respectively. In each panel, the upper green
curve is the survival fixed point $\mathbf{f}_{4}$, while the lower horizontal
green line corresponds to the extinction state $\mathbf{f}_{2}$. The blue dot-
dashed ellipse indicates the interval of $\kappa$ in which two stable
equilibria coexist (bistability), whose right edge marks a tipping point.
Figure 4: Basin boundary between the extinction and survival fixed-point
attractors for different rate of parameter change. Shown are a series of basin
boundaries for $r\in[0\,1]$. As $r$ increases from zero, the boundary moves in
the direction of large species abundances. The basin boundaries are calculated
from the 2D model of the empirical network ${\rm M}\\_{\rm PL}\\_036$ in Tab.
1. Other parameters are the same as those in Figs. 3(c) and 3(d). Figure 5:
Analysis of the dynamical mechanism responsible for R-tipping. The dynamics of
the unstable fixed point $\mathbf{f}_{5}$ together with those of a rectangle
region of the initial conditions in its neighborhood are illustrated for two
values of the rate of parameter change: (a) $r=r_{1}$ and (b) $r=r_{2}>r_{1}$.
At $t=0$ ($\kappa=\kappa_{\rm min}$, a segment of the basin boundary in the
vicinity of $\mathbf{f}_{5}$ is illustrated as the solid green line. As the
rate of parameter change begins to increase, the basin boundary moves upward
and eventually accumulates at that for $\kappa=\kappa_{\rm max}$ \- the solid
orange lines. For a small rate, the time required for the green boundary to
reach the orange boundary is longer. All the initial conditions in the light
green region above the green boundary belong to the basin of the survival
attractor for $\kappa=\kappa_{\rm min}$, whose trajectories are determined by
the local exponentially contracting and expanding dynamics of
$\mathbf{f}_{5}$. The fate of these initial conditions is the result of a
“race” to go above the orange basin boundary within the “allowed” time
duration [$T_{1}$ in (a) and $T_{2}<T_{1}$ in (b)]: those managing to go above
will approach the survival attractor, but those that stay below will lead to
extinction. Since $T_{1}>T_{2}$, the fraction of initial conditions that can
go above in (a) is larger than that in (b), leading to a higher probability of
R-tipping for case (b). The precise fraction of those initial conditions is
indicated by the red rectangular region in (b), whose initial height stretches
to the distance between the green and orange boundaries in the permissible
time. This gives the probability of R-tipping as quantified by the scaling law
(1). See text for more details.
What will happen to the dynamics when the parameter $\kappa$ becomes time
dependent? Without loss of generality, we focus on the interval of $\kappa$ as
exemplified by the horizontal range of the dot-dashed ellipse in Figs. 3(a)
and 3(b), defined as $[\kappa_{\rm min},\kappa_{\rm max}]$, in which there is
bistability in the original high-dimensional network and in the 2D reduced
model as well. Note that, for different high-dimensional empirical networks,
the values of $\kappa_{\rm min}$ and $\kappa_{\rm max}$ in the 2D effective
models are different, as illustrated in the fifth column of Tab. 1. Because of
the coexistence of two stable fixed points, for every parameter value in the
range $[\kappa_{\rm min},\kappa_{\rm max}]$, there are two basins of
attraction, as illustrated in Figs. 3(c) and 3(d) for the 2D effective model
of an empirical network ${\rm M}\\_{\rm PL}\\_036$ for $\kappa=0.74$ and
$\kappa=0.88$, respectively, where the pink and gray regions correspond to the
basins of the extinction and survival attractors, respectively. The basin
boundary is the stable manifold of the unstable fixed point $\mathbf{f}_{5}$.
It can be seen that, as $\kappa$ increases, the basin of the extinction
attractor increases, accompanied by a simultaneous decrease in the basin area
of the survival attractor. This can be understood by comparing the positions
of the equilibria in Fig. 3 where, as $\kappa$ increases, the position of
survival fixed point $\mathbf{f}_{4}$ moves towards lower plant and pollinator
abundances, but the unstable fixed point $\mathbf{f}_{5}$ moves in the
opposite direction: the direction of larger species abundances.
The boundary separating the basins of the extinction and survival fixed-point
attractors is the stable manifold of $\mathbf{f}_{5}$. As $r$ increases from
zero, $\mathbf{f}_{5}$ moves in the direction of larger plant and pollinator
abundances, so must the basin boundary, as exemplified in Fig. 4 (the various
dashed curves) for $r\in[0,1]$. Note that, since the decay parameter $\kappa$
increases from $\kappa_{\rm min}$ to $\kappa_{\rm max}$ at the linear rate
$r$, we have $\kappa(r=0)=\kappa_{\rm min}$ and $\kappa(r=\infty)=\kappa_{\rm
max}$. As $r$ increases, the basin boundaries accumulate at the one for
$\kappa_{\rm max}$.
## Theory
Figure 4 provides a physical base for deriving the scaling law (1). Consider
the small neighborhood of the unstable fixed point $\mathbf{f}_{5}$, where the
basin boundary is approximately straight, as shown schematically in Fig. 5.
Consider two cases: one with rate $r_{1}$ of parameter increase and another
with rate $r_{2}$, where $r_{2}>r_{1}$, as shown in Figs. 5(a) and 5(b),
respectively. For any given rate $r$, at the beginning and end of the
parameter variation, we have $\kappa=\kappa_{\rm min}$ and $\kappa=\kappa_{\rm
max}$, respectively, where the time it takes to complete this process is
$T=(\kappa_{\rm max}-\kappa_{\rm min})/r$. Since $r_{2}>r_{1}$, we have
$T_{2}<T_{1}$. For both Figs. 5(a) and 5(b), the unstable fixed point
$\mathbf{f}_{5}$ is marked by the filled green circles at $t=0$ and filled
orange circles at the end of the parameter variation, and the blue dashed line
with an arrow indicates the direction of change in the location of
$\mathbf{f}_{5}$ in the phase space as the parameter varies with time.
Likewise, the solid green (orange) line segments through $\mathbf{f}_{5}$
denote the boundary separating the extinction basin from the survival basin of
attraction at $t=0$ ($t=T_{1}$ or $T_{2}$). That is, before the parameter
variation is turned on for $\kappa=\kappa_{\rm min}$, the initial conditions
below (above) the solid green lines belong to the basin of the extinction
(survival) attractor. After the process of parameter variation ends so that
$\kappa=\kappa_{\rm max}$, the initial conditions below (above) the solid
orange lines belong to the basin of the extinction (survival) attractor.
During the process of parameter variation, $\mathbf{f}_{5}$ moves from the
position of the green circle to that of the orange circle, and its stable
manifold (the basin boundary) moves accordingly. Now consider the initial
conditions in the light shaded green area, which belong to the basin of the
survival attractor for $\kappa=\kappa_{\rm min}$. Without parameter variation,
as time goes, this green rectangular area will be stretched along the unstable
direction of $\mathbf{f}_{5}$ exponentially according to its unstable
eigenvalue $\lambda$ and compressed exponentially in the stable direction,
evolving into an orange rectangle that is long in the unstable direction.
Since $T_{1}>T_{2}$, the orange rectangle for $r=r_{1}$ is longer and thinner
than that for $r=r_{2}$.
The dynamical mechanism responsible for R-tipping can now be understood based
on the schematic illustration in Figs. 5(a) and 5(b). In particular, because
of the movement of $\mathbf{f}_{5}$ and the basin boundary as the parameter
variation is turned on, the dark shaded orange part of the long rectangle now
belongs to the basin of the extinction attractor. The initial conditions in
the original green rectangle which evolve into this dark shaded orange region
are nothing but the initial conditions that switch their destinations from the
survival to the extinction attractor as the result of the time variation of
the parameter. That is, these initial conditions will experience R-tipping, as
indicated by the red rectangle inside the green area in Fig. 5(b). For any
given rate $r$, the fraction of such initial conditions determines the
R-tipping probability. Let $d(0)$ denote the fraction of R-tipping initial
conditions and let $D$ be the distance between the basin boundaries at the
beginning and end of parameter variation along the unstable direction of
$\mathbf{f}_{5}$. We have $d(T)=D=d(0)\exp{(\lambda T)}$. Our argument for
$\Phi(r)\sim d(0)$ and the use of $T=(\kappa_{\rm max}-\kappa_{\rm min})/r$
lead to the scaling law (1).
## Discussion
To summarize, nonlinear dynamical systems in nature such as ecosystems and
climate systems on different scales are experiencing parameter changes due to
increasing human activities, and it is of interest to understand how the
“pace” or rate of parameter change might lead to catastrophic consequences. To
this end, we studied high-dimensional mutualistic networks, as motivated by
the following general considerations. In ecosystems, mutualistic interactions,
broadly defined as a close, interdependent, mutually beneficial relationship
between two species, are one of the most fundamental interspecific
relationships. Mutualistic networks contribute to biodiversity and ecosystem
stability. As species within these networks rely on each other for essential
services, such as pollination, seed dispersal, or nutrient exchange, they
promote species coexistence and reduce competitive exclusion. This coexistence
enhances the overall diversity of the ecosystem, making it more resilient to
disturbances and less susceptible to the dominance of a few species. They can
drive co-evolutionary processes between interacting species. As species
interact over time, they may evolve in response to each other’s adaptations,
leading to reciprocal changes that strengthen the mutualistic relationship.
Disruptions to these networks, such as the decline of pollinator populations,
can have cascading effects on ecosystem functions and the survival of
dependent species. By studying mutualistic interactions, conservationists can
design more effective strategies to protect and restore these vital
relationships and the ecosystems they support.
This paper focuses on the phenomenon of rate-induced tipping or R-tipping,
where the rate of parameter change can cause the system to experience a
tipping point from normal functioning to collapse. The main accomplishments
are three. First, we went beyond the existing local approaches to R-tipping by
taking a global approach of dynamical analysis based on consideration of
basins of attraction of coexisting attractors. This allows us to introduce the
probability of R-tipping with respect to initial conditions taken from the
whole phase space. Second, most previous works on R-tipping analyzed low-
dimensional toy models but our study focused on high-dimensional mutualistic
networks constructed from empirical data. Third, we developed a geometric
analysis and derived a scaling law governing the probability of R-tipping with
respect to the rate of parameter change. The scaling law contains two
parameters which, for two-dimensional systems, can be determined theoretically
from the dynamics. For high-dimensional systems, the two scaling parameters
can be determined through numerical fitting. For all ten empirical networks
studied, the scaling law agrees well with the results from direct numerical
simulations. To our knowledge, no such quantitative law characterizing
R-tipping has been uncovered previously.
The effective 2D systems in our study were previously derived (17), which
capture the bipartite and mutualistic nature of the ecological interactions in
the empirical, high-dimensional networks. It employs one collective variable
to account for the dynamical behavior of the pollinators and another for the
plants. The effective 2D models allow us to mathematically analyze the global
R-tipping dynamics, with results and the scaling law verified by direct
simulations of high-dimensional networks. The key reason that the dimension-
reduced 2D models can be effective lies, again, in the general setting of the
ecological systems under consideration: coexistence of a survival and an
extinction state. While our present work focused on mutualistic networks, we
expect the global approach to R-tipping and the scaling law to be generally
applicable to ecological systems in which large-scale extinction from a
survival state is possible due to even a small, nonzero rate of parameter
change.
It is worth noting that the setting under which the scaling law of the
probability of R-tipping with the rate of parameter change holds is the
coexistence of two stable steady states in the phase space, one associated
with survival and another corresponding to extinction. This setting is general
for studying tipping, system collapse and massive extinction in ecological
systems. Our theoretical analysis leading to the scaling law requires minimal
conditions: two coexisting basins of attraction separated by a basin boundary.
The scaling law is not the result of some specific parameterization of the
mutualistic systems but is a generic feature in systems with two coexisting
states. Insofar as the system can potentially undergo a transition from the
survival to the extinction state, we expect our R-tipping scaling law to hold.
In a broad context, coexisting stable steady states or attractors are
ubiquitous in nonlinear physical and biological systems.
The scaling law stipulates that, as the rate increases from zero, the
R-tipping probability increases rapidly first and then saturates. This has a
striking and potentially devastating consequence: in order to reduce the
probability of R-tipping, the parameter change must be slowed down to such an
extent that its rate of change is practically zero. This has serious
implications. For example, to avoid climate-change induced species extinction,
it would be necessary to ensure that no parameters change with time, and this
may pose an extremely significant challenge in our efforts to protect and
preserve the natural environment.
This work was supported by the Office of Naval Research under Grant No.
N00014-21-1-2323. Y. Do was supported by the National Research Foundation of
Korea (NRF) grant funded by the Korean government (MSIP) (Nos.
NRF-2022R1A5A1033624 and 2022R1A2C3011711).
## References
* Scheffer (2004) Marten Scheffer. _Ecology of Shallow Lakes_. Springer Science & Business Media, 2004.
* Scheffer et al. (2009) Marten Scheffer, Jordi Bascompte, William A Brock, Victor Brovkin, Stephen R Carpenter, Vasilis Dakos, Hermann Held, Egbert H Van Nes, Max Rietkerk, and George Sugihara. Early-warning signals for critical transitions. _Nature_ , 461(7260):53–59, 2009.
* Scheffer (2010) M. Scheffer. Complex systems: foreseeing tipping points. _Nature_ , 467:411–412, 2010.
* Wysham and Hastings (2010) D. B. Wysham and A. Hastings. Regime shifts in ecological systems can occur with no warning. _Ecol. Lett._ , 13:464–472, 2010.
* Drake and Griffen (2010) John M Drake and Blaine D Griffen. Early warning signals of extinction in deteriorating environments. _Nature_ , 467(7314):456–459, 2010.
* Chen et al. (2012) Luonan Chen, Rui Liu, Zhi-Ping Liu, Meiyi Li, and Kazuyuki Aihara. Detecting early-warning signals for sudden deterioration of complex diseases by dynamical network biomarkers. _Sci. Rep._ , 2:342, 2012.
* Boettiger and Hastings (2012) C. Boettiger and Alan Hastings. Quantifying limits to detection of early warning for critical transitions. _J. R. Soc. Interface_ , 9:2527–2539, 2012.
* Dai et al. (2012) Lei Dai, Daan Vorselen, Kirill S Korolev, and Jeff Gore. Generic indicators for loss of resilience before a tipping point leading to population collapse. _Science_ , 336(6085):1175–1177, 2012.
* Ashwin et al. (2012) Peter Ashwin, Sebastian Wieczorek, Renato Vitolo, and Peter Cox. Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system. _Philo Trans. R. Soc. A Math. Phys. Eng. Sci._ , 370(1962):1166–1184, 2012.
* Lenton et al. (2012) T. M. Lenton, V. N. Livina, V. Dakos, E. H. van Nes, and M. Scheffer. Early warning of climate tipping points from critical slowing down: comparing methods to improve robustness. _Phil. Trans. Roy. Soc. A_ , 370:1185–1204, 2012.
* Barnosky et al. (2012) A. D. Barnosky, E. A. Hadly, J. Bascompte, E. L. Berlowand J. H. Brown, M. Fortelius, W. M. Getz, J. Harte, A. Hastings, P. A. Marquet, N. D. Martinez, A. Mooers, P. Roopnarine, G. Vermeij, J. W. Williams, R. Gillespie, J. Kitzes, C. Marshall, N. Matzke, D. P. Mindell, E. Revilla, and A. B. Smith. Approaching a state shift in earth’s biosphere. _Nature_ , 486:52–58, 2012.
* Boettiger and Hastings (2013) C. Boettiger and A. Hastings. Tipping points: From patterns to predictions. _Nature_ , 493:157–158, 2013.
* Tylianakis and Coux (2014) Jason M Tylianakis and Camille Coux. Tipping points in ecological networks. _Trends. Plant. Sci._ , 19(5):281–283, 2014.
* Lever et al. (2014) J Jelle Lever, Egbert H Nes, Marten Scheffer, and Jordi Bascompte. The sudden collapse of pollinator communities. _Ecol. Lett._ , 17(3):350–359, 2014.
* Lontzek et al. (2015) T. S. Lontzek, Y.-Y. Cai, K. L. Judd, and T. M. Lenton. Stochastic integrated assessment of climate tipping points indicates the need for strict climate policy. _Nat. Clim. Change_ , 5:441–444, 2015.
* Gualdia et al. (2015) S. Gualdia, M. Tarziaa, F. Zamponic, and J.-P. Bouchaudd. Tipping points in macroeconomic agent-based models. _J. Econ. Dyn. Contr._ , 50:29–61, 2015.
* Jiang et al. (2018) Junjie Jiang, Zi-Gang Huang, Thomas P Seager, Wei Lin, Celso Grebogi, Alan Hastings, and Ying-Cheng Lai. Predicting tipping points in mutualistic networks through dimension reduction. _Proc. Nat. Acad. Sci. (UsA)_ , 115(4):E639–E647, 2018.
* Yang et al. (2018) Biwei Yang, Meiyi Li, Wenqing Tang, Si Liu, Weixinand Zhang, Luonan Chen, and Jinglin Xia. Dynamic network biomarker indicates pulmonary metastasis at the tipping point of hepatocellular carcinoma. _Nat. Commun._ , 9:678, 2018.
* Jiang et al. (2019) Junjie Jiang, Alan Hastings, and Ying-Cheng Lai. Harnessing tipping points in complex ecological networks. _J. R. Soc. Interface_ , 16(158):20190345, 2019\.
* Scheffer (2020) Marten Scheffer. _Critical Transitions in Nature and Society_ , volume 16. Princeton University Press, 2020.
* Meng et al. (2020a) Yu Meng, Junjie Jiang, Celso Grebogi, and Ying-Cheng Lai. Noise-enabled species recovery in the aftermath of a tipping point. _Phys. Rev. E_ , 101(1):012206, 2020a.
* Meng et al. (2020b) Yu Meng, Ying-Cheng Lai, and Celso Grebogi. Tipping point and noise-induced transients in ecological networks. _J. R. Soc. Interface_ , 17(171):20200645, 2020b.
* Meng and Grebogi (2021) Yu Meng and Celso Grebogi. Control of tipping points in stochastic mutualistic complex networks. _Chaos_ , 31(2):023118, 2021.
* Meng et al. (2022) Yu Meng, Ying-Cheng Lai, and Celso Grebogi. The fundamental benefits of multiplexity in ecological networks. _J. R. Soc. Interface_ , 19:20220438, 2022.
* O’Keeffe and Wieczorek (2020) Paul E O’Keeffe and Sebastian Wieczorek. Tipping phenomena and points of no return in ecosystems: beyond classical bifurcations. _SIAM J. Appl. Dyn. Syst._ , 19(4):2371–2402, 2020.
* Trefois et al. (2015) Christophe Trefois, Paul MA Antony, Jorge Goncalves, Alexander Skupin, and Rudi Balling. Critical transitions in chronic disease: transferring concepts from ecology to systems medicine. _Cur. Opin. Biotechnol._ , 34:48–55, 2015.
* Albrich et al. (2020) Katharina Albrich, Werner Rammer, and Rupert Seidl. Climate change causes critical transitions and irreversible alterations of mountain forests. _Global Change Biol._ , 26(7):4013–4027, 2020\.
* Bayani et al. (2017) Atiyeh Bayani, Fatemeh Hadaeghi, Sajad Jafari, and Greg Murray. Critical slowing down as an early warning of transitions in episodes of bipolar disorder: A simulation study based on a computational model of circadian activity rhythms. _Chronobiol. Int._ , 34(2):235–245, 2017.
* Synodinos et al. (2022) Alexis D Synodinos, Rajat Karnatak, Carlos A Aguilar-Trigueros, Pierre Gras, Tina Heger, Danny Ionescu, Stefanie Maaß, Camille L Musseau, Gabriela Onandia, Aimara Planillo, et al. The rate of environmental change as an important driver across scales in ecology. _Oikos_ , page e09616, 2022.
* Morris et al. (2002) James T Morris, PV Sundareshwar, Christopher T Nietch, Björn Kjerfve, and Donald R Cahoon. Responses of coastal wetlands to rising sea level. _Ecology_ , 83(10):2869–2877, 2002.
* Ritchie et al. (2023) Paul DL Ritchie, Hassan Alkhayuon, Peter M Cox, and Sebastian Wieczorek. Rate-induced tipping in natural and human systems. _Earth Syst. Dynam_ , 14(3):669–683, 2023.
* Wieczorek et al. (2011) Sebastian Wieczorek, Peter Ashwin, Catherine M Luke, and Peter M Cox. Excitability in ramped systems: the compost-bomb instability. _Proc. R. Soc. A Math. Phys. Eng. Sci._ , 467(2129):1243–1269, 2011.
* Mitry et al. (2013) John Mitry, Michelle McCarthy, Nancy Kopell, and Martin Wechselberger. Excitable neurons, firing threshold manifolds and canards. _J. Math. Neurosci._ , 3(1):1–32, 2013.
* Alexander et al. (2011) NA Alexander, O Oddbjornsson, CA Taylor, HM Osinga, and David E Kelly. Exploring the dynamics of a class of post-tensioned, moment resisting frames. _J. Sound Vib._ , 330(15):3710–3728, 2011.
* Hsu and Shih (2015) Sheng-Yi Hsu and Mau-Hsiang Shih. The tendency toward a moving equilibrium. _SIAM J. Appl. Dyn. Sys._ , 14(4):1699–1730, 2015\.
* Ashwin et al. (2017) Peter Ashwin, Clare Perryman, and Sebastian Wieczorek. Parameter shifts for nonautonomous systems in low dimension: bifurcation-and rate-induced tipping. _Nonlinearity_ , 30(6):2185, 2017.
* Vanselow et al. (2019) Anna Vanselow, Sebastian Wieczorek, and Ulrike Feudel. When very slow is too fast-collapse of a predator-prey system. _J. Theo. Biol._ , 479:64–72, 2019.
* Lande et al. (2003) Russell Lande, Steinar Engen, and Bernt-Erik Saether. _Stochastic population dynamics in ecology and conservation_. Oxford University Press on Demand, 2003.
* Hansen et al. (2019) Brage B Hansen, Marlène Gamelon, Steve D Albon, Aline M Lee, Audun Stien, R Justin Irvine, Bernt-Erik Sæther, Leif E Loe, Erik Ropstad, Vebjørn Veiberg, et al. More frequent extreme climate events stabilize reindeer population dynamics. _Nat. Commun_ , 10(1):1616, 2019.
* Hastings et al. (2018) Alan Hastings, Karen C Abbott, Kim Cuddington, Tessa Francis, Gabriel Gellner, Ying-Cheng Lai, Andrew Morozov, Sergei Petrovskii, Katherine Scranton, and Mary Lou Zeeman. Transient phenomena in ecology. _Science_ , 361(6406), 2018.
* Morozov et al. (2020) Andrew Morozov, Karen Abbott, Kim Cuddington, Tessa Francis, Gabriel Gellner, Alan Hastings, Ying-Cheng Lai, Sergei Petrovskii, Katherine Scranton, and Mary Lou Zeeman. Long transients in ecology: theory and applications. _Phys. Life Rev._ , 32:1–40, 2020.
* Bascompte et al. (2003) J. Bascompte, P. Jordano, C. J. Melián, and J. M. Olesen. The nested assembly of plant-animal mutualistic networks. _Proc. Natl. Acad. Sci. (USA)_ , 100:9383–9387, 2003.
* Guimaraes et al. (2011) P. R. Guimaraes, P. Jordano, and J. N. Thompson. Evolution and coevolution in mutualistic networks. _Ecol. Lett._ , 14:877–885, 2011.
* Nuismer et al. (2013) S. L. Nuismer, P. Jordano, and J. Bascompte. Coevolution and the architecture of mutualistic networks. _Evolution_ , 67:338–354, 2013.
* Rohr et al. (2014) Rudolf P Rohr, Serguei Saavedra, and Jordi Bascompte. On the structural stability of mutualistic systems. _Science_ , 345(6195):1253497, 2014.
* Dakos and Bascompte (2014) Vasilis Dakos and Jordi Bascompte. Critical slowing down as early warning for the onset of collapse in mutualistic communities. _Proc. Natl. Acad. Sci. (USA)_ , 111(49):17546–17551, 2014.
* Guimaraes et al. (2017) P. R. Guimaraes, M. M. Pires, P. Jordano, J. Bascompte, and J. N. Thompson. Indirect effects drive coevolution in mutualistic networks. _Nature_ , 550:511–514, 2017.
* Ohgushi et al. (2012) T. Ohgushi, O. Schmitz, and R. D. Holt. _Trait-Mediated Indirect Interactions: Ecological and Evolutionary Perspectives_. Cambridge Univ. Press, Cambridge UK, 2012.
* Bastolla et al. (2009) Ugo Bastolla, Miguel A Fortuna, Alberto Pascual-García, Antonio Ferrera, Bartolo Luque, and Jordi Bascompte. The architecture of mutualistic networks minimizes competition and increases biodiversity. _Nature_ , 458(7241):1018–1020, 2009.
* Bascompte and Jordano (2007) Jordi Bascompte and Pedro Jordano. Plant-animal mutualistic networks: the architecture of biodiversity. _Annu. Rev. Ecol. Evol. Syst._ , 38:567–593, 2007.
* Menck et al. (2013) Peter J Menck, Jobst Heitzig, Norbert Marwan, and Jürgen Kurths. How basin stability complements the linear-stability paradigm. _Nature physics_ , 9(2):89–92, 2013.
* Holling (1973) Crawford S Holling. Resilience and stability of ecological systems. _Ann. Rev. Ecol. Systema_ , pages 1–23, 1973.
* Schleuning et al. (2016) Matthias Schleuning, Jochen Fründ, Oliver Schweiger, Erik Welk, Jörg Albrecht, Matthias Albrecht, Marion Beil, Gita Benadi, Nico Blüthgen, Helge Bruelheide, et al. Ecological networks are more sensitive to plant than to animal extinction under climate change. _Nat. Commun_ , 7(1):13965, 2016.
* Vanbergen et al. (2017) Adam J Vanbergen, Ben A Woodcock, Matthew S Heard, and Daniel S Chapman. Network size, structure and mutualism dependence affect the propensity for plant–pollinator extinction cascades. _Funct. Ecol._ , 31(6):1285–1293, 2017.
* Lewinsohn et al. (2006) Thomas M Lewinsohn, Paulo Inácio Prado, Pedro Jordano, Jordi Bascompte, and Jens M. Olesen. Structure in plant–animal interaction assemblages. _Oikos_ , 113(1):174–184, 2006.
* Vázquez et al. (2009) Diego P. Vázquez, Nico Blüthgen, Luciano Cagnolo, and Natacha P. Chacoff. Uniting pattern and process in plant–animal mutualistic networks: a review. _Ann. Bot._ , 103(9):1445–1457, 03 2009. ISSN 0305-7364. 10.1093/aob/mcp057. URL https://doi.org/10.1093/aob/mcp057.
* Aslan et al. (2013) Clare E Aslan, Erika S Zavaleta, Bernie Tershy, and Donald Croll. Mutualism disruption threatens global plant biodiversity: a systematic review. _PLoS one_ , 8(6):e66993, 2013.
* Sheykhali et al. (2020) Somaye Sheykhali, Juan Fernández-Gracia, Anna Traveset, Maren Ziegler, Christian R Voolstra, Carlos M Duarte, and Víctor M Eguíluz. Robustness to extinction and plasticity derived from mutualistic bipartite ecological networks. _Sci. Rep._ , 10(1):9783, 2020.
* Pires et al. (2020) Mathias M Pires, James L O’Donnell, Laura A Burkle, Cecilia Díaz-Castelazo, David H Hembry, Justin D Yeakel, Erica A Newman, Lucas P Medeiros, Marcus AM de Aguiar, and Paulo R Guimarães Jr. The indirect paths to cascading effects of extinctions in mutualistic networks, 2020.
* Young et al. (2021) Jean-Gabriel Young, Fernanda S Valdovinos, and MEJ Newman. Reconstruction of plant–pollinator networks from observational data. _Nat. Commun_ , 12(1):3911, 2021.
* Gao et al. (2016) Jianxi Gao, Baruch Barzel, and Albert-László Barabási. Universal resilience patterns in complex networks. _Nature_ , 530(7590):307–312, 2016.
|
# End-to-End Optimization of JPEG-Based Deep Learning Process for Image
Classification ††thanks: This material is based upon work supported by Google
Cloud.
Siyu Qi, Lahiru D. Chamain, and Zhi Ding Department of Electrical and
Computer Engineering
University of California, Davis, CA, 95616
Email: {syqi, hdchamain<EMAIL_ADDRESS>
###### Abstract
Among major deep learning (DL) applications, distributed learning involving
image classification require effective image compression codecs deployed on
low-cost sensing devices for efficient transmission and storage. Traditional
codecs such as JPEG designed for perceptual quality are not configured for DL
tasks. This work introduces an integrative end-to-end trainable model for
image compression and classification consisting of a JPEG image codec and a
DL-based classifier. We demonstrate how this model can optimize the widely
deployed JPEG codec settings to improve classification accuracy in
consideration of bandwidth constraint. Our tests on CIFAR-100 and ImageNet
also demonstrate improved validation accuracy over preset JPEG configuration.
###### Index Terms:
JPEG, joint compression and classification, end-to-end optimization.
## I Introduction
In recent years, deep convolutional neural networks (CNNs) have demonstrated
successes in learning tasks such as image classification and recognition,
owing to their capability of extracting image features among adjacent pixels.
The emergence of residual network (ResNet)[1] further enhanced image
classification without introducing extra computational complexity.
In the era of IoT and cloud computing, many practical applications rely on
widely deployed low-cost cameras and sensors for data collection before
transmitting sensor data to powerful cloud or edge servers that host pre-
trained deep classifiers. As most (RF) network links usually are severely
band-limited and must prioritize heavy data traffic, image compression
techniques are vital for efficient and effective utilization of limited
network bandwidth and storage resources. JPEG [2] is a highly popular codec
standard for lossy image compression, widely used to conserve bandwidth in
source data transmission and storage. The JPEG encoding process includes
discrete cosine transform (DCT) and quantization. The quantized integer DCT
coefficients are encoded via run-length encoding (RLE) and Huffman coding. Due
to RLE, total bit rate of an image cannot be predicted straightforwardly[3, 4,
5]. The JPEG encoding achieves substantial image compression ratio with little
human perception quality sacrifice. These encoded bits are then transmitted
over a channel of limited capacity (bit rate) before decoding and recovery for
various applications.
Targeting human users, the parameters in JPEG configuration are selected
according to visualization subjective tests. However, in CNN-based image
classifications, naïve adoption of the lossy JPEG image encoding, designed
primarily for human visualization needs, can lead to unexpected accuracy loss
because the traditional CNN models are agnostic of the compression distortion.
To tackle this issue, this work is motivated by the obvious and important
question in distributed AI: How to optimally (re)configure standardized JPEG
for image compression to improve DL-based image classification.
Motivated by the strong need to conserve network bandwidth and local storage
for remote image classification, we present an end-to-end trainable DL model
for joint image compression and classification that can optimize the widely
deployed JPEG codec to improve classification accuracy over current JPEG
settings. We formulate this dual-objective problem as a constrained
optimization problem which maximizes classification accuracy subject to a
compression ratio constraint. We incorporate trainable JPEG compression blocks
and JPEG decoding blocks together with the trainable CNN classifier in our
end-to-end learning model. Our proposed DL model can configure JPEG encoding
parameters to achieve high classification accuracy.
We organize the rest of the paper as follows. Section II introduces the basics
of JPEG codec and summarizes related works. Section III proposes the novel
end-to-end DL architecture. We provide experimental results in Section IV,
before concluding and discussing potential future directions in Section V.
## II JPEG Codec and Learning over JPEG
### II-A JPEG Codec in View of Deep Learning
In JPEG compression with 4:2:0 chroma subsampling, an RGB source image is
first converted to YCBCR color space through linear transformations.
Chrominance channels (CB and CR) are subsampled by 2 both vertically and
horizontally. After subsampling, each of the 3 YCBCR channels is split into
non-overlapping $8\times 8$ blocks before applying blockwise DCT. The
2-dimensional (2-D) DCT of an image block $\boldsymbol{I}$ of size $N\times N$
with entries $\boldsymbol{I}(k,l)$ is defined by $N\times N$ block
$\boldsymbol{F}=\boldsymbol{DID}^{T}$, where $\boldsymbol{D}$ is a constant
matrix. The DCT is capable of compacting image features with a small number of
DCT coefficients with little perceptible loss after compression.
For compression, each block of the $8\times 8$ frequency-domain coefficients
$\boldsymbol{F}$ is quantized using pre-defined quantization matrices, or
“Q-tables” $\boldsymbol{Q}$ with entries $\boldsymbol{Q}(j,k)$ at JPEG encoder
to obtain quantized block $\boldsymbol{F}_{q}$ whose entries are
$\boldsymbol{F}_{q}(j,k)=\mbox{round}[{\boldsymbol{F}(j,k)}/{\boldsymbol{Q}(j,k)}]$.
The decoder reconstructs from the compressed block $\boldsymbol{F}_{q}$ and
the Q-table to form Hadamard product
$\boldsymbol{F}_{q}=\boldsymbol{F}\circ\boldsymbol{Q}$. Decoder would then use
inverse DCT (IDCT) to recover spatial RGB images. Parameters in
$\boldsymbol{Q}$ can be adjusted to achieve different compression levels and
visual effects. JPEG standard provides two Q-tables to adjust compression
loss, one for the Y channel and another for CB and CR channels.
In networked image applications, training using full resolution images would
make little practical sense and would be prone to accuracy loss because only
codec-compressed image data are available at the cloud/edge processing node.
Thus, deep learning networks should directly use compressed DCT coefficients
as inputs for both training and inference instead of full resolution RGB
images for training. Image classification (labeling) directly based on DCT
coefficients can further reduce decoder computation during both training and
inference by skipping the IDCT and potentially achieve better robustness under
dynamic levels of JPEG compression.
Importantly, ResNets that were successfully developed for recognition of fully
reconstructed JPEG images tend to exhibit performance loss if they are
directly used on image data in DCT domain. Motivated by the need to improve
image processing performance in networked environments under channel bandwidth
and storage constraints, this work investigates deep learning architecture
designs suitable for optimizing standard compliant JPEG configurations to
achieve high classification accuracy and low bandwidth consumption by directly
applying DCT input data. Our joint optimization of the JPEG configuration is
achieved by optimizing both the JPEG Q-table parameters and the deep learning
classifier to achieve end-to-end deep learning framework spanning from the IoT
source encoder to the cloud classifier. Our experiments include tests on the
high resolution ImageNet dataset.
### II-B Related Works
For bandwidth and storage conservation, DL architectures such as auto-encoders
have been effectively trained[6, 7, 8] for image compression with little
degradation of classification accuracy or perceptual quality. Previous works
[9, 10] also revealed direct training of DL models on the DCT coefficients
using faster CNN structure with a modified ResNet-50 architecture. The authors
of [11] developed a joint compression and classification network model based
on JPEG2000 encoding. These works suggest the benefit of DL-based end-to-end
optimization of image codecs.
Previous studies [12, 13, 14, 5] have also recognized the importance of
Q-tables in JPEG codecs and seek to optimize them for DL-based image
classification. [13, 12] propose to design JPEG Q-tables based on the
importance of DCT coefficients, evaluated by the relative frequency [13] or
the standard deviation [12] of the coefficients. Both [14, 5] offer end-to-end
DL models to estimate a set of optimized Q-tables for each input image, where
models are pre-trained to predict the bandwidth of each image. In contrast,
targeting low-cost sensing nodes, our proposed model learns a single set of
Q-tables for all images during training, which can be pre-configured within
the JPEG codec after training for inference tasks. This reduces the required
computational power at the sensing nodes. Moreover, our proposed training
tunes Q-tables and does not require a separate entropy estimation model.
To our best knowledge, there exists no published work on JPEG Q-table
optimization for distributed learning to target low-cost sensing devices.
Since JPEG continues to be a commonly used image coding methods in massive
number of low-cost devices, we focus our investigation on the rate-accuracy
trade-off to facilitate their widespread applications in distributed learning.
## III Joint DL Architectures
### III-A Wide ResNet (WRN) for CIFAR-100 and Tiny ImageNet
For CIFAR-100 and Tiny ImageNet, we propose the WRN model of Fig. 1. Following
JPEG , our preprocessing steps include level shifting, color transformation,
subsampling, and DCT. The invertible color transformation can also be trained.
Figure 1: Proposed architecture for CIFAR-100 and Tiny ImageNet. The detailed
structure in WRN-28 is omitted here.
#### III-A1 Compression Layers
The trainable quantization layer in our proposed WRN model is similar to the
“quan block” in [11], as shown in Fig. 1. JPEG can take three distinct
Q-tables $\boldsymbol{Q}_{1}$, $\boldsymbol{Q}_{2}$, and $\boldsymbol{Q}_{3}$,
respectively, for each of the YCBCR color channels, leading to 192 parameters
in this layer. Same as [11], we replace the trainable quantization parameters
by element-wise reciprocal of the Q-table entries
$\boldsymbol{q}_{i}(j,k)=\boldsymbol{Q}_{i}(j,k)^{-1}.$ The matrices
$\boldsymbol{q}_{i}$, referred to as the “compression kernels”, allows the
model to learn to discard any frequency domain information by setting the
corresponding entry $\boldsymbol{q}_{i}(j,k)=0$. Smaller $\boldsymbol{q}_{i}$
values lead to smaller range of quantized DCT coefficients and consequently
generates fewer encoded bits.
The quantization layer includes a non-differentiable rounding operation
$a(\boldsymbol{F})=\mbox{round}(\boldsymbol{F})$, which cannot be used in a
gradient-based training framework, as its activation function. Following [15],
we address this problem by substituting a smooth approximation
$\hat{a}(\boldsymbol{F})=\boldsymbol{F}$ for the rounding function in
backpropagation. Together, preprocessing and quantization layers form a JPEG-
based encoder.
The dequantization layer only needs to multiply the encoded DCT blocks
$\boldsymbol{F}_{q}$ element-wise by their respective Q-table matrices
$\boldsymbol{Q}_{1}$, $\boldsymbol{Q}_{2}$ and $\boldsymbol{Q}_{3}$
corresponding to the encoder quantization layer. The quantization and
dequantization layers jointly form a pair of “compression layers”.
#### III-A2 Reconstruction Layer
As described in Section II, the reconstruction layer performs IDCT via
$\boldsymbol{I}=\boldsymbol{D}^{T}\boldsymbol{F}_{q}\boldsymbol{D}$ for each
quantized DCT coefficient block $\boldsymbol{F}_{q}$ and rearranges the
reconstructed blocks $\boldsymbol{I}$. CB and CR channels are upsampled via
bilinear interpolation. The outputs of this layer are YCBCR spatial images.
Together, the dequantization layer and the reconstruction layer form the JPEG
decoder.
#### III-A3 Classifier
WRNs[16] achieve impressive classification performance on the CIFAR-100[17]
and Tiny ImageNet[18] datasets. Without loss of generality, we adopt a
28-layer WRN as the classifier. For CIFAR-100, we set the convolutional layer
width multiplier $k=10$, same as that used in [16]. For Tiny ImageNet, we set
$k=1$ to further simplify training.
### III-B Loss Function During Training
To jointly reconfigure the JPEG parameters in compression layers and to
optimize the deep learning classifier, we design the following loss function
during training:
$\displaystyle\mathcal{L}=\mathcal{L}_{\rm CLA}+\lambda\mathcal{L}_{\rm
Quan},\vspace*{-2mm}$
where $\mathcal{L}_{\rm CLA}$ is the cross entropy classification loss,
$\mathcal{L}_{\rm Quan}$ is a penalty term for the compression kernels and
$\lambda$ is an adjustable hyper-parameter to govern the importance of the
rate. Since there is no simple or standard metric to directly control JPEG
encoded image size, which further involves RLE and Huffman coding, we propose
the following surrogate penalty function:
$\displaystyle\mathcal{L}_{\rm
Quan}=\sum_{i=1}^{3}\sum_{j=1}^{8}\sum_{k=1}^{8}$
$\displaystyle\left[\mbox{max}(\boldsymbol{q}_{i}^{2}(j,k)-c,0)+\lambda_{1}|\boldsymbol{q}_{i}(j,k)|\right]$
where $c$ and $\lambda_{1}$ are tunable hyper-parameters. The $\ell_{1}$ loss
term promotes sparsity whereas the $\ell_{2}$ loss term regulates the
compression kernels. The hyper-parameter $c$ acts as a constraint on the
squared magnitude of $\boldsymbol{q}_{i}$, shall be appropriately selected
based on the values of $\lambda$ and $\lambda_{1}$. Larger $\lambda$ and
$\lambda_{1}$ and a smaller $c$ leads to higher compression ratio and lower
classification accuracy. We propose the current form of the surrogate penalty
function after testing both logarithm and sigmoid functions without witnessing
performance benefits.
### III-C Modified ResNet for ImageNet
For ImageNet, we adopt the same preprocessing steps and quantization layer
from III-A and utilize the Deconvolution-RFA architecture in [10], which is
inspired by ResNet-50, as the classifier. As demonstrated in Fig. 2, the
quantized DCT coefficients of CB and CR channels are augmented to the same
spatial size as Y channel by two separate transposed convolutional layers. The
three channels are concatenated as input of the deconvolution-RFA model.
Considering the higher complexity of this model, we suggest a single
$\ell_{2}$ regularization for compression kernels for optimizing the
quantization parameters via quantization loss $\mathcal{L}_{\rm
Quan}=\sum_{i=1}^{3}\sum_{j=1}^{8}\sum_{k=1}^{8}\boldsymbol{q}_{i}^{2}(j,k).$
Figure 2: Architecture of our proposed model for ImageNet. The detailed
structure of Deconvolution-RFA[10] is omitted here.
### III-D Implementation
We test the learning framework in Fig. 1 with CIFAR-100 and Tiny ImageNet
datasets. The images in CIFAR-100 dataset are of $32\times 32$ pixels, while
images in Tiny ImageNet are of $64\times 64$ pixels.
There are 192 trainable parameters in the compression kernels, all initialized
to 1. The randomly-initialized WRN-28 classifier is trained jointly with the
parameters in compression kernels, as well as the color transformation
coefficients if needed, using Adam optimizer with a batch size of 100. The
training of WRN-28 proceeds alternatively: the classifier is trained for 2
epochs while JPEG-based layers are frozen, followed by the compression layer
being trained for 1 epoch while freezing the classifier. The training takes
150 such alternations. The learning rate starts from 0.05, and is scaled by
0.1 and 0.01, at alternation 50 and 100, respectively.
We implement the larger learning framework in Fig. 2 with ImageNet in which
the images are of $224\times 224$ pixels. Similarly, color transformation
coefficients are initialized to JPEG standard and all 192 quantization
parameters are initialized to 1. The color transformation, compression
kernels, and Deconvolution-RFA classifier are trained from end to end by using
Adam optimizer with a batch size of 32. The learning rate of compression
kernels starts from $1\times 10^{-8}$ while that for other parameters starts
from 0.001. Both learning rates are scaled by 0.1, 0.01, and 0.001, at epoch
30, 60 and 80, respectively. The training takes 90 epochs.
For the quantized DCT coefficients using the new Q-tables, we customize
Huffman tables for bandwidth saving by randomly choosing 50k images from the
corresponding training set to generate Huffman tables. The bandwidth of the
validation data set is measured by combining the principles of RLE and Huffman
coding.
## IV Experiments
Our experiments are conducted on Keras and TensorFlow. We utilize the two
metrics to evaluate perceptual quality of images: peak signal-to-noise ratio
(PSNR) and structural similarity index measure (SSIM) index. We test the
proposed joint compression and classification (JCC) frameworks on three
datasets: (a) CIFAR-100 dataset based on 50k training images and 10k test
images belonging to 100 categories; (b) Tiny ImageNet dataset based on 100k
training images and 10k validation images belonging to 200 categories; (c)
ImageNet dataset based on approximately 1.3M training images and 50k test
images belonging to 1000 categories.
### IV-A JPEG Standard Baseline
We first present baseline results of images using standard JPEG algorithm with
4:2:0 chroma subsampling. We initialize compression kernels
$\boldsymbol{q}_{i}$, $i=1,\ 2,\ 3$ using the Q-tables given in JPEG standard.
In this baseline scenario, we only adapt the classifier parameters during
training.
For CIFAR-100, we consider 9 different JPEG image qualities between $12.5$%
and $100\%$. For Tiny ImageNet, we select 4 different image qualities between
$10$% and $80\%$. For ImageNet, our experiments consider 5 different image
qualities between $12.5$% and $100\%$. The classification results are shown in
Figs. 3-4. These baseline results reveal that the classification accuracy
correlates positively with image qualities and average image bandwidth (rate).
### IV-B Joint Compression and Classification (JCC)
For JCC, we train and optimize the compression kernels and the classifier. For
CIFAR-100, we selected 8 values of $\lambda$ from $10^{-6}$ to $5$ with
$\lambda_{1}=1$ and $c=0.01/\lambda$. The classification accuracy, PSNR and
SSIM results can be found in Fig. 3. Compared with the JPEG baseline, JCC
achieves clear improvement of up to 2.4% in accuracy at bandwidth between 0.75
and 1.5 KB per image. The PSNR and SSIM of JCC-compressed images are similar
to those using JPEG.
(a)
(b)
(c)
Figure 3: Comparison between JCC and JPEG standard on CIFAR-100 and Tiny
ImageNet datasets with WRN-28 Model, with respect to classification accuracy,
PSNR and SSIM versus average bandwidth.
For Tiny ImageNet, we select 6 values of $c$ from $5\times 10^{-3}$ to $0.8$.
As shown in Fig. 3, when comparing with the JPEG baseline, we observe accuracy
gain of up to 4% by the proposed JCC model at low bandwidth between 0.9 and
1.6 KB per image while maintaining similar visual quality. Overall, the PSNR
and SSIM of JCC-optimized and JPEG standard quantization tables are similar.
For ImageNet, we consider 9 values of $\lambda$ between $25$ and $100$. The
resulting top-5 classification accuracy, PSNR, and SSIM are given in Fig. 4.
For encoding rates below 11 KB per image, the JCC model outperforms the
baseline by up to $3.7\%$ in terms of classification accuracy. For bandwidths
above 11 KB per image, the classification accuracy difference between JPEG and
JCC is quite insignificant. Furthermore, PSNR and SSIM of JCC-compressed
images outperform those of standard JPEG encoded images.
(a)
(b)
(c)
Figure 4: Comparison between JCC and JPEG baseline on ImageNet dataset with
Deconvolution-RFA Model, with respect to top-5 classification accuracy, PSNR
and SSIM versus average bandwidth.
From these experimental results, we observe that the JCC model can effectively
optimize the JPEG compression kernels for better rate-accuracy trade-off,
especially at moderate image bit rates. It is intuitive that the performance
edge of JCC diminishes for very high image sizes because most image features
can be preserved when given sufficient number of bits and JCC and JPEG no
longer need to delicately balance the rate-accuracy trade-off. Similarly, for
very low image sizes, very few bits can be used to encode vital information in
DCT coefficients. Hence, the encoders have less flexibility to further
optimize the rate-accuracy trade-off, thereby making it difficult for even the
JCC model to find better parameter settings.
### IV-C JCC and Color Transformation
Considering JCC and color transformation (JCC-color), color transformation
coefficients, compression kernels and classifier are jointly trained. We
initialize the color conversion coefficients according to the settings in
JPEG. According to our experimental results, tuning the color transformation
coefficients offers no performance gain over JCC. Theoretically, invertible
color space transformation does not lead to information loss and can be
subsumed by first dense-layer in the neural network. In fact, we observe that
the resulting color transform rarely move from their initial values and
further optimization is unnecessary.
### IV-D Further Analysis and Discussions
(a) Figure 5: JCC optimized compression kernels based on CIFAR-100 for Y
(left), CB (middle) and CR (right) channels, with
$\lambda=0.001,\lambda_{1}=1$ and $c=10$.
Our experimental results suggest that trainable JCC model can extract critical
DCT features for classification among categories of CIFAR-100, Tiny ImageNet,
and ImageNet datasets. Furthermore, the perceptual quality of images are
preserved. Fig. 5 shows the resulting compression kernels with which WRN-28
achieves classification accuracy of 75.20% at an average rate of 0.979
KB/image. Darker grids imply low compression or higher importance of the
corresponding DCT coefficient. The encoder clearly favors lower frequency
bands. Since there are longer consecutive 0’s in the zig-zag order in the end-
to-end learned compression kernels, the correspondingly compressed DCT
coefficients require fewer bits via RLE. Furthermore, the trainable model
learns to discard some higher frequency DCT components less critical to
classification.
Together, these experimental results show performance enhancement on
standardized JPEG codec for cloud-based image classification. The optimized Q
tables can be distributed in pre-installed JPEG encoders of low-cost devices
through software updates for different encoding sizes.
## V Conclusions
We present an end-to-end deep learning (DL) architecture to jointly optimize
JPEG image compression and classification for low-cost sensors in distributed
learning systems. Results on CIFAR-100, Tiny ImageNet, and ImageNet datasets
demonstrate successful training of the end-to-end DL framework for better
image compression and classification performance without perceptual quality
loss. Optimized JPEG Q-tables can be readily incorporated within deployed
codecs in practice. Future works may explore the broad appeal of this end-to-
end learning principle in other bandwidth-constrained distributed DL tasks
such as object detection, segmentation, and tracking.
## References
* [1] K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” pp. 770–778, 2016.
* [2] C. R. T.81, “Information technology - Digital compression and coding of continuous-tone still images - Requirements and guidelines.” Standard, 1992\.
* [3] S.-W. Wu and A. Gersho, “Rate-constrained picture-adaptive quantization for JPEG baseline coders,” in _Int. Conf. on Acoustics, Speech, and Signal Processing_. IEEE, 1993, pp. 389–392.
* [4] E. Tuba, M. Tuba, D. Simian, and R. Jovanovic, “JPEG quantization table optimization by guided fireworks algorithm,” in _Int. Workshop on Combinatorial Image Analysis_. Springer, 2017, pp. 294–307.
* [5] X. Luo, H. Talebi, F. Yang, M. Elad, and P. Milanfar, “The rate-distortion-accuracy tradeoff: JPEG case study,” _arXiv:2008.00605_ , 2020.
* [6] S. Singh, S. Abu-el Haija, N. Johnston, J. Ballé, A. Shrivastava, and G. Toderici, “End-to-end learning of compressible features,” _arXiv:2007.11797_ , 2020.
* [7] L. Theis, W. Shi, A. Cunningham, and F. Husz$\acute{a}$r, “Lossy image compression with compressive autoencoders,” 2017.
* [8] S. Diamond, V. Sitzmann, S. Boyd, G. Wetzstein, and F. Heide, “Dirty pixels: Optimizing image classification architectures for raw sensor data,” 2017.
* [9] X. Zou, X. Xu, C. Qing, and X. Xing, “High speed deep networks based on discrete cosine transformation,” in _IEEE Int. Conf. on Image Proc._ , 2014, pp. 5921–5925.
* [10] L. Gueguen, A. Sergeev, B. Kadlec, R. Liu, and J. Yosinski, “Faster neural networks straight from JPEG,” in _Advances in Neural Information Processing Systems_ , 2018, pp. 3937–3948.
* [11] L. D. Chamain, S. S. Cheung, and Z. Ding, “Quannet: Joint image compression and classification over channels with limited bandwidth,” in _2019 IEEE Int. Conf. on Multimedia and Expo (ICME)_ , 2019, pp. 338–343.
* [12] Z. Liu, T. Liu, W. Wen, L. Jiang, J. Xu, Y. Wang, and G. Quan, “DeepN-JPEG: A deep neural network favorable JPEG-based image compression framework,” _Proc. 55th Annual Design Automation Conf._ , 2018.
* [13] Z. Li, C. D. Sa, and A. Sampson, “Optimizing JPEG quantization for classification networks,” _arXiv:2003.02874_ , 2020.
* [14] J. Choi and B. Han, “Task-aware quantization network for JPEG image compression,” in _European Conf. on Computer Vision_. Springer, 2020, pp. 309–324.
* [15] L. Theis, W. Shi, A. Cunningham, and F. Huszár, “Lossy image compression with compressive autoencoders,” _arXiv:1703.00395_ , 2017.
* [16] S. Zagoruyko and N. Komodakis, “Wide residual networks,” _arXiv:1605.07146_ , 2016.
* [17] A. Krizhevsky, “Learning multiple layers of features from tiny images,” Tech. Rep., 2009.
* [18] Y. Le and X. Yang, “Tiny imagenet visual recognition challenge,” _CS 231N_ , vol. 7, 2015.
|
# Double semidirect products and
skew left braces of size $np$
Teresa Crespo(1), Daniel Gil-Muñoz(2), Anna Rio(3)
and Montserrat Vela(3)
###### Abstract
We define the double semidirect product of skew left braces and prove that if
$p$ is an odd prime and $n$ is an integer such that $p\nmid n$ and each group
of order $np$ has a unique $p$-Sylow subgroup, then any skew left brace of
size $np$ is a double semidirect product of the trivial brace of size $p$ and
a skew brace of size $n$. We develop an algorithm to obtain all braces of size
$np$ from the set of braces of size $n$ and provide a formula to count them.
We use the result to describe all braces of size $12p$ for $p\geq 7$ and prove
a conjecture of Bardakov, M.V. Neshchadim and M.K. Yadav.
(1) Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran
Via de les Corts Catalanes 585, 08007, Barcelona (Spain)
(2) Charles University, Faculty of Mathematics and Physics, Department of
Algebra, Sokolovska 83, 18600 Praha 8, Czech Republic
(3) Departament de Matemàtiques, Universitat Politècnica de Catalunya, Edifici
Omega, Jordi Girona, 1-3, 08034, Barcelona (Spain)
Keywords: Left braces, Holomorphs, regular subgroups, Hopf Galois structures.
2020MSC: Primary: 16T05, 16T25, 12F10; Secondary: 20B35, 20B05, 20D20, 20D45.
††footnotetext: The first author was supported by grant PID2019-107297GB-I00
(Ministerio de Ciencia, Innovación y Universidades). The second author was
supported by Czech Science Foundation, grant 21-00420M, and by Charles
University Research Centre program UNCE/SCI/022. Third and fourth authors are
supported by 2021 SGR 01468. Email addresses<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
## 1 Introduction
In [7] Rump introduced an algebraic structure called brace to study set-
theoretic solutions of the Yang-Baxter equation. Guarnieri and Vendramin in
[6] gave the more general concept of skew brace. A skew (left) brace is a set
$B$ with two operations $\cdot$ and $\circ$ such that $(B,\cdot)$ and
$(B,\circ)$ are groups and the brace relation is satisfied, namely,
$a\circ(b\cdot c)=a\circ b\cdot a^{-1}\cdot a\circ c,$
for all $a,b,c\in B$, where $a^{-1}$ denotes the inverse of $a$ with respect
to $\cdot$. We call $(B,\cdot$) the additive group and $(B,\circ)$ the
multiplicative group of the skew left brace. We say that $B$ is a classical
brace if its additive group is abelian.
Let $B_{1}$ and $B_{2}$ be skew left braces. A map $f:B_{1}\to B_{2}$ is said
to be a skew brace morphism if $f(a\cdot b)=f(a)\cdot f(b)$ and $f(a\circ
b)=f(a)\circ f(b)$ for all $a,b\in B_{1}$. If $f$ is bijective, we say that
$f$ is an isomorphism. In that case we say that the skew braces $B_{1}$ and
$B_{2}$ are isomorphic.
In [6] Guarnieri and Vendramin proved that given a group $N$, there is a
bijective correspondence between isomorphism classes of skew left braces with
additive group $N$ and conjugacy classes of regular subgroups of
$\mathrm{Hol}(N)$.
In [3], we classified classical left braces of size $8p$, for $p$ a prime
number, $p\neq 3,7$. In [4], we gave a method to determine all braces of size
$np$ from the braces of size $n$ and certain classes of morphisms from the
multiplicative group of these braces of size $n$ to $\mathbf{Z}_{p}^{*}$. For
a prime number $p\geq 7$, we applied the obtained results to classify
classical left braces of size $12p$. In particular we proved the conjectures
in [1] on the numbers $b(8p),b(12p)$ of isomorphism classes of classical left
braces of size $8p$ and $12p$, respectively.
Our aim in this paper is to generalize the method obtained in [4] to skew
nonclassical left braces of size $np$, where $p$ is a prime number and $n$ an
integer not divisible by $p$ and such that every group of order $np$ has a
normal subgroup of order $p$. We shall apply our method to classify skew left
braces of size $12p$, for $p$ a prime number, $p\geq 7$. In particular we will
prove the following conjecture in [1] on the number $s(12p)$ of isomorphism
classes of skew left braces of size $12p$, $p\geq 7$.
$s(12p)=\left\\{\begin{array}[]{ll}324&\text{\ if }p\equiv 11\pmod{12},\\\
410&\text{\ if }p\equiv 5\pmod{12},\\\ 606&\text{\ if }p\equiv 7\pmod{12},\\\
782&\text{\ if }p\equiv 1\pmod{12}.\end{array}\right.$ (1)
We note that $s(24)=855,s(36)=400$ and $s(60)=418$ (see [9]).
## 2 Skew braces of size $np$
We recall the definition of direct and semidirect product of braces as defined
in [2] and [8].
Let $A$ and $B$ be skew left braces. Then $A\times B$ together with
$(a,b)\cdot(a^{\prime},b^{\prime})=(a\cdot a^{\prime},b\cdot
b^{\prime})\quad(a,b)\circ(a^{\prime},b^{\prime})=(a\circ a^{\prime},b\circ
b^{\prime})$
is a skew left brace called the direct product of the braces $A$ and $B$.
Now, let $\tau:(B,\circ)\to\operatorname{Aut}(A,\cdot,\circ)$ be a group
homomorphism. Consider in $A\times B$ the additive structure of the direct
product $(A,\cdot)\times(B,\cdot)$
$(a,b)\cdot(a^{\prime},b^{\prime})=(a\cdot a^{\prime},b\cdot b^{\prime})$
and the multiplicative structure of the semidirect product
$(A,\circ)\rtimes_{\tau}(B,\circ)$
$(a,b)\circ(a^{\prime},b^{\prime})=(a\circ\tau(b)(a^{\prime}),b\circ
b^{\prime})$
Then, we obtain a skew left brace, which is called the semidirect product of
the skew left braces $A$ and $B$ via $\tau$.
We observe in particular that the additive group of the semidirect product of
two left braces is the direct product of the corresponding additive groups. In
[4] we proved that a classical brace of size $np$, where $p$ is a prime number
and $n$ an integer not divisible by $p$ and such that every group of order
$np$ has a normal subgroup of order $p$, is a direct or semidirect product of
the trivial brace of size $p$ and a brace of size $n$. In this case, the
additive group was a direct product of $C_{p}$ and an abelian group of order
$n$. However we are now interested in skew braces of size $np$, for which both
additive and multiplicative groups may be strictly semidirect products of
$C_{p}$ and a group of order $n$. We want then to determine the relationship
between these skew braces of order $np$ and skew braces of order $n$. To this
end we shall define the double semidirect product of two skew braces.
Let us recall first that for a skew left brace $(A,\cdot,\circ)$ and each
$a\in A$, the map
$\begin{array}[]{llll}\lambda_{a}:&A&\rightarrow&A\\\ &b&\mapsto&a^{-1}\cdot
a\circ b\end{array}$
is bijective with inverse
$\begin{array}[]{llll}\lambda_{a}^{-1}:&A&\rightarrow&A\\\
&b&\mapsto&\overline{a}\circ(a\cdot b)\end{array},$
where $\overline{a}$ denotes the inverse of $a$ with respect to $\circ$. It
follows that $a\circ b=a\cdot\lambda_{a}(b),a\cdot
b=a\circ\lambda_{a}^{-1}(b)$.
###### Proposition 1.
Let $A$ and $B$ be skew left braces and let
$\sigma:(B,\cdot)\to\operatorname{Aut}(A,\cdot,\circ),\quad\tau:(B,\circ)\to\operatorname{Aut}(A,\cdot,\circ)$
be morphisms of groups. Consider in $A\times B$ the additive structure of the
semidirect product $(A,\cdot)\rtimes_{\sigma}(B,\cdot)$
$(a,b)\cdot(a^{\prime},b^{\prime})=(a\cdot\sigma(b)(a^{\prime}),b\cdot
b^{\prime})$
and the multiplicative structure of the semidirect product
$(A,\circ)\rtimes_{\tau}(B,\circ)$
$(a,b)\circ(a^{\prime},b^{\prime})=(a\circ\tau(b)(a^{\prime}),b\circ
b^{\prime})$
We assume that the following equality holds for any $a\in A$ and all
$b_{1},b_{2}\in B$
$\sigma(b_{1}\circ b_{2}\cdot
b_{1}^{-1})\lambda_{a}=\lambda_{a}\tau(b_{1})\sigma(b_{2})\tau(\overline{b_{1}}).$
(2)
Then $(A\times B,\cdot,\circ)$ is a skew brace.
###### Proof.
We check the brace condition for $(A\times B,\cdot,\circ)$, i.e., the equality
$(a_{1},b_{1})\circ\left((a_{2},b_{2})\cdot(a_{3},b_{3})\right)=(a_{1},b_{1})\circ(a_{2},b_{2})\cdot(a_{1},b_{1})^{-1}\cdot(a_{1},b_{1})\circ(a_{3},b_{3})$
for $(a_{1},b_{1}),(a_{2},b_{2}),(a_{3},b_{3})\in A\times B$. The term on the
left is
$\begin{array}[]{lll}(a_{1},b_{1})\circ\left((a_{2},b_{2})\cdot(a_{3},b_{3})\right)&=&(a_{1},b_{1})\circ(a_{2}\cdot\sigma(b_{2})(a_{3}),b_{2}\cdot
b_{3})\\\
&=&(a_{1}\circ\tau(b_{1})(a_{2}\cdot\sigma(b_{2})(a_{3})),b_{1}\circ(b_{2}\cdot
b_{3}))\end{array}$
and the term on the right is
$\begin{array}[]{l}(a_{1},b_{1})\circ(a_{2},b_{2})\cdot(a_{1},b_{1})^{-1}\cdot(a_{1},b_{1})\circ(a_{3},b_{3})\\\
=(a_{1}\circ\tau(b_{1})(a_{2}),b_{1}\circ
b_{2})\cdot(\sigma(b_{1}^{-1})(a_{1}^{-1}),b_{1}^{-1})\cdot(a_{1}\circ\tau(b_{1})(a_{3}),b_{1}\circ
b_{3})\\\ =(a_{1}\circ\tau(b_{1})(a_{2})\cdot\sigma(b_{1}\circ
b_{2})((\sigma(b_{1}^{-1})(a_{1}^{-1})),b_{1}\circ b_{2}\cdot
b_{1}^{-1})\cdot(a_{1}\circ\tau(b_{1})(a_{3}),b_{1}\circ b_{3})\\\
=(a_{1}\circ\tau(b_{1})(a_{2})\cdot\sigma(b_{1}\circ b_{2}\cdot
b_{1}^{-1})(a_{1}^{-1})\cdot\sigma(b_{1}\circ b_{2}\cdot
b_{1}^{-1})(a_{1}\circ\tau(b_{1})(a_{3})),b_{1}\circ b_{2}\cdot
b_{1}^{-1}\cdot b_{1}\circ b_{3})\end{array}$
Since $B$ is a brace, the two second components are equal. Now, the first
component of the term on the left is
$\begin{array}[]{lll}a_{1}\circ\tau(b_{1})(a_{2}\cdot\sigma(b_{2})(a_{3}))&=&a_{1}\circ\left(\tau(b_{1})(a_{2})\cdot\tau(b_{1})(\sigma(b_{2})(a_{3})\right)\\\
&=&a_{1}\circ\tau(b_{1})(a_{2})\cdot a_{1}^{-1}\cdot
a_{1}\circ\tau(b_{1})(\sigma(b_{2})(a_{3}))\\\
&=&a_{1}\circ\tau(b_{1})(a_{2})\cdot\lambda_{a_{1}}(\tau(b_{1})(\sigma(b_{2})(a_{3})))\end{array}$
where the second equality follows from the brace condition for $A$. The first
component of the term on the right is
$\begin{array}[]{l}a_{1}\circ\tau(b_{1})(a_{2})\cdot\sigma(b_{1}\circ
b_{2}\cdot b_{1}^{-1})(a_{1}^{-1})\cdot\sigma(b_{1}\circ b_{2}\cdot
b_{1}^{-1})(a_{1}\circ\tau(b_{1})(a_{3}))\\\
=a_{1}\circ\tau(b_{1})(a_{2})\cdot\sigma(b_{1}\circ b_{2}\cdot
b_{1}^{-1})(a_{1}^{-1}\cdot a_{1}\circ\tau(b_{1})(a_{3}))\\\
=a_{1}\circ\tau(b_{1})(a_{2})\cdot\sigma(b_{1}\circ b_{2}\cdot
b_{1}^{-1})(\lambda_{a_{1}}(\tau(b_{1})(a_{3}))).\end{array}$
The equality of the first components is then equivalent to
$\lambda_{a_{1}}(\tau(b_{1})(\sigma(b_{2})(a_{3})))=\sigma(b_{1}\circ
b_{2}\cdot b_{1}^{-1})(\lambda_{a_{1}}(\tau(b_{1})(a_{3})))$
for all $a_{3}\in A$, hence to equality (2). ∎
###### Definition 2.
We call double semidirect product of the skew braces $A$ and $B$ via $\sigma$
and $\tau$ the skew brace $(A\times B,\cdot,\circ)$ in Proposition 1.
###### Remarks 3.
1. 1)
If $\sigma$ is the trivial morphism, then (2) is satisfied and we recover the
notion of semidirect product of skew braces.
2. 2)
If $A$ is a trivial brace and $\operatorname{Aut}A$ is abelian, then (2) is
equivalent to $\sigma(b_{1}\circ b_{2})=\sigma(b_{1})\sigma(b_{2})$ for all
$b_{1},b_{2}\in B$, i.e. to $\sigma$ being a group morphism with respect to
$\circ$.
Turning back to skew braces of size $np$, we establish the following
hypothesis on $n$ and $p$ which will be maintained all along the paper.
###### Hypothesis.
Let $p$ be an odd prime and $n$ an integer such that $p$ does not divide $n$
and each group of order $np$ has a normal subgroup of order $p$.
In particular, we may consider $n=12$ and $p\geq 7$, which satisfy this
hypothesis.
###### Proposition 4.
Any skew brace of size $np$ is a double semidirect product of the trivial
brace of size $p$ and a skew brace of size $n$.
###### Proof.
Let $(B_{np},\cdot,\circ)$ be a left brace of size $np$ with additive group
$N$ and multiplicative group $G$. Then, by the Schur-Zassenhaus theorem,
$N=\mathbf{Z}_{p}\rtimes_{\sigma}E$ with $(E,\cdot)$ a group of order $n$ and
$\sigma:E\to\operatorname{Aut}(\mathbf{Z}_{p})$ a group morphism and
$G=\mathbf{Z}_{p}\rtimes_{\tau}F$ with $(F,\circ)$ a group of order $n$ and
$\tau:F\to\operatorname{Aut}(\mathbf{Z}_{p})$ a group morphism. We note that
$\operatorname{Aut}(\mathbf{Z}_{p})$ is the group of automorphisms of the
trivial brace of size $p$, which is the unique brace of size $p$. We write
$\mathbf{Z}_{p}$ additively and identify $\operatorname{Aut}(\mathbf{Z}_{p})$
with $\mathbf{Z}_{p}^{*}$. As a set, $B_{np}$ is the cartesian product of
$\mathbf{Z}_{p}$ and a set $B_{n}$ with $n$ elements endowed with two
operations $\cdot$ and $\circ$ such that $(B_{n},\cdot)$ and $(B_{n},\circ)$
are groups isomorphic to $(E,\cdot)$ and $(F,\circ)$, respectively.
For $(a,b),(a^{\prime},b^{\prime})\in B_{np}$, we have then
$\begin{array}[]{rcl}(a,b)\cdot(a^{\prime},b^{\prime})&=&(a+\sigma(b)a^{\prime},bb^{\prime})\\\
(a,b)\circ(a^{\prime},b^{\prime})&=&(a+\tau(b)a^{\prime},b\circ
b^{\prime})\end{array}.$
We write the brace relation for $B_{np}$
$(a,b)\circ\left((a^{\prime},b^{\prime})\cdot(a^{\prime\prime},b^{\prime\prime})\right)=(a,b)\circ(a^{\prime},b^{\prime})\cdot(a,b)^{-1}\cdot(a,b)\circ(a^{\prime\prime},b^{\prime\prime}).$
(3)
The lefthand side is
$\begin{array}[]{rcl}(a,b)\circ\left((a^{\prime},b^{\prime})\cdot(a^{\prime\prime},b^{\prime\prime})\right)&=&(a,b)\circ\left(a^{\prime}+\sigma(b^{\prime})a^{\prime\prime},b^{\prime}\cdot
b^{\prime\prime}\right)\\\
&=&\left(a+\tau(b)(a^{\prime}+\sigma(b^{\prime})a^{\prime\prime}),b\circ(b^{\prime}\cdot
b^{\prime\prime})\right)\end{array}$
and the righthand side is
$\begin{array}[]{l}(a,b)\circ(a^{\prime},b^{\prime})\cdot(a,b)^{-1}\cdot(a,b)\circ(a^{\prime\prime},b^{\prime\prime})\\\
=(a+\tau(b)a^{\prime},b\circ
b^{\prime})\cdot(-\sigma(b^{-1})a,b^{-1})\cdot(a+\tau(b)a^{\prime\prime},b\circ
b^{\prime\prime})\\\ =(a+\tau(b)a^{\prime}-\sigma(b\circ
b^{\prime})\sigma(b^{-1})a,(b\circ b^{\prime})\cdot
b^{-1})\cdot(a+\tau(b)a^{\prime\prime},b\circ b^{\prime\prime})\\\
=(a+\tau(b)a^{\prime}-\sigma(b\circ b^{\prime})\sigma(b^{-1})a+\sigma((b\circ
b^{\prime})\cdot b^{-1})(a+\tau(b)a^{\prime\prime}),(b\circ b^{\prime})\cdot
b^{-1}\cdot(b\circ b^{\prime\prime})).\end{array}$
The equality of the second components gives a brace $B_{n}$ of size $n$ with
additive group $E$ and multiplicative group $F$. The first component in the
lefthand side is
$a+\tau(b)(a^{\prime}+\sigma(b^{\prime})a^{\prime\prime})=a+\tau(b)a^{\prime}+\tau(b)\sigma(b^{\prime})a^{\prime\prime}$
and the first component in the righthand side is
$\begin{array}[]{l}a+\tau(b)a^{\prime}-\sigma(b\circ
b^{\prime})\sigma(b^{-1})a+\sigma((b\circ b^{\prime})\cdot
b^{-1})(a+\tau(b)a^{\prime\prime})\\\ =a+\tau(b)a^{\prime}-\sigma((b\circ
b^{\prime})\cdot b^{-1})a+\sigma((b\circ b^{\prime})\cdot
b^{-1})a+\sigma((b\circ b^{\prime})\cdot b^{-1})\tau(b)a^{\prime\prime}\\\
=a+\tau(b)a^{\prime}+\sigma((b\circ b^{\prime})\cdot
b^{-1})\tau(b)a^{\prime\prime}\end{array}$
The equality of the first components is then equivalent to
$\tau(b)\sigma(b^{\prime})a^{\prime\prime}=\sigma((b\circ b^{\prime})\cdot
b^{-1})\tau(b)a^{\prime\prime}$
for all $a^{\prime\prime}\in\mathbf{Z}_{p}$, hence to
$\sigma(b^{\prime})=\sigma((b\circ b^{\prime})\cdot b^{-1})=\sigma(b\circ
b^{\prime})\sigma(b^{-1})$
which is equivalent to
$\sigma(b\circ b^{\prime})=\sigma(b)\sigma(b^{\prime}),$ (4)
for all $b,b^{\prime}\in B_{n}$, hence to $\sigma$ being also a morphism with
respect to $\circ$.
We have then obtained that $B_{np}$ is the double semidirect product of the
trivial brace $\mathbf{Z}_{p}$ and the skew brace $B_{n}$. ∎
We recall now precisely the correspondence between braces and regular
subgroups of the holomorph given by Guarnieri and Vendramin. We note that, if
$G$ is a regular subgroup of $\mathrm{Hol}(A,\cdot)$, then
$\begin{array}[]{lcll}{\pi_{1}}_{|G}:&G&\rightarrow&A\\\
&(a,f)&\mapsto&a\end{array}$
is bijective.
###### Proposition 5 ([6] th. 4.2).
Let $A$ be a skew left brace. Then
$\\{(a,\lambda_{a})\,:\,a\in A\\}$
is a regular subgroup of $\mathrm{Hol}(A,\cdot)$, isomorphic to $(A,\circ)$.
Conversely, if $(A,\cdot)$ is a group and $G$ is a regular subgroup of
$\mathrm{Hol}(A,\cdot)$, then $A$ is a skew left brace with $(A,\circ)\simeq
G$, where
$a\circ b=a\cdot f(b),$
and $({\pi_{1}}_{|G})^{-1}(a)=(a,f)\in G$.
These assignments give a bijective correspondence between the set of
isomorphism classes of skew left braces $(A,\cdot,\circ)$ and the set of
conjugation classes of regular subgroups of $\mathrm{Hol}(A,\cdot)$.
Using this correspondence we can see how braces of size $np$ give braces of
size $n$ and how to obtain braces of size $np$ from the trivial brace of size
of size $p$ and braces of size $n$.
Let $(B_{np},\cdot,\circ)$ be a brace of size $np$ with additive group
$N=\mathbf{Z}_{p}\rtimes_{\sigma}E$, as above, and multiplicative group
$G=\mathbf{Z}_{p}\rtimes_{\tau}F$. By applying Proposition 5, we obtain that
$\\{(x,\lambda_{x})\,:\,x\in N\\}$ is a regular subgroup of $\mathrm{Hol}(N)$,
isomorphic to $G$. Now, for $x=(m,a),y=(n,b)\in N$, we have
$\lambda_{x}(y)=x^{-1}\cdot(x\circ
y)=(-\sigma(a^{-1})m,a^{-1})\cdot(m+\tau(a)n,a\circ
b)=(\sigma(a^{-1})\tau(a)n,\lambda_{a}(b)),$
hence $\pi_{2}\lambda_{x}=\lambda_{a}\pi_{2}$, where
$\pi_{2}:N=\mathbf{Z}_{p}\rtimes_{\sigma}E\rightarrow E$ is the projection on
the second component.
Let us see these regular groups explicitly in $\mathrm{Hol}(N)$. Following
[5], the automorphisms of $N$ can be described as matrices
$\begin{pmatrix}k&\gamma\\\ 0&\lambda\end{pmatrix}$
with $k\in\mathbf{Z}_{p}^{*}=\operatorname{Aut}(\mathbf{Z}_{p})$,
$\lambda\in\operatorname{Aut}(E)$, such that $\sigma\lambda=\sigma$ and
$\gamma:E\to\mathbf{Z}_{p}$ a $1$-cocycle. The action on $N$ is given by
$\begin{pmatrix}k&\gamma\\\ 0&\lambda\end{pmatrix}\begin{pmatrix}m\\\
a\end{pmatrix}=\begin{pmatrix}km+\gamma(a)\\\ \lambda(a)\end{pmatrix}.$
Under our hypothesis, Schur Zassenhaus condition on subgroups of order $|E|$
being conjugates of $E$ by elements in $\mathbf{Z}_{p}$ leaves us with the
coboundaries
$\gamma_{i}(a)=i-\sigma(a)i,\quad i\in\mathbf{Z}_{p}.$
For a trivial $\sigma$, we get the direct product
$\operatorname{Aut}(\mathbf{Z}_{p})\times\operatorname{Aut}(E)$ and for a non
trivial $\sigma$ the group $\operatorname{Aut}(N)$ has order $p(p-1)s$ with
$s=\\#\\{\lambda\in\operatorname{Aut}(E):\sigma\lambda=\sigma\\}$. This
subgroup is the stabilizer $\Sigma_{\sigma}$ of $\sigma$ under the action
$\begin{array}[]{ccl}\operatorname{Aut}(E)\times\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})&\longrightarrow&\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})\\\
(g,\sigma)&\longrightarrow&\sigma g\end{array}.$
The orbit of $\sigma$ gives the $\sigma^{\prime}$ such that
$\mathbf{Z}_{p}\rtimes_{\sigma^{\prime}}E\simeq N$.
All together, if we write
$\operatorname{Aut}(\mathbf{Z}_{p}\rtimes_{\sigma}E)=\left\\{\begin{pmatrix}k&\gamma_{i}\\\
0&\lambda\end{pmatrix}\qquad k\in\mathbf{Z}_{p}^{*},\ i\in\mathbf{Z}_{p},\
\lambda\in\Sigma_{\sigma}\right\\}$
then
$\mathrm{Hol}(\mathbf{Z}_{p}\rtimes_{\sigma}E)=\left\\{[\begin{pmatrix}m\\\
a\end{pmatrix},M]\qquad m\in\mathbf{Z}_{p},\ a\in
E,M\in\operatorname{Aut}(\mathbf{Z}_{p}\rtimes_{\sigma}E)\right\\}$
with
$[u,M_{u}][v,M_{v}]=[u\rtimes M_{u}v,M_{u}M_{v}]$
where $\rtimes$ indicates operation in $\mathbf{Z}_{p}\rtimes_{\sigma}E$. For
a brace $B_{np}$ as before, this is the $\cdot$ operation. The set
$\\{(0,a)\mid a\in E\\}$ is stable under this operation. For the
multiplicative group $G=\\{(x,\lambda_{x})\mid\ x\in N\\}$, if $x=(m,a)$ we
have
$\lambda_{x}=\begin{pmatrix}\sigma(a)^{-1}\tau(a)&0\\\
0&\lambda_{a}\end{pmatrix}$
The set $\\{(a,\lambda_{a})\mid a\in E\\}$ is a group isomorphic to $F$, and a
regular subgroup of $\mathrm{Hol}(E)$. This describes the subbbrace $B_{n}$ of
$B_{np}$ with additive structure isomorphic to $E$ and multiplicative
structure isomorphic to $F$. Notice that in the description of automorphisms
of $N$ we got the condition $\sigma\in\mathrm{Hom}(B_{n},\mathbf{Z}_{p}^{*})$,
namely $\sigma\lambda_{a}=\sigma$ for all $a\in E$.
From the above, we have also obtained a way to construct a brace of size $np$
from a brace of size $n$. If $E$ is a group of order $n$,
$\sigma\in\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})$ and $F$ is a regular subgroup of
$\mathrm{Hol}(E)$ such that $\pi_{2}(F)$ is inside the stabilizer of $\sigma$
by the action of $\operatorname{Aut}(E)$, then for any
$\tau\in\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$
$G(\sigma,\tau)=\left\\{[\begin{pmatrix}m\\\
a\end{pmatrix},\begin{pmatrix}\sigma(a)^{-1}\tau(a,\lambda_{a})&0\\\
0&\lambda_{a}\end{pmatrix}]\quad m\in\mathbf{Z}_{p},a\in E\right\\}.$
is a regular subgroup of $\mathrm{Hol}(\mathbf{Z}_{p}\rtimes_{\sigma}E)$
isomorphic to $\mathbf{Z}_{p}\rtimes_{\tau}F.$
But we can also consider the opposite group operation on $N$, so that elements
in $N$ are $(\sigma(a)m,a)$, with $m\in\mathbf{Z}_{p}$ and $a\in E$. Then,
$\begin{array}[]{ccl}N&\longrightarrow&\mathrm{Hol}(N)\\\
(m,a)&\to&[\begin{pmatrix}\sigma(a)m\\\
a\end{pmatrix},\begin{pmatrix}\sigma(a)&\gamma_{-m}\\\
0&\lambda_{a}\end{pmatrix}]\end{array}$
is a regular embedding which restricted to $E$ provides an embedding of $F$:
$[\begin{pmatrix}\sigma(a)m\\\
a\end{pmatrix},\begin{pmatrix}\sigma(a)&\gamma_{-m}\\\
0&\lambda_{a}\end{pmatrix}][\begin{pmatrix}\sigma(b)n\\\
b\end{pmatrix},\begin{pmatrix}\sigma(b)&\gamma_{-n}\\\
0&\lambda_{b}\end{pmatrix}]=$ $=[\begin{pmatrix}\sigma(ab)(m+\sigma(a)n)\\\
a\lambda_{a}(b)\end{pmatrix},\begin{pmatrix}\sigma(ab)&\gamma_{-(m+\sigma(a)n)}\\\
0&\lambda_{a}\lambda_{b}\end{pmatrix}].$
Using again [6] we get the regular subgroup
$G(\sigma,\tau)^{\prime}=\left\\{[\begin{pmatrix}\sigma(a)m\\\
a\end{pmatrix},\begin{pmatrix}\tau(a,\lambda_{a})&\gamma_{-m}\\\
0&\lambda_{a}\end{pmatrix}]\quad m\in\mathbf{Z}_{p},a\in E\right\\}$
isomorphic to $\mathbf{Z}_{p}\rtimes_{\tau}F$.
###### Proposition 6.
Let $p$ be a prime and $n$ a positive integer not divisible by $p$ and such
that every group of order $np$ has a normal subgroup of order $p$. Let $B_{n}$
a brace of order $n$ with additive structure $(E,\cdot)$ and multiplicative
structure $(F,\circ)$. For every
$\sigma\in\mathrm{Hom}(B_{n},\mathbf{Z}_{p}^{*})$ and every
$\tau\in\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$,
$G(\sigma,\tau)=\left\\{[\begin{pmatrix}m\\\
a\end{pmatrix},\begin{pmatrix}\sigma(a)^{-1}\tau(a,\lambda_{a})&0\\\ 0&\
\lambda_{a}\end{pmatrix}]\quad m\in\mathbf{Z}_{p},(a,\lambda_{a})\in
F\right\\}$ $G(\sigma,\tau)^{\prime}=\left\\{[\begin{pmatrix}\sigma(a)m\\\
a\end{pmatrix},\begin{pmatrix}\tau(a,\lambda)&\gamma_{-m}\\\
0&\lambda_{a}\end{pmatrix}]\quad m\in\mathbf{Z}_{p},(a,\lambda_{a})\in
F\right\\}$
are regular subgroups of $\mathrm{Hol}(\mathbf{Z}_{p}\rtimes_{\sigma}E)$
isomorphic to $\mathbf{Z}_{p}\rtimes_{\tau}F$.
We have $G(1,\tau)=G(1,\tau)^{\prime}$ for any $\tau$. For a nontrivial
$\sigma$, $G(\sigma,\tau)$ and $G(\sigma,\tau)^{\prime}$ are not conjugate in
$\mathrm{Hol}(N)$.
Any regular subgroup of $\mathrm{Hol}(\mathbf{Z}_{p}\rtimes_{\sigma}N)$
isomorphic to $\mathbf{Z}_{p}\rtimes_{\tau}F$ is conjugate to $G(\sigma,\tau)$
or $G(\sigma,\tau)^{\prime}$ by an element of $\operatorname{Aut}(N)$.
When $\sigma=1$, the coboundaries are also trivial, and both subgroups are
equal. For a nontrivial $\sigma$, since $\mathbf{Z}_{p}^{*}$ is abelian, the
first coefficient of the matrix is invariant under conjugation, and this shows
that the groups are not conjugate in $\mathrm{Hol}(N)$ for any $\tau$.
Let us prove the last statement. If $G\subseteq\mathrm{Hol}(N)$ is a regular
subgroup isomorphic to $\mathbf{Z}_{p}\rtimes_{\tau}F$, then via
$\mathbf{Z}_{p}\rtimes_{\tau}F\simeq G\hookrightarrow\mathrm{Hol}(N)$, using
[6] and $\mathrm{Hom}(\mathbf{Z}_{p},\mathbf{Z}_{p}^{*})=1$, we get a
semiregular embedding of $\mathbf{Z}_{p}$ in $\mathrm{Hol}(N)$,
$\ \begin{array}[]{rcl}\quad\mathbf{Z}_{p}&\longrightarrow&\mathrm{Hol}(N)\\\
m&\to&[\begin{pmatrix}f(m)\\\ 1\end{pmatrix},\begin{pmatrix}1&\gamma_{h(m)}\\\
0&1\end{pmatrix}]\\\ \end{array}.$
so that $f\in\operatorname{Aut}(\mathbf{Z}_{p})$ and
$h\in\mathrm{Hom}(\mathbf{Z}_{p},\mathbf{Z}_{p})$, and the embedding is
$\
\begin{array}[]{rcl}\nu_{k,l}:\quad\mathbf{Z}_{p}&\longrightarrow&\mathrm{Hol}(N)\\\
m&\to&[\begin{pmatrix}km\\\ 1\end{pmatrix},\begin{pmatrix}1&\gamma_{lm}\\\
0&1\end{pmatrix}]\\\ \end{array}$
for some $k\in\mathbf{Z}_{p}^{*},\ l\in\mathbf{Z}_{p}$. It is easy to check
that conjugation by elements in $\operatorname{Aut}(N)$ splits this family
into two classes represented by $\nu_{1,0}$ and $\nu_{1,l}$ for any
$l\in\mathbf{Z}_{p}^{*}$, and that conjugation is given by an element in
$\operatorname{Aut}(\mathbf{Z}_{p})$.
On the other hand, we get a semiregular embedding of $F$ in $\mathrm{Hol}(N)$.
By [6] and using $\mathrm{Hom}(E,\mathbf{Z}_{p})=1$, it has to be
$\begin{array}[]{rcl}\Psi_{\alpha}:\quad F&\longrightarrow&\mathrm{Hol}(N)\\\
(a,\lambda_{a})&\to&[\begin{pmatrix}0\\\
a\end{pmatrix},\begin{pmatrix}\alpha(a)&0\\\ 0&\lambda_{a}\end{pmatrix}]\\\
\end{array}$
with $\alpha\in\mathrm{Hom}(B_{n},\mathbf{Z}_{p}^{*})$. The conjugacy class by
elements of $\operatorname{Aut}(N)$ of the image of $\Psi_{\alpha}$ is
$[\begin{pmatrix}\gamma_{i_{M}}(a)\\\
\lambda_{M}(a)\end{pmatrix},\begin{pmatrix}\alpha(a)&\gamma_{i_{M}-\alpha(a)i_{M}}\\\
0&\lambda_{M}\lambda_{a}\lambda_{M}^{-1}\end{pmatrix}]$
so that $\Psi_{\alpha}(F)$ and $\Psi_{\alpha^{\prime}}(F)$ are not conjugate
when $\alpha\neq\alpha^{\prime}$. Notice that
$\\{(\lambda_{M}(x),\lambda_{M}\lambda_{a}\lambda_{M}^{-1})\mid(a,\lambda_{a})\in
F\\}=\Phi_{\lambda_{M}}(F).$ It is equal to $F$ if and only if $\lambda_{M}$
is a brace automorphism. Notice also that the image is invariant under
conjugation by $\operatorname{Aut}(\mathbf{Z}_{p})$. This gives the following
result.
###### Proposition 7.
Let $F,F^{\prime}$ be regular subgroups of $\mathrm{Hol}(E)$ giving
multiplicative structures for a brace of order $n$. If $\Psi_{\alpha}(F)$ and
$\Psi_{\alpha}(F^{\prime})$ are conjugate in $\mathrm{Hol}(N)$ then they are
conjugate in $\mathrm{Hol}(E)$.
Back to the proof of the previous proposition, up to conjugation by elements
in $\operatorname{Aut}(\mathbf{Z}_{p})$, the group $G$ has to be the
semidirect product of $\Psi_{\alpha}(F)$ and either
$\nu_{1,0}(\mathbf{Z}_{p})$ or $\nu_{1,l}(\mathbf{Z}_{p})$. An easy
computation shows that the only possibilities are those corresponding to
$G(\sigma,\tau)$ and $G(\sigma,\tau)^{\prime}$, namely $\nu_{1,0}$ and
$\alpha(a)=\tau(a,\lambda_{a})\sigma(a)^{-1}$ or $\nu_{1,-1}$ and
$\alpha(a)=\tau(a,\lambda_{a})$. Indeed,
$[\begin{pmatrix}m\\\ a\end{pmatrix},\begin{pmatrix}\alpha(a)&0\\\
0&\lambda_{a}\end{pmatrix}][\begin{pmatrix}n\\\
b\end{pmatrix},\begin{pmatrix}\alpha(b)&0\\\
0&\lambda_{b}\end{pmatrix}]=[\begin{pmatrix}m+\sigma(a)\alpha(a)n\\\
a\lambda_{a}(b)\end{pmatrix}\begin{pmatrix}\alpha(ab)&0\\\
0&\lambda_{a}\lambda_{b}\end{pmatrix}]$
gives the image of the product
$(m+\tau(a,\lambda_{a})n,(a\lambda_{a}(b),\lambda_{a}\lambda_{b}))$ if and
only if $\alpha(a)=\tau(a,\lambda_{a})\sigma(a)^{-1}$, and
$[\begin{pmatrix}\sigma(a)m\\\
a\end{pmatrix},\begin{pmatrix}\alpha(a)&\gamma_{lm}\\\
0&\lambda_{a}\end{pmatrix}][\begin{pmatrix}\sigma(b)n\\\
b\end{pmatrix},\begin{pmatrix}\alpha(b)&\gamma_{ln}\\\
0&\lambda_{b}\end{pmatrix}]=$
$=[\begin{pmatrix}\sigma(a)(m+lm)+\sigma(ab)(-lm+\alpha(a)n)\\\
a\lambda_{a}(b)\end{pmatrix}\begin{pmatrix}\alpha(ab)&\gamma_{l(m+\alpha(a)n)}\\\
0&\lambda_{a}\lambda_{b}\end{pmatrix}]$
gives the image of the product if and only if $l=-1$ and
$\alpha(a)=\tau(a,\lambda_{a})$.
In order to count different brace structures, we have to determine conjugation
orbits in the families $\\{G(\sigma,\tau)\\}_{\sigma,\tau},$ and
$\\{G(\sigma,\tau)^{\prime}\\}_{\sigma,\tau}$ or equivalently, isomorphism
classes of braces structures. Therefore, we should take $\sigma$ up to brace
automorphisms of $(B_{n},\cdot,\circ)$ and $\tau$ up to brace automorphisms of
$(B_{n},\circ)$.
### 2.1 Algorithm
Our aim is to determine all skew braces of size $np$ from skew braces of size
$n$. Therefore we need a precomputation
Step 0
Determine isomorphism classes of groups $E$ of order $n$ and the number of
braces of size $n$ of each type $(E,F)$, identifying $F$ with a representative
for a conjugacy class of regular subgroups of $\mathrm{Hol}(E)$.
For every $E$ compute the stabilisers $\Sigma_{\sigma}$ for the action
$\begin{array}[]{ccc}\operatorname{Aut}(E)\times\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})&\longrightarrow&\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})\\\
(g,\sigma)&\longrightarrow&\sigma g\end{array}$
namely, $\Sigma_{\sigma}=\\{g\in\operatorname{Aut}(E):\ \sigma g=\sigma\\}$.
Next, we consider as an input a pair $(E,F)$, namely a brace $B_{n}$. The
steps to count all the braces of size $np$ having additive group isomorphic to
$\mathbf{Z}_{p}\rtimes E$ and multiplicative group isomorphic to
$\mathbf{Z}_{p}\rtimes F$ are the following:
Step 1
Determine
$\mathrm{Hom}(B_{n},\mathbf{Z}_{p}^{*})=\\{\sigma\in\mathrm{Hom}(E,\mathbf{Z}_{p}^{*}):\
\pi_{2}(F)\subseteq\Sigma_{\sigma}\ \\}$
Step 2
Determine $\operatorname{Aut}(B_{n})=\\{g\in\operatorname{Aut}(E):\
\Phi_{g}(F)=F\ \\}$
Step 3
Compute the number of orbits of the action
$\begin{array}[]{ccc}\operatorname{Aut}(B_{n})\times\mathrm{Hom}(B_{n},\mathbf{Z}_{p}^{*})&\longrightarrow&\mathrm{Hom}(B_{n},\mathbf{Z}_{p}^{*})\\\
(g,\sigma)&\longrightarrow&\sigma g\end{array}$
Notice that the orbit of $\sigma=1$, which corresponds to the direct product,
has a single element. The remaining orbits will give rise to two different
braces.
The number of additive structures for $F$ is the number of orbits. Since
orbits preserve order and isomorphism class of kernel, we can keep track of
the number of structures according to these parameters.
Step 4
For each $\sigma$ in a system of representatives of the above orbits, compute
orbits of the action
$\begin{array}[]{ccc}(\operatorname{Aut}(B_{n})\cap\Sigma_{\sigma})\times\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})&\longrightarrow&\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})\\\
(g,\tau)&\longrightarrow&\tau\Phi_{g}\end{array}$
Notice that $\operatorname{Aut}(B_{n})\cap\Sigma_{\sigma}$ is a subgroup of
the normaliser of $\pi_{2}(F)$ in $\Sigma_{\sigma}$.
The number of multiplicative structures for $F$ is twice the number of orbits
except for $\sigma$ trivial, when we get a single one. Since orbits preserve
order and isomorphism class of kernel, we can keep track of the number of
structures according to these parameters.
### 2.2 Number of braces of size $np$
Taking into account the description given in terms of orbits, using the
Burnside lemma we can give formulas for the number of different braces of size
$np$ in terms of the number of braces of size $n$.
###### Theorem 8.
Let $p$ be a prime and $n$ a positive integer not divisible by $p$ and such
that every group of order $np$ has a normal subgroup of order $p$. For braces
$B_{np}$ of size $np$ the number of additive structures is
${\sum}_{(B_{n},\cdot)}\,\dfrac{1}{|\operatorname{Aut}(B_{n})|}\sum_{\sigma\in\mathrm{Hom}(B_{n},\mathbf{Z}_{p}^{*})}|\mathrm{Stab}_{\operatorname{Aut}(B_{n})}(\sigma)|$
where $B_{n}$ runs over the braces of size $n$. The total amount of braces of
size $np$ is
${\sum}_{(B_{n},\cdot)}\,\left(\dfrac{1}{|A_{1}|}\sum_{\tau\in
H}|\mathrm{Stab}_{A_{1}}(\tau))|+2\sum_{\sigma\neq
1}\dfrac{1}{|A_{\sigma}|}\sum_{\tau\in
H}|\mathrm{Stab}_{A_{\sigma}}(\tau)|\right)$
where $H=\mathrm{Hom}((B_{n},\circ),\mathbf{Z}_{p}^{*})$,
$\sigma\in\mathrm{Hom}(B_{n},\mathbf{Z}_{p}^{*})$ runs over a system of
representatives for additive structures and
$A_{\sigma}=\operatorname{Aut}(B_{n})\cap\mathrm{Stab}_{\operatorname{Aut}(B_{n},\cdot)}(\sigma)$.
The case $\sigma=1$, where $A_{1}=\operatorname{Aut}(B_{n})$, corresponds to
the semidirect product of $B_{n}$ and the trivial brace of order $p$. The
remaining cases are double semidirect products of these two braces.
Equivalently, the number of additive structures can be expressed as
${\sum}_{(B_{n},\cdot)}\,\dfrac{1}{|\operatorname{Aut}(B_{n})|}\sum_{g\in\operatorname{Aut}(B_{n})}|\operatorname{Fix}_{\mathrm{Hom}(B_{n},\mathbf{Z}_{p}^{*})}(g)|$
and the total amount as
${\sum}_{(B_{n},\cdot)}\,\left(\dfrac{1}{|A_{1}|}\sum_{g\in
A_{1}}|\operatorname{Fix}_{H}(g)|+2\sum_{\sigma\neq
1}\dfrac{1}{|A_{\sigma}|}\sum_{g\in
A_{\sigma}}|\operatorname{Fix}_{H}(g)|\right),$
where $\operatorname{Fix}$ denotes the number of fixed points.
## 3 Groups of order 12 and skew braces of size 12
There are five groups of order 12, up to isomorphism, two abelian ones
$C_{12}$ and $C_{6}\times C_{2}$ and three non-abelian ones, the alternating
group $A_{4}$, the dihedral group $D_{2\cdot 6}$ and the dicyclic group
$\operatorname{Dic}_{12}$. By computation with Magma, we obtain that the
number of skew left braces with additive group $E$ and multiplicative group
$F$ is as shown in the following table.
$E\backslash F$ | $C_{12}$ | $C_{6}\times C_{2}$ | $A_{4}$ | $D_{2\cdot 6}$ | $\operatorname{Dic}_{12}$
---|---|---|---|---|---
$C_{12}$ | 1 | 1 | 0 | 2 | 1
$C_{6}\times C_{2}$ | 1 | 1 | 1 | 1 | 1
$A_{4}$ | 0 | 2 | 4 | 0 | 2
$D_{2\cdot 6}$ | 2 | 2 | 0 | 4 | 2
$\operatorname{Dic}_{12}$ | 2 | 2 | 0 | 4 | 2
We determine now for each group $E$ of order 12, its automorphism group
$\operatorname{Aut}E$, the group $\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})$ and
stabilizers of the corresponding group action.
Since $\mathbf{Z}_{p}^{*}=\langle\zeta\rangle$ is a cyclic group of order
$p-1$, for each $d\mid p-1$ it has a unique subgroup
$\mu_{d}=\langle\zeta^{\frac{p-1}{d}}\rangle$ of order $d$. We write
$\zeta_{d}$ for $\zeta^{\frac{p-1}{d}}$. In order to have elements in
$\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})$ with image $\mu_{d}$ it is necessary that
$d\mid\gcd(\exp(E),p-1)$, where $\exp(E)$ is the exponent of the group $E$.
### 3.1 $E=C_{12}$
Let $E=\langle c\rangle\simeq C_{12}$. Then, $\operatorname{Aut}E=\langle
g_{5}\rangle\times\langle
g_{7}\rangle\simeq\mathbf{Z}_{12}^{*}\simeq\mathbf{Z}_{2}\times\mathbf{Z}_{2}$
with $g_{i}(c)=c^{i}$. If $D=\gcd(12,p-1)$, then
$\begin{array}[]{ccc}\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})&\xrightarrow{\sim}&\mu_{D}=\langle\zeta_{D}\rangle\\\
\sigma&\to&\sigma(c)\end{array}.$
If $|\operatorname{Ker}\sigma|=k$, then the order of $\sigma$ in
$\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})$ is $d=\frac{12}{k}$. The Euler totient
function gives the number $\varphi(d)$ of $\sigma$ having order $d$. Such a
sigma can be identified as a pair $(d,j)$ meaning $\sigma(c)=\zeta_{d}^{j}$
for $j$ relatively prime to $d$. Elements of the same order are in the same
orbit under the action of $\operatorname{Aut}(E)$
$(d,j)g_{i}=(d,j^{\prime})\mbox{ for }i=j^{\prime}j^{-1}\bmod d$
and the size of the kernel parametrizes the orbits. As for stabilisers, since
$\sigma
g_{i}=\sigma\iff\sigma(g_{i}(c))=\sigma(c)\iff\sigma(c)^{i}=\sigma(c)\iff
i\equiv 1\bmod d,$
we have $\Sigma_{\sigma}=\\{g_{i}\in\operatorname{Aut}(E):i\equiv 1\bmod
d\\},$ which identifies as subgroup of $\mathbf{Z}_{12}^{*}=\\{1,5,7,11\\}$.
In the following table we compile all the information:
$k$ | $d$ | $\varphi(d)$ | Orbit of $\sigma=(d,j\bmod d)$ | $\Sigma_{\sigma}$
---|---|---|---|---
12 | 1 | 1 | 1 | $\operatorname{Aut}(E)$
6 | 2 | 1 | 1 | $\operatorname{Aut}(E)$
4 | 3 | 2 | $1\xrightarrow{5}2$ | $\\{1,7\\}$
3 | 4 | 2 | $1\xrightarrow{7}3$ | $\\{1,5\\}$
2 | 6 | 2 | $1\xrightarrow{5}5$ | $\\{1,7\\}$
1 | 12 | 4 | $1\xrightarrow{5}5\xrightarrow{11}7\xrightarrow{5}11$ | $\\{1\\}$
The number of isomorphism classes of semidirect products
$\mathbf{Z}_{p}\rtimes C_{12}$ is equal to the number of divisors of
$\gcd(12,p-1)$, namely $\sigma_{0}(D)$, and can be labelled by the size $k$ of
$\operatorname{Ker}\sigma$ (size 12 means direct product):
$p\bmod 12$ | $D$ | $\sigma_{0}(D)$ | $k$
---|---|---|---
1 | 12 | 6 | 12,6,4,3,2,1
5 | 4 | 3 | 12,6,3
7 | 6 | 4 | 12,6,4,2
11 | 2 | 2 | 12,6
### 3.2 $E=C_{6}\times C_{2}$
Let us write $E=\langle a\rangle\times\langle b\rangle$, with $a$ of order 6,
$b$ of order 2. The automorphism group is $\operatorname{Aut}(C_{6}\times
C_{2})\simeq D_{2\cdot 6}=\langle g_{1},\ g_{2}\rangle$, where
$\begin{array}[]{llll}g_{1}:&a&\mapsto&ab\\\
&b&\mapsto&b\end{array},\quad\begin{array}[]{llll}g_{2}:&a&\mapsto&a^{2}b\\\
&b&\mapsto&a^{3}b\end{array}$
$g_{1}$ has order 2, $g_{2}$ has order 6 and $g_{1}g_{2}g_{1}=g_{2}^{-1}$.
Since the exponent of $E$ is $6$, elements
$\sigma\in\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})$ have order dividing $6$. If
$D=\gcd(6,p-1)$, we have
$\begin{array}[]{rcccl}\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})&\xrightarrow{\sim}&\mu_{D}\times\mu_{2}=\langle\zeta_{D}\rangle\times\langle-1\rangle&\xrightarrow{\sim}&\mathbf{Z}_{D}\times\mathbf{Z}_{2}\\\
\end{array}$
For each $d\mid D$, if we write $(j\bmod d,i\bmod 2)$ for
$\sigma(a)=\zeta_{d}^{j},\ \sigma(b)=(-1)^{i},$
we get the following orbits for the action of $\operatorname{Aut}(E)$
$k$ | $d$ | Orbit of $\sigma=(j\bmod d,i\bmod 2)$
---|---|---
12 | 1 | $(1,0)$
6 | 2 | $(1,0)\xrightarrow{g_{2}}(0,1)\xrightarrow{g_{2}}(1,1)$
4 | 3 | $(1,0)\xrightarrow{g_{2}}(2,0)$
2 | 6 | $(1,0)\xrightarrow{g_{2}}(2,1)\xrightarrow{g_{2}}(1,1)\xrightarrow{g_{2}}(5,0)\xrightarrow{g_{2}}(4,1)\xrightarrow{g_{2}}(5,1)$
and again the order determines the orbit. Therefore, the size of kernel
parameterise isomorphism classes of semidirect products.
$p\bmod 12$ | $D$ | $k$
---|---|---
1,7 | 6 | 2,4,6,12
5,11 | 2 | 6,12
Finally, let us compute stabilizers for the first element in each orbit and
take the corresponding conjugates. If $\sigma(b)=1$, then
$\sigma(g_{1}(a))=\sigma(ab)=\sigma(a),$ $\sigma(g_{1}(b)=\sigma(b)$ and we
have $g_{1}\in\mathrm{Stab}_{\sigma}$. Since $\langle g_{1}\rangle$ has index
6, this is the stabilizer for $d=6$. For the other values, the (cyclic) orbits
above show the power of $g_{2}$ that we have to take: $g_{2}^{3}$ for $d=2$
and $g_{2}^{2}$ for $d=3$.
$k$ | $d$ | $\Sigma_{\sigma}$ | | |
---|---|---|---|---|---
12 | 1 | $\operatorname{Aut}(E)$ | | |
6 | 2 | $g_{2}^{m}\langle g_{2}^{3},g_{1}\rangle g_{2}^{-m}$ | $=\langle g_{2}^{3},\ g_{2}^{2m}g_{1}\rangle$ | $m=0,1,2$ | $C_{2}\times C_{2}$
4 | 3 | $g_{2}^{m}\langle g_{2}^{2},g_{1}\rangle g_{2}^{-m}$ | $=\langle g_{2}^{2},\ g_{2}^{2m}g_{1}\rangle$ | $m=0,1$ | $S_{3}$
2 | 6 | $g_{2}^{m}\langle g_{1}\rangle g_{2}^{-m}$ | $=\langle\ g_{2}^{2m}g_{1}\rangle$ | $m=0,1,2$ | $C_{2}$
### 3.3 $E=A_{4}$
Let $E$ be the alternating group $A_{4}$, and take generators $(1,2,3)$ and
$(1,2)(3,4)$. We have $\operatorname{Aut}A_{4}\simeq S_{4}$, the isomorphism
given by restriction to $A_{4}$ of inner automorphisms of $S_{4}$. Therefore
the action is
$\phi_{s}(1,2,3)=(s(1),s(2),s(3)),\quad\phi_{s}(1,2)(3,4)=(s(1),s(2)),(s(3),s(4))$
for any $s\in S_{4}$.
Since the exponent of $E$ is $3$, nontrivial elements
$\sigma\in\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})$ have order $3$, which forces
them to be trivial on $(1,2)(3,4)$ and every element of order 2, namely,
$\operatorname{Ker}\sigma\simeq V_{4}$. For $p\not\equiv 1\bmod 3$,
$\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})=1$ and for $p\equiv 1\bmod 3$
$\begin{array}[]{cclc}\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})&\xrightarrow{\sim}&\mu_{3}=\langle\zeta_{3}\rangle\simeq\mathbf{Z}_{3}\\\
\sigma_{j}&\to&\sigma_{j}(1,2,3)=\zeta_{3}^{j}\simeq j\end{array}$
For any $s\in S_{4}$ such that $(1,2,3)(s(1),s(2),s(3))$ has order 1 or 2, for
example $s=(1,2)$, we have $\sigma_{1}=\sigma_{2}\phi_{s}$ so that they are in
the same orbit for the action of $\operatorname{Aut}(E)$. The stabiliser is a
normal subgroup of $S_{4}$ of index $2$, namely
$\Sigma_{\sigma_{1}}=\Sigma_{\sigma_{2}}=A_{4}$.
$k$ | $d$ | Orbit of $\sigma=j\bmod 3$ | Stabiliser $\Sigma_{\sigma}$
---|---|---|---
12 | 1 | $0$ | $S_{4}$
4 | 3 | $1\xrightarrow{\phi_{(1,2)}}2$ | $A_{4}$
### 3.4 $E=D_{2\cdot 6}$
Let us write $E=\langle r,s\mid r^{6}=\operatorname{Id},\
s^{2}=\operatorname{Id},\ srs=r^{5}\rangle$. We note that
$\operatorname{Aut}E$ is generated by
$\begin{array}[]{llll}g_{1}:&r&\mapsto&r^{-1}\\\
&s&\mapsto&s\end{array},\quad\begin{array}[]{llll}g_{2}:&r&\mapsto&r\\\
&s&\mapsto&rs\end{array}$
$g_{1}$ has order 2, $g_{2}$ has order 6 and $g_{1}g_{2}g_{1}=g_{2}^{-1}$,
hence $\operatorname{Aut}D_{2\cdot 6}\simeq D_{2\cdot 6}$.
Since $E$ has exponent $6$ and the unique normal subgroups with cyclic kernel
are those of índex 2, we have that for every $p$,
$\begin{array}[]{ccl}\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})&\xrightarrow{\sim}&\mu_{2}\times\mu_{2}=\\{\pm
1\\}\times\\{\pm 1\\}\\\ \sigma&\to&(\sigma(r),\sigma(s))\end{array}$
$(1,-1)$ has cyclic kernel $\langle r\rangle$ while $(-1,-1)$ and $(-1,1)$
have kernel isomorphic to dihedral group $D_{2\cdot 3}$. Therefore, they give
at have at least two orbits under the action of $\operatorname{Aut}(E)$. Since
$(-1,-1)g_{2}(r)=(-1,-1)(r)=-1$ and $(-1,-1)g_{2}(s)=(-1,-1)(rs)=1$, namely
$(-1,-1)g_{2}=(-1,1)$, we have three orbits for the action, which means three
isomorphism classes of semidirect (or direct) products $\mathbf{Z}_{p}\rtimes
E$, parameterised by the isomorphism class of the kernel. For $\sigma=(1,-1)$,
we denote $\mathbb{Z}_{p}\rtimes_{\sigma}E=\mathbb{Z}_{p}\rtimes_{6}^{c}E$
(cyclic kernel), while for $\sigma=(-1,-1)$ we write
$\mathbb{Z}_{p}\rtimes_{\sigma}E=\mathbb{Z}_{p}\rtimes_{6}^{d}E$ (dihedral
kernel).
As for stabilisers, the size is determined by the length of the orbit and we
have
$(-1,-1)g_{1}=(-1,-1),\ (-1,-1)g_{2}^{2}=(-1,1)g_{2}=(-1,-1).$
Therefore, the action of $\operatorname{Aut}(E)$ can be described as follows:
$k$ | $d$ | Orbit of $\sigma$ | Stabiliser $\Sigma_{\sigma}$
---|---|---|---
12 | 1 | $(1,1)$ | $\operatorname{Aut}(E)$
6 | 2 | $(1,-1)$ | $\operatorname{Aut}(E)$
6 | 2 | $(-1,-1)\xrightarrow{g_{2}}(-1,1)$ | $\langle g_{1},g_{2}^{2}\rangle\simeq D_{2\cdot 3}$
### 3.5 $E=\operatorname{Dic}_{12}$
Let us write $E=\langle x,y\mid x^{3}=1,y^{4}=1,yxy^{-1}=x^{2}\rangle$. The
group $\operatorname{Aut}E$ is generated by
$\begin{array}[]{llll}g_{1}:&x&\mapsto&x^{2}\\\
&y&\mapsto&y^{3}\end{array},\quad\begin{array}[]{llll}g_{2}:&x&\mapsto&x\\\
&y&\mapsto&xy^{3}\end{array}$
$g_{1}$ has order 2, $g_{2}$ has order 6 and $g_{1}g_{2}g_{1}=g_{2}^{-1}$,
hence $\operatorname{Aut}\operatorname{Dic}_{12}\simeq D_{2\cdot 6}$.
The exponent of the dicyclic group is $6$ (the element $xy$ has order $6$) and
the normal subgroups with cyclic quotient are $\langle x^{2}\rangle$ and the
derived subgroup $\langle x\rangle$. Therefore, $\sigma(x)=1$ for all
$\sigma\in\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})$ and the image of $y$ determines
$\sigma$. If $D=\gcd(4,p-1)$, we have
$\begin{array}[]{ccl}\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})&\xrightarrow{\sim}&\mu_{D}\\\
\sigma&\to&\sigma(y)=\zeta_{d}^{j}\end{array}$
Again, we can identify the elements by a pair $(d,j)$. For $d=2$ we have a
single element, and for $d=4$, since $g_{1}(y)=y^{-1}$, we have $(4,1)$ and
$(4,3)$ in the same orbit. In this case, the stabiliser is a normal subgroup
of $\operatorname{Aut}(E)$ of index $2$. We have
$(4,1)g_{1}g_{2}=(4,3)g_{2}=(4,1),\quad(4,1)g_{2}^{2}=(4,3)g_{2}=(4,1)$
and $(4,1)g_{2}^{3}=(4,1)g_{2}=(4,3).$ Therefore, the action goes as follows:
$k$ | $d$ | Orbit of $\sigma=(d,j)$ | Stabiliser $\Sigma_{\sigma}$
---|---|---|---
12 | 1 | $(1,1)$ | $\operatorname{Aut}(E)$
6 | 2 | $(2,1)$ | $\operatorname{Aut}(E)$
3 | 4 | $(4,1)\xrightarrow[g_{2}^{3}]{g_{1}}(4,3)$ | $\langle g_{2}^{2},g_{1}g_{2}\rangle\simeq D_{2\cdot 3}$
## 4 Skew braces of size $12p$
Once we have done the precomputations for all groups $E$, we bring in the
multiplicative group $F$ of the brace $B_{12}$. In all the sub-cases we follow
the procedure described at the end of Section 2 and the notations in Section 3
for $E$, $\operatorname{Aut}(E)$ and $\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})$.
### 4.1 $F=C_{12}$
According to the table in Section 3 and Proposition 4, for a brace of size 12
with multiplicative group $C_{12}$, the additive group can be any group except
for $A_{4}$. In all the sub-cases the group
$\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$ is as in 3.1 so that $\tau$ identifies as
a pair $(d,j)$ meaning that the given generator of $F$ has image
$\zeta_{d}^{j}.$
#### Case $E=C_{12}$
Let $E=\langle c\rangle\simeq C_{12}$. There is a unique conjugacy class of
regular subgroups of $\mathrm{Hol}(E)$ isomorphic to $C_{12}$, given by the
normal subgroup
$F=\langle\ (c,\operatorname{Id})\ \rangle\subseteq\mathrm{Hol}(E).$
We have stabilisers $\Sigma_{\sigma}=\\{g_{i}\in\operatorname{Aut}(E):i\equiv
1\bmod\mbox{ ord}(\sigma)\\}.$ Since $\pi_{2}(F)=\\{\operatorname{Id}\\}$ we
have
$\mathrm{Hom}(B_{12},\mathbf{Z}_{p}^{*})=\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})$.
Since $F$ is a normal subgroup of $\mathrm{Hol}(E)$, we have $\Phi_{g}(F)=F$
for all $g\in\operatorname{Aut}(E)$ and
$\operatorname{Aut}(B_{12})=\operatorname{Aut}(E).$ Therefore, the orbits of
the action of $\operatorname{Aut}(B_{12})$ on
$\mathrm{Hom}(B_{12},\mathbf{Z}_{p}^{*})$ are those computed in 3.1 and we
have one orbit for each size of kernel. The number of addditive structures is
the number of divisors $\sigma_{0}(\gcd(12,p-1))$.
Finally we have to compute orbits of the action of $\Sigma_{\sigma}$ on
$\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$. Since
$\Phi_{g_{i}}(c,\operatorname{Id})=(g_{i}(c),g_{i}\operatorname{Id}g_{i}^{-1})=(c^{i},\operatorname{Id})$
we have $(d,j)$ and $(d,j^{\prime})$ in the same orbit if and only if $j\equiv
ij^{\prime}\bmod d$ for some $i\in\Sigma_{\sigma}$. Therefore, we have the
following representatives for the orbits
$k_{\sigma}$ | $\Sigma_{\sigma}$ | $(d,j)$
---|---|---
$12,6$ | $\langle 5,7\rangle$ | $(d,1)$, $\quad d|12$
$4,2$ | $\langle 7\rangle$ | $(1,1)$, $(2,1)$, $(4,1)$ (3,1), (3,2), (6,1), (6,5), (12,1) (12,5)
$3$ | $\langle 5\rangle$ | $(1,1)$, $(2,1)$, $(3,1)$ (4,1), (4,3), (6,1), (12,1) (12,7)
$1$ | $\\{1\\}$ | $(d,j)$, $\quad d|12$, $\quad j\in\mathbf{Z}_{d}^{*}$
If we count the number of classes according to the size $\frac{12}{d}$ of
$\operatorname{Ker}\tau$, we obtain
$k_{\sigma}$ | $\Sigma_{\sigma}$ | 12 | 6 | 4 | 3 | 2 | 1
---|---|---|---|---|---|---|---
$12,6$ | $\langle 5,7\rangle$ | 1 | 1 | 1 | 1 | 1 | 1
$4,2$ | $\langle 7\rangle$ | 1 | 1 | 2 | 1 | 2 | 2
$3$ | $\langle 5\rangle$ | 1 | 1 | 1 | 2 | 1 | 2
$1$ | $\\{1\\}$ | 1 | 1 | 2 | 2 | 2 | 4
For $k=12$, the trivial $\sigma$, we have one structure for each $\tau$,
otherwise we have two multiplicative structures for each additive one.
###### Proposition 9.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N:=\mathbf{Z}_{p}\rtimes_{k}C_{12}$ and multiplicative group
$G:=\mathbf{Z}_{p}\rtimes_{k^{\prime}}C_{12}$ is as shown in the following
table, where we need $p\equiv 1\pmod{\frac{12}{k}}$ for a kernel of size $k$
to occur.
$N\backslash G=\mathbf{Z}_{p}...$ | $\times C_{12}$ | $\rtimes_{6}C_{12}$ | $\rtimes_{4}C_{12}$ | $\rtimes_{3}C_{12}$ | $\rtimes_{2}C_{12}$ | $\rtimes_{1}C_{12}$
---|---|---|---|---|---|---
$\mathbf{Z}_{p}\times C_{12}$ | 1 | 1 | 1 | 1 | 1 | 1
$\mathbf{Z}_{p}\rtimes_{6}C_{12}$ | 2 | 2 | 2 | 2 | 2 | 2
$\mathbf{Z}_{p}\rtimes_{4}C_{12}$ | 2 | 2 | 4 | 2 | 4 | 4
$\mathbf{Z}_{p}\rtimes_{3}C_{12}$ | 2 | 2 | 2 | 4 | 2 | 4
$\mathbf{Z}_{p}\rtimes_{2}C_{12}$ | 2 | 2 | 4 | 2 | 4 | 4
$\mathbf{Z}_{p}\rtimes_{1}C_{12}$ | 2 | 2 | 4 | 4 | 4 | 8
#### Case $E=C_{6}\times C_{2}$
We write $E=\langle a\rangle\times\langle b\rangle$. There is one brace with
additive group $C_{6}\times C_{2}$ and multiplicative group $C_{12}$, hence
one conjugation class of regular subgroups of $\mathrm{Hol}(E)$ isomorphic to
$C_{12}$. We may consider as representative the following regular subgroup
$F=\langle c=(a,g_{1})\rangle$
where $(a,g_{1})^{m}$ is $(a^{m}b^{\frac{m}{2}},\operatorname{Id})$ if $m$ is
even and $(a^{m}b^{\frac{m-1}{2}},g_{1})$ if $m$ is odd.
Since $\pi_{2}(F)=\langle g_{1}\rangle$, computation of stabilisers
$\Sigma_{\sigma}$ in 3.2 gives that the $\sigma$’s to consider are $(1\bmod
d,0\bmod 2)$ for every $d|6$, meaning $\sigma(a)=\zeta_{d}$, $\sigma(b)=1$.
Since there is only one for each order, they give a set of representatives for
the action of $\operatorname{Aut}(B_{12})$. Therefore, we have 4 possible
additive structures.
We consider now $\tau\in\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$ and the action by
elements $\Phi_{g}$ such that $\Phi_{g}(F)=F$. We need $gg_{1}g^{-1}=g_{1}$,
that is, $g$ belongs to the centraliser of $g_{1}$, which is the Klein group
$\langle g_{2}^{3},g_{1}\rangle$. We have
$\Phi_{g_{1}}(c)=(ab,g_{1})=c^{7},\ \Phi_{g_{2}^{3}}(c)=c^{5},\
\Phi_{g_{2}^{3}g_{1}}(c)=(a^{5}b,g_{1})=c^{11}.$
For $k=12,6$ we have all these automorphisms in $\Sigma_{\sigma}$, therefore,
$(d,j)$ and $(d,j^{\prime})$ are conjugate. If $k=4,2$, the intersection
between $\Sigma_{\sigma}$ and the centralizer is $\langle g_{1}\rangle$ and
the orbit consists just in $(d,j)$ and $(d,7j\bmod d)$. Therefore, we have the
following representatives for the orbits:
$k$ | $\langle g_{2}^{3},g_{1}\rangle{\cap}\Sigma$ | $(d,j)$
---|---|---
$12$ | $\langle g_{2}^{3},g_{1}\rangle$ | $(d,1)$, $\quad d|12$
$6$ | $\langle g_{2}^{3},g_{1}\rangle$ | $(d,1)$, $\quad d|12$
$4$ | $\langle g_{1}\rangle$ | $(1,1)$, $(2,1)$, $(3,1)$, $(3,2)$ (4,1), (6,1), (6,5), (12,1) (12,5)
$2$ | $\langle g_{1}\rangle$ | $(1,1)$, $(2,1)$, $(3,1)$, $(3,2)$ (4,1), (6,1), (6,5), (12,1) (12,5)
###### Proposition 10.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N:=\mathbf{Z}_{p}\rtimes(C_{6}\times C_{2})$ and multiplicative group
$G:=\mathbf{Z}_{p}\rtimes C_{12}$ is as shown in the following table, where we
need $p\equiv 1\pmod{\frac{12}{k}}$ for a kernel of size $k$ to occur.
$N\backslash G=\mathbf{Z}_{p}...$ | $\times C_{12}$ | $\rtimes_{6}C_{12}$ | $\rtimes_{4}C_{12}$ | $\rtimes_{3}C_{12}$ | $\rtimes_{2}C_{12}$ | $\rtimes_{1}C_{12}$
---|---|---|---|---|---|---
$\mathbf{Z}_{p}\times(C_{6}\times C_{2})$ | 1 | 1 | 1 | 1 | 1 | 1
$\mathbf{Z}_{p}\rtimes_{6}(C_{6}\times C_{2})$ | 2 | 2 | 2 | 2 | 2 | 2
$\mathbf{Z}_{p}\rtimes_{4}(C_{6}\times C_{2})$ | 2 | 2 | 4 | 2 | 4 | 4
$\mathbf{Z}_{p}\rtimes_{2}(C_{6}\times C_{2})$ | 2 | 2 | 4 | 2 | 4 | 4
#### Case $E=D_{2\cdot 6}$
We write $E=\langle r,s\rangle$. There are two braces with additive group
$D_{2\cdot 6}$ and multiplicative group $C_{12}$, hence two conjugation
classes of regular subgroups of $\mathrm{Hol}(D_{2\cdot 6})$ isomorphic to
$C_{12}$. We may consider the following regular subgroups of $\mathrm{Hol}(E)$
isomorphic to $C_{12}$,
$F_{1}=\langle\ c_{1}=(s,g_{2})\ \rangle$
where $c_{1}^{m}$ is $(r^{6-m/2},g_{2}^{m})$ for $m$ even and
$(r^{\frac{k-1}{2}}s,g_{2}^{m})$ for $m$ odd, and
$F_{2}=\langle\ c_{2}=(rs,g_{2}g_{1})\ \rangle$
where $c_{2}^{m}$ is $(r^{m/2},\operatorname{Id})$ for $m$ even and
$(r^{\frac{k+1}{2}}s,g_{2}g_{1})$ for $m$ odd. The groups
$\pi_{2}(F_{1})=\langle g_{2}\rangle\simeq C_{6}$ and $\pi_{2}(F_{2})=\langle
g_{2}g_{1}\rangle\simeq C_{2}$ are not conjugate in $\operatorname{Aut}(E)$,
hence $F_{1}$ and $F_{2}$ are not conjugate in $\mathrm{Hol}(E)$. These groups
show also that the unique nontrivial element in
$\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$ to consider is $\sigma=(1,-1)$, meaning
$\sigma(r)=1,\sigma(s)=-1$, whose stabiliser is $\operatorname{Aut}(E)$.
Therefore, we have 2 additive structures for each $F$. The nontrivial one has
cyclic kernel and we denoted it by $\mathbf{Z}_{p}\rtimes_{6}^{c}D_{2\cdot
6}$.
Regarding orbits on $\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$, since
$\pi_{2}(F_{1})=\langle g_{2}\rangle$ is a normal subgroup, we can take any
$g$ in $\operatorname{Aut}(E)$ to satisfy
$g\pi_{2}(F_{1})g^{-1}=\pi_{2}(F_{1})$. But
$\Phi_{g_{2}^{m}}(c_{1})=(r^{m}s,g_{2})\in F_{1}\iff m=0,3$
and
$\Phi_{g_{2}^{m}g_{1}}(c_{1})=(r^{m}s,g_{2}^{-1})\in F_{1}\iff m=2,5$
Therefore, we are left with the automorphisms in $\langle
g_{2}^{3},g_{2}^{2}g_{1}\rangle\simeq V_{4}$ which happens to be the
normaliser of $\pi_{2}(F_{2})=\langle g_{2}g_{1}\rangle$. For automorphisms of
this group we have
$\Phi_{g_{2}^{3}}(c_{1})=c_{1}^{7},\quad\Phi_{g_{2}^{2}g_{1}}(c_{1})=c_{1}^{5},\quad\Phi_{g_{2}^{5}g_{1}}(c_{1})=c_{1}^{11}$
$\Phi_{g_{2}^{3}}(c_{2})=c_{2}^{7},\quad\Phi_{g_{2}^{2}g_{1}}(c_{2})=c_{2}^{11},\quad\Phi_{g_{2}^{5}g_{1}}(c_{2})=c_{2}^{5}.$
Therefore, in $\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$ we have just orbit for each
$d$.
###### Proposition 11.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N:=\mathbf{Z}_{p}\rtimes D_{2.6}$ and multiplicative group
$G:=\mathbf{Z}_{p}\rtimes C_{12}$ is as shown in the following table, where we
need $p\equiv 1\pmod{\frac{12}{k}}$ for a kernel of size $k$ to occur.
$N\backslash G=\mathbf{Z}_{p}...$ | $\times C_{12}$ | $\rtimes_{6}C_{12}$ | $\rtimes_{4}C_{12}$ | $\rtimes_{3}C_{12}$ | $\rtimes_{2}C_{12}$ | $\rtimes_{1}C_{12}$
---|---|---|---|---|---|---
$\mathbf{Z}_{p}\times D_{2\cdot 6}$ | 2 | 2 | 2 | 2 | 2 | 2
$\mathbf{Z}_{p}\rtimes_{6}^{c}D_{2\cdot 6}$ | 4 | 4 | 4 | 4 | 4 | 4
#### Case $E=\operatorname{Dic}_{12}$
We write $E=\langle\ x,y\ \rangle$. There are two braces with additive group
$\operatorname{Dic}_{12}$ and multiplicative group $C_{12}$, hence two
conjugation classes of regular subgroups of
$\mathrm{Hol}(\operatorname{Dic}_{12})$ isomorphic to $C_{12}$. We may
consider representatives
$\begin{array}[]{l}F_{1}=\langle\ c_{1}=(y,g_{2}g_{1})\ \rangle,\\\
F_{2}=\langle\ c_{2}=(y,g_{2}^{2})\ \rangle,\end{array}$
with $\pi_{2}(F_{1})=\langle g_{2}g_{1}\rangle\simeq C_{2}$ and
$\pi_{2}(F_{2})=\langle g_{2}^{2}\rangle\simeq C_{3}$. Since
$\pi_{2}(F_{1}),\pi_{2}(F_{2})\subseteq\langle g_{2}^{2},g_{1}g_{2}\rangle$,
we have $\pi_{2}(F_{1}),\pi_{2}(F_{2})\subseteq\Sigma_{\sigma}$ for all
$\sigma\in\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})$.
In order to compute brace automorphisms we consider first the normalizer of
$\pi_{2}(F_{1})$ in $\operatorname{Aut}(E)$, which is $\langle
g_{2}^{3},g_{2}g_{1}\rangle$, and the normalizer of $\pi_{2}(F_{2})$, which is
$\langle g_{2}\rangle$. Since
$\Phi_{g_{2}^{3}}(F_{1})=\langle\ (y^{3},g_{2}g_{1})\ \rangle=\langle\
c_{1}^{7}\ \rangle=F_{1},$ $\Phi_{g_{2}^{3}}(F_{2})=\langle\
(y^{3},g_{2}^{2})\ \rangle=\langle\ c_{2}^{7}\ \rangle=F_{2},$
we have one orbit for each order and 3 possible additive structures for each
$F$. On the other hand, we have
$\begin{array}[]{lcl}\Phi_{g_{2}}(c_{2})=(xy^{3},g_{2})\not\in
F_{2},&&\Phi_{g_{2}^{2}}(c_{2})=(x^{2}y,g_{2}^{2})\not\in
F_{2},\\\\[2.15277pt]
\Phi_{g_{2}^{3}}(c_{2})=(y^{3},g_{2}^{2})=c_{2}^{7},&&\Phi_{g_{2}^{3}}(c_{1})=(y^{3},g_{2}g_{1})=c_{1}^{7},\\\\[2.15277pt]
\Phi_{g_{2}g_{1}}(c_{2})=(xy,g_{2}^{4})=c_{2}^{5},&&\Phi_{g_{2}g_{1}}(c_{1})=(xy,g_{2}g_{1})=c_{1}^{5},\end{array}$
and this shows that in both cases the automorphism group of the brace is the
maximal subgroup $\langle g_{2}^{3},g_{2}g_{1}\rangle\simeq V_{4}$.
For $\sigma=(1,1)$ or $(2,1)$, we have
$\operatorname{Aut}(B_{12})\cap\Sigma_{\sigma}=\operatorname{Aut}(B_{12})$ and
$\operatorname{Aut}(B_{12})$ acts on
$\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})=\\{(d,j)\\}$ giving a single orbit for
each order $d$. But, for $\sigma=(4,1)$, we have
$\operatorname{Aut}(B_{12})\cap\Sigma_{\sigma}=\langle g_{2}g_{1}\rangle$.
Then, $\tau=(d,j)$ and $\tau^{\prime}=(d,j^{\prime})$ are in the same orbit
if, and only if, $j\equiv 5j^{\prime}\bmod d$, and we have orbits represented
by $(1,1),(2,1),(3,1),(4,1),(4,3),(6,1),(12,1),(12,7).$
###### Proposition 12.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N:=\mathbf{Z}_{p}\rtimes\operatorname{Dic}_{12}$ and multiplicative group
$G:=\mathbf{Z}_{p}\rtimes C_{12}$ is as shown in the following table, where we
need where we need $p\equiv 1\pmod{\frac{12}{k}}$ for a kernel of size $k$ to
occur.
$N\backslash G=\mathbf{Z}_{p}...$ | $\times C_{12}$ | $\rtimes_{6}C_{12}$ | $\rtimes_{4}C_{12}$ | $\rtimes_{3}C_{12}$ | $\rtimes_{2}C_{12}$ | $\rtimes_{1}C_{12}$
---|---|---|---|---|---|---
$\mathbf{Z}_{p}\times\operatorname{Dic}_{12}$ | 2 | 2 | 2 | 2 | 2 | 2
$\mathbf{Z}_{p}\rtimes_{6}\operatorname{Dic}_{12}$ | 4 | 4 | 4 | 4 | 4 | 4
$\mathbf{Z}_{p}\rtimes_{3}\operatorname{Dic}_{12}$ | 4 | 4 | 4 | 8 | 4 | 8
### 4.2 $F=C_{6}\times C_{2}$
According to the table in Section 3 and Proposition 4, for a brace of size 12
with multiplicative group $C_{12}$, the additive group can be any group. In
all the sub-cases the group $\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$ is as in 3.2
so that $\tau$ identifies as a pair $(j\bmod d,i\bmod 2)$ meaning
$\tau(a)=\zeta_{d}^{j},\ \tau(b)=(-1)^{i}$ for $F=\langle
a\rangle\times\langle b\rangle$.
#### Case $E=C_{12}$
There is one brace with additive group $C_{12}$ and multiplicative group
$C_{6}\times C_{2}$, hence one conjugation class of regular subgroups of
$\mathrm{Hol}(C_{12})$ isomorphic to $C_{6}\times C_{2}$. We may consider the
following representative,
$F=\langle a=(c,g_{7}),\ b=(c^{6},\operatorname{Id})\rangle,$
where $a$ has order 6, $b$ has order 2. Since $\pi_{2}(F)=\langle
g_{7}\rangle$ we cannot consider $\sigma$ of order 4 or 12. Since $F$ is a
normal subgroup of $\mathrm{Hol}(E)$ all automorphisms of $E$ are brace
automorphisms and we have a single orbit for each order. Therefore, we have 4
possible additive structures.
Regarding the action on $\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$, since
$\begin{array}[]{lcl}\Phi_{g_{5}}(a)=(c^{5},g_{7})=a^{5},&&\Phi_{g_{5}}(b)=(c^{6},\operatorname{Id})=b,\\\\[2.15277pt]
\Phi_{g_{7}}(a)=(c^{7},g_{7})=ab,&&\Phi_{g_{7}}(b)=(c^{6},\operatorname{Id})=b,\\\\[2.15277pt]
\Phi_{g_{11}}(a)=(c^{11},g_{7})=a^{5}b,&&\Phi_{g_{11}}(b)=(c^{6},\operatorname{Id})=b.\end{array}$
we have the following orbits, acording to the order of $\tau$:
$\begin{array}[]{lccccc}d=1&&&(1,0)\ \bullet\circlearrowleft&&\\\\[8.61108pt]
d=2&&&(1,0)\ \bullet\circlearrowleft&&(0,1)\
\bullet\xleftrightarrow[11]{7}\bullet(1,1)\\\\[12.91663pt]
d=3&&&(1,0)\bullet\xleftrightarrow[11]{5}\bullet(2,0)&&\\\\[12.91663pt]
d=6&&&(1,0)\bullet&&(1,1)\bullet\xleftrightarrow{7}\bullet(4,1)\\\ &&&\
\qquad{{}_{5}}\Big{\updownarrow}{{}_{11}}&&{{}_{5}}\Big{\updownarrow}\quad\quad\Big{\updownarrow}{{}_{5}}\\\
&&&(5,0)\bullet&&(5,1)\bullet\xleftrightarrow{7}\bullet(2,1)\\\ \end{array}$
In the last two cases, when $\Sigma_{\sigma}=\langle g_{7}\rangle$ some points
get disconnected and we have 2 and 4 orbits, respectively.
###### Proposition 13.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N=\mathbf{Z}_{p}\rtimes C_{12}$ and multiplicative group
$G=\mathbf{Z}_{p}\rtimes(C_{6}\times C_{2})$ is as shown in the following
table, where we need $p\equiv 1\pmod{\frac{12}{k}}$ for a kernel of size $k$
to occur.
$N\backslash G=\mathbf{Z}_{p}...$ | $\times(C_{6}\times C_{2})$ | $\rtimes_{6}(C_{6}\times C_{2})$ | $\rtimes_{4}(C_{6}\times C_{2})$ | $\rtimes_{2}(C_{6}\times C_{2})$
---|---|---|---|---
$\mathbf{Z}_{p}\times C_{12}$ | 1 | 2 | 1 | 2
$\mathbf{Z}_{p}\rtimes_{6}C_{12}$ | 2 | 4 | 2 | 4
$\mathbf{Z}_{p}\rtimes_{2}C_{12}$ | 2 | 4 | 4 | 8
$\mathbf{Z}_{p}\rtimes_{4}C_{12}$ | 2 | 4 | 4 | 8
#### Case $E=C_{6}\times C_{2}$
Let $E=\langle a\rangle\times\langle b\rangle$. There is one brace with
additive group $C_{6}\times C_{2}$ and multiplicative group $C_{6}\times
C_{2}$, hence one conjugation class of regular subgroups of
$\mathrm{Hol}(C_{6}\times C_{2})$ isomorphic to $C_{6}\times C_{2}$, which has
to be that of the normal subgroup
$F=\langle(a,\operatorname{Id}),(b,\operatorname{Id})\rangle.$
Clearly, we have a single orbit for each order of
$\sigma\in\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})$ and 4 possible additive
structures.
Regarding the action on $\tau\in\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$, for
$\sigma=1$ we have one orbit for each order and for $\tau=1$ the orbit has a
single point. According to the stabilisers and orbits computed in 3.2, in the
remaining cases we have
$\Sigma_{(1,0)}$ | $d$ | Orbits
---|---|---
$\langle g_{2}^{3},g_{1}\rangle$ | 2 | $(1,0)\ \bullet\circlearrowleft\quad(0,1)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (1,1)$
| 3 | $(1,0)\ \bullet\xleftrightarrow{g_{2}^{3}}\bullet\ (2,0)$
| 6 | $(1,0)\ \bullet\xleftrightarrow{g_{2}^{3}}\bullet\ (5,0)$
| | $(2,1)\ \bullet\xleftrightarrow{g_{2}^{3}}\bullet\ (4,1)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (1,1)\ \bullet\xleftrightarrow{g_{2}^{3}}\bullet(5,1)$
$\langle g_{2}^{2},g_{1}\rangle$ | 2 | $(1,0)\ \bullet\xleftrightarrow{g_{2}}\bullet\ (1,1)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (0,1)$
| 3 | $(1,0)\ \bullet\circlearrowleft\qquad\bullet\ (2,0)\circlearrowleft$
| 6 | $(1,0)\ \bullet\xleftrightarrow{g_{2}^{2}}\bullet\ (1,1)\ \bullet\xleftrightarrow{g_{2}^{2}}\bullet\ (4,1)$
| | $(5,1)\ \bullet\xleftrightarrow{g_{2}^{2}}\bullet\ (2,1)\ \bullet\xleftrightarrow{g_{2}^{2}}\bullet\ (5,0)$
$\langle g_{1}\rangle$ | 2 | $(1,0)\ \bullet\circlearrowleft\quad(0,1)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (1,1)$
| 3 | $(1,0)\ \bullet\circlearrowleft\qquad\bullet\ (2,0)\circlearrowleft$
| 6 | $(1,0)\ \bullet\circlearrowleft\qquad\bullet\ (5,0)\circlearrowleft$
| | $(1,1)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (4,1)$ $(2,1)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (5,1)$
###### Proposition 14.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N=\mathbf{Z}_{p}\rtimes(C_{6}\times C_{2})$ and multiplicative group
$G=\mathbf{Z}_{p}\rtimes(C_{6}\times C_{2})$ is as shown in the following
table, where we need where we need $p\equiv 1\pmod{\frac{12}{k}}$ for a kernel
of size $k$ to occur.
$N\backslash G=\mathbf{Z}_{p}...$ | $\times C_{6}\times C_{2}$ | $\rtimes_{6}(C_{6}\times C_{2})$ | $\rtimes_{4}(C_{6}\times C_{2})$ | $\rtimes_{2}(C_{6}\times C_{2})$
---|---|---|---|---
$\mathbf{Z}_{p}\times C_{6}\times C_{2}$ | 1 | 1 | 1 | 1
$\mathbf{Z}_{p}\rtimes_{6}(C_{6}\times C_{2})$ | 2 | 4 | 2 | 4
$\mathbf{Z}_{p}\rtimes_{4}(C_{6}\times C_{2})$ | 2 | 2 | 4 | 4
$\mathbf{Z}_{p}\rtimes_{2}(C_{6}\times C_{2})$ | 2 | 4 | 4 | 8
#### Case $E=A_{4}$
There are two braces with additive group $A_{4}$ and multiplicative group
$C_{6}\times C_{2}$, hence two conjugation classes of regular subgroups of
$\mathrm{Hol}(A_{4})$ isomorphic to $C_{6}\times C_{2}$. We may consider the
following representatives,
$\begin{array}[]{l}F_{1}=\langle a_{1}=((1,2,3),\phi_{(1,2,4)}),\
b_{1}=((1,2)(3,4),\operatorname{Id})\rangle,\\\ F_{2}=\langle
a_{2}=((1,3,4),\phi_{(1,2)(3,4)}),\
b_{2}=((1,3)(2,4),\phi_{(1,3)(2,4)})\rangle,\end{array}$
where $a_{1},a_{2}$ and have order 6 and $b_{1},b_{2}$ have order 2. We have
$\pi_{2}(F_{1})=\langle\phi_{(1,2,4)}\rangle\simeq C_{3}$ and
$\pi_{2}(F_{1})\simeq V_{4}$. Therefore, on one hand we see that the groups
are not conjugate in $\mathrm{Hol}(A_{4})$ and one the other hand that we can
consider all $\sigma\in\mathrm{Hom}(E,\mathbf{Z}_{p}^{*}).$
Since $\pi_{2}(F_{1})=\langle\phi_{(1,2,4)}\rangle$, in order to find brace
automorphisms we need $\phi_{s}$ in its normaliser, namely
$s\in\langle(2,4),(1,2,4)\rangle\simeq S_{3}$. We have
$\begin{array}[]{lcl}\phi_{\phi_{(2,4)}}(a_{1})=a_{1}^{5},&&\phi_{(2,4)}(b_{1})=a_{1}^{3}b_{1},\\\
\phi_{\phi_{(1,2,4)}}(a_{1})=b_{1}a_{1},&&\phi_{(1,2,4)}(b_{1})=a_{1}^{3}.\end{array}$
Since $\phi_{(2,4)}(1,2,3)=(1,4,3)=(1,3,2)(1,4)(2,3)$, we have that for
$F_{1}$ it is enough to consider $\sigma_{1}$. The normaliser of
$\pi_{2}(F_{1})=V_{4}$ is all $S_{4}$ and we find
$\Phi_{\phi_{(1,2)}}(a_{2})=a_{2}^{5},\quad\phi_{(1,2)}(b_{2})=a_{2}^{3}b_{2}.$
Since $\phi_{(1,2)}(1,2,3)=(1,3,2)(1,4)$ we are also left with $\sigma_{1}$
for $F_{2}$. We have two additive structures for each $F$.
We complete the description of the group of brace automorphisms, since
$\Phi_{\phi_{(1,2,3,4)}}(a_{2})\notin
F_{2},\quad\Phi_{\phi_{(1,2,3)}}(a_{2})=a_{2}b_{2},\quad\Phi_{\phi_{(1,2,3)}}(b_{2})=a_{2}^{3}$
shows that it is not $S_{4}$ and contains the maximal group
$\langle(1,2),(1,2,3)\rangle\simeq S_{3}$. All together,
$\\{s\in
S_{4}:\Phi_{\phi_{s}}(F_{i})=F_{i}\\}=\left\\{\begin{array}[]{l}\langle(2,4),(1,2,4)\rangle\
\mbox{ for }i=1\\\ \langle(1,2),(1,2,3)\rangle\ \mbox{ for
}i=2\end{array}\right.\ \simeq S_{3}.$
The intersection with $\Sigma_{\sigma_{1}}$ is
$A_{4}\cap\\{s\in
S_{4}:\Phi_{\phi_{s}}(F_{i})=F_{i}\\}=\left\\{\begin{array}[]{l}\langle(1,2,4)\rangle\
\mbox{ for }i=1\\\ \langle(1,2,3)\rangle\ \mbox{ for }i=2.\end{array}\right.$
To get the multiplicative structures we need the orbits of
$(s,\tau)\longrightarrow\tau\Phi_{\phi_{s}}$. We have the same action on
generators $\phi_{c}(a)=ab$, $\phi_{c}(b)=a^{3}$, so that
$(j\bmod d,i\bmod 2)\Phi_{\phi_{c}}(a)=(-1)^{i}\zeta_{d}^{j}\quad(j\bmod
d,i\bmod 2)\Phi_{\phi_{c}}(b)=\zeta_{d}^{3j}$
and we obtain the following orbits:
$d=2$ | | $(1,0)\ \bullet\xleftrightarrow{}\bullet\ (1,1)\ \bullet\xleftrightarrow{}\bullet\ (0,1)$
---|---|---
$d=3$ | | $(1,0)\ \bullet\circlearrowleft\qquad\bullet\ (2,0)\circlearrowleft$
$d=6$ | | $(1,0)\ \bullet\xleftrightarrow{}\bullet\ (1,1)\ \bullet\xleftrightarrow{}\bullet\ (4,1)$
| | $(5,0)\ \bullet\xleftrightarrow{}\bullet\ (5,1)\ \bullet\xleftrightarrow{}\bullet\ (2,1)$
###### Proposition 15.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N:=\mathbf{Z}_{p}\rtimes A_{4}$ and multiplicative group
$G:=\mathbf{Z}_{p}\rtimes(C_{6}\times C_{2})$ is as shown in the following
table, where we need $p\equiv 1\pmod{\frac{12}{k}}$ for a kernel of size $k$
to occur.
$N\backslash G=\mathbf{Z}_{p}...$ | $\times(C_{6}\times C_{2})$ | $\rtimes_{6}(C_{6}\times C_{2})$ | $\rtimes_{4}(C_{6}\times C_{2})$ | $\rtimes_{2}(C_{6}\times C_{2})$
---|---|---|---|---
$\mathbf{Z}_{p}\times A_{4}$ | 2 | 2 | 2 | 2
$\mathbf{Z}_{p}\rtimes A_{4}$ | 4 | 4 | 8 | 8
#### Case $E=D_{2\cdot 6}$
We write $E=\langle r,s\rangle$. There are two braces with additive group
$D_{2\cdot 6}$ and multiplicative group $C_{6}\times C_{2}$, hence two
conjugation classes of regular subgroups of $\mathrm{Hol}(D_{2\cdot 6})$
isomorphic to $C_{6}\times C_{2}$. We may consider the following
representatives:
$\begin{array}[]{l}F_{1}=\langle a_{1}=(r,\operatorname{Id}),\
b_{1}=(s,g_{1})\rangle,\\\ F_{2}=\langle a_{2}=(r,g_{2}^{4}),\
b_{2}=(s,\operatorname{Id})\rangle,\end{array}$
where $a_{1},a_{2}$ have order 6, and $b_{1},b_{2}$ have order 2. We have
$\pi_{2}(F_{2})=\langle g_{1}\rangle\simeq C_{2}$ and $\pi_{2}(F_{1})=\langle
g_{2}^{4}\rangle\simeq C_{3}$. This shows that they are not conjugate and that
they are subgroups of all the stabilisers $\Sigma_{\sigma}$. Moreover, since
$g_{2}^{3}$ normalizes both $F_{1}$ and $F_{2}$ we have only one orbit in
order $6$ with dihedral kernel and we have 3 additive structures.
Since the normaliser of $\langle g_{1}\rangle$ is $\langle
g_{2}^{3},g_{1}\rangle$ and
$\begin{array}[]{lll}\Phi_{g_{1}}(a_{1})=(r^{5},\operatorname{Id})=a_{1}^{5},&&\Phi_{g_{1}}(b_{1})=(s,g_{1})=b_{1},\\\
\Phi_{g_{2}^{3}}(a_{1})=(r,\operatorname{Id})=a_{1},&&\Phi_{g_{2}^{3}}(b_{1})=(r^{3},g_{1})=a_{1}^{3}b_{1}\end{array}$
we have that for $F_{1}$ the group of brace automorphisms is $\langle
g_{2}^{3},g_{1}\rangle\simeq V_{4}$. On the other hand,
$\begin{array}[]{lll}\Phi_{g_{1}}(a_{2})=(r^{5},g_{2}^{2})=a_{2}^{5},&&\Phi_{g_{1}}(b_{2})=(s,\operatorname{Id})=b_{2},\\\
\Phi_{g_{2}^{3}}(a_{2})=(r,g_{2}^{4})=a_{2},&&\Phi_{g_{2}^{3}}(b_{2})=(r^{3}s,\operatorname{Id})=a_{2}^{3}b_{2}\\\
\Phi_{g_{2}}(a_{2})=(r,g_{2}^{4})=a_{2},&&\Phi_{g_{2}}(b_{2})=(rs,\operatorname{Id})\not\in\mathbf{F}_{2}\\\
\end{array}$
shows that for $F_{2}$ the group of brace automorphisms is also the maximal
subgroup $\langle g_{2}^{3},g_{1}\rangle$. Since the action of generators is
the same, there is no difference in computation of orbits for $F_{1}$ or
$F_{2}$. We obtain
$d=2$ | | $(0,1)\ \bullet\circlearrowleft\qquad(1,0)\ \bullet\xleftrightarrow{g_{2}^{3}}\bullet\ (1,1)$ |
---|---|---|---
$d=3$ | | $(1,0)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (2,0)$ |
$d=6$ | | $(2,1)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (4,1)$ |
| | $(1,0)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (5,0)$ |
| | $\quad\ {{}_{g_{2}^{3}}}\Big{\updownarrow}\qquad\ \Big{\updownarrow}{{}_{g_{2}^{3}}}$ |
| | $(1,1)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (5,1)$ |
For $\sigma$ of order $6$ with dihedral kernel
$g_{2}^{3}\not\in\Sigma_{\sigma}$ and in first and last case orbits split.
###### Proposition 16.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N:=\mathbf{Z}_{p}\rtimes D_{2.6}$ and multiplicative group
$G:=\mathbf{Z}_{p}\rtimes(C_{6}\times C_{2})$ is as shown in the following
table, where we need $p\equiv 1\pmod{\frac{12}{k}}$ for a kernel of size $k$
to occur.
$N\backslash G=\mathbf{Z}_{p}...$ | $\times(C_{6}\times C_{2})$ | $\rtimes_{6}(C_{6}\times C_{2})$ | $\rtimes_{4}(C_{6}\times C_{2})$ | $\rtimes_{2}(C_{6}\times C_{2})$
---|---|---|---|---
$\mathbf{Z}_{p}\times D_{2\cdot 6}$ | 2 | 4 | 2 | 4
$\mathbf{Z}_{p}\rtimes_{6}^{c}D_{2\cdot 6}$ | 4 | 8 | 4 | 8
$\mathbf{Z}_{p}\rtimes_{6}^{d}D_{2\cdot 6}$ | 4 | 12 | 4 | 12
#### Case $E=\operatorname{Dic}_{12}$
We write $E=\langle x,y\rangle$. There are two braces with additive group
$\operatorname{Dic}_{12}$ and multiplicative group $C_{6}\times C_{2}$, hence
two conjugation classes of regular subgroups of
$\mathrm{Hol}(\operatorname{Dic}_{12})$ isomorphic to $C_{6}\times C_{2}$. We
may consider the following representatives
$\begin{array}[]{l}F_{1}=\langle a_{1}=(xy,g_{1}),\
\,b_{1}=(y^{2},\operatorname{Id})\rangle,\\\ F_{2}=\langle
a_{2}=(xy^{2},\operatorname{Id}),\ b_{2}=(y,g_{1})\rangle,\end{array}$
where $a_{1},a_{2}$ have order 6, $b_{1},b_{2}$ have order 2. We note that
since the second component of $a_{2}$ is $\operatorname{Id}$ and the second
components of $a_{1}$ and $a_{1}^{-1}$ are equal to $g_{1}$, they are not
conjugated. We have $\pi_{2}(F_{1})=\pi_{2}(F_{2})=\langle g_{1}\rangle$ and
therefore we cannot consider $\sigma$ order $4$ and then we have
$\Sigma_{\sigma}=\operatorname{Aut}(E)$. We need to determine brace
automorphisms in order to determine the orbits of the action on
$\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$
The normaliser of $\langle g_{1}\rangle$ in $\operatorname{Aut}(E)$ is
$\langle g_{1},g_{2}^{3}\rangle$ and we have
$\begin{array}[]{lll}\Phi_{g_{1}}(a_{1})=(x^{2}y^{3},g_{1})=a_{1}^{5}b_{1},&&\Phi_{g_{1}}(b_{1})=(y^{2},\operatorname{Id})=b_{1}\\\
\Phi_{g_{2}^{3}}(a_{1})=(xy^{3},g_{1})=a_{1}b_{1}&&\Phi_{g_{2}^{3}}(b_{1})=(y^{2},\operatorname{Id})=b_{1}\\\
\Phi_{g_{1}}(a_{2})=(x^{2}y^{2},\operatorname{Id})=a_{2}^{5}&&\Phi_{g_{1}}(b_{2})=(y^{3},g_{1})=a_{2}^{3}b_{2}\\\
\Phi_{g_{2}^{3}}(a_{2})=(xy^{2},\operatorname{Id})=a_{2}&&\Phi_{g_{2}^{3}}(b_{2})=(y^{3},g_{1})=a_{2}^{3}b_{2}.\\\
\end{array}$
This group of brace automorphism gives raise to the following orbits
$d=2$ | | $(1,0)\ \bullet\circlearrowleft\qquad(0,1)\ \bullet\xleftrightarrow[g_{1}]{g_{2}^{3}}\bullet\ (1,1)$ | | $F_{1}$
---|---|---|---|---
| | $(0,1)\ \bullet\circlearrowleft\qquad(1,0)\ \bullet\xleftrightarrow[g_{1}]{g_{2}^{3}}\bullet\ (2,0)$ | | $F_{2}$
$d=3$ | | $(1,0)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (2,0)$ | | $F_{1},F_{2}$
$d=6$ | | $(1,0)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (5,0)$ | $(1,1)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (2,1)$ | $F_{1}$
| | | $\quad\ {{}_{g_{2}^{3}}}\Big{\updownarrow}\qquad\ \Big{\updownarrow}{{}_{g_{2}^{3}}}$ |
| | | $(4,1)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (5,1)$ |
| | $(2,1)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (4,1)$ | $(1,0)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (5,1)$ | $F_{2}$
| | | $\quad\ {{}_{g_{2}^{3}}}\Big{\updownarrow}\qquad\ \Big{\updownarrow}{{}_{g_{2}^{3}}}$ |
| | | $(1,1)\ \bullet\xleftrightarrow{g_{1}}\bullet\ (5,0)$ |
and both groups give raise to the same number of multiplicative structures.
###### Proposition 17.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N=\mathbf{Z}_{p}\rtimes\operatorname{Dic}_{12}$ and multiplicative group
$G=\mathbf{Z}_{p}\rtimes(C_{6}\times C_{2})$ is as shown in the following
table, where we need $p\equiv 1\pmod{\frac{12}{k}}$ for a kernel of size $k$
to occur.
$N\backslash G=\mathbf{Z}_{p}...$ | $\times(C_{6}\times C_{2})$ | $\rtimes_{{}^{6}}(C_{6}\times C_{2})$ | $\rtimes_{4}(C_{6}\times C_{2})$ | $\rtimes_{2}(C_{6}\times C_{2})$
---|---|---|---|---
$\mathbf{Z}_{p}\times\operatorname{Dic}_{12}$ | 2 | 4 | 2 | 4
$\mathbf{Z}_{p}\rtimes_{6}\operatorname{Dic}_{12}$ | 4 | 8 | 4 | 8
### 4.3 $F=A_{4}$
According to the table in Section 3 and Proposition 4, for a brace with
multiplicative group a semidirect product of $\mathbf{Z}_{p}$ and $A_{4}$, the
additive group is either a semidirect product of $\mathbf{Z}_{p}$ and
$C_{6}\times C_{2}$ or a semidirect product of $\mathbf{Z}_{p}$ and $A_{4}$.
In both sub-cases the group $\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$ is as in 3.3
so that its elements are $\tau_{j}$, with $j=0,1,2$, defined by
$\tau_{j}(1,2,3)=\zeta_{3}^{j}$ and $\tau_{j}$ trivial on elements of order 2.
#### Case $E=C_{6}\times C_{2}$
There is one brace with additive group $E=\langle a\rangle\times\langle
b\rangle\simeq C_{6}\times C_{2}$ and multiplicative group $A_{4}$, hence one
conjugation class of regular subgroups of $\mathrm{Hol}(C_{6}\times C_{2})$
isomorphic to $A_{4}$. We may consider representative
$F=\langle(a,g_{2}^{2}),(b,\operatorname{Id})\rangle.$
Indeed, $(a,g_{2}^{2})$ has order 3, $(b,\operatorname{Id})$ has order 2, and
$(a,g_{2}^{2})(b,\operatorname{Id})=(a^{4},g_{2}^{2})$ which has order 3. The
Klein subgroup is
$\langle(b,\operatorname{Id}),(a^{3},\operatorname{Id})\rangle$.
We have $\pi_{2}(F)=<g_{2}^{2}>$ and therefore it is a subgroup of
$\Sigma_{\sigma}$ only when $\sigma$ has order $1$ or $3$. For $\sigma=(1,0)$
of order $3$, we have $(1,0)g_{1}g_{2}=(1,0)g_{2}=(2,0)$. Since
$\Phi_{g_{1}g_{2}}(a,g_{2}^{2})=(a^{2}b,g_{2}^{4})=(a,g_{2}^{2})^{2},\quad\Phi_{g_{1}g_{2}}(b,\operatorname{Id})=(a^{3},\operatorname{Id})\in
F$
there is one orbit for each order and two different additive structures.
Let us check if elements of $\Sigma_{\sigma}=\langle g_{2}^{2},g_{1}\rangle$
are brace automorphisms. Since
$\Phi_{g_{2}^{2}}(a)=(a^{2}b,g_{2}^{2})\not\in
F,\quad\Phi_{g_{1}}(a)=(ab,g_{2}^{4})\not\in F$
the action on $\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$ is just given by the
identity and orbits consist on single elements. For a non trivial $\sigma$,
$\tau_{1}$ and $\tau_{2}$ give different multiplicative structures.
###### Proposition 18.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N:=\mathbf{Z}_{p}\rtimes(C_{6}\times C_{2})$ and multiplicative group
$G:=\mathbf{Z}_{p}\rtimes A_{4}$ is as shown in the following table, where we
need $p\equiv 1\pmod{\frac{12}{k}}$ for a kernel of size $k$ to occur.
$N\backslash G$ | $\mathbf{Z}_{p}\times A_{4}$ | $\mathbf{Z}_{p}\rtimes_{4}A_{4}$
---|---|---
$\mathbf{Z}_{p}\times(C_{6}\times C_{2})$ | 1 | 1
$\mathbf{Z}_{p}\rtimes_{4}(C_{6}\times C_{2})$ | 2 | 4
#### Case $E=A_{4}$
There are four braces with additive group $A_{4}$ and multiplicative group
$A_{4}$, hence four conjugation classes of regular subgroups of
$\mathrm{Hol}(A_{4})$ isomorphic to $A_{4}$. If we write $\rho=(1,2,3)$ and
$\mu=(1,2)(3,4)$, we may consider the following representatives
$\begin{array}[]{l}F_{1}=\langle(\rho,\operatorname{Id}),(\mu,\operatorname{Id})\rangle,\quad
F_{2}=\langle(\rho,\phi_{\rho}),(\mu,\operatorname{Id})\rangle,\\\\[4.30554pt]
F_{3}=\langle(\rho,\phi_{\rho^{-1}}),(\mu,\phi_{\mu})\rangle,\quad
F_{4}=\langle(\rho,\phi_{\rho}),(\mu,\phi_{\mu})\rangle,\\\ \end{array}$
We have $\pi_{2}(F_{1})={\operatorname{Id}}$,
$\pi_{2}(F_{2})=\langle\phi_{\rho}\rangle\simeq C_{3}$, and
$\pi_{2}(F_{3})=\pi_{2}(F_{4})\simeq A_{4}$. But they are not conjugate, since
in $F_{3}$ all elements of order $3$ are of the form $(x,\phi_{x^{-1}})$. On
the other hand, we see that $\pi_{2}(F_{i})\subseteq\Sigma_{\sigma}$ for all
$i=1..4$ and all $\sigma\in\mathrm{Hom}(E,\mathbf{Z}_{p}^{*})$.
Since
$\begin{array}[]{l}\Phi_{\phi_{(1,2)}}((1,2,3),\operatorname{Id})=((1,3,2),\operatorname{Id})=((1,2,3),\operatorname{Id})^{2},\\\
\Phi_{\phi_{(1,2)}}((1,2,3),\phi_{(1,3,2)})=((1,3,2),\phi_{(1,2,3)})=((1,2,3),\phi_{(1,3,2)})^{2},\\\
\Phi_{\phi_{(1,2)}}((1,2,3),\phi_{(1,2,3)})=((1,3,2),\phi_{(1,3,2)})=((1,2,3),\phi_{(1,2,3)})^{2},\end{array}$
and $((1,2)(3,4),\operatorname{Id})$ and $((1,2)(3,4),\phi_{(1,2)(3,4)}))$ are
invariant under $\Phi_{\phi_{(1,2)}}$ we have $\Phi_{2}(F_{i})=F_{i}$ for all
$i$ and elements of order 3 of $\mathrm{Hom}(E,\mathbf{Z}_{p})^{*}$ are in the
same orbit. We have two possible additive structures for each $F_{i}$.
As for morphisms $\tau\in\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$, since
$\Sigma_{\sigma}$ stabilizes them, the action give orbits of a single elememt,
so that we have one orbit in order 1 and two orbits in order 3.
###### Proposition 19.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N=\mathbf{Z}_{p}\rtimes A_{4}$ and multiplicative group
$G=\mathbf{Z}_{p}\rtimes A_{4}$ is as shown in the following table, where we
need $p\equiv 1\pmod{3}$ for the group $\mathbf{Z}_{p}\rtimes_{4}A_{4}$ to
occur.
$N\backslash G$ | $\mathbf{Z}_{p}\times A_{4}$ | $\mathbf{Z}_{p}\rtimes_{4}A_{4}$
---|---|---
$\mathbf{Z}_{p}\times A_{4}$ | 4 | 4
$\mathbf{Z}_{p}\rtimes_{4}A_{4}$ | 8 | 16
### 4.4 $F=D_{2\cdot 6}$
Let us assume that $F$ is isomorphic to the dihedral group $D_{2\cdot 6}$.
From the table in Section 3 we know that for a brace of size $12$ with
multiplicative group $F$, the additive group can be in any isomorphism class
of groups of order $12$ except $A_{4}$. Moreover, in all the sub-cases the
group $\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$ is as in 3.4, so $\tau$ identifies
with any of the pairs $(\pm 1,\pm 1)$. Let us call $\tau_{1}=(1,-1)$,
$\tau_{2}=(-1,-1)$ and $\tau_{3}=(-1,1)$, so that $\tau_{1}$ has kernel
isomorphic to $C_{6}$ and $\tau_{2}$, $\tau_{3}$ have kernel isomorphic to
$D_{2\cdot 3}$.
#### Case $E=C_{12}$
There are two braces with additive group $C_{12}$ and multiplicative group
$D_{2\cdot 6}$, hence two conjugation classes of regular subgroups of
$\mathrm{Hol}(C_{12})$ isomorphic to $D_{2\cdot 6}$. These are represented by
$\begin{array}[]{l}F_{1}=\langle
a_{1}=(c^{2},\mathrm{Id}),b_{1}=(c,g_{11})\rangle,\\\ F_{2}=\langle
a_{2}=(c,g_{7}),b_{2}=(c^{3},g_{11})\rangle,\end{array}$
respectively. Here $a_{1}$, $a_{2}$ have order $6$ and $b_{1}$, $b_{2}$ have
order $2$. Since $\pi_{2}(F_{1})=\langle g_{11}\rangle$ and
$\pi_{2}(F_{2})=\mathrm{Aut}(E)$, in both cases we have
$\pi_{2}(F_{i})\subseteq\Sigma_{\sigma}$ only for $\sigma=(d,1)$, $d=1,2$. On
the other hand, conjugating $a_{1}$ and $b_{1}$ by the three nontrivial
automorphisms of $C_{12}$, we obtain
$\begin{array}[]{lcl}\Phi_{g_{5}}(a_{1})=(c^{10},\operatorname{Id})=a_{1}^{5},&&\Phi_{g_{5}}(b_{1})=(c^{5},g_{11})=a_{1}^{2}b_{1},\\\
\Phi_{g_{7}}(a_{1})=(c^{2},\operatorname{Id})=a_{1},&&\Phi_{g_{7}}(b)=(c^{7},g_{11})=a_{1}^{3}b_{1},\\\
\Phi_{g_{11}}(a_{1})=(c^{10},\operatorname{Id})=a_{1}^{5},&&\Phi_{g_{11}}bg_{11}^{-1}=(c^{11},g_{11})=a_{1}^{5}b_{1}.\end{array}$
Hence, $\Phi_{g}(F_{1})=F_{1}$ for every $g$ and
$\mathrm{Aut}(B_{12})=\mathrm{Aut}(E)$. As for $F_{2}$, the only
$g\in\mathrm{Aut}(E)$ such that $\Phi_{g}(F_{2})=F_{2}$ is $g=g_{5}$, so
$\mathrm{Aut}(B_{12})=\langle g_{5}\rangle$. In any case, each morphism
$\sigma=(1,1),(2,1)$ give rise to two orbits (the ones at the first two rows
at the first table in 3.1). For each of $F_{1}$ and $F_{2}$, the first
morphism gives an additive structure, while the second one gives two additive
structures.
Now, we consider the morphisms $\tau\colon
F\longrightarrow\mathbb{Z}_{p}^{*}$. For $F_{1}$, we have that
$\tau_{2}\Phi_{g_{7}}=\tau_{3}$ with
$g_{7}\in\Sigma_{\sigma}=\mathrm{Aut}(E)$. If we now take $F_{2}$, we need to
consider the action of $\langle g_{5}\rangle$ on
$\mathrm{Hom}(F,\mathbb{Z}_{p}^{*})$, and we see that
$\tau_{2}\Phi_{g_{5}}\neq\tau_{3}$. Therefore, for each $\sigma$, the number
of orbits of the action of $\mathrm{Aut}(B_{12})\cap\Sigma_{\sigma}$ over
$\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$ is as follows:
$F_{i}\backslash\operatorname{Ker}\tau$ | $C_{6}$ | $D_{2\cdot 3}$
---|---|---
$F_{1}$ | 1 | 1
$F_{2}$ | 1 | 2
###### Proposition 20.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N:=\mathbf{Z}_{p}\rtimes C_{12}$ and multiplicative group
$G:=\mathbf{Z}_{p}\rtimes D_{2\cdot 6}$ is as shown in the following table.
$N\backslash G$ | $\mathbf{Z}_{p}\times D_{2\cdot 6}$ | $\mathbf{Z}_{p}\rtimes_{6}^{c}D_{2\cdot 6}$ | $\mathbf{Z}_{p}\rtimes_{6}^{d}D_{2\cdot 6}$
---|---|---|---
$\mathbf{Z}_{p}\times C_{12}$ | 2 | 2 | 3
$\mathbf{Z}_{p}\rtimes_{6}C_{12}$ | 4 | 4 | 6
#### Case $E=C_{6}\times C_{2}$
There is one brace with additive group $C_{6}\times C_{2}$ and multiplicative
group $D_{2\cdot 6}$, and hence one conjugacy class of regular subgroups of
$\mathrm{Hol}(C_{6}\times C_{2})$ isomorphic to $D_{2\cdot 6}$. This is
represented by
$F=\langle r=(a,\mathrm{Id}),s=(b,g_{2}^{3})\rangle\cong D_{2\cdot 6}.$
Here $r$ has order $6$, $s$ has order $2$ and $srs=r^{-1}$.
Since $\pi_{2}(F)=\langle g_{2}^{3}\rangle$, we have that
$\pi_{2}(F)=\Sigma_{\sigma}$ only for $\sigma=(0,0)$ and
$\sigma=(1,0),(0,1),(1,1)$ (corresponding to the first two rows of the final
table at 3.2). On the other hand, it is easily checked that $\Phi_{g}(F)=F$ if
and only if $g\in\langle g_{2}^{3},g_{1}g_{2}^{2}\rangle$. None of these
automorphisms satisfy that $(0,1)=(1,0)g$ or $(0,1)=(1,1)g$, while
$(1,0)=(1,1)g_{1}g_{2}^{2}$. Hence, we obtain two orbits (other than the
trivial one, given by $(0,0)$).
As for the morphisms $\tau\colon F\longrightarrow\mathbb{Z}_{p}^{*}$, we have
that $\tau_{2}=\tau_{3}\Phi_{g}$ if and only if
$g\in\\{g_{1}g_{2}^{2},g_{1}g_{2}^{5}\\}$, with
$g_{1}g_{2}^{5}\in\Sigma_{(0,1)}$, while for $\sigma=(1,0),(1,1)$, no
$g\in\mathrm{Aut}(B_{12})$ satisfies that $\tau_{2}\Phi_{g}=\tau_{3}$. We
collect the number of orbits for the action corresponding to each $\sigma$ in
the following table.
$\Sigma\backslash\operatorname{Ker}\tau$ | $C_{6}$ | $D_{2\cdot 3}$
---|---|---
$\operatorname{Aut}(E)$ | 1 | 1
$\Sigma_{(0,1)}$ | 1 | 1
$\Sigma_{(1,0)}$ | 1 | 2
$\Sigma_{(1,1)}$ | 1 | 2
###### Proposition 21.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N=\mathbf{Z}_{p}\rtimes C_{6}\times C_{2}$ and multiplicative group
$G=\mathbf{Z}_{p}\rtimes D_{2\cdot 6}$ is as shown in the following table.
$N\backslash G$ | $\mathbf{Z}_{p}\times D_{2\cdot 6}$ | $\mathbf{Z}_{p}\rtimes_{6}^{c}D_{2\cdot 6}$ | $\mathbf{Z}_{p}\rtimes_{6}^{d}D_{2\cdot 6}$
---|---|---|---
$\mathbf{Z}_{p}\times C_{6}\times C_{2}$ | 1 | 1 | 1
$\mathbf{Z}_{p}\rtimes_{6}C_{6}\times C_{2}$ | 4 | 4 | 6
#### Case $E=D_{2\cdot 6}$
There are four braces with additive group $D_{2\cdot 6}$ and multiplicative
group $D_{2\cdot 6}$, and hence four conjugacy classes of regular subgroups of
$\mathrm{Hol}(D_{2\cdot 6})$ isomorphic to $D_{2\cdot 6}$. These are
represented by
$F_{1}=\langle\varphi_{1}=(r,\mathrm{Id}),\psi_{1}=(s,\mathrm{Id})\rangle,$
$F_{2}=\langle\varphi_{2}=(rs,g_{1}),\psi_{2}=(s,\mathrm{Id})\rangle,$
$F_{3}=\langle\varphi_{3}=(r,g_{2}^{4}),\psi_{3}=(s,g_{1})\rangle,$
$F_{4}=\langle\varphi_{4}=(s,g_{2}^{2}),\psi_{4}=(r^{3}s,g_{1})\rangle.$
For $j\in\\{1,2,3,4\\}$, $\varphi_{j}$ has order $6$, $\psi_{j}$ has order
$2$, and $\psi_{j}\varphi_{j}\psi_{j}^{-1}=\varphi_{j}^{-1}$.
Let us call $\sigma_{1}=(1,-1)$, $\sigma_{2}=(-1,-1)$ and $\sigma_{3}=(-1,1)$.
We see immediately that $\pi_{2}(F_{1})=\\{\mathrm{Id}\\}$,
$\pi_{2}(F_{2})=\langle g_{1}\rangle$ and
$\pi_{2}(F_{3})=\pi_{2}(F_{4})=\langle g_{1},g_{2}^{2}\rangle$. All of these
are contained in $\Sigma_{\sigma_{i}}$ for $i\in\\{1,2,3\\}$, so
$\mathrm{Aut}(B_{12})=\\{1,\sigma_{1},\sigma_{2},\sigma_{3}\\}$. On the other
hand, we find that:
* •
For $j=1,3$, $\Phi_{g}(F_{j})=F_{j}$ for every $g\in\mathrm{Aut}(E)$.
* •
$\Phi_{g}(F_{2})=F_{2}$ if and only if $g\in\langle g_{1}\rangle$.
* •
$\Phi_{g}(F_{4})=F_{4}$ if and only if $g\in\langle g_{1}g_{2}^{2}\rangle$.
Now, we have that $\sigma_{3}=\sigma_{2}g_{1}g_{2}^{i}$ if and only if $i$ is
odd, but this is a brace automorphism only for $F_{1}$ and $F_{3}$. Thus,
other than the trivial one, there are two orbits for $F_{1}$ and $F_{3}$ and
three orbits for $F_{2}$ and $F_{4}$.
We examine the classes of morphisms $\tau\colon
F_{i}\longrightarrow\mathbb{Z}_{p}^{*}$. For $F_{1}$, we have that
$\tau_{3}\Phi_{g_{1}g_{2}^{i}}=\tau_{2}$ if and only if $i$ is odd. Hence,
$\tau_{2}\simeq\tau_{3}$ under $\mathrm{Aut}(E)$ but define two different
classes under $\Sigma_{\sigma_{2}}=\Sigma_{\sigma_{3}}$. For $F_{2}$, since
$\tau_{3}\Phi_{g_{1}}\neq\tau_{2}$, we have that $\tau_{2}\not\simeq\tau_{3}$.
For $F_{3}$, we have that $\tau_{3}\Phi_{g_{2}^{i}}=\tau_{2}$ if and only if
$i$ is odd, so we obtain the same conclusions as with $F_{1}$. Finally, for
$F_{4}$ we have that $\tau_{3}\Phi_{g_{1}g_{2}^{2}}\neq\tau_{2}$, so
$\tau_{2}\not\simeq\tau_{3}$.
We collect the numbers of orbits corresponding to each $\sigma$ in the
following tables.
$\Sigma\backslash\operatorname{Ker}\tau$ | $C_{6}$ | $D_{2\cdot 3}$
---|---|---
$\operatorname{Aut}(E)$ | 1 | 1
$\Sigma_{2}=\Sigma_{3}$ | 1 | 2
$\Sigma\backslash\operatorname{Ker}\tau$ | $C_{6}$ | $D_{2\cdot 3}$
---|---|---
$\operatorname{Aut}(E)$ | 1 | 2
$\Sigma_{2}=\Sigma_{3}$ | 1 | 2
$F_{1},F_{3}\qquad\qquad\qquad\qquad\qquad\qquad F_{2},F_{4}$
###### Proposition 22.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N=\mathbf{Z}_{p}\rtimes D_{2\cdot 6}$ and multiplicative group
$G=\mathbf{Z}_{p}\rtimes D_{2\cdot 6}$ is as shown in the following table.
$N\backslash G$ | $\mathbf{Z}_{p}\times D_{2\cdot 6}$ | $\mathbf{Z}_{p}\rtimes_{6}^{c}D_{2\cdot 6}$ | $\mathbf{Z}_{p}\rtimes_{6}^{d}D_{2\cdot 6}$
---|---|---|---
$\mathbf{Z}_{p}\times D_{2\cdot 6}$ | 4 | 4 | 6
$\mathbf{Z}_{p}\rtimes_{6}^{c}D_{2\cdot 6}$ | 8 | 8 | 12
$\mathbf{Z}_{p}\rtimes_{6}^{d}D_{2\cdot 6}$ | 12 | 12 | 24
#### Case $E=\operatorname{Dic}_{12}$
There are four braces with additive group $\operatorname{Dic}_{12}$ and
multiplicative group $D_{2\cdot 6}$, and hence four conjugacy classes of
regular subgroups of $\mathrm{Hol}(\operatorname{Dic}_{12})$ isomorphic to
$D_{2\cdot 6}$. These are represented by
$\begin{array}[]{l}F_{1}=\langle r_{1}=(xy,g_{1}),\
\,s_{1}=(y^{2},g_{1}g_{2}^{3})\rangle,\\\ F_{2}=\langle r_{2}=(xy,g_{2}),\
\,s_{2}=(y^{3},g_{1})\rangle,\\\ F_{3}=\langle
r_{3}=(xy^{2},\operatorname{Id}),\ s_{3}=(y,g_{2}^{3})\rangle,\\\
F_{4}=\langle r_{4}=(xy^{2},g_{2}^{4}),\ \,s_{4}=(y,g_{1})\rangle.\end{array}$
We have that $\pi_{2}(F_{1})=\langle g_{1},g_{2}^{3}\rangle$,
$\pi_{2}(F_{2})=\mathrm{Aut}(E)$, $\pi_{2}(F_{3})=\langle g_{2}^{3}\rangle$
and $\pi_{2}(F_{4})=\langle g_{1},g_{2}^{2}\rangle$. Hence, for each
$j=1,2,3,4$, $\pi_{2}(F_{j})\subseteq\Sigma_{\sigma}$ only for $\sigma=(d,1)$
with $d=1,2$, in which case $\Sigma_{\sigma}=\mathrm{Aut}(E)$. On the other
hand, we find that
$\Phi_{g}(F_{1})=F_{1}\iff g\in\langle
g_{1}g_{2}^{3}\rangle,\quad\Phi_{g}(F_{2})=F_{2}\iff g\in\langle
g_{1}g_{2}^{5}\rangle$
and for $j\in\\{3,4\\}$, $\Phi_{g}(F_{j})=F_{j}$ for every
$g\in\mathrm{Aut}(E)$.
Let us examine the classes of morphisms $\tau\colon
F_{j}\longrightarrow\mathbb{Z}_{p}^{*}$ for each $j$. For $F_{1}$ we have that
$\tau_{2}(s_{1})\neq\tau_{3}\Phi_{g_{1}g_{2}^{3}}(s_{1})$, so
$\tau_{2}\not\simeq\tau_{3}$. For $F_{2}$,
$\tau_{2}(s_{2})\neq\tau_{3}\Phi_{g_{1}g_{2}^{5}}(s_{2})$, whence
$\tau_{2}\not\simeq\tau_{3}$ neither. For $F_{3}$, we see that
$\tau_{2}=\tau_{3}\Phi_{g_{1}g_{2}^{2}}$ with $g_{1}g_{2}^{2}$, hence
$\tau_{2}\simeq\tau_{3}$. Finally, for $F_{4}$, we see that
$\tau_{2}=\tau_{3}\Phi_{g_{1}}$, so $\tau_{2}\simeq\tau_{3}$. Then, the orbits
for each $\sigma$ are as in the following table.
$F_{j}\backslash\operatorname{Ker}\tau$ | $C_{6}$ | $D_{2\cdot 3}$
---|---|---
$F_{1}$ | 1 | 2
$F_{2}$ | 1 | 2
$F_{3}$ | 1 | 1
$F_{4}$ | 1 | 1
###### Proposition 23.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N=\mathbf{Z}_{p}\rtimes Dic_{12}$ and multiplicative group
$G=\mathbf{Z}_{p}\rtimes D_{2\cdot 6}$ is as shown in the following table.
$N\backslash G$ | $\mathbf{Z}_{p}\times D_{2\cdot 6}$ | $\mathbf{Z}_{p}\rtimes_{6}^{c}D_{2\cdot 6}$ | $\mathbf{Z}_{p}\rtimes_{6}^{d}D_{2\cdot 6}$
---|---|---|---
$\mathbf{Z}_{p}\times\operatorname{Dic}_{12}$ | 4 | 4 | 6
$\mathbf{Z}_{p}\rtimes_{6}\operatorname{Dic}_{12}$ | 8 | 8 | 12
### 4.5 $F=\operatorname{Dic}_{12}$
We take for $F$ the dicyclic group of order 12. According to the table in
Section 3 and Proposition 4, for a brace of size 12 with multiplicative group
$F=\operatorname{Dic}_{12}$, the additive group can be any group of order 12.
In all the sub-cases the group $\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$ is as in
3.5 so that $\tau$ identifies as a pair $(1,1),(2,1),(4,1)$ or $(4,3)$.
#### Case $E=C_{12}$
Let $E=\langle c\rangle\simeq C_{12}$. There is one brace with additive group
$C_{12}$ and multiplicative group $\operatorname{Dic}_{12}$, hence one
conjugation class of regular subgroups of $\mathrm{Hol}(C_{12})$ isomorphic to
$\operatorname{Dic}_{12}$. We may consider representative
$F=\langle x=(c^{4},\operatorname{Id}),\ y=(c^{3},g_{5})\rangle,$
where $x$ has order 3, $y$ has order 4 and $y\,x=x^{2}\,y$.
Since $\pi_{2}(F)=\langle g_{5}\rangle$ we have
$\pi_{2}(F)\subseteq\Sigma_{\sigma}$ only for $\sigma=(d,j)$ with $d=1,2,4$.
Since $F$ is a normal subgroup of $\mathrm{Hol}(E)$, we have
$\Phi_{g}(F)=F,\,\forall g\in\operatorname{Aut}(E)$. Since $\sigma=(4,1)$ and
$\sigma^{\prime}=(4,3)$ are in the same orbit, the action of
$\operatorname{Aut}(B_{12})$ gives one orbit for each order. All together, we
have 3 additive structures for $F$ if $p\equiv 1\pmod{4}$ and 2 otherwise.
For $\sigma$ of order $1$ or two,
$\Sigma_{\sigma}=\operatorname{Aut}(E)=\operatorname{Aut}(B_{12})$ and the
action on $\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$ gives one orbit for each order.
For $\sigma$ of order $4$, to consider the action of
$\operatorname{Aut}(B_{12})\cap\Sigma_{\sigma}$ we just have to check the
action of $g_{5}$ on $y$. Since
$\Phi_{g_{5}}(y)=(c^{3},g_{5})=y$
$\tau=(4,1)$ and $\tau^{\prime}=(4,3)$ are not in the same orbit and this
gives two multiplicative structures with kernel of order 3 for $\sigma$.
###### Proposition 24.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N=\mathbf{Z}_{p}\rtimes C_{12}$ and multiplicative group
$G=\mathbf{Z}_{p}\rtimes\operatorname{Dic}_{12}$ is as shown in the following
table, where we need $p\equiv 1\pmod{\frac{12}{k}}$ for a kernel of size $k$
to occur.
$N\backslash G$ | $\mathbf{Z}_{p}\times\operatorname{Dic}_{12}$ | $\mathbf{Z}_{p}\rtimes_{6}\operatorname{Dic}_{12}$ | $\mathbf{Z}_{p}\rtimes_{3}\operatorname{Dic}_{12}$
---|---|---|---
$\mathbf{Z}_{p}\times C_{12}$ | 1 | 1 | 1
$\mathbf{Z}_{p}\rtimes_{6}C_{12}$ | 2 | 2 | 2
$\mathbf{Z}_{p}\rtimes_{3}C_{12}$ | 2 | 2 | 4
#### Case $E=C_{6}\times C_{2}$
We write $E=\langle a\rangle\times\langle b\rangle$. There is one brace with
additive group $C_{6}\times C_{2}$ and multiplicative group
$\operatorname{Dic}_{12}$, hence one conjugation class of regular subgroups of
$\mathrm{Hol}(C_{6}\times C_{2})$ isomorphic to $\operatorname{Dic}_{12}$ for
which we may consider as representative the following subgroup
$F=\langle x=(a^{2},\operatorname{Id}),y=(b,g_{1}g_{2})\rangle.$
Since $\pi_{2}(F)=\langle g_{1}g_{2}\rangle$ there is only one nontrivial
$\sigma$ to consider, namely that of order 2 identified as $(1,1)$ and defined
by $\sigma(a)=\sigma(b)=\sigma_{2}=-1$, which has stabiliser $\langle
g_{2}^{3},g_{2}^{4}g_{1}\rangle\simeq V_{4}$. Since we only have $1$ and
$\sigma$, it is clear that they are fixed points for the action of
$\operatorname{Aut}(B_{12)}$ and we have two orbits, namely 2 additive
structures for $F$.
Since $F$ is normal in $\mathrm{Hol}(E)$, the elements of $\langle
g_{2}^{3},g_{2}^{4}g_{1}\rangle$ are brace automorphisms and to determine its
action on $\mathrm{Hom}(F,\mathbf{Z}_{p}^{*})$, we just have to check the
action on $y$. From $\Phi_{g_{1}g_{2}}(y)=y^{3}$, we obtain that
$\tau=(4,3)=(4,1)\Phi_{g_{1}g_{2}}$ and there is just one orbit in order 4.
###### Proposition 25.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N=\mathbf{Z}_{p}\rtimes(C_{6}\times C_{2})$ and multiplicative group
$G=\mathbf{Z}_{p}\rtimes\operatorname{Dic}_{12}$ is as shown in the following
table, where we need $p\equiv 1\pmod{4}$ for the group
$\mathbf{Z}_{p}\rtimes_{3}\operatorname{Dic}_{12}$ to occur.
$N\backslash G$ | $\mathbf{Z}_{p}\times\operatorname{Dic}_{12}$ | $\mathbf{Z}_{p}\rtimes_{6}\operatorname{Dic}_{12}$ | $\mathbf{Z}_{p}\rtimes_{3}\operatorname{Dic}_{12}$
---|---|---|---
$\mathbf{Z}_{p}\times(C_{6}\times C_{2})$ | 1 | 1 | 1
$\mathbf{Z}_{p}\rtimes_{6}(C_{6}\times C_{2})$ | 2 | 2 | 2
#### Case $E=A_{4}$
There are two braces with additive group $A_{4}$ and multiplicative group
$\operatorname{Dic}_{12}$, hence two conjugation classes of regular subgroups
of $\mathrm{Hol}(A_{4})$ isomorphic to $\operatorname{Dic}_{12}$. We may
consider the following representatives
$\begin{array}[]{l}F_{1}=\langle x_{1}=((1,3,2),\phi_{(1,2,3)}),\
y_{1}=((1,3)(2,4),\phi_{(1,2)})\rangle,\\\\[4.30554pt] F_{2}=\langle
x_{2}=((1,3,4),\operatorname{Id}),\
y_{2}=((1,3)(2,4),\phi_{(1,3,2,4)})\rangle,\end{array}$
where $x_{1},x_{2}$ have order 3, $y_{1},y_{2}$ have order 4 and
$y_{i}x_{i}=x_{i}^{2}y_{i}$ for $i=1,2$. We have
$\pi_{2}(F_{1})=\langle\phi_{(1,2,3)}\phi_{(1,2)}\rangle\simeq S_{3}$ and
$\pi_{2}(F_{1})=\langle\phi_{(1,3,2,4)}\rangle$ which shows that $F_{1},F_{2}$
are not conjugate in $\mathrm{Hol}(A_{4})$ and that their projections are not
subgroups of $A_{4}$, so that we obtain only one additive structure for each
$F_{1},F_{2}$, corresponding to the direct product.
Since $\Sigma_{1}=\operatorname{Aut}(E)$ in order to determine multiplicative
structures, we have one orbit for order 1 and one for order 2, and we only
need to check if $\tau=(4,1)$ and $\tau^{\prime}=(4,3)$ differ in a brace
automorphism. We have
$\Phi_{\phi_{(1,2)}}(x_{1})=x_{1}^{2},\quad\Phi_{\phi_{(1,2)}}(y_{1})=y_{1}^{3},\quad\Phi_{\phi_{(1,2)}}(F_{1})=F_{1},\quad(4,3)=(4,1)\Phi_{\phi_{(1,2)}}$
$\Phi_{\phi_{(3,4)}}(x_{2})=x_{2}^{2},\quad\Phi_{\phi_{(3,4)}}(y_{2})=y_{2}^{3},\quad\Phi_{\phi_{(3,4)}}(F_{2})=F_{2},\quad(4,3)=(4,1)\Phi_{\phi_{(3,4)}}.$
So, for $F_{1}$ and $F_{2}$ we obtain one class of morphisms $\tau$ with
kernel of order 6 and, if $p\equiv 1\pmod{4}$, there is another class of
morphisms $\tau$ with kernel of order 3.
###### Proposition 26.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N=\mathbf{Z}_{p}\rtimes A_{4}$ and multiplicative group
$G=\mathbf{Z}_{p}\rtimes\operatorname{Dic}_{12}$ is as shown in the following
table, where we need $p\equiv 1\pmod{4}$ for the group
$\mathbf{Z}_{p}\rtimes_{3}\operatorname{Dic}_{12}$ to occur.
$N\backslash G$ | $\mathbf{Z}_{p}\times\operatorname{Dic}_{12}$ | $\mathbf{Z}_{p}\rtimes_{6}\operatorname{Dic}_{12}$ | $\mathbf{Z}_{p}\rtimes_{3}\operatorname{Dic}_{12}$
---|---|---|---
$\mathbf{Z}_{p}\times A_{4}$ | 2 | 2 | 2
#### Case $E=D_{2\cdot 6}$
There are two braces with additive group $D_{2\cdot 6}$ and multiplicative
group $\operatorname{Dic}_{12}$, hence two conjugation classes of regular
subgroups of $\mathrm{Hol}(D_{2\cdot 6})$ isomorphic to
$\operatorname{Dic}_{12}$. We take representatives
$\begin{array}[]{l}F_{1}=\langle x_{1}=(r^{2},\operatorname{Id}),\
y_{1}=(sr^{5},g_{2}^{3})\rangle,\\\ F_{2}=\langle x_{2}=(r^{2},g_{2}^{2}),\
y_{2}=(r^{2}s,g_{2}g_{1})\rangle,\end{array}$
where $x_{1},x_{2}$ have order 3, $y_{1},y_{2}$ have order 4 and
$y_{i}x_{i}=x_{i}^{2}y_{i}$ for $i=1,2$.
Since $\pi_{2}(F_{1})=\langle g_{2}^{3}\rangle$ and $\pi_{2}(F_{2})=\langle
g_{2}^{2},g_{2}g_{1}\rangle$, we see that the groups are not conjugate and
that the only nontrivial $\sigma$ to consider is that of order $2$ with cyclic
kernel. So, the number of additive structures for each $F_{i}$ is $2$.
Again, to determine multiplicative structures it is enough to check if $(4,3)$
and $(4,1)$ differ in a brace automorphism. We have
$\Phi_{g_{2}^{3}}(x_{1})=x_{1},\quad\Phi_{g_{2}^{3}}(y_{1})=(r^{4}s,g_{2}^{3})=y_{1}^{3},\quad\Phi_{g_{2}^{3}}(F_{1})=F_{1},\quad(4,3)=(4,1)\Phi_{g_{2}^{3}}$
$\Phi_{g_{2}^{3}}(x_{2})=x_{2},\quad\Phi_{g_{2}^{3}}(y_{2})=(r^{5}s,g_{2}g_{1})=y_{2}^{3},\quad\Phi_{g_{2}^{3}}(F_{2})=F_{2},\
(4,3)=(4,1)\Phi_{g_{2}^{3}}$
So, for each $F_{1},F_{2}$ we obtain one orbit for each order.
###### Proposition 27.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N=\mathbf{Z}_{p}\rtimes D_{2\cdot 6}$ and multiplicative group
$G=\mathbf{Z}_{p}\rtimes\operatorname{Dic}_{12}$ is as shown in the following
table, where we need $p\equiv 1\pmod{4}$ for the group
$\mathbf{Z}_{p}\rtimes_{3}\operatorname{Dic}_{12}$ to occur.
$N\backslash G$ | $\mathbf{Z}_{p}\times\operatorname{Dic}_{12}$ | $\mathbf{Z}_{p}\rtimes_{6}\operatorname{Dic}_{12}$ | $\mathbf{Z}_{p}\rtimes_{3}\operatorname{Dic}_{12}$
---|---|---|---
$\mathbf{Z}_{p}\times D_{2\cdot 6}$ | 2 | 2 | 2
$\mathbf{Z}_{p}\rtimes_{6}^{c}D_{2\cdot 6}$ | 4 | 4 | 4
#### Case $E=\operatorname{Dic}_{12}$
There are two braces with additive group $\operatorname{Dic}_{12}$ and
multiplicative group $\operatorname{Dic}_{12}$, hence two conjugation classes
of regular subgroups of $\mathrm{Hol}(\operatorname{Dic}_{12})$ isomorphic to
$\operatorname{Dic}_{12}$. We may consider the following representatives
$\begin{array}[]{l}F_{1}=\langle x_{1}=(x,\operatorname{Id}),\
y_{1}=(y,\operatorname{Id})\rangle,\\\ F_{2}=\langle x_{2}=(x^{2},g_{2}^{2}),\
y_{2}=(x^{2}y,g_{2}g_{1})\rangle,\end{array}$
where $x_{1},x_{2}$ have order 3, $y_{1},y_{2}$ have order 4 and
$y_{i}x_{i}=x_{i}^{2}y_{i}$ for $i=1,2$. Since
$\pi_{2}(F_{1})=\\{\operatorname{Id}\\}$ and $\pi_{2}(F_{2})=\langle
g_{2}^{2},g_{2}g_{1}\rangle=\langle g_{2}^{2},g_{1}g_{2}\rangle$ we see that
groups are not conjugate and that we can consider all possible $\sigma$. We
have
$\Phi_{g_{1}}(x_{1})=x_{1}^{2},\quad\Phi_{g_{1}}(y_{1})=y_{1}^{3},\quad\Phi_{g_{1}}(F_{1})=F_{1},$
$\Phi_{g_{1}}(x_{2})=x_{2}^{2},\quad\Phi_{g_{1}}(y_{2})=y_{2}^{3},\quad\Phi_{g_{1}}(F_{2})=F_{2},$
and $(4,3)=(4,1)g_{1}\mbox{ in }\mathrm{Hom}(E,\mathbf{Z}_{p}^{*}).$ Then, the
number of additive structures for each $F_{i}$ is $3$ if $p\equiv 1\pmod{4}$
and $2$ otherwise.
For $\sigma=(2,1)$, since $\Sigma_{\sigma}=\operatorname{Aut}(E)$, the above
computation shows that $(4,3)=(4,1)\Phi_{g_{1}}$ and we have just one orbit of
order 4. But for $\sigma$ order 4, since $\Sigma_{\sigma}$ is also the
stabiliser of the $\tau$’s of order 4, we have fixed points and 2 orbits.
###### Proposition 28.
Let $p$ be a prime number, $p\geq 7$. The number of braces with additive group
$N:=\mathbf{Z}_{p}\rtimes\operatorname{Dic}_{12}$ and multiplicative group
$G:=\mathbf{Z}_{p}\rtimes\operatorname{Dic}_{12}$ is as shown in the following
table, where we need $p\equiv 1\pmod{\frac{12}{k}}$ for a kernel of size $k$
to occur.
$N\backslash G$ | $\mathbf{Z}_{p}\times\operatorname{Dic}_{12}$ | $\mathbf{Z}_{p}\rtimes_{6}\operatorname{Dic}_{12}$ | $\mathbf{Z}_{p}\rtimes_{3}\operatorname{Dic}_{12}$
---|---|---|---
$\mathbf{Z}_{p}\times\operatorname{Dic}_{12}$ | 2 | 2 | 2
$\mathbf{Z}_{p}\rtimes_{6}\operatorname{Dic}_{12}$ | 4 | 4 | 4
$\mathbf{Z}_{p}\rtimes_{3}\operatorname{Dic}_{12}$ | 4 | 4 | 8
### 4.6 Total numbers
For a prime number $p\geq 7$ we compile in the following tables the total
number of skew left braces of size $12p$.
The additive group is a semidirect product $\mathbf{Z}_{p}\rtimes E$ and the
multiplicative group is a semidirect product $\mathbf{Z}_{p}\rtimes F$. In the
first column we have the possible $E$’s and in the first row the possible
$F$’s.
* •
If $p\equiv 11\pmod{12}$
| $C_{12}$ | $C_{6}\times C_{2}$ | $A_{4}$ | $D_{2\cdot 6}$ | $\operatorname{Dic}_{12}$ |
---|---|---|---|---|---|---
$C_{12}$ | 6 | 9 | 0 | 21 | 6 |
$C_{6}\times C_{2}$ | 6 | 8 | 1 | 17 | 6 |
$A_{4}$ | 0 | 4 | 4 | 0 | 4 |
$D_{2\cdot 6}$ | 12 | 34 | 0 | 90 | 12 |
$\operatorname{Dic}_{12}$ | 12 | 18 | 0 | 42 | 12 |
| 36 | 73 | 5 | 170 | 40 | $\mathbf{324}$
* •
If $p\equiv 5\pmod{12}$
| $C_{12}$ | $C_{6}\times C_{2}$ | $A_{4}$ | $D_{2\cdot 6}$ | $\operatorname{Dic}_{12}$ |
---|---|---|---|---|---|---
$C_{12}$ | 17 | 9 | 0 | 21 | 17 |
$C_{6}\times C_{2}$ | 9 | 8 | 1 | 17 | 9 |
$A_{4}$ | 0 | 4 | 4 | 0 | 6 |
$D_{2\cdot 6}$ | 18 | 34 | 0 | 90 | 18 |
$\operatorname{Dic}_{12}$ | 34 | 18 | 0 | 42 | 34 |
| 78 | 73 | 5 | 170 | 84 | $\mathbf{410}$
* •
If $p\equiv 7\pmod{12}$
| $C_{12}$ | $C_{6}\times C_{2}$ | $A_{4}$ | $D_{2\cdot 6}$ | $\operatorname{Dic}_{12}$ |
---|---|---|---|---|---|---
$C_{12}$ | 36 | 54 | 0 | 21 | 6 |
$C_{6}\times C_{2}$ | 36 | 46 | 8 | 17 | 6 |
$A_{4}$ | 0 | 32 | 32 | 0 | 4 |
$D_{2\cdot 6}$ | 24 | 68 | 0 | 90 | 12 |
$\operatorname{Dic}_{12}$ | 24 | 36 | 0 | 42 | 12 |
| 120 | 236 | 40 | 170 | 40 | $\mathbf{606}$
* •
If $p\equiv 1\pmod{12}$
| $C_{12}$ | $C_{6}\times C_{2}$ | $A_{4}$ | $D_{2\cdot 6}$ | $\operatorname{Dic}_{12}$ |
---|---|---|---|---|---|---
$C_{12}$ | 94 | 54 | 0 | 21 | 17 |
$C_{6}\times C_{2}$ | 54 | 46 | 8 | 17 | 9 |
$A_{4}$ | 0 | 32 | 32 | 0 | 6 |
$D_{2\cdot 6}$ | 36 | 68 | 0 | 90 | 18 |
$\operatorname{Dic}_{12}$ | 68 | 36 | 0 | 42 | 34 |
| 252 | 236 | 40 | 170 | 84 | $\mathbf{782}$
With the results summarized in the above tables, the validity of conjecture
(1) is then established.
## References
* [1] V.G. Bardakov, M.V. Neshchadim, M.K. Yadav, _Computing skew left braces of small orders_ , Internat. J. Algebra Comput. 30 (2020), no. 4, 839–851.
* [2] F. Cedó, _Left Braces: Solutions of the Yang-Baxter Equation_ , Advances in Group Theory and Applications, 5 (2018), 3–90.
* [3] T. Crespo, D. Gil-Muñoz, A. Rio, M. Vela, _Left braces of size $8p$_, Journal of Algebra Volume 617, (2023), 317-339.
* [4] T. Crespo, D. Gil-Muñoz, A. Rio, M. Vela, _Inducing braces and Hopf Galois structures_ , to appear in J. Pure Appl. Algebra (2023)
* [5] M. J. Curran, _Automorphisms of semidirect products_ , Mathematical Proceedings of the Royal Irish Academy Vol. 108A, No. 2 (2008), pp. 205-210
* [6] L. Guarnieri, L. Vendramin, _Skew braces and the Yang-Baxter equation_ , Math. Comp. 86 (2017), 2519-2534.
* [7] W. Rump, _Braces, radical rings, and the quantum Yang–Baxter equation_ , Journal of Algebra, 307 (2007), 153-170.
* [8] A. Smoktunowicz, L. Vendramin, _On skew braces (with an appendix by N. Byott and L. Vendramin)_ , J. Comb. Algebra 2 no. 1 (2018), 47–86
* [9] L. Vendramin, _Problems on skew left braces_ , Advances in Group Theory and Applications, 7 (2019), 15–37.
|
# AI-Assisted Design Concept Exploration
Through Character Space Construction
Shin Sano 1,3∗ and Seiji Yamada 1,2
1Department of Informatics, School of Multidisciplinary Sciences,
Graduate University for Advanced Studies (SOKENDAI), Hayama , Japan
2Digital Content and Media Sciences Research Division, National Institute of
Informatics , Tokyo, Japan
3Institute for Creative Integration , Oakland, California, USA
###### Abstract
We propose an AI-assisted design concept exploration tool, the “Character
Space Construction” (“CSC”). Concept designers explore and articulate the
target product aesthetics and semantics in language, which is expressed using
“Design Concept Phrases” (“DCPs”), that is, compound adjective phrases, and
contrasting terms that convey what are not their target design concepts.
Designers often utilize this dichotomy technique to communicate the nature of
their aesthetic and semantic design concepts with stakeholders, especially in
an early design development phase. The CSC assists this designers’ cognitive
activity by constructing a “Character Space” (“CS”), which is a semantic
quadrant system, in a structured manner. A CS created by designers with the
assistance of the CSC enables them to discern and explain their design
concepts in contrast with opposing terms. These terms in a CS are retrieved
and combined in the CSC by using a knowledge graph. The CSC presents terms and
phrases as lists of candidates to users from which users will choose in order
to define the target design concept, which is then visualized in a CS. The
participants in our experiment, who were in the “arts and design” profession,
were given two conditions under which to create DCPs and explain them. One
group created and explained the DCPs with the assistance of the proposed CSC,
and the other did the same task without this assistance, given the freedom to
use any publicly available web search tools instead. The result showed that
the group assisted by the CSC indicated their tasks were supported
significantly better, especially in exploration, as measured by the Creativity
Support Index (CSI).
_K_ eywords Intelligent interactive system $\cdot$ Industrial design $\cdot$
Concept design $\cdot$ Creativity support tool $\cdot$ Product semantics
$\cdot$ Design aesthetics $\cdot$ Human-computer interaction $\cdot$ Lexical
semantics
## 1 Introduction
Our research is motivated by an observation in the professional industrial
design domain and a research question derived from it: “could concept
designers’ creative activities, especially verbalizing novel design concept
phrase, be modeled and computationally supported?” The concept design used in
industrial design is primarily concerned with developing the aesthetics and
semantics of products[1]. The visual appearance and meaning behind a product
is frequently a key attribute of a product’s appeal to consumers [2, 3, 4], as
the functionality of a product is increasingly taken for granted, and users
are looking for different levels of appreciation [5]. Consequently, designers
are in charge of not only creating the visual appearance of products but also
verbally articulating the product semantics and emotional feelings attached to
them [6]. Design concepts articulated using this verbal mode often have to be
explored, communicated, understood, and approved by the project stakeholders
during a design project. Krippendorff [1] called this practice a “design
discourse”, describing, “Designers have to justify the aesthetics of mass-
produced products, products that would ideally be of use to everyone, not
individual works of art.” This characterizes the special role industrial
designers play, as distinguished from engineering designers, and this role is
most clearly observable in the automobile industry [7].
While emotional components in industrial design are important [8, 9, 10],
automotive customers especially value emotional experiences when they own and
use products, such as those related to their self-image and brand messages
reflected in vehicles [11, 12]. Designers and other stakeholders exchange
views on the form of a vehicle design as part of the design process. Tovey [7]
characterized the language they use as “idiosyncratic and atypical,” requiring
them to explore and communicate different shapes and features in words, such
as “slippery,” “exciting,” “fluid,” “tailored,” and “sheer.” Such language
used in car design studios describes particular forms or connotes a “feeling.”
Bouchard et al. [13] described these verbal expressions as “intermediate
representations” that will eventually be translated into visual aesthetic and
semantic features of product designs. Designers in practice often feel that
they run out of these words to express different design concepts in different
projects.
The present research attempts to uncover professional concept designers’
cognitive activities, that are traditionally highly empirical and
undocumented, for when they explore and form such verbal representations of
design concepts and also attempts to computationally support the process by an
AI algorithm. The contributions of this research are as follows:
* •
We defined concept designers’ cognitive activities in exploring, generating,
and explaining a design concept with a “Design Concept Phrase” (“DCP”) and
contrasting concepts, and modeled this process as constructing what we call a
“Character Space” (“CS”).
* •
We implemented this model of constructing a CS into an AI algorithm and
created an interactive system, which supports concept designers’ creative
process described above.
* •
We defined a DCP as a compound phrase using an adjective-“noun-form-of”
adjective formula.
In this paper, we will first discuss the background of the topic space
together with related works. We will highlight how professional concept
designers utilize a verbal mode of exploration to generate and communicate
novel design concepts in the early stage of the design process, especially
with an emphasis on their process of using a dichotomy technique, taking as an
example the development of an automotive design concept. Then, we propose an
AI-assisted interactive system, “Character Space Construction” (“CSC”), and
its methodologies, utilizing the quadrant system, that is, the CS, for
designers to generate and use DCPs. Last, we introduce our experiment with
participants in design profession to measure the effectiveness of the tool,
followed by discussions.
We illustrate a case study of how they use DCPs from an automotive design,
however, these adjective phrases are used in the practice of industrial design
in general [14], fashion design[15], furniture design [16], or even in the
tourism industry [17]. Therefore, the proposed methods can contribute to
various design and marketing fields.
## 2 Background
Figure 1: (A) left: Character space (CS) for “kinetic warmth” (B) right:
Toyota Concept-i design derived from “kinetic warmth” (©2021 Toyota Motor
Sales, U.S.A., Inc.).
### 2.1 Defining Design Concept Phrase and Character Space
First, we define the “Design Concept Phrase” (“DCP”) as a compound adjective
phrase that conveys product aesthetics and semantics. In the industry, such
phrases are sometimes called “keywords” [18]; however, we call them DCP in
this research since the term “keywords” is used differently in the natural
language processing (NLP) context. A DCP, “kinetic warmth,” for example, was
created by the designers at Toyota’s North American design studio for
Concept-i (Fig. 1-(B)). Note that the noun “warmth” was converted into a noun
from the adjective “warm.” Ian Cartabiano, the chief designer of Concept-i at
Toyota’s Calty Design Research studio explained, “There’s a school of design
at the moment which makes a car feel just like a laptop on wheels, an
impersonal transport unit. We wanted to do something more soulful, intriguing,
lively, and human.” When they identified “kinetic” and “warmth,” the design
team explored other terms, such as “dynamic,” “engaging,” “intriguing,”
“soulful,” “human,” “lively,” and “passionate” 111Car Design News (2017). CES
2017: Toyota Concept-i in depth. https://www.cardesignnews.com/cdn-
live/ces-2017-toyota-concept-i-in-depth/22991.article [Accessed October 22,
2021]. . Notice that Cartabiano expressed a type of aesthetics and semantics
to be avoided, that is, “a laptop on wheels, an impersonal transport unit.”
They later put those feelings into two other contrasting terms, “calm” and
“cold,” and placed them onto a quadrant system, which we define in this
research as the “Character Space” (“CS”, Fig. 1(A)). These contrasting terms
make “kinetic warmth” understanable by stating what is NOT “kinetic warmth.”
A CS visually resembles a dimensionality reduction on 2D semantic space
employing multidimensional scaling (MDS); however, the way a CS is formed in
practice fundamentally differs from how an MDS is formed. A CS is informally
utilized as a synthesis tool employing a quadrant system, on which a user
identifies each end of the axis through creative exploration and speculation.
The primary focus is the upper-right quadrant, which represents the target
concept expressed by a compound adjective; then, the other quadrants will be
defined as the user identifies opposite terms to each of the compound
adjectives (e.g., “kinetic-calm” and “warm-cold”). These polar terms on the
axes consequently form semantic differentials [19], however in a CS, the two
terms in an orthogonal relationship (e.g., “kinetic” on the vertical axis and
“warm” on the horizontal axis) are identified first before a pair of polar
terms on a semantic differential is defined. Also, that these polar terms can
be defined by the designers as a part of creation depending on what aspect of
a particular adjective they want to emphasize. For instance, an adjective,
“beautiful,” can be perceived differently depending on the term at the other
end (e.g., “beautiful”$\leftrightarrow$“ugly” or
“beautiful”$\leftrightarrow$“practical”). A CS is completed when all four ends
on the axes are defined. In this research, we label the top end of the
vertical axis as word 1 ($w_{1}$), and the rest are then labeled in clockwise
order as word 2 ($w_{2}$), word 3 ($w_{3}$), and word 4 ($w_{4}$). The upper-
right quadrant, represented by the combination of $w_{1}$ and $w_{2}$, is the
target design-concept phrase. All the other quadrants, represented by
$w_{2}$-$w_{3}$, $w_{3}$-$w_{4}$, and $w_{4}$-$w_{1}$, are contrasting
concepts to be used to explain the target DCP in terms of how and by which
attributes they differ and what already exists or what is to be avoided.
Meanwhile, a multidimensional scaling with MDSs has been widely used as an
analytical tool since first being introduced by Ramsay [20] and became a
popular data visualization tool when Zimet et al. [21] used them in the form
of the Multidimensional Scale of Perceived Social Support (MSPSS). MDSs have
been successfully implemented in new product development domains, including
concept mapping [22] and kansei engineering [23]. They visualize the spatial
distance between individual cases of a data set according to similarities. The
usage of MDSs first involves collecting data on the basis of ratings of
statements or semantic differentials for each case, followed by dimensionality
reduction done by using statistical techniques, such as principal component
analysis (PCA), to yield a scatter plot visualization of the cases on a two-
dimensional space. After a visualization with an MDS users usually have to
define the labels of the axes by interpreting the collective meanings of the
distantly-plotted clusters on an MDS. Therefore, a dimensionality reduction is
a methodologically different process from the CS, which we attempt to define
in this research.
### 2.2 Compound adjective phrase
While our goal is to address product aesthetics and semantics that evoke end
users’ emotional feelings attached to them, we focused on adjectives that
describe the emotional aspects of product designs. Hsiao and Chen [24]
conducted surveys using product images and adjectives on semantic
differentials. The factor analyses identified four main dimensions of the
participants’ emotional responses to product shapes, characterized by the
pairs of polar affective adjectives. They demonstrated the role of adjectives
in product designs, yet their focus was not to explore a variety of nuanced
adjectives to describe a unique design concept.
In order to describe distinct quality of a design aesthetics and semantics, we
chose to use adjective-adjective combinations. This is called a compound
adjective in a coordinated sequence - a sequence in which two or more
adjectives severally restrict the head noun (e.g., a tall thin woman). In this
example, the phrase therefore means a “woman who is both tall and thin.” The
criteria for being coordinated and in a sequence is that the adjectives in a
coordinated sequence may, but need not, be linked by “and.” Also, a change in
the order of adjectives in a coordinated sequence may result in a shifting of
the emphasis, or even in a stylistic flaw, but will not change the meaning of
the phrase [25]. In addition, the second adjective takes the noun form (e.g.,
“kinetic warm”$\rightarrow$“kinetic warmth,” “effortless
elegant”$\rightarrow$“effortless elegance,” “fluid
beautiful”$\rightarrow$“fluid beauty”) in order for the phrases to sound
complete by themselves. In the case of “kinetic warm(th),” the complete phrase
would be “kinetic and warm style” or “kinetic and warm car”; however, since
the head nouns “style” or “car” are obvious in the context, they are omitted.
Therefore, the type of phrase we use in this research can be defined as a
compound adjective (adjective + adjective) phrase in a coordinated sequence,
which is modified to an adjective-noun form. This type of compound phrase is
underexplored in the research topic called “combinational creativity.”
Combinational creativity attempts to produce new ideas by combining existing
ideas in unfamiliar ways [26]. A number of studies have identified the effect
of combinational creativity as a tool for generating creative design concepts
[27, 28, 29]. Nagai et al. [30], in a collaboration with designers, suggested
that a new concept can be created by synthesizing two ideas in noun-noun
combination phrases. Chiu and Shu [29] used oppositely and similarly related
pairs of words as stimuli for design-concept generation and observed that
oppositely related verb-verb combination stimuli could increase concept
creativity. The Combinator [31] imitates the way the human brain achieves
combinational creativity and suggests combined ideas in both visual and
textual representations. In these works, combinational creativity focused
either on combined product categories or objects derived from noun-noun
combinations (e.g., “desk + elevator” or “pen + ruler”) or functional features
derived from verb-verb combinations (e.g., “fill + insert”), and they did not
address product aesthetics.
### 2.3 Lexico-semantic exploration tool
In constructing a CS in our system, we use the ConceptNet knowledge graph [32]
for exploring compound phrases. ConceptNet is a knowledge graph that connects
words and phrases with labeled edges. It is designed to represent varieties of
general knowledge to allow users to make word associations and make sense of
the meanings behind them. Its knowledge is collected from a wide range of
sources, such as expert-created resources, crowd-sourcing, and games with a
purpose. Among creativity support tools, Spinneret [33] uses ConceptNet and
provides idea suggestions as nodes on a graph. It aims to support divergent
thinking, and it explores ideas and provides “suggestions” to the user to add
to a mind map. Mini-Map [34] also uses ConceptNet, and it gamifies
collaborations with an intelligent agent in creating a mind map. The both
studies chose the ConceptNet as to be suitable for enhancing explorations and
mitigating fixation during an ideation process.
### 2.4 Metrics of creativity support tools
Unlike productivity support tools that can be evaluated on the basis of
objective metrics, the evaluation of creativity support tools lacks obvious
metrics, and there is no one-size-fits-all approach [35]. In one of the
several dimensions of creativity that Sternberg [36] discussed, creativity can
be a property of people, a property of products, and a property of a set of
cognitive processes. With respect to a property of products, “originality” and
“effectiveness” are popular criteria [37, 38, 39]. Both terms have versions of
labels, such as “novelty,” “unusuality,” or “uniqueness” for the former and
“usefulness,” “practicality,” “appropriateness,” or “relevancy” for the latter
[40, 41, 42]. They are measured either on the basis of self-report evaluation
or rated by experts with the precaution of the inter-rater disagreement. With
respect to evaluating a property of people, it tends to lead to a concern with
individual differences between people.
In this research, we used the Creativity Support Index (CSI)[43], which covers
the two properties of creativity above, that are, a property of products, and
a property of a set of cognitive processes, both in terms of users’ own
perceptions. This would be suitable for the present research, especially when
the objective is to cater a tool, which is designed with certain functional
requirements in mind, to support the effectiveness of designers’ cognitive
processes. We will discuss the further details of the CSI, its benefits and
application in section 4.3.
## 3 Method
### 3.1 Preliminary study for adjective-adjective compound phrases
As discussed in section 2.2, we intend to use compound adjective phrases for
the DCP. Despite the fact that some related works demonstrated the role of
adjectives in conveying users’ emotional responses to product aesthetics[24],
or the effectiveness of combinational creativity[24], this particular type of
combinational phrase is under-researched in the context of supporting
designers’ creativity, there is no prior work that demonstrated the
effectiveness of adjective-adjective compound phrase compare to different
types with other parts-of-speech(PoS), such as noun-noun or verb-verb. Thus,
we conducted a preliminary experiment comparing phrases with different PoS to
confirm that adjective phrase combinations are suitable for communicating
product aesthetics.
We used Survey Monkey to recruit 55 participants whose job function is “arts
and design.” Six sample phrases were randomly generated for each of four
different combinations of PoS: adjective-adjective (AA), adjective-noun (AN),
noun-noun (NN), and verb-verb (VV). For the selection of these words, we
randomly extracted the words for all combinations from a corpus, which we
created with English automotive design articles containing about 1.4 million
words on Sketch Engine [44]. The following filtration was performed equally
for all PoS combinations. First, we removed non-English terms and proper nouns
and then removed words whose relative frequency per a million tokens was less
than 1.0, compared with enTenTen15 [45], a large web text corpus containing
about 1.6 billion words. This made the list of all words, regardless of PoS,
more suitable for designing concept expressions that participants can readily
recognize and make a judgment about. The participants rated 24 randomly
generated two-word combinational phrases in randomized order on a 7-point
Likert scale on the basis of the degree to which they agreed with the
statement “I can imagine the product aesthetics, characters, mood, and
emotional quality that are conveyed by the product design.” An ANOVA-Tukey
test identified homogeneous subsets as [AA/AN/NN] and [VV/NN], where the mean
score of the former was significantly higher $(p=0.05^{*})$. Therefore, using
adjective-“noun-form-of” adjective phrases, as opposed to using noun-noun or
verb-verb phrases, would yield optimized effect on addressing product
aesthetics and semantics, as well as emotional qualities attached to them we
are trying to accomplish in this research.
Further details on the individual comparisons are as follows. There was a
significant difference $(p=.038^{*})$ in mean scores between AA
$(3.890,\sigma=1.121)$ and VV $(3.300,\sigma=1.205)$, a significant difference
$(p=.040^{*})$ between AN $(3.885,\sigma=1.025)$ and VV, no significant
difference $(p=.581)$ between AA and NN $(3.612,\sigma=1.232)$, no significant
difference $(p=.599)$ between AN and NN, no significant difference $(p=.485)$
between NN and VV, and no significant difference $(p=1.000)$ between AA and
AN.
### 3.2 System overview
Figure 2: A Process forming and explaining a DCP (A), which is modeled as
constructing a CS (B) Figure 3: An example of a moodboard, which is inspired
by a DCP “flavorous tranquility”
The present system, Character Space Construction (CSC), is an AI-assisted
interactive system. A new contribution of the CSC is that it helps designers
to construct a 2-dimensional CS as a quadrant system, which yields a DCP, a
compound adjective phrase, that portrays a design concept and explains it in
semantic comparisons. The CSC is a web application that has all the functions
above in one system and is operated in an interactive manner. The algorithm of
the system, which constructs a CS by letting the user set four words
($w_{1}$-$w_{4}$), will be discussed in detail in section 3.3. The CSC
application consists of a front-end web application written in JavaScript,
HTML, and CSS and a back-end web server written in Python with a MySQL
database hosted on PythonAnywhere, which is a hosting service for Python web
applications. The MySQL database was built on PythonAnywhere and holds the
data of word embeddings from the ConceptNet Numberbatch [32].
### 3.3 Character Space Construction algorithm
As discussed in Section 2.1., some concept designers empirically use a CS as a
synthesis tool, not as an analytical tool, in a process of forming a DCP. On a
quadrant system, two terms in the upper right quadrant are identified first
then the opposite term on each vertical and horizontal axis are identified.
This unique process has not been formalized in the past to the best of our
knowledge. Fig. 2(A) illustrates the sequence of designers’ cognitive
activities. It starts with searching and selecting a few candidates of the
first words of a compound adjective, followed by searching and identifying the
second adjective to be combined with the candidates of word 1. Once word 1 and
word 2 are identified as a DCP, designers then attempt to find contrasting
words for each to explain the DCP in comparison to those terms, with or
without CS. Fig. 2(B) shows the process model of constructing a CS in
sequence, replicating the process in Fig. 2(A) in a structured manner. A
formalization of constructing a CS not only clarifies the designers’ process
of verbal design concept explorations, but also signifies the next phase of
creative activities, that is, creating a visual mood board. Although creating
a mood board is outside of this research’s scope, it should be noted that this
quadrant system on a CS will eventually help them explore the visual images
for a target design concept, followed by finding visual images that should be
avoided. Fig. 3 is an example of what a mood board would look like with a DCP,
“flavorous tranquility.”222All the images in Fig. 4 are licensed under
Creative Commons in either of the following conditions: CC PDM 1.0, CC BY-NC-
ND 2.0, CC BY 2.0, or CC BY-SA 2.0
Adhering to this model, the CSC tool consists of three steps in a UI (Fig. 4).
The top section is step 1, where users can input a design brief in sentence
form to start searching for word 1 $w_{1}$. A design brief is a written
description of a project that requires some form of design, containing a
project overview, its objectives, tasks, target audience, and expected
outcomes [46, 47]. In design competitions, design briefs are usually
explicitly written material that calls for designers’ entries. In industry
practice, design briefs may be implicitly formed through discussions between
stakeholders in a project. In the top right section above step 2, the system
provides a search window in which the participant can input queries in case
they did not find any word that they wanted to use in the Explorer, after
candidate words are displayed. The lower left space is allocated for
“Explorer,” in which ranked search results are shown either as $w_{1}$ or
combinations of $w_{1}$ and $w_{2}$, depending on the phase of the
exploration.
The middle right section is step 2, where users can choose and pool candidates
for $w_{1}$ and search for candidates for $w_{2}$ that are combined with the
$w_{1}$ candidates. The users at this point will see a variety of adjective
phrases ($w_{1}$-$w_{2}$) in the Explorer as ranked lists in columns, grouped
by $w_{1}$.
Figure 4: Web application user interface of the Character Space Construction
(CSC). The top sections are the input windows for “Design Brief” and
individual query words. The largest section is allocated to “Explorer”, where
candidates for word 1 and word1 - word 2 compound phrases are presented. On
the right sidebar, the “Candidates for Word 1” section shows the words a user
selected and the “Search for Word 2” button. “Character Space” shows a
quadrant system as a user chooses word 1 and word 2, that trigger to show the
candidates of Word 3 and Word 4 as opposing concepts.
Finally, the lower right section is the CS, which is a quadrant system with
all four words set at the ends of each axis. The first quadrant of the CS is
set by dragging and dropping an adjective phrase chosen by the user from the
lists in the Explorer. Once the user finalizes the first quadrant of the CS
with adjective phrases ($w_{1}$-$w_{2}$) and clicks the “Search for word 3/4”
button, the system will search for antonyms to $w_{1}$ and $w_{2}$, as $w_{3}$
and $w_{4}$, respectively. The candidates for $w_{3}$ and $w_{4}$ are shown in
a pull-down menu at the end of each axis so that the user can choose a
suitable contrasting word for each $w_{1}$ and $w_{2}$ to complete the CS. The
upper right quadrant, represented by the combination of $w_{1}$ and $w_{2}$,
is the target design-concept phrase. All the other quadrants, represented by
$w_{2}$-$w_{3}$, $w_{3}$-$w_{4}$, and $w_{4}$-$w_{1}$, are contrasting
concepts to be used by the participants to explain the target design concepts
in comparison with opposing concepts. After clicking on the “Finish” button,
the system generates an explanation of the DCP as a template matched sentence,
like “My design concept is $w_{1}$-$w_{2}$. It has a sense of $w_{2}$ yet is
$w_{1}$, not $w_{3}$. It is $w_{1}$ but not $w_{4}$. In this design, $w_{1}$
and $w_{2}$ can go together.” Algorithm 1 illustrates the procedure described
above. Table 1 shows brief explanations of each sub-function in Algorithm 1.
Algorithm 1 Character Space Construction
function construct_character_space
$\mathit{query\\_words}$:= GenerateQuery(design_brief)
$w_{1}\\_initial\\_\mathit{candidates}$:=[]
$\mathit{phrases}$:=[]
$phrase\\_scores$:=[]
for each $w$ in $\mathit{query\\_words}$ do
$w_{1}\\_\mathit{initial\\_candidates}$.append(SearchCandidates($w$))
sort $w_{1}\\_\mathit{initial\\_candidates}$ by CalculateWordScore($w_{1}$)
display $w_{1}\\_\mathit{initial\\_candidates}$
$w_{1}\\_\mathit{candidates}$:= let user set 5 words from
$w_{1}\\_\mathit{initial\\_candidates}$
end for
for each $w_{1}$ in $w_{1}\\_\mathit{candidate}$ do
$w_{2}\\_\mathit{candidates}$:= SearchCandidates($w_{1}$)
for each $w_{2}$ in $w_{2}\\_\mathit{candidates}(w_{1})$ do
if CalculateWordScore($w_{2})>=1.7$ then
$\mathit{phrases}$.append($w_{1},w_{2}$)
$\mathit{phrase\\_scores}$.append(CalculatePhraseScore($w_{1},w_{2}$))
end if
end for
end for
sort $\mathit{phrases}$($w_{1}$, $w_{2}$) by $phrase\\_scores$
display $\mathit{phrases}$
($w_{1},w_{2}$):= let user set a phrase from $\mathit{phrases}$
display CharacterSpace($w_{1},w_{2}$)
$w_{3}\\_candidates$:= SearchAntomnyms($w_{1}$)
$w_{4}\\_candidates$:= SearchAntomnyms($w_{2}$)
display CharacterSpace($w_{1},w_{2},w_{3},w_{4}$)
display GenerateExplanation($w_{1},w_{2},w_{3},w_{4}$)
end function
Table 1: List of functions in CSC algorithm Function | Description
---|---
GenerateQuery | Extract adjectives from design brief
SearchCandidates | Search related words from ConceptNet
CalculateWordScore | Calculate score of word according to usefulness
CalculatePhraseScore | Calculate score of adjective phrase according to creativity
SearchAntonyms | Search antonyms from ConceptNet
CharacterSpace | Draw a quadrant with set words
GenerateExplanation | Generate an explanation with template matching
For the function CalculateWordScore, the preliminary study gave us an
opportunity to observe a list of randomly chosen adjectives. Despite being
extracted from the domain-specific corpus we created, the list included a
considerable number of unusable adjectives. These unusable adjectives included
ones with negative connotations (e.g., amateurish, costly, detrimental),
temporality (e.g., first, latest, recent, subsequent), distance (e.g., far,
near, close), or utilities (e.g., guest, sourced, takeaway) as well as those
that were too concrete in terms of expressing visuals (e.g., square, yellow,
golden), to name a few.
To rule out such unusable adjectives, we used a binary labeling
(usable/unusable) for a design concept by five professional automotive
designers, who were independent from this research, and we used the count of
usable labels and the word embedding vectors from the ConceptNet Numberbatch
[32] to form a model that predicts the usefulness score of unknown adjectives
using gradient boosting [48], which is a decision-tree-based ensemble machine
learning algorithm. We used a Python software library called XGBoost
(v0.90)[49], which uses a gradient boosting framework. We implemented a linear
regression predictive model that was trained to minimize the prediction errors
(RMSE) in a 10-fold cross validation that yielded the max depth of the
decision tree and the learning rate as $6$ and $0.05$, respectively. The
average RMSE of the prediction with the best hyperparameters on test data was
$1.406$. We used this model to predict usefulness scores for new adjectives,
which were used to display the candidate adjectives for $w_{1}$ in descending
order.
For the function CalculatePhraseScore, we tried to replicate the predictor of
the degree of creativity, which is the threshold of the similarity of two
words, from the related works on combinational creativity [50, 51], which
demonstrated that noun-noun combinations that represent good (award-winning)
design concepts fell into a certain range of distance between two nouns.
However, our test with five professional automotive designers on adjective
phrases in terms of creativity with pre-calculated similarity did not show a
clear normal distribution in the scores to determine a threshold. Yet, some
brackets for similarity showed better or worse scores than the others, so we
assigned these scores to unknown adjective phrases in accordance with the
calculated similarity values for each instance. These phrase scores were used
to display the candidate phrases for $w_{1}$-$w_{2}$ in descending order.
## 4 Experiment Design
### 4.1 Participants and independent variables
The study protocols below have been approved by the Institutional Review Board
of the National Institute of Informatics, Tokyo, Japan on February 15th, 2021
(Approval number 0042).
172 participants, whose job function was arts and design and who were fluent
in English, were recruited via Prolific, an online user test participant
recruiting service. 17 duplicate participants were removed. Of the 155 non-
duplicate participants, 15 ($9.68\%$) withdrew due to system trouble, and 37
($33.55\%$) did not complete the study for unknown reasons. The withdrawal
rate was high due to the complex and time-consuming nature of the task. That
left us with a total of 103 participants (54 M, 49 F) who completed the study,
with a mean age of 28.1 years ($\sigma=8.51$).
We recruited professional designers who would utilize the tool, instead of
recruiting potential end users who would consume the product, because as
discussed in Section 2.4, the present research focuses on an evaluation of a
creative support tool (CST), most commonly defined as tools to make more
people more creative more often [52].
All of the participants who completed the study used either laptops or desktop
computers of their own. The participants who completed the study were paid
US$15. The independent variables differed by whether or not the participants
used the CSC system (Fig. 4). When not using the CSC, the participants were
given complete freedom to use any publicly available web search tools, such as
Google search, an online thesaurus, or Wikipedia. All of the participants were
asked to perform the same task twice with (experiment) and without (control)
the CSC system. The order of the tool they used in two tasks was assigned
randomly in a counterbalanced order. They were given different design briefs
(Table 2) for each task to mitigate a learning effect, however, we deemed
those two design briefs equivalent in terms of influences on the results for
the following rationals. First, both briefs share the the domain of the
products (automotive products), and the basic component of the design briefs:
target audience, the context of the product in use, and expected features.
Second, the textual stimuli in both design briefs are described in the same
abstraction levels. Gonalves et al. [53] reported although different visual
stimuli strongly provoked different outcomes, difference in textual stimuli
did not show significant difference in the outcomes, unless they are
substantially different in specificity of domain, level of abstraction, and
the distance between the baseline and the supplemental briefs.
We did not offer a blank CS for the control participants because using the CS
is a part of our proposed method. The participants in the control group were
given instructions and an example of how to generate an explanation of their
DCPs. They participated in the experiment online using Survey Monkey, an
online platform, to which the participants were redirected from the recruiting
service, Prolific. Each participant was given video instructions explaining
the goal of the task and how to use the CSC tool (experiment group only)
before the task. The CSC tool was given to the experiment group with a URL
link.
Table 2: Two variations of design brief Design brief A: | The owner of this car will be retired, married, a countryside dweller, and a dog owner. The customer’s main requirement is that the car be economical and spacious. The car should be easy to get in and out of and easy to load. It should be functional and smart.
---|---
Design brief B: | Imagine a car for a family with small kids in a suburban community of 2030, where technologies are ubiquitous. Consider three important aspects of the car: harmony with local charm, fun activities, and safety.
### 4.2 Stimuli and tasks
After reading a brief, the participants were prompted to start the task. Those
with the experiment condition (CSC tool) was instructed to start the task by
copying and pasting the design brief to the CSC tool, and as described in
section 3.3, the CSC system provides a search window in which the participant
can input queries in case they did not find any word that they wanted to use
in the Explorer.
For both experiment and control conditions, there were two tasks. The first
task was “to create a unique and original DCP as an adjective-noun phrase form
that will inspire the audience (described in the design brief) to imagine the
character of the design.” “Effortless elegance” was given to the participants
as an example of an adjective phrase in the instruction video. The second task
was to explain their own DCP with contrasting words. As an example, the
adjective “uneasy” and the noun “clumsiness” were presented as contrasting
words for “effortless” and “elegance,” respectively. An example of an
explanation sentence was also given as follows: “My design concept is
‘effortless elegance.’ It has a sense of elegance, yet it is effortless, not
uneasy. It is effortless but not clumsy. In this design, effortless and
elegance can go together.”
For the experiment condition, this explanation was generated automatically
using all $w_{1}$ through $w_{4}$ using template matching as long as they
completed the CSC properly, and the CSC provided a “copy” button that allowed
the participants to copy the automatically generated explanation to the
clipboard so that they could paste it into the survey. However, they were also
given the freedom of editing it in the survey questionnaire. The participants
with the control condition had to generate and write an explanation down in
the survey questionnaire on their own, however, they were also provided
exactly the same example paragraph described above. In the experiment
condition. participants were also given a unique session ID by the CSC system
so that we could match the system log to examine how each participant
interacted with the words and phrases.
### 4.3 Evaluation method
As discussed in section 2.4, we used the CSI [43] as a post-task psychometric
measurement to compare two conditions, with (experiment) or without (control)
the CSC, in terms of creativity support. The CSI enables us to quantify the
cognitive processes of users using psychometric scales for six factors:
Exploration, Expressiveness, Immersion, Enjoyment, Results Worth Effort, and
Collaboration (Table 4). The CSI evaluates a creativity support tool itself,
and focuses on the experience of using it to create rather than trying to
directly evaluate a property of products. Instead, the CSI evaluates the
result of a creation in relation to an effort a user made, such as “I was
satisfied with what I got out of the system or tool.” This is suitable for
tools designed for experienced users, who know what the creative outcomes are
and what the ideal experiences in creation are, as opposed to the tools
designed for novice users where creativity is not sufficient and thus needs
support. The CSI also offers flexibility, such that it can be applied to
various tools and scenarios over time and provides standardized measurement.
It is robustly developed as each of the six factors has two different
statements (Table 3) that improve the statistical power of a survey, which is
deployed in the CSI. In combination with paired factor comparisons for each of
the six factors (Table 4), it also provides insight into what aspects of
creativity support may need attention.
The CSI has a rigorous protocol that the researcher should follow in order for
the measurement and analysis to be universal and reliable. For instance, our
CSC tool is not designed for collaborative tasks; however, the CSI protocol
discourages researchers from skipping the statements in the Collaboration
factor, still allowing participants to rate the collaboration statements.
Instead, the CSI protocol allows adding “N/A” responses to statements that
belong to the collaboration factor, which we incorporated in our survey.
Table 3: The 12 Agreement Statements on the CSI. Each agreement statement is
answered on a scale of “Highly Disagree” (1) to “Highly Agree” (10). In
deployment,the factor names are not shown, and the participant does not see
the statements grouped by factor[43]. Collaboration:
---
1\. The system or tool allowed other people to work with me easily.
2\. It was really easy to share ideas and designs with other people inside
this system or tool.
Enjoyment:
1\. I would be happy to use this system or tool on a regular basis.
2\. I enjoyed using the system or tool.
Exploration:
1\. It was easy for me to explore many different ideas, options, designs,or
outcomes, using this system or tool.
2\. The system or tool was helpful in allowing me to track different ideas,
outcomes, or possibilities.
Expressiveness:
1\. I was able to be very creative while doing the activity inside this system
or tool.
2.The system or tool allowed me to be very expressive.
Immersion:
1\. My attention was fully tuned to the activity, and I forgot about the
system or tool that I was using.
2\. I became so absorbed in the activity that I forgot about the system or
tool that I was using.
Results Worth Effort:
1\. I was satisfied with what I got out of the system or tool.
2\. What I was able to produce was worth the effort I had to exert to produce
it.
Table 4: The Paired-Factor Comparison Test has 15 comparisons for each pair, a
user will choose a factor description in response to the following statement:
“When doing this task, it’s most important that I’m able to…”[43] 1\. Be
creative and expressive.
---
2\. Become immersed in the activity.
3\. Enjoy using the system or tool
4\. Explore many different ideas, outcomes, or possibilities.
5\. Produce results that are worth the effort I put in.
6\. Work with other people.
Figure 5: Equation for scoring the CSI [43]
Also, the final CSI score will be weighted by the scores of paired-factor
comparisons, so if a participant does not think of the collaboration factor as
important, it is reflected in the final score accordingly. The only other
modification we made was to replace “the system or tool” with “The CSC” in
each statement (e.g., “I enjoyed using the CSC.”) that we were able to do so
in. We asked 12 questions on how strongly they agreed or disagreed with each
statement on a 10-point Likert scale after each task was completed and 15
paired-factor comparison questions at the end, emphasizing that they were
comparing the factors that they thought were important for the tasks, not for
the different tools. To check the reliability of each factor, we examined the
similarity of the scores (Table 5) across the two different statements.
### 4.4 Participant profile
Since we designed the CSC with professional designers in mind, we were
interested in examining the CSI ratings depending on the years of experience
and how often the participants performed the particular task we attempted to
support with the CSC, that is, generating DCPs especially in the aesthetic
sense and explaining them to stakeholders. For the former, we bracketed the
years-of-experience groups as: less than 1 year, 1 to 3 years, 3 to 5 years, 5
to 10 years, and 10+ years. For the latter, the groups were divided as: less
than quarterly or never, quarterly, monthly, weekly, and daily.
## 5 Results
### 5.1 Creativity Support Index
Table 5: CSI and factor scores | Cronbach’s Alpha per Factor | Average Factor Count | Factor Score | Weighted Factor Score
---|---|---|---|---
| | | CSC (N=93) | Control (N=95) | CSC (N=93) | Control (N=95) |
| | Mean ($\mathit{\sigma}$) | Mean ($\mathit{\sigma}$) | Mean ($\mathit{\sigma}$) | Mean ($\mathit{\sigma}$) | Mean ($\mathit{\sigma}$) | $p$ (Weighted Factor Score)
Collaboration | $0.829$ | $0.38$ ($0.8$) | $4.96$ ($6.07$) | $5.37$ ($6.53$) | $2.69$ ($8.24$) | $3.06$ ($8.39$) | $0.758$
Enjoyment | $0.894$ | $2.26$ ($1.36$) | $15.48$ ($4.13$) | $14.11$ ($3.88$) | $35.63$ ($24.84$) | $30.8$ ($20.76$) | $0.149$
Exploration | $0.787$ | $3.89$ ($0.88$) | $15.87$ ($3.48$) | $13.35$ ($3.94$) | $61.91$ ($20.6$) | $51.32$ ($18.73$) | $<0.010^{**}$
Expressiveness | $0.843$ | $3.69$ ($1.04$) | $14.45$ ($3.85$) | $13.63$ ($3.25$) | $52.56$ ($19.71$) | $50.61$ ($19.48$) | $0.496$
Immersion | $0.846$ | $1.78$ ($1.07$) | $11.06$ ($4.76$) | $11.37$ ($4.41$) | $19.95$ ($16.12$) | $21.15$ ($15.7$) | $0.605$
Result Worth Effort | $0.811$ | $3.01$ ($1.25$) | $14.66$ ($3.75$) | $14.03$ ($3.11$) | $43.35$ ($20.67$) | $42.03$ ($21.9$) | $0.671$
CSI | | | | | 72.03 (16.67) | 66.32 (14.22) | $0.012^{*}$
Table 5 shows the total CSI scores and factor scores between the experiment
(CSC) group and the control group. The mean CSI score of the experiment group,
which used our proposed CSC ($72.03,\sigma=16.67)$, was significantly higher
($p=.012<.05^{*}$, Cohen’s d $=.369)$ than that of the control group
($66.32,\sigma=14.22$)(Fig. 6, error bars show standard error). Note that 4
participants out of 103 were excluded due to their DCP being invalid because
the compound phrase ($w_{1}$-$w_{2}$) was impossible to interpret, hence
leaving us with 198 cases from 99 participants. Ten cases were excluded as
outliers whose CSI scores were more than two standard deviations away from
both sides of the mean, which left us with 188 cases (93 experiments, 95
control) for the final analysis.
For the reliability of the ratings within each factor, the Cronbach’s alpha
for Exploration, Expressiveness, Immersion, Enjoyment, Results Worth Effort,
and Collaboration was $.787,.843,.846,.894,.811$, and $.829$, respectively. As
for the factors of the CSC the participants found important, we compared the
mean factor counts from paired factor comparisons, which were
$3.8876(\sigma=0.88)$, $3.6854(\sigma=1.04)$, $1.7753(\sigma=1.07)$, $2.2584$
($\sigma=1.36)$, $3.0112(\sigma=1.25)$, and $.3820(\sigma=0.80)$,
respectively.
Weighted factor scores were calculated by multiplying a participant’s factor
agreement rating on the statement that belongs to the factor by the factor
count (Fig.5) [43]. This was done in order to make the weighted factor score
more sensitive to the factors that were the most important to the given task.
The weighted factor score of Exploration for the experiment group ($61.91$,
$\sigma=20.6$) was significantly higher ($p<.001^{**}$, Cohen’s d $=.679)$
than that of the control group ($51.32,\sigma=18.73$). The other weighted
factor scores did not indicate a significant difference between the experiment
and control groups. As for the perceived importance of the factors by the
participants, the mean count of the factors for Exploration [$3.89$
($\sigma=1.04$)] was among the highest, and that of Collaboration [$0.38$
($\sigma=0.8$)] was among the lowest.
A partly-repeated two-way ANOVA was performed in order to check the order
effects. The within-participants factor is the tool they used (CSC or
control), and the between-participants factor is the order they used the tool
(CSC first or control first). We report a significant order effect
($p=.007^{**},F=7.70$) between CSC-first or control-first groups. As for the
within-participants test, there were no significant main effect or significant
interaction between the tool and the order.
### 5.2 CSI score distribution by participants profile
Fig. 7 shows the distribution of CSI score between the CSI(experiment) group
and control group by years of experience brackets in “arts and design”
profession. While the years of experience was asked as a categorical choice we
examined a Pearson rank correlation between the CSI score and the median rank
in each bracket with respect to the years of professional experiences for each
group. Weak correlations were observed between the CSI score and the years of
experience for both CSC ($r=.235,p=0.023^{*}$ and the
Control($r=.237,p=0.021^{*}$ groups. This indicates that the longer the
experience, the more they value the benefit of the tool they used, which is
consistent in both CSC and control groups. The mean CSI scores of CSC group
for: less than 1 year, 1 to 3 years, 3 to 5 years, 5 to 10 years, and 10+
years were $65.27$, $69.9$0, $69.61$, $76.41$, and $76.87$, respectively. The
mean CSI scores of the control group for the same years-of-experience brackets
were $61.54$, $63.01$, $65.54$, $69.62$, $71.98$, and $66.32$, respectively.
As for how often the participants performed the particular task, we did not
observe any such tendencies in the CSI score by the frequency of the DCP
creation task they engaged in. Table 6 shows the illustrative examples of
randomly selected DCPs generated by the participants per tools and design
briefs.
Figure 6: CSI scores between experiment (CSC) group and Control group Figure 7: CSI score distribution between groups by years of experience bracket in “arts & design” profession Table 6: Examples of DCPs generated by participants per tools Tool | Des. Brief | $w_{1}+w_{2}$ | Explanation
---|---|---|---
CSC | A | efficient intelligence | My design concept is efficient-intelligence. It has a sense of intelligence yet it is efficient, not incompetent. It is efficient, but not ignorance. In this design, efficient and intelligence can go together.
CSC | B | harmonious interaction | My design concept is harmonious interaction. It has a sense of interactivity yet it is harmonious, not incongruous. It is harmonious, but not incompatibility. In this design, harmony and interaction can go together.
Control | B | practical smart | The design is practical and smart, yet is not non-rational. It is practical, not useless.
Control | A | comfortable utility | This design focuses on comfortable utility, it foregoes the roughness and harshness of typical utility vehicles, yet also avoids the impracticality and compromises of vehicles that put aesthetics over function.
## 6 Discussion
### 6.1 Contributions and applications
The present research showed that our proposed system, CSC, demonstrated its
advantage for the goal we aimed to achieve, which is to support the creativity
of professional designers. Our research is motivated by an empirical
observation on the characteristics of the concept designer task, that is,
exploring, identifying, and communicating a design concept, especially with
respect to product aesthetics and semantics, as opposed to product features
and functionalities.
We took the example of the automotive industry to describe a case study
because the particular characteristics of the design conceptualization task
are most apparent in automotive design studios. Yet, these characteristics are
common in industrial design in general, where the skills of industrial
designers are highly specialized and focused. As Tovey [7] discussed , the
tasks of industrial designers are distinguished from those of their
engineering colleagues, and they are specialized in determining the appearance
and identity of a product. However, these practices are traditionally highly
empirical, informal, undocumented, and sometimes even secretive due to the
strict confidentiality of the industry. It also tends to be dependent upon the
individual experiences of designers. We modeled this process as generating a
compound adjective phrase that best conceptualizes the nature of the design
aesthetics and semantics and then constructing a CS to explain them in
contrast with an opposing or distinct concept in a quadrant system (Fig.2
(B)). With respect to adjective phrases, as discussed in section 1, these
adjective phrases can also be used in design practices on other industries
[15, 16, 17], in computer poetry [54], or even in the advertisements in
tourism industry [17] where the verbal expressions of aesthetics and the
semantics of the products matter. Therefore, the proposed methods can be
applied to various design and new product development fields, and are likely
to be optimized by introducing domain-specialized corpora for the function
CalculateWordScore and the function CalculatePhraseScore. In regards to the
quadrant system, there has been a vague distinction between analysis using a
dimensionality reduction on multidimensional scaling (MDS) visualization
technique and what designers use as a synthesis tool. We defined the latter as
the CS, translated it into a computational algorithm, and implemented it in an
interactive web application so that designers can effectively trace the
process in an orderly sequence. Designers know what the important factors of
this task are, and evidently, we confirmed that designers value the
Exploration factor most in this task.
In this research, we shed light on concept designers’ cognitive activities
(Fig. 2-(A)), that were previously indefinite, formalized them (Fig. 2-(B)),
translated them into the CSC algorithm (Algorithm 1), and created the
interactive user interface (Fig. 3). The theoretical contribution of the
present research is based on a formalization of a designer’s concept
exploration activities, which involves searching and selecting a compound
adjective as a DCP. This formalization of constructing a CS not only signifies
the designers’ process of verbal design concept explorations but also leads to
subsequent visual creations (Fig.3).
As for a practical implication of the present research, experienced designers
seem to have valued the structured process of forming and explaining the DCPs
in the experiment. As some of the Weighted Factor Scores of CSI, such as
Enjoyment and Immersion, did not seem to matter to them, the participants
probably appraised the CSC as to offer them a certain efficiency and
effectiveness as a tool for professional use. This factor evaluation result,
with less emphasis on Enjoyment and Immersion, came in with a bit of surprise
to us, however, it actually gave us an assurance that we should spend more
resources to improve more Exploration factors, which we put many efforts into
this research by offering a variety of words and phrases presented on the
Explorer, enabled by the ConceptNet knowledge graph. Also, while a new tool
often involves a novelty effect [55], we can claim that the CSC did not
introduce a notable novelty effect, as it did not show a significant
difference in Enjoyment and Immersion compared with the control condition,
where participants used generic tools.
Regarding the level of expertise of the target audience, about half of
creativity support tools (CSTs) have no clear target level of expertise,
followed by experts (33%) and novices (17%) according to Frich et al. [56].
Our proposed method is clearly targeted for the tasks of experienced designers
since we observed the need for support in a particular task. In that sense, as
discussed in Fig. 7, though consistent between two groups, there was a
tendency indicating that the more experienced the participants were, the more
aware they were of the importance of the task, which suggests the necessity or
usefulness of this type of creativity support tool. Meanwhile, it can be also
said that, if the CSC were successfully emulating the process that an expert
designer would use, we should have observed a bottom-up effect for the
designers with fewer experiences. In the future, we may need to promote a
method to raise the awareness of process-oriented practices for less
experienced designers.
As reported in section 5.1, we observed an order effect between the
participants who used CSC first or second. The CSI score for the control tool
rated by the participants who used the CSC first substantially dropped.
Meanwhile, the CSI score for the CSC tool rated by the participants who used
the control tool first practically stayed the same. While this may have been
cased by a “fatigue effect” on repeated task [57] as the tasks given to the
participants were fairly complex and time consuming, it is probably a positive
sign to see the CSI score which is maintained the same level when used in the
second task.
### 6.2 Limitation
As the focus of this research was to identify and implement the design concept
generation process used by designers, that is, creating DCPs and communicating
them, into an algorithm for constructing the CS, there are some limitations on
the CSC tool. First, we did not put much emphasis on generating the “good”
results achieved by some of the related works mentioned in section 2.3., which
would have been reflected in the Result Worth Effort factor or Expressiveness
of the CSI. For example, the function, CalculateWordScore and the function
CalculatePhraseScore could be iterated more in optimizing the algorithms. In
fact, how to optimize suggested words and phrases can be another research
question, while the features and characteristics of adjectives can be
identified and utilized in order to suggest words and phrases that users can
find more creative. In the course of the research we have examined several
potential predictors, such as adjective supersense [58], and psycholinguistic
attributes of words [59], however, none appeared promising. There should be
more potential predictors we can investigate in the future. Second, the CSC
tool focuses on verbal concept design activities and does not involve visual
activities at this time. Although the verbal activity can be a starting point
to conceptualize and manipulate the product aesthetics and semantics,
designers will engage in visual ideation activities that can be also
computationally supported. We would like to implement this visual process into
the system in the future, that is, creating a mood board as we previewed an
example (Fig. 4), into the CS.
As for the limitation of our experiment, the participants were not given the
identical design brief for each condition, however, we carefully composed both
design briefs in consistent manner described in the section 4.1., and deemed
those two design briefs equivalent in terms of influences on the results. Our
intention of the experiment was to compare different tools, not to compare the
different stimuli though it may leave a room for discussions. Some may argue
it might have been better to set up a between-participants arrangement, yet,
we had to take a within-participants arrangement for two reasons: first, we
were still trying to test the relative effectiveness between two conditions -
the proposed method and what they usually practice in their regular works.
Second, recruiting a sufficient body of participants who are in “arts and
design” profession for a between-participants arrangement was prohibitively
difficult.
We did not ask the participants with control conditions what web search
tool(s) they actually used in creating and explaining their DCPs, in order to
avoid overloading the participants with additional questions. Also, those
publicly available web search tools do not provide the log data of the
participants’ behaviors during the search. Thus, how many and what queries the
participants tried, or how many words the participant actually looked at
before deciding $w1$ or $w1-w2$ combinations, for example, is unknown and we
were unable to compare such interactions between the conditions.
Another factor that may affect the CSI results is the graphic user interface.
Our proposed CSC tool did not suggest novelty effect indicated by rather
humble Enjoyment and Immersion scores. However, it can also be said that there
is a room to introduce a novelty bias if we adopt a more entertaining graphic
user interface. In the future, we would also like to compare different graphic
user interfaces to examine how Enjoyment and Immersion factors would interact
with Exploration, Result Worth Effort, or overall CSI scores.
Lastly, despite the fact that the CSI is most commonly used to evaluate
creative support tools, we acknowledge that it does not evaluate the
“effectiveness” of the tool in terms of how strongly the potential end-users
are influenced by the generated design concepts. As there is no one-size-fits-
all approach [35] we may also need to develop our original tool that
appropriately measure the effectiveness of the creation while we implement the
visual component discussed earlier in this section.
## 7 Conclusion
We focused on the process of verbal design concept exploration practiced by
concept designers, that is, generating and verbally communicating product
aesthetics and semantics, and we attempted to implement them into an
algorithm. We created a system, Character Space Construction (CSC), that
enables concept designers to explore and choose words to create a compound
adjective phrase, which we call the Design Concept Phrase (DCP). This is a
part of the process of constructing a quadrant system, which we call the
Character Space (CS). In a CS, a DCP can be represented in the upper right
quadrant, whereas other quadrants can be described as what to stay away from
or avoid in a semantic space. Our experiment, which was done with 103
participants using a within-participants arrangement, compared our proposed
methods, CSC and the general process designers use, and it was shown that our
proposed method, the CSC, has a significantly positive effect using Creativity
Support Index (CSI).
## 8 list of non-standard abbreviations
* •
CSC: Character Space Construction
* •
CS: Character Space
* •
DCP: design concept phrase
* •
CSI: Creativity Support Index
* •
MDS: Multidimensional scaling
## 9 Acknowledgments
We appreciate the support of the automotive designers at an anonymous company
for cooperating with the preliminary studies on labeling adjectives and
adjective phrases. This work was partially supported by JST, CREST
(JPMJCR21D4), Japan.
## 10 Data Availability Statement
The following data, supporting the conclusions of this article, will be made
available by the authors, without undue reservation.
* •
Participants responses to the experiment.
* •
Expert labeling on adjectives and adjective phrases.
* •
The generated design corpus from which we randomly selected the words and
phrases for PoS comparisons
* •
Participants responses to the PoS ratings on the preliminary study
* •
The source code of the CSC system
The following datasets presented in this study can be found in online
repositories. The names of the repositories can be found in the article.
* •
The ConceptNet API
* •
The word embedding vectors of the ConceptNet Numberbatch
* •
EnTenTen15 English web corpus
## 11 Conflict of Interest Statement
Author Shin Sano was employed by Toyota Motor Corporation and its North
American subsidiary, Calty Design Research, Inc. between April 1992 and July
2007. Author Shin Sano was hired as a design consultant between January 2015
and July 2020 for the Concept-i project, which is introduced as an industry
case study in the introduction section of this paper. This research is
independent of any part of that project and the authors received no funding
for the cost of this research. The remaining author declared that the research
was conducted in the absence of any commercial or financial relationships that
could be construed as a potential conflict of interest.
## References
* Krippendorff [2005] Klaus Krippendorff. _The semantic turn: A new foundation for design_. crc Press, 2005.
* Bloch et al. [2003] Peter H Bloch, Frederic F Brunel, and Todd J Arnold. Individual differences in the centrality of visual product aesthetics: Concept and measurement. _Journal of consumer research_ , 29(4):551–565, 2003.
* Alcaide-Marzal et al. [2020] Jorge Alcaide-Marzal, Jose Antonio Diego-Mas, and Gonzalo Acosta-Zazueta. A 3D shape generative method for aesthetic product design. _Design Studies_ , 66:144–176, 2020.
* Han et al. [2021] Ji Han, Hannah Forbes, and Dirk Schaefer. An exploration of how creativity, functionality, and aesthetics are related in design. _Research in Engineering Design_ , pages 1–19, 2021.
* Demirbilek and Sener [2003] Oya Demirbilek and Bahar Sener. Product design, semantics and emotional response. _Ergonomics_ , 46(13-14):1346–1360, 2003.
* Dong [2005] Andy Dong. The latent semantic approach to studying design team communication. _Design studies_ , 26(5):445–461, 2005.
* Tovey [1992] Michael Tovey. Intuitive and objective processes in automotive design. _Design Studies_ , 13(1):23–41, 1992.
* Desmet and Hekkert [2002] Pieter MA Desmet and Paul Hekkert. The basis of product emotions. _Pleasure with products, beyond usability_ , pages 60–68, 2002.
* McDonagh et al. [2002] Deana McDonagh, Anne Bruseberg, and Cheryl Haslam. Visual product evaluation: exploring users’ emotional relationships with products. _Applied Ergonomics_ , 33(3):231–240, 2002.
* Norman [2004] Donald A Norman. _Emotional design: Why we love (or hate) everyday things_. Basic Civitas Books, 2004.
* Hekkert [2006] Paul Hekkert. Design aesthetics: principles of pleasure in design. _Psychology science_ , 48(2):157, 2006.
* Helander et al. [2013] Martin G Helander, Halimahtun M Khalid, Tek Yong Lim, Hong Peng, and Xi Yang. Emotional needs of car buyers and emotional intent of car designers. _Theoretical Issues in Ergonomics Science_ , 14(5):455–474, 2013.
* Bouchard et al. [2005] Carole Bouchard, Roger Camous, and Ameziane Aoussat. Nature and role of intermediate representations (ir) in the design process: Case studies in car design. _International journal of vehicle design_ , 38(1):1–25, 2005.
* Shieh et al. [2011] Meng Dar Shieh, Tsung Hsing Wang, and Chih Chieh Yang. Affective response dimension selection for product design: A comparison of cluster analysis and procrustes analysis. _International Journal of Digital Content Technology and its Applications_ , 5(1):305–318, 2011.
* Choo and Kim [2003] Sunhyung Choo and Youngin Kim. Effect of color on fashion fabric image. _Color Research & Application: Endorsed by Inter-Society Color Council, The Colour Group (Great Britain), Canadian Society for Color, Color Science Association of Japan, Dutch Society for the Study of Color, The Swedish Colour Centre Foundation, Colour Society of Australia, Centre Français de la Couleur_, 28(3):221–226, 2003.
* Khalaj and Pedgley [2014] Javad Khalaj and Owain Pedgley. Comparison of semantic intent and realization in product design: A study on high-end furniture impressions. _International Journal of Design_ , 8(3), 2014.
* Durán-Muñoz [2019] Isabel Durán-Muñoz. Adjectives and their keyness: a corpus-based analysis of tourism discourse in english. _Corpora_ , 14(3):351–378, 2019.
* Nagai and Noguchi [2003] Y Nagai and H Noguchi. An experimental study on the design thinking process started from difficult keywords: Modeling the thinking process of creative design. _Journal of engineering design_ , 14(4):429–437, 2003.
* Osgood et al. [1957] Charles Egerton Osgood, George J Suci, and Percy H Tannenbaum. _The measurement of meaning_. Number 47. University of Illinois press, 1957.
* Ramsay [1975] James O Ramsay. Solving implicit equations in psychometric data analysis. _Psychometrika_ , 40(3):337–360, 1975.
* Zimet et al. [1988] Gregory D Zimet, Nancy W Dahlem, Sara G Zimet, and Gordon K Farley. The multidimensional scale of perceived social support. _Journal of personality assessment_ , 52(1):30–41, 1988.
* Trochim [1989] William MK Trochim. An introduction to concept mapping for planning and evaluation. _Evaluation and program planning_ , 12(1):1–16, 1989.
* Nagamachi [1995] Mitsuo Nagamachi. Kansei engineering: a new ergonomic consumer-oriented technology for product development. _International Journal of industrial ergonomics_ , 15(1):3–11, 1995.
* Hsiao and Chen [2006] Kun-An Hsiao and Lin-Lin Chen. Fundamental dimensions of affective responses to product shapes. _International Journal of Industrial Ergonomics_ , 36(6):553–564, 2006.
* Sopher [1962] H Sopher. Sequence of adjectives. _ELT Journal_ , 16(4):192–198, 1962.
* Boden [2001] Margaret Boden. Creativity and knowledge. _Creativity in education_ , pages 95–102, 2001.
* Han et al. [2020] Ji Han, Min Hua, Dongmyung Park, Pan Wang, and PRN Childs. Computational conceptual distances in combinational creativity. In _Proceedings of the Design Society: DESIGN Conference_ , volume 1, pages 177–186. Cambridge University Press, 2020.
* Georgiev et al. [2010] Georgi V. Georgiev, Yukari Nagai, and Toshiharu Taura. A method for the evaluation of meaning structures and its application in conceptual design. _Journal of Design Research_ , 8(3):214–234, 2010\. ISSN 15691551. doi: 10.1504/JDR.2010.032607.
* Chiu and Shu [2012] I. Chiu and L. H. Shu. Investigating effects of oppositely related semantic stimuli on design concept creativity. _Journal of Engineering Design_ , 23(4):271–296, 2012. ISSN 09544828. doi: 10.1080/09544828.2011.603298.
* Nagai et al. [2009] Yukari Nagai, Toshiharu Taura, and Futoshi Mukai. Concept blending and dissimilarity: factors for creative concept generation process. _Design Studies_ , 30(6):648–675, 2009.
* Han et al. [2016] J Han, Feng Shi, PRN Childs, et al. The combinator: A computer-based tool for idea generation. In _DS 84: Proceedings of the DESIGN 2016 14th International Design Conference_ , pages 639–648, 2016.
* Speer et al. [2017] Robyn Speer, Joshua Chin, and Catherine Havasi. ConceptNet 5.5: An open multilingual graph of general knowledge. In _Thirty-first AAAI conference on artificial intelligence_ , 2017\.
* Bae et al. [2020] Suyun Sandra Bae, Oh-Hyun Kwon, Senthil Chandrasegaran, and Kwan-Liu Ma. Spinneret: Aiding creative ideation through non-obvious concept associations. In _Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems_ , pages 1–13, 2020.
* Chen et al. [2019] Ting-Ju Chen, Sai Ganesh Subramanian, and Vinayak R Krishnamurthy. Mini-map: Mixed-initiative mind-mapping via contextual query expansion. In _AIAA Scitech 2019 Forum_ , page 2347, 2019.
* Hewett et al. [2005] Tom Hewett, Mary Czerwinski, Michael Terry, Jay Nunamaker, Linda Candy, Bill Kules, and Elisabeth Sylvan. Creativity support tool evaluation methods and metrics. _Creativity Support Tools_ , pages 10–24, 2005.
* Sternberg [1999] Robert J Sternberg. _Handbook of creativity_. Cambridge University Press, 1999.
* Barron [1955] Frank Barron. The disposition toward originality. _The Journal of Abnormal and Social Psychology_ , 51(3):478, 1955.
* Stein [1953] Morris I Stein. Creativity and culture. _The journal of psychology_ , 36(2):311–322, 1953\.
* Runco and Jaeger [2012] Mark A Runco and Garrett J Jaeger. The standard definition of creativity. _Creativity research journal_ , 24(1):92–96, 2012\.
* Bruner [1962] Jerome S Bruner. The conditions of creativity. In _Contemporary Approaches to Creative Thinking, 1958, University of Colorado, CO, US; This paper was presented at the aforementioned symposium._ Atherton Press, 1962.
* Cropley [1967] AJ Cropley. Creativity, intelligence, and achievement. _Alberta Journal of Educational Research_ , 1967.
* Kneller [1965] George Frederick Kneller. _The art and science of creativity_. Holt, Rinehart and Winston, 1965.
* Cherry and Latulipe [2014] Erin Cherry and Celine Latulipe. Quantifying the creativity support of digital tools through the creativity support index. _ACM Transactions on Computer-Human Interaction (TOCHI)_ , 21(4):1–25, 2014.
* Kilgarriff et al. [2004] Adam Kilgarriff, Pavel Rychly, Pavel Smrz, and David Tugwell. Itri-04-08 the sketch engine. _Information Technology_ , 105:116, 2004.
* Jakubíček et al. [2013] Miloš Jakubíček, Adam Kilgarriff, Vojtěch Kovář, Pavel Rychlỳ, and Vít Suchomel. The tenten corpus family. In _7th International Corpus Linguistics Conference CL_ , pages 125–127, 2013.
* Phillips [2004] Peter L Phillips. _Creating the Perfect Design Brief: How to manage design for strategic advantage_. Skyhorse Publishing Inc., 2004.
* Koronis et al. [2018] Georgios Koronis, Arlindo Silva, Jacob Kang, et al. The impact of design briefs on creativity: A study on measuring student designers outcomes. In _DS 92: Proceedings of the DESIGN 2018 15th International Design Conference_ , pages 2461–2472, 2018.
* Friedman [2001] Jerome H Friedman. Greedy function approximation: a gradient boosting machine. _Annals of statistics_ , pages 1189–1232, 2001.
* Chen and Guestrin [2016] Tianqi Chen and Carlos Guestrin. Xgboost: A scalable tree boosting system. In _Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining_ , pages 785–794, 2016.
* Han et al. [2019] Ji Han, Dongmyung Park, Feng Shi, Liuqing Chen, Min Hua, and Peter RN Childs. Three driven approaches to combinational creativity: Problem-, similarity-and inspiration-driven. _Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science_ , 233(2):373–384, 2019.
* Han et al. [2018] Ji Han, Feng Shi, Dongmyung Park, Liuqing Chen, Peter Childs, et al. The conceptual distances between ideas in combinational creativity. In _DS 92: Proceedings of the DESIGN 2018 15th International Design Conference_ , pages 1857–1866, 2018.
* Shneiderman [2002] Ben Shneiderman. Creativity support tools. _Communications of the ACM_ , 45(10):116–120, 2002.
* Gonalves et al. [2012] M Gonalves, Carlos Cardoso, and Petra Badke-Schaub. Find your inspiration: exploring different levels of abstraction in textual stimuli. In _DS 73-1 Proceedings of the 2nd International Conference on Design Creativity Volume 1_ , 2012.
* Toivanen et al. [2012] Jukka Toivanen, Hannu Toivonen, Alessandro Valitutti, Oskar Gross, et al. Corpus-based generation of content and form in poetry. In _Proceedings of the third international conference on computational creativity_. University College Dublin, 2012.
* Tsay et al. [2020] Crystal Han-Huei Tsay, Alexander K Kofinas, Smita K Trivedi, and Yang Yang. Overcoming the novelty effect in online gamified learning systems: An empirical evaluation of student engagement and performance. _Journal of Computer Assisted Learning_ , 36(2):128–146, 2020.
* Frich et al. [2019] Jonas Frich, Lindsay MacDonald Vermeulen, Christian Remy, Michael Mose Biskjaer, and Peter Dalsgaard. Mapping the landscape of creativity support tools in hci. In _Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems_ , pages 1–18, 2019.
* Bradley and Daly [1994] Mark Bradley and Andrew Daly. Use of the logit scaling approach to test for rank-order and fatigue effects in stated preference data. _Transportation_ , 21(2):167–184, 1994.
* Tsvetkov et al. [2014] Yulia Tsvetkov, Nathan Schneider, Dirk Hovy, Archna Bhatia, Manaal Faruqui, and Chris Dyer. Augmenting english adjective senses with supersenses. 2014\.
* Wilson [1988] Michael Wilson. Mrc psycholinguistic database: Machine-usable dictionary, version 2.00. _Behavior research methods, instruments, & computers_, 20(1):6–10, 1988.
|
# Domain-Agnostic Prior for Transfer Semantic Segmentation
Xinyue Huo1,2 Lingxi Xie2 Hengtong Hu2,3 Wengang Zhou1 Houqiang Li1 Qi Tian2
1University of Science and Technology of China 2Huawei Inc. 3Hefei University
of Technology
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Unsupervised domain adaptation (UDA) is an important topic in the computer
vision community. The key difficulty lies in defining a common property
between the source and target domains so that the source-domain features can
align with the target-domain semantics. In this paper, we present a simple and
effective mechanism that regularizes cross-domain representation learning with
a domain-agnostic prior (DAP) that constrains the features extracted from
source and target domains to align with a domain-agnostic space. In practice,
this is easily implemented as an extra loss term that requires a little extra
costs. In the standard evaluation protocol of transferring synthesized data to
real data, we validate the effectiveness of different types of DAP, especially
that borrowed from a text embedding model that shows favorable performance
beyond the state-of-the-art UDA approaches in terms of segmentation accuracy.
Our research reveals that UDA benefits much from better proxies, possibly from
other data modalities.
## 1 Introduction
Figure 1: Top: The goal is to transfer a segmentation model trained in the
source domain to the target domain, but some semantically similar classes (see
the left of (a) for examples of bike and motorbike) are difficult to
distinguish due to domain shift. Bottom: segmentation results of the upper-
right image without and with DAP, where (b) shows incorrect segmentation
(bike$\rightarrow$motorbike and sidewalk$\rightarrow$road) of the baseline
(DACS [49]), and (c) shows how DAP improves segmentation.
In the deep learning era, the most powerful approach for visual recognition is
to train deep neural networks with abundant, labeled data. Such a data-driven
methodology suffers the difficulty of transferring across domains, which
raises an important research field named domain adaptation. This paper focuses
on the setting of unsupervised domain adaptation (UDA), which assumes that the
source domain offers full supervision but the target domain has no annotations
available. In particular, we investigate semantic segmentation – provided the
increasing amount of unlabeled data and the expensiveness of annotation, it
becomes increasingly important to gain the ability of transferring models from
a known domain (e.g., labeled or synthesized data).
The goal of semantic segmentation is to assign each pixel with a class label.
In the scenarios that annotations are absent, this is even more challenging
than image-level prediction (e.g., classification) because the subtle
differences at the pixel level can be easily affected by the change of data
distribution. Note that this factor can bring in the risk that pseudo labels
are inaccurate and thus deteriorate transfer performance, and the risk becomes
even higher when the target domain lacks sufficient data for specific classes
or class pairs. Fig 1 shows an example that such approaches [64, 49] are
difficult in distinguishing semantically similar classes (e.g., motorbike vs.
bike, road vs. sidewalk) from each other.
To alleviate the above issue, we first offer a hypothesis about the confusion
– the proportion of similar categories in the two domains varies too much or
they often appear adjacent to each other and the border is difficult to find (
there are limited pixels around the boundary). Consequently, it is difficult
for the deep network to learn the discrimination boundary based on the
transferred image features. To compensate, we propose to add a domain-agnostic
prior (DAP) to force the features from the source and target domains to align
with an auxiliary space that is individual to both domains. According to the
Bayesian theory, a properly designed prior relieves the instability of
likelihood (provided by the limited training data with similar classes co-
occurring) and leads to more accurate inference.
We implement our algorithm upon DACS, a recent approach built upon an advanced
data augmentation named ClassMix [37]. The training procedure of DACS involves
sampling a pair of source and target images and making use of pseudo labels to
generate a mixed image with partly-pseudo labels, and feeding the source and
mixed images into the deep network. We introduce DAP into the framework by
defining a high-dimensional, domain-agnostic embedding vector for each class,
and force the features extracted from both the source and mixed images to
align with the embedding vectors through an auxiliary module. We investigate
two types of domain-agnostic embedding, namely, one-hot vectors and the
word2vec features [35], and show that both of them bring consistent gain in
transferring. As a side note, DAP is efficient to carry out – the auxiliary
module is lightweight that requires around $7\%$ extra training computation
and is removed during the inference stage.
We evaluate DAP on two standard UDA segmentation benchmarks, in which the
source domain is defined by a synthesized dataset (i.e., GTAv [39] or SYNTHIA
[40]) while the target domain involves Cityscapes [10], a dataset captured
from the real world. With the word2vec features as prior, DAP achieves
segmentation mIOU scores of $55.0\%$ and $50.2\%$ from GTAv and SYNTHIA,
respectively, with absolute gains of $2.9\%$ and $1.3\%$ over the DACS
baseline, setting the new state-of-the-art among single-model, single-round
approaches for UDA segmentation.
The main contribution of this work lies in the proposal and implementation of
domain-agnostic prior for UDA segmentation. With such a simple and effective
approach, we reveal that much room is left behind UDA. We expect more
sophisticated priors and/or more effective constraints to be explored in the
future.
Figure 2: The proposed framework of that involves building DAP (the blue-
shaded region) upon DACS [49] (the yellow-shaded region). We omit the
illustration of using ClassMix [37] to generate
$\mathbf{x}_{n,m}^{\mathbb{T}}$. The details of producing the embedding map
(i.e., the $\mathrm{proj}(\cdot)$ function) is shown in Fig 3. This figure is
best viewed in color.
## 2 Related Work
Unsupervised domain adaptation (UDA) aims to transfer models trained in a
known (labeled) source domain to an unknown (unlabeled) target domain [15, 32,
64]. It differs from semi-supervised learning [63, 27] mainly in the
potentially significant difference between the labeled and unlabeled data. The
past years have witnessed a fast development of UDA , extending the studied
task from image classification [42, 5] to fine-scaled visual recognition
including detection [8, 25], segmentation [50, 4], person re-identification
[14, 45], etc. Specifically, segmentation is a good testbed for UDA for at
least two reasons. First, compared to classification, pixel-level segmentation
requires more sophisticated manipulation of domain transfer. Second, UDA can
reduce the annotation cost for segmentation which is often expensive. The
recent approaches of UDA segmentation is roughly categorized into three parts,
adversarial learning, self-training, and data generation.
$\bullet$ Adversarial learning is an important tool for domain adaptation [52,
2]. When it was applied to UDA segmentation [50, 48], the segmentation modules
are regarded as the generator that produce prediction, and an auxiliary
discriminator is trained to judge which domain the inputs are from. By
confusing the discriminator, the purpose of domain adaptation is achieved.
Extensions beyond this idea include [24] that facilitated the domain alignment
from the input image and transferred the low-level representation (e.g. the
texture and brightness) from the target domain to the source domain, [56] that
applied a Fourier transform to narrow the low-frequency gap between the two
domains, and others.
$\bullet$ The idea of self-training was borrowed from semi-supervised learning
[47, 41, 21] and few-shot learning [28, 31], and was applied well to domain
adaptation [44]. The key to self-training is to produce high-quality pseudo
labels [27], but prediction errors are inevitable especially when the domain
gap is significant. Many efforts have been made to alleviate the inaccuracy,
including entropy minimization [65, 54, 7] that tried to increase the
confidence on unlabeled data, class-balance regulation [64] that improved the
prediction probability for hard categories and thus alleviated the burden that
minor categories may be suppressed, and [17, 60] that reduced the uncertainty
of prediction to rectify pseudo labels.
$\bullet$ The data generation branch tried to integrate the advantages of
adversarial learning and self-training. The idea is to bridge the gap between
the source and target domains using a generated domain in which source and
target data are mixed. The images can be generated either by performing GAN
[62, 19] or pixel-level mixing [37, 57]. Our approach follows the second path
which is verified powerful in UDA segmentation [49, 16, 61], and we introduce
a domain-agnostic prior to constrain representation learning on the target
domain.
There have been studies of combining visual data with other types of
information, in particular, with text data [23, 36, 26]. Powered by pre-
training on a large amount of image and text data, either paired [38] or
unpaired [22], deep neural networks gain the ability of aligning visual and
linguistic features in a shared space, hence facilitating cross-modal
understanding [46, 29] and retrieval [55, 13]. Recently, the rapid development
of Transformers [53, 12] brings the possibility of unifying image and text
data within one framework. Transferring the learned knowledge from one
modality to another may cause the setting somewhat similar to zero-shot
learning [3, 59]. There are also some discussions on safe and/or efficient
cross-modal learning, including [20] that introduced knowledge distillation
into cross-modal retrieval, and [1] that introduced boundary-aware regression
and semantic consistency constraint and improved the discrimination for the
unlabeled classes.
## 3 Our Approach
### 3.1 Problem Setting and Baselines
Unsupervised domain adaptation (UDA) starts with defining two domains
corresponding to different data distributions. In our setting for semantic
segmentation, complete pixel-level annotations are available for the source
domain, $\mathbb{S}$, but unavailable for the target domain, $\mathbb{T}$, yet
we hold an assumption that the class sets on $\mathbb{S}$ and $\mathbb{T}$ are
identical. From the data perspective, let the training samples on $\mathbb{S}$
from a set of
$\mathcal{D}^{\mathbb{S}}=\\{(\mathbf{x}_{n}^{\mathbb{S}},\mathbf{y}_{n}^{\mathbb{S}})\\}_{n=1}^{N}$
where both $\mathbf{x}_{n}^{\mathbb{S}}$ is a high-resolution image and
$\mathbf{y}_{n}^{\mathbb{S}}$ of the same size offers pixel-wise semantic
labels. Similarly, we have a training set on $\mathbb{T}$ being
$\mathcal{D}^{\mathbb{T}}=\\{(\mathbf{x}_{m}^{\mathbb{T}})\\}_{m=1}^{M}$ where
ground-truth labels are not provided. The goal is to train a model
$\mathbf{y}=f(\mathbf{x};\boldsymbol{\theta})$ on both
$\mathcal{D}^{\mathbb{S}}$ and $\mathcal{D}^{\mathbb{T}}$ so that it works
well on a hidden test set $\mathcal{D}^{\mathbb{T}\prime}$ which is also
sampled from $\mathbb{T}$.
The major difficulty of UDA comes from the domain gap between $\mathbb{S}$ and
$\mathbb{T}$, e.g., in our testing environment, $\mathbb{S}$ corresponds to
synthesized data [39, 40] but $\mathbb{T}$ corresponds to real-world data
[10]. The potential differences in lighting, object styles, etc., can
downgrade prediction confidence as well as quality on the target domain. One
of the popular solutions to bridge the domain gap is to use the source model
to generate pseudo labels $\hat{\mathbf{y}}_{m}^{\mathbb{T}}$ for
$\mathbf{x}_{m}^{\mathbb{T}}$ by a mean-teacher model [47], and uses both
$\mathcal{D}^{\mathbb{S}}$ and
$\hat{\mathcal{D}}^{\mathbb{T}}=\\{(\mathbf{x}_{m}^{\mathbb{T}},\hat{\mathbf{y}}_{m}^{\mathbb{T}})\\}_{m=1}^{M}$,
the extended target set, for updating the online student model. We denote the
teacher and student models as
$f^{\mathrm{te}}(\mathbf{x};\boldsymbol{\theta}^{\mathrm{te}})$ and
$f^{\mathrm{st}}(\mathbf{x};\boldsymbol{\theta}^{\mathrm{st}})$, respectively,
where in the mean-teacher algorithm, $f^{\mathrm{te}}(\cdot)$ is the moving
average of $f^{\mathrm{st}}(\cdot)$.
The above mechanism, though simple and elegant, suffers the unreliability of
$\hat{\mathbf{y}}_{m}^{\mathbb{T}}$. To alleviate this issue, in a recently
published work named DACS [49], researchers proposed to replace the training
data from the target domain with that from a mixed domain, $\mathbb{M}$, where
the samples are generated by mixing images at the pixel level guided by class
labels [37]. In each training iteration, a pair of source and target images
with (true or pseudo) labels are sampled and cropped into the same resolution,
denoted as
$(\mathbf{x}_{n}^{\mathbb{S}},\mathbf{y}_{n}^{\mathbb{S}},\mathbf{x}_{m}^{\mathbb{T}},\hat{\mathbf{y}}_{m}^{\mathbb{T}})$.
Next, a subset of classes is randomly chosen from
$\mathbf{y}_{n}^{\mathbb{S}}$ and a binary mask $\mathbf{M}_{n,m}$ of the same
size as $\mathbf{x}_{n}^{\mathbb{S}}$ is made, with all pixels that
$\mathbf{y}_{n}^{\mathbb{S}}$ belongs to the subset being $1$ and otherwise
$0$. Upon $\mathbf{M}_{n,m}$, a mixed image with its label is defined:
$\left\\{\begin{array}[]{l}\mathbf{x}_{n,m}^{\mathbb{M}}=\mathbf{x}_{n}^{\mathbb{S}}\odot\mathbf{M}_{n,m}+\mathbf{x}_{m}^{\mathbb{T}}\odot(\mathbf{1}-\mathbf{M}_{n,m})\\\
\mathbf{y}_{n,m}^{\mathbb{M}}=\mathbf{y}_{n}^{\mathbb{S}}\odot\mathbf{M}_{n,m}+\hat{\mathbf{y}}_{m}^{\mathbb{T}}\odot(\mathbf{1}-\mathbf{M}_{n,m})\end{array}\right.,$
(1)
where $\odot$ denotes element-wise multiplication. The student model,
$f^{\mathrm{st}}(\mathbf{x};\boldsymbol{\theta}^{\mathrm{st}})$, is trained
with $(\mathbf{x}_{n}^{\mathbb{S}},\mathbf{y}_{n}^{\mathbb{S}})$ and
$(\mathbf{x}_{n}^{\mathbb{M}},\mathbf{y}_{n}^{\mathbb{M}})$:
$\mathcal{L}_{\mathrm{seg}}=\mathcal{L}_{\mathrm{CE}}(f^{\mathrm{st}}(\mathbf{x}_{n}^{\mathbb{S}};\boldsymbol{\theta}^{\mathrm{st}}),\mathbf{y}_{n}^{\mathbb{S}})+\mathcal{L}_{\mathrm{CE}}(f^{\mathrm{st}}(\mathbf{x}_{n,m}^{\mathbb{M}};\boldsymbol{\theta}^{\mathrm{st}}),\mathbf{y}_{n,m}^{\mathbb{M}}),$
(2)
where $\mathcal{L}_{\mathrm{CE}}(\cdot,\cdot)$ is the pixel-wise cross-entropy
loss. The teacher model,
$f^{\mathrm{te}}(\mathbf{x};\boldsymbol{\theta}^{\mathrm{te}})$, is updated
with the student model using the exponential moving average (EMA) mechanism,
namely,
$\boldsymbol{\theta}^{\mathrm{te}}\leftarrow\boldsymbol{\theta}^{\mathrm{te}}\cdot\lambda+\boldsymbol{\theta}^{\mathrm{st}}\cdot(1-\lambda)$,
where $\lambda$ controls the window of EMA and is often close to $1.0$. The
entire flowchart of DACS is illustrated in the yellow-shaded part of Fig 2.
### 3.2 Confusion of Semantically Similar Classes
| | $\mathrm{A}$: motor, $\mathrm{B}$: bike | $\mathrm{A}$: road, $\mathrm{B}$: sidewalk
---|---|---|---
Methods | Data | $\mathrm{cos}\langle\boldsymbol{\mu}_{\mathrm{A}}^{\cdot},\boldsymbol{\mu}_{\mathrm{B}}^{\cdot}\rangle$ | IOU (%) | $\mathrm{cos}\langle\boldsymbol{\mu}_{\mathrm{A}}^{\cdot},\boldsymbol{\mu}_{\mathrm{B}}^{\cdot}\rangle$ | IOU (%)
Source Only | $\mathbb{S}$ | 0.42 | 6.36 | 0.35 | 1.43
$\mathbb{T}$ | 0.60 | 27.76 | 0.43 | 5.53
DACS | $\mathbb{T}$ | 0.60 | 26.62 | 0.38 | 1.95
DAP | $\mathbb{T}$ | 0.56 | 22.19 | 0.23 | 1.29
Table 1: Statistics of features extracted two pairs of semantically similar
classes, where $\mathbb{S}$ and $\mathbb{T}$ corresponds to GTAv and
Cityscapes, respectively. Please refer to the main text for details.
Despite the effectiveness in stabilizing self-learning, DACS still has
difficulties in distinguishing semantically similar classes, especially when
these classes do not appear frequently in the target domain, e.g., motorbike
accounts for only $0.1\%$ of the total number of pixels. Fig 1 shows an
example that the class pair of motorbike and bike can easily confuse the
model, and so can the pair of road and sidewalk. According to experimental
results, such confusion contributes significantly to segmentation error, e.g.,
$20.8\%$ mis-classification of bike pixels goes to motorbike.
We offer a hypothesis for the above phenomenon. Since data from the target
domain, say $\mathbf{x}_{m}^{\mathbb{T}}$, are not labeled, the semantic
correspondence is learned by mapping $\mathbf{x}_{m}^{\mathbb{T}}$ to the
source domain, e.g., by image-level style transfer for GAN-based approaches
[62, 19] and by label-level simulation for DACS. Mathematically, this is to
learn a transfer function (which transfers visual features from the target
domain to the source domain) in a weakly-supervised manner. This causes
approximation of visual representation and consequently incurs inaccuracy of
recognition.
In addition, we consider two semantically similar classes denoted as
$\mathrm{A}$ and $\mathrm{B}$, and the corresponding features extracted from
these classes form two sets of $\mathcal{F}_{\mathrm{A}}^{\cdot}$ and
$\mathcal{F}_{\mathrm{B}}^{\cdot}$, respectively, where the superscript can be
either $\mathbb{S}$ or $\mathbb{T}$. Let us assume that each feature set
follows a multi-variate Gaussian distribution denoted by
$\mathcal{N}(\boldsymbol{\mu}_{\mathrm{A}}^{\cdot},\boldsymbol{\Sigma}_{\mathrm{A}}^{\cdot})$
and
$\mathcal{N}(\boldsymbol{\mu}_{\mathrm{B}}^{\cdot},\boldsymbol{\Sigma}_{\mathrm{B}}^{\cdot})$,
abbreviated as $\mathcal{N}_{\mathrm{A}}^{\cdot}$ and
$\mathcal{N}_{\mathrm{B}}^{\cdot}$, respectively. Since $\mathrm{A}$ and
$\mathrm{B}$ are semantically similar, we assume that
$\boldsymbol{\mu}_{\mathrm{A}}^{\mathbb{S}}$ and
$\boldsymbol{\mu}_{\mathrm{B}}^{\mathbb{S}}$ are close in the feature space,
and the source model learns to distinguish $\mathrm{A}$ from $\mathrm{B}$ by
reducing
$\cos\langle\boldsymbol{\mu}_{\mathrm{A}}^{\mathbb{S}},\boldsymbol{\mu}_{\mathrm{B}}^{\mathbb{S}}\rangle$
and hence the IOU between $\mathcal{N}_{\mathrm{A}}^{\mathbb{S}}$ and
$\mathcal{N}_{\mathrm{B}}^{\mathbb{S}}$ 111To calculate the IOU between
$\mathcal{N}_{\mathrm{A}}^{\mathbb{S}}$ and
$\mathcal{N}_{\mathrm{B}}^{\mathbb{S}}$, we sample an equal amount of points
from $\mathcal{N}_{\mathrm{A}}^{\mathbb{S}}$ and
$\mathcal{N}_{\mathrm{B}}^{\mathbb{S}}$, and calculate $r_{\mathrm{A}}$ as the
probability that a point sampled from $\mathcal{N}_{\mathrm{A}}^{\mathbb{S}}$
actually has a higher density at $\mathcal{N}_{\mathrm{B}}^{\mathbb{S}}$, and
$r_{\mathrm{B}}$ vice versa. The IOU is then approximated as
$(r_{\mathrm{A}}+r_{\mathrm{B}})/(2-r_{\mathrm{A}}-r_{\mathrm{B}})$.. However,
in the target domain, these conditions are not necessarily satisfied since no
strong supervision is present – the distance between
$\boldsymbol{\mu}_{\mathrm{A}}^{\mathbb{T}}$ and
$\boldsymbol{\mu}_{\mathrm{B}}^{\mathbb{T}}$ gets smaller (i.e.,
$\cos\langle\boldsymbol{\mu}_{\mathrm{A}}^{\mathbb{T}},\boldsymbol{\mu}_{\mathrm{B}}^{\mathbb{T}}\rangle$
gets larger), and, consequently, the IOU between
$\mathcal{N}_{\mathrm{A}}^{\mathbb{T}}$ and
$\mathcal{N}_{\mathrm{B}}^{\mathbb{T}}$ is larger. Tab 1 offers quantitative
results from transferring a segmentation model from GTAv to Cityscapes. The
features of two semantically similar class pairs, namely, motorbike vs. bike,
and road vs. sidewalk, are both made easier to confuse the network.
### 3.3 Domain-Agnostic Prior for UDA Segmentation
To obtain more accurate estimation for $\mathcal{N}_{\mathrm{A}}^{\mathbb{T}}$
and $\mathcal{N}_{\mathrm{B}}^{\mathbb{T}}$, we refer to the Bayesian theory
that the posterior distribution is composed of a prior and a likelihood. In
our setting, the likelihood comes from the target dataset,
$\mathcal{D}^{\mathbb{T}}$, where there are insufficient data to guarantee
accurate estimation. The solution lies in introducing an informative prior for
the elements, namely,
$\mathbf{z}_{\mathrm{A}}^{\mathbb{T}}\sim\mathcal{N}_{\mathrm{A}}^{\mathbb{T}}$
and
$\mathbf{z}_{\mathrm{B}}^{\mathbb{T}}\sim\mathcal{N}_{\mathrm{B}}^{\mathbb{T}}$.
This involves defining constraints as follows:
$\mathbf{z}_{\mathrm{A}}^{\mathbb{T}}=g(\mathbf{e}_{\mathrm{A}}),\quad\mathbf{z}_{\mathrm{B}}^{\mathbb{T}}=g(\mathbf{e}_{\mathrm{B}}),$
(3)
where $\mathbf{e}_{\mathrm{A}}$ and $\mathbf{e}_{\mathrm{B}}$ are not related
to any specific domain – we name them domain-agnostic priors (DAP), and
$g(\cdot)$ is a learnable function that projects the priors to the semantic
space. In practice, it is less likely to strictly satisfy Eqn (3), so we
implement it as a loss term to minimize.
We instantiate DAP into two examples. The first one is a set of one-hot
vectors, i.e., for a target domain with $C$ classes, the $c$-th class is
encoded into $\mathbf{I}_{c}$, a $C$-dimensional vector in which all entries
are $0$ except for the $c$-th dimension being $1$. The second one is borrowed
from word2vec [35], a pre-trained language model that represents each word
using a $300$-dimensional vector. Note that both cases are completely
independent from the vision domain, i.e., reflecting the principle of being
domain-agnostic. We denote the set of embedding vectors as
$\mathcal{E}=\\{\mathbf{e}_{c}\\}_{c=1}^{C}$, where $\mathbf{e}_{c}$ is the
embedding vector of the $c$-th class.
Back to the main story, DAP and the embedded vectors are easily integrated
into DACS, the baseline framework. Besides the segmentation loss
$\mathcal{L}_{\mathrm{seg}}$ defined in Eqn (2), we introduce another loss
term $\mathcal{L}_{\mathrm{DAP}}$ that measures the distance between the
embedded domain-agnostic priors and visual features extracted from training
images,
$\displaystyle\mathcal{L}_{\mathrm{DAP}}=$
$\displaystyle\|g_{\mathrm{vi}}(\tilde{f}^{\mathrm{st}}(\mathbf{x}_{n}^{\mathbb{S}};\boldsymbol{\theta}^{\mathrm{st}}))-g_{\mathrm{pr}}(\mathrm{proj}(\tilde{\mathbf{y}}_{n}^{\mathbb{S}};\mathcal{E}))\|_{2}^{2}+$
(4)
$\displaystyle\|g_{\mathrm{vi}}(\tilde{f}^{\mathrm{st}}(\mathbf{x}_{n,m}^{\mathbb{M}};\boldsymbol{\theta}^{\mathrm{st}}))-g_{\mathrm{pr}}(\mathrm{proj}(\tilde{\mathbf{y}}_{n,m}^{\mathbb{M}};\mathcal{E}))\|_{2}^{2},$
where $\tilde{f}^{\mathrm{st}}(\cdot)$ is the backbone part of
$f^{\mathrm{st}}(\cdot)$ that extracts mid-resolution features (i.e., visual
features), $\tilde{\mathbf{y}}_{n}^{\mathbb{S}}$ and
$\tilde{\mathbf{y}}_{n,m}^{\mathbb{M}}$ denote the supervisions adjusted to
the same size of the backbone outputs, $\mathrm{proj}(\cdot)$ projects the
domain-agnostic embedding vectors to the image plane based on the labels (to
be detailed later), and $g_{\mathrm{vi}}(\cdot)$ and $g_{\mathrm{pr}}(\cdot)$
are each a learnable convolution layer that maps initial features from the
vision and prior spaces to the common space.
We elaborate the construction of $\mathrm{proj}(\cdot)$ in Fig 3. The
computation is simple: for each position on the image plane, we refer to the
ground-truth label $\tilde{\mathbf{y}}_{n}^{\mathbb{S}}$ or partly-pseudo
label $\tilde{\mathbf{y}}_{n,m}^{\mathbb{M}}$, and directly paste the class-
wise embedding vectors to the corresponding positions. Since $\mathcal{E}$ is
fixed, this procedure does not need any gradient back-propagation. We are
aware that advanced versions of $\mathrm{proj}(\cdot)$ (e.g., performing local
smoothing) are available, and we leave these properties to be learned by
$g_{\mathrm{vi}}(\cdot)$ and $g_{\mathrm{pr}}(\cdot)$.
Figure 3: An illustration of constructing the domain-agnostic embedding map
with prior vectors. Each number in the label map stands for a class ID. The
total number of classes can be arbitrary.
Finally, the overall loss function is written as
$\mathcal{L}_{\mathrm{overall}}=\mathcal{L}_{\mathrm{seg}}+\alpha\cdot\mathcal{L}_{\mathrm{DAP}},$
(5)
where $\alpha$ is the balancing coefficient which is by default set to be
$1.0$ – an analysis of the effect of $\alpha$ is provided in the experimental
part (see Tab 6).
Back to Tab 1, one can observe how DAP reduces the IOU between
$\mathcal{N}_{\mathrm{A}}^{\mathbb{T}}$ and
$\mathcal{N}_{\mathrm{B}}^{\mathbb{T}}$, and thus alleviate the confusion
between the semantically similar classes. Interestingly, the gains of two
class pairs are consistent and significant, even though the training samples
of road vs. sidewalk being more abundant. In experiments, we shall see how the
ability of discriminating these class pairs is improved.
### 3.4 Discussions
To the best of our knowledge, this is the first work that integrates text
embedding into UDA segmentation and producing considerable accuracy gain,
which demonstrating the effectiveness of linguistic cues assisting visual
recognition. However, it is yet a preliminary solution, and some possible
directions can be discovered.
$\bullet$ Enhancing text embedding. The currently used word2vec features it
does not consider different words that correspond to the same semantics (e.g.,
person can be pedestrian). Interestingly, we tried to enhance the prior by
searching for semantically similar words, but obtained little accuracy gain.
This may call for a complicated mechanism of exploring the text world.
$\bullet$ Constructing domain-agnostic yet vision-aware priors. This is to
answer the question: what kind of image data is considered to offer domain-
free information? The answer may lie in generalized datasets like ImageNet
[11] or Conceptual Captioning [43], or even the pre-trained image-text models
such as CLIP [38] that absorbed $400$ million image-text pairs. Note that it
is a major challenge to disentangle domain-related information to avoid over-
fitting, and we will continue exploring the possibility in the future.
## 4 Experiments
GTAv$\rightarrow$Cityscapes
---
Method | Road | Sidewalk | Building | Wall | Fence | Pole | Light | Sign | Veg | Terrain | Sky | Person | Rider | Car | Truck | Bus | Train | Motor | Bike | mIOU
AdaptSegNe [50] | 86.5 | 36.0 | 79.9 | 23.4 | 23.3 | 23.9 | 35.2 | 14.8 | 83.4 | 33.3 | 75.6 | 58.5 | 27.6 | 73.7 | 32.5 | 35.4 | 3.9 | 30.1 | 28.1 | 42.4
PatchAligne [51] | 92.3 | 51.9 | 82.1 | 29.2 | 25.1 | 24.5 | 33.8 | 33.0 | 82.4 | 32.8 | 82.2 | 58.6 | 27.2 | 84.3 | 33.4 | 46.3 | 2.2 | 29.5 | 32.3 | 46.5
LTIR [24] | 92.9 | 55.0 | 85.3 | 34.2 | 31.1 | 34.9 | 40.7 | 34.0 | 85.2 | 40.1 | 87.1 | 61.0 | 31.1 | 82.5 | 32.3 | 42.9 | 0.3 | 36.4 | 46.1 | 50.2
PIT [33] | 87.5 | 43.4 | 78.8 | 31.2 | 30.2 | 36.3 | 39.9 | 42.0 | 79.2 | 37.1 | 79.3 | 65.4 | 37.5 | 83.2 | 46.0 | 45.6 | 25.7 | 23.5 | 49.9 | 50.6
FDA [56] | 92.5 | 53.3 | 82.4 | 26.5 | 27.6 | 36.4 | 40.6 | 38.9 | 82.3 | 39.8 | 78.0 | 62.6 | 34.4 | 84.9 | 34.1 | 53.1 | 16.9 | 27.7 | 46.4 | 50.5
MetaCorrect [17] | 92.8 | 58.1 | 86.2 | 39.7 | 33.1 | 36.3 | 42.0 | 38.6 | 85.5 | 37.8 | 87.6 | 62.8 | 31.7 | 84.8 | 35.7 | 50.3 | 2.0 | 36.8 | 48.0 | 52.1
DACS [49] | 89.9 | 39.7 | 87.9 | 39.7 | 39.5 | 38.5 | 46.4 | 52.8 | 88.0 | 44.0 | 88.8 | 67.2 | 35.8 | 84.5 | 45.7 | 50.2 | 0.0 | 27.3 | 34.0 | 52.1
IAST [34] | 94.1 | 58.8 | 85.4 | 39.7 | 29.2 | 25.1 | 43.1 | 34.2 | 84.8 | 34.6 | 88.7 | 62.7 | 30.3 | 87.6 | 42.3 | 50.3 | 24.7 | 35.2 | 40.2 | 52.2
DPL [9] | 92.8 | 54.4 | 86.2 | 41.6 | 32.7 | 36.4 | 49.0 | 34.0 | 85.8 | 41.3 | 86.0 | 63.2 | 34.2 | 87.2 | 39.3 | 44.5 | 18.7 | 42.6 | 43.1 | 53.3
Source only | 75.6 | 17.1 | 69.8 | 10.7 | 16.1 | 21.1 | 27.0 | 10.6 | 77.3 | 15.1 | 71.1 | 53.4 | 20.5 | 73.9 | 28.6 | 31.1 | 1.62 | 32.4 | 21.5 | 35.5
DACS (rep) | 93.1 | 48.1 | 87.3 | 36.7 | 35.1 | 38.7 | 42.5 | 49.3 | 87.5 | 41.9 | 87.9 | 64.8 | 30.7 | 88.3 | 40.2 | 51.0 | 0.0 | 25.1 | 42.6 | 52.1
DACS+DAP | 93.5 | 53.9 | 87.5 | 30.0 | 36.4 | 39.0 | 43.9 | 49.5 | 87.5 | 45.4 | 88.8 | 66.6 | 36.8 | 89.4 | 49.1 | 51.4 | 0.0 | 42.2 | 53.1 | 55.0
ProDA [58] | 87.8 | 56.0 | 79.7 | 46.3 | 44.8 | 45.6 | 53.5 | 53.5 | 88.6 | 45.2 | 82.1 | 70.7 | 39.2 | 88.8 | 45.5 | 50.4 | 1.0 | 48.9 | 56.4 | 57.5
Chao et al. [4] | 94.4 | 60.9 | 88.1 | 39.5 | 41.8 | 43.2 | 49.1 | 56.0 | 88.0 | 45.8 | 87.8 | 67.6 | 38.1 | 90.1 | 57.6 | 51.9 | 0.0 | 46.6 | 55.3 | 58.0
DAP + ProDA | 94.5 | 63.1 | 89.1 | 29.8 | 47.5 | 50.4 | 56.7 | 58.7 | 89.5 | 50.2 | 87.0 | 73.6 | 38.6 | 91.3 | 50.2 | 52.9 | 0.0 | 50.2 | 63.5 | 59.8
Table 2: Segmentation accuracy (IOU, %) of different UDA approaches from GTAv
to Cityscapes. DACS (rep) indicates our re-implementation of DACS. For each
class, we mark the highest number with bold and the second highest with
underline. The top and bottom parts are achieved without and with multi-stage
training or multi-model fusion.
SYNTHIA$\rightarrow$Cityscapes
---
Method | Road | Sidewalk | Building | Wall* | Fence* | Pole* | Light | Sign | Veg | Sky | Person | Rider | Car | Bus | Motor | Bike | mIOU | mIOU*
PatchAlign [51] | 82.4 | 38.0 | 78.6 | 8.7 | 0.6 | 26.0 | 3.9 | 11.1 | 75.5 | 84.6 | 53.5 | 21.6 | 71.4 | 32.6 | 19.3 | 31.7 | 40.0 | 46.5
AdaptSegNe [50] | 84.3 | 42.7 | 77.5 | – | – | – | 4.7 | 7.0 | 77.9 | 82.5 | 54.3 | 21.0 | 72.3 | 32.2 | 18.9 | 32.3 | – | 46.7
FDA [56] | 79.3 | 35.0 | 73.2 | – | – | – | 19.9 | 24.0 | 61.7 | 82.6 | 61.4 | 31.1 | 83.9 | 40.8 | 38.4 | 51.1 | – | 52.5
LTIR [24] | 92.6 | 53.2 | 79.2 | – | – | – | 1.6 | 7.5 | 78.6 | 84.4 | 52.6 | 20.0 | 82.1 | 34.8 | 14.6 | 39.4 | – | 49.3
PIT [33] | 83.1 | 27.6 | 81.5 | 8.9 | 0.3 | 21.8 | 26.4 | 33.8 | 76.4 | 78.8 | 64.2 | 27.6 | 79.6 | 31.2 | 31.0 | 31.3 | 44.0 | 51.8
MetaCorrect [17] | 92.6 | 52.7 | 81.3 | 8.9 | 2.4 | 28.1 | 13.0 | 7.3 | 83.5 | 85.0 | 60.1 | 19.7 | 84.8 | 37.2 | 21.5 | 43.9 | 45.1 | 52.5
DPL [9] | 87.5 | 45.7 | 82.8 | 13.3 | 0.6 | 33.2 | 22.0 | 20.1 | 83.1 | 86.0 | 56.6 | 21.9 | 83.1 | 40.3 | 29.8 | 45.7 | 47.0 | 54.2
DACS [49] | 80.6 | 25.1 | 81.9 | 21.5 | 2.9 | 37.2 | 22.7 | 24.0 | 83.7 | 90.8 | 67.6 | 38.3 | 82.9 | 38.9 | 28.5 | 47.6 | 48.3 | 54.8
IAST [34] | 81.9 | 41.5 | 83.3 | 17.7 | 4.6 | 32.3 | 30.9 | 28.8 | 83.4 | 85.0 | 65.6 | 30.8 | 86.5 | 38.2 | 33.1 | 52.7 | 49.8 | 57.0
Source only | 24.7 | 12.1 | 75.4 | 11.3 | 0.1 | 22.4 | 7.5 | 16.6 | 71.7 | 78.3 | 52.9 | 10.1 | 56.6 | 23.3 | 4.0 | 13.0 | 30.0 | 34.3
DACS (rep) | 82.1 | 31.0 | 82.4 | 22.1 | 1.2 | 33.1 | 32.7 | 25.1 | 84.4 | 88.2 | 65.2 | 34.3 | 83.4 | 42.9 | 24.1 | 50.8 | 48.9 | 55.9
DACS+DAP | 83.9 | 33.3 | 80.2 | 24.1 | 1.2 | 33.4 | 30.8 | 33.8 | 84.3 | 88.5 | 65.7 | 36.2 | 84.3 | 43.3 | 33.3 | 46.3 | 50.2 | 57.2
Chao et al. [4] | 88.7 | 46.7 | 83.8 | 22.7 | 4.1 | 35.0 | 35.9 | 36.1 | 82.8 | 81.4 | 61.6 | 32.1 | 87.9 | 52.8 | 32.0 | 57.7 | 52.6 | 60.0
ProDA [58] | 87.8 | 45.7 | 84.6 | 37.1 | 0.6 | 44.0 | 54.6 | 37.0 | 88.1 | 84.4 | 74.2 | 24.3 | 88.2 | 51.1 | 40.5 | 45.6 | 55.5 | 62.0
DAP + ProDA | 84.2 | 46.5 | 82.5 | 35.1 | 0.2 | 46.7 | 53.6 | 45.7 | 89.3 | 87.5 | 75.7 | 34.6 | 91.7 | 73.5 | 49.4 | 60.5 | 59.8 | 64.3
Table 3: Segmentation accuracy (IOU, %) of different UDA approaches from
SYNTHIA to Cityscapes. DACS (rep) indicates our re-implementation of DACS. For
each class, we mark the highest number with bold and the second highest with
underline. The top and bottom parts are achieved without and with multi-stage
training or multi-model fusion. mIOU* denotes the average of 13 classes, with
the classes marked with * not computed.
### 4.1 Datasets and Implementation Details
$\bullet$ Datasets. We evaluate our method on a popular scenario transferring
the information from a synthesis domain to a real domain. We use GTAv [39] and
SYNTHIA [40] as composite domain datasets and Cityscapes [10] as the real
domain. GTAv [39] is a synthetic dataset extracted from the game of Grand
Theft Auto V. There are $24\rm{,}966$ images with pixel-level semantic
segmentation ground truth. The resolution of these images is $1914\times 1052$
and we resize them into $1280\times 720$ in our experiments. GTAv shares $19$
common classes with Cityscapes. SYNTHIA [40] contains $9\rm{,}400$ virtual
European-style urban images whose resolution is $1280\times 760$ and we keep
the original size in our experiments. We evaluate two settings (13 and 16
categories) in SYNTHIA. Cityscapes is a large-scale dataset with a resolution
of $2048\times 1024$. There are $2\rm{,}975$ and $500$ images in the training
and validation sets, respectively.
$\bullet$ Implementation Details. To be consistent with other methods, we use
the Deeplabv2 [6] framework with a RseNet101 [18] backbone as our image
encoder and an ASPP classifier as segmentation head. The output map is up-
sampled and operated by a softmax layer to match the size of the inputs. The
pre-trained model on ImageNet [11] and MSCOCO [30] is applied to initialize
the backbone. The visual and prior feature projectors,
$g_{\mathrm{vi}}(\cdot)$ and $g_{\mathrm{pr}}(\cdot)$, are both convolution
layers with a $1\times 1$ kernel to adjust the channel to $256$. The batch
size of one GPU is set to $2$. We use SGD with Nesterov acceleration as the
optimizer which is decreased based on a polynomial decay policy with exponent
$0.9$. The initial learning rate of backbone and feature projectors is
$2.5\times 10^{-4}$, that of segmentation head is $10\times$ larger. The
momentum and weight decay of the optimizer are $0.9$ and $5\times 10^{-4}$.
During the training process, we apply color jittering and Gaussian blurring on
the mixed data and only resize operation on source and target data. Teacher
model is updated with an EMA decay and $\lambda$ equals $0.99$. We train the
model for $250\mathrm{K}$ iterations on a single NVIDIA Tesla-V100 GPU and
adopt an early stop setting. The background and invalid categories are ignored
during training. The weight of $\mathcal{L}_{\mathrm{DAP}}$, i.e. $\alpha$, is
set to be $1.0$. We adapt bilinear interpolation to down-sample the embedding
map from the input size to that of the visual feature map – this is an
important step for DAP, which is ablated in Section 4.3.
Figure 4: An example of transfer segmentation from GTAv to Cityscapes. The top
row shows the ground-truth and three segmentation results, while the input and
legend is in the bottom row. The yellow boxes indicate the regions that the
segmentation quality is largely improved, and the number corresponds to
single-image mIOU (over the existing classes). The bottom row also shows the
t-SNE of visual features colored by the predicted class. The blue boxes locate
the features of bike and motorbike, in which DAP achieves a favorable ability
to distinguish them. This figure is best viewed in color and we suggest the
reader to zoom in for details.
### 4.2 Quantitative Results and Visualization
We first evaluate DAP on the transfer segmentation task from GTAv to
Cityscapes. The comparison against recent approaches are shown in Tab 2. To
ensure reliability, we run DACS (the baseline) and DAP three times and report
the average accuracy. DAP achieves a mean IOU of $55.0\%$ over 19 classes,
which claims a $2.9\%$ gain beyond the baseline and also outperforms all other
competitors except for Chao et al. [4] and ProDA [58]. Specifically, Chao et
al. [4] used ensemble learning to integrate the prediction from four
complementarily-trained models, including DACS, but DAP used a single model;
ProDA [58] improved the segmentation accuracy significantly via multi-stage
training, yet its first stage reported a $53.7\%$ mIOU.What is more, the
results on transferring SYNTHIA to Cityscapes, as shown in Tab 3, demonstrate
the similar trend – DAP outperforms all the competitors, except for ProDA and
RED, in terms of either 13-class or 16-class mIOU. To show that DAP offers
complementary benefits, we feed the output of DAP as the pseudo labels to the
1st stage of ProDA, and the 2nd and 3rd stages remain unchanged. As shown in
Tables 2 and 3, the segmentation mIOUs of ProDA in GTAv$\rightarrow$Cityscapes
and SYNTHIA$\rightarrow$Cityscapes are improved by $2.3\%$ and $4.3\%$,
respectively, setting new records in these two scenarios.
Next, we investigate the ability of DAP in distinguishing semantically similar
classes. From GTAv to Cityscapes, the segmentation mIOUs of bike and motorbike
are improved from $42.6\%$ and $25.1\%$ to $53.1\%$ and $42.2\%$, with
absolute gains of $10.5\%$ and $17.1\%$, respectively. From SYNTHIA to
Cityscapes, the mIOU of bike drops by $4.5\%$ and that of motorbike increases
by $9.2\%$, achieving an average improvement of $2.4\%$. We visualize an
example of segmentation in Fig 4. Besides a qualitative observation on the
improvement of distinguishing bike from motorbike and road from sidewalk, we
also notice the reason behind the improvement being a scattered feature
distribution of these similar classes. This aligns with the statistics shown
in Tab 1, indicating that DAP reduces the IOU between the estimated
distributions of bike and motorbike as well as that between road from
sidewalk.
From another perspective, we study how the language-based prior assists visual
recognition. In Fig 5, we show the relationship matrix of the features learned
from the source (GTAv) and target (Cityscapes) domains as well as the word2vec
features. There is an interesting example that person and rider are
semantically similar in both GTAv and Cityscapes but not so correlated
according to word2vec. This may cause a strong yet harmful correlation of
these two feature sets, leading to confusion in segmentation. The relatively
weaker correlation of word2vec features alleviates the bias and improves the
IOU of both classes (refer to Tab 2). Similar phenomena are also observed for
the class pairs of road vs. sidewalk, and bike vs. motorbike.
Figure 5: The relationship matrix of all 19 classes, where the left and middle
are generated by the class-averaged feature vectors extracted from the models
directly trained on GTAv and Cityscapes, and the right uses the word2vec
features. Each cell is the inner product of two normalized features. The order
of class is identical to that in Tab 2 – for easier reference: 1 for road, 2
for sidewalk, 12 for person, 13 for rider, 18 for motorbike, 19 for bike.
Source Dataset | GTAv | SYNTHIA
---|---|---
mIOU (%) | gain | std(%) | mIOU (%) | gain | std(%)
Baseline | 52.1 | – | 1.6 | 48.9 | – | 0.3
w/ random vectors | 52.3 | 0.2$\uparrow$ | 2.3 | 48.0 | 0.9$\downarrow$ | 0.8
w/ one-hot vectors | 53.0 | 0.9$\uparrow$ | 0.8 | 49.8 | 0.9$\uparrow$ | 0.4
w/ CLIP [38] | 54.6 | 2.5$\uparrow$ | 0.6 | 50.6 | 1.7$\uparrow$ | 0.5
w/ word2vec [35] | 55.0 | 2.9$\uparrow$ | 0.5 | 50.2 | 1.3$\uparrow$ | 0.4
Table 4: The results of different vectors choices include the adaptation from
GTAv and SYNTHIA
### 4.3 Diagnostic Studies
$\bullet$ The choice of domain-agnostic prior. We investigate three other
options except for word2vec embedding [35], namely, (1) that generating a
$300$-dimensional, normalized random vector for each class, (2) directly
creating a one-hot vector for each class, where the dimensionality equals to
the number of classes, (3) classes embeddings generated by the language branch
of CLIP [38]. Similar to the main experiments, we run each option three times
and report the averaged accuracy. Results are summarized in Tab 4.
One can see that random vectors achieve a slight $0.2\%$ accuracy gain on
GTAv$\rightarrow$Cityscapes, but incurs a $0.9\%$ accuracy drop on
SYNTHIA$\rightarrow$Cityscapes, implying the instability. Throughout the three
individual runs on GTAv$\rightarrow$Cityscapes, the best run achieves a
$54.4\%$ mIOU, just $0.6\%$ lower than using word2vec, but the worst one
reports $49.2\%$ which is even significantly lower than the baseline.
Regarding the one-hot vectors, the results over three runs are less
diversified, and the average improvement is consistent, (i.e., $0.9\%$ on both
GTAv$\rightarrow$Cityscapes and SYNTHIA$\rightarrow$Cityscapes), though
smaller than that brought by word2vec embedding. The prior from CLIP [38]
reports 54.6% and 50.6% mIOUs on the GTAv and SYNTHIA experiments,
respectively. Despite the fact that CLIP is stronger than word2vec, the mIOUs
are just comparable to that using word2vec (55.0% and 50.2%). From these
results, we learn the lesson that (1) even a naive prior alleviates the inter-
class confusion caused by domain shift, however, (2) it would be better if the
inter-class relationship is better captured so that the model is aware of
semantically similar classes – text embedding offers a safe and effective
option. (3) The limited number and diversity of target categories may have
diminished the advantages of a stronger language model (e.g., CLIP), and we
still pursue for a vision-aware yet domain-agnostic embedding method.
$\bullet$ Different Backbones. To verify the effectiveness of DAP on different
network structures, we replace the convolution backbone (ResNet101) with a
transformer network (ViT-Base [12]). The numbers of the block layers, token
size, and heads are $12,768$ and $12$ respectively in the transformer encoder.
The input size of the training data is set $768\times 768$. And we initiate
the ViT encoder with a model pre-trained on ImageNet-21k then fine-tune the
segmentation network with a base learning rate of $0.01$ adopted with the
’poly’ learning rate decay and use SGD as the optimizer. The $g_{\mathrm{pr}}$
is one layer transformer structure and weight of $\mathcal{L}_{\mathrm{DAP}}$
is $0.25$. On GTAv, source-only, DACS, DAP report $49.4\%$, $58.4\%$, $61.1\%$
mIOUs. As for the transferring from SYNTHIA, the numbers of the three settings
are $42.7\%$, $53.2\%$, $59.1\%$ when evaluating on 16 classes and $48.1\%$,
$60.9\%$, $66.1\%$ on 13 classes. We can see that DAP still obtains consistent
accuracy gain. To the best of our knowledge, the DAP numbers are SOTA.
$\bullet$ The importance of feature interpolation. From the model that only
uses the source domain, we gradually add mixed data (by DACS [49]), introduce
DAP (using the word2vec embedding), and perform feature interpolation to down-
sample the embedding map. As shown in Tab 5, for the both two transferring
scenarios, feature interpolation contributes nearly half accuracy gain of DAP
over DACS. Intuitively, feature interpolation enables the features on small-
area classes to be preserved, yet nearest-neighbor down-sampling can cause
these features to be ignored.
Setting | source only | DACS | DAP w/o interp | DAP w/ interp
---|---|---|---|---
GTAv | 35.5 | 52.1 | 53.8 | 55.0
SYNTHIA | 30.0 | 48.9 | 49.6 | 50.2
Table 5: Ablation on the contribution of each module in both
GTAv$\rightarrow$Cityscapes and SYNTHIA$\rightarrow$Cityscapes experiments.
$\bullet$ Parameter Analysis. Lastly, we study the impact of the coefficient
$\alpha$ that balances between $\mathcal{L}_{\mathrm{seg}}$ and
$\mathcal{L}_{\mathrm{DAP}}$. The results shown in Tab 6 suggest that
$\alpha=1.0$ is the best option. In addition, increasing $\alpha$ causes a
larger accuracy drop compared to decreasing it, which may indicate that
$\mathcal{L}_{\mathrm{seg}}$ is the essential goal and
$\mathcal{L}_{\mathrm{DAP}}$ serves as an auxiliary term.
$\alpha$ | 0.50 | 0.75 | 1.00 | 1.25 | 1.50
---|---|---|---|---|---
GTAv | 54.5 | 54.1 | 55.0 | 54.3 | 54.7
SYNTHIA | 50.0 | 49.4 | 50.2 | 49.6 | 49.1
Table 6: Impact of tuning the balancing coefficient, $\alpha$. All numbers are
segmentation mIOU (%).
## 5 Conclusions
In this paper, we investigate the UDA segmentation problem and observe that
semantically similar classes are easily confused during the transfer
procedure. We formulate this problem using the Bayesian theory and owe such
confusion to the weakness of likelihood, e.g., insufficient training data. To
alleviate the issue, we introduce domain-agnostic priors to compensate the
likelihood. Experiments on two standard benchmarks for UDA segmentation in
urban scenes verify its effectiveness both quantitatively and qualitatively
Limitations of this work. Currently, the best option of DAP is to leverage the
text embedding vectors. We are looking forward to more powerful priors, e.g.,
from the cross-modal pre-trained models [38, 22]. This may call for more
sophisticated designs of prior embedding, projection, alignment, etc., which
we leave for future work.
Acknowledge This work was supported in part by the National Natural Science
Foundation of China under Contract U20A20183 and 62021001, and in part by the
Youth Innovation Promotion Association CAS under Grant 2018497. It was also
supported by the GPU cluster built by MCC Lab of Information Science and
Technology Institution, USTC.
## References
* [1] Donghyeon Baek, Youngmin Oh, and Bumsub Ham. Exploiting a joint embedding space for generalized zero-shot semantic segmentation. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 9536–9545, 2021.
* [2] Konstantinos Bousmalis, Nathan Silberman, David Dohan, Dumitru Erhan, and Dilip Krishnan. Unsupervised pixel-level domain adaptation with generative adversarial networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3722–3731, 2017.
* [3] Maxime Bucher, Tuan-Hung Vu, Matthieu Cord, and Patrick Pérez. Zero-shot semantic segmentation. Advances in Neural Information Processing Systems, 32:468–479, 2019\.
* [4] Chen-Hao Chao, Bo-Wun Cheng, and Chun-Yi Lee. Rethinking ensemble-distillation for semantic segmentation based unsupervised domain adaption. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2610–2620, 2021.
* [5] Chaoqi Chen, Weiping Xie, Wenbing Huang, Yu Rong, Xinghao Ding, Yue Huang, Tingyang Xu, and Junzhou Huang. Progressive feature alignment for unsupervised domain adaptation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 627–636, 2019.
* [6] Liang-Chieh Chen, George Papandreou, Iasonas Kokkinos, Kevin Murphy, and Alan L Yuille. Deeplab: Semantic image segmentation with deep convolutional nets, atrous convolution, and fully connected crfs. IEEE transactions on pattern analysis and machine intelligence, 40(4):834–848, 2017.
* [7] Minghao Chen, Hongyang Xue, and Deng Cai. Domain adaptation for semantic segmentation with maximum squares loss. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 2090–2099, 2019.
* [8] Yuhua Chen, Wen Li, Christos Sakaridis, Dengxin Dai, and Luc Van Gool. Domain adaptive faster r-cnn for object detection in the wild. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3339–3348, 2018.
* [9] Yiting Cheng, Fangyun Wei, Jianmin Bao, Dong Chen, Fang Wen, and Wenqiang Zhang. Dual path learning for domain adaptation of semantic segmentation. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 9082–9091, 2021.
* [10] Marius Cordts, Mohamed Omran, Sebastian Ramos, Timo Rehfeld, Markus Enzweiler, Rodrigo Benenson, Uwe Franke, Stefan Roth, and Bernt Schiele. The cityscapes dataset for semantic urban scene understanding. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3213–3223, 2016.
* [11] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei. Imagenet: A large-scale hierarchical image database. In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.
* [12] Alexey Dosovitskiy, Lucas Beyer, Alexander Kolesnikov, Dirk Weissenborn, Xiaohua Zhai, Thomas Unterthiner, Mostafa Dehghani, Matthias Minderer, Georg Heigold, Sylvain Gelly, et al. An image is worth 16x16 words: Transformers for image recognition at scale. arXiv preprint arXiv:2010.11929, 2020.
* [13] Fangxiang Feng, Xiaojie Wang, and Ruifan Li. Cross-modal retrieval with correspondence autoencoder. In Proceedings of the 22nd ACM international conference on Multimedia, pages 7–16, 2014.
* [14] Yang Fu, Yunchao Wei, Guanshuo Wang, Yuqian Zhou, Honghui Shi, and Thomas S Huang. Self-similarity grouping: A simple unsupervised cross domain adaptation approach for person re-identification. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 6112–6121, 2019.
* [15] Yaroslav Ganin and Victor Lempitsky. Unsupervised domain adaptation by backpropagation. In International conference on machine learning, pages 1180–1189. PMLR, 2015.
* [16] Li Gao, Jing Zhang, Lefei Zhang, and Dacheng Tao. Dsp: Dual soft-paste for unsupervised domain adaptive semantic segmentation. arXiv preprint arXiv:2107.09600, 2021.
* [17] Xiaoqing Guo, Chen Yang, Baopu Li, and Yixuan Yuan. Metacorrection: Domain-aware meta loss correction for unsupervised domain adaptation in semantic segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3927–3936, 2021.
* [18] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
* [19] Judy Hoffman, Eric Tzeng, Taesung Park, Jun-Yan Zhu, Phillip Isola, Kate Saenko, Alexei Efros, and Trevor Darrell. Cycada: Cycle-consistent adversarial domain adaptation. In International conference on machine learning, pages 1989–1998. PMLR, 2018.
* [20] Hengtong Hu, Lingxi Xie, Richang Hong, and Qi Tian. Creating something from nothing: Unsupervised knowledge distillation for cross-modal hashing. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 3123–3132, 2020.
* [21] Xinyue Huo, Lingxi Xie, Jianzhong He, Zijie Yang, Wengang Zhou, Houqiang Li, and Qi Tian. Atso: Asynchronous teacher-student optimization for semi-supervised image segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 1235–1244, 2021.
* [22] Chao Jia, Yinfei Yang, Ye Xia, Yi-Ting Chen, Zarana Parekh, Hieu Pham, Quoc V Le, Yunhsuan Sung, Zhen Li, and Tom Duerig. Scaling up visual and vision-language representation learning with noisy text supervision. arXiv preprint arXiv:2102.05918, 2021.
* [23] Aishwarya Kamath, Mannat Singh, Yann LeCun, Gabriel Synnaeve, Ishan Misra, and Nicolas Carion. Mdetr-modulated detection for end-to-end multi-modal understanding. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 1780–1790, 2021.
* [24] Myeongjin Kim and Hyeran Byun. Learning texture invariant representation for domain adaptation of semantic segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12975–12984, 2020.
* [25] Seunghyeon Kim, Jaehoon Choi, Taekyung Kim, and Changick Kim. Self-training and adversarial background regularization for unsupervised domain adaptive one-stage object detection. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 6092–6101, 2019.
* [26] Hugo Larochelle, Dumitru Erhan, and Yoshua Bengio. Zero-data learning of new tasks. In AAAI, volume 1, page 3, 2008.
* [27] Dong-Hyun Lee et al. Pseudo-label: The simple and efficient semi-supervised learning method for deep neural networks. In Workshop on challenges in representation learning, ICML, volume 3, page 896, 2013.
* [28] Xinzhe Li, Qianru Sun, Yaoyao Liu, Qin Zhou, Shibao Zheng, Tat-Seng Chua, and Bernt Schiele. Learning to self-train for semi-supervised few-shot classification. Advances in Neural Information Processing Systems, 32:10276–10286, 2019.
* [29] Xiujun Li, Xi Yin, Chunyuan Li, Pengchuan Zhang, Xiaowei Hu, Lei Zhang, Lijuan Wang, Houdong Hu, Li Dong, Furu Wei, et al. Oscar: Object-semantics aligned pre-training for vision-language tasks. In European Conference on Computer Vision, pages 121–137. Springer, 2020.
* [30] Tsung-Yi Lin, Michael Maire, Serge Belongie, James Hays, Pietro Perona, Deva Ramanan, Piotr Dollár, and C Lawrence Zitnick. Microsoft coco: Common objects in context. In European conference on computer vision, pages 740–755. Springer, 2014.
* [31] Jinlu Liu, Liang Song, and Yongqiang Qin. Prototype rectification for few-shot learning. In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part I 16, pages 741–756. Springer, 2020.
* [32] Mingsheng Long, Yue Cao, Jianmin Wang, and Michael Jordan. Learning transferable features with deep adaptation networks. In International conference on machine learning, pages 97–105. PMLR, 2015.
* [33] Fengmao Lv, Tao Liang, Xiang Chen, and Guosheng Lin. Cross-domain semantic segmentation via domain-invariant interactive relation transfer. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4334–4343, 2020.
* [34] Ke Mei, Chuang Zhu, Jiaqi Zou, and Shanghang Zhang. Instance adaptive self-training for unsupervised domain adaptation. In Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XXVI 16, pages 415–430. Springer, 2020.
* [35] Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In Advances in neural information processing systems, pages 3111–3119, 2013.
* [36] Roozbeh Mottaghi, Xianjie Chen, Xiaobai Liu, Nam-Gyu Cho, Seong-Whan Lee, Sanja Fidler, Raquel Urtasun, and Alan Yuille. The role of context for object detection and semantic segmentation in the wild. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 891–898, 2014.
* [37] Viktor Olsson, Wilhelm Tranheden, Juliano Pinto, and Lennart Svensson. Classmix: Segmentation-based data augmentation for semi-supervised learning. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 1369–1378, 2021.
* [38] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al. Learning transferable visual models from natural language supervision. arXiv preprint arXiv:2103.00020, 2021.
* [39] Stephan R Richter, Vibhav Vineet, Stefan Roth, and Vladlen Koltun. Playing for data: Ground truth from computer games. In European conference on computer vision, pages 102–118. Springer, 2016.
* [40] German Ros, Laura Sellart, Joanna Materzynska, David Vazquez, and Antonio M Lopez. The synthia dataset: A large collection of synthetic images for semantic segmentation of urban scenes. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3234–3243, 2016.
* [41] Chuck Rosenberg, Martial Hebert, and Henry Schneiderman. Semi-supervised self-training of object detection models. 2005\.
* [42] Kuniaki Saito, Kohei Watanabe, Yoshitaka Ushiku, and Tatsuya Harada. Maximum classifier discrepancy for unsupervised domain adaptation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3723–3732, 2018.
* [43] Piyush Sharma, Nan Ding, Sebastian Goodman, and Radu Soricut. Conceptual captions: A cleaned, hypernymed, image alt-text dataset for automatic image captioning. In Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 2556–2565, 2018.
* [44] Inkyu Shin, Sanghyun Woo, Fei Pan, and In So Kweon. Two-phase pseudo label densification for self-training based domain adaptation. In European conference on computer vision, pages 532–548. Springer, 2020.
* [45] Liangchen Song, Cheng Wang, Lefei Zhang, Bo Du, Qian Zhang, Chang Huang, and Xinggang Wang. Unsupervised domain adaptive re-identification: Theory and practice. Pattern Recognition, 102:107173, 2020.
* [46] Hao Tan and Mohit Bansal. Lxmert: Learning cross-modality encoder representations from transformers. arXiv preprint arXiv:1908.07490, 2019.
* [47] Antti Tarvainen and Harri Valpola. Mean teachers are better role models: Weight-averaged consistency targets improve semi-supervised deep learning results. arXiv preprint arXiv:1703.01780, 2017.
* [48] Marco Toldo, Umberto Michieli, Gianluca Agresti, and Pietro Zanuttigh. Unsupervised domain adaptation for mobile semantic segmentation based on cycle consistency and feature alignment. Image and Vision Computing, 95:103889, 2020.
* [49] Wilhelm Tranheden, Viktor Olsson, Juliano Pinto, and Lennart Svensson. Dacs: Domain adaptation via cross-domain mixed sampling. In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision, pages 1379–1389, 2021.
* [50] Yi-Hsuan Tsai, Wei-Chih Hung, Samuel Schulter, Kihyuk Sohn, Ming-Hsuan Yang, and Manmohan Chandraker. Learning to adapt structured output space for semantic segmentation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7472–7481, 2018.
* [51] Yi-Hsuan Tsai, Kihyuk Sohn, Samuel Schulter, and Manmohan Chandraker. Domain adaptation for structured output via discriminative patch representations. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 1456–1465, 2019.
* [52] Eric Tzeng, Judy Hoffman, Kate Saenko, and Trevor Darrell. Adversarial discriminative domain adaptation. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 7167–7176, 2017.
* [53] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pages 5998–6008, 2017.
* [54] Tuan-Hung Vu, Himalaya Jain, Maxime Bucher, Matthieu Cord, and Patrick Pérez. Advent: Adversarial entropy minimization for domain adaptation in semantic segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2517–2526, 2019.
* [55] Bokun Wang, Yang Yang, Xing Xu, Alan Hanjalic, and Heng Tao Shen. Adversarial cross-modal retrieval. In Proceedings of the 25th ACM international conference on Multimedia, pages 154–162, 2017.
* [56] Yanchao Yang and Stefano Soatto. Fda: Fourier domain adaptation for semantic segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 4085–4095, 2020.
* [57] Hongyi Zhang, Moustapha Cisse, Yann N Dauphin, and David Lopez-Paz. mixup: Beyond empirical risk minimization. arXiv preprint arXiv:1710.09412, 2017.
* [58] Pan Zhang, Bo Zhang, Ting Zhang, Dong Chen, Yong Wang, and Fang Wen. Prototypical pseudo label denoising and target structure learning for domain adaptive semantic segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 12414–12424, 2021.
* [59] Hang Zhao, Xavier Puig, Bolei Zhou, Sanja Fidler, and Antonio Torralba. Open vocabulary scene parsing. In Proceedings of the IEEE International Conference on Computer Vision, pages 2002–2010, 2017.
* [60] Qianyu Zhou, Zhengyang Feng, Qiqi Gu, Guangliang Cheng, Xuequan Lu, Jianping Shi, and Lizhuang Ma. Uncertainty-aware consistency regularization for cross-domain semantic segmentation. arXiv preprint arXiv:2004.08878, 2020.
* [61] Qianyu Zhou, Zhengyang Feng, Qiqi Gu, Jiangmiao Pang, Guangliang Cheng, Xuequan Lu, Jianping Shi, and Lizhuang Ma. Context-aware mixup for domain adaptive semantic segmentation. arXiv preprint arXiv:2108.03557, 2021.
* [62] Jun-Yan Zhu, Taesung Park, Phillip Isola, and Alexei A Efros. Unpaired image-to-image translation using cycle-consistent adversarial networks. In Proceedings of the IEEE international conference on computer vision, pages 2223–2232, 2017.
* [63] Xiaojin Jerry Zhu. Semi-supervised learning literature survey. 2005\.
* [64] Yang Zou, Zhiding Yu, BVK Kumar, and Jinsong Wang. Unsupervised domain adaptation for semantic segmentation via class-balanced self-training. In Proceedings of the European conference on computer vision (ECCV), pages 289–305, 2018.
* [65] Yang Zou, Zhiding Yu, Xiaofeng Liu, BVK Kumar, and Jinsong Wang. Confidence regularized self-training. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pages 5982–5991, 2019.
|
# On Computing Probabilistic Abductive Explanations
Yacine Izza<EMAIL_ADDRESS>
University of Toulouse, Toulouse, France
Monash University, Melbourne, Australia Xuanxiang Huang xuanxiang.huang@univ-
toulouse.fr
University of Toulouse, Toulouse, France Alexey Ignatiev
<EMAIL_ADDRESS>
Monash University, Melbourne, Australia Nina Narodytska<EMAIL_ADDRESS>
VMware Research, Palo Alto, CA, U.S.A. Martin C. Cooper<EMAIL_ADDRESS>
IRIT, UPS, Toulouse, France Joao Marques-Silva<EMAIL_ADDRESS>
IRIT, CNRS, Toulouse, France
###### Abstract
The most widely studied explainable AI (XAI) approaches are unsound. This is
the case with well-known model-agnostic explanation approaches, and it is also
the case with approaches based on saliency maps. One solution is to consider
intrinsic interpretability, which does not exhibit the drawback of
unsoundness. Unfortunately, intrinsic interpretability can display unwieldy
explanation redundancy. Formal explainability represents the alternative to
these non-rigorous approaches, with one example being PI-explanations.
Unfortunately, PI-explanations also exhibit important drawbacks, the most
visible of which is arguably their size. Recently, it has been observed that
the (absolute) rigor of PI-explanations can be traded off for a smaller
explanation size, by computing the so-called relevant sets. Given some
positive $\delta$, a set ${\mathcal{S}}$ of features is $\delta$-relevant if,
when the features in ${\mathcal{S}}$ are fixed, the probability of getting the
target class exceeds $\delta$. However, even for very simple classifiers, the
complexity of computing relevant sets of features is prohibitive, with the
decision problem being $\textnormal{NP}^{\textnormal{PP}}$-complete for
circuit-based classifiers. In contrast with earlier negative results, this
paper investigates practical approaches for computing relevant sets for a
number of widely used classifiers that include Decision Trees (DTs), Naive
Bayes Classifiers (NBCs), and several families of classifiers obtained from
propositional languages. Moreover, the paper shows that, in practice, and for
these families of classifiers, relevant sets are easy to compute. Furthermore,
the experiments confirm that succinct sets of relevant features can be
obtained for the families of classifiers considered.
###### Contents
1. 1 Introduction
2. 2 Preliminaries
1. 2.1 Logic Foundations
2. 2.2 Classification Problems
3. 2.3 Families of Classifiers
4. 2.4 Running Examples
5. 2.5 Formal Explainability
6. 2.6 $\delta$-Relevant Sets
3. 3 Relevant Sets – Probabilistic Abductive Explanations
1. 3.1 Definitions of Probabilistic AXp’s
2. 3.2 Computing Locally-Minimal PAXp’s
4. 4 Probabilistic Explanations for Decision Trees
1. 4.1 Path Probabilities for DTs
2. 4.2 Computing Locally-Minimal PAXp’s for DT’s
3. 4.3 Computing Minimum-Size PAXp’s for DTs
4. 4.4 Deciding Whether a Locally-Minimal PAXp is a Plain PAXp for DTs
5. 4.5 Instance-Based vs. Path-Based Explanations
5. 5 Probabilistic Explanations for Naive Bayes Classifiers
1. 5.1 Explaining NBCs in Polynomial Time
2. 5.2 Counting Models of XLCs
3. 5.3 Computing Locally-Minimal PAXp’s for NBCs
6. 6 Probabilistic Explanations for Other Families of Classifiers
1. 6.1 Propositional Classifiers
2. 6.2 Graph-Based Classifiers
7. 7 Experiments
1. 7.1 Case Study 1: Decision Trees
2. 7.2 Case Study 2: Naive Bayes Classifiers
3. 7.3 Case Study 3: Graph-Based Classifiers
8. 8 Related Work
9. 9 Conclusions
## 1 Introduction
The advances in Machine Learning (ML) in recent years motivated an ever
increasing range of practical applications of systems of Artificial
Intelligence (AI). Some uses of ML models are deemed _high-risk_ given the
impact that their operation can have on people [28]. (Other authors refer to
_high-stakes_ applications [84].) In some domains, with high-risk applications
representing one key example, the deployment of AI systems is premised on the
availability of mechanisms for explaining the often opaque operation of ML
models [28]. The need for explaining the opaque operation of ML models
motivated the emergence of eXplainable AI (XAI). Moreover, the rigor of
explanations is a cornerstone to delivering trustworthy AI [64]. For example,
rigorous explanations can be mathematically proven correct, and so a human
decision maker can independently validate that an explanation is (logically)
rigorous. The rigor of explanations is even more significant in high-risk AI
systems, where human decision makers must be certain that provided
explanations are sound.
Motivated by the importance of understanding the operation of blackbox ML
models, recent years have witnessed a growing interest in XAI [72, 33, 86, 87,
71, 85]. The best-known XAI approaches can be broadly categorized as either
model-agnostic methods, that include for example LIME [81], SHAP [60] and
Anchor [82], or intrinsic interpretability [84, 71], for which the explanation
is represented by the actual (interpretable) ML model. For specific ML models,
e.g. neural networks, there are dedicated explainability approaches, including
those based on saliency maps [91, 8]; these exhibit limitations similar to
model-agnostic approaches [1, 55, 99, 92]. Moreover, intrinsic
interpretability may not represent a viable option in some uses of AI systems.
Also, it has been shown, both in theory and in practice, that intrinsic
interpretable models (such as decision trees or sets) also require being
explained [45, 41, 46]. On the other hand, model-agnostic methods, even if
locally accurate, can produce explanations that are unsound [39], in addition
to displaying several other drawbacks [16, 93, 57, 25]. Unsound explanations
are hopeless whenever rigor is a key requirement; thus, model-agnostic
explanations ought not be used in high-risk settings. Indeed, it has been
reported [39] that an explanation ${\mathcal{X}}$ can be consistent with
different predicted classes. For example, for a bank loan application,
${\mathcal{X}}$ might be consistent with an approved loan application, but
also with a declined loan application. An explanation that is consistent with
both a declined and an approved loan applications offers no insight to why one
of the loan applications was declined. There have been recent efforts on
rigorous XAI approaches [89, 43, 22, 7, 61, 13, 32], most of which are based
on feature selection, namely the computation of so-called abductive
explanations (AXp’s). However, these efforts have mostly focused on the
scalability of computing rigorous explanations, with more recent work
investigating input distributions [32]. Nevertheless, another important
limitation of rigorous XAI approaches is the often unwieldy size of
explanations. Recent work studied probabilistic explanations, as a mechanism
to reduce the size of rigorous explanations [102, 101, 100]. Some approaches
for computing probabilistic explanations have extended model-agnostic
approaches [102], and so can suffer from unsoundness. Alternatively, more
rigorous approaches to computing probabilistic explanations have been shown to
be computationally hard, concretely hard for
$\textnormal{NP}^{\textnormal{PP}}$ [101, 100], and so are all but certain to
fall beyond the reach of modern automated reasoners.
This paper builds on recent work [101, 100] on computing rigorous
probabilistic explanations, and investigates their practical scalability.
However, instead of considering classifiers represented as boolean circuits
(as in [101]), the paper studies families of classifiers that are shown to be
amenable to the practical computation of relevant sets. As the paper shows,
such examples include decision trees (DTs), naive Bayes classifiers (NBCs),
but also several propositional classifiers, graph-based classifiers, and their
multi-valued variants 111This paper aggregates and extends recent preprints
that compute relevant sets for concrete families of classifiers [48, 50]..
The paper revisits the original definition of $\delta$-relevant set, and
considers different variants, in addition to the one proposed in earlier work
[101, 100], i.e. smallest relevant sets. These include subset-minimal relevant
sets, and locally-minimal relevant sets. Throughout the paper, relevant sets
will be referred to as _probabilistic abductive explanations_ (PAXp’s).
Whereas computing a smallest and a subset-minimal PAXp’s is shown to be in NP
for decision trees and different propositional classifiers, computing locally-
minimal PAXp’s is shown to be in P for decision trees and different
propositional classifiers, and in pseudo-polynomial time for naive Bayes
classifiers. Furthermore, the experiments confirm that locally-minimal PAXp’s
most often match subset-minimal PAXp’s. As a result, locally-minimal relevant
sets are shown to represent a practically efficient approach for computing (in
polynomial or pseudo-polynomial time) PAXp’s that are most often subset-
minimal.
Complexity of
PAXpFP$\textnormal{LmPAXp}(\textnormal{d-DDNF})$$\textnormal{LmPAXp}(\textnormal{GDF})$$\textnormal{LmPAXp}(\textnormal{SDD})$$\textnormal{LmPAXp}(\textnormal{DG})$$\textnormal{LmPAXp}(\textnormal{DT})$$\textbf{FP}^{\textbf{NP}}$$\textnormal{PAXp}(\textnormal{d-DDNF})$$\textnormal{PAXp}(\textnormal{GDF})$$\textnormal{PAXp}(\textnormal{SDD})$$\textnormal{PAXp}(\textnormal{DG})$$\textnormal{PAXp}(\textnormal{DT})$$\textbf{FP}^{\textbf{NP}^{\textbf{PP}}}$$\textnormal{PAXp}(\textnormal{BC})$$\textbf{FP}^{\textbf{pseudo-
NP}}$$\textnormal{PAXp}(\textnormal{NBC})$pseudo-
FP$\textnormal{LmPAXp}(\textnormal{NBC})$ Figure 1: Summary of results
The paper’s contributions are summarized in Figure 1. Figure 1 overviews
complexity class membership results for several families of classifiers, most
of which are established in this paper. For several families of classifiers,
the paper proves that computing one locally-minimal PAXp is either in
polynomial time or pseudo-polynomial time. Similarly, computing one subset-
minimal (and also one cardinality-minimal) PAXp is shown to be in NP (or
pseudo-NP) for several families of classifiers.
The paper is organized as follows. Section 2 introduces the definitions and
notation used throughout. Section 3 offers an overview of probabilistic
abductive explanations (or relevant sets). Section 4 investigates the
computation of relevant sets in the case of decision trees. Sections 5 and 6
replicate the same exercise, respectively in the case of naive Bayes
classifiers, and also for graph-based classifiers and classifiers based on
propositional languages. Section 7 presents experimental results on computing
relevant sets for the classifiers studied in the earlier sections. The paper
concludes in Section 9.
## 2 Preliminaries
#### Complexity classes.
The paper addresses a number of well-known classes of decision and function
problems, that include P, NP, $\textnormal{FP}^{\textnormal{NP}}$,
$\textnormal{NP}^{\textnormal{PP}}$, among others. The interested reader is
referred to a standard reference on the computational complexity [5].
### 2.1 Logic Foundations
#### Propositional logic.
The paper assumes the notation and definitions that are standard when
reasoning about the decision problem for propositional logic, i.e. the Boolean
Satisfiability (SAT) problem [12]. SAT is well-known to be an NP-complete [17]
decision problem. A propositional formula $\varphi$ is defined over a finite
set of Boolean variables $X=\\{x_{1},x_{2},\ldots,x_{n}\\}$. Formulas are most
often represented in _conjunctive normal form_ (CNF). A CNF formula is a
conjunction of clauses, a clause is a disjunction of literals, and a literal
is a variable ($x_{i}$) or its negation ($\neg{x_{i}}$). A term is a
conjunction of literals. Whenever convenient, a formula is viewed as a set of
sets of literals. A Boolean interpretation $\mu$ of a formula $\varphi$ is a
total mapping of $X$ to $\\{0,1\\}$ ($0$ corresponds to _False_ and $1$
corresponds to _True_). Interpretations can be extended to literals, clauses
and formulas with the usual semantics; hence we can refer to $\mu(l)$,
$\mu(\omega)$, $\mu(\varphi)$, to denote respectively the value of a literal,
clause and formula given an interpretation. Given a formula $\varphi$, $\mu$
is a _model_ of $\varphi$ if it makes $\varphi$ _True_ , i.e. $\mu(\phi)=1$. A
formula $\varphi$ is _satisfiable_ ($\varphi\operatorname*{\nvDash}\perp$) if
it admits a model, otherwise, it is _unsatisfiable_
($\varphi\operatorname*{\vDash}\perp$). Given two formulas $\varphi$ and
$\psi$, we say that $\varphi$ entails $\psi$ (denoted
$\varphi\operatorname*{\vDash}\psi$) if all models of $\varphi$ are also
models of $\psi$. $\varphi$ and $\psi$ are equivalent (denoted
$\varphi\equiv\psi$) if $\varphi\operatorname*{\vDash}\psi$ and
$\psi\operatorname*{\vDash}\varphi$.
#### First Order Logic (FOL) and SMT.
When necessary, the paper will consider the restriction of FOL to
Satisfiability Modulo Theories (SMT). These represent restricted (and often
decidable) fragments of FOL [10]. All the definitions above apply to SMT. A
SMT-solver reports whether a formula is satisfiable, and if so, may provide a
model of this satisfaction. (Other possible features include dynamic addition
and retraction of constraints, production of proofs, and optimization.)
### 2.2 Classification Problems
This paper considers classification problems, which are defined on a set of
features (or attributes) ${\mathcal{F}}=\\{1,\ldots,m\\}$ and a set of classes
${\mathcal{K}}=\\{c_{1},c_{2},\ldots,c_{K}\\}$. Each feature
$i\in{\mathcal{F}}$ takes values from a domain $\mathbb{D}_{i}$. In general,
domains can be categorical or ordinal, with values that can be boolean or
integer. (Although real-valued could be considered for some of the classifiers
studied in the paper, we opt not to specifically address real-valued
features.) Feature space is defined as
$\mathbb{F}=\mathbb{D}_{1}\times{\mathbb{D}_{2}}\times\ldots\times{\mathbb{D}_{m}}$;
$|\mathbb{F}|$ represents the total number of points in $\mathbb{F}$. For
boolean domains, $\mathbb{D}_{i}=\\{0,1\\}=\mathbb{B}$, $i=1,\ldots,m$, and
$\mathbb{F}=\mathbb{B}^{m}$. The notation $\mathbf{x}=(x_{1},\ldots,x_{m})$
denotes an arbitrary point in feature space, where each $x_{i}$ is a variable
taking values from $\mathbb{D}_{i}$. The set of variables associated with
features is $X=\\{x_{1},\ldots,x_{m}\\}$. Moreover, the notation
$\mathbf{v}=(v_{1},\ldots,v_{m})$ represents a specific point in feature
space, where each $v_{i}$ is a constant representing one concrete value from
$\mathbb{D}_{i}$. With respect to the set of classes ${\mathcal{K}}$, the size
of ${\mathcal{K}}$ is assumed to be finite; no additional restrictions are
imposed on ${\mathcal{K}}$. Nevertheless, with the goal of simplicity, the
paper considers examples where $|{\mathcal{K}}|=2$, concretely
${\mathcal{K}}=\\{\ominus,\oplus\\}$, or alternatively
${\mathcal{K}}=\\{0,1\\}$. An ML classifier $\mathbb{M}$ is characterized by a
(non-constant) _classification function_ $\kappa$ that maps feature space
$\mathbb{F}$ into the set of classes ${\mathcal{K}}$, i.e.
$\kappa:\mathbb{F}\to{\mathcal{K}}$. An _instance_ (or observation) denotes a
pair $(\mathbf{v},c)$, where $\mathbf{v}\in\mathbb{F}$ and
$c\in{\mathcal{K}}$, with $c=\kappa(\mathbf{v})$.
### 2.3 Families of Classifiers
Throughout the paper, a number of families of classifiers will be studied in
detail. These include decision trees [14, 80], naive Bayes classifiers [29],
graph-based classifiers [76, 77, 35] and classifiers based on propositional
languages [23, 21].
#### Decision trees.
A decision tree ${\mathcal{T}}=(V,E)$ is a directed acyclic graph, with
$V=\\{1,\ldots,|V|\\}$, having at most one path between every pair of nodes.
${\mathcal{T}}$ has a root node, characterized by having no incoming edges.
All other nodes have one incoming edge. We consider univariate decision trees
where each non-terminal node is associated with a single feature $x_{i}$. Each
edge is labeled with a literal, relating a feature (associated with the edge’s
starting node) with some values (or range of values) from the feature’s
domain. We will consider literals to be of the form $x_{i}\in\mathbb{E}_{i}$.
$x_{i}$ is a variable that denotes the value taken by feature $i$, whereas
$\mathbb{E}_{i}\subseteq\mathbb{D}_{i}$ is a subset of the domain of feature
$i\in{\mathcal{F}}$. The type of literals used to label the edges of a DT
allows the representation of the DTs generated by a wide range of decision
tree learners (e.g. [96]). The set of paths of ${\mathcal{T}}$ is denoted by
${\mathcal{R}}$. $\mathrm{\Phi}(R_{k})$ denotes the set of features associated
with path $R_{k}\in{\mathcal{R}}$, one per node in the tree, with repetitions
allowed. It is assumed that for any $\mathbf{v}\in\mathbb{F}$ there exists
_exactly_ one path in ${\mathcal{T}}$ that is consistent with $\mathbf{v}$. By
_consistent_ we mean that the literals associated with the path are satisfied
(or consistent) with the feature values in $\mathbf{v}$. Given $\mathbf{v}$,
the set of paths ${\mathcal{R}}$ is partitioned into ${\mathcal{P}}$ and
${\mathcal{Q}}$, such that each of the paths in ${\mathcal{P}}$ yields the
prediction $c=\kappa(\mathbf{v}$, whereas each of the paths in ${\mathcal{Q}}$
yields a prediction in ${\mathcal{K}}\setminus\\{c\\}$. For the purposes of
this paper, the path consistent with $\mathbf{v}$ as $P_{t}\in{\mathcal{P}}$,
i.e. the target path. (A more in-depth analysis of explaining decision trees
is available in [46].)
#### Naive Bayes Classifiers (NBCs).
An NBC [26] is a Bayesian Network model [29] characterized by strong
conditional independence assumptions among the features. Given some
observation $\mathbf{x}\in\mathbb{F}$, the predicted class is given by:
$\kappa(\mathbf{x})=\operatorname*{\mathsf{argmax}}\nolimits_{c\in{\mathcal{K}}}\left(\textnormal{Pr}(c|\mathbf{x})\right)$
(1)
Using Bayes’s theorem, $\textnormal{Pr}(c|\mathbf{x})$ can be computed as
follows:
$\textnormal{Pr}(c|\mathbf{x})=\nicefrac{{\textnormal{Pr}(c,\mathbf{x})}}{{\textnormal{Pr}(\mathbf{x})}}$.
In practice, we compute only the numerator of the fraction, since the
denominator $\textnormal{Pr}(\mathbf{x})$ is constant for every
$c\in{\mathcal{K}}$. Moreover, given the conditional mutual independency of
the features, we have:
$\textnormal{Pr}(c,\mathbf{x})=\textnormal{Pr}(c)\times\prod\nolimits_{i}\textnormal{Pr}(x_{i}|c)$
Furthermore, it is also common in practice to apply logarithmic
transformations on probabilities of $\textnormal{Pr}(c,\mathbf{x})$, thus
getting:
$\log\textnormal{Pr}(c,\mathbf{x})=\log{\textnormal{Pr}(c)}+\sum\nolimits_{i}\log{\textnormal{Pr}(x_{i}|c)}$
Therefore, (1) can be rewritten as follows:
$\kappa(\mathbf{x})=\operatorname*{\mathsf{argmax}}\nolimits_{c\in{\mathcal{K}}}\left(\log{\textnormal{Pr}(c)}+\sum\nolimits_{i}\log{\textnormal{Pr}(x_{i}|c)}\right)$
(2)
For simplicity, and following the notation used in earlier work [62],
$\operatorname*{lPr}$ denotes a logarithmic probability. Thus, we get:
$\kappa(\mathbf{x})=\operatorname*{\mathsf{argmax}}\nolimits_{c\in{\mathcal{K}}}\left(\operatorname*{lPr}(c)+\sum\nolimits_{i}\operatorname*{lPr}(x_{i}|c)\right)$
(3)
(Note that also for simplicity, it is common in practice to add a sufficiently
large positive constant $T$ to each probability, which allows using only
positive values.)
#### Graph-based classifiers.
The paper considers the class of graph-based classifiers (referred to as
Decision Graphs) studied in earlier work [35]. A Decision Graph (DG) is a
Directed Acyclic Graph (DAG) consisting of two types of nodes, non-terminal
nodes and terminal nodes, and has a single root node. Each non-terminal node
is labeled with a feature and has at least one child node, each terminal node
is labeled with a class and has no child node. Moreover, for an arbitrary non-
terminal node $p$ labeled with feature $i$, we use
$\mathbb{C}_{i}\subseteq\mathbb{D}_{i}$ to denote the set of values of feature
$i$ which are consistent with any path connecting the root to $p$. Each
outgoing edge of node $p$ represents a literal of the form
$x_{i}\in{\mathbb{E}_{i}}$ ($\mathbb{E}_{i}\neq\emptyset$) where
$\mathbb{E}_{i}\subseteq\mathbb{C}_{i}$. Furthermore, the following
constraints are imposed on DGs:
1. 1.
The literals associated with the outgoing edges of $p$ represent a partition
of $\mathbb{C}_{i}$.
2. 2.
Any path connecting the root node to a terminal node contains no inconsistent
literals.
3. 3.
No dead-ends.
A DG is _read-once_ if each feature tested at most once on any path. A DG is
_ordered_ , if features are tested in the same order on all paths. An Ordered
Multi-valued Decision Diagrams (OMDD) is a _read-once_ and _ordered_ DG such
that every $\mathbb{E}_{i}$ is a singleton, which means multiple-edges between
two nodes may exist. Furthermore, the OMDDs considered in this paper are
_reduced_ [53, 94], i.e.
1. 1.
No node $p$ such that all child nodes of $p$ are isomorphic; and
2. 2.
No isomorphic subgraphs.
When the domains of features are all boolean and the set of classes is binary,
then OMDDs corresponds to Ordered Binary Decision Diagrams (OBDDs) [15].
#### Propositional languages & classifiers.
For classification problems for which the feature space $\mathbb{F}$ is
restricted to $\mathbb{B}^{m}$, then boolean circuits can be used as binary
classifiers. Each boolean circuit is a sentence of some propositional
language. We briefly review some well-known propositional languages and
queries/transformations that these languages support in polynomial time.
The language _negation normal form_ (NNF) is the set of all directed acyclic
graphs, where each terminal node is labeled with either $\top$, $\bot$,
$x_{i}$ or $\neg{x_{i}}$, for $x_{i}\in{X}$. Each non-terminal node is labeled
with either $\land$ (or AND) or $\lor$ (or OR). The language _decomposable_
NNF (DNNF) [19, 23] is the set of all NNFs, where for every node labeled with
$\land$, $\alpha=\alpha_{1}\land\cdots\land\alpha_{k}$, no variables are
shared between the conjuncts $\alpha_{j}$. A _deterministic_ DNNF (d-DNNF)
[23, 73] is a DNNF, where for every node labeled with $\lor$,
$\beta=\beta_{1}\lor\cdots\lor\beta_{k}$, each pair $\beta_{p},\beta_{q}$,
with $p\not=q$, is inconsistent, i.e.
$\beta_{p}\land\beta_{q}\operatorname*{\vDash}\bot$. A _Smooth_ d-DNNF (sd-
DNNF) [20] is a d-DNNF, where for every node labeled with $\lor$,
$\beta=\beta_{1}\lor\cdots\lor\beta_{k}$, each pair $\beta_{p},\beta_{q}$ is
defined on the same set of variables. We focus in this paper on d-DNNF, but
for simplicity of algorithms, sd-DNNF is often considered. Furthermore,
_sentential decision diagrams_ (SDDs) [21, 98] represent a well-known subset
of the d-DNNF. (Furthermore, it should be noted that OBDD is a proper subset
of SDD.) SDDs are based on a strongly deterministic decomposition [21], which
is used to decompose a Boolean function into the form: $(p_{1}\land
s_{1})\lor\dots\lor(p_{n}\land s_{n})$, where each $p_{i}$ is called a prime
and each $s_{i}$ is called a sub (both primes and subs are sub-functions).
Furthermore, the process of decomposition is governed by a variable tree
(vtree) which stipulates the variable order [21].
The languages d-DNNF, sd-DNNF and SDD satisfy the query _polytime model
counting_ (CT), and the transformation _polytime conditioning_ (CD). Let
$\Delta$ represent a propositional formula and let $\rho$ denote a consistent
term ($\rho\operatorname*{\nvDash}\bot$). The _conditioning_ [23] of $\Delta$
on $\rho$, denoted $\Delta|_{\rho}$ is the formula obtained by replacing each
variable $x_{i}$ by $\top$ (resp. $\bot$) if $x_{i}$ (resp. $\neg{x_{i}}$) is
a positive (resp. negative) literal of $\rho$. A propositional language L
satisfies CT if there exists a polynomial-time algorithm that maps every
formula $\Delta$ from L into a non-negative integer denoting the number of
models of $\Delta$. L satisfies CD iff there exists a polynomial-time
algorithm that maps every formula $\Delta$ from L and every consistent term
$\rho$ into a formula in L that is logically equivalent to $\Delta|_{\rho}$.
There are additional queries and transformations of interest [23], but these
are beyond the goals of this paper. It is important to note that OMDD and OBDD
also satisfy CT and CD [23, 75].
### 2.4 Running Examples
#### Example Decision Tree.
Figure 2 shows the example DT used throughout the paper. This example DT also
illustrates the notation used to represent DTs.
BDT/.style= for tree= l=1.35cm,s sep=1.5cm, if n children=0circle, draw, edge=
my edge , if n=1 edge+=0 my edge, , , BDT [$x_{1}$, label=1 [$x_{2}$, s
sep=0.75cm, label=2, edge label=node[near start,left,xshift=-0.75pt]
$\in\\{1\\}$ [$\ominus$, label=[xshift=0.25ex,yshift=1.875ex]4, edge
label=node[near start,left,xshift=-0.5pt] $\in\\{1\\}$] [$\oplus$,
label=[xshift=0.25ex,yshift=1.875ex]5, edge label=node[near
start,right,xshift=-1pt] $\in\\{2..4\\}$] ] [$x_{2}$, s sep=0.75cm, label=3,
edge label=node[near start,right,xshift=1pt] $\in\\{2..4\\}$ [$\oplus$,
label=[xshift=0.25ex,yshift=1.875ex]6, edge label=node[near
start,left,xshift=-1pt] $\in\\{1\\}$] [$x_{3}$, label=7, edge label=node[near
start,right,xshift=-1pt] $\in\\{2..4\\}$ [$\ominus$,
label=[xshift=0.25ex,yshift=1.875ex]8, edge
label=node[pos=0.6,left,xshift=-1pt] $\in\\{1\\}$] [$\oplus$,
label=[xshift=0.25ex,yshift=1.875ex]9, edge
label=node[pos=0.6,right,xshift=0.25pt] $\in\\{2\\}$] ] ] ]
(a) Decision tree
Paths
---
$P_{1}=\langle 1,2,5\rangle$
$P_{2}=\langle 1,3,6\rangle$
$P_{3}=\langle 1,3,7,9\rangle$
$Q_{1}=\langle 1,2,4\rangle$
$Q_{2}=\langle 1,3,7,8\rangle$
(b) Paths in the DT
Figure 2: Example DT.
The set of paths ${\mathcal{R}}$ is partitioned into two sets ${\mathcal{P}}$
and ${\mathcal{Q}}$, such that the paths in
${\mathcal{P}}=\\{P_{1},P_{2},P_{3}\\}$ yield a prediction of $\oplus$, and
such that the paths in ${\mathcal{Q}}=\\{Q_{1},Q_{2}\\}$ yield a prediction of
$\ominus$. (In general, ${\mathcal{P}}$ denotes the paths with prediction
$c\in{\mathcal{K}}$, and ${\mathcal{Q}}$ denotes the paths with prediction
other than $c$, i.e. any class in ${\mathcal{K}}\setminus\\{c\\}$.)
#### Example Naive Bayes Classifier.
Consider the NBC depicted graphically in Figure 3 222This example of an NBC is
adapted from [62], and it was first studied in [9, Ch.10]..
$G$$R_{2}$$R_{3}$$R_{1}$$R_{4}$$R_{5}$ $G$ | $\textnormal{Pr}(G)$
---|---
$\ominus$ | 0.90
$G$ | $\textnormal{Pr}(R_{1}|G)$
---|---
$\oplus$ | 0.95
$\ominus$ | 0.03
$G$ | $\textnormal{Pr}(R_{2}|G)$
---|---
$\oplus$ | 0.05
$\ominus$ | 0.95
$G$ | $\textnormal{Pr}(R_{3}|G)$
---|---
$\oplus$ | 0.02
$\ominus$ | 0.34
$G$ | $\textnormal{Pr}(R_{4}|G)$
---|---
$\oplus$ | 0.20
$\ominus$ | 0.75
$G$ | $\textnormal{Pr}(R_{5}|G)$
---|---
$\oplus$ | 0.95
$\ominus$ | 0.03
Figure 3: Example NBC.
The features are the boolean random variables $R_{1}$, $R_{2}$, $R_{3}$,
$R_{4}$ and $R_{5}$. Each $R_{i}$ can take values $\mathbf{t}$ or $\mathbf{f}$
denoting, respectively, whether a listener likes or not that radio station.
The boolean random variable $G$ corresponds to an age class: the target class
$\oplus$ denotes the prediction that the listener is young and $\ominus$
denotes the prediction that the listener is old. Thus,
${\mathcal{K}}=\\{\ominus,\oplus\\}$. Let us consider
$\mathbf{v}=(R_{1},R_{2},R_{3},R_{4},R_{5})=(\mathbf{t},\mathbf{f},\mathbf{f},\mathbf{f},\mathbf{t})$.
We associate $r_{i}$ to each literal ($R_{i}=\mathbf{t}$) and $\neg{r_{i}}$ to
literals ($R_{i}=\mathbf{f}$). Using (3), we get the values shown in Figure 4.
(Note that to use positive values, we added $T=+4$ to each
$\operatorname*{lPr}(\cdot)$.). As can be seen by comparing the values of
$\operatorname*{lPr}(\oplus|\mathbf{v})$ and
$\operatorname*{lPr}(\ominus|\mathbf{v})$, the classifier will predict
$\oplus$.
| $\textnormal{Pr}(\oplus)$ | $\textnormal{Pr}(r_{1}|\oplus)$ | $\textnormal{Pr}(\neg{r_{2}}|\oplus)$ | $\textnormal{Pr}(\neg{r_{3}}|\oplus)$ | $\textnormal{Pr}(\neg{r_{4}}|\oplus)$ | $\textnormal{Pr}(r_{5}|\oplus)$ | $\operatorname*{lPr}(\oplus|\mathbf{v})$
---|---|---|---|---|---|---|---
$\textnormal{Pr}(\cdot)$ | 0.10 | 0.95 | 0.95 | 0.98 | 0.80 | 0.95 |
$\operatorname*{lPr}(\cdot)$ | 1.70 | 3.95 | 3.95 | 3.98 | 3.78 | 3.95 | 21.31
(a) Computing $\operatorname*{lPr}(\oplus|\mathbf{v})$
| $\textnormal{Pr}({\ominus})$ | $\textnormal{Pr}(r_{1}|{\ominus})$ | $\textnormal{Pr}(\neg{r_{2}}|{\ominus})$ | $\textnormal{Pr}(\neg{r_{3}}|{\ominus})$ | $\textnormal{Pr}(\neg{r_{4}}|{\ominus})$ | $\textnormal{Pr}(r_{5}|{\ominus})$ | $\operatorname*{lPr}(\ominus|\mathbf{v})$
---|---|---|---|---|---|---|---
$\textnormal{Pr}(\cdot)$ | 0.90 | 0.03 | 0.05 | 0.66 | 0.25 | 0.03 |
$\operatorname*{lPr}(\cdot)$ | 3.89 | 0.49 | 1.00 | 3.58 | 2.61 | 0.49 | 12.06
(b) Computing $\operatorname*{lPr}(\ominus|\mathbf{v})$
Figure 4: Deciding prediction for
$\mathbf{v}=(\mathbf{t},\mathbf{f},\mathbf{f},\mathbf{f},\mathbf{t})$. (Note
that to use positive values, $T=+4$ was added to each
$\operatorname*{lPr}(\cdot)$.)
#### Example graph-based classifiers.
Figure 5 shows two examples of graph-based classifiers. Figure 5(a) shows an
OBDD, and Figure 5(b) an OMDD. Figure 5(b) represents a function defined on
${\mathcal{F}}=\\{1,2,3\\}$ and ${\mathcal{K}}=\\{\ominus,\oplus,\otimes\\}$,
with the domains of features being $\mathbb{D}_{1}=\\{0,1\\}$,
$\mathbb{D}_{2}=\mathbb{D}_{3}=\\{0,1,2\\}$. If we consider the instance
$\mathbf{v}=\\{1,1,2\\}$, the classifier predicts the class $\otimes$.
$x_{1}$$x_{2}$$x_{3}$$\ominus$$\oplus$ 1 2 3 4 5 010101
(a) An OBDD
$x_{1}$$x_{2}$$x_{3}$$x_{3}$$\oplus$$\ominus$$\otimes$ 1 2 3 4 5 6 7
01012201012
(b) An OMDD
Figure 5: Example DD.
### 2.5 Formal Explainability
In contrast with well-known model-agnostic approaches to XAI [81, 60, 82, 33],
formal explanations are model-precise, i.e. their definition reflects the
model’s computed function.
#### Abductive explanations.
Prime implicant (PI) explanations [89] denote a minimal set of literals
(relating a feature value $x_{i}$ and a constant $v_{i}\in\mathbb{D}_{i}$)
that are sufficient for the prediction. PI-explanations are related with
abduction, and so are also referred to as abductive explanations (AXp’s)
[43]333PI-explanations were first proposed in the context of boolean
classifiers based on restricted bayesian networks [89]. Independent work [43]
studied PI-explanations in the case of more general classification functions,
i.e. not necessarily boolean, and related instead explanations with abduction.
This paper follows the formalizations used in more recent work [62, 49, 41,
63, 35, 18, 34, 40, 64]. . Formally, given
$\mathbf{v}=(v_{1},\ldots,v_{m})\in\mathbb{F}$ with $\kappa(\mathbf{v})=c$, a
set of features ${\mathcal{X}}\subseteq{\mathcal{F}}$ is a _weak abductive
explanation_ [18] (or weak AXp) if the following predicate holds true444Each
predicate associated with a given concept will be noted in sans-serif
letterform. When referring to the same concept in the text, the same acronym
will be used, but in standard letterform. For example, the predicate name
$\mathsf{AXp}$ will be used in logic statements, and the acronym AXp will be
used throughout the text.:
$\begin{array}[]{lcr}\mathsf{WeakAXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c)&\>{:=}&\forall(\mathbf{x}\in\mathbb{F}).\left[\bigwedge\nolimits_{i\in{{\mathcal{X}}}}(x_{i}=v_{i})\right]\operatorname*{\rightarrow}(\kappa(\mathbf{x})=c)\end{array}$
(4)
Moreover, a set of features ${\mathcal{X}}\subseteq{\mathcal{F}}$ is an
_abductive explanation_ (or (plain) AXp) if the following predicate holds
true:
$\displaystyle\mathsf{AXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c)\quad{:=}\quad$
$\displaystyle\mathsf{WeakAXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c)\leavevmode\nobreak\
\land$
$\displaystyle\forall({\mathcal{X}}^{\prime}\subsetneq{\mathcal{X}}).\neg\mathsf{WeakAXp}({\mathcal{X}}^{\prime};\mathbb{F},\kappa,\mathbf{v},c)$
(5)
Clearly, an AXp is any weak AXp that is subset-minimal (or irrreducible). It
is straightforward to observe that the definition of predicate
$\mathsf{WeakAXp}$ is monotone, and so an AXp can instead be defined as
follows:
$\displaystyle\mathsf{AXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c)\quad{:=}\quad$
$\displaystyle\mathsf{WeakAXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c)\leavevmode\nobreak\
\land$
$\displaystyle\forall(j\in{\mathcal{X}}).\neg\mathsf{WeakAXp}({\mathcal{X}}\setminus\\{j\\};\mathbb{F},\kappa,\mathbf{v},c)$
(6)
This alternative equivalent definition of abductive explanation is at the core
of most algorithms for computing one AXp. (Throughout the paper, we will drop
the parameterization associated with each predicate, and so we will write
$\mathsf{AXp}({\mathcal{X}})$ instead of
$\mathsf{AXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c)$, when the
parameters are clear from the context.)
Figure 6: A schematic representation of relationships between different types
of explanations. $E_{1}({\mathcal{X}};\ldots)\Rightarrow
E_{2}({\mathcal{X}};\ldots)$ means that if ${\mathcal{X}}$ is an $E_{1}$
explanation then ${\mathcal{X}}$ must be an $E_{2}$ explanation.
###### Example 1.
The computation of (weak) AXp’s is illustrated with the DT from Figure 2(a).
The instance considered throughout is
$\mathbf{v}=(v_{1},v_{2},v_{3})=(4,4,2)$, with $c=\kappa(\mathbf{v})=\oplus$.
The point $\mathbf{v}$ is consistent with $P_{3}$, and
$\mathrm{\Phi}(P_{3})=\\{1,2,3\\}$. Table 1 (columns 1 to 4) analyzes three
sets of features $\\{1,2,3\\}$, $\\{1,3\\}$ and $\\{3\\}$ in terms of being a
weak AXp or an AXp. The decision on whether each set is a weak AXp or an AXp
can be obtained by analyzing all the 32 points in feature space, or by using
an off-the-shelf algorithm. (The analysis of all points in feature space is
ommited for brevity.)
${\mathcal{S}}$ | ${\mathcal{U}}$ | $\mathsf{WeakAXp}$? | $\mathsf{AXp}$? | $\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c|(\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}}))$ | $\mathsf{WeakPAXp}$? | $\mathsf{PAXp}$? | $\\#({\mathcal{S}})$ | $\\#(P_{1})$ | $\\#(P_{2})$ | $\\#(P_{3})$ | $\\#(Q_{1})$ | $\\#(Q_{2})$
---|---|---|---|---|---|---|---|---|---|---|---|---
$\\{1,2,3\\}$ | $\emptyset$ | Yes | No | $1\geq\delta$ | Yes | No | 1 | 0 | 0 | 1 | 0 | 0
$\\{1,3\\}$ | $\\{2\\}$ | Yes | Yes | $1\geq\delta$ | Yes | No | 4 | 0 | 1 | 3 | 0 | 0
$\\{3\\}$ | $\\{1,2\\}$ | No | – | $\nicefrac{{15}}{{16}}=0.9375\geq\delta$ | Yes | Yes | 16 | 3 | 3 | 9 | 1 | 0
Table 1: Examples of sets of fixed features given $\mathbf{v}=(4,4,2)$ and
$\delta=0.93$
It is apparent that (4), (2.5), and (2.5) can be viewed as representing a
(logic) _rule_ of the form:
$\textbf{IF\leavevmode\nobreak\ \leavevmode\nobreak\
}\left[\land_{i\in{\mathcal{X}}}(x_{i}=v_{i})\right]\textbf{\leavevmode\nobreak\
\leavevmode\nobreak\ THEN\leavevmode\nobreak\ \leavevmode\nobreak\
}\left[\kappa(\mathbf{x})=c\right]$ (7)
Unless otherwise noted, this interpretation of explanations will be assumed
throughout the paper.
Abductive explanations can be viewed as answering a ‘Why?’ question, i.e. why
is some prediction made given some point in feature space. A different view of
explanations is a contrastive explanation [70], which answers a ‘Why Not?’
question, i.e. which features can be changed to change the prediction. The
formalization of contrastive explanations revealed a minimal hitting set
duality relationship between abductive and contrastive explanations [42]. The
paper does not detail further contrastive explanations, as the focus is solely
on abductive explanations.
Figure 6 shows relationships between different classes of explanations that we
investigate in the paper.
$\mathsf{AXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c)$ and
$\mathsf{WeakAXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c)$ are
deterministic classes of explanations. These classes are shown in yellow
boxes. Other classes of explanations, shown in blue boxes, represent
probabilistic counterparts of AXp’s. We study them in the following sections.
#### Progress in formal explainability.
The introduction of abductive explanations [89, 43] also revealed practical
limitations in the case of bayesian network classifiers [89, 90] and neural
networks [43]. However, since then there has been a stream of results, that
demonstrate the practical applicability of formal explainability. These
results can be broadly organized as follows (a more detailed overview is
available in [64]):
* •
Tractable explanations.
Recent work showed that computing one explanation is tractable for naive Bayes
classifiers [62], decision trees [45, 35, 46], graph-based classifiers [35],
monotonic classifiers [63, 18], and classifiers represented with well-known
classes of propositional languages [34]. Additional tractability results were
obtained in [18].
* •
Efficient explanations.
For some other families of classifiers, recent work showed that computing one
explanation is computationally hard, but it is nevertheless efficient in
practice. This is the case with decision lists and sets [41], random forests
[49], and tree ensembles in general [43, 39, 40].
* •
Explainability queries.
There has been interest in understanding the complexity of answering different
queries related with reasoning about explainability [7, 36, 35, 6]. For
example, the feature membership problem is the decision problem of deciding
whether some (possibly sensitive) feature occurs in some explanation. Although
computationally hard in general [35], it has been shown to be solved
efficiently in theory and in practice for specific families of classifiers
[35, 37]. Queries related with enumeration of explanations have been
extensively studied [62, 42, 41, 63, 35, 34, 40].
* •
Properties of explanations.
A number of works studied the connections between explanations and robustness
[44], and connections between different types of explanations [42].
Despite the observed progress, formal explainability still faces several
important challenges. First, for some widely used families of classifiers,
e.g. neural networks, formal explainability does not scale in practice [43].
Second, input distributions are not taken into account, since these are not
readily available. There is however recent work on accounting for input
constraints [18, 32, 104]. Third, the size of explanations may exceed the
cognitive limits of human decision makers [69], and computing smallest
explanations does not offer a computationally realistic alternative [43].
Recent work studied $\delta$-relevant sets [101, 100], and these are also the
focus of this paper.
Finally, we note that there have been different approaches to formal
explainability based on the study of the formal logic or the axiomatics of
explainers [103, 2, 59]. This paper studies exclusively those approaches for
which there is practical supporting evidence of observed progress, as attested
above.
### 2.6 $\delta$-Relevant Sets
$\delta$-relevant sets were proposed in more recent work [101, 100] as a
generalized formalization of PI-explanations (or AXp’s). $\delta$-relevant
sets can be viewed as _probabilistic_ PIs [100], with $\mathsf{AXp}$’s
representing a special case of $\delta$-relevant sets where $\delta=1$, i.e.
probabilistic PIs that are actual PIs. We briefly overview the definitions
related with relevant sets. The assumptions regarding the probabilities of
logical propositions are those made in earlier work [101, 100]. Let
$\textnormal{Pr}_{\mathbf{x}}(A(\mathbf{x}))$ denote the probability of some
proposition $A$ defined on the vector of variables
$\mathbf{x}=(x_{1},\ldots,x_{m})$, i.e.
$\begin{array}[]{rcl}\textnormal{Pr}_{\mathbf{x}}(A(\mathbf{x}))&=&\frac{|\\{\mathbf{x}\in\mathbb{F}:A(\mathbf{x})=1\\}|}{|\\{\mathbf{x}\in\mathbb{F}\\}|}\\\\[8.0pt]
\textnormal{Pr}_{\mathbf{x}}(A(\mathbf{x})\,|\,B(\mathbf{x}))&=&\frac{|\\{\mathbf{x}\in\mathbb{F}:A(\mathbf{x})=1\land{B(\mathbf{x})=1}\\}|}{|\\{\mathbf{x}\in\mathbb{F}:B(\mathbf{x})=1\\}|}\end{array}$
(8)
(Similar to earlier work, it is assumed that the features are independent and
uniformly distributed [101]. Moreover, the definitions above can be adapted in
case some of the features are real-valued. As noted earlier, the present paper
studies only non-continuous features.)
###### Definition 1 ($\delta$-relevant set [101]).
Consider $\kappa:\mathbb{B}^{m}\to{\mathcal{K}}=\mathbb{B}$,
$\mathbf{v}\in\mathbb{B}^{m}$, $\kappa(\mathbf{v})=c\in\mathbb{B}$, and
$\delta\in[0,1]$. ${\mathcal{S}}\subseteq{\mathcal{F}}$ is a $\delta$-relevant
set for $\kappa$ and $\mathbf{v}$ if,
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c\,|\,\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}})\geq\delta$
(9)
(Where the restriction of $\mathbf{x}$ to the variables with indices in
${\mathcal{S}}$ is represented by
$\mathbf{x}_{{\mathcal{S}}}=(x_{i})_{i\in{\mathcal{S}}}$. Concretely, the
notation $\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}}$ represents
the constraint $\land_{i\in{\mathcal{S}}}x_{i}=v_{i}$.)
(Moreover, observe that
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c\,|\,\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}})$
is often referred to as the _precision_ of ${\mathcal{S}}$ [82, 74].) Thus, a
$\delta$-relevant set represents a set of features which, if fixed to some
pre-defined value (taken from a reference vector $\mathbf{v}$), ensures that
the probability of the prediction being the same as the one for $\mathbf{v}$
is no less than $\delta$.
###### Definition 2 (Min-$\delta$-relevant set).
Given $\kappa$, $\mathbf{v}\in\mathbb{B}^{m}$, and $\delta\in[0,1]$, find the
smallest $k$, such that there exists ${\mathcal{S}}\subseteq{\mathcal{F}}$,
with $|{\mathcal{S}}|={k}$, and ${\mathcal{S}}$ is a $\delta$-relevant set for
$\kappa$ and $\mathbf{v}$.
With the goal of proving the computational complexity of finding a minimum-
size set of features that is a $\delta$-relevant set, earlier work [101]
restricted the definition to the case where $\kappa$ is represented as a
boolean circuit. (Boolean circuits were restricted to propositional formulas
defined using the operators $\lor$, $\land$ and $\neg$, and using a set of
variables representing the inputs; this explains the choice of _inputs_ over
_sets_ in earlier work [101].) The main complexity result from earlier work is
that the computation of $\delta$-relevant sets is hard for
$\textnormal{NP}^{\textnormal{PP}}$ [101]. Hence, as noted in earlier work
[101, 100], it is unlikely that exact computation of $\delta$-relevant sets
will be practically feasible.
## 3 Relevant Sets – Probabilistic Abductive Explanations
In contrast with Min-$\delta$-relevant sets, whose focus are smallest-size
explanations, this section investigates alternative definitions of relevant
sets (which we will also and indistinguishably refer to as _probabilistic
abductive explanations_).
### 3.1 Definitions of Probabilistic AXp’s
Conceptually, Definition 1 does not need to impose a restriction on the
classifier considered (although this is done in earlier work [101]), i.e. the
logical representation of $\kappa$ need not be a boolean circuit. As a result,
Definition 1 can also be considered in the case of multi-class classifiers
defined on categorical or ordinal (non-continuous) features.
Given the above, a _weak probabilistic_ AXp (or weak PAXp) is a pick of fixed
features for which the conditional probability of predicting the correct class
$c$ exceeds $\delta$, given $c=\kappa(\mathbf{v})$. Thus,
${\mathcal{X}}\subseteq{\mathcal{F}}$ is a weak PAXp if the following
predicate holds true,
$\displaystyle\mathsf{WeakPAXp}$
$\displaystyle({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c,\delta)$
$\displaystyle:=\,\>$
$\displaystyle\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c\,|\,\mathbf{x}_{{\mathcal{X}}}=\mathbf{v}_{{\mathcal{X}}})\geq\delta$
(10) $\displaystyle:=\,\>$
$\displaystyle\frac{|\\{\mathbf{x}\in\mathbb{F}:\kappa(\mathbf{x})=c\land(\mathbf{x}_{{\mathcal{X}}}=\mathbf{v}_{{\mathcal{X}}})\\}|}{|\\{\mathbf{x}\in\mathbb{F}:(\mathbf{x}_{{\mathcal{X}}}=\mathbf{v}_{{\mathcal{X}}})\\}|}\geq\delta$
which means that the fraction of the number of points predicting the target
class and consistent with the fixed features (represented by ${\mathcal{X}}$),
given the total number of points in feature space consistent with the fixed
features, must exceed $\delta$. (Observe that the difference to (1) is solely
that features and classes are no longer required to be boolean. Hence, weak
PAXp’s can be viewed as generalized $\delta$-relevant sets.) Moreover, a set
${\mathcal{X}}\subseteq{\mathcal{F}}$ is a _probabilistic_ AXp (or (plain)
PAXp) if the following predicate holds true,
$\displaystyle\mathsf{PAXp}$
$\displaystyle({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c,\delta)\>:=$
$\displaystyle\mathsf{WeakPAXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c,\delta)\>\>\land$
(11)
$\displaystyle\forall({\mathcal{X}}^{\prime}\subsetneq{\mathcal{X}}).\neg\mathsf{WeakPAXp}({\mathcal{X}}^{\prime};\mathbb{F},\kappa,\mathbf{v},c,\delta)$
Thus, ${\mathcal{X}}\subseteq{\mathcal{F}}$ is a PAXp if it is a weak PAXp
that is also subset-minimal,
As can be observed, the definition of weak PAXp (see (3.1)) does not guarantee
monotonicity. In turn, this makes the computation of (subset-minimal) PAXp’s
harder. With the purpose of identifiying classes of weak PAXp’s that are
easier to compute, it will be convenient to study _locally-minimal_ PAXp’s. A
set of features ${\mathcal{X}}\subseteq{\mathcal{F}}$ is a locally-minimal
PAXp if,
$\displaystyle\mathsf{LmPAXp}$
$\displaystyle({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c,\delta)\>:=$
$\displaystyle\mathsf{WeakPAXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c,\delta)\>\>\land$
(12)
$\displaystyle\forall(j\in{\mathcal{X}}).\neg\mathsf{WeakPAXp}({\mathcal{X}}\setminus\\{j\\};\mathbb{F},\kappa,\mathbf{v},c,\delta)$
As observed earlier in Section 2.5, because the predicate $\mathsf{WeakAXp}$
is monotone, subset-minimal AXp’s match locally-minimal AXp’s. An important
practical consequence is that most algorithms for computing one subset-minimal
AXp, will instead compute a locally-minimal AXp, since these will be the same.
Nevertheless, a critical observation is that in the case of probabilistic
AXp’s (see (3.1)), the predicate $\mathsf{WeakPAXp}$ is _not_ monotone. Thus,
there can exist locally-minimal PAXp’s that are not subset-minimal PAXp’s. (As
shown in the experiments, computed locally minimal APXp’s are most often
PAXp’s. However, exceptions do exist, even though these are rarely observed).
Furthermore, the fact that a set of features
${\mathcal{X}}\subseteq{\mathcal{F}}$ may satisfy (3.1) but not (3.1) imposes
that subset-minimal PAXp’s must be computed by using (3.1); as shown later,
this requires more complex algorithms.
Finally, minimum-size PAXp’s (or a smallest PAXp’s) generalize
Min-$\delta$-relevant sets in Definition 2. A set of features
${\mathcal{X}}\subseteq{\mathcal{F}}$ is a minimum-size AXp if,
$\displaystyle\mathsf{MinPAXp}$
$\displaystyle({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c,\delta)\>:=$
$\displaystyle\mathsf{WeakPAXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c,\delta)\>\>\land$
(13)
$\displaystyle\forall({\mathcal{X}}^{\prime}\subseteq{\mathcal{F}}).\left[(|{\mathcal{X}}^{\prime}|<|{\mathcal{X}}|)\operatorname*{\rightarrow}\neg\mathsf{WeakPAXp}({\mathcal{X}}^{\prime};\mathbb{F},\kappa,\mathbf{v},c,\delta)\right]$
(As stated earlier, throughout the paper, we will drop the parameterization
associated with each predicate, and so we will write
$\mathsf{PAXp}({\mathcal{X}})$ instead of
$\mathsf{PAXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c,\delta)$, when the
parameters are clear from the context. Although the parameterization on
$\delta$ is paramount, we opt instead for simpler notation.)
###### Example 2.
The computation of (probabilistic) AXp’s is illustrated with the DT from
Figure 2. The instance considered throughout is
$\mathbf{v}=(v_{1},v_{2},v_{3})=(4,4,2)$, with $c=\kappa(\mathbf{v})=\oplus$.
Clearly, $\mathbf{v}$ is consistent with $P_{3}$. The goal is to compute a
$\delta$-relevant set given $\delta=0.93$. Let $\\#(R_{k})$ denote the number
of points in feature space that are consistent with path $R_{k}$. Moreover,
let $\\#({\mathcal{X}})$ denote the total number of points in feature space
that are consistent with the set of _fixed_ features
${\mathcal{X}}\in{\mathcal{F}}$. Table 1 summarizes the computation of
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c|\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}})$
for different sets ${\mathcal{S}}$. The table also includes information on
whether each set is a weak AXp, an AXp, a weak PAXp, or a PAXp. The set
$\\{1,3\\}$ represents an AXp, since for any point consistent with the
assignments $x_{1}=4$ and $x_{3}=2$, the prediction is $\oplus$. However, by
setting ${\mathcal{S}}=\\{3\\}$, the probability of predicting $\oplus$ given
a point consistent with $x_{3}=2$ still exceeds $\delta$, since
$\nicefrac{{15}}{{16}}=93.75\%$. Hence, $\\{3\\}$ is a PAXp for
$\mathbf{v}=(4,4,2)$ when $\delta=0.93$.
#### Properties of locally-minimal PAXp’s.
Let ${\mathcal{X}}$ denote an AXp. Clearly, ${\mathcal{X}}$ is also a PAXp.
Then, for any locally-minimal ${\mathcal{A}}\subseteq{\mathcal{X}}$ (i.e.
${\mathcal{A}}$ is computed using ${\mathcal{X}}$ as a seed), we have the
following properties, which follow from the definition:
1. 1.
${\mathcal{A}}\subseteq{\mathcal{X}}$ (by hypothesis);
2. 2.
${\mathcal{A}}$ is a weak PAXp (by definition); and
3. 3.
There exists at least one PAXp ${\mathcal{E}}$ such that
${\mathcal{E}}\subseteq{\mathcal{A}}$.
Thus, given some AXp ${\mathcal{X}}$, we can compute a locally-minimal PAXp
${\mathcal{A}}$ that is both a subset of ${\mathcal{X}}$ and a a superset of
some PAXp, and such that ${\mathcal{A}}$ exhibits the strong probabilistic
properties of relevant sets. Although any locally-minimal PAXp is a subset of
a weak AXp, there can exist locally-minimal PAXp’s that are not subsets of
some (plain) AXp.
### 3.2 Computing Locally-Minimal PAXp’s
Algorithm 1 shows one approach for computing a locally-minimal PAXp555This
simple algorithm is often referred to as the deletion-based algorithm, namely
in settings related with solving function problems in propositional logic and
constraint programming [66]. However, the same general algorithm can be traced
at least to the work of Valiant [97], and some authors [51] argue that it is
implicit in works from the $\text{19}^{\text{th}}$ century [68]. . As shown,
to compute one locally-minimal PAXp, one starts from
${\mathcal{F}}=\\{1,\ldots,m\\}$ and iterately removes features while it is
safe to do so, i.e. while (3.1) holds for the resulting set. Beside Algorithm
1, one could consider for example variants of the QuickXplain [52] and the
Progression [65, 66] algorithms. Both of which also allow computing preferred
(locally-minimal) sets subject to anti-lexicographic preferences [52, 67].
Furthermore, we note that the same algorithms (i.e. Deletion, Progression and
QuickXplain, among others) can also be used for computing one AXp. Moreover,
observe that these algorithms can also be applied to _any classifier_ with
respect to which we seek to compute one locally-minimal PAXp. Furthermore,
another simple observation is that explanations can be enumerated by
exploiting hitting set dualization [42], e.g. using a MARCO-like algorithm
[58].
Input: Features $\\{1,\ldots,m\\}$; feature space $\mathbb{F}$, classifier
$\kappa$, instance $(\mathbf{v},c)$, threshold $\delta$
Output: Locally-minimal PAXp ${\mathcal{S}}$
1:procedure
$\mathsf{findLmPAXp}$($\\{1,\dots,m\\};\mathbb{F},\kappa,\mathbf{v},c,\delta$)
2: ${\mathcal{S}}\leftarrow\\{1,\ldots,m\\}$
3: for $i\in\\{1,\ldots,m\\}$ do
4: if
$\mathsf{WeakPAXp}({\mathcal{S}}\setminus\\{i\\};\mathbb{F},\kappa,\mathbf{v},c,\delta)$
then
5: ${\mathcal{S}}\leftarrow{\mathcal{S}}\setminus\\{i\\}$
6: return ${\mathcal{S}}$
Algorithm 1 Computing one locally-minimal PAXp
#### Practically efficient computation of relevant sets.
Further to the computation of locally-minimal PAXp’s, the next few sections
show that the computation of relevant sets (PAXp’s) can be achieved
efficiently in practice, for several families of classifiers. Concretely, over
the next few sections we analyze decision trees, naive Bayes classifiers, but
also several families of propositional and graph-based classifiers, studied in
recent work [35, 34].
## 4 Probabilistic Explanations for Decision Trees
This section shows that the problem of deciding whether a set
${\mathcal{X}}\subseteq{\mathcal{F}}$ is a PAXp is in NP when $\kappa$ is
represented by a decision tree666As noted earlier, and for simplicity, the
paper considers the case of non-continuous features. However, in the case of
DTs, the results generalize to continuous features.. As a result, a minimum-
size PAXp can be computed with at most a logarithmic number of calls to an NP
oracle. (This is a consequence that optimizing a linear cost function, subject
to a set of constraints for which deciding satisfiability is in NP, can be
achieved with a logarithmic number of calls to a NP oracle.) An SMT
formulation of the problem is proposed and the empirical evaluation confirms
its practical effectiveness. This section also proposes a polynomial time
algorithm to compute one locally-minimal PAXp, thus offering an alternative to
computing one PAXp. The results in Section 7 confirm that in practice computed
locally-minimal PAXp’s are often subset-minimal, i.e. a locally-minimal PAXp
actually represents a (plain) PAXp.
### 4.1 Path Probabilities for DTs
This section investigates how to compute, in the case of DTs, the conditional
probability,
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c\,|\,\mathbf{x}_{{\mathcal{X}}}=\mathbf{v}_{{\mathcal{X}}})$
(14)
where ${\mathcal{X}}$ is a set of _fixed_ features (whereas the other features
are not fixed, being deemed _universal_), and $P_{t}$ is a path in the DT
consistent with the instance $(\mathbf{v},c)$. (Also, note that (14) is the
left-hand side of (9).) To motivate the proposed approach, let us first
analyze how we can compute
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c)$, where
${\mathcal{P}}\subseteq{\mathcal{R}}$ is the set of paths in the DT with
prediction $c$. Let $\mathrm{\Lambda}(R_{k})$ denote the set of literals (each
of the form $x_{i}\in\mathbb{E}_{i}$) in some path $R_{k}\in{\mathcal{R}}$. If
a feature $i$ is tested multiple times along path $R_{k}$, then
$\mathbb{E}_{i}$ is the intersection of the sets in each of the literals of
$R_{k}$ on $i$. The number of values of $\mathbb{D}_{i}$ consistent with
literal $x_{i}\in\mathbb{E}_{i}$ is $|\mathbb{E}_{i}|$. Finally, the features
_not_ tested along $R_{k}$ are denoted by $\mathrm{\Psi}(R_{k})$. For path
$R_{k}$, the probability that a randomly chosen point in feature space is
consistent with $R_{k}$ (i.e. the _path probability_ of $R_{k}$) is given by,
$\textnormal{Pr}(R_{k})=\nicefrac{{\left[\prod_{(x_{i}\in\mathbb{E}_{i})\in\mathrm{\Lambda}(R_{k})}|\mathbb{E}_{i}|\times\prod_{i\in\mathrm{\Psi}(R_{k})}|\mathbb{D}_{i}|\right]}}{{|\mathbb{F}|}}$
(15)
As a result, we get that,
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c)={\textstyle\sum\nolimits}_{R_{k}\in{\mathcal{P}}}\textnormal{Pr}(R_{k})$
(16)
Given an instance $(\mathbf{v},c)$ and a set of fixed features ${\mathcal{X}}$
(and so a set of universal features ${\mathcal{F}}\setminus{\mathcal{X}}$), we
now detail how to compute (14). Since some features will now be declared
universal, multiple paths with possibly different conditions can become
consistent. For example, in Figure 2 if feature 1 and 2 are declared
universal, then (at least) paths $P_{1}$, $P_{2}$ and $Q_{1}$ are consistent
with some of the possible assignments. Although universal variables might seem
to complicate the computation of the conditional probability, this is not the
case.
A key observation is that the feature values that make a path consistent are
disjoint from the values that make other paths consistent. This observation
allows us to compute the models consistent with each path and, as a result, to
compute (9). Let $R_{k}\in{\mathcal{R}}$ represent some path in the decision
tree. (Recall that $P_{t}\in{\mathcal{P}}$ is the target path, which is
consistent with $\mathbf{v}$.) Let $n_{ik}$ represent the (integer) number of
assignments to feature $i$ that are consistent with path
$R_{k}\in{\mathcal{R}}$, given $\mathbf{v}\in\mathbb{F}$ and
${\mathcal{X}}\subseteq{\mathcal{F}}$. For a feature $i$, let $\mathbb{E}_{i}$
denote the set of domain values of feature $i$ that is consistent with path
$R_{k}$. Hence, for path $R_{k}$, we consider a literal
$(x_{i}\in{\mathbb{E}_{i}})$. Given the above, the value of $n_{ik}$ is
defined as follows:
1. 1.
If $i$ is fixed:
1. (a)
If $i$ is tested along $R_{k}$ and the value of $x_{i}$ is inconsistent with
$\mathbf{v}$, i.e. there exists a literal
$(x_{i}\in\mathbb{E}_{i})\in\mathrm{\Lambda}(R_{k})$ and
$\\{v_{i}\\}\cap{\mathbb{E}_{i}}=\emptyset$, then $n_{ik}=0$;
2. (b)
If $i$ is tested along $R_{k}$ and the value of $x_{i}$ is consistent with
$R_{k}$, i.e. there exists a literal
$(x_{i}\in\mathbb{E}_{i})\in\mathrm{\Lambda}(R_{k})$ and
$\\{v_{i}\\}\cap{\mathbb{E}_{i}}\not=\emptyset$, then $n_{ik}=1$;
3. (c)
If $i$ is not tested along $R_{k}$, then $n_{ik}=1$.
2. 2.
Otherwise, $i$ is universal:
1. (a)
If $i$ is tested along $R_{k}$, with some literal $x_{i}\in\mathbb{E}_{i}$,
then $n_{ik}=|\mathbb{E}_{i}|$;
2. (b)
If $i$ is not tested along $R_{k}$, then $n_{ik}=|\mathbb{D}_{i}|$.
Using the definition of $n_{ik}$, we can then compute the number of
assignments consistent with $R_{k}$ as follows:
$\\#(R_{k};\mathbf{v},{\mathcal{X}})={\textstyle\prod\nolimits}_{i\in{\mathcal{F}}}n_{ik}$
(17)
Finally, (14) is given by,
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c\,|\,\mathbf{x}_{{\mathcal{X}}}=\mathbf{v}_{{\mathcal{X}}})\,=\,\nicefrac{{\sum_{P_{k}\in{\mathcal{P}}}\\#(P_{k};\mathbf{v},{\mathcal{X}})}}{{\sum_{R_{k}\in{\mathcal{R}}}\\#(R_{k};\mathbf{v},{\mathcal{X}})}}$
(18)
As can be concluded, and in the case of a decision tree, both
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c\,|\,\mathbf{x}_{{\mathcal{X}}}=\mathbf{v}_{{\mathcal{X}}})$
and $\mathsf{WeakPAXp}({\mathcal{X}};\mathbb{F},\kappa,\mathbf{v},c,\delta)$
are computed in polynomial time on the size of the DT.
###### Example 3.
With respect to the DT in Figure 2, and given the instance $((4,4,2),\oplus)$,
the number of models for each path is shown in Table 1. For example, for set
$\\{3\\}$, we immediately get that
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c\,|\,\mathbf{x}_{{\mathcal{X}}}=\mathbf{v}_{{\mathcal{X}}})=\nicefrac{{15}}{{(15+1)}}=\nicefrac{{15}}{{16}}$.
### 4.2 Computing Locally-Minimal PAXp’s for DT’s
Recent work showed that, for DTs, one AXp can be computed in polynomial time
[45, 35, 46]. A simple polynomial-time algorithm can be summarized as follows.
The AXp ${\mathcal{X}}$ is initialized with all the features in
${\mathcal{F}}$. Pick the path consistent with a given instance
$(\mathbf{v},c)$. The features not in the path are removed from
${\mathcal{X}}$. Then, iteratively check whether
${\mathcal{X}}\setminus\\{i\\}$ guarantees that all paths to a prediction in
${\mathcal{K}}\setminus\\{c\\}$ are still inconsistent. If so, then update
${\mathcal{X}}$. As argued in Section 3, we can use a similar (deletion-based)
approach for computing one locally-minimal PAXp for DTs. Such an approach
builds on Algorithm 1. In the case of DTs, (3.1) is computed using (18) on
some given set ${\mathcal{S}}\setminus\\{i\\}$. to decide whether the
precision of the approximation ${\mathcal{S}}\setminus\\{i\\}$ is no smaller
than the threshold $\delta$. As stated earlier, (18) is computed in polynomial
time. Hence, Algorithm 1 runs in polynomial time for DTs.
### 4.3 Computing Minimum-Size PAXp’s for DTs
For computing a minimum-size PAXp, we propose two SMT encodings, thus showing
that the decision problem is in NP, and that finding a smallest set requires a
logarithmic number of calls to an NP-oracle. Regarding the two SMT encodings,
one involves the multiplication of integer variables, and so it involves non-
linear arithmetic. Given the structure of the problem, we also show that
linear arithmetic can be used, by proposing a (polynomially) larger encoding.
#### A multiplication-based SMT encoding.
Taking into account the definition of path probabilities (see Section 4.1), we
now devise a model that computes path probabilities based on the same ideas.
Let $j\in{\mathcal{F}}$ denote a given feature. Let $n_{jk}$ denote the number
of elements in $\mathbb{D}_{j}$ consistent with path $R_{k}$ (for simplicity,
we just use the path index $k$). Also, $u_{j}$ is a boolean variable that
indicates whether feature $j$ is fixed ($u_{j}=0$) or universal ($u_{j}=1$).
If feature $j$ is not tested along path $R_{k}$, then if $j$ is fixed, then
$n_{jk}=1$. If not, then $n_{jk}=|\mathbb{D}_{j}|$. Otherwise, $j$ is tested
along path $R_{k}$. $n_{jk}$ is 0 if $j$ is fixed (i.e. $u_{j}=0$) and
inconsistent with the values of $\mathbb{D}_{j}$ allowed for path $R_{k}$.
$n_{jk}$ is 1 if $j$ is fixed and consistent with the values of
$\mathbb{D}_{j}$ allowed for path $R_{k}$. If feature $j$ is not fixed (i.e.
it is deemed universal and so $u_{j}=1$), then $n_{jk}$ denotes the number of
domain values of $j$ consistent with path $R_{k}$. Let the fixed value of
$n_{jk}$ be $n_{0jk}$ and the _universal_ value of $n_{jk}$ be $n_{1jk}$.
Thus, $n_{jk}$ is defined as follows,
$n_{jk}=\textnormal{ite}(u_{j},n_{1jk},n_{0jk})$ (19)
Moreover, let $\eta_{k}$ denote the number of models of path $R_{k}$. Then,
$\eta_{k}$ is defined as follows:
$\eta_{k}={\textstyle\prod\nolimits}_{i\in\mathrm{\Phi}(k)}n_{ik}$ (20)
If the domains are boolean, then we can use a purely boolean formulation for
the problem. However, if the domains are multi-valued, then we need this
formulation.
Recall what we must ensure that (3.1) holds true. In the case of DTs, since we
can count the models associated with each path, depending on which features
are fixed or not, then the previous constraint can be translated to:
${\textstyle\sum\nolimits}_{R_{k}\in{\mathcal{P}}}\eta_{k}\geq\delta\times{\textstyle\sum\nolimits}_{R_{k}\in{\mathcal{P}}}\eta_{k}+\delta\times{\textstyle\sum\nolimits}_{R_{k}\in{\mathcal{Q}}}\eta_{k}$
(21)
Recall that ${\mathcal{P}}$ are the paths with the matching prediction, and
${\mathcal{Q}}$ are the rest of the paths.
Finally, the soft constraints are of the form $(u_{i})$, one for each feature
$i\in{\mathcal{F}}\setminus\Psi(R_{k})$, i.e. for the features tested along
path $R_{k}$. (For each feature $i$ not tested along $R_{k}$, i.e.
$i\in\Psi(R_{k})$, enforce that the feature is universal by adding a hard
clause $(u_{i})$.) The solution to the optimization problem will then be a
_smallest_ weak PAXp, and so also a (plain) PAXp. (The minimum-cost solution
is well-known to be computed with a worst-case logarithmic number of calls (on
the number of features) to an SMT solver.)
Feature | Attr. | $P_{1}$ | $P_{2}$ | $P_{3}$ | $Q_{1}$ | $Q_{2}$
---|---|---|---|---|---|---
1 | $n_{01k}$ | 0 | 1 | 1 | 0 | 1
$n_{11k}$ | 1 | 3 | 3 | 1 | 3
$n_{1k}$ | $n_{1k}=\textnormal{ite}(u_{1},n_{11k},n_{01k})$
2 | $n_{02k}$ | 1 | 0 | 1 | 0 | 1
$n_{12k}$ | 3 | 1 | 3 | 1 | 3
$n_{2k}$ | $n_{2k}=\textnormal{ite}(u_{2},n_{12k},n_{02k})$
3 | $n_{03k}$ | 1 | 1 | 1 | 1 | 0
$n_{13k}$ | 2 | 2 | 1 | 2 | 1
$n_{3k}$ | $n_{3k}=\textnormal{ite}(u_{3},n_{13k},n_{03k})$
Path counts | $\eta_{k}=n_{1k}\times{n_{2k}}\times{n_{3k}}$
Table 2: SMT encoding for multiplication-based encoding
###### Example 4.
For the running example, let us consider ${\mathcal{X}}=\\{3\\}$. This means
that $u_{1}=u_{2}=1$. As a result, given the instance and the proposed
encoding, we get Table 2 and Table 3.
Path | $n_{1k}$ | $n_{2k}$ | $n_{3k}$ | $\eta_{k}$
---|---|---|---|---
$R_{1}$ | 1 | 3 | 1 | 3
$R_{2}$ | 1 | 3 | 1 | 3
$R_{3}$ | 3 | 3 | 1 | 9
$R_{4}$ | 1 | 1 | 1 | 1
$R_{5}$ | 3 | 3 | 0 | 0
Table 3: Concrete values for the multiplication-based encoding for the case
${\mathcal{X}}=\\{3\\}$, i.e. $u_{1}=u_{2}=1$ and $u_{3}=0$
Finally, by plugging into (21) the values from Table 3, we get: $15\geq
0.93\times(15+1)$. Thus, ${\mathcal{X}}$ is a weak PAXp, and we can show that
it is both a plain PAXp and a smallest PAXp. Indeed, with
${\mathcal{Y}}=\emptyset$, we get
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c\,|\,\mathbf{x}_{{\mathcal{Y}}}=\mathbf{v}_{{\mathcal{Y}}})=21/32=0.65625<\delta$.
Hence, ${\mathcal{X}}=\\{3\\}$ is subset-minimal. Since there can be no PAXp’s
of smaller size, then ${\mathcal{X}}$ is also a smallest PAXp.
#### An alternative addition-based SMT encoding.
A possible downside of the SMT encoding described above is the use of
multiplication of variables in (20); this causes the SMT problem formulation
to involve different theories (which may turn out to be harder to reason about
in practice). Given the problem formulation, we can use an encoding that just
uses linear arithmetic. This encoding is organized as follows. Let the order
of features be: $\langle 1,2,\ldots,m\rangle$. Define $\eta_{j,k}$ as the sum
of models of path $R_{k}$ taking into account features 1 up to $j$, with
$\eta_{0,k}=1$. Given $\eta_{{j-1},{k}}$, $\eta_{{j},{k}}$ is computed as
follows:
* •
Let the domain of feature $j$ be $\mathbb{D}_{j}=\\{v_{j1},\ldots,v_{jr}\\}$,
and let $s_{j,l,k}$ denote the number of models taking into account features 1
up to $j-1$ and domain values $v_{j1}$ up to $v_{j{l-1}}$. Also, let
$s_{j,0,k}=0$.
* •
For each value $v_{jl}$ in $\mathbb{D}_{j}$, for $l=1,\ldots,r$:
* –
If $j$ is tested along path $R_{k}$: (i) If $v_{jl}$ is inconsistent with path
$R_{k}$, then $s_{j,l,k}=s_{j,l-1,k}$; (ii) If $v_{jl}$ is consistent with
path $R_{k}$ and with $\mathbf{v}$, then
$s_{j,l,k}=s_{j,l-1,k}+\eta_{{j-1},{k}}$; (iii) If $v_{jl}$ is consistent with
path $R_{k}$ but not with $\mathbf{v}$, or if feature $j$ is not tested in
path $R_{k}$, then
$s_{j,l,k}=s_{j,l-1,k}+\textnormal{ite}(u_{j},\eta_{{j-1},{k}},0)$.
* –
If $j$ is not tested along path $R_{k}$: (i) If $v_{jl}$ is consistent with
$\mathbf{v}$, then $s_{j,l,k}=s_{j,l-1,k}+\eta_{{j-1},{k}}$; (ii) Otherwise,
$s_{j,l,k}=s_{j,l-1,k}+\textnormal{ite}(u_{j},\eta_{{j-1},{k}},0)$.
* •
Finally, define $\eta_{{j},{k}}=s_{j,r,k}$.
After considering all the features in order, $\eta_{m,k}$ represents the
number of models for path $R_{k}$ given the assigment to the $u_{j}$
variables. As a result, we can re-write (21) as follows:
${\textstyle\sum\nolimits}_{R_{k}\in{\mathcal{P}}}\eta_{m,k}\geq\delta\times{\textstyle\sum\nolimits}_{R_{k}\in{\mathcal{P}}}\eta_{m,k}+\delta\times{\textstyle\sum\nolimits}_{R_{k}\in{\mathcal{Q}}}\eta_{m,k}$
(22)
As with the multiplication-based encoding, the soft clauses are of the form
$(u_{i})$ for $i\in{\mathcal{F}}$.
Var. | $R_{1}\cong{P_{1}}$ | $R_{2}\cong{P_{2}}$ | $R_{3}\cong{P_{3}}$ | $R_{4}\cong{Q_{1}}$ | $R_{5}\cong{Q_{2}}$
---|---|---|---|---|---
$s_{1,0,k}$ | $s_{1,0,1}=0$ | $s_{1,0,2}=0$ | $s_{1,0,3}=0$ | $s_{1,0,4}=0$ | $s_{1,0,5}=0$
$s_{1,1,k}$ | $s_{1,0,1}+\textnormal{ite}(u_{1},\eta_{0,1},0)$ | $s_{1,0,2}$ | $s_{1,0,3}$ | $s_{1,0,4}+\textnormal{ite}(u_{1},\eta_{0,4},0)$ | $s_{1,0,5}$
$s_{1,2,k}$ | $s_{1,1,1}$ | $s_{1,1,2}+\textnormal{ite}(u_{1},\eta_{0,2},0)$ | $s_{1,1,3}+\textnormal{ite}(u_{1},\eta_{0,3},0)$ | $s_{1,1,4}$ | $s_{1,1,5}+\textnormal{ite}(u_{1},\eta_{0,5},0)$
$s_{1,3,k}$ | $s_{1,2,1}$ | $s_{1,2,2}+\textnormal{ite}(u_{1},\eta_{0,2},0)$ | $s_{1,2,3}+\textnormal{ite}(u_{1},\eta_{0,3},0)$ | $s_{1,2,4}$ | $s_{1,2,5}+\textnormal{ite}(u_{1},\eta_{0,5},0)$
$s_{1,4,k}$ | $s_{1,3,1}$ | $s_{1,3,2}+\eta_{0,2}$ | $s_{1,3,3}+\eta_{0,3}$ | $s_{1,3,4}$ | $s_{1,3,5}+\eta_{0,5}$
$\eta_{1,k}$ | $s_{1,4,1}$ | $s_{1,4,2}$ | $s_{1,4,3}$ | $s_{1,4,4}$ | $s_{1,4,5}$
$s_{2,0,k}$ | $s_{2,0,1}=0$ | $s_{2,0,2}=0$ | $s_{2,0,3}=0$ | $s_{2,0,4}=0$ | $s_{2,0,5}=0$
$s_{2,1,k}$ | $s_{2,0,1}$ | $s_{2,0,2}+\eta_{1,2}$ | $s_{2,0,3}$ | $s_{2,0,4}+\textnormal{ite}(u_{2},\eta_{1,4},0)$ | $s_{2,0,5}+\textnormal{ite}(u_{2},\eta_{1,5},0)$
$s_{2,2,k}$ | $s_{2,1,1}+\textnormal{ite}(u_{2},\eta_{1,1},0)$ | $s_{2,1,2}$ | $s_{2,1,3}+\textnormal{ite}(u_{2},\eta_{1,3},0)$ | $s_{2,1,4}$ | $s_{2,1,5}$
$s_{2,3,k}$ | $s_{2,2,1}+\textnormal{ite}(u_{2},\eta_{1,1},0)$ | $s_{2,2,2}$ | $s_{2,2,3}+\textnormal{ite}(u_{2},\eta_{1,3},0)$ | $s_{2,2,4}$ | $s_{2,2,5}$
$s_{2,4,k}$ | $s_{2,3,1}+\eta_{1,1}$ | $s_{2,3,2}$ | $s_{2,3,3}+\eta_{1,3},0$ | $s_{2,3,4}$ | $s_{2,3,5}$
$\eta_{2,k}$ | $s_{2,4,1}$ | $s_{2,4,2}$ | $s_{2,4,3}$ | $s_{2,4,4}$ | $s_{2,4,5}$
$s_{3,0,k}$ | $s_{3,0,1}=0$ | $s_{3,0,2}=0$ | $s_{3,0,3}=0$ | $s_{3,0,4}=0$ | $s_{3,0,5}=0$
$s_{3,1,k}$ | $s_{3,0,1}+\textnormal{ite}(u_{3},\eta_{2,1},0)$ | $s_{3,0,2}+\textnormal{ite}(u_{3},\eta_{2,2},0)$ | $s_{3,0,3}$ | $s_{3,0,4}+\textnormal{ite}(u_{3},\eta_{2,4},0)$ | $s_{3,0,5}+\textnormal{ite}(u_{3},\eta_{2,5},0)$
$s_{3,2,k}$ | $s_{3,1,1}+\eta_{2,1}$ | $s_{3,1,2}+\eta_{2,2}$ | $s_{3,1,3}+\eta_{2,3}$ | $s_{3,1,4}+\eta_{2,4}$ | $s_{3,1,5}$
$\eta_{3,k}$ | $s_{3,4,1}$ | $s_{3,4,2}$ | $s_{3,4,3}$ | $s_{3,2,4}$ | $s_{3,2,5}$
Table 4: Partial addition-based SMT encoding for paths with prediction
$\oplus$, with $(\mathbf{v},c)=((4,4,2),\oplus)$, and with
$\eta_{0,1}=\eta_{0,2}=\eta_{0,3}=1$
###### Example 5.
Table 4 summarizes the SMT encoding based on iterated summations for paths
with either prediction $\oplus$ or $\ominus$. The final computed values are
then used in the linear inequality (22), as follows,
$\eta_{3,1}+\eta_{3,2}+\eta_{3,2}\geq\delta\times(\eta_{3,1}+\eta_{3,2}+\eta_{3,2})+\delta\times(\eta_{3,4}+\eta_{3,5})$
The optimization problem also includes
${\mathcal{B}}=\\{(\neg{u_{1}}),(\neg{u_{2}}),(\neg{u_{3}})\\}$ as the soft
clauses. For the counting-based encoding, and from Table 4, we get the values
shown in Table 5. Moreover, we can then confirm that $15\geq 0.93\times 16$,
as intended.
Var. | $R_{1}\cong{P_{1}}$ | $R_{2}\cong{P_{2}}$ | $R_{3}\cong{P_{3}}$ | $R_{4}\cong{Q_{1}}$ | $R_{5}\cong{Q_{2}}$
---|---|---|---|---|---
$s_{1,0,k}$ | 0 | 0 | 0 | 0 | 0
$s_{1,1,k}$ | 1 | 0 | 0 | 1 | 0
$s_{1,2,k}$ | 1 | 2 | 1 | 1 | 1
$s_{1,3,k}$ | 1 | 2 | 2 | 1 | 2
$s_{1,4,k}$ | 1 | 3 | 3 | 1 | 3
$\eta_{1,k}$ | 1 | 3 | 3 | 1 | 3
$s_{2,0,k}$ | 0 | 0 | 0 | 0 | 0
$s_{2,1,k}$ | 0 | 3 | 0 | 0 | 3
$s_{2,2,k}$ | 1 | 3 | 3 | 1 | 3
$s_{2,3,k}$ | 2 | 3 | 6 | 1 | 3
$s_{2,4,k}$ | 3 | 3 | 9 | 1 | 3
$\eta_{2,k}$ | 3 | 3 | 9 | 1 | 3
$s_{3,0,k}$ | 0 | 0 | 0 | 0 | 0
$s_{3,1,k}$ | 0 | 2 | 0 | 0 | 0
$s_{3,2,k}$ | 3 | 3 | 9 | 1 | 0
$\eta_{3,k}$ | 3 | 3 | 9 | 1 | 0
Table 5: Assignment to variables of addition-based SMT encoding, given
${\mathcal{X}}=\\{3\\}$, i.e. $u_{1}=u_{2}=1$ and $u_{3}=0$
#### Discussion.
In this as in the following sections, one might consider the use of a model
counter as a possible alternative. However, a model counter would have to be
used for each pick of features. Given the complexity of exactly computing the
number of models, such approaches are all but assured to be impractical in
practice.
### 4.4 Deciding Whether a Locally-Minimal PAXp is a Plain PAXp for DTs
The problem of deciding whether a set of features ${\mathcal{X}}$,
representing an $\mathsf{LmPAXp}$, is subset-minimal can be achieved by using
one of the models above, keeping the features that are already universal, and
checking whether additional universal features can be made to exist. In
addition, we need to add constraints forcing universal features to remain
universal, and at least one of the currently fixed features to also become
universal. Thus, if ${\mathcal{X}}$ is the set of fixed features, the SMT
models proposed in earlier sections is extended with the following
constraints:
${\textstyle\bigwedge\nolimits}_{j\in{\mathcal{F}}\setminus{\mathcal{X}}}(u_{j}){\textstyle\bigwedge}\left({\textstyle\bigvee\nolimits}_{j\in{\mathcal{X}}}u_{j}\right)$
(23)
which allow checking whether some set of fixed features can be declared
universal while respecting the other constraints.
### 4.5 Instance-Based vs. Path-Based Explanations
The standard definitions of abductive explanations consider a concrete
instance $(\mathbf{v},c)$. As argued earlier (see (7)), each (weak) AXp
${\mathcal{X}}$ can then be viewed as a rule of the form:
$\textbf{IF\leavevmode\nobreak\ \leavevmode\nobreak\
}\left[\land_{i\in{\mathcal{X}}}(x_{i}=v_{i})\right]\textbf{\leavevmode\nobreak\
\leavevmode\nobreak\ THEN\leavevmode\nobreak\ \leavevmode\nobreak\
}\left[\kappa(\mathbf{x})=c\right]$
In the case of DTs, a given $\mathbf{v}$ is consistent with a concrete path
$P_{t}$. As argued in recent work [46], this enables studying instead
generalizations of AXp’s, concretely to so-called _path-based explanations_ ,
each of which can be viewed as representing instead a rule of the form:
$\textbf{IF\leavevmode\nobreak\ \leavevmode\nobreak\
}\left[\land_{i\in{\mathcal{X}}}(x_{i}\in{\mathbb{E}_{i}})\right]\textbf{\leavevmode\nobreak\
\leavevmode\nobreak\ THEN\leavevmode\nobreak\ \leavevmode\nobreak\
}\left[\kappa(\mathbf{x})=c\right]$ (24)
where $\mathbb{E}_{i}\subseteq{\mathbb{D}_{i}}$ and where each literal
$x_{i}\in{\mathbb{E}_{i}}$ is one of the literals in the path $P_{t}$
consistent with the instance $(\mathbf{v},c)$.
Clearly, the literals associated with a path $R_{k}$ offer more information
than those associated with a concrete point $\mathbf{v}$ in feature space. As
a result, in the case of DTs, we consider a generalization of the definition
of relevant set, and seek instead to compute:
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c\,|\,\mathbf{x}_{{\mathcal{X}}}\in\mathbb{E}_{{\mathcal{X}}})$
(25)
where the notation $\mathbf{x}_{{\mathcal{X}}}\in\mathbb{E}_{{\mathcal{X}}}$
represents the constraint $\land_{i\in{\mathcal{X}}}x_{i}\in{E_{i}}$, and
where $\mathbb{E}_{i}$ denotes the set of values consistent with feature $i$
in path $R_{k}$.) Thus, the condition of weak PAXp considers instead the
following probability:
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c\,|\,\mathbf{x}_{{\mathcal{X}}}\in\mathbb{E}_{{\mathcal{X}}})\geq\delta$
(26)
The rest of this section investigates the computation of path probabilities in
the case of path-based explanations. For instance-based explanations, the
definition of $n_{ik}$ needs to be adapted. Let $P_{t}\in{\mathcal{P}}$ be the
target path. (For example, $P_{t}$ can be the path consistent with
$\mathbf{v}$.) Moreover, let $R_{K}\in{\mathcal{R}}$ by some path in the
decision tree. For a feature $i$, let $E_{ik}$ denote the set of domain values
of feature $i$ that is consistent with path $R_{k}$. Hence, for path $R_{k}$,
we consider a literal $(x_{i}\in{E_{ik}})$. Similarly, let $E_{it}$ denote the
set of domain values of feature $i$ that is consistent with path $P_{t}$.
Thus, for path $P_{t}$, we consider a literal $(x_{i}\in{E_{it}})$. Given the
above, the value of $n_{ik}$ is now defined as follows:
1. 1.
If $i$ is fixed:
1. (a)
If $i$ is tested along $R_{k}$ and the value of $x_{i}$ is inconsistent with
$E_{it}$, i.e. there exists a literal
$(x_{i}\in\mathbb{E}_{i})\in\mathrm{\Lambda}(R_{k})$ and
${\mathbb{E}_{it}}\cap{\mathbb{E}_{ik}}=\emptyset$, then $n_{ik}=0$;
2. (b)
If $i$ is tested along $R_{k}$ and the value of $x_{i}$ is consistent with
$R_{k}$, i.e. there exists a literal
$(x_{i}\in\mathbb{E}_{i})\in\mathrm{\Lambda}(R_{k})$ and
${\mathbb{E}_{it}}\cap{\mathbb{E}_{ik}}\not=\emptyset$, then $n_{ik}=1$.
3. (c)
If $i$ is not tested along $R_{k}$, then $n_{ik}=1$.
2. 2.
Otherwise, $i$ is universal:
1. (a)
If $i$ is tested along $R_{k}$, with some literal $x_{i}\in\mathbb{E}_{ik}$,
then $n_{ik}=|\mathbb{E}_{ik}|$;
2. (b)
If $i$ is not tested along $R_{k}$, then $n_{ik}=|\mathbb{D}_{i}|$.
Using the modified definition of $n_{ik}$, we can now compute (25) as follows:
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c\,|\,\mathbf{x}_{{\mathcal{X}}}\in\mathbb{E}_{{\mathcal{X}}})\,=\,\nicefrac{{\sum_{P_{k}\in{\mathcal{P}}}\\#(P_{k};{\mathcal{F}}\setminus{\mathcal{X}},\mathbf{v})}}{{\sum_{R_{k}\in{\mathcal{R}}}\\#(R_{k};{\mathcal{F}}\setminus{\mathcal{X}},\mathbf{v})}}$
(27)
The computation of probabilistic explanations proposed in the previous
sections can either assume instance-based or path-based explanations. For
consistency with the rest of the paper, we opted to investigate instance-based
explanations. Changing the proposed algorithms to consider instead path-based
explanations would be straightforward, but that is beyond the scope of this
paper.
## 5 Probabilistic Explanations for Naive Bayes Classifiers
This section investigates the computation of relevant sets in the concrete
case of NBCs.
### 5.1 Explaining NBCs in Polynomial Time
This section overviews the approach proposed in [62] for computing AXp’s for
binary NBCs. The general idea is to reduce the NBC problem into an Extended
Linear Classifier (XLC) and then explain the resulting XLC. Our purpose is to
devise a new approach that builds on the XLC formulation to compute
$\delta$-relevant sets for NBCs. Hence, it is useful to recall first the
translation of NBCs into XLCs and the extraction of AXp’s from XLCs.
#### Extended Linear Classifiers.
We consider an XLC with categorical features. (Recall that the present paper
considers NBCs with binary classes and categorical data.) Each feature
$i\in{\mathcal{F}}$ has $x_{i}\in\\{1,\dots,d_{i}\\}$, (i.e.
$\mathbb{D}_{i}=\\{1,\dots,d_{i}\\}$). Let,
$\nu(\mathbf{x})\triangleq
w_{0}+\sum\nolimits_{i\in{\mathcal{F}}}\sigma(x_{i},v_{i}^{1},v_{i}^{2},\ldots,v_{i}^{d_{i}})$
(28)
$\sigma$ is a selector function that picks the value $v_{i}^{r}$ iff $x_{i}$
takes value $r$. Moreover, let us define the decision function,
$\kappa(\mathbf{x})=\oplus$ if $\nu(\mathbf{x})>0$ and
$\kappa(\mathbf{x})=\ominus$ if $\nu(\mathbf{x})\leq 0$.
The reduction of a binary NBC, with categorical features, to an XLC is
completed by setting:
$w_{0}\triangleq\operatorname*{lPr}(\oplus)-\operatorname*{lPr}(\ominus)$,
$v_{i}^{1}\triangleq\operatorname*{lPr}(x_{i}=1|\oplus)-\operatorname*{lPr}(x_{i}=1|\ominus)$,
$v_{i}^{2}\triangleq\operatorname*{lPr}(x_{i}=2|\oplus)-\operatorname*{lPr}(x_{i}=2|\ominus)$,
$\dots$,
$v_{i}^{d_{i}}\triangleq\operatorname*{lPr}(x_{i}=d_{i}|\oplus)-\operatorname*{lPr}(x_{i}=d_{i}|\ominus)$.
Hence, the argmax in (3) is replaced with inequality to get the following:
$\displaystyle\operatorname*{lPr}(\oplus)-\operatorname*{lPr}(\ominus)+\sum\nolimits_{i=1}^{m}\sum\nolimits_{k=1}^{k=d_{i}}$
$\displaystyle(\operatorname*{lPr}(x_{i}=k|\oplus)-\operatorname*{lPr}(x_{i}=k|\ominus))(x_{i}=k)>0$
(29)
$w_{0}$ | $v_{1}^{1}$ | $v_{1}^{2}$ | $v_{2}^{1}$ | $v_{2}^{2}$ | $v_{3}^{1}$ | $v_{3}^{2}$ | $v_{4}^{1}$ | $v_{4}^{2}$ | $v_{5}^{1}$ | $v_{5}^{2}$
---|---|---|---|---|---|---|---|---|---|---
-2.19 | -2.97 | 3.46 | 2.95 | -2.95 | 0.4 | -2.83 | 1.17 | -1.32 | -2.97 | 3.46
(a) Example reduction of NBC to XLC (Example 6)
$\Gamma$ | $\delta_{1}$ | $\delta_{5}$ | $\delta_{2}$ | $\delta_{3}$ | $\delta_{4}$ | $\Phi$
---|---|---|---|---|---|---
9.25 | 6.43 | 6.43 | 5.90 | 3.23 | 2.49 | 15.23
(b) Computing $\delta_{j}$’s for the XLC (Example 7)
Figure 7: Values used in the running example (Example 6 and Example 7)
###### Example 6.
7(a) shows the resulting XLC formulation for the example in Figure 4. We also
let $\mathbf{f}$ be associated with value 1 and $\mathbf{t}$ be associated
with value 2, and $d_{i}=2$.
#### Explaining XLCs.
We now describe how AXp’s can be computed for XLCs. For a given instance
$\mathbf{x}=\mathbf{a}$, define a _constant_ slack (or gap) value $\Gamma$
given by,
$\Gamma\triangleq\nu(\mathbf{a})=\sum\nolimits_{i\in{\mathcal{F}}}\sigma(a_{i},v_{i}^{1},v_{i}^{2},\ldots,v_{i}^{d_{i}})$
(30)
Computing an AXp corresponds to finding a subset-minimal set of literals
${\mathcal{S}}\subseteq{\mathcal{F}}$ such that (2.5) holds, or alternatively,
$\forall(\mathbf{x}\in\mathbb{F}).\bigwedge\nolimits_{i\in{\mathcal{S}}}(x_{i}=a_{i})\
\operatorname*{\rightarrow}\ \left(\nu(\mathbf{x})>0\right)$ (31)
under the assumption that $\nu(\mathbf{a})>0$. Thus, the purpose is to find
the _smallest_ slack that can be achieved by allowing the feature not in
${\mathcal{S}}$ to take any value (i.e. _universal_ /_free_ features), given
that the literals in ${\mathcal{S}}$ are fixed by $\mathbf{a}$ (i.e.
$\bigwedge\nolimits_{i\in{\mathcal{S}}}(x_{i}=a_{i})$).
Let $v_{i}^{\omega}$ denote the _smallest_ (or _worst-case_) value associated
with $x_{i}$. Then, by letting every $x_{i}$ take _any_ value, the _worst-
case_ value of $\nu(\mathbf{e})$ is,
$\Gamma^{\omega}=w_{0}+\sum\nolimits_{i\in{\mathcal{F}}}v_{i}^{\omega}$ (32)
Moreover, from (30), we have:
$\Gamma=w_{0}+\sum_{i\in{\mathcal{F}}}v_{i}^{a_{i}}$. The expression above can
be rewritten as follows,
$\begin{array}[]{rcl}\Gamma^{\omega}&=&w_{0}+\sum\nolimits_{i\in{\mathcal{F}}}v_{i}^{a_{i}}-\sum\nolimits_{i\in{\mathcal{F}}}(v_{i}^{a_{i}}-v_{i}^{\omega})\\\\[3.0pt]
&=&\Gamma-\sum\nolimits_{i\in{\mathcal{F}}}\delta_{i}=-\Phi\\\ \end{array}$
(33)
where $\delta_{i}\triangleq{v_{i}^{a_{i}}}-{v_{i}^{\omega}}$, and
$\Phi\triangleq\sum_{i\in{\mathcal{F}}}\delta_{i}-\Gamma=-\Gamma^{\omega}$.
Recall the goal is to find a subset-minimal set ${\mathcal{S}}$ such that the
prediction is still $c$ (whatever the values of the other features):
$w_{0}+\sum\nolimits_{i\in{\mathcal{S}}}v_{i}^{a_{i}}+\sum\nolimits_{i\notin{\mathcal{S}}}v_{i}^{\omega}=-\Phi+\sum\nolimits_{i\in{\mathcal{S}}}\delta_{i}>0$
(34)
In turn, (34) can be represented as the following knapsack problem [54]:
$\begin{array}[]{lcl}\textnormal{min}&&\sum_{i=1}^{m}p_{i}\\\\[4.5pt]
\textnormal{such that}&&\sum_{i=1}^{m}\delta_{i}p_{i}>\Phi\\\\[2.5pt]
&&p_{i}\in\\{0,1\\}\\\ \end{array}$ (35)
where the variables $p_{i}$ assigned value 1 denote the indices included in
${\mathcal{S}}$. Note that, the fact that the coefficients in the cost
function are all equal to 1 makes the problem solvable in log-linear time
[62].
###### Example 7.
7(b) shows the values used for computing explanations for the example in
Figure 4. For this example, the sorted $\delta_{j}$’s become
$\langle\delta_{1},\delta_{5},\delta_{2},\delta_{4},\delta_{3}\rangle$. By
picking $\delta_{1}$, $\delta_{2}$ and $\delta_{5}$, we ensure that the
prediction is $\oplus$, independently of the values assigned to features 3 and
4. Thus $\\{1,2,5\\}$ is an AXp for the NBC shown in Figure 3, with the input
instance
$(v_{1},v_{2},v_{3},v_{4},v_{5})=(\mathbf{t},\mathbf{f},\mathbf{f},\mathbf{f},\mathbf{t})$.
(It is easy to observe that
$\kappa(\mathbf{t},\mathbf{f},\mathbf{f},\mathbf{t},\mathbf{t})=\kappa(\mathbf{t},\mathbf{f},\mathbf{t},\mathbf{f},\mathbf{t})=\kappa(\mathbf{t},\mathbf{f},\mathbf{t},\mathbf{t},\mathbf{t})=\oplus$.)
The next section introduces a pseudo-polynomial time algorithm for computing
locally-minimal PAXp’s. Although locally-minimal PAXp’s are not necessarily
subset/cardinality minimal, the experiments (see Section 7) show that the
proposed approach computes (in pseudo-polynomial time) succinct [69] and
highly precise locally-minimal explanations.
### 5.2 Counting Models of XLCs
Earlier work [27, 31, 30, 83] proposed the use of dynamic programming (DP) for
approximating the number of feasible solutions of the 0-1 knapsack constraint,
i.e. the #knapsack problem. Here we propose an extension of the basic
formulation, to allow counting feasible solutions of XLCs.
We are interested in the number of solutions of,
$\sum\nolimits_{j\in{\mathcal{F}}}\sigma(x_{j},v_{j}^{1},v_{j}^{2},\ldots,v_{j}^{d_{j}})>-w_{0}$
(36)
where we assume all $v^{i}_{j}$ to be integer-valued and non-negative (e.g.
this is what our translation from NBCs to XLCs yields after scaling and
rounding). Moreover, (36) can be written as follows:
$\sum\nolimits_{j\in{\mathcal{F}}}\sigma(x_{j},-v_{j}^{1},-v_{j}^{2},\ldots,-v_{j}^{d_{j}})<{w_{0}}$
(37)
which reveals the relationship with the standard knapsack constraint.
For each $j$, let us sort the $-v_{j}^{i}$ in non-decreasing order, collapsing
duplicates, and counting the number of duplicates, obtaining two sequences:
$\begin{array}[]{l}\langle
w^{1}_{j},\ldots,w^{d^{{}^{\prime}}_{j}}_{j}\rangle\\\\[4.5pt] \langle
n^{1}_{j},\ldots,n^{d^{{}^{\prime}}_{j}}_{j}\rangle\\\ \end{array}$
such that $w^{1}_{j}<w^{2}_{j}<\ldots<w^{d^{{}^{\prime}}_{j}}_{j}$ and each
$n^{i}_{j}\geq 1$ gives the number of repetitions of weight $w^{i}_{j}$.
#### Counting.
Let $C(k,r)$ denote the number of solutions of (37) when the subset of
features considered is $\\{1,\ldots,k\\}$ and the sum of picked weights is at
most $r$. To define the solution for the first $k$ features, taking into
account the solution for the first $k-1$ features, we must consider that the
solution for $r$ can be obtained due to _any_ of the possible values of
$x_{j}$. As a result, for an XLC the general recursive definition of $C(k,r)$
becomes,
$C(k,r)=\sum\nolimits_{i=1}^{d^{{}^{\prime}}_{k}}n^{i}_{k}\times{C}(k-1,r-w^{i}_{k})$
Moreover, $C(1,r)$ is given by,
$C(1,r)=\left\\{\begin{array}[]{lcl}0&&\textnormal{if
$r<w^{1}_{1}$}\\\\[4.5pt] n^{1}_{1}&&\textnormal{if $w^{1}_{1}\leq
r<w^{2}_{1}$}\\\\[5.5pt] n^{1}_{1}+n^{2}_{1}&&\textnormal{if $w^{2}_{1}\leq
r<w^{3}_{1}$}\\\\[2.0pt] \ldots\\\\[2.0pt]
\sum\nolimits_{i=1}^{d^{{}^{\prime}}_{1}}n^{i}_{1}&&\textnormal{if
$w^{d^{{}^{\prime}}_{1}}_{1}\leq r$}\\\\[2.0pt] \end{array}\right.$
In addition, if $r<0$, then $C(k,r)=0$, for $k=1,\ldots,m$. Finally, the
dimensions of the $C(k,r)$ table are as follows:
1. 1.
The number of rows is $m$.
2. 2.
The (worst-case) number of columns is given by:
$W^{\prime}=\sum_{j\in{\mathcal{F}}}{n^{d^{\prime}_{j}}_{j}}\times{w^{d^{\prime}_{j}}_{j}}$
(38)
$W^{\prime}$ represents the largest possible value, in theory. However, in
practice, it suffices to set the number of columns to $W=w_{0}+T$, which is
often much smaller than $W^{\prime}$.
###### Example 8.
Consider the following problem. There are 4 features,
${\mathcal{F}}=\\{1,2,3,4\\}$. Each feature $j$ takes values in $\\{1,2,3\\}$,
i.e. $x_{j}\in\\{1,2,3\\}$. The prediction should be 1 when the sum of the
values of the $x_{j}$ variables is not less than 8. We set $w_{0}=-7$, and get
the formulation,
$\sum_{j\in\\{1,2,3,4\\}}\sigma(x_{j},1,2,3)>7$
where each $x_{j}$ picks value in $\\{1,2,3\\}$. We translate to the extended
knapsack formulation and obtain:
$\sum_{j\in\\{1,2,3,4\\}}\sigma(x_{j},-1,-2,-3)<-7$
We require the weights to be integer and non-negative, and so we add to each
$w_{j}^{k}$ the complement of the most negative $w_{j}^{k}$ plus 1. Therefore,
we add +4 to each $j$ and +16 to right-hand side of the inequality. Thus, we
get
$\sum_{j\in\\{1,2,3,4\\}}\sigma(x_{j},3,2,1)<9$
For this formulation, $x_{j}=1$ picks value $3$. (For example, we can pick two
(but not three) $x_{j}$ with value 1, which is as expected.)
In this case, the DP table size will be $4\times 12$, even though we are
interested in entry $C(4,8)$.
$k$ | $r$
---|---
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
1 | 0 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | – | – | – | –
2 | 0 | 0 | 1 | 3 | 6 | 8 | 9 | 9 | 9 | – | – | – | –
3 | 0 | 0 | 0 | 1 | 4 | 10 | 17 | 23 | 16 | – | – | – | –
4 | 0 | 0 | 0 | 0 | 1 | 5 | 15 | 31 | 50 | – | – | – | –
Table 6: DP table for Example 8
Table 6 shows the DP table, and the number of solutions for the starting
problem, i.e. there are 50 combinations of values whose sum is no less than 8.
By default, the dynamic programming formulation assumes that features can take
any value. However, the same formulation can be adapted when features take a
given (fixed) value. Observe that this will be instrumental for computing
$\mathsf{LmPAXp}$’s.
Consider that feature $k$ is fixed to value $l$. Then, the formulation for
$C(k,r)$ becomes:
$C(k,r)=n^{l}_{k}\times{C}(k-1,r-w^{l}_{k})={C}(k-1,r-w^{l}_{k})$
Given that $k$ is fixed, then we have $n^{l}_{k}=1$.
###### Example 9.
For Example 8, assume that $x_{2}=1$ and $x_{4}=3$. Then, the constraint we
want to satisfy is:
$\sum_{j\in\\{1,3\\}}\sigma(x_{j},1,2,3)>3$
Following a similar transformation into knapsack formulation, we get
$\sum_{j\in\\{1,3\\}}\sigma(x_{j},3,2,1)<5$
After updating the DP table, with fixing features 2 and 4, we get the DP table
shown in Table 7. As a result, we can conclude that the number of solutions is
6.
$k$ | $r$
---|---
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12
1 | 0 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | – | – | – | –
2 | 0 | 0 | 0 | 0 | 1 | 2 | 3 | 3 | 3 | – | – | – | –
3 | 0 | 0 | 0 | 0 | 0 | 1 | 3 | 6 | 8 | – | – | – | –
4 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 | 6 | – | – | – | –
Table 7: DP table for Example 9
The table $C(k,r)$ can be filled out in pseudo-polynomial time. The number of
rows is $m$. The number of columns is $W$ (see (38)). Moreover, the
computation of each entry uses the values of at most $m$ other entries. Thus,
the total running time is: $\Theta(m^{2}\times{W})$.
#### From NBCs to positive integer knapsacks.
To assess heuristic explainers, we consider NBCs, and use a standard
transformation from probabilities to positive real values [78]. Afterwards, we
convert the real values to integer values by scaling the numbers. However, to
avoid building a very large DP table, we implement the following optimization.
The number of decimal places of the probabilities is reduced while there is no
decrease in the accuracy of the classifier both on training and on test data.
In our experiments, we observed that there is no loss of accuracy if four
decimal places are used, and that there is a negligible loss of accuracy with
three decimal places.
#### Assessing explanation precision.
Given a Naive Bayes classifier, expressed as an XLC, we can assess explanation
accuracy in pseudo-polynomial time. Given an instance $\mathbf{v}$, a
prediction $\kappa(\mathbf{v})=\oplus$, and an approximate explanation
$\mathbf{S}$, we can use the approach described in this section to count the
number of instances consistent with the explanation for which the prediction
remains unchanged (i.e. number of points $\mathbf{x}\in\mathbb{F}$ s.t.
$(\kappa(\mathbf{x})=\kappa(\mathbf{v})\land(\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}}))$).
Let this number be $n_{\oplus}$ (given the assumption that the prediction is
$\oplus$). Let the number of instances with a different prediction be
$n_{\ominus}$ 777Recall that we are assuming that
${\mathcal{K}}=\\{\ominus,\oplus\\}$.. Hence, the conditional probability (8)
can be defined, in the case of NBCs, as follow:
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=\oplus\,|\,\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}})=\frac{n_{\oplus}}{|\\{\mathbf{x}\in\mathbb{F}:(\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}})\\}|}$
Observe that the numerator
$|\\{\mathbf{x}\in\mathbb{F}:\kappa(\mathbf{x})=\oplus\land(\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}})\\}|$
is expressed by the number of models $n_{\oplus}$, i.e. the points
$\mathbf{x}$ in feature space that are consistent with $\mathbf{v}$ given
${\mathcal{S}}$ and with prediction $\oplus$. Further, we have
$\displaystyle\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=\oplus\,|\,\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}})=$
$\displaystyle
1-\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=\ominus\,|\,\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}})$
$\displaystyle=$ $\displaystyle
1-\frac{n_{\ominus}}{|\\{\mathbf{x}\in\mathbb{F}:(\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}})\\}|}$
where
$n_{\ominus}=|\\{\mathbf{x}\in\mathbb{F}:\kappa(\mathbf{x})=\ominus\land(\mathbf{x}_{{\mathcal{S}}}=\mathbf{v}_{{\mathcal{S}}})\\}|$.
### 5.3 Computing Locally-Minimal PAXp’s for NBCs
Similarly to the case of DTs, we can also use Algorithm 1 for computing
locally-minimal PAXp’s in the case of NBCs. The only difference is in the
definition of the predicate $\mathsf{WeakPAXp}$. For NBCs, the procedure
$\mathsf{isWeakPAXp}$ implements the pseudo-polynomial approach, described in
the previous section, for model counting. Hence, in the case of NBCs, it is
implicit that the DP table is updated at each iteration of the main loop of
Algorithm 1. More specifically, when a feature $i$ is just set to universal,
its associated cells $C(i,r)$ are recalculated such that
$C(k,r)=\sum\nolimits_{i=1}^{d^{{}^{\prime}}_{k}}n^{i}_{k}\times{C}(k-1,r-w^{i}_{k})$;
and when $i$ is fixed, i.e. $i\in{\mathcal{S}}$, then
$C(i,r)=C(i-1,r-v^{j}_{i})$ where
$v^{j}_{i}\triangleq\operatorname*{lPr}(v_{i}=j|c)-\operatorname*{lPr}(v_{i}=j|\neg
c)$. Furthermore, we point out that in our experiments, ${\mathcal{S}}$ is
initialized to an AXp ${\mathcal{X}}$ that we compute initially for all tested
instances using the outlined (polynomial) algorithm in Section 5.1. It is easy
to observe that features not belonging to ${\mathcal{X}}$ do not contribute in
the decision of $\kappa(\mathbf{v})$ (i.e. their removal does not change the
value of $n_{\ominus}$ that is equal to zero) and thus can be set universal at
the initialisation step, which allows us to improve the performance of
Algorithm 1.
Moreover, we apply a heuristic order over ${\mathcal{S}}$ that aims to remove
earlier less relevant features and thus to produce shorter approximate
explanations. Typically, we order ${\mathcal{S}}$ following the increasing
order of $\delta_{i}$ values, namely the reverse order applied to compute the
AXp. Preliminary experiments conducted using a (naive heuristic) lexicographic
order over the features produced less succinct explanations.
Finally, notice that Algorithm 1 can be used to compute an AXp, i.e. locally-
minimal PAXp when $\delta=1$. Nevertheless, the polynomial time algorithm for
computing AXp’s proposed in [62] remains a better choice to use in case of
AXp’s than Algorithm 1 which runs in pseudo-polynomial time.
###### Example 10.
Let us consider again the NBC of the running example (Example 6) and
$\mathbf{v}=(\mathbf{t},\mathbf{f},\mathbf{f},\mathbf{f},\mathbf{t})$. The
corresponding XLC is shown in 7(b) (Example 7). Also, consider the AXp
$\\{1,2,5\\}$ of $\mathbf{v}$ and $\delta=0.85$. The resulting DP table for
${\mathcal{S}}=\\{1,2,5\\}$ is shown in Table 8. Note that for illustrating
small tables, we set the number of decimal places to zero (greater number of
decimal places, i.e. 1,2, etc, were tested and returned the same results).
(Also, note that the DP table reports “—” if the cell is not calculated during
the running of Algorithm 1.) Moreover, we convert the probabilities into
positive integers, so we add to each $w_{j}^{k}$ the complement of the most
negative $w_{j}^{k}$ plus 1. The resulting weights are shown in Figure 8.
Thus, we get
$\sum\nolimits_{i\in\\{1,2,3,4,5\\}}\sigma(x_{i},w_{i}^{1},w_{i}^{2})<17$.
Observe that the number of models $n_{\oplus}=C(5,16)$, and $C(5,16)$ is
calculated using $C(4,16-w^{2}_{5})=C(4,15)$, i.e. $C(4,15)=C(5,16)$ (feature
5 is fixed, so it is allowed to take only the value $w^{2}_{5}=1$). Next,
$C(4,15)=C(3,15-w^{1}_{4})+C(3,15-w^{2}_{4})=C(3,12)+C(3,14)$ (feature 4 is
free, so it is allowed to take any value of $\\{w^{1}_{4},w^{2}_{4}\\}$); the
recursion ends when k=1, namely for $C(1,5)=C(2,6)=n^{2}_{1}=1$,
$C(1,7)=C(2,7)=n^{2}_{1}=1$, $C(1,8)=C(2,8)=n^{2}_{1}=1$ and
$C(1,10)=C(2,11)=n^{2}_{1}=1$ (feature 1 is fixed and takes value
$w^{2}_{1}$). Next, Table 9 (resp. Table 10 and Table 11) report the resulting
DP table for ${\mathcal{S}}=\\{2,5\\}$ (resp. ${\mathcal{S}}=\\{1,5\\}$ and
${\mathcal{S}}=\\{1\\}$). It is easy to confirm that after dropping feature 2,
the precision of ${\mathcal{S}}=\\{1,5\\}$ becomes $87.5\%$, i.e.
$\frac{7}{8}=0.875>\delta$. Furthermore, observe that the resulting
${\mathcal{S}}$ when dropping feature 1 or 2 and 5, are not weak PAXp’s,
namely, the precision of $\\{2,5\\}$ is $\frac{6}{8}=0.75<\delta$ and the
precision of $\\{1\\}$ is $\frac{9}{16}=0.5625<\delta$. In summary, Algorithm
1 starts with ${\mathcal{S}}=\\{1,2,5\\}$, then at iteration #1, feature 1 is
tested and since $\\{2,5\\}$ is not a weak PAXp then 1 is kept in
${\mathcal{S}}$; at iteration #2, feature 2 is tested and since $\\{1,5\\}$ is
a weak PAXp, then ${\mathcal{S}}$ is updated (i.e. ${\mathcal{S}}=\\{1,5\\}$);
at iteration #3, feature 5 is tested and since $\\{1\\}$ is not a weak PAXp,
then 5 is saved in ${\mathcal{S}}$. As a result, the computed locally-minimal
PAXp is $\\{1,5\\}$.
We underline that we could initialize ${\mathcal{S}}$ to ${\mathcal{F}}$, in
which case the number of models would be 1. However, we opt instead to always
start from an AXp. In the example, the AXp is $\\{1,2,5\\}$ which, because it
is an AXp, the number of models must be 4 (i.e. $2^{2}$, since two features
are free).
For any proper subset of the AXp, with $r$ free variables, it must be the case
that the number of models is strictly less than $2^{r}$. Otherwise, we would
have an AXp as a proper subset of another AXp; but this would contradict the
definition of AXp. The fact that the number of models is strictly less than
$2^{r}$ is confirmed by the examples of subsets considered. It must also be
the case that if ${\mathcal{S}}^{\prime}\subseteq{\mathcal{S}}$, then the
number of models of ${\mathcal{S}}^{\prime}$ must not exceed the number of
models of ${\mathcal{S}}$. So, we can argue that there is monotonicity in the
number of models, but not in the precision.
$W$ | $w_{1}^{1}$ | $w_{1}^{2}$ | $w_{2}^{1}$ | $w_{2}^{2}$ | $w_{3}^{1}$ | $w_{3}^{2}$ | $w_{4}^{1}$ | $w_{4}^{2}$ | $w_{5}^{1}$ | $w_{5}^{2}$
---|---|---|---|---|---|---|---|---|---|---
16 | 7 | 1 | 1 | 6 | 3 | 6 | 1 | 3 | 7 | 1
Figure 8: #knapsack problem of Example 10 $k$ | $r$
---|---
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16
1 | 0 | — | — | — | — | 1 | — | 1 | 1 | — | 1 | — | — | — | — | — | —
2 | 0 | — | — | — | — | — | 1 | — | 1 | 1 | — | 1 | — | — | — | — | —
3 | 0 | — | — | — | — | — | — | — | — | — | — | — | 2 | — | 2 | — | —
4 | 0 | — | — | — | — | — | — | — | — | — | — | — | — | — | — | 4 | —
5 | 0 | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | 4
Table 8: DP table for ${\mathcal{S}}=\\{1,2,5\\}$ (Example 10) $k$ | $r$
---|---
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16
1 | 0 | — | — | — | — | 1 | — | 1 | 2 | — | 2 | — | — | — | — | — | —
2 | 0 | — | — | — | — | — | 1 | — | 1 | 2 | — | 2 | — | — | — | — | —
3 | 0 | — | — | — | — | — | — | — | — | — | — | — | 3 | — | 3 | — | —
4 | 0 | — | — | — | — | — | — | — | — | — | — | — | — | — | — | 6 | —
5 | 0 | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | 6
Table 9: DP table for ${\mathcal{S}}=\\{2,5\\}$ (Example 10) $k$ | $r$
---|---
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16
1 | 0 | — | 1 | 1 | — | 1 | — | 1 | 1 | — | 1 | — | — | — | — | — | —
2 | 0 | — | — | — | — | — | 1 | — | 2 | 2 | — | 2 | — | — | — | — | —
3 | 0 | — | — | — | — | — | — | — | — | — | — | — | 3 | — | 4 | — | —
4 | 0 | — | — | — | — | — | — | — | — | — | — | — | — | — | — | 7 | —
5 | 0 | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | 7
Table 10: DP table for ${\mathcal{S}}=\\{1,5\\}$ (Example 10) $k$ | $r$
---|---
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16
1 | 0 | 0 | 1 | 1 | 1 | 1 | — | 1 | 1 | — | 1 | — | — | — | — | — | —
2 | 0 | — | 0 | 1 | — | 1 | 1 | — | 2 | 2 | — | 2 | — | — | — | — | —
3 | 0 | — | — | — | — | — | 1 | — | 1 | — | — | — | 3 | — | 4 | — | —
4 | 0 | — | — | — | — | — | — | — | — | 2 | — | — | — | — | — | 7 | —
5 | 0 | — | — | — | — | — | — | — | — | — | — | — | — | — | — | — | 9
Table 11: DP table for ${\mathcal{S}}=\\{1\\}$ (Example 10)
## 6 Probabilistic Explanations for Other Families of Classifiers
This section investigates additional families of classifiers, namely those
either based on propositional languages [34] or based on graphs [35]. It
should be noted that some families of classifiers can be viewed as both
propositional languages and as graphs.
### 6.1 Propositional Classifiers
This section considers the families of classifiers represented as
propositional languages and studied in [34]. Examples include classifiers
based on d-DNNFs and SDDs.
###### Proposition 1.
For any language that allows conditioning (CD) in polynomial time, a locally-
minimal PAXp can be found in polynomial time (for all $\delta\in[0,1]$) if and
only if the language allows _counting_ (CT) in polynomial time.
###### Proof.
If the language allows CD and CT in polynomial time, Algorithm 1 finds a
locally-minimal PAXp in polynomial time. Conversely, testing whether
${\mathcal{X}}=\emptyset$ is a locally-minimal PAXp amounts to determining
whether the number of models of the classifier $\kappa$ is at least $\delta
2^{m}$, and hence a binary search on $\delta$ computes the number of models
(i.e. CT) in polynomial time. ∎
###### Proposition 2.
For any language that allows CT and CD in polynomial time, given an integer
$k$, the problem of deciding the existence of a weak PAXp of size at most $k$
is in NP.
###### Proof.
Let ${\mathcal{P}}\subseteq{\mathcal{F}}$ denote a guessed set of picked
features of size ${k^{\prime}}\leq{k}$, to be fixed. Since CT and CD run in
polynomial time, the left-hand side of (3.1), i.e.
$\textnormal{Pr}_{\mathbf{x}}(\kappa(\mathbf{x})=c\,|\,\mathbf{x}_{{\mathcal{X}}}=\mathbf{v}_{{\mathcal{X}}})$
can be computed in polynomial time. Consequently, we can decide in polynomial
time whether or not ${\mathcal{P}}$ is a weak PAXp, and so the decision
problem is in NP. ∎
###### Corollary 1.
For any language that allows CT and CD in polynomial time, the problem of
finding a minimum-size PAXp belongs to
$\textnormal{FP}^{\textnormal{NP}}[{\mathcal{O}}(\log m)]$.
###### Proof.
It suffices to run a binary search, on the number $k$ of features, using an NP
oracle that decides the decision problem introduced in the proof of
Proposition 2. This algorithm finds a smallest weak PAXp ${\mathcal{P}}$,
which is necessarily a PAXp (and hence a smallest PAXp) since no proper subset
of ${\mathcal{P}}$ can be a weak PAXp. ∎
Corollary 1 notably applies to the following languages:
* •
the language EADT of extended affine decision trees [56] which are a strict
generalisation of decision trees.
* •
the language d-DNNF and sublanguages of d-DNNF, such as SDD and OBDD.
### 6.2 Graph-Based Classifiers
This section considers the DG classifiers introduced in Section 2.3, and in
particular two restricted types of DGs: OMDDs and OBDDs. For OMDDs and OBDDs,
it is well-known that there are poly-time algorithms for model counting [23,
75]. Hence, we can compute a minimum-size PAXp with a logarithmic number of
calls to an NP oracle. We will describe one such algorithm in this section.
Moreover, a general method for counting models of DGs will be described as
well.
#### Counting models of OBDD/OMDD.
We describe next a dedicated algorithm for counting models of an OMDD, which
can also be used with OBDDs.
We associate an indicator $n_{p}$ to each node $p$, the indicator of the root
node being $n_{1}$, such that the value of $n_{1}$ represents the number of
models consistent with the class $c$. And let $s_{i}$ be the boolean selector
of feature $i$ such that $s_{i}=1$ if feature $i$ is included in PAXp,
otherwise it is free. Besides, we assume that the feature-index order is
increasing top-down, and the leftmost outgoing edge of a non-terminal node is
consistent with the given instance $\mathbf{v}$.
1. 1.
For a terminal node $p$, $n_{p}=1$ if the label of node $p$ is consistent with
the class $c$; otherwise $n_{p}=0$.
2. 2.
For a non-terminal node $p$ label with feature $i$, that has $k$ child nodes
$q_{1},\ldots,q_{k}$, let $q$ be an arbitrary child node of $p$ labeled with
feature $j$, and edge $(p,q_{1})$ be the leftmost edge. By assumption, since
$i<j$, there may exist some untested features $i<l<j$ along the outgoing edge
$(p,q)$. Let $b_{p,q}$ be the indicator of the edge $(p,q)$, we have:
$b_{p,q}=n_{q}\times\prod_{i<l<j}ite(s_{l},1,\mathbb{D}_{l})\times\lvert(p,q)\rvert$,
so the indicator of node $p$ is $n_{p}=ite(s_{i},b_{p,q_{1}},\sum_{q_{1}\leq
l\leq q_{k}}b_{p,l})$. It should be noted that multiple edges may exist; we
use $\lvert(p,q)\rvert$ to denote the number of edges between node $p$ and
$q$.
3. 3.
The number of models consistent with the class $c$ is $n_{1}$.
In the case of $\lvert{\mathcal{K}}\rvert=2$ and $\mathbb{F}=\mathbb{B}^{m}$,
this algorithm is still applicable to OBDDs with some minor modifications.
Note that each non-terminal node of an OBDD has exactly two outgoing edges and
multiple edges between two nodes are not allowed (otherwise, the OBDD is not
_reduced_); this means that $\mathbb{D}_{l}=\mathbb{B}$, and
$\lvert(p,q)\rvert=1$.
#### Counting models for unrestricted DGs.
For unrestricted DG classifiers, given Section 2.3, any path connecting the
root to a terminal node is consistent with some input, and no two paths are
consistent with the same input. As a result, in the case of unrestricted DG
classifiers, it suffices to enumerate all paths, and count the models
associated with the path. Of course, the downside is that the number of paths
is worst-case exponential on the size of the graph. To alleviate this, an
approach similar to the one outlined above can be considered.
## 7 Experiments
This section reports the experimental results on computing relevant sets for
the classifiers studied in the earlier sections, namely: decision trees, naive
Bayes classifiers and graph-based classifiers (in our case OMDDs). For each
case study, we describe the prototype implementation and the used benchmarks;
we show a table that summarizes the results and then we discuss the results.
Furthermore, for the case of DTs, the evaluation includes a comparison with
the model-agnostic explainer Anchor [82], aiming at assessing not only the
succinctness and precision of computed explanations but also the scalability
of our solution. (Observe that for the case of NBCs, earlier work on computing
AXp’s [62] have compared their approach with the heuristic methods, e.g.
Anchor, SHAP, and show that the latter are slower and do not show a strong
correlation between features of their explanations and common features
identified from AXp’s. As a result, the comparison with Anchor is restricted
to the case of DTs.)
All experiments were conducted on a MacBook Air with a 1.1GHz Quad-Core Intel
Core i5 CPU with 16 GByte RAM running macOS Monterey.
Dataset | | | | $\bm{\mathsf{MinPAXp}}$ | $\bm{\mathsf{LmPAXp}}$ | Anchor
---|---|---|---|---|---|---
DT | Path | $\bm{\delta}$ | Length | Prec | Time | Length | Prec | m⊆ | Time | D | Length | Prec | Time
N | A | M | m | avg | | M | m | avg | avg | avg | M | m | avg | avg | | avg | | M | m | avg | F／∈P | avg | avg
| | | | | | $100$ | $11$ | $3$ | $6.8$ | 100 | $2.34$ | $11$ | $3$ | $6.9$ | 100 | 100 | $0.00$ | d | $12$ | $2$ | $7.0$ | $26.8$ | 76.8 | $0.96$
adult | $1241$ | $89$ | $14$ | $3$ | $10.7$ | $95$ | $11$ | $3$ | $6.2$ | 98.4 | $5.36$ | $11$ | $3$ | $6.3$ | 98.6 | 99.0 | $0.01$ | u | $12$ | $3$ | $10.0$ | $29.4$ | 93.7 | $2.20$
| | | | | | $90$ | $11$ | $2$ | $5.6$ | 94.6 | $4.64$ | $11$ | $2$ | $5.8$ | 95.2 | 96.4 | $0.01$ | | | | | | |
| | | | | | $100$ | $12$ | $1$ | $4.4$ | 100 | $0.35$ | $12$ | $1$ | $4.4$ | 100 | 100 | $0.00$ | d | $31$ | $1$ | $4.8$ | $58.1$ | 32.9 | $3.10$
dermatology | $71$ | $100$ | $13$ | $1$ | $5.1$ | $95$ | $12$ | $1$ | $4.1$ | 99.7 | $0.37$ | $12$ | $1$ | $4.1$ | 99.7 | 99.3 | $0.00$ | u | $34$ | $1$ | $13.1$ | $43.2$ | 87.2 | $25.13$
| | | | | | $90$ | $11$ | $1$ | $4.0$ | 98.8 | $0.35$ | $11$ | $1$ | $4.0$ | 98.8 | 100 | $0.00$ | | | | | | |
| | | | | | $100$ | $12$ | $2$ | $4.8$ | 100 | $0.93$ | $12$ | $2$ | $4.9$ | 100 | 100 | $0.00$ | d | $36$ | $2$ | $7.9$ | $44.8$ | 69.4 | $1.94$
kr-vs-kp | $231$ | $100$ | $14$ | $3$ | $6.6$ | $95$ | $11$ | $2$ | $3.9$ | 98.1 | $0.97$ | $11$ | $2$ | $4.0$ | 98.1 | 100 | $0.00$ | u | $12$ | $2$ | $3.6$ | $16.6$ | 97.3 | $1.81$
| | | | | | $90$ | $10$ | $2$ | $3.2$ | 95.4 | $0.92$ | $10$ | $2$ | $3.3$ | 95.4 | 99.0 | $0.00$ | | | | | | |
| | | | | | $100$ | $12$ | $4$ | $8.2$ | 100 | $16.06$ | $11$ | $4$ | $8.2$ | 100 | 100 | $0.00$ | d | $16$ | $3$ | $13.2$ | $43.1$ | 71.3 | $12.22$
letter | $3261$ | $93$ | $14$ | $4$ | $11.8$ | $95$ | $12$ | $4$ | $8.0$ | 99.6 | $18.28$ | $11$ | $4$ | $8.0$ | 99.5 | 100 | $0.00$ | u | $16$ | $3$ | $13.7$ | $47.3$ | 66.3 | $10.15$
| | | | | | $90$ | $12$ | $4$ | $7.7$ | 97.7 | $16.35$ | $10$ | $4$ | $7.8$ | 97.8 | 100 | $0.00$ | | | | | | |
| | | | | | $100$ | $14$ | $3$ | $6.4$ | 100 | $0.92$ | $14$ | $3$ | $6.5$ | 100 | 100 | $0.00$ | d | $35$ | $2$ | $8.6$ | $55.4$ | 33.6 | $5.43$
soybean | $219$ | $100$ | $16$ | $3$ | $7.3$ | $95$ | $14$ | $3$ | $6.4$ | 99.8 | $0.95$ | $14$ | $3$ | $6.4$ | 99.8 | 100 | $0.00$ | u | $35$ | $3$ | $19.2$ | $66.0$ | 75.0 | $38.96$
| | | | | | $90$ | $14$ | $3$ | $6.1$ | 98.1 | $0.94$ | $14$ | $3$ | $6.1$ | 98.2 | 98.5 | $0.00$ | | | | | | |
| | | | | | $0$ | $12$ | $3$ | $7.4$ | 100 | $1.23$ | $12$ | $3$ | $7.5$ | 100 | 100 | $0.01$ | d | $38$ | $2$ | $6.3$ | $65.3$ | 63.3 | $24.12$
spambase | $141$ | $99$ | $14$ | $3$ | $8.5$ | $95$ | $9$ | $1$ | $3.7$ | 96.1 | $2.16$ | $9$ | $1$ | $3.8$ | 96.5 | 100 | $0.01$ | u | $57$ | $3$ | $28.0$ | $86.2$ | 65.3 | $834.70$
| | | | | | $90$ | $6$ | $1$ | $2.4$ | 92.4 | $2.15$ | $8$ | $1$ | $2.4$ | 92.2 | 100 | $0.01$ | | | | | | |
| | | | | | $100$ | $12$ | $3$ | $6.2$ | 100 | $2.01$ | $11$ | $3$ | $6.2$ | 100 | 100 | $0.01$ | d | $40$ | $2$ | $16.5$ | $80.6$ | 32.2 | $532.42$
texture | $257$ | $100$ | $13$ | $3$ | $6.6$ | $95$ | $11$ | $3$ | $5.4$ | 99.3 | $2.19$ | $11$ | $3$ | $5.4$ | 99.4 | 100 | $0.01$ | u | $40$ | $5$ | $17.5$ | $84.4$ | 31.6 | $402.07$
| | | | | | $90$ | $11$ | $3$ | $5.4$ | 98.5 | $2.20$ | $11$ | $3$ | $5.4$ | 99.4 | 100 | $0.01$ | | | | | | |
Table 12: Assessing explanations of $\mathsf{MinPAXp}$, $\mathsf{LmPAXp}$ and
Anchor for DTs. (For each dataset, we run the explainers on 500 samples
randomly picked or all samples if there are less than 500.) In column DT, N
and A denote, resp., the number of nodes and the training accuracy of the DT.
Column $\bm{\delta}$ reports (in %) the value of the threshold $\delta$. In
column Path, avg (resp. M and m) denotes the average (resp. max. and min.)
depth of paths consistent with the instances. In column Length, avg (resp. M
and m) denotes the average (resp. max. and min.) length of the explanations;
and F／∈P denotes the avg. % of features in Anchor’s explanations that do not
belong to the consistent paths. Prec reports (in %) the average precision
(defined in (9)) of resulting explanations. m⊆ shows the number in (%) of
LmPAXp’s that are subset-minimal, i.e. PAXp’s. Time reports (in seconds) the
average runtime to compute an explanation. Finally, D indicates which
distribution is applied on data given to Anchor: either data distribution
(denoted by d) or uniform distribution (denoted by u).
Dataset | (#F #I) | NBC | $\bm{\mathsf{AXp}}$ | | $\bm{\mathsf{LmPAXp}}_{\leq 9}$ | $\bm{\mathsf{LmPAXp}}_{\leq 7}$ | $\bm{\mathsf{LmPAXp}}_{\leq 4}$
---|---|---|---|---|---|---|---
A% | Length | $\delta$ | Length | Precision | W% | Time | Length | Precision | W% | Time | Length | Precision | W% | Time
adult | (13 | 200) | 81.37 | 6.8$\pm$ 1.2 | $98$ | 6.8$\pm$ 1.1 | 100$\pm$ 0.0 | $100$ | $0.003$ | 6.3$\pm$ 0.9 | 99.61$\pm$ 0.6 | $96$ | $0.023$ | 4.8$\pm$ 1.3 | 98.73$\pm$ 0.5 | $48$ | $0.059$
$95$ | 6.8$\pm$ 1.1 | 99.99$\pm$ 0.2 | $100$ | $0.074$ | 5.9$\pm$ 1.0 | 98.87$\pm$ 1.8 | $99$ | $0.058$ | 3.9$\pm$ 1.0 | 96.93$\pm$ 1.1 | $80$ | $0.071$
$93$ | 6.8$\pm$ 1.1 | 99.97$\pm$ 0.4 | $100$ | $0.104$ | 5.7$\pm$ 1.3 | 98.34$\pm$ 2.6 | $100$ | $0.086$ | 3.4$\pm$ 0.9 | 95.21$\pm$ 1.6 | $90$ | $0.093$
$90$ | 6.8$\pm$ 1.1 | 99.95$\pm$ 0.6 | $100$ | $0.164$ | 5.5$\pm$ 1.4 | 97.86$\pm$ 3.4 | $100$ | $0.100$ | 3.0$\pm$ 0.8 | 93.46$\pm$ 1.5 | $94$ | $0.103$
agaricus | (23 | 200) | 95.41 | 10.3$\pm$ 2.5 | $98$ | 7.7$\pm$ 2.7 | 99.12$\pm$ 0.8 | $92$ | $0.593$ | 6.4$\pm$ 3.0 | 98.75$\pm$ 0.6 | $87$ | $0.763$ | 6.0$\pm$ 3.1 | 98.67$\pm$ 0.5 | $29$ | $0.870$
$95$ | 6.9$\pm$ 3.1 | 97.62$\pm$ 2.1 | $95$ | $0.954$ | 5.3$\pm$ 3.2 | 96.59$\pm$ 1.6 | $92$ | $1.273$ | 4.8$\pm$ 3.3 | 96.24$\pm$ 1.2 | $55$ | $1.217$
$93$ | 6.5$\pm$ 3.1 | 96.65$\pm$ 2.8 | $95$ | $1.112$ | 4.8$\pm$ 3.1 | 95.38$\pm$ 1.9 | $93$ | $1.309$ | 4.3$\pm$ 3.1 | 94.92$\pm$ 1.3 | $64$ | $1.390$
$90$ | 5.9$\pm$ 3.3 | 94.95$\pm$ 4.1 | $96$ | $1.332$ | 4.0$\pm$ 3.0 | 92.60$\pm$ 2.8 | $95$ | $1.598$ | 3.6$\pm$ 2.8 | 92.08$\pm$ 1.7 | $76$ | $1.830$
chess | (37 | 200) | 88.34 | 12.1$\pm$ 3.7 | $98$ | 8.1$\pm$ 4.1 | 99.27$\pm$ 0.6 | $64$ | $0.383$ | 5.9$\pm$ 4.9 | 98.70$\pm$ 0.4 | $64$ | $0.454$ | 5.7$\pm$ 5.0 | 98.65$\pm$ 0.4 | $46$ | $0.457$
$95$ | 7.7$\pm$ 3.8 | 98.51$\pm$ 1.4 | $68$ | $0.404$ | 5.5$\pm$ 4.4 | 97.90$\pm$ 0.9 | $64$ | $0.483$ | 5.3$\pm$ 4.5 | 97.85$\pm$ 0.8 | $46$ | $0.478$
$93$ | 7.3$\pm$ 3.5 | 97.56$\pm$ 2.4 | $68$ | $0.419$ | 5.0$\pm$ 4.1 | 96.26$\pm$ 2.2 | $64$ | $0.485$ | 4.8$\pm$ 4.1 | 96.21$\pm$ 2.1 | $64$ | $0.493$
$90$ | 7.3$\pm$ 3.5 | 97.29$\pm$ 2.9 | $70$ | $0.413$ | 4.9$\pm$ 4.0 | 95.99$\pm$ 2.6 | $64$ | $0.483$ | 4.8$\pm$ 4.0 | 95.93$\pm$ 2.5 | $64$ | $0.543$
vote | (17 | 81) | 89.66 | 5.3$\pm$ 1.4 | $98$ | 5.3$\pm$ 1.4 | 100$\pm$ 0.0 | $100$ | $0.000$ | 5.3$\pm$ 1.3 | 99.95$\pm$ 0.2 | $100$ | $0.007$ | 4.6$\pm$ 1.1 | 99.60$\pm$ 0.4 | $64$ | $0.014$
$95$ | 5.3$\pm$ 1.4 | 100$\pm$ 0.0 | $100$ | $0.000$ | 5.3$\pm$ 1.3 | 99.93$\pm$ 0.3 | $100$ | $0.008$ | 4.1$\pm$ 1.0 | 98.25$\pm$ 1.7 | $64$ | $0.018$
$93$ | 5.3$\pm$ 1.4 | 100$\pm$ 0.0 | $100$ | $0.000$ | 5.2$\pm$ 1.3 | 99.78$\pm$ 1.1 | $100$ | $0.012$ | 4.1$\pm$ 0.9 | 98.10$\pm$ 1.9 | $64$ | $0.018$
$90$ | 5.3$\pm$ 1.4 | 100$\pm$ 0.0 | $100$ | $0.000$ | 5.2$\pm$ 1.3 | 99.78$\pm$ 1.1 | $100$ | $0.012$ | 4.0$\pm$ 1.2 | 97.24$\pm$ 3.1 | $64$ | $0.022$
kr-vs-kp | (37 | 200) | 88.07 | 12.2$\pm$ 3.9 | $98$ | 7.8$\pm$ 4.2 | 99.19$\pm$ 0.5 | $64$ | $0.387$ | 6.5$\pm$ 4.7 | 98.99$\pm$ 0.4 | $64$ | $0.427$ | 6.1$\pm$ 4.9 | 98.88$\pm$ 0.3 | $43$ | $0.457$
$95$ | 7.3$\pm$ 3.9 | 98.29$\pm$ 1.4 | $64$ | $0.416$ | 6.0$\pm$ 4.3 | 97.89$\pm$ 1.1 | $64$ | $0.453$ | 5.5$\pm$ 4.5 | 97.79$\pm$ 0.9 | $43$ | $0.462$
$93$ | 6.9$\pm$ 3.5 | 97.21$\pm$ 2.5 | $69$ | $0.422$ | 5.6$\pm$ 3.8 | 96.82$\pm$ 2.2 | $64$ | $0.448$ | 5.2$\pm$ 4.0 | 96.71$\pm$ 2.1 | $43$ | $0.468$
$90$ | 6.8$\pm$ 3.5 | 96.65$\pm$ 3.1 | $69$ | $0.418$ | 5.4$\pm$ 3.8 | 95.69$\pm$ 3.0 | $64$ | $0.468$ | 5.0$\pm$ 4.0 | 95.59$\pm$ 2.8 | $61$ | $0.487$
mushroom | (23 | 200) | 95.51 | 10.7$\pm$ 2.3 | $98$ | 7.5$\pm$ 2.4 | 98.99$\pm$ 0.7 | $90$ | $0.641$ | 6.5$\pm$ 2.6 | 98.74$\pm$ 0.5 | $83$ | $0.751$ | 6.3$\pm$ 2.7 | 98.70$\pm$ 0.4 | $18$ | $0.828$
$95$ | 6.5$\pm$ 2.6 | 97.35$\pm$ 1.8 | $96$ | $1.011$ | 5.1$\pm$ 2.5 | 96.52$\pm$ 1.0 | $90$ | $1.130$ | 5.0$\pm$ 2.5 | 96.39$\pm$ 0.8 | $54$ | $1.113$
$93$ | 5.8$\pm$ 2.8 | 95.77$\pm$ 2.7 | $96$ | $1.257$ | 4.4$\pm$ 2.5 | 94.67$\pm$ 1.6 | $94$ | $1.297$ | 4.2$\pm$ 2.4 | 94.48$\pm$ 1.3 | $65$ | $1.324$
$90$ | 5.3$\pm$ 3.0 | 94.01$\pm$ 3.9 | $97$ | $1.455$ | 3.8$\pm$ 2.3 | 92.36$\pm$ 2.2 | $96$ | $1.543$ | 3.6$\pm$ 2.2 | 92.07$\pm$ 1.6 | $76$ | $1.650$
threeOf9 | (10 | 103) | 83.13 | 4.2$\pm$ 0.4 | $98$ | 4.2$\pm$ 0.4 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.2$\pm$ 0.4 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.2$\pm$ 0.4 | 100$\pm$ 0.0 | $78$ | $0.001$
$95$ | 4.2$\pm$ 0.4 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.2$\pm$ 0.4 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.0$\pm$ 0.2 | 99.23$\pm$ 1.4 | $100$ | $0.002$
$93$ | 4.2$\pm$ 0.4 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.2$\pm$ 0.4 | 100$\pm$ 0.0 | $100$ | $0.000$ | 3.9$\pm$ 0.2 | 99.20$\pm$ 1.5 | $100$ | $0.002$
$90$ | 4.2$\pm$ 0.4 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.2$\pm$ 0.4 | 100$\pm$ 0.0 | $100$ | $0.000$ | 3.8$\pm$ 0.4 | 98.29$\pm$ 3.3 | $100$ | $0.003$
xd6 | (10 | 176) | 81.36 | 4.5$\pm$ 0.9 | $98$ | 4.5$\pm$ 0.8 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.5$\pm$ 0.8 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.5$\pm$ 0.8 | 100$\pm$ 0.0 | $73$ | $0.001$
$95$ | 4.5$\pm$ 0.8 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.5$\pm$ 0.8 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.5$\pm$ 0.8 | 100$\pm$ 0.0 | $73$ | $0.001$
$93$ | 4.5$\pm$ 0.8 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.5$\pm$ 0.8 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.3$\pm$ 0.4 | 98.30$\pm$ 2.7 | $73$ | $0.001$
$90$ | 4.5$\pm$ 0.8 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.5$\pm$ 0.8 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.3$\pm$ 0.4 | 98.30$\pm$ 2.7 | $73$ | $0.002$
mamo | (14 | 53) | 80.21 | 4.9$\pm$ 0.8 | $98$ | 4.9$\pm$ 0.7 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.9$\pm$ 0.7 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.6$\pm$ 0.6 | 99.66$\pm$ 0.5 | $53$ | $0.007$
$95$ | 4.9$\pm$ 0.7 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.9$\pm$ 0.7 | 100$\pm$ 0.0 | $100$ | $0.000$ | 3.9$\pm$ 0.6 | 97.80$\pm$ 1.6 | $85$ | $0.009$
$93$ | 4.9$\pm$ 0.7 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.9$\pm$ 0.7 | 100$\pm$ 0.0 | $100$ | $0.000$ | 3.9$\pm$ 0.6 | 97.68$\pm$ 1.7 | $85$ | $0.009$
$90$ | 4.9$\pm$ 0.7 | 100$\pm$ 0.0 | $100$ | $0.000$ | 4.9$\pm$ 0.7 | 100$\pm$ 0.0 | $100$ | $0.000$ | 3.6$\pm$ 0.8 | 96.18$\pm$ 3.2 | $96$ | $0.011$
tumor | (16 | 104) | 83.21 | 5.3$\pm$ 0.9 | $98$ | 5.3$\pm$ 0.8 | 100$\pm$ 0.0 | $100$ | $0.000$ | 5.2$\pm$ 0.7 | 99.96$\pm$ 0.2 | $100$ | $0.008$ | 4.1$\pm$ 0.7 | 99.41$\pm$ 0.5 | $91$ | $0.012$
$95$ | 5.3$\pm$ 0.8 | 100$\pm$ 0.0 | $100$ | $0.000$ | 5.2$\pm$ 0.6 | 99.83$\pm$ 0.7 | $100$ | $0.012$ | 3.2$\pm$ 0.6 | 96.02$\pm$ 1.5 | $94$ | $0.016$
$93$ | 5.3$\pm$ 0.8 | 100$\pm$ 0.0 | $100$ | $0.000$ | 5.2$\pm$ 0.6 | 99.74$\pm$ 1.2 | $100$ | $0.014$ | 3.1$\pm$ 0.7 | 95.50$\pm$ 1.4 | $95$ | $0.016$
$90$ | 5.3$\pm$ 0.8 | 100$\pm$ 0.0 | $100$ | $0.000$ | 5.1$\pm$ 0.7 | 99.67$\pm$ 1.4 | $100$ | $0.016$ | 3.0$\pm$ 0.6 | 95.30$\pm$ 1.6 | $95$ | $0.017$
Table 13: Assessing $\mathsf{LmPAXp}$ explanations for NBCs. Columns #F and
#I show, respectively, number of features and tested instances in the Dataset.
Column A% reports (in %) the training accuracy of the classifier. Column
$\delta$ reports (in %) the value of the parameter $\delta$.
$\bm{\mathsf{LmPAXp}}_{\leq 9}$, $\bm{\mathsf{LmPAXp}}_{\leq 7}$ and
$\bm{\mathsf{LmPAXp}}_{\leq 4}$ denote, respectively, LmPAXp’s of (target)
length 9, 7 and 4. Columns Length and Precision report, respectively, the
average explanation length and the average explanation precision ($\pm$
denotes the standard deviation). W% shows (in %) the number of success/wins
where the explanation size is less than or equal to the target size. Finally,
the average runtime to compute an explanation is shown (in seconds) in Time.
(Note that the reported average time is the mean of runtimes for instances for
which we effectively computed an approximate explanation, namely instances
that have AXp’s of length longer than the target length; whereas for the
remaining instances the trimming process is skipped and the runtime is 0 sec,
thus we exclude them when calculating the average.)
### 7.1 Case Study 1: Decision Trees
#### Prototype implementation.
A prototype implementation888All sources implemented in these experiments will
be publicly available after the paper gets accepted. of the proposed
algorithms for DTs was developed in Python; whenever necessary, it instruments
oracle calls to the well-known SMT solver z3999https://github.com/Z3Prover/z3/
[24] as described in Section 4 . Hence, the prototype implements the
$\mathsf{LmPAXp}$ procedure outlined in Algorithm 1 and augmented with a
heuristic that orders the features in ${\mathcal{X}}$. The idea consists in
computing the precision loss of the overapproximation of each
${\mathcal{X}}\setminus\\{j\\}$ and then sorting the features from the less to
the most important one. This strategy often allows us to obtain the closest
superset to a PAXp, in contrast to the simple lexicographic order applied over
${\mathcal{X}}$. (Recall that ${\mathcal{X}}$ is initialized to the set of
features involved in the decision path.) Algorithm $\mathsf{MinPAXp}$ outlined
in Section 4.3 implements the two (multiplication- and addition-based) SMT
encodings. Nevertheless, preliminary results show that both encodings perform
similarly, with some exceptions where the addition-based encoding is much
larger and so slower. Therefore, the results reported below refer only to the
multiplication-based encoding.
#### Benchmarks.
The benchmarks used in the experiments comprise publicly available and widely
used datasets obtained from the UCI ML Repository [95]. All the DTs are
trained using the learning tool _IAI_ (_Interpretable AI_) [11, 38]. The
maximum depth parameter in IAI is set to 16. As the baseline, we ran Anchor
with the default explanation precision of 0.95. Two assessments are performed
with Anchor: (i) with the original training data101010The same training set
used to learn the model. that follows the data distribution; (ii) with using
sampled data that follows a uniform distribution. Our setup assumes that all
instances of the feature space are equally possible, and so there is no
assumed probability distribution over the features. Therefore in order to be
fair with Anchor, we further assess Anchor with uniformly sampled data. (Also,
we point out that the implementation of Anchor demonstrates that it can
generate samples that do not belong to the input distribution. Thus, there is
no guarantee that these samples come from the input distribution.) Also, the
prototype implementation was tested with varying the threshold $\delta$ while
Anchor runs guided by its own metric.
#### Results.
Table 12 summarizes the results of our experiments for the case study of DTs.
One can observe that $\mathsf{MinPAXp}$ and $\mathsf{LmPAXp}$ compute succinct
explanations (i.e. of average size $7\pm 2$ [69]), for the majority of tested
instances across all datasets, noticeably shorter than consistent-path
explanations. More importantly, the computed explanations are trustworthy and
show good quality precision, e.g. dermatology, soybean and texture show
average precisions greater than 98% for all values of $\delta$. Additionally,
the results clearly demonstrate that our proposed SMT encoding scales for deep
DTs with runtimes on average less than 20 sec for the largest encodings while
the runtimes of $\mathsf{LmPAXp}$ are negligible, never exceeding 0.01 sec.
Also, observe from the table (see column m⊆) that the over-approximations
computed by $\mathsf{LmPAXp}$ are often subset-minimal PAXp’s, and often as
short as computed MinPAXp’s. This confirms empirically the advantages of
computing LmPAXp’s, i.e. in practice one may rely on the computation of
LmPAXp’s, which pays off in terms of (1) performance, (2) sufficiently high
probabilistic guarantees of precision, and (3) good quality over-approximation
of subset-minimal PAXp’s. In contrast, Anchor is unable to provide precise and
succinct explanations in both settings of data and uniform distribution.
Moreover, we observe that Anchor’s explanations often include features that
are not involved in the consistent path, e.g. for texture less than 20% of an
explanation is shared with the consistent path. (This trend was also pointed
out by [39].) In terms of average runtime, Anchor is overall slower, being
outperformed by the computation of $\mathsf{LmPAXp}$ by several orders of
magnitude.
Overall, the experiments demonstrate that our approach efficiently computes
succinct and provably precise explanations for large DTs. The results also
substantiate the limitations of model-agnostic explainers, both in terms of
explanation quality and computation time.
### 7.2 Case Study 2: Naive Bayes Classifiers
#### Prototype implementation.
A prototype implementation of the proposed approach for computing relevant
sets for NBCs was developed in Python. To compute AXp’s, we use the Perl
script implemented by [62]111111Publicly available from:
https://github.com/jpmarquessilva/expxlc. The prototype implementation was
tested with varying thresholds $\delta\in\\{0.90,0.93,0.95,0.98\\}$. When
converting probabilities from real values to integer values, the selected
number of decimal places is 3. (As outlined earlier, we observed that there is
a negligible accuracy loss from using three decimal places). In order to
produce explanations of size admissible for the cognitive capacity of human
decision makers [69], we selected three different target sizes for the
explanations to compute: 9, 7 and 4, and we computed a LmPAXp for the input
instance when its AXp ${\mathcal{X}}$ is larger than the target size (recall
that ${\mathcal{S}}$ is initialized to ${\mathcal{X}}$); otherwise we consider
that the AXp is succinct and the explainer returns ${\mathcal{X}}$. For
example, assume the target size is 7, an instance $\mathbf{v}_{1}$ with an
$\mathsf{AXp}$ ${\mathcal{S}}_{1}$ of 5 features and a second instance
$\mathbf{v}_{2}$ with an $\mathsf{AXp}$ ${\mathcal{S}}_{2}$ of 8 features,
then for $\mathbf{v}_{1}$ the output will be ${\mathcal{S}}_{1}$ and for
$\mathbf{v}_{2}$ the output will be a subset of ${\mathcal{S}}_{2}$.
#### Benchmarks.
The benchmarks used in this evaluation originate from the UCI ML Repository
[95] and Penn ML Benchmarks [79]. The number of training data (resp. features)
in the target datasets varies from 336 to 14113 (resp. 10 to 37) and on
average is 3999.1 (resp. 20.0). All the NBCs are trained using the learning
tool _scikit-learn_ [88]. The data split for training and test data is set to
80% and 20%, respectively. Model accuracies are above 80% for the training
accuracy and above 75% for the test accuracy. For each dataset, we run the
explainer on 200 instances randomly picked from the test data or on all
instances if there are less than 200.
#### Results.
Table 13 summarizes the results of our experiments for the case study of NBCs.
For all tested values for the parameter threshold $\delta$ and target size,
the reported results show the sizes and precisions of the computed
explanations. As can be observed for all considered settings, the approximate
explanations are succinct, in particular the average sizes of the explanations
are invariably lower than the target sizes. Moreover, theses explanations
offer strong guarantees of precision, as their average precisions are strictly
greater than $\delta$ with significant gaps (e.g. above 97%, in column
$\mathsf{LmPAXp}_{\leq 7}$, for datasets _adult_ , _vote_ , _threeOf9_ , _xd6_
, _mamo_ and _tumor_ and above 95% for _chess_ and _kr-vs-kp_). An important
observation from the results, is the gain of succinctness (explanation size)
|
# Histories without collapse
Anthony Sudbery1
Department of Mathematics, University of York,
Heslington, York, England YO10 5DD
1<EMAIL_ADDRESS>
###### Abstract
This paper is a comparison of two theories of the probability of a history in
quantum mechanics. One is derived from Copenhagen quantum mechanics using the
projection postulate and is the basis of the “consistent histories”
interpretation; the other is based on a proposal by Bell, originally for the
“pilot state” theory but here applied to pure unitary quantum mechanics. The
first can be used for a wider class of histories but depends on the projection
postulate, or “collapse”, which is widely held to be an unsatisfactory feature
of the theory; the second can be used in a theory of the universal state
vector without collapse. We examine a simple model based on Wigner’s friend,
in which Bell’s model and the projection postulate give different
probabilities for the histories of a sentient system. We also examine the
Frauchiger-Renner extension of this model, in which comparison of the two
calculations of histories throws light on the contradiction found by
Frauchiger and Renner. By extending the model to equip the observer with a
memory, we reduce the probability of histories to the use of the Born rule at
a single time, and show that the Born rule, with the memory, gives the same
result as applying projection in the course of the history, because of
entanglement with the memory. Entanglement implements collapse. We discuss the
implications of this for the use of histories in quantum cosmology.
Keywords: quantum mechanics, foundations, histories
Declarations
Funding: This research was not supported by any funding.
Conflicts of interest: I am not aware of any conflict of interest.
Availabiilty of data and material: There is no data or material to be made
available.
Code availability: There is no relevant code.
## 1 Introduction
In quantum mechanics one often wants to know the probability of a sequence of
measurements, or the conditional probability of a measurement result at one
time given a result at an earlier time. In other words, one wants the
probability of a history. In the form of quantum mechanics presented in most
textbooks (the “Copenhagen interpretation”), this is managed by assuming that
the unitary evolution is interrupted by a projection, or “collapse”, at the
times of the sequence. Other interpretations of quantum mechanics, usually
associated with the name of Everett, deny that such collapses take place at a
fundamental level, and assert that all evolution follows a continuous unitary
process. If this is so, however, it is not obvious how to calculate the
probability of a history: the only point at which probability appears to enter
the theory is via the Born rule at one time. However, in 1984 John Bell [3],
in a study of quantum field theory in the context of Bohmian “pilot state”
theory, proposed a process which can be generalised, detached from the Bohmian
context of “beables” (or “hidden variables”), and applied to any history in
the different ontology of Everettian quantum mechanics. In this paper the
generalisation of Bell’s process is presented and applied to some examples
deriving from the scenario of Wigner’s friend, which bring out particularly
clearly the issues related to collapse. The probabilities derived from Bell’s
process are compared with those obtained from Copenhagen quantum mechanics.
Finally, we consider how the presence of memory makes it possible to reduce
the probability of a history to a single-time probability, using the Born
rule.
The paper is organised as follows. In Section 2 we set up the general
formalism for a discussion of histories. The next three sections set out the
two methods (Copenhagen and Bell) for calculating of the probability of a
history, with a discussion (Section 4) of the problems of histories and
probability in pure unitary quantum mechanics. In Section 5 the solution of
these problems based on Bell’s process is introduced and applied to a variant
of the Wigner’s friend scenario due to Brukner; the results are compared with
those of the Copenhagen calculation. In Section 6 the two methods are applied
to a more elaborate version of the Wigner’s friend scenario due to Frauchiger
and Renner. Section 7 shows the effects of memory in this scenario, and the
paper concludes with a discussion of the results.
## 2 Histories and probabilities
A _history_ is a sequence of events $E_{0},\ldots,E_{n}$ at times
$t_{0},\ldots,t_{n}$. In quantum mechanics, as in statistics and law (and
unlike the study of human history, relativity theory or high-energy physics)
an event is defined by a proposition and therefore, in quantum mechanics, by a
closed subspace of Hilbert space, or equivalently by the orthogonal projection
onto this subspace. A history is therefore given by a time-labelled sequence
$h=\big{(}(\Pi_{0},t_{0}),\ldots,(\Pi_{n},t_{n})\big{)}.$ (2.1)
of events $(\Pi_{j},t_{j})$ each defined by a projection $\Pi_{j}$ and a time
$t_{j}$. For simplicity, we will assume that the system starts in a pure state
$|\Psi_{0}\rangle$ at time $t=0$, so that
$\Pi_{0}=|\Psi_{0}\rangle\langle\Psi_{0}|$ and $t_{0}=0$. The theory should
give a probability for this history, i.e. for the truth of the proposition
represented by the succession of projections $\Pi_{j}$ at the times $t_{j}$. A
closely related quantity is the _conditional_ probability of an event at one
time conditional on an event at an earlier time. However, there are competing
accounts of this probability. We now consider two of these accounts.
## 3 The Copenhagen probability
In the version of quantum mechanics presented in most textbooks, the
probability of a history like $h$ in (2.1) is obtained by applying the Born
rule to successive measurements of the projectors $\Pi_{j}$, assuming
orthogonal projection of the state after each measurement and unitary
evolution according to the Schrödinger equation between the measurements. This
gives the probability
$\displaystyle P_{\text{C}}(h)$
$\displaystyle=\langle\Psi_{0}|\widetilde{\Pi}_{1}\ldots\widetilde{\Pi}_{n}\ldots\widetilde{\Pi}_{1}|\Psi_{0}\rangle$
(3.1) $\displaystyle=\langle K(h)|K(h)\rangle$ (3.2)
where
$\widetilde{\Pi}_{j}=\mathrm{e}^{\mathrm{i}Ht_{j}}\Pi_{j}\mathrm{e}^{-\mathrm{i}Ht_{j}}$
(taking $\hbar=1$) and
$|K(h)\rangle=\widetilde{\Pi}_{n}\ldots\widetilde{\Pi}_{1}|\Psi_{0}\rangle.$
(3.3)
In the consistent-histories [12] and temporal-logic [19] interpretations of
quantum mechanics the formula (3.1) is extended to include conjunctions and
disjunctions of histories (though this is not essential in the latter
interpretation [21]). Logically desirable properties of probability then
require a consistency condition on the set of histories, namely that
$|K(h_{1})\rangle$ and $|K(h_{2})\rangle$ are orthogonal if $h_{1}$ and
$h_{2}$ are distinct histories. However, if this condition is not satisfied
the formula (3.1) for single histories is still meaningful provided that the
evolution in each time interval $[t_{j},t_{j+1}]$ is purely unitary.
Recent discussions of conditional probabilities and histories ([8, 11, 2])
have used the Page-Wootters mechanism to deal with time. The resulting
calculations of conditional probabilities are either equivalent to assuming
collapse at the events of a history, or proceed by examining the contents of a
memory at a single time after all the times of the history. The latter
procedure, as will be discussed in Section 7, gives the same results as
assuming collapse.
## 4 Probability in collapse-free quantum mechanics
The derivation of $P_{\text{C}}$ assumes that, in addition to unitary
evolution according to the Schrödinger equation, the quantum state undergoes
projection at the events in the history. Such projection is denied in some
influential interpretations of quantum mechanics, in particular the relative-
state interpretation of Everett and Wheeler [9, 23]. In this interpretation
the truth about the world and its development in time is given by a universal,
time-dependent state vector $|\Psi(t)\rangle$, and nothing else. It is
possible to use the Born rule to deduce that, given a projector $\Pi$, the
probability that the corresponding proposition is true at time $t$ is
$|\langle\Psi(t)|\Pi|\Psi(t)\rangle|^{2}$, though we need to discuss what this
probability means. But if projection processes are removed from the time
evolution, the theory offers no connection between events at different times.
This might appear to suggest that the probability of the history $h$ of (2.1)
should be obtained by treating the $(\Pi_{j},t_{j})$ as independent events, so
that the probability of the history $h$ is
$P_{\text{E}}=\langle\Psi(t_{1})|\Pi_{1}|\Psi(t_{1})\rangle\langle\Psi(t_{2})|\Pi_{2}|\Psi(t_{2})\rangle\cdots\langle\Psi(t_{n})|\Pi_{n}|\Psi(t_{n})\rangle.$
(4.1)
But this makes little sense. It does not reflect the continuity of experience,
implying as it does that in a very short time interval, no matter how short,
there is a finite probability, independent of the length of the interval, of
the state changing to a very different one; and it gives a non-zero
probability for a history (decayed at $t_{1}$, undecayed at $t_{2}$) of an
unstable particle, where $t_{2}>t_{1}$. These defects are remedied [17] in the
proposal of John Bell which is explained in the next section. First, however,
we will look more closely at the meaning of probability in the Everett
interpretation.
Probability enters the formalism of quantum mechanics only in the Born rule
for the results of measurements and the projection postulate which is
associated with measurements. This is rejected in Everettian theory (and
indeed, according to Bell [5], any respectable physical theory) in which there
are no preferred physical processes which have the special status of
“measurement”. In Everettian quantum mechanics there is only deterministic
evolution of the universal state vector, so there is no absolute role for
probability. However, the universe includes sentient systems (or “observers”,
or “organisms”) and the universal state vector must incorporate the experience
of such systems, including the changing experience in time. Each possible
experience of a sentient system must be defined by a set of values of physical
quantities relating to the organism, so the experience corresponds to a set of
eigenvalues of these quantities and therefore to a subspace of the sentient
organism’s state space. The orthogonal projection onto this experience
subspace extends to a projection of the universal state space, corresponding
to the proposition “the organism has the given experience”. Now probability
applies to the change in time of the organism’s experience: while having
experience $E_{1}$ at time $t_{1}$, the organism wants to know the probability
that it will experience $E_{2}$ at a later time $t_{2}$, and then $E_{3}$ at
time $t_{3}$, and so on. Thus probability is not absolute but _relative_ to a
particular (appropriately constituted) subsystem, to a particular experience
of that subsystem, and to a particular time.
Note that “relative” does not mean “subjective”; this probability is an
objective fact about the organism’s situation and does not refer to its
beliefs, just as the velocity of a particle is relative to a frame of
reference but is an objective property of the particle.
## 5 The Bell probability
Let $\Pi_{j}$ be the projector onto the subspace $\mathcal{S}_{j}$ of the
universal state space describing experience $\eta_{j}$ of the sentient being,
or “observer”. We will call these subspaces _sensible_ subspaces because of
their relation to the sense organs of the observer; the same word will be
applied to the corresponding subspaces of the observer’s state space (a tensor
factor of the universal state space). The experience label $j$ is assumed
discrete for convenience. Adopting the observer’s perspective, we assume that
at each time $t$ the observer has a definite experience $\eta_{j}$ (this,
however, is not true in the external perspective in which the whole truth is
the universal state vector $|\Psi(t)\rangle$, a superposition of different
experience states). To describe the perceived change of this experience in
time we assume, following Bell [3], that it follows a stochastic process
governed by the universal Hamiltonian $H$ together with the universal state
vector $|\Psi(t)\rangle$, as follows. If, at time $t$, the observer’s
experience is $\eta_{j}$, then the probability that at time $t+\delta t$ their
experience has changed to $\eta_{k}$ is $w_{jk}\delta t$ where the transition
probability $w_{jk}$ is given by
$\text{\normalsize$w_{jk}$}=\begin{cases}\frac{2\text{Im}[\hbar^{-1}\langle\Psi(t)|\Pi_{k}H\Pi_{j}|\Psi(t)\rangle]}{\langle\Psi(t)|\Pi_{j}|\Psi(t)\rangle}&\text{\normalsize
if this is $\geq 0$},\\\ \text{\normalsize$0$}&\text{\normalsize if it is
negative.}\end{cases}$ (5.1)
The probabilities for the observer’s changing experience are then compatible
with the Born rule, in the following sense.
###### Theorem 1.
([3], [18] p.215)
If the experience state moves between the sensible subspaces with transition
probabilities given by (5.1), then the probability $P_{j}(t)$ that it lies in
the subspace $\mathcal{S}_{j}$ at time $t$ is given by the Born rule
$P_{j}(t)=\langle\Psi(t)|\Pi_{j}|\Psi(t)\rangle$
at all positive times $t$, if the probabilities are so given at the initial
time $t=0$.
This theorem is the justification for the transition probabilities (5.1).
Bell’s formula is not uniquely determined by the requirement that the theorem
should hold: there is a range of possible transition probabilities with the
same property [1]. However, Bell’s formula is uniquely natural in applications
to decay [17] and measurement processes [20, 13]: it ensures that the
underlying direction of change in such processes is always forwards, without
intermittent reversals (decay products, for example, recombining to
reconstitute the unstable decaying state). This means that the theory is not
invariant under time reversal. I regard this as a virtue; for those with
different predilections, the range of transition probabilities found in [1]
include one which is time-symmetric. However, it should be emphasised that in
the theory presented here, the probabilities arise only in the description of
the subjective experience of a sentient system; the existence of a preferred
direction of time for such a system has been explained within a time-symmetric
theory [7].
We use these transition probabilities to define the Bell probability
$P_{\text{B}}(h)$, relative to a particular observer, of a history of
experiences of that observer. More generally, Bell’s process will give
probabilities for any set of histories in which the projectors $\Pi_{j}$ are
chosen from a fixed complete set of projectors. There is no need for the
histories to satisfy the decoherence condition of the consistent-histories
formulation of quantum mechanics.
Note that our ontology is different from Bell’s. Bell intended the subspaces
$\mathcal{S}_{j}$ to have an absolute meaning, being the subspaces in which
certain preferred observables, called “beables”, had definite values, and that
it was absolutely true that the state of the world was in one of those
subspaces. In the theory adopted here, nothing has absolute existence or truth
except the universal state vector. The subspaces $\mathcal{S}_{j}$ are only
defined _if_ the universal state space admits a tensor product decomposition
in which some factors have the structure of sentient systems (whatever that
may be); the focus on a particular one of those systems is arbitrary. (This is
somewhat like some other recently expressed interpretations, such as those
called “convivial solipsism” [24] and “minimalism” [16].) In Bell’s exposition
of Bohmian theory these subspaces are called “beable” [3] or “viable” ([18] p.
215).
Theorem 1 usually gives the probabilities for the first stage of a history,
but often is not helpful in subsequent stages since the condition that the
initial probabilities are given by the Born rule is not satisfied. To
illustrate this, we now give the calculation of history probabilities in a
version of the “Wigner’s friend” scenario considered by Brukner [6].
In this scenario an experimenter (“Wigner”) observes a closed laboratory
containing another experimenter (“Wigner’s friend”) who makes measurements on
a quantum system $S$. We take $S$ to be a qubit (e.g. an electron spin) with
orthogonal states $|1\rangle$ and $|2\rangle$, and we simplify Wigner and his
friend so that each of them has just three orthogonal states: a ready state
$|0\rangle$ and two states registering the result of a measurement:
$|+\rangle$ and $|-\rangle$ for Wigner, $|1\rangle$ and $|2\rangle$ for his
friend (let us call her Frieda). The scenario is as follows.
At $t=0$ the qubit $S$ is in a general state $c_{1}|1\rangle+c_{2}|2\rangle$
and both Wigner and Frieda are in their ready states. The universal state is
therefore
$|\Psi(0)\rangle=\big{(}c_{1}|1\rangle+c_{2}|2\rangle\big{)}_{S}|0\rangle_{F}|0\rangle_{W}.$
Between $t=0$ and $t=1$ Frieda measures $S$ in the basis
$\\{|1\rangle_{S},|2\rangle_{S}\\}$, so that $S$ and $F$ become entangled and
the universal state at $t=1$ is
$\displaystyle|\Psi(1)\rangle$
$\displaystyle=\big{(}c_{1}|1\rangle_{S}|1\rangle_{F}+c_{2}|2\rangle_{S}|2\rangle_{F}\big{)}|0\rangle_{W}$
$\displaystyle=\big{(}c_{1}|1\rangle_{SF}+c_{2}|2\rangle_{SF}\big{)}|0\rangle_{W}\quad\text{in
an obvious notation.}$
Between $t=1$ and $t=2$ Wigner measures the state of Frieda and her laboratory
in a basis which includes the states
$|\pm\rangle_{SF}=\frac{1}{\sqrt{2}}\big{(}|1\rangle\pm|2\rangle\big{)}_{SF},$
so that the universal state at $t=2$ is
$|\Psi(2)\rangle=\frac{1}{\sqrt{2}}(c_{1}+c_{2})|+\rangle_{SF}|+\rangle_{W}+\frac{1}{\sqrt{2}}(c_{1}-c_{2})|-\rangle_{SF}|-\rangle_{W}.$
Thus the sensible state of $F$ is $|0\rangle$ at $t=0$ and either $|1\rangle$
or $|2\rangle$ at both $t=1$ and $t=2$. For $F$, there are four possible
histories
$(0jk)=\big{(}(\Pi_{0},t=0),(\Pi_{j},t=1),(\Pi_{k},t=2)\big{)}$
with $j,k\in\\{1,2\\}$. The probabilities of transition from $|0\rangle$ at
$t=0$ are given by Theorem 1 since the probabilities at $t=0$, namely
$P_{0}=1,P_{1}=P_{2}=0$, are indeed given by the Born rule; hence the
probability that Frieda finds herself in state $|j\rangle$ at $t=1$ is
$|c_{j}|^{2}$. To find the probability that she experiences a change of state
between $t=1$ and $t=2$, however, we cannot use this theorem, since her
probability for the state $|j\rangle$ at $t=1$ is either $0$ or $1$, which is
not the probability given by the Born rule: her experience of her state is at
odds with the universal state. To calculate the probability of a change of
state we need to know the Hamiltonian governing the change in the universal
state in this time interval, when Wigner is measuring Frieda and her
laboratory. Let us assume that the measurement process takes place by a simple
rotation, so that the state in the time interval $1<t<2$ is given by
$|\Psi(1+\tau)\rangle=\frac{1}{\sqrt{2}}(c_{1}+c_{2})|+\rangle_{SF}\big{(}\cos\tfrac{1}{2}\pi\tau|0\rangle_{W}+\sin\tfrac{1}{2}\pi\tau|+\rangle_{W}\big{)}\\\
+\frac{1}{\sqrt{2}}(c_{1}-c_{2})|-\rangle_{SF}\big{(}\cos\tfrac{1}{2}\pi\tau|0\rangle_{W}+\sin\tfrac{1}{2}\pi\tau|-\rangle_{W}\big{)}$
and the components in Frieda’s sensible subspaces are
$\displaystyle\Pi_{1}|\Psi(1+\tau)\rangle$
$\displaystyle=|1\rangle_{SF}\Big{(}c_{1}\cos\tfrac{1}{2}\pi\tau|0\rangle_{W}+\sin\tfrac{1}{2}\pi\tau\big{[}\tfrac{1}{2}(c_{1}+c_{2})|+\rangle+\tfrac{1}{2}(c_{1}-c_{2})|-\rangle\big{]}_{W}\Big{)},$
$\displaystyle\Pi_{2}|\Psi(1+\tau)\rangle$
$\displaystyle=|2\rangle_{SF}\Big{(}c_{2}\cos\tfrac{1}{2}\pi\tau|0\rangle_{W}+\sin\tfrac{1}{2}\pi\tau\big{[}\tfrac{1}{2}(c_{1}+c_{2})|+\rangle-\tfrac{1}{2}(c_{1}-c_{2})|-\rangle\big{]}_{W}\Big{)}.$
The Hamiltonian driving this evolution is, taking $\hbar=1$,
$H=\frac{\mathrm{i}\pi}{2}\sum_{z=\pm}|z\rangle\langle
z|_{SF}\otimes\big{[}|z\rangle\langle 0|-|0\rangle\langle z|\big{]}_{W}.$
(5.2)
The matrix element relevant to the transition
$|1\rangle_{F}\rightarrow|2\rangle_{F}$ is
$\langle\Psi(1+\tau)|\Pi_{2}H\Pi_{1}|\Psi(1+\tau)\rangle=\frac{\mathrm{i}\pi}{2}\big{(}|c_{1}|^{2}-|c_{2}|^{2}\big{)}\cos\tfrac{1}{2}\pi\tau\sin\tfrac{1}{2}\pi\tau.$
(5.3)
Suppose $|c_{1}|^{2}\geq|c_{2}|^{2}$; then the transition rate for
$|1\rangle_{F}\rightarrow|2\rangle_{F}$ at time $1+\tau$ is, according to
(5.1),
$\displaystyle w_{12}(1+\tau)$
$\displaystyle=\frac{2\text{Re}[\langle\Psi(1+\tau)|\Pi_{2}H\Pi_{1}|\Psi(1+\tau)\rangle]}{\langle\Psi(t)|\Pi_{1}|\Psi(t)\rangle}$
$\displaystyle=\frac{\pi}{2}\big{(}|c_{1}|^{2}-|c_{2}|^{2}\big{)}\frac{\cos\tfrac{1}{2}\pi\tau\sin\tfrac{1}{2}\pi\tau}{|c_{1}|^{2}+\big{(}|c_{1}|^{2}-|c_{2}|^{2}\big{)}\sin^{2}\tfrac{1}{2}\pi\tau},$
while the rate for the opposite transition is $w_{21}=0$. Hence the
probabilities $P_{1}(1+\tau),P_{2}(1+\tau)$ for Frieda’s experience at time
$1+\tau$ satisfy
$\frac{\text{d}}{\text{d}\tau}P_{1}=-w_{12}P_{1},\quad\frac{\text{d}}{\text{d}\tau}P_{2}=w_{12}P_{1},$
the solution of which is
$P_{1}(1+\tau)=C\Big{[}|c_{1}|^{2}-\tfrac{1}{2}\big{(}|c_{1}|^{2}-|c_{2}|^{2}\big{)}\sin^{2}\tfrac{1}{2}\pi\tau\big{]},\quad
P_{2}=1-P_{1}$
where $C$ is a constant of integration. By taking $C=1/|c_{1}|^{2}$ or $C=0$,
corresponding to initial conditions $P_{1}(1)=1$ and $P_{1}(1)=0$
respectively, we get the conditional probabilities for Frieda’s experience:
$\displaystyle P\big{(}|1\rangle_{F}\text{ at
}t=2\;\big{|}\;|1\rangle_{F}\text{ at }t=1)$
$\displaystyle=\frac{1}{2|c_{1}|^{2}},$ (5.4) $\displaystyle
P\big{(}|1\rangle_{F}\text{ at }t=2\;\big{|}\;|2\rangle_{F}\text{ at }t=1)$
$\displaystyle=0,$ (5.5) $\displaystyle P\big{(}|2\rangle_{F}\text{ at
}t=2\;\big{|}\;|1\rangle_{F}\text{ at }t=1)$
$\displaystyle=\frac{|c_{1}|^{2}-|c_{2}|^{2}}{2|c_{1}|^{2}},$ (5.6)
$\displaystyle P\big{(}|2\rangle_{F}\text{ at
}t=2\;\big{|}\;|2\rangle_{F}\text{ at }t=1)$ $\displaystyle=1.$ (5.7)
These, together with the probabilities $|c_{1}|^{2}$ and $|c_{2}|^{2}$ for the
results of Frieda’s measurement at $t=1$, give the non-zero Bell probabilities
for her histories as set out in Table 1, which also shows the corresponding
Copenhagen probabilities.
$h$ | $P_{B}(h)$ | $P_{C}(h)$
---|---|---
(011) | $\tfrac{1}{2}$ | $\tfrac{1}{2}|c_{1}|^{2}$
(012) | $\tfrac{1}{2}\big{(}|c_{1}|^{2}-|c_{2}|^{2})$ | $\tfrac{1}{2}|c_{1}|^{2}$
(021) | $0$ | $\tfrac{1}{2}|c_{2}|^{2}$
(022) | $|c_{2}|^{2}$ | $\tfrac{1}{2}|c_{2}|^{2}$
Table 1: Probabilities for the histories of Wigner’s friend in Brukner’s
scenario
Bell and Copenhagen agree that the probability that Frieda is in state
$|1\rangle$ at $t=1$ is $|c_{1}|^{2}$, while the probability that she is in
$|2\rangle$ is $|c_{2}|^{2}$, so she is more likely to be in the state
$|1\rangle$, while at $t=2$ she is equally likely to be in either state.
According to Bell, this change is caused entirely by a transition from
$|1\rangle$ to $|2\rangle$, so she will only feel the effect of Wigner’s
measurement if she is in the more probable state $|1\rangle$; but according to
Copenhagen there is a flow of probability in both directions.
It is illuminating to examine the two sets of probabilities in the special
case $c_{1}=c_{2}=1/\sqrt{2}$. for which Bell and Copenhagen agree that the
two states $|1\rangle_{F}$ and $|2\rangle_{F}$ each have probability
$\tfrac{1}{2}$ at both $t=1$ and $t=2$. According to Bell there are no
transitions between these two states in the time period $1\leq t\leq 2$,
whereas Copenhagen gives equiprobable transitions in both directions. For
these values of the coefficients Frieda and her laboratory are already in the
state $|+\rangle_{SF}$ at $t=1$, so Wigner’s measurement simply registers this
fact, changing Wigner’s state from $|0\rangle_{W}$ to $|+\rangle_{W}$ but
leaving Frieda’s state unchanged. This is reflected in the fact that the
Hamiltonian (5.2) has zero matrix elements (5.3) between the two components
$\Pi_{i}|\Psi(1+\tau)\rangle$ in which Frieda has a definite value $i=1$ or
$2$. The Copenhagen calculation of probabilities, however, assumes that
Frieda’s state collapses at $t=1$, despite the lack of physical mechanism for
this collapse; if this happens then Wigner’s measurement does have an effect
on the resulting states, leading to transitions between them.
In the general case the Bell calculation gives the minimal transitions
necessary for the change in probabilities from $t=1$ to $t=2$.
## 6 The Frauchiger-Renner universe
A more elaborate variant of the Wigner’s friend thought experiment has been
devised by Frauchiger and Renner [10]. It is instructive to compare the
different probabilities for this scenario. This takes place in a universe
consisting of six subsystems $C,S,F_{1},F_{2},W_{1}$ and $W_{2}$. Of these $C$
is a quantum coin with two orthogonal states $|\text{head}\rangle_{C}$ and
$|\text{tail}\rangle_{C}$; $S$ is a qubit with orthogonal states
$|\uparrow\rangle_{S}$ and $|\downarrow\rangle_{S}$; and the four systems
$F_{1},F_{2},W_{1},W_{2}$ are sentient agents, each agent $X$ having three
orthogonal states $|0\rangle_{X},|a\rangle_{X},|b\rangle_{X}$ where
$|0\rangle$ is a state of readiness before $X$ makes a measurement and
$|a\rangle$ and $|b\rangle$ are states recording the results $a,b$ of the
measurement. We refer to the agents $F_{1}$ and $F_{2}$ as “experimenters”,
since they perform operations on the physical systems $C$ and $S$, and to
$W_{1}$ and $W_{2}$ as “para-experimenters” since they operate on the
experimenters $F_{1}$ and $F_{2}$ as well as $C$ and $S$. The experiment
proceeds as follows.
Initially, at time $t=0$, the system is in the state
$|\Psi(0)\rangle=\left(\sqrt{\tfrac{1}{3}}|\text{head}\rangle+\sqrt{\tfrac{2}{3}}|\text{tail}\rangle\right)_{C}|0\rangle_{S}|0\rangle_{F_{1}}|0\rangle_{F_{2}}|0\rangle_{W_{1}}|0\rangle_{W_{2}}.$
(6.1)
Between times $t=0$ and $t=1$ the experimenter $F_{1}$ measures the state of
the coin, so that at time $t=1$ the universal state is
$|\Psi(1)\rangle=\left(\sqrt{\tfrac{1}{3}}|\text{head}\rangle+\sqrt{\tfrac{2}{3}}|\text{tail}\rangle\right)_{F_{1}C}|0\rangle_{S}|0\rangle_{F_{2}}|0\rangle_{W_{1}}|0\rangle_{W_{2}},$
(6.2)
where
$|\text{head}\rangle_{F_{1}C}=|\text{head}\rangle_{F_{1}}|\text{head}\rangle_{C}$
and similarly for “tail”.
Between $t=1$ and $t=2$ the experimenter $F_{1}$ prepares the qubit $S$ in a
way which depends on the result of their measurement of $C$: if the coin came
down showing “head”, $F_{1}$ prepares $|\downarrow\rangle_{S}$; if it showed
“tail”, $F_{1}$ prepares
$|\rightarrow\rangle_{S}=\tfrac{1}{\sqrt{2}}\big{(}|\uparrow\rangle+|\downarrow\rangle\big{)}_{S}$.
The universal state at $t=2$ is therefore
$|\Psi(2)\rangle=\left(\sqrt{\tfrac{1}{3}}|\text{head}\rangle_{F_{1}C}|\downarrow\rangle_{S}+\sqrt{\tfrac{2}{3}}|\text{tail}\rangle_{F_{1}C}|\rightarrow\rangle_{S}\right)|0\rangle_{F_{2}}|0\rangle_{W_{1}}|0\rangle_{W_{2}}.$
(6.3)
Between $t=2$ and $t=3$ the qubit $S$ is passed to the second experimenter
$F_{2}$, who measures it in the basis
$\\{|\uparrow\rangle,|\downarrow\rangle\\}$, recording the result $+,-$
respectively; we will use the same symbol $\pm$ to denote the joint state of
$F_{2}$ and $S$. The universal state becomes
$|\Psi(3)\rangle=\sqrt{\tfrac{1}{3}}\Big{(}|\text{head}\rangle_{F_{1}C}|-\rangle_{F_{2}S}+|\text{tail}\rangle_{F_{1}C}|+\rangle_{F_{2}S}+|\text{tail}\rangle_{F_{1}C}|-\rangle_{F_{2}S}\Big{)}|0\rangle_{W_{1}}|0\rangle_{W_{2}}.$
(6.4)
Between $t=3$ and $t=4$ the first para-experimenter $W_{1}$ measures the state
of $F_{1}$ and the coin $C$ in the basis
$\displaystyle|\text{ok}\rangle_{F_{1}C}$
$\displaystyle=\tfrac{1}{\sqrt{2}}\big{(}|\text{head}\rangle_{F_{1}C}-|\text{tail}\rangle_{F_{1}C}\big{)},$
$\displaystyle|\text{fail}\rangle_{F_{1}C}$
$\displaystyle=\tfrac{1}{\sqrt{2}}\big{(}|\text{head}\rangle_{F_{1}C}+|\text{tail}\rangle_{F_{1}}C\big{)},$
recording the result in their own state $|\text{ok}\rangle_{W_{1}}$ or
$|\text{fail}\rangle_{W_{1}}$. The universal state becomes, at $t=4$,
$|\Psi(4)\rangle=\Big{(}\sqrt{\tfrac{1}{6}}\big{[}-|\text{ok}\rangle_{F_{1}C}|\text{ok}\rangle_{W_{1}}+|\text{fail}\rangle_{F_{1}C}|\text{fail}\rangle_{W_{1}}\big{]}|+\rangle_{F_{2}S}\\\
+\sqrt{\tfrac{2}{3}}|\text{fail}\rangle_{F_{1}C}|\text{fail}\rangle_{W_{1}}|-\rangle_{F_{2}S}\Big{)}|0\rangle_{W_{2}}.$
Between $t=4$ and $t=5$ the second para-experimenter $W_{2}$ measures the
state of $F_{2}$ and the qubit $S$ in the basis
$\displaystyle|\text{ok}\rangle_{F_{2}S}$
$\displaystyle=\tfrac{1}{\sqrt{2}}\big{(}|-\rangle_{F_{2}S}-|+\rangle_{F_{2}S}\big{)},$
$\displaystyle|\text{fail}\rangle_{F_{2}S}$
$\displaystyle=\tfrac{1}{\sqrt{2}}\big{(}|-\rangle_{F_{2}S}+|+\rangle_{F_{2}S}\big{)},$
recording the result in their own state $|\text{ok}\rangle_{W_{2}}$ or
$|\text{fail}\rangle_{W_{2}}$. The universal state becomes, at $t=5$,
$|\Psi(5)\rangle=\sqrt{\tfrac{1}{12}}\Big{(}|\text{ok}\rangle_{F_{1}C}|\text{ok}\rangle_{W_{1}}+|\text{fail}\rangle_{F_{1}C}|\text{fail}\rangle_{W_{1}}\Big{)}|\text{ok}\rangle_{F_{2}S}|\text{ok}\rangle_{W_{2}}\\\
+\sqrt{\tfrac{1}{12}}\Big{(}-|\text{ok}\rangle_{F_{1}C}|\text{ok}\rangle_{W_{1}}+3|\text{fail}\rangle_{F_{1}C}|\text{fail}\rangle_{W_{1}}\Big{)}|\text{fail}\rangle_{F_{2}S}|\text{fail}\rangle_{W_{2}}.$
Now consider the history of this experiment as seen by the experimenter
$F_{2}$ from $t=1$ to $t=5$. The possible histories are
$(abcde)=h(a,b,c,d,e)=\big{(}(\Pi_{a},t=1),\ldots,(\Pi_{e},t=5)\big{)}$
where each of $a,b,c,d,e$ is taken from the set $\\{0,+,-\\}$, and $\Pi_{x}$
is the projector onto the 32-dimensional subspace
$\mathcal{S}_{x}=\mathcal{H}_{C}\otimes\mathcal{H}_{S}\otimes\mathcal{H}_{F_{1}}\otimes\text{span}\left(|x\rangle_{F_{2}}\right)\otimes\mathcal{H}_{W_{1}}\otimes\mathcal{H}_{W_{2}}.$
The Copenhagen probabilities $P_{\text{C}}(abcde)$ can be calculated in the
same way as in Section 3. The result is that the only non-zero probabilities
are
$P_{\text{C}}(00+++)=\tfrac{1}{6},\quad P_{\text{C}}(00--+)=\tfrac{1}{3},\quad
P_{\text{C}}(00++-)=\tfrac{1}{6},\quad P_{\text{C}}(00---)=\tfrac{1}{3}.$
To calculate the Bell probabilities, we need to know the Hamiltonian governing
the evolution of the system, i.e. the succession of measurements made by the
four agents. In each time interval $[n,n+1]$ ($n=1,\ldots,4$) the system
evolves from a product state to an entangled state of the experimenter and the
measured system. The Bell probabilities can be calculated, as in Section 5, by
assuming that the Hamiltonian in this interval is a generator of rotations in
the subspace spanned by these two states, operating for a time such that
$\mathrm{e}^{-\mathrm{i}H}$ is a rotation through $\pi/2$. The resulting non-
zero probabilities are
$P_{\text{B}}(00+++)=\tfrac{1}{3},\quad P_{\text{B}}(00--+)=\tfrac{1}{6},\quad
P_{\text{C}}(00---)=\tfrac{1}{2}.$
These Bell probabilities are set out beside the Copenhagen probabilities in
Table 2.
$abcde$ | $P_{\text{B}}(abcde)$ | $P_{\text{C}}(abcde)$
---|---|---
($00+++$) | $\tfrac{1}{3}$ | $\tfrac{1}{6}$
($00++-$) | $0$ | $\tfrac{1}{6}$
($00--+$) | $\tfrac{1}{6}$ | $\tfrac{1}{3}$
($00---$) | $\tfrac{1}{2}$ | $\tfrac{1}{3}$
Table 2: Probabilities for the histories of $F_{2}$ in the Frauchiger-Renner
experiment
The difference between the Bell and Copenhagen probabilities in Table 2 is
similar to what we have already seen in Table 1 for Brukner’s version of the
Wigner’s friend experiment.
To be relevant to the conclusion that Frauchiger and Renner drew from their
thought experiment, we must extend our concept of a sentient observer. The
Frauchiger-Renner argument assumes communication between $F_{1}$, $W_{1}$ and
$W_{2}$ and therefore requires that they should be treated as a single
observer whose experience covers the measurement results of all three. The
histories of this three-headed observer from $t=0$ to $t=5$ are of the form
$h(a\,a^{\prime}\,a^{\prime\prime},\ldots,f\,f^{\prime}\,f^{\prime\prime})=\big{(}(a,a^{\prime},a^{\prime\prime};t=0),\ldots,(f,f^{\prime},f^{\prime\prime};t=5)\big{)}$
where
$a,\ldots,f\in\\{0,\text{head},\text{tail}\\},\,a^{\prime}\ldots,f^{\prime}\in\\{0,\text{ok},\text{fail}\\}$
and
$a^{\prime\prime},\ldots,f^{\prime\prime}\in\\{0,\text{ok},\text{fail}\\}$.
However, the Bell and Copenhagen probabilities are both zero for any history
in which there is any change in the experience of $F_{1}$ between $t=1$ and
$t=3$; the relevant projection operators commute with the evolution operators
$\mathrm{e}^{-\mathrm{i}H(t-1)}$ for $1<t<3$ since all the action in this
interval involves only $S$ and $F_{2}$. We therefore only need to consider
histories
$h(a\,a^{\prime}\,a^{\prime\prime},\;b\,b^{\prime}\,b^{\prime\prime},\;c\,c^{\prime}\,c^{\prime\prime},\;d\,d^{\prime}\,d^{\prime\prime})\\\
=\big{(}(a,a^{\prime},a^{\prime\prime};t=0),(b,b^{\prime},b^{\prime\prime};t=1),(c,c^{\prime},c^{\prime\prime};t=4),(d,d^{\prime},d^{\prime\prime};t=5)\big{)}$
The two sets of probabilities for these histories are shown in Table 3, in
which “head”, “tail” and “fail” are abbreviated to their initial letters. All
histories not shown in this table have $P_{\text{B}}=P_{\text{C}}=0$.
$h$ | $P_{\text{B}}(h)$ | $P_{\text{C}}(h)$
---|---|---
$(0\,0\,0,\;\text{h}\,0\,0,\;\text{h}\,\text{f}\,0,\;\text{h}\,\text{f}\,\text{f})$ | $\tfrac{3}{20}$ | $\tfrac{1}{24}$
$(0\,0\,0,\;\text{h}\,0\,0,\;\text{h}\,\text{f}\,0,\;\text{h}\,\text{f}\,\,\text{ok})$ | $\tfrac{1}{60}$ | $\tfrac{1}{24}$
$(0\,0\,0,\;\text{h}\,0\,0,\;\text{h}\,\text{ok}\,0,\;\text{h}\,\text{f}\,\text{ok})$ | $0$ | $\tfrac{1}{24}$
$(0\,0\,0,\;\text{h}\,0\,0,\;\text{h}\,\text{ok}\,0,\;\text{h}\,\text{ok}\,\text{ok})$ | $0$ | $\tfrac{1}{24}$
$(0\,0\,0,\;\text{h}\,0\,0,\;\text{t}\,\text{f}\,0,\;\text{t}\,\text{f}\,0$ | $\tfrac{3}{20}$ | $\tfrac{1}{24}$
$(0\,0\,0.\;\text{h}\,0\,0,\;\text{t}\,\text{f}\,0,\;\text{t}\,\text{f}\,\text{ok})$ | $\tfrac{1}{60}$ | $\tfrac{1}{24}$
$(0\,0\,0,\;\text{h}\,0\,0,\;\text{t}\,\text{ok}\,0,\;\text{t}\,\text{ok}\,\text{f})$ | $0$ | $\tfrac{1}{24}$
$(0\,0\,0,\;\text{h}\,0\,0,\;\text{t}\,\text{ok}\,0,\;\text{t}\,\text{ok}\,\text{ok})$ | $0$ | $\tfrac{1}{24}$
$(0\,0\,0,\;\text{t}\,0\,0,\;\text{h}\,\text{f}\,0,\;\text{h}\,\text{f}\,\text{f})$ | $\tfrac{9}{40}$ | $\tfrac{1}{6}$
$(0\,0\,0,\;\text{t}\,0\,0,\;\text{h}\,\text{f}\,0,\;\text{h}\,\text{f}\,\,\text{ok})$ | $\tfrac{1}{40}$ | $0$
$(0\,0\,0,\;\text{t}\,0\,0,\;\text{h}\,\text{ok}\,0,\;\text{h}\,\text{ok}\,\text{f})$ | $\tfrac{1}{24}$ | $\tfrac{1}{6}$
$(0\,0\,0,\;\text{t}\,0\,0,\;\text{h}\,\text{ok}\,0,\;\text{h}\,\text{ok}\,\text{ok})$ | $\tfrac{1}{24}$ | $0$
$(0\,0\,0,\;\text{t}\,0\,0,\;\text{t}\,\text{f}\,0,\;\text{t}\,\text{f}\,\text{f})$ | $\tfrac{9}{40}$ | $\tfrac{1}{6}$
$(0\,0\,0,\;\text{t}\,0\,0,\;\text{t}\,\text{f}\,0,\;\text{t}\,\text{f}\,\,\text{ok})$ | $\tfrac{1}{40}$ | $0$
$(0\,0\,0,\;\text{t}\,0\,0,\;\text{t}\,\text{ok}\,0,\;\text{t}\,\text{ok}\,\text{f})$ | $\tfrac{1}{24}$ | $\tfrac{1}{6}$
$(0\,0\,0,\;\text{t}\,0\,0,\;\text{t}\,\text{ok}\,0,\;\text{t}\,\text{ok}\,\text{ok})$ | $\tfrac{1}{24}$ | $0$
Table 3: Probabilities for the histories of $F_{1}$, $W_{1}$ and $W_{2}$ in
the Frauchiger-Renner experiment
As before, there are pairs of histories which are equally probable in the
Copenhagen calculation, while Bell assigns zero probability to one of them.
But now there are also histories which have $P_{\text{C}}=0$ but
$P_{\text{B}}\neq 0$, in particular those in which $F_{1}$ observes “tails”
when she measures the coin at $t=2$ and $W_{1}$ and $W_{2}$ both obtain the
result “ok” in their measurements of $F_{1}$ and $F_{2}$. This difference is
the core of the Frauchiger-Renner argument, and the substance of their claim
of a problem in the description of this thought experiment by standard quantum
mechanics. The zero probability in the Copenhagen calculation corresponds to
$F_{1}$ applying the projection postulate to the universal state after seeing
the result of her measurement. The non-zero probability in the Bell
calculation arises if $F_{1}$ applies Vedral’s dictum “Unobserved results can
affect future measurements” [22]. Knowing how the experiment is proceeding,
she knows that what she observes is registered in just one component of the
universal state. The whole of the universal state continues to evolve and
affects the future measurements, as is explicitly included in Bell’s process:
the transition probabilities at any time depend on the value of the universal
state at that time. The agents $F_{1}$, $W_{1}$ and $W_{2}$ are entitled to
regard their experience as “reality”; then in their perspective the universal
state vector is not a description of reality but an influence, or force, that
affects the development of reality.
We note that the history states $|K(h)\rangle$, as defined in (3.3), for the
histories $h$ in table 3 are not all orthogonal; in fact there are four cases
in which two different histories give rise to the same history state, for
example
$\displaystyle|K(0\,0\,0,\;\text{h}\,0\,0,\;\text{h}\,\text{f}\,0,\;\text{h}\,\text{f}\,\text{f})\rangle$
$\displaystyle=|K(0\,0\,0,\;\text{t}\,0\,0,\;\text{t}\,\text{f}\,0,\;\text{t}\,\text{f}\,\text{f})\rangle$
$\displaystyle=\frac{1}{\sqrt{6}}|\text{h}\rangle_{F_{1}C}|\text{f}\rangle_{F_{2}S}|\text{f}\rangle_{W_{1}}|\text{f}\rangle_{W_{2}}.$
It follows, as has been pointed out before [14], that the histories in Table 3
do not satisfy the consistency condition of [12]. This does not mean that
there is anything inconsistent in these histories; it only means that the
formula (3.1) cannot consistently be applied to disjunctions of them. This can
be seen, for example, by using (3.1) to calculate the Copenhagen probability
of a three-time history with times $t=0$, $t=4$ and $t=5$; the result is not
the sum of the corresponding histories in Table 3 — as is to be expected in
standard quantum mechanics when one ignores the fact that a measurement has
taken place at $t=1$. But if the histories are considered as atoms in a
probability space111Confusingly, this would normally be called an “event
space.”, the usual (Kolmogorov) rules can be applied.
## 7 Observation of History
If an agent is equipped with a memory which records their momentary experience
at each stage of a history, they can reduce their experience of history to a
single measurement at the end of the history. Let us return to $F_{2}$ and
suppose she has a memory with a basis $|x_{1}x_{2}x_{3}x_{4}x_{5}\rangle_{M}$
where $x_{t}$ can take values $0,+,-$ or $\times$, the final value $\times$
being recorded at times when $t$ lies in the future. Then the universal state
space has an extra tensor factor $\mathcal{H}_{M}$. We denote the universal
state vector in this enlarged space by $|\widetilde{\Psi}(t)\rangle$, with
$|\Psi(t)\rangle$ continuing to denote the universal state in Section 6. Then
the state with memory is of the form
$|\widetilde{\Psi}(t)\rangle=\sum_{a\in\\{0,+,-\\}}|a\rangle_{F_{2}}|x_{1}x_{2}x_{3}x_{4}x_{5}\rangle_{M}|\psi(t)\rangle\\\
\text{with
}|\psi(t)\rangle\in\mathcal{H}_{S}\otimes\mathcal{H}_{C}\otimes\mathcal{H}_{F_{1}}\otimes\mathcal{H}_{W_{1}}\otimes\mathcal{H}_{W_{2}}$
where $x_{t}=a$ and $x_{s}=\times$ if $s>t$. This develops as follows:
$\displaystyle|\widetilde{\Psi}(0)\rangle$
$\displaystyle=|\Psi(0)\rangle|0\times\times\times\times\rangle_{M},$
$\displaystyle|\widetilde{\Psi}(1)\rangle$
$\displaystyle=|\Psi(1)\rangle|00\times\times\times\rangle_{M},$
$\displaystyle|\widetilde{\Psi}(2)\rangle$
$\displaystyle=\Big{[}\sqrt{\tfrac{1}{6}}\big{(}|\text{fail}\rangle-|\text{ok}\rangle\big{)}_{F_{1}C}|+\rangle_{F_{2}S}|00+\times\times\rangle_{M}$
$\displaystyle\quad+\sqrt{\tfrac{2}{3}}|\text{fail}\rangle_{F_{1}C}|-\rangle_{F_{2}S}|00-\times\times\rangle_{M}\Big{]}|0\rangle_{W_{1}}|0\rangle_{W_{2}},$
$\displaystyle|\widetilde{\Psi}(3)\rangle$
$\displaystyle=\sqrt{\tfrac{1}{6}}|00++\times\rangle\Big{(}|\text{fail}\rangle_{F_{1}C}|\text{fail}\rangle_{W_{1}}-|\text{ok}\rangle_{F_{1}C}|\text{ok}\rangle_{W_{1}}\Big{)}|+\rangle_{F_{2}}|0\rangle_{W_{2}}$
$\displaystyle\quad+\sqrt{\tfrac{2}{3}}|00--\times\rangle|\text{fail}\rangle_{F_{1}C}|\text{fail}\rangle_{W_{1}}|-\rangle_{F_{2}}|0\rangle_{W_{2}}$
$\displaystyle|\widetilde{\Psi}(4)\rangle$
$\displaystyle=\tfrac{1}{\sqrt{24}}|00+++\rangle_{M}\left(|\text{fail}\rangle_{F_{1}C}|\text{fail}\rangle_{W_{1}}-|\text{ok}\rangle_{F_{1}C}|\text{ok}\rangle_{W_{1}}\right)|+\rangle_{F_{2}S}\big{(}|\text{fail}\rangle+|\text{ok}\rangle\big{)}_{W_{2}}$
$\displaystyle+\tfrac{1}{\sqrt{24}}|00++-\rangle_{M}\left(|\text{fail}\rangle_{F_{1}C}|\text{fail}\rangle_{W_{1}}-|\text{ok}\rangle_{F_{1}C}|\text{ok}\rangle_{W_{1}}\right)|+\rangle_{F_{2}S}\big{(}|\text{fail}\rangle-|\text{ok}\rangle\big{)}_{W_{2}}$
$\displaystyle+\tfrac{1}{\sqrt{6}}|00--+\rangle_{M}|\text{fail}\rangle_{F_{1}C}|\text{fail}\rangle_{W_{1}}|+\rangle_{F_{2}S}\big{(}|\text{fail}\rangle-|\text{ok}\rangle\big{)}_{W_{2}}$
$\displaystyle+\tfrac{1}{\sqrt{6}}|00---\rangle_{M}|\text{fail}\rangle_{F_{1}C}|\text{fail}\rangle_{W_{1}}|-\rangle_{F_{2}S}\big{(}|\text{fail}\rangle+|\text{ok}\rangle\big{)}_{W_{2}}.$
From this we can read off the probabilities of the memory readings as
$P(00+++)=P(00++-)=\tfrac{1}{6},\quad P(00--+)=P(00---)=\tfrac{1}{3},$
the same as the Copenhagen probabilities in Table 1. The reason for this is
clear: the entanglement of the original system with the memory serves to
separate the sensible states of $F_{2}$, so that the time development of each
such state proceeds as if the other states had been annihilated by a
projection operator. This prevents subsequent interference between these
components: entanglement implements collapse.
This calculation illustrates a general fact: if a system is linked to a memory
which keeps a permanent record of a set $B$ of basis states of the system,
then the probabilities which will be observed in the memory are the same as
those which would be calculated in a Copenhagen calculation assuming that the
system (without the memory) undergoes collapse at each time stage onto a state
of the basis $B$. Thus a direct attempt to establish the probabilities of
various histories of the system will lead to the Copenhagen probabilities.
This might be seen as a justification of the probabilities assumed in the
“consistent histories” version of quantum mechanics. However, this does not
rule out the possibility of indirect effects of Bell probabilities if the
system is not coupled to a memory, in a similar way to the quantum effects of
superposition which are not visible if the system undergoes measurement.
Moreover, this justification of Copenhagen probabilities is only valid if the
system whose histories are in question is not the whole universe; it depends
on there being something (the memory) external to the system. This undercuts
the claim of consistent-histories theory to be a version of quantum mechanics
which is specially adapted to cosmology.
## 8 Conclusion
We have explored an alternative theory of histories — sequences of events,
such as measurements — in quantum mechanics. These have usually been treated
by assuming the Dirac/von Neumann/Lüders projection postulate of the
Copenhagen formulation of quantum mechanics. We have shown that this is not
necessary; there is an alternative formulation in which this postulate is not
assumed, but histories can still be treated. We have shown that this gives
different results for the probability of a history. However, this difference
is not directly observable: a simple empirical procedure for observing
histories is bound to give the same results as if the projection postulate was
valid. Nevertheless, this means that any theory that assumes this postulate,
such as the “consistent histories” interpretation of quantum mechanics, cannot
be a theory of the whole universe and cannot be a basis for cosmology.
Acknowledgement
I am grateful to Flavio del Santo and his co-workers for a valuable
discussion, and to an anonymous referee for suggestions which have greatly
improved the presentation of these ideas.
## References
* [1] G. Bacciagaluppi and M. Dickson. Dynamics for density operator interpretations of quantum theory. Found. Phys. 29:1165, 1999. arXiv:quant-ph/9711048
* [2] Veronika Baumann, Flavio Del Santo, Alexander R. H. Smith, Flaminia Giacomini, Esteban Castro-Ruiz, and Caslav Brukner. Generalized probability rules from a timeless formulation of Wigner’s friend scenarios. 2019\. arXiv:1911.09696.
* [3] J. S. Bell. Beables for quantum field theory. In B. J. Hiley and F. D. Peat, editors, Quantum Implications, pages 227–234. Routledge and Kegan Paul, London, 1984. Reprinted in [4].
* [4] J. S. Bell. Speakable and unspeakable in quantum mechanics. Cambridge University Press, 1987.
* [5] J. S. Bell. Against “measurement”. Physics World, 3(8):33–40, August 1990.
* [6] C. Brukner. On the quantum measurement problem. In Quantum [Un]Speakables II: 50 Years of Bell’s Theorem, pages 95–117. Springer, 2017.
* [7] Andrea Di Biagio, Pietro Donà and Carlo Rovelli. The arrow of time in operational formulations of quantum theory. 2021\. arXiv:2010.05734.
* [8] Carl E. Dolby. The conditional probability interpretation of the Hamiltonian constraint. 2004\. arXiv:gr-qc.0406034.
* [9] Hugh Everett III. “Relative state” formulation of quantum mechanics. Rev. Mod. Phys., 29:141–153, 1957.
* [10] Daniela Frauchiger and Renato Renner. Quantum theory cannot consistently describe the use of itself. Nature Communications, 9:3711, 2018.
* [11] Vittorio Giovannetti, Seth Lloyd, and Lorenzo Maccone. Quantum time. Phys. Rev. D, 92:045033, 2015. arXiv:1504.04215.
* [12] R. B. Griffiths. Consistent Quantum Theory. Cambridge University Press, 2002.
* [13] Timothy J. Hollowood. Classical from quantum. 2018\. arXiv:1803.04700
* [14] Marcelo Losada, Roberto Laura, and Olimpia Lombardi. Frauchiger-Renner argument and quantum histories. Physical Review A, 100:052114–052119, 2019. arXiv:1907.10095.
* [15] Carlo Rovelli. Relational quantum mechanics. Int. J. Theor. Phys., 35:1637–1677, 1996.
* [16] D. Sokolovski. A minimalist’s view of quantum mechanics. Euro. Phys. Lett., 128:50001, 2019. arXiv:2005.12674.
* [17] A. Sudbery. Diese verdammte Quantenspringerei. Stud. Hist. Phil. Mod. Phys., 33B:387–411, 2002. arXiv:quant-ph/0011082.
* [18] Anthony Sudbery. Quantum Mechanics and the Particles of Nature. Cambridge University Press, 1986.
* [19] Anthony Sudbery. The logic of the future in quantum theory. Synthese, 194:4429–4453, 2017. arXiv:1409.0755.
* [20] Anthony Sudbery. Single-world theory of the extended Wigner’s friend experiment. Found. Phys., 47:658-669, 2017. arXiv:1608.05873.
* [21] Anthony Sudbery. John Bell and the great enterprise. Quanta, 7:68–73, 2018. arXiv:1808.06845.
* [22] Vlatko Vedral. Local quantum reality. 2020\. arXiv:2011.01039.
* [23] J. A. Wheeler. Assessment of Everett’s “relative state” formulation of quantum theory. Rev. Mod. Phys., 29:463–465, 1957.
* [24] Hervé Zwirn. Non locality versus modified realism. Found. Phys., 50:1–26, 2020. arXiv:1812.06451.
|
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>to party $\mathsf{P}_{i}$.
* •
Receive the teleportation measurement result $(z_{i},x_{i})$ from party
$\mathsf{P}_{i}$.
* •
Extract the input
$\rho^{\prime}_{i}\leftarrow\mathsf{TP.Receive}(z_{i},x_{i},\tilde{e}_{R,i})$
and send it to $\mathsf{T}$.
3. 3.
Output recovery:
* •
Upon receiving $\sigma_{j}$ from $\mathsf{T}$, teleport $\sigma_{j}$ into the
first component $\hat{e}_{S,j}^{(0)}$.
* •
Obtain
$(z^{\prime}_{j},x^{\prime}_{j})\leftarrow\mathsf{TP.Send}(\sigma_{j},\hat{e}_{S,j}^{(0)})$.
* •
Send
$P^{\prime\prime}_{j}=\hat{P}_{j}(X^{x^{\prime}_{j}}Z^{z^{\prime}_{j}}\otimes\mathbbm{1}^{(\ell-\ell^{\prime})(1+t)})$
to $\mathsf{P}_{j}$.
4. 4.
For each measurement step of $C$, do the following (with $l$ from $\ell$ to
$\ell^{\prime}+1$):
* •
Send to $\mathsf{P}_{j}$ the gate $\hat{E}_{j}^{(l)}$.
* •
Receive the measurement result $r_{j}\in\\{0,1\\}^{1+t}$ from party
$\mathsf{P}_{j}$.
* •
Measure $\hat{e}_{S,j}^{(l)}$ and compare the result
$\tilde{r}_{j}=\mathsf{\Pi}(\hat{e}_{S,j}^{(l)})$ with $r_{j}$.
* •
If $\tilde{r}_{j}\neq r_{j}$, let $\mathsf{P}_{j}$ send abort to $\mathsf{T}$.
Output: the final output of ${\mathcal{A}}$.
We prove by induction on the number of circuit steps $|C|$ that the real
protocol can be simulated by the simulator $\mathsf{Sim}^{\mathsf{RQC}}$. We
also prove in the induction that the end state of both the real and ideal
world when $\mathsf{P}_{j}$ is not being marked as corrupted are
indistinguishable from a state where ${\mathsf{cMPC}}$ holds random Clifford
keys $\\{E_{j}\\}_{j\in[n]}$ and the adversary holds
$\displaystyle A^{\Delta}\big{(}\underset{i\in[n]}{\cup}\rho_{i}\text{
};\text{
}{(\underset{j\in[n]}{\otimes}E_{j})({\mathsf{QECC.Enc}}(C(\cdot,\phi_{h})),0^{lnt})}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}R_{j}}},{\rho_{{\mathsf{aux}}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}W}}\big{)}$
(9)
for some map $A$. For the base case $C=\mathbbm{1}$, the protocol is equal to
$(\Sigma^{\mathsf{IE}},\Pi^{\mathsf{IE}})$. From the analysis of the previous
section, it remains to show that $\mathsf{Sim}^{\mathsf{RQC}}(\mathbbm{1})$
ends in state (7). Using notation (5), the state at the end of simulator step
$2$ is
$\underset{\Delta}{\Sigma}\text{
}\big{(}{\ket{z_{i}}\\!\bra{z_{i}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}I_{i}}},{\ket{x_{i}}\\!\bra{x_{i}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}S_{i}}},{\underset{i\in[n]}{\cup}\rho_{i}^{\Delta}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}R^{\prime}_{i}}},{U_{P_{I_{i}},P_{S_{i}}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}R_{j},W}}({\hat{P}_{j}\cdot\hat{e}_{R,j}^{(0)}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}R_{j}}},{\rho_{{\mathsf{aux}}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}W}}){U_{P^{\prime}_{I_{i}},P_{S^{\prime}_{i}}}^{\dagger}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}R_{j},W}}\big{)}$
After step $3$, the values in registers
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>is correctly QECC-encoded, securely authenticated, and being teleported. The
end state of the simulation becomes
$\displaystyle A^{\Delta}\big{(}\underset{i\in[n]}{\cup}\rho_{i}\text{
};\text{
}{(\underset{j\in[n]}{\otimes}\hat{P}_{j}X^{x^{\prime}_{j}}Z^{z^{\prime}_{j}}E_{j})({\mathsf{QECC.Enc}}(\cdot,\phi_{h}),0^{lnt})}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}R_{j}}},{\rho_{{\mathsf{aux}}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}W}}\big{)}$
$\displaystyle=$ $\displaystyle
A^{\Delta}\big{(}\underset{i\in[n]}{\cup}\rho_{i}\text{ };\text{
}{(\underset{j\in[n]}{\otimes}P^{\prime\prime}_{j}E_{j})({\mathsf{QECC.Enc}}(\cdot,\phi_{h}),0^{lnt})}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}R_{j}}},{\rho_{{\mathsf{aux}}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}W}}\big{)}$
which is equal to (7). (When $C=\mathbbm{1}$, the simulator does nothing in
step $4$.)
Suppose the parties are going to run a further computation step under our
induction hypothesis. For the protocol of performing a Clifford gate $G$,
${\mathsf{cMPC}}$ updates the keys to
$\\{E^{\prime}_{j}=E_{j}(G\otimes\mathbbm{1}^{lt})^{\dagger}\\}_{j\in[n]}$.
These keys are still random Clifford keys. We can rewrite the state of the
real-protocol adversary in terms of $\\{E^{\prime}_{j}\\}_{j\in[n]}$ as
$\displaystyle A^{\Delta}\big{(}\underset{i\in[n]}{\cup}\rho_{i}\text{
};\text{
}{(\underset{j\in[n]}{\otimes}E^{\prime}_{j}(G\otimes\mathbbm{1}^{lt}))({\mathsf{QECC.Enc}}(C(\cdot,\phi_{h})),0^{lnt})}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}R_{j}}},{\rho_{{\mathsf{aux}}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}W}}\big{)}$
(10) $\displaystyle=$ $\displaystyle
A^{\Delta}\big{(}\underset{i\in[n]}{\cup}\rho_{i}\text{ };\text{
}{(\underset{j\in[n]}{\otimes}E^{\prime}_{j})({\mathsf{QECC.Enc}}(G\circ
C(\cdot,\phi_{h})),0^{lnt})}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}R_{j}}},{\rho_{{\mathsf{aux}}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}W}}\big{)}$
since the QECC supports transversal computation on these Clifford gates. As
the only difference between $\mathsf{Sim}^{\mathsf{RQC}}(C)$ and
$\mathsf{Sim}^{\mathsf{RQC}}(G\circ C)$ is the ideal functionalities they
interact with, we see that $\mathsf{Sim}^{\mathsf{RQC}}(G\circ C)$ also ends
with state (10).
For the protocol of measuring a logical qubit shared over registers
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}Q_{j}}\subseteq{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>the adversary receives $V_{j}=(E^{\prime}_{j}\otimes
Z^{c^{z}_{j}}X^{c^{x}_{j}}\mathsf{CNOT}_{c^{t}_{j}})E_{j}^{\dagger}$, so its
state can be described as
$\displaystyle A^{\prime\Delta}\big{(}\underset{i\in[n]}{\cup}\rho_{i}\text{
};\text{ }{(\underset{j\in[n]}{\otimes}(E^{\prime}_{j}\otimes
Z^{c^{z}_{j}}X^{c^{x}_{j}}\mathsf{CNOT}_{c^{t}_{j}}))({\mathsf{QECC.Enc}}(C(\cdot,\phi_{h})),0^{lnt})}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}R_{j}}},{\rho_{{\mathsf{aux}}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}W}}\big{)}$
for some $A^{\prime}$ that can depend on $V_{j}$. Next, the adversary measures
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>To analyze ${\mathsf{cMPC}}$’s examination, we decompose the attack
$A^{\prime}$ according to Pauli basis on registers
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>That is, we write
$A^{\prime}={\Sigma_{x_{j}}}{A^{\prime}_{x_{j}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}R_{j},W}}{X^{x_{j}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>where
$A^{\prime}_{x_{j}}={\Sigma_{z_{j}}}{A^{\prime}_{x_{j},z_{j}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>Q_{j},W}}{Z^{z_{j}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>Similar to [DGJ+20], all Pauli $P$ with $x(P)\neq 0$ would fail
${\mathsf{cMPC}}$’s test with overwhelming probability over random
$(c^{z}_{j},c^{x}_{j},c^{t}_{j})$. Thus, the residual state passing the test
would be indistinguishable from
$\displaystyle
A^{\prime\Delta}_{0}\big{(}\underset{i\in[n]}{\cup}\rho_{i}\text{ };\text{
}{(\underset{j\in[n]}{\otimes}E^{\prime}_{j})({\mathsf{QECC.Enc}}(C(\cdot,\phi_{h}))_{[:l-1]},0^{(l-1)nt})}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>Q_{j}}},{\rho_{{\mathsf{aux}}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}W}}\big{)}$
(11)
while the string $r_{j}$ provided by $\mathsf{P}_{j}$ is
$(\mathsf{\Pi}({\mathsf{QECC.Enc}}(C(\cup_{i}\rho_{i},\phi_{h})_{[l]})_{[j]})(1,c^{t}_{j})\oplus
x_{j})$. Then the bit $b$ recovered by ${\mathsf{cMPC}}$ is
$\mathsf{\Pi}(C(\cup_{i}\rho_{i},\phi_{h}){[l]})$ because measurement is
transversal on our QECC. Now we turn to the simulated end state. There are two
main differences between $\mathsf{Sim}^{\mathsf{RQC}}(C)$ and
$\mathsf{Sim}^{\mathsf{RQC}}((\mathbbm{1}\otimes\mathsf{\Pi})\circ C)$. First,
the latter simulator encrypts part of the receiving portals under key
$\hat{E}_{j}^{(l)}$. Second, the latter simulator performs an additional
measurement step. As neither simulators give $\hat{e}_{S,j}^{(l)}$ to the
adversary, both $\hat{e}_{R,j}^{(l)}$ and
$\hat{E}_{j}^{(l)}\cdot\hat{e}_{R,j}^{(l)}$ are maximally mixed states and are
indistinguishable from the view of ${\mathcal{A}}$. Next, we decompose
$A^{\prime}={\Sigma_{x_{j}}}{A^{\prime}_{x_{j}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color@gray@fill{.5}R_{j},W}}{X^{x_{j}}}^{{\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>The only possible Pauli operators on
${\color[rgb]{.5,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{.5,.5,.5}\pgfsys@color@gray@stroke{.5}\pgfsys@color<EMAIL_ADDRESS>that can yield equal measurements on $\hat{e}_{S,j}^{(l)}$ and
$\hat{e}_{R,j}^{(l)}$ are those without any bit-flips. Hence, conditioned that
the simulator does not make $\mathsf{P}_{j}$ send abort, the attack reduces to
$A^{\prime}_{0}$, and the end state is equal to (11).
Finally, we analyze the case when some corruptions are detected. If party
$\mathsf{P}_{j}$ is identified as corrupted, the output key $E_{j}$ will be
replaced with $\bot$ in both the real and simulated worlds, so that the $j$-th
share in (9) becomes the maximally mixed state in both worlds. When
$\mathsf{P}_{j}$ is added into the list Corr because the measurement step
failed, one easily sees that both worlds have residual states
(indistinguishably) described by
$(A^{\prime\Delta}-A^{\prime\Delta}_{0})(\cdot)$.
∎
### 6.3 Multiparty Quantum Computation
The final step is to transit from RQC to MPQC. After the computation, the
evaluators must send the encrypted and encoded outputs back to their owners.
Apparently, the outputs could turn out ruined when the owner unpacks the
messages. This situation is where AQA comes into play, which guarantees public
verifiable identifiable abort when some sender modifies the message. If no
such problem happens, the parties have accomplished their MPQC task. Our MPQC
protocol will be a direct concatenation of RQC and AQA.
Protocol 4 $(\Sigma^{{{\mathsf{BoBW}}_{0}}},\Pi^{{{\mathsf{BoBW}}_{0}}})$ for
the Preprocessing Best-of-Both-Worlds Multiparty Quantum Computation with
Public Verifiable Identifiable Abort
Parameters:
$n,t,\mathsf{t},(\ell_{1},\cdots,\ell_{n}),\ell_{{\mathsf{anc}}},\ell^{\prime}$
Preprocessing $\Sigma^{{\mathsf{BoBW}}_{0}}$:
1. 1.
Run $\Sigma^{\mathsf{RQC}}$. Then run
$\Sigma^{\mathsf{AQA}}(\mathsf{P}_{j},\mathsf{P}_{i})$ for every pair of
parties $i,j\in[n]$.
Online Input:
1. 1.
Everyone holds the circuit description $C$.
2. 2.
$\mathsf{P}_{i}$ holds private input $\rho_{i}\in\mathcal{D}^{\ell_{i}}$.
Protocol $\Pi^{{\mathsf{BoBW}}_{0}}$:
1. 1.
All parties run the protocol $\Pi^{\mathsf{RQC}}$ to compute $C$.
At any time if $|{\texttt{Corr}}|>\mathsf{t}$, ${\mathsf{cMPC}}$ publicly
aborts to all players in Corr.
2. 2.
Suppose $\sigma_{j}$ that $\mathsf{P}_{j}$ obtains from step 1 is
authenticated under key $E_{j}\neq\bot$.
${\mathsf{cMPC}}$ retrieves key pair
$(E^{\prime}_{i,j},E^{\prime\prime}_{i,j})$ that was given by
$\Sigma^{\mathsf{AQA}}(\mathsf{P}_{j},\mathsf{P}_{i})$.
${\mathsf{cMPC}}$ sends $V_{j}=(E^{\prime}_{1,j}\otimes\cdots\otimes
E^{\prime}_{1,n})E^{{\dagger}}_{j}$ to $\mathsf{P}_{j}$.
$\mathsf{P}_{j}$ calculates $V_{j}\cdot\sigma_{j}$ and parses the result into
$(\sigma^{\prime}_{1,j},\cdots,\sigma^{\prime}_{n,j})$.
3. 3.
AQA from party $\mathsf{P}_{j}$ to party $\mathsf{P}_{i}$: run
$\Pi^{\mathsf{AQA}}(\mathsf{P}_{j}(\sigma^{\prime}_{i,j}),\mathsf{P}_{i},{\mathsf{cMPC}})$
for $i,j\in[n]$.
$\mathsf{P}_{i}$ collects what it received
$\\{\sigma^{\prime\prime}_{i,j}\\}_{j\in[n]}$ from the other parties.
If AQA identifies $\mathsf{P}_{j}$ as malicious, put $\mathsf{P}_{j}$ into
Corr.
4. 4.
${\mathsf{cMPC}}$ sends keys $\\{E^{\prime\prime}_{i,j}\\}_{j\in[n]}$ to
$\mathsf{P}_{i}$ and publicly outputs classical output $r^{\prime}$.
Output:
1. 1.
$\mathsf{P}_{i}$ outputs
$\rho^{\prime}_{i}\leftarrow\mathsf{QECC.Dec}\big{(}E^{\prime\prime{\dagger}}_{i,1}\cdot\sigma^{\prime\prime}_{i,1},\cdots,E^{\prime\prime{\dagger}}_{i,n}\cdot\sigma^{\prime\prime}_{i,n}\big{)}$
and $r^{\prime}$.
###### Theorem 6.3.
$(\Sigma^{{\mathsf{BoBW}}_{0}},\Pi^{{{\mathsf{BoBW}}_{0}}})$ is a best-of-
both-worlds multi-party quantum computation secure with public verifiable
identifiable abort of threshold $\mathsf{t}$ in the preprocessing MPC-hybrid
model as defined in 4.1.
###### Proof.
In short, the simulator runs a concatenation of $\mathsf{Sim}^{\mathsf{RQC}}$
with $\mathsf{Sim}^{\mathsf{AQA}}$. ${\mathsf{Ideal}}^{{\mathsf{MPQC}}}$ is
invoked in $\mathsf{Sim}^{\mathsf{RQC}}$ to get $\\{\rho^{\prime}_{i}\\}_{i\in
I}$. In the output recovery phase of $\mathsf{Sim}^{\mathsf{RQC}}$, the
simulator teleports $\mathsf{TP}^{\dagger}\cdot\tilde{e}_{S}$ prepared by
$\mathsf{Sim}^{\mathsf{AQA}}$ instead of $\hat{e}_{S}^{(0)}$. Whenever
$|{\texttt{Corr}}|>\mathsf{t}$, the simulator sends abort message from parties
in $|{\texttt{Corr}}|$ to ${\mathsf{Ideal}}^{\mathsf{MPQC}}$. In the output
recovery phase of $\mathsf{Sim}^{\mathsf{AQA}}$, the simulator authenticates
$\rho^{\prime}_{i}$ with fresh random keys, teleports the result to
$\mathsf{P}_{i}$, and sends the key out.
When $|I|\leq\mathsf{t}$, the protocol never reaches the abort condition. By
Lemma 6.2 and Lemma 5.1, the honest parties will certainly receive at least a
proportion $1-\frac{\mathsf{t}}{n}$ of their output codewords. Our requirement
on QECC then implies that the honest parties can recover their outputs. From
Lemma 6.2 and Lemma 5.1, we deduce that the protocol is indistinguishable from
simulation using ${\mathsf{Ideal}}^{{\mathsf{MPQC-Full}}}$ when
$|I|\leq\mathsf{t}$ and using ${\mathsf{Ideal}}^{\mathsf{MPQC}}$ when $|I|\leq
n-1-\mathsf{t}$.
∎
## 7 BoBW-MPQC-PVIA without Trusted Setup
So far, we have shown an MPQC-PVIA protocol in the preprocessing model. This
section presents how we discard the trusted setup and obtain an MPQC-PVIA
protocol without preprocessing. The idea here is to instantiate the
preprocessing phase with MPQC-SWIA e.g., [ACC+21].
##### Hierarchical MPQC-SWIA
We quickly recall the SWIA property of [ACC+21]. When the [ACC+21] protocol
aborts, their classical MPC outputs a partition of all parties into two
mutually distrusting groups. By observation, one of the two groups comprises
only malicious players, and the other group contains all honest players, with
some malicious players pretending to be honest. We can take advantage of this
information to get an MPQC-PVIA protocol. First, we replace the trusted setup
of our preprocessing MPQC-PVIA by running MPQC-SWIA, where it can create
preprocessing data (including portals, magic states, and ancilla) for each
party. Then, if the protocol fails, we allow those two groups to run their
MPQC-SWIA protocols independently. The parties will ultimately obtain reliable
preprocessing data within subgroups by iterating this procedure. Note that we
do not publicly identify anyone during this stage.
After generating enough reliable preprocessing data, every party can run the
protocol $\Pi^{{\mathsf{BoBW}}_{0}}$ within its group to obtain the output.
Since the honest parties are always in the same group, they will jointly
evaluate their outputs in the $\Pi^{{\mathsf{BoBW}}_{0}}$. Our approach
circumvents the false accusation problem encountered in [ACC+21] because the
parties in our protocol no longer accuse between groups. Instead, each group
runs its own $\Pi^{{\mathsf{BoBW}}_{0}}$, which only aborts dishonest members.
We emphasize that the parties may divide into groups when the protocol
executes successfully. In this situation, if the circuit of interest involves
public output, the legitimacy of each group must be determined, and the public
output selected accordingly. Our protocol does not attempt to address this. We
take a step back and consider only circuits with private outputs. At the end
of our protocol, either a public verifiable identifiable abort occurs, or each
party is satisfied with its private outcome. Dealing with circuits with public
outputs remains an intriguing open problem for one who would like to achieve a
stronger PVIA notion.
Now, we formulate our protocol. Let $\Pi^{\mathsf{SWIA}}$ be [ACC+21]’s MPQC-
SWIA protocol.
Protocol 5 $\Pi^{{\mathsf{BoBW}}_{1}}$ for the Best-of-Both-Worlds MPQC-PVIA
Input:
1. 1.
Everyone holds the circuit description $C$ which has only private outputs.
2. 2.
Party $\mathsf{P}_{i}$ holds private input
$\rho_{i}\in\mathcal{D}^{\ell_{i}}$.
Protocol:
1. 1.
Put $G=\\{\\{\mathsf{P}_{1},\cdots,\mathsf{P}_{n}\\}\\}$ as the initial
partition of parties (i.e., no partition).
Mark the set $\\{\mathsf{P}_{1},\cdots,\mathsf{P}_{n}\\}\in G$ as unfinished.
2. 2.
Repeat the following until all sets in $G$ are marked as finished:
For unfinished $S\in G$, the parties in $S$ run $\Pi^{\mathsf{SWIA}}$ together
on the circuit $\Sigma^{{\mathsf{BoBW}}_{0}}(C_{S})$, where $C_{S}$ is the
circuit that prepares default inputs for parties $\not\in S$ and runs $C$.
1. (a)
If the protocol succeeds, parties in $S$ obtain the required preprocessing
data. In this case, mark $S$ as finished.
2. (b)
Otherwise, the MPQC-SWIA protocol outputs a partition $\\{S_{0},S_{1}\\}$ of
$S$.
In this case, replace $S$ with the unfinished $S_{0}$ and $S_{1}$ in partition
$G$.
3. 3.
Run $\Pi^{{\mathsf{BoBW}}_{0}}$ within every set $S\in G$.
Abort to all identified parties if any execution of
$\Pi^{{\mathsf{BoBW}}_{0}}$ aborts.
Output:
1. 1.
Upon abort, all parties output the identities collected from all
$\Pi^{{\mathsf{BoBW}}_{0}}$.
2. 2.
Otherwise, $\mathsf{P}_{i}$ outputs the result obtained from his/her execution
of $\Pi^{{\mathsf{BoBW}}_{0}}$.
###### Theorem 7.1.
$\Pi^{{\mathsf{BoBW}}_{1}}$ is a best-of-both-worlds multi-party quantum
computation secure with public verifiable identifiable abort over circuits
with only private outputs in the MPC-hybrid model as defined in 4.1.
###### Proof.
Since our protocol makes only sequential calls to MPQC-SWIA protocols, the
security of MPQC-SWIA allows one to replace these protocol executions with the
ideal functionality of MPQC-SWIA. The resulting hybrid world $\mathcal{H}_{1}$
is indistinguishable from the real world. The ideal functionality of MPQC-SWIA
in $\mathcal{H}_{1}$ translates to the following game. At the beginning of
$\mathcal{H}_{1}$, the adversary chooses how to partition all parties step by
step, but cannot partition the honest parties. Afterward, there is a trusted
setup on $\Sigma^{{\mathsf{BoBW}}_{0}}$ followed by
$\Pi^{{\mathsf{BoBW}}_{0}}$ for each group of the partition. Since these
preprocessing BoBW protocols are independent instances, 6.3 implies that they
can be replaced with the ideal world of BoBW-MPQC-PVIA. It is easy to see that
the resulting hybrid $\mathcal{H}_{2}$ preserves the PVIA property. To argue
full security when $|I|\leq\mathsf{t}$, one observes that the group containing
all the honest parties has at least a $1-\frac{\mathsf{t}}{n}$ proportion of
honest parties, so the full security of BoBW-MPQC-PVIA is also preserved. ∎
## 8 Extentions
### 8.1 Constant Round MPQC-PVIA in the Preprocessing Model
When aiming only for MPQC-PVIA, our BoBW-MPQC-PVIA protocol in the
preprocessing model can be made round efficient. The reason is that we no
longer need to keep the data distributed across parties, which eliminates the
need for information exchange during computation. Instead, a designated party
can hold all the data, and the entire process can become constant-round using
quantum garbled circuits for computation as in [BCKM21].
###### Theorem 8.1.
There is a constant-round multi-party quantum computation secure with public
verifiable identifiable abort in the preprocessing MPC-hybrid model.
Protocol 6
$(\Sigma^{{{\mathsf{Const}}{\mathsf{MPQC}}}},\Pi^{{{\mathsf{Const}}{\mathsf{MPQC}}}})$
for the Preprocessing Constant-Round MPQC-PVIA
Preprocessing $\Sigma^{{{\mathsf{Const}}{\mathsf{MPQC}}}}$:
1. 1.
Run $\Sigma^{\mathsf{IE}}$. Then run
$\Sigma^{\mathsf{AQA}}(\mathsf{P}_{i},\mathsf{P}_{1})$ for every party
$i\in[n]$.
Protocol $\Pi^{{{\mathsf{Const}}{\mathsf{MPQC}}}}$:
1. 1.
All parties run the input encoding protocol $\Pi^{\mathsf{IE}}$.
The designated evaluator $\mathsf{P}_{1}$ obtains ciphertext $\sigma$ under
key $E$.
2. 2.
${\mathsf{cMPC}}$ generates quantum garbled circuit:
1. (a)
Retrieve key pair $(E^{\prime}_{i},E^{\prime\prime}_{i})$ that was created by
$\Sigma^{\mathsf{AQA}}(\mathsf{P}_{i},\mathsf{P}_{1})$.
2. (b)
Define the augmented circuit $C^{\prime}$ as:
* •
Measure all the traps. If any trap is non-zero, end by outputting $\bot$.
* •
Evaluate $C$ on the inputs and ancilla states, and apply
$E^{\prime}_{1}\otimes\cdots\otimes E^{\prime}_{n}$.
3. (c)
Run $(\hat{E},\hat{C})\leftarrow\mathsf{QG.Garble}(C^{\prime})$.
4. (d)
Send the gate $V=\hat{E}E^{\dagger}$ and the garbled circuit $\hat{C}$ to the
evaluator $\mathsf{P}_{1}$.
3. 3.
$\mathsf{P}_{1}$ evaluates the garbled circuit
$\sigma^{\prime}\leftarrow\mathsf{QG.Eval}(V\cdot\sigma,\hat{C})$.
$\mathsf{P}_{1}$ parses the result into
$(\sigma^{\prime}_{1},\cdots,\sigma^{\prime}_{n})$.
4. 4.
AQA from party $\mathsf{P}_{1}$ to party $\mathsf{P}_{i}$ for $i\in[n]$: run
$\Pi^{\mathsf{AQA}}(\mathsf{P}_{1}(\sigma^{\prime}_{i}),\mathsf{P}_{i},{\mathsf{cMPC}})$.
If any $\Pi^{\mathsf{AQA}}$ fails, ${\mathsf{cMPC}}$ publicly aborts to
$\mathsf{P}_{1}$.
Otherwise, $\mathsf{P}_{i}$ receives the ciphertext
$\sigma^{\prime\prime}_{i}$ successfully.
5. 5.
${\mathsf{cMPC}}$ sends key $E^{\prime\prime}_{i}$ to $\mathsf{P}_{i}$.
Output: $\mathsf{P}_{i}$ outputs
$\rho^{\prime}_{i}\leftarrow\mathsf{C.Dec}_{E^{\prime\prime}_{i}}(\sigma^{\prime\prime}_{i})$.
The security follows from adapting the proof of 6.3 with the proofs [BCKM21,
BY22] that use garbled circuits to perform the computation.
### 8.2 MPQC-PVIA from Trap-code AQA
We also discover the possibility of constructing MPQC-PVIA using Trap-code
form AQA (Figure 4). Existing MPQC protocols against a dishonest majority
[DGJ+20, ACC+21, BCKM21] only consider Clifford authentication code because of
the obstruction of the Trap authentication code’s checking step. For example,
if we replace the Clifford code in [ACC+21] with Trap code, the player will
not know if the errors are caused by the previous player or other prior
players. This is because trap code requires a permutation to hide the input
qubits’ information. By only considering the Trap authentication code, one
cannot guarantee the secrecy of the qubits’ location while still identifying
the suspect.
Thanks to our AQA, by slightly twisting Trap-code and plugging it into our
AQA, we can construct the first MPQC-PVIA protocol from Trap-code form AQA in
the preprocessing model. The protocol will be very similar to the Clifford-
form protocol (Section 6.3) but with Trap-form IE and AQA (Figure 4). Similar
to [ACC+21], we make use of a verifiable quantum fully homomorphic encryption
scheme (vQFHE) [ADSS17] for computation on Trap code.
Protocol 7
$(\Sigma^{{\mathsf{BoBW}}_{\mathsf{TP}}},\Pi^{{\mathsf{BoBW}}_{\mathsf{TP}}})$
for the Trap-form Preprocessing MPQC-PVIA
Preprocessing: $\Sigma^{{\mathsf{BoBW}}_{\mathsf{TP}}}$
1. 1.
Run $\Sigma^{\mathsf{IE}}$, distributing not only Trap-code ciphertexts on EPR
pairs and ancilla states, but also gadgets required for vQFHE computation
under secret key $\mathsf{sk}$ given to ${\mathsf{cMPC}}$.
2. 2.
Run $\Sigma^{\mathsf{AQA}}(\mathsf{P}_{j},\mathsf{P}_{i})$ for all pairs
$i,j\in[n]$.
Protocol: $\Pi^{{\mathsf{BoBW}}_{\mathsf{TP}}}$
1. 1.
All parties run $\Pi^{\mathsf{IE}}$. so that each $\mathsf{P}_{i}$ obtains
several Trap-code ciphertexts $\\{\sigma_{i,l}\\}$ corresponding to QECC
codewords of inputs, under the same permutation key $\pi_{i}$.
2. 2.
${\mathsf{cMPC}}$ initializes the set ${\texttt{Corr}}=\\{\\}$.
At any time if $|{\texttt{Corr}}|>\mathsf{t}$, ${\mathsf{cMPC}}$ publicly
abort to all parties in Corr.
3. 3.
$\mathsf{P}_{i}$ computes via
$(\\{\tilde{\sigma}_{i,l}\\},\mathsf{log}_{i})\leftarrow\mathsf{TrapTP.Eval}(C,\\{\sigma_{i,l}\\})$
and sends $\mathsf{log}_{i}$ to ${\mathsf{cMPC}}$.
4. 4.
${\mathsf{cMPC}}$ runs $\mathsf{CheckLog}(\mathsf{log}_{i},\mathsf{sk})$.If
the output is $\bot$, ${\mathsf{cMPC}}$ publicly abort $\mathsf{P}_{i}$.
5. 5.
$\mathsf{P}_{i}$ sends $\tilde{\sigma}_{i,l}$ to the respective receiver
$\mathsf{P}_{j}$ via $\Pi^{\mathsf{AQA}}$. The receiver $\mathsf{P}_{j}$ gets
$\sigma^{\prime}_{j}$.
6. 6.
${\mathsf{cMPC}}$ sends the decoding instruction $V_{\mathsf{Dec},i}$ to the
respective party $\mathsf{P}_{i}$.
Output: $\mathsf{P}_{i}$ outputs $\rho^{\prime}_{i}\leftarrow
V_{\mathsf{Dec},i}(\sigma^{\prime}_{i})$.
The Trap-form protocol
$(\Sigma^{{\mathsf{BoBW}}_{\mathsf{TP}}},\Pi^{{\mathsf{BoBW}}_{\mathsf{TP}}})$
satisfies the statement of 6.3. We omit the proof because of its similarity to
the Clifford-code form MPQC-PVIA while basing on the security guarantee of
Trap-code form AQA.
## Acknowledgement
We would like to thank Kai-Min Chung and Jiapeng Zhang for their helpful
discussions. This research is partially supported by NSTC QC project under
Grant no. NSTC 111-2119-M-001-004- and partially supported by NSF CAREER award
2141536.
## References
* [ABDR04] Andris Ambainis, Harry Buhrman, Yevgeniy Dodis, and Hein Rohrig. Multiparty quantum coin flipping. In Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004., pages 250–259. IEEE, 2004.
* [ABOEM17] Dorit Aharonov, Michael Ben-Or, Elad Eban, and Urmila Mahadev. Interactive proofs for quantum computations. arXiv preprint arXiv:1704.04487, 2017.
* [ACC+21] Bar Alon, Hao Chung, Kai-Min Chung, Mi-Ying Huang, Yi Lee, and Yu-Ching Shen. Round efficient secure multiparty quantum computation with identifiable abort. In Annual International Cryptology Conference, pages 436–466. Springer, 2021.
* [ADSS17] Gorjan Alagic, Yfke Dulek, Christian Schaffner, and Florian Speelman. Quantum fully homomorphic encryption with verification. In International Conference on the Theory and Application of Cryptology and Information Security, pages 438–467. Springer, 2017.
* [BCKM21] James Bartusek, Andrea Coladangelo, Dakshita Khurana, and Fermi Ma. On the round complexity of secure quantum computation. In Annual International Cryptology Conference, pages 406–435. Springer, 2021.
* [BDO14] Carsten Baum, Ivan Damgård, and Claudio Orlandi. Publicly auditable secure multi-party computation. In International Conference on Security and Cryptography for Networks, pages 175–196. Springer, 2014.
* [BDSW96] Charles H. Bennett, David P. DiVincenzo, John A. Smolin, and William K. Wootters. Mixed-state entanglement and quantum error correction. Phys. Rev. A, 54:3824–3851, Nov 1996.
* [BGS13] Anne Broadbent, Gus Gutoski, and Douglas Stebila. Quantum one-time programs. In Annual Cryptology Conference, pages 344–360. Springer, 2013\.
* [BJSW16] Anne Broadbent, Zhengfeng Ji, Fang Song, and John Watrous. Zero-knowledge proof systems for qma. In 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS), pages 31–40. IEEE, 2016.
* [BLOO11] Amos Beimel, Yehuda Lindell, Eran Omri, and Ilan Orlov. 1/p-secure multiparty computation without honest majority and the best of both worlds. In Annual Cryptology Conference, pages 277–296. Springer, 2011\.
* [BOCG+06] Michael Ben-Or, Claude Crepeau, Daniel Gottesman, Avinatan Hassidim, and Adam Smith. Secure multiparty quantum computation with (only) a strict honest majority. In 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06). IEEE, 2006.
* [BOSSV20] Carsten Baum, Emmanuela Orsini, Peter Scholl, and Eduardo Soria-Vazquez. Efficient constant-round mpc with identifiable abort and public verifiability. In Annual International Cryptology Conference, pages 562–592. Springer, 2020.
* [BW16] Anne Broadbent and Evelyn Wainewright. Efficient simulation for quantum message authentication. In International Conference on Information Theoretic Security, pages 72–91. Springer, 2016.
* [BY22] Zvika Brakerski and Henry Yuen. Quantum garbled circuits. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 804–817, 2022.
* [CGL99] Richard Cleve, Daniel Gottesman, and Hoi-Kwong Lo. How to share a quantum secret. Physical review letters, 83(3):648, 1999.
* [CGS02] Claude Crépeau, Daniel Gottesman, and Adam Smith. Secure multi-party quantum computation. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing - STOC '02. ACM Press, 2002.
* [CLLW22] Kai-Min Chung, Yi Lee, Han-Hsuan Lin, and Xiaodi Wu. Constant-round blind classical verification of quantum sampling. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 707–736. Springer, 2022.
* [DGJ+20] Yfke Dulek, Alex B Grilo, Stacey Jeffery, Christian Majenz, and Christian Schaffner. Secure multi-party quantum computation with a dishonest majority. Advances in Cryptology - EUROCRYPT 2020., 2020.
* [DNS12] Frédéric Dupuis, Jesper Buus Nielsen, and Louis Salvail. Actively secure two-party evaluation of any quantum operation. In Annual Cryptology Conference, pages 794–811. Springer, 2012\.
* [IKK+11] Yuval Ishai, Jonathan Katz, Eyal Kushilevitz, Yehuda Lindell, and Erez Petrank. On achieving the “best of both worlds” in secure multiparty computation. SIAM journal on computing, 40(1):122–141, 2011.
* [IOZ14] Yuval Ishai, Rafail Ostrovsky, and Vassilis Zikas. Secure multi-party computation with identifiable abort. In Advances in Cryptology – CRYPTO 2014, pages 369–386. Springer Berlin Heidelberg, 2014.
* [Kat07] Jonathan Katz. On achieving the” best of both worlds” in secure multiparty computation. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 11–20, 2007.
* [Mah18] Urmila Mahadev. Classical verification of quantum computations. In 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS), pages 259–267. IEEE, 2018.
* [MGW87] Silvio Micali, Oded Goldreich, and Avi Wigderson. How to play any mental game. In Proceedings of the Nineteenth ACM Symp. on Theory of Computing, STOC, pages 218–229. ACM, 1987.
* [RBO89] Tal Rabin and Michael Ben-Or. Verifiable secret sharing and multiparty protocols with honest majority. In Proceedings of the twenty-first annual ACM symposium on Theory of computing, pages 73–85, 1989.
* [Unr10] Dominique Unruh. Universally composable quantum multi-party computation. In Annual International Conference on the Theory and Applications of Cryptographic Techniques, pages 486–505. Springer, 2010.
* [Yao86] Andrew Chi-Chih Yao. How to generate and exchange secrets. In 27th Annual Symposium on Foundations of Computer Science (sfcs 1986), pages 162–167, 1986.
|
# Greybody Radiation of scalar and Dirac perturbations of NUT Black Holes
Ahmad Al-Badawi<EMAIL_ADDRESS>Department of Physics, Al-Hussein Bin
Talal University, P. O. Box: 20, 71111, Ma’an, Jordan. Sara Kanzi
<EMAIL_ADDRESS>İzzet Sakallı<EMAIL_ADDRESS>Physics
Department, Eastern Mediterranean University, Famagusta, North Cyprus via
Mersin 10, Turkey.
###### Abstract
We consider the spinorial wave equations, namely the Dirac and the Klein-
Gordon equations, and greybody radiation in the NUT black hole spacetime. To
this end, we first study the Dirac equation in NUT spacetime by using a null
tetrad in the Newman-Penrose (NP) formalism. Next, we separate the Dirac
equation into radial and angular sets. The angular set is solved in terms of
associated Legendre functions. With the radial set, we obtain the decoupled
radial wave equations and derive the one-dimensional Schrödinger wave
equations together with effective potentials. Then, we discuss the potentials
by plotting them as a function of radial distance in a physically acceptable
region. We also study the Klein-Gordon equation to compute the greybody
factors (GFs) for both bosons and fermions. The influence of the NUT parameter
on the GFs of the NUT spacetime is investigated in detail.
Hawking Radiation, NUT, Black Hole, Dirac Equation, Klein-Gordon Equation,
Greybody Factor, Effective Potential, Perturbation
††preprint: HEP/123-qed
###### Contents
1. I Introduction
2. II Dirac equation in NUT BH spacetime
3. III Solutions of angular and radial equations
1. III.1 Angular equations
2. III.2 Radial equations
4. IV Greybody Radiation from NUT BH
1. IV.1 Klein-Gordon equation
2. IV.2 GFs of bosons
3. IV.3 GFs of fermions
5. V Conclusion
## I Introduction
NUT spacetimes are special case of the family of electrovac spacetimes of
Petrov type-D given by the Plebański-Demiański class of solutions 1 . These
axially symmetric spacetimes are characterized by the seven parameters: mass,
rotation, acceleration, cosmological constant, magnetic charge, electric
charge, and NUT parameter. We are interested in the solution, which is
characterized by two physical parameters: the gravitational mass and the NUT
parameter. From now on, we will refer to this spacetime [see metric (1)] as
the NUT black hole (BH). In fact, the NUT BH is a solution to the vacuum
Einstein equation that was found by Newman–Unti–Tamburino (NUT) in 1963 2 .
Since the discovery of the NUT solution, its physical meaning is a matter of
debate. The interpretations of the NUT solution and thus the NUT parameter are
still a controversy and it has attracted lots of attention over the years.
Misner approach 3 to the interpretation of the NUT solution aims at
eliminating singular regions at the expense of introducing the “periodic” time
and, consequently, the $S^{3}$ topology for the hypersurfaces $r=const$. In 4
the NUT solution was interpreted as representing the exterior field of a mass
located at the origin together with a semi-infinite massless source of angular
momentum. Whereas in 5 the solution was interpreted as the presence of two
semi-infinite counter-rotating singular regions endowed with negative masses
and infinitely large angular momenta.
Regarding to the NUT parameter, in general it was interpreted as a
gravitomagnetic charge bestowed upon the central mass 6 . However, the exact
physical meaning has not been attained yet. In 7 it was explained that if the
NUT parameter dominates the rotation parameter, then this leaves the spacetime
free of curvature singularities. But if the rotation parameter dominates the
NUT parameter, the solution is Kerr-like, and a ring curvature singularity
arises. Conversely, exact interpretation of the NUT parameter becomes possible
when a static Schwarzschild mass is immersed in a stationary, source free
electromagnetic universe 8 . In this case, the NUT parameter is interpreted as
a twist parameter of the electromagnetic universe. However, in the absence of
this field, it reduces to the twist of the vacuum spacetime. A different
interpretation of the NUT spacetime has been clarified via its thermodynamics.
In 9 , it was shown that if the NUT parameter is interpreted as possessing
simultaneously rotational and electromagnetic features then the thermodynamic
quantities satisfy the Bekenstein-Smarr mass formula.
In article 10 , Misner called the NUT metric a ’counter-example to almost
anything’: it means that it has so many peculiar properties. Following this
line of thought, the aim of the present paper is to study the scalar and Dirac
perturbations of the NUT BH and analyze their corresponding greybody
radiation. Studies of this type have always been important in the field of
quantum gravity. Furthermore, an effective way to understand and analyze
characteristics of a spacetime is to study its behavior under different kinds
of perturbations including spinor and gauge fields. Also, one can pave the way
to study the GFs of the NUT spacetime in order to investigate the evolution of
perturbations in the exterior region of the BH 11 ; 12 . The analytical
expressions of the solution obtained in this paper could be useful for the
study of the thermodynamical properties of the spinor fields in NUT geometry.
In the literature, the scalar/Dirac equation and greybody radiation in BH
geometry have been extensively studied. For example, wave analysis of the
Schwarzschild BH was performed in 13 ; 14 ; 15 ; 16 ; 17 ; 18 , for the Kerr
BH in 19 ; 20 ; 21 ; 22 , for the Kerr Taub-NUT BH in 23 , for the Kerr-Newman
AdS BH in 24 , and for the Reissner-Nordsrtöm de Sitter BH in 25 . Meanwhile,
it is worth noting that the GFs measure the deviation of the thermal spectra
from the black body or the so-called Hawking radiation 26 ; 27 . Essentially,
GF is a physical quantity that relates to the quantum nature of a BH. A high
value of GF shows a high probability that Hawking radiation can reach to
spatial infinity (or an observer) 28 ; 29 . Today, there are different methods
to compute the GF such as the matching method 30 ; 31 , rigorous bound method
32 , the WKB approximation 33 ; 34 , and analytical method for the various of
spin fields 35 ; 36 ; 37 ; 38 ; 38n ; 39 ; 40 . Here, we shall employ the
general method of semi-analytical bounds for the GFs. To this end, we shall
examine the characteristic bosonic and fermionic quantum radiation (i.e., the
GF) of NUT BH based on the current studies in the literature (see for example
Harmark:2007jy ) and our recent studies 37 ; 38 ; 38n . Although the
methodology followed is parallel to the procedure seen in 37 ; 38 ; 38n , due
to the unique characteristics of each BH solution, the calculations have their
own difficulties and therefore the results obtained will differ significantly.
In those recent studies 37 ; 38 ; 38n , for example, we have revealed the
effects of quintessence, negative cosmological constant, and Lorentz symmetry
breaking parameters on the quantum thermal radiation of black holes. However,
in this study, we aim to reveal the effect of the NUT parameter on the GF and
thus present a theoretical study to the literature that will contribute to the
black hole classification studies according to GF measurements that might be
obtained in the (distant) future.
The paper is organized as follows: In the next section, we obtain the Dirac
equation in the NUT spacetime and separate the coupled equations into the
angular and radial parts. In Sect. 3, analytical solutions to the angular
equations are sought. We also study the radial wave equations and examine the
behaviors of the effective potential that emerge in the transformed radial
equations. In Sect. 4, we study the Klein–Gordon equation in the NUT BH
spacetime. Then, we derive the associated effective potential of bosons. Then,
we compute the GFs of the NUT BH spacetime for both fermions and bosons in
Sect. 5. We draw our conclusions in Sect. 6.
## II Dirac equation in NUT BH spacetime
The NUT metric is a solution to the vacuum Einstein equation that depends on
two parameters: mass $M$ and NUT parameter $l$ 1 . A detailed discussion on
the NUT spacetime can be found in the famous monograph of Griffiths and
Podolsky 41 and in the specific studies given in 42 ; 43 ; 44 . In spherical
coordinates $(t,r,\theta,\phi)$ the NUT BH metric reads
$ds^{2}=f(r)\left[dt-2l\cos\theta
d\phi\right]^{2}-\frac{1}{f(r)}dr^{2}-\left(r^{2}+l^{2}\right)\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right),$ (1)
where
$f(r)=1-\frac{2\left(Mr+l^{2}\right)}{r^{2}+l^{2}}>0.$ (2)
The distinctive feature of the NUT metric is the presence of the NUT parameter
$l$ that causes the spacetime to be asymptotically non-flat. The horizons are
defined by $f(r)=0$ and hence they will occur at
$r_{\pm}=M\pm\sqrt{M^{2}+l^{2}}$.
Following the NP formalism, we introduce null tetrads $(l,n,m,\overline{m})$
to satisfy orthogonality relations: $l.n=-m.\overline{m}=1.$ Thus, we write
the basis vectors of null tetrad in terms of elements of the NUT geometry as
follows
$l_{\mu}=(1,-\frac{1}{f\left(r\right)},0,-2l\cos\theta),$
$n_{\mu}=\frac{1}{2}(f\left(r\right),1,0,-2lf(r)\cos\theta),$
$m_{\mu}=\sqrt{\frac{r^{2}+l^{2}}{2}}(0,0,-1,-i\sin\theta),$
$\overline{m}_{\mu}=\sqrt{\frac{r^{2}+l^{2}}{2}}(0,0,-1,i\sin\theta),$ (3)
where a bar over a quantity denotes complex conjugation and the dual co-tetrad
of Eq. (3) are given by
$l^{\mu}=(\frac{1}{f\left(r\right)},1,0,0),$
$n^{\mu}=\frac{1}{2}(1,-f(r),0,0),$
$m^{\mu}=\frac{1}{\sqrt{2\left(r^{2}+l^{2}\right)}}(2il\cot\theta,0,1,\frac{i}{\sin\theta}),$
$\overline{m}^{\mu}=\frac{1}{\sqrt{2\left(r^{2}+l^{2}\right)}}(-2il\cot\theta,0,1,\frac{-i}{\sin\theta}).$
(4)
Using the above null tetrad one can find the non-zero spin coefficients in
terms of NUT metric:
$\epsilon=\frac{il}{2\left(r^{2}+l^{2}\right)};\gamma=\frac{1}{4}\left(f^{\prime}(r)+\frac{ilf(r)}{r^{2}+l^{2}}\right);\alpha=-\beta=\frac{-\cot\theta}{2\sqrt{2\left(r^{2}+l^{2}\right)}}$
$\rho=\frac{-r+il}{r^{2}+l^{2}};\mu=\frac{f(r)}{2}\frac{-r+il}{r^{2}+l^{2}}.$
(5)
Their corresponding directional derivatives become
$D=l^{\mu}\partial_{\mu}=\frac{1}{f(r)}\frac{\partial}{\partial
t}+\frac{\partial}{\partial r},$
$\Delta=n^{\mu}\partial_{\mu}=\frac{1}{2}\frac{\partial}{\partial
t}-\frac{f(r)}{2}\frac{\partial}{\partial r},$
$\delta=m^{\mu}\partial_{\mu}=\frac{1}{\sqrt{2\left(r^{2}+l^{2}\right)}}(2il\cot\theta\frac{\partial}{\partial
t}+\frac{\partial}{\partial\theta}+\frac{i}{\sin\theta}\frac{\partial}{\partial\phi}),$
$\bar{\delta}=\overline{m}^{\mu}\partial_{\mu}=\frac{1}{\sqrt{2\left(r^{2}+l^{2}\right)}}(-2il\cot\theta\frac{\partial}{\partial
t}+\frac{\partial}{\partial\theta}-\frac{i}{\sin\theta}\frac{\partial}{\partial\phi}),$
(6)
The Dirac equations in the curved spacetime are given by 45
$\left(D+\epsilon-\rho\right)F_{1}+\left(\overline{\delta}+\pi-\alpha\right)F_{2}=i\mu_{\ast}G_{1},$
$\left(\Delta+\mu-\gamma\right)F_{2}+\left(\delta+\beta-\tau\right)F_{1}=i\mu_{\ast}G_{2},$
$\left(D+\overline{\epsilon}-\overline{\rho}\right)G_{2}-\left(\delta+\overline{\pi}-\overline{\alpha}\right)G_{1}=i\mu_{\ast}F_{2},$
$\left(\Delta+\overline{\mu}-\overline{\gamma}\right)G_{1}-\left(\overline{\delta}+\overline{\beta}-\overline{\overline{\tau}}\right)G_{2}=i\mu_{\ast}F_{1},$
(7)
where $\mu_{\ast}$ is the mass of the Dirac particle.
The form of the Dirac equation suggests that we define the spinor fields as
follows 45 ,
$F_{1}=F_{1}\left(r,\theta\right)e^{i\left(kt+n\phi\right)},$
$F_{2}=F_{2}\left(r,\theta\right)e^{i\left(kt+n\phi\right)},$
$G_{1}=G_{1}\left(r,\theta\right)e^{i\left(kt+n\phi\right)},$
$G_{2}=G_{2}\left(r,\theta\right)e^{i\left(kt+n\phi\right)}.$ (8)
where $k$ is the frequency of the incoming wave and $n$ is the azimuthal
quantum number of the wave.
Substituting the appropriate spin coefficients (5) and the spinors (8) into
the Dirac equation (7) , we obtain
$\sqrt{r^{2}+l^{2}}\left(\emph{D}-\frac{il}{2\left(r^{2}+l^{2}\right)}\right)F_{1}+\frac{1}{\sqrt{2}}\mathit{L}F_{2}=i\mu_{0}\sqrt{r^{2}+l^{2}}G_{1},$
$\frac{f}{2}\sqrt{r^{2}+l^{2}}\left(\mathit{D}^{{\dagger}}+\frac{f^{\prime}}{2f}-\frac{il}{2\left(r^{2}+l^{2}\right)}\right)F_{2}-\frac{1}{\sqrt{2}}\mathit{L}^{{\dagger}}F_{1}=-i\mu_{0}\sqrt{r^{2}+l^{2}}G_{2},$
$\sqrt{r^{2}+l^{2}}\left(\emph{D}+\frac{il}{2\left(r^{2}+l^{2}\right)}\right)G_{2}-\frac{1}{\sqrt{2}}\mathit{L}^{{\dagger}}G_{1}=i\mu_{0}\sqrt{r^{2}+l^{2}}F_{2},$
$\frac{f}{2}\sqrt{r^{2}+l^{2}}\left(\mathit{D}^{{\dagger}}+\frac{f^{\prime}}{2f}+\frac{il}{2\left(r^{2}+l^{2}\right)}\right)G_{1}+\frac{1}{\sqrt{2}}\mathit{L}G_{2}=-i\mu_{0}\sqrt{r^{2}+l^{2}}F_{1},$
(9)
where, $\mu_{0}=\sqrt{2}\mu_{\ast}$ and the above operators are given by
$\emph{D}=\frac{\partial}{\partial r}+\frac{r}{r^{2}+l^{2}}+\frac{ik}{f},$
$\mathit{D}^{{\dagger}}=\frac{\partial}{\partial
r}+\frac{r}{r^{2}+l^{2}}-\frac{ik}{f}$
$\mathit{L}=\frac{\partial}{\partial\theta}+\left(\frac{1}{2}+2lk\right)\cot\theta+\frac{n}{\sin\theta},$
$\mathit{L}^{{\dagger}}=\frac{\partial}{\partial\theta}+\left(\frac{1}{2}-2lk\right)\cot\theta-\frac{n}{\sin\theta}.$
(10)
Further, we define
$F_{1}\left(r,\theta\right)=R_{+1/2}\left(r\right)A_{+1/2}\left(\theta\right),F_{2}\left(r,\theta\right)=R_{-1/2}\left(r\right)A_{-1/2}\left(\theta\right),G_{1}\left(r,\theta\right)=R_{-1/2}\left(r\right)A_{+1/2}\left(\theta\right)$
,$G_{2}\left(r,\theta\right)=R_{+1/2}\left(r\right)A_{-1/2}\left(\theta\right)$
hence, the set (9) can be separated into angular and radial parts as
$\sqrt{r^{2}+l^{2}}\emph{D}R_{+1/2}=\left(\lambda+i\mu_{0}\sqrt{r^{2}+l^{2}}\right)R_{-1/2},$
(11)
$\frac{f}{2}\sqrt{r^{2}+l^{2}}\mathit{D}^{{\dagger}}R_{-1/2}=\left(\lambda-i\mu_{0}\sqrt{r^{2}+l^{2}}\right)R_{+1/2},$
(12)
$\frac{1}{\sqrt{2}}\mathit{L}A_{-1/2}=\lambda A_{+1/2},$ (13)
$\frac{1}{\sqrt{2}}\mathit{L}^{{\dagger}}A_{+1/2}=-\lambda A_{-1/2},$ (14)
where $\lambda$ is the separation constant.
## III Solutions of angular and radial equations
### III.1 Angular equations
Angular Eqs. (14) and (13) can be written as
$\frac{dA_{+1/2}}{d\theta}+\left(\left(\frac{1}{2}+2lk\right)\cot\theta+\frac{n}{\sin\theta}\right)A_{+1/2}=-\lambda
A_{-1/2},$ (15)
$\frac{dA_{-1/2}}{d\theta}+\left(\left(\frac{1}{2}-2lk\right)\cot\theta-\frac{n}{\sin\theta}\right)A_{-1/2}=\lambda
A_{+1/2}.$ (16)
We note that the structure of the angular equations admits the similar
solutions for both $A_{+1/2}$ and $A_{-1/2}$. Therefore, it is sufficient to
focus on one of the angular equations. To this end, we apply the following
transformations
$A_{+1/2}\left(\theta\right)=\cos\left(\frac{\theta}{2}\right)S_{1}(\theta)+\sin\left(\frac{\theta}{2}\right)S_{2}(\theta),$
(17)
$A_{-1/2}\left(\theta\right)=-\sin\left(\frac{\theta}{2}\right)S_{1}(\theta)+\cos\left(\frac{\theta}{2}\right)S_{2}(\theta).$
(18)
By setting $x=\cos\theta$, one can write Eq. (15) as a decoupled second order
differential equation:
$\left(1-x^{2}\right)\frac{d^{2}S_{1}}{dx^{2}}-2x\frac{dS_{1}}{dx}+\left[\lambda\left(\lambda+1\right)-\frac{\left(n+\frac{1}{2}\right)^{2}}{1-x^{2}}\right]S_{1}=0,$
(19)
with $\left(n+\frac{1}{2}\right)^{2}\leq\lambda^{2}$. The solution to Eq. (19)
can be expressed in terms of the associated Legendre functions 46 :
$P_{\lambda}^{n}\left(x\right)=\left(1-x^{2}\right)^{\tau/2}\frac{d^{n}}{dx^{n}}P_{\lambda}\left(x\right).$
(20)
### III.2 Radial equations
The radial equations (11-12) can be rearranged as
$\sqrt{r^{2}+l^{2}}\left(\frac{d}{dr}+\frac{r}{r^{2}+l^{2}}+\frac{ik}{f}\right)R_{+1/2}=\left(\lambda-i\mu_{0}\sqrt{r^{2}+l^{2}}\right)R_{-1/2},$
(21)
$\frac{f}{2}\sqrt{r^{2}+l^{2}}\left(\frac{d}{dr}+\frac{r}{r^{2}+l^{2}}+\frac{f^{\prime}}{2f}-\frac{ik}{f}\right)R_{-1/2}=\left(\lambda+i\mu_{0}\sqrt{r^{2}+l^{2}}\right)R_{+1/2}.$
(22)
To obtain the radial equations in the form of one-dimensional Schrödinger like
wave equations, we perform the following transformations
$P_{+1/2}=\sqrt{r^{2}+l^{2}}R_{+1/2},\qquad
P_{-1/2}=\sqrt{\frac{f}{2}}\sqrt{r^{2}+l^{2}}R_{-1/2}.$ (23)
Then, Eqs. (21) and (22) transform to
$\left(\frac{d}{dr}+i\frac{k}{f}\right)P_{+1/2}=\sqrt{\frac{2}{f}}\left(\frac{\lambda}{\sqrt{r^{2}+l^{2}}}-i\mu_{0}\right)P_{-1/2},$
(24)
$\left(\frac{d}{dr}-i\frac{k}{f}\right)P_{-1/2}=\sqrt{\frac{2}{f}}\left(\frac{\lambda}{\sqrt{r^{2}+l^{2}}}+i\mu_{0}\right)P_{+1/2},$
(25)
and defining the tortoise coordinate $r_{\ast}$ as
$\frac{dr_{\ast}}{dr}=\frac{1}{f}$, Eqs. (24) and (25) become
$\left(\frac{d}{dr_{\ast}}+ik\right)P_{+1/2}=\sqrt{2f}\left(\frac{\lambda}{\sqrt{r^{2}+l^{2}}}-i\mu_{0}\right)P_{-1/2}$
(26)
$\left(\frac{d}{dr_{\ast}}-ik\right)P_{-1/2}=\sqrt{2f}\left(\frac{\lambda}{\sqrt{r^{2}+l^{2}}}+i\mu_{0}\right)P_{+1/2}$
(27)
In order to write Eqs. (26) and (27) in a more compact form, we combine the
solutions as $\psi_{+}=P_{+1/2}+P_{-1/2}$ and $\psi_{-}=P_{+1/2}-P_{-1/2}$ .
After making some computations, we end up with a pair of one-dimensional
Schrödinger like equations:
$\frac{d^{2}\psi_{+}}{dr_{\ast}^{2}}+\left(4k^{2}-V_{+}\right)\psi_{+}=0,$
(28)
$\frac{d^{2}\psi_{-}}{dr_{\ast}^{2}}+\left(4k^{2}-V_{-}\right)\psi_{-}=0,$
(29)
with the following effective potentials:
$V_{\pm}=\frac{2L^{3/2}\sqrt{\Delta}}{D^{2}}\left[2L^{3/2}\sqrt{\Delta}\pm
3\mu_{0}^{2}r\Delta\pm L\left(r-M\right)\mp\frac{\Delta
L}{D}\left(2\mu_{0}^{2}r\left(r^{2}+l^{2}\right)-\frac{\lambda\mu_{0}\left(r-M\right)}{2k}+2rL\right)\right],$
$L\left(r-M\right)\mp\frac{\Delta
L}{D}\left(2\mu_{0}^{2}r\left(r^{2}+l^{2}\right)-\frac{\lambda\mu_{0}\left(r-M\right)}{2k}+2rL\right)],$
(30)
where
$\Delta=r^{2}-2Mr-l^{2},L=\left(\lambda^{2}+\mu_{0}^{2}r^{2}\right),D=L\left(r^{2}+l^{2}\right)-\frac{\lambda\mu_{0}}{4k}.$
(31)
We see that the potentials (30) depend on the NUT parameter $l$. To obtain the
potentials for massless fermions (neutrino), we simply set $\mu_{0}=0$ in Eq.
(30), which yields
$V_{\pm}=\frac{2\lambda\sqrt{\Delta}}{\left(r^{2}+l^{2}\right)^{2}}\left[2\lambda\sqrt{\Delta}\pm\left(r-M\right)\mp\frac{2\Delta
r}{r^{2}+l^{2}}\right].$ (32)
To study the asymptotic behavior of the potentials (30), we can expand it up
to order $O\left(\frac{1}{r^{3}}\right)$. Thus, the potentials behave as
$V_{\pm}\simeq
4\mu_{0}^{2}+8M\mu_{0}^{2}\frac{1}{r}\pm\left(6l^{2}\mu_{0}^{2}-2\lambda^{2}+\frac{\lambda\mu_{0}}{2k}\right)\frac{1}{r^{2}}+O\left(\frac{1}{r^{3}}\right).$
(33)
From Eq. (33), we see that the first term corresponds to the constant value of
the potential at the asymptotic infinity. In the second term, the mass
produces the usual monopole type attractive potential. The NUT parameter in
the third term represents the dipole type potential.
Figure 1: Graph of $V_{+}$ (30) for various values of the NUT parameter $l$.
Here, $\lambda=k=1,\mu=0.12,$ and $M=0.5$.
We will now investigate the effect of the NUT parameter on the effective
potentials (30) by plotting them as a function of radial distance. Figs. 1 and
2 describe the effective potentials (30) for massive particles where we obtain
potential curves for some specific values of the NUT parameter. As can be seen
from Figs. 1 and 2, for sufficiently small values of the NUT parameter, the
potentials have sharp peaks in the physical region. When the NUT parameter
$l=0$, the peak is seen to be maximum. We also observe that while the NUT
parameter increases, the sharpness of the peaks decreases, and the peaks tend
to disappear after a specific value of the NUT parameter. This implies that
for a large values of the NUT parameter, a massive Dirac particle moving in
the physical region experiences a potential barrier without peaks and its
kinetic energy increases. But for a small value of the NUT parameter, it feels
a sharp potential barrier.
Figure 2: Graph of $V_{-}$ (30) for various values of NUT parameter $l$. Here,
$\lambda=k=1,\mu=0.12,$ and $M=0.5$. Figure 3: Three-dimensional plot of
$V_{+}$ for various values of NUT parameter $l$. Here, $\lambda=k=1,\mu=0.12,$
and $M=0.5$.
Explicitly, the influence of the NUT parameter is shown in the three-
dimensional plots of the potentials with respect to the NUT parameter and the
radial distance. As can be seen from Fig. (3), we observe a peak for small
values of the NUT parameter and as the value of the NUT parameter increases,
the potentials decrease. Let us note that, for massless Dirac particle
$\left(\mu_{0}=0\right)$, the potentials have similar behaviors as for the
massive case.
## IV Greybody Radiation from NUT BH
### IV.1 Klein-Gordon equation
The Klein-Gordon equation for the massless scalar field in curved coordinates
is given by isepjc
$\square\Psi=\frac{1}{\sqrt{-g}}\partial_{\mu}\sqrt{-g}g^{\mu\nu}\partial_{\nu}\Psi=0,$
(34)
where $g$ is the determinant of the spacetime metric (1):
$\sqrt{-g}=\left(r^{2}+l^{2}\right)\sin\theta$. To separate the variables in
equation (34), we assume the solutions to the wave equation in the form
$\Psi=R\left(r\right)\Theta\left(\theta\right)\exp\left(-i\omega
t\right)\exp\left(im\phi\right),$ (35)
where $\omega$ denotes the frequency of the wave and $m$ is the azimuthal
quantum number. Therefore, the Klein-Gordon equation (34) becomes
$\left(\frac{\left(r^{2}+l^{2}\right)^{2}}{\Delta}-\frac{4l^{2}\cos^{2}\theta}{\sin^{2}\theta}\right)\omega^{2}+\frac{1}{R\left(r\right)}\frac{d}{dr}\left(\Delta\frac{dR\left(r\right)}{dr}\right)+$
$\frac{1}{\Theta\left(\theta\right)}\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{\partial\Theta\left(\theta\right)}{\partial\theta}\right)-\frac{m^{2}}{\sin^{2}\theta}-\frac{4l\cos\theta}{\sin^{2}\theta}\omega
m=0.$ (36)
The angular part satisfies
$\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{\partial\Theta\left(\theta\right)}{\partial\theta}\right)-\left(\frac{4l^{2}\cos^{2}\theta+4l\omega
m\cos\theta+m^{2}}{\sin^{2}\theta}\right)\Theta\left(\theta\right)=-\lambda\Theta\left(\theta\right),$
(37)
where it has solutions in terms of the oblate spherical harmonic functions
having eigenvalues $\lambda$ 46 . Therefore, the Klein–Gordon equation (36) is
left with the radial part
$\frac{d}{dr}\left(\Delta\frac{d}{dr}R\left(r\right)\right)+\left(\left(r^{2}+l^{2}\right)^{2}\omega^{2}-\Delta\lambda\right)R\left(r\right)=0.$
(38)
By changing the variable in a new form as
$R\left(r\right)=\frac{U\left(r\right)}{\sqrt{r^{2}+l^{2}}},$ the radial wave
equation (38) recasts into a one-dimensional Schrödinger like equation as
follows
$\frac{d^{2}U}{dr_{\ast}^{2}}+\left(\omega^{2}-V_{eff}\right)U=0,$ (39)
in which $r_{\ast}$ is the tortoise coordinate:
$\frac{dr_{\ast}}{dr}=\frac{r^{2}+l^{2}}{\Delta},$ and $V_{eff}$ is the
effective potential given by
$V_{eff}=\frac{\Delta}{\left(r^{2}+l^{2}\right)^{3}}\left(\frac{-3r^{2}\Delta}{r^{2}+l^{2}}-\lambda\left(r^{2}+l^{2}\right)-3r^{2}-4Mr-l^{2}\right).$
(40)
Figure 4: Plots of $V_{eff}$ (40) versus $r$ for the spin-$0$ particles for
various values of NUT parameter $l$. The physical parameters are chosen as;
$\lambda=k=1,\mu=0.12,$ and $M=0.5$.
It is seen from (40) that the effective potential depends on the NUT parameter
$l$. Fig. 4 indicates the behavior of potential (40) under the effect of NUT
parameter for massless particles. As can be seen from Fig. (4), when the NUT
parameter increases, peaks of the potential goes down.
### IV.2 GFs of bosons
It is known that nothing can escape a BH when approaching it. But when a
quantum effect is considered, the BH can radiate. This radiation is known as
the Hawking radiation. We now assume that Hawking radiation is a massless
scalar field (bosons) that satisfies the Klein–Gordon equation. To evaluate
the GF of the NUT metric for bosons, we use 47
$\sigma_{l}\left(\omega\right)\geq\sec h^{2}\left(\int_{-\infty}^{+\infty}\wp
dr_{\ast}\right),$ (41)
in which $r_{\ast}$ is the tortoise coordinate and
$\wp=\frac{1}{2h}\sqrt{\left(\frac{dh\left(r_{\ast}\right)}{dr_{\ast}}\right)^{2}+(\omega^{2}-V_{eff}-h^{2}\left(r_{\ast}\right))^{2}},$
(42)
where $h(r\ast)$ is a positive function satisfying
$h\left(-\infty\right)=h\left(-\infty\right)=\omega$: see Refs. 47 ; 48 for
more details. Without loss of generality, we select $h=\omega$. Therefore,
$\sigma_{l}\left(\omega\right)\geq\sec
h^{2}\left(\int_{r_{h}}^{+\infty}\frac{V_{eff}}{2\omega}dr_{\ast}\right).$
(43)
We use the potential derived in (40) to obtain the GFs for bosons, namely
$\sigma^{s}_{l}\left(\omega\right)\geq\sec
h^{2}\frac{1}{2\omega}\int_{r_{h}}^{+\infty}\frac{1}{\left(r^{2}+l^{2}\right)}\left(\frac{-3r^{2}\Delta}{\left(r^{2}+l^{2}\right)^{2}}+\lambda+\frac{\Delta^{\prime}{r}+\Delta}{r^{2}+l^{2}}\right)dr.$
(44)
After integration, we obtain
$\sigma^{s}_{l}\left(\omega\right)\geq\sec
h^{2}\frac{1}{2\omega}{\left(\frac{-\lambda}{r}-\frac{M}{r^{2}}+\frac{l^{2}\left(\lambda-5\right)}{3r^{3}}+\frac{5Ml^{2}}{2r^{4}}+\frac{l^{4}\left(16-\lambda\right)}{5r^{5}}+\frac{l^{6}\left(\lambda-3\right)}{7r^{7}}\right)}.$
(45)
The behaviors of the obtained bosonic GFs of the NUT BH are depicted in Fig. 5
in which the plots of $\sigma\left(\omega\right)$ for various values of NUT
parameter. As can be seen from those graphs, the NUT has an effect on the
bosonic GFs. Remarkably, $\sigma\left(\omega\right)$ clearly decreases with
the increasing NUT parameter. In other words, the NUT plays a kind of booster
role for the GFs of the spin-0 particles.
Figure 5: Plots of $\sigma^{s}_{l}\left(\omega\right)$ (45) versus $\omega$
for the zero spin particles for various values of NUT parameter $l$. The
physical parameters are chosen as; $\lambda=2$ and $M=1$.
### IV.3 GFs of fermions
Here, we shall derive the fermionic GF of the neutrinos emitted from the NUT
BH. To this end, we consider the potentials (32). Following the procedure
described above [see Eqs. (41)-(43)], we obtain,
$\sigma_{\pm}\left(\omega\right)\geq\sec
h^{2}\frac{\lambda}{\omega}\left(-\frac{2\lambda\mp
1}{r}\mp\left(\frac{M}{r^{2}}+\frac{M^{2}-l^{2}\mp
4l^{2}\lambda}{6r^{3}}\right)\pm\frac{3M^{3}-Ml^{2}}{8r^{4}}\right).$ (46)
The fermionic GFs of the NUT BH are depicted in Figs. 6 and 7 in which the
plots of $\sigma_{\pm}\left(\omega\right)$ for various values of NUT parameter
are obtained. Similar to the bosonic GFs, the NUT has an intensifier effect on
the GFs of the spin-$\frac{1}{2}$ particles.
Figure 6: Plots of $\sigma_{+}\left(\omega\right)$ (45) versus $\omega$ for
the Non-zero spin particles for various values of NUT parameter $l$. The
physical parameters are chosen as; $\lambda=-0.5$ and $M=1$. Figure 7: Plots
of $\sigma_{-}\left(\omega\right)$ (46) versus $\omega$ for the Non-zero spin
particles for various values of NUT parameter $l$. The physical parameters are
chosen as; $\lambda=-0.5$ and $M=1$.
## V Conclusion
We have studied the perturbation and greybody radiation in the NUT spacetime.
The NUT metric describes the vacuum spacetime around a source that is
characterized by two parameters, the mass of the central object and the NUT
parameter. The perturbations studied here are for the bosons and fermions,
namely the Klein-Gordon and Dirac equations are considered. For the Dirac
equation, we explicitly work out the separability of the equations into radial
and angular parts by using a null tetrad in the NP formalism. It is shown that
solutions to the angular equations could be given in terms of the associated
Legendre functions. We also discussed the radial equations and obtain a wave
equation with an effective potential. The radial part involves both mass and
NUT parameters. It is shown in the asymptotic expansion of the potentials (33)
that the mass produces the usual $1/r$ attractive potential while the NUT
parameter manifests itself in the $1/r^{2}$ asymptotically. Thus, the effect
of gravity is as expected stronger than the effect of the NUT parameter in the
NUT geometry. To understand the physical interpretations of the potentials
(30), we make two- and three-dimensional plots of the potentials for different
values of the NUT parameter. It is seen from the plots that the potential
barriers are higher for small values of the NUT parameter. If the NUT
parameter decreases, the peak of the potential barriers distinctly increases.
Potentials become limited regardless of the value of the NUT parameter and
tend to a constant value for large values of distances.
For the GFs of bosons, we have considered the massless scalar wave equation.
To this end, we have reduced the radial part of the Klein-Gordon equation to
the one-dimensional Schrödinger like wave equation. The behaviors of the
effective potential (40) under the effect of the NUT parameter for the scalar
waves are depicted. It is seen from Fig. 4 that as the NUT parameter increases
then the peak of the barrier goes down. In the sequel, we have computed the
GFs, one of the fundamental information that can be obtained from the BHs, for
both bosons and fermions. It has been observed from Figs. 5, 6 and 7 that
increment of the NUT value significantly increases the GF radiation of both
bosons and fermions.
In the cosmology, the NUT parameter measures the amount of anisotropy that a
metric has at large times [50]. Therefore, the GF analyses of the rotating NUT
BHs, like Kerr-NUT-de Sitter BH [50], in the AdS background shall be a
significant future extension of the present work. This may also be important
to understand the AdS/CFT conjecture with the NUT similar to the quasinormal
modes [51] analysis of planar Taub-NUT BHs [52]. This will be investigated in
a near future work.
## References
* (1) J. Plebański and M. Demiański, Ann. Phys. (N.Y.) 98, 98 (1976).
* (2) Newman E T, Tamburino L and Unti T W J, J. Math. Phys. 4 915-23, (1963).
* (3) C. W. Misner, Relativity Theory and Astrophysics ed J Ehlers (Amer. Math. Soc., Providence, Rhode Island) 1, 160 (1967).
* (4) W.B. Bonnor, Proc. Camb. Philos. Soc. 66, 145 (1969).
* (5) V.S. Manko, E. Ruiz, Class. Quantum Gravity 22, 3555 (2005).
* (6) D. Lynden-Bell, M. Nouri-Zonoz, Rev. Mod. Phys. 70, 427 (1998).
* (7) Griffiths. J.B. and Padolsky.J, Class. Quant. Grav., 24, 1687. (2007).
* (8) A. Al-Badawi, M. Halilsoy, Gen. Relativ. Gravit. 38, 1729 (2006).
* (9) S.-Q. Wu, D. Wu, Phys. Rev. D 100, 101501(R) (2019).
* (10) c. Misner, Relativity Theory and Astrophysics. I. ed J Ehlers (Providence, RI: American Mathematical Society), 160 (1967).
* (11) K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel. 2, 2 (1999).[gr-qc/9909058].
* (12) H. P. Nollert, Class. Quant. Grav. 16, R159 (1999).
* (13) B. Mukhopadhyay, S.K. Chakrabarti, Class. Quantum Grav. 16, 3165 (1999).
* (14) J. Jing, Phys. Rev. D 70 065004 (2004); Phys. Rev. D 71 124006 (2005).
* (15) A. Al-Badawi, M.Q. Owaidat, Gen Relativ Gravit 49,110 (2017).
* (16) I.I. Cotaescu, Mod. Phys. Lett. A 22, 2493 (2007).
* (17) W. G. Unruh, Phys. Rev. D 14, 3251 (1976).
* (18) A. Zecca, Nuovo Cimento 30, 13091315 (1998).
* (19) H. Schmid, Math. Nachr. 274275, 117129 (2004).
* (20) D. Page, Phys. Rev. D 14, 1509 (1976).
* (21) E. G. Kalnins and W. Miller, J. Math. Phys. 33, 286 (1992).
* (22) F. Belgiorno, S.L. Cacciatori, J. Phys. A 42, 135207 (2009)
* (23) H. Cebeci, N. Ozdemir, Class. Quantum Grav. 30, 175005 (2013).
* (24) F. Belgiorno, S.L. Cacciatori, J. Math. Phys. 51, 033517 (2010).
* (25) Yan Lyu and Song Cui, Phys. Scr. 79, 025001 (2009).
* (26) D.N. Page, Phys. Rev. D 13, 198 (1976).
* (27) D.N. Page, Phys. Rev. D 14, 3260 (1976).
* (28) R. Jorge, E.S. de Oliveira, J.V. Rocha, Class. Quantum Gravity 32, 065008 (2015).
* (29) J. Abedi, H. Arfaei, Class. Quantum Gravity 31, 195005 (2014).
* (30) S. Fernando, Gen. Relativ. Gravit. 37, 461 (2005).
* (31) W. Kim, J.J. Oh, JKPS 52, 986 (2008).
* (32) T. Harmark, J. Natario, R. Schiappa, Adv. Theor. Math. Phys. 14, 727 (2010).
* (33) K. Jusufi, M. Amir, M. Sabir Ali, S.D. Maharaj, Phys. Rev. D 102, 064020 (2020)
* (34) M.K. Parikh, F. Wilczek, Phys. Rev. Lett. 85, 5049 (2000).
* (35) P. Boonserm, T. Ngampitipan, P.Wongjun, Eur. Phys. J. C 79, 330 (2019).
* (36) I. Sakalli, Phys. Rev. D 94, 084040 (2016). arXiv:1606.00896 [grqc].
* (37) A. Al-Badawi, I. Sakalli, S. Kanzi, Ann. Phys. 412, 168026 (2020).
* (38) A. Al-Badawi, S. Kanzi, I. Sakalli, Eur. Phys. J. Plus 135, 219 (2020).
* (39) H. Gursel, I. Sakalli, Eur. Phys. J. C 80, 234 (2020).
* (40) S. Kanzi, S.H.Mazharimousavi, I. Sakalli, Ann. Phys. 422, 168301 (2020).
* (41) S. Kanzi, İ. Sakallı, Eur. Phys. J. C 81, 501 (2021).
* (42) T. Harmark, J. Natario, R. Schiappa, Adv. Theor. Math. Phys. 14, no.3, 727-794 (2010).
* (43) J. B. Griffiths, J. Podolsky, Exact Space-Times in Einstein’s General Relativity (Cambridge: Cambridge University Press), (2009).
* (44) Jefremov, P.I., Perlick, V.: Circular motion in NUT space-time, Class. Quantum Gravity 33, 179501 (2016).
* (45) M. Nouri-Zonoz, D. Lynden-Bell, Mon. Not. Roy. Astron. Soc. 292, 714 (1998).
* (46) M. Halla, V. Perlick, Gen. Relativ. Gravit. 52, 112 (2020).
* (47) S. Chandrasekhar, The Mathematical Theory of Black Holes Clarendon, London (1983).
* (48) M. Abramowitz, I. A. Stegun (ed.), Handbook of Mathematical Functions (Dover, New York, 1970) and [S. Detweiler, Phys. Rev. D 22, 2323 (1980)].
* (49) I. Sakalli, Eur. Phys. J. C 75, no.4, 144 (2015).
* (50) Y.G. Miao, Z.M. Xu, Phys. Lett. B 772, 542 (2017).
* (51) M. Visser, Phys. Rev. A 59, 427438 (1999). arXiv: quant-ph/9901030.
* (52) A. Anabalón, S. F. Bramberger and J. L. Lehners, JHEP 09, 096 (2019).
* (53) I. Sakalli, Mod. Phys. Lett. A 28, 1350109 (2013).
* (54) P. A. Cano and D. Pereñiguez, [arXiv:2101.10652 [hep-th]].
|
Given a conical affine special Kähler (CASK) manifold together with a compatible mutually local variation of BPS structures, one can construct a quaternionic-Kähler (QK) manifold. We call the resulting QK manifold an instanton corrected c-map space. Our main aim is to study the isometries of a subclass of instanton corrected c-map spaces associated to projective special real (PSR) manifolds with a compatible mutually local variation of BPS structures. We call the latter subclass instanton corrected q-map spaces. In the setting of Calabi-Yau compactifications of type IIB string theory, instanton corrected q-map spaces are related to the hypermultiplet moduli space metric with perturbative corrections, together with worldsheet, D(-1) and D1 instanton corrections. In the physics literature, it has been shown that the hypermultiplet metric with such corrections must have an $\mathrm{SL}(2,\mathbb{Z})$ acting by isometries, related to S-duality. We give a mathematical treatment of this result, specifying under which conditions instanton corrected q-map spaces carry an action by isometries by $\mathrm{SL}(2,\mathbb{Z})$ or some of its subgroups. We further study the universal isometries of instanton corrected q-map spaces, and compare them to the universal isometries of tree-level q-map spaces. Finally, we give an explicit example of a non-trivial instanton corrected q-map space with full $\mathrm{SL}(2,\mathbb{Z})$ acting by isometries and admitting a quotient of finite volume by a discrete group of isometries.
§ INTRODUCTION
The supergravity c-map assigns to a projective special Kähler (PSK) manifold $(\overline{M},g_{\overline{M}})$ of complex dimension $n$, a quaternionic-Kähler (QK) manifold $(\overline{N},g_{\overline{N}})$ of real dimension $4n+4$. In the context of Calabi-Yau compactifications of type IIA/B string theory on a Calabi-Yau threefold $X$, the c-map takes the vector multiplet moduli space $\mathcal{M}_{\text{VM}}^{\text{IIA/B}}(X)$ to the hypermultiplet moduli space $\mathcal{M}_{\text{HM}}^{\text{IIB/A}}(X)$ with its string-tree-level metric $g_{\text{FS}}$, also known as the Ferrara-Sabharwal metric [22, 32]. Such a construction receives quantum corrections in the string coupling $g_s$ of several kinds:
* Perturbative corrections: these produce the so-called 1-loop corrected Ferrara-Sabharwal metric on $\mathcal{M}_{\text{HM}}^{\text{IIA/B}}(X)$ [38, 7]. In a purely mathematical setting, this construction can be understood as a way of assigning to a PSK manifold $(\overline{M},g_{\overline{N}})$ a 1-parameter family of QK manifolds $\{(\overline{N}_{c_{\ell}},g_{\overline{N},c_{\ell}})\}_{c_{\ell}\in \mathbb{R}}$ [6], where $c_{\ell}\in \mathbb{R}$ corresponds to the 1-loop correction.
* Non-perturbative quantum corrections: these are divided in D-instanton and NS5-instanton corrections. These have been extensively studied in the physics literature via twistor methods, see for example the reviews [8, 10] and the references therein. The inclusion of all D-instanton corrections is understood in the physics literature [12, 13], while the NS5-instanton corrections are still work in progress (see for example [1]).
When restricted to the simpler setting of mutually local D-instanton corrections, a fairly explicit local formula for the QK metric was given in the physics literature via twistor methods in [2][Here by “fairly explicit" we mean that the expression is explicit except for a certain function $\mathcal{R}$, which is implicitly defined in terms of the field variables.]. On the other hand, in [25] a mathematical treatment (based on the geometric approach of [6]) of a class of QK metrics related to the above mutually local case was given. Namely, if $(\overline{M},g_{\overline{M}})$ is a PSK manifold and $(M,g_M,\omega_M,\nabla,\xi)$ the associated conical affine special Kähler (CASK) manifold, then one can complement this data with a mutually local variation of BPS structures $(M,\Gamma,Z,\Omega)$ to construct a new QK metric $(\overline{N},g_{\overline{N}})$ (we suppress the choice of $1$-loop parameter $c_{\ell}$ from the notation). The general notion of variation of BPS structure can be found in [17] (see also Definition <ref> below for the specific case to be used throughout this paper). Here $(M,\Gamma,Z,\Omega)$ is assumed to satisfy certain compatibility conditions with the CASK structure $(M,g_M,\omega_M,\nabla,\xi)$ (see Section <ref> below), and encodes the analog of “mutually local D-instanton" corrections when compared to the string theory setting.
On the other hand, type IIB string theory carries an $\mathrm{SL}(2,\mathbb{Z})$-symmetry called S-duality. In the physics literature, it has been shown that S-duality descends to an action by isometries on $\mathcal{M}_{\text{HM}}^{\text{IIB}}(X)$ when one includes the appropriate amount of quantum corrections for the metric. For example:
* When one drops all corrections in the string coupling $g_s$ and takes a large volume limit, one obtains the classical QK metric on $\mathcal{M}_{\text{HM}}^{\text{IIB}}(X)$. This metric has been shown to have an $\mathrm{SL}(2,\mathbb{R})$ acting by isometries [16], which comes from the $\mathrm{SL}(2,\mathbb{Z})$ S-duality action. Furthermore, it has been shown in [16] that the $\mathrm{SL}(2,\mathbb{R})$ (and even $\mathrm{SL}(2,\mathbb{Z})\subset \mathrm{SL}(2,\mathbb{R}))$ is broken when one includes world-sheet instanton corrections and/or the 1-loop correction in $g_s$.
* On the other hand, it has been shown in [15, 11] that when one includes world-sheet instanton corrections, the 1-loop correction, and D(-1) and D1 instanton corrections, one recovers again an $\mathrm{SL}(2,\mathbb{Z})$ action by isometries coming from the S-duality action. As a consequence of their result, it also follows that only including perturbative world-sheet instanton corrections, the 1-loop correction in $g_s$ and D(-1) instanton corrections also preserves the isometric S-duality action. For some other extensions of this result in the physics literature see for example [11, 1, 3, 4].
* The QK metric of $\mathcal{M}_{\text{HM}}^{\text{IIB}}(X)$ is also expected to retain the $\mathrm{SL}(2,\mathbb{Z})$ S-duality action by isometries when one includes all quantum corrections, but such a metric has not been constructed or understood (as far as the authors know) in the physics literature.
The classical metric obtained in the case when one drops all corrections in $g_s$ and take a large volume limit lies in a subset of c-map metrics called q-map metrics [27]. Mathematically, the q-map assign to a $n-1\geq 0$ dimensional projective special real (PSR) manifold a $4n+4$ dimensional QK manifold [23, 18]. A purely differential geometric proof that q-map metrics carry an $\mathrm{SL}(2,\mathbb{R})$ of isometries was given in [26], while more traditional supergravity arguments can be found for example in [29, 28]. On the other hand, including world-sheet instanton corrections and the 1-loop correction takes the q-map metric to the class of 1-loop corrected c-map metrics; while including D(-1) and D1 instanton corrections takes the 1-loop corrected c-map metric to the class of mutually local instanton corrected QK metrics studied in [2, 25].
Our main objective in this paper is to do a mathematical treatment of the S-duality results in [15, 11], and to study the universal isometries of instanton corrected q-map spaces, i.e. those isometries that are independent of the initial PSR manifold and the form of the quantum corrections that we restrict ourselves to. Namely, among the class of instanton corrected c-map metrics $(\overline{N},g_{\overline{N}})$ constructed in [25] we restrict to a subclass, which we call instanton corrected q-map metrics, and show under which conditions they carry an $\mathrm{SL}(2,\mathbb{Z})$-action (or an action by some of its subgroups) by isometries. This $\mathrm{SL}(2,\mathbb{Z})$-action is furthermore related to the S-duality symmetry in the string theory setting. Furthermore, we study how the universal group of isometries of q-map spaces described in [26] gets modified for instanton corrected q-map spaces (see Section <ref> below for more details).
§.§ Summary of main results
The main differences and new results compared to the works in the physics literature [15, 11], from which we take a lot of inspiration from, are the following:
* If the S-duality $\mathrm{SL}(2,\mathbb{Z})$ action defined in (<ref>) restricts to an action on the domain of definition of the instanton corrected q-map space, then the twistor space argument of [15, 11] follows and the action is also by isometries. However, verifying that the domain of definition actually carries such an action seems non-trivial. In Theorem <ref> (explained in more detail below) we collect results regarding this point. In particular we find that, even in a case where we do not have an $\mathrm{SL}(2,\mathbb{Z})$-invariant domain of definition, one can always find either an $S\in \mathrm{SL}(2,\mathbb{Z})$ or $T\in \mathrm{SL}(2,\mathbb{Z})$-invariant one (where $T$ and $S$ are the usual generators of $\mathrm{SL}(2,\mathbb{Z})$ given in (<ref>)). Furthermore, in Section <ref> we give an explicit non-trivial (albeit simple) example where one can find such an $\mathrm{SL}(2,\mathbb{Z})$-invariant neighborhood of definition.
* Assuming that the domain of definition of the instanton corrected q-map space carries an action by the S-duality $\mathrm{SL}(2,\mathbb{Z})$, we show that this action must be by isometries by following an argument similar but slightly different from [15]. Using their twistor description of the relevant QK manifolds, it is shown in [15] that certain “type IIA" Darboux coordinates for the contact structure of the twistor space can be Poisson resummed, and then it is shown that the later can be related via a gauge transformation to certain “type IIB" Darboux coordinates. The type IIB coordinates then make transparent that a certain lift of the $\mathrm{SL}(2,\mathbb{Z})$ action to the twistor space acts by twistor space automorphisms, and hence that $\mathrm{SL}(2,\mathbb{Z})$ acts by isometries on the QK metric. In our work, we find it simpler to do a direct Poisson resummation of the contact structure corresponding to the QK metric constructed in [25], and then use the resulting expression to show that the “type IIB" coordinates from [15] are Darboux coordinates for the contact structure.
* We further study certain universal isometry groups of instanton corrected q-map spaces, and compare with with what happens in the case where no quantum corrections are considered (see Section <ref> and Theorem <ref>). In particular, while S-duality is a universal isometry for tree-level q-map spaces, we are not able to guarantee that the the same is true for instanton corrected q-map spaces (see Remark <ref>). Furthermore, in the example of Section <ref> we use the $\mathrm{SL}(2,\mathbb{Z})$ action by isometries together with the universal isometry group to show that our example admits a quotient of finite volume by a discrete group of isometries.
In the following, we explain in more detail the setting and the aforementioned results. As previously mentioned, our main results concerns a class of QK metrics called instanton corrected q-map spaces, defined in Section <ref>. Roughly speaking, these are QK manifolds associated to a CASK manifold described by a holomorphic prepotential $(M,\mathfrak{F})$, together with a variation of mutually local BPS structures $(M,\Gamma,Z,\Omega)$ (see [17] for the general definition of variation of BPS structures) of the following form:
* The holomorphic prepotential $\mathfrak{F}:M\subset \mathbb{C}^{n+1}\to \mathbb{C}$ has the form
\begin{equation}
\mathfrak{F}(Z^i)=-\frac{1}{6}k_{abc}\frac{Z^aZ^bZ^c}{Z^0}+\chi\frac{(Z^0)^2\zeta(3)}{2(2\pi \I )^3}-\frac{(Z^0)^2}{(2\pi \I)^3}\sum_{\hat{\gamma}=q_a\gamma^a \in \Lambda^+}n_{\hat{\gamma}}\mathrm{Li}_3(e^{2\pi \I q_aZ^a/Z^0}),
\end{equation}
where $i=0,...,n$; $a,b,c=1,...,n$; $k_{abc}\in \mathbb{R}$ are symmetric in the indices, $\chi\in \mathbb{Z}$, $n_{\hat{\gamma}}\in \mathbb{Z}$, $\mathrm{Li}_n(x)$ denote the n-th polylogarithms, $\zeta(x)$ is the Riemann zeta function, and $\Lambda^{+}:=\text{span}_{\mathbb{Z}_{\geq 0}}\{\gamma^a\}_{a=1}^n-\{0\}$ is a commutative semigroup freely generated by $n$ elements $\{\gamma^a\}_{a=1}^n$.
This choice of $\mathfrak{F}$ is motivated by Calabi-Yau compactifications of string theory. Namely, if $X$ denotes a Calabi-Yau threefold, and if $k_{abc}$ are taken to be the triple intersection numbers of $X$, $\chi=\chi(X)$ the Euler characteristic, and for $\hat{\gamma}\in H_2^{+}(X,\mathbb{Z})$ we have that $n_{\hat{\gamma}}=n_{\hat{\gamma}}^{(0)}$ are the genus zero Gopakumar-Vafa invariants, then $\mathfrak{F}$ denotes the prepotential specifying the PSK geometry of $\mathcal{M}_{\text{VM}}^{\text{IIA}}(X)$ with all worldsheet corrections. Applying the 1-loop corrected c-map, one obtains from $\mathcal{M}_{\text{VM}}^{\text{IIA}}(X)$ the $1$-loop corrected metric on $\mathcal{M}_{\text{HM}}^{\text{IIB}}(X)$.
* The charge lattice $\Gamma$ and the central charge $Z$ of $(M,\Gamma,Z,\Omega)$ are canonically determined by $\mathfrak{F}$ (see Section <ref>), while the BPS indices $\Omega(\gamma)$ are also determined by $\mathfrak{F}$ as follows: with respect to a canonical Darboux basis $(\widetilde{\gamma}_i,\gamma^i)_{i=0}^n$ of $\Gamma$ with respect to its symplectic pairing, we have
\begin{equation}\label{BPSindintro}
\begin{cases}
\Omega(q_0\gamma^0)=-\chi, \quad q_0\in \mathbb{Z}-\{0\}\\
\Omega(q_0\gamma^0\pm q_a\gamma^a)=\Omega(\pm q_a\gamma^a)=n_{q_a\gamma^a} \quad \text{for $q_a\gamma^a \in \Lambda^+$, $q_0\in \mathbb{Z}$}\\
\Omega(\gamma)=0 \quad \text{else}.\\
\end{cases}
\end{equation}
The prescription (<ref>) has previously appeared in the physics literature (see for example <cit.>), and is the data required to add the D(-1) and D1 instanton corrections to $\mathcal{M}_{\text{HM}}^{\text{IIB}}(X)$ in an “S-duality compatible"-way. Furthermore, in the case of a non-compact Calabi-Yau threefold $X$ without compact divisors the same structure for the BPS indices is expected. Indeed, in [34] the appropriate $\Omega(\gamma)$ determining generalized Donaldson-Thomas invariants are constructed. It is then shown that $\Omega(q_0\gamma^0)=-\chi(X)$ <cit.>, and conjectured that $\Omega(q_0\gamma^0\pm \hat{\gamma})=n_{\hat{\gamma}}^{(0)}$<cit.>.
To the above data $(M,\mathfrak{F})$ and $(M,\Gamma,Z,\Omega)$ we can apply the construction of [25] and obtain a QK manifold $(\overline{N},g_{\overline{N}})$, which we call an instanton corrected q-map space. $(\overline{N},g_{\overline{N}})$ depends on a choice of projective special real (PSR) manifold $(\mathcal{H},g_{\mathcal{H}})$ (determining the first term in $\mathfrak{F}$), the choice of $\chi\in \mathbb{Z}$ and $n_{\hat{\gamma}} \in \mathbb{Z}$, and the choice of 1-loop parameter $c_{\ell}\in \mathbb{R}$ (see Section <ref>). Our main results concern the isometries of a lift $(\widetilde{N},g_{\overline{N}})\to (\overline{N},g_{\overline{N}})$ on which we have no periodic directions (see Definition <ref> for a more precise statement). In order to state the main results, we consider the following subgroups of the Heisenberg group $\text{Heis}_{2n+3}(\mathbb{R})$ (endowed with standard global coordinates $(\eta^i,\widetilde{\eta}_i,\kappa )$, $i=0,\ldots, n$):
\begin{equation}
\begin{split}
H_{2n+2}&:=\{ (\eta^i,\widetilde{\eta}_i,\kappa)\in \text{Heis}_{2n+3}(\mathbb{R}) \; |\quad \; \eta^0=0 \}\\
H_{2n+2,D}&:=\{ (\eta^i,\widetilde{\eta}_i,\kappa)\in H_{2n+2} \; |\quad \; \text{$\eta^a \in \mathbb{Z}$ for $a=1,...,n$}\}\,.
\end{split}
\end{equation}
The following theorem collects our main results:
Theorem <ref>: Consider an instanton corrected q-map space $(\widetilde{N},g_{\overline{N}})$ of dimension $4n+4$ as defined in Section <ref> (in particular $\widetilde{N}$ here is the maximal domain of definition). Furthermore, let $T,S\in \mathrm{SL}(2,\mathbb{Z})$ be as in (<ref>), where $\mathrm{SL}(2,\mathbb{Z})$ acts on the ambient manifold $\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}\supset \overline{\mathcal{N}}_{\text{IIB}}\cong \overline{\mathcal{N}}_{\text{IIA}}\supset \widetilde{N}$ as described in (<ref>). Then:
* $(\widetilde{N},g_{\overline{N}})$ has a group acting by isometries of the form
\begin{equation}\label{breakisointroduction}
\langle T \rangle \ltimes (\mathbb{Z}^n \ltimes H_{2n+2,D})
\end{equation}
where $\langle T \rangle \cong \mathbb{Z}$ denotes the subgroup of $\mathrm{SL}(2,\mathbb{Z})$ generated by $T$.
* Assume that we take the one-loop parameter to be $c_{\ell}=\frac{\chi}{192\pi}$. Then we can always find a non-empty open subset $\widetilde{N}_S\subset \widetilde{N}$ where $(\widetilde{N}_S,g_{\overline{N}})$ has a group acting by isometries of the form
\begin{equation}
\langle S \rangle \ltimes (\mathbb{Z}^n \ltimes H_{2n+2,D}),
\end{equation}
where $\langle S \rangle \cong \mathbb{Z}/4\mathbb{Z}$ is the subgroup generated by $S$. Furthermore, if $\widetilde{N}_{\mathrm{SL}(2,\mathbb{Z})}\subset \widetilde{N}$
is an open subset, which is $\mathrm{SL}(2,\mathbb{Z})$-invariant under the S-duality action (<ref>), then $\mathrm{SL}(2,\mathbb{Z})$ acts by isometries on $(\widetilde{N}_{\mathrm{SL}(2,\mathbb{Z})},g_{\overline{N}})$. In particular, if $\widetilde{N}$ is already invariant under $\mathrm{SL}(2,\mathbb{Z})$ then (<ref>) can be enhanced to
\begin{equation}
\mathrm{SL}(2,\mathbb{Z})\ltimes (\mathbb{Z}^n\ltimes H_{2n+2,D})\,.
\end{equation}
* Finally, if $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}\in \Lambda^{+}$, then in the previous statements we can replace $\mathbb{Z}^n$ and $H_{2n+2,D}$ by $\mathbb{R}^n$ and $H_{2n+2}$.
If furthermore we take $\chi=c_{\ell}=0$ and $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}\in \Lambda^{+}$, then we return to the tree-level q-map space case, where there is a connected $3n+6$ dimensional Lie group $G$ acting by isometries on $(\widetilde{N},g_{\overline{N}})$, see <cit.>. The group $G$ in particular contains the S-duality action by $\mathrm{SL}(2,\mathbb{R})$, an action by $\mathbb{R}^{n}\ltimes H_{2n+2}$, and a dilation action by $\mathbb{R}_{>0}$.
* Note that from Theorem <ref> one finds that
$\langle T \rangle \ltimes (\mathbb{Z}^n \ltimes H_{2n+2,D})$ is a universal group of isometries, in the sense that it is always an isometry group for any instanton corrected q-map space (provided one takes the maximal domain of definition $\widetilde{N}$ of the metric $g_{\overline{N}}$). On the other hand, even in the case of $c_{\ell}=\frac{\chi}{192\pi}$, Theorem <ref> does not guarantee in general that $\mathrm{SL}(2,\mathbb{Z})$ is an isometry group for $(\widetilde{N},g_{\overline{N}})$, but rather one must first check that $\widetilde{N}$ (or an open subset) carries an action by S-duality. In particular, Theorem <ref> does not let us conclude that $\mathrm{SL}(2,\mathbb{Z})$ is a universal group of isometries. This should be contrasted to the tree-level q-map space case, where $\mathrm{SL}(2,\mathbb{R})$ is known to always act by isometries.
* We remark that the action of $S\in \mathrm{SL}(2,\mathbb{Z})$ is perhaps the most interesting and non-trivial within $\mathrm{SL}(2,\mathbb{Z})$, and corresponds to interchanging weak coupling and strong coupling in the type IIB string theory setting. On the other hand, the action by $T\in \mathrm{SL}(2,\mathbb{Z})$ generates the discrete Heisenberg isometry that is missing from $H_{2n+2,D}$.
In Section <ref> we give an explicit example where we can achieve full $\mathrm{SL}(2,\mathbb{Z})$ acting by isometries. More precisely, we consider the case where $\mathfrak{F}$ is given simply by
\begin{equation}\label{exint}
\mathfrak{F}=-\frac{1}{6}\frac{(Z^1)^3}{Z^0}+\chi \frac{(Z^0)^2 \zeta(3)}{2(2\pi \mathrm{i})^3}, \quad \chi>0\,,
\end{equation}
with variation of BPS structures having BPS indices of the form
\begin{equation}\label{exint2}
\Omega(\gamma)=\begin{cases}
\Omega(q_0\gamma^0)=-\chi, \quad q_0\neq 0 \\
\Omega(\gamma)=0 \quad \text{else}.\\
\end{cases}
\end{equation}
From this data one obtains an 8-dimensional instanton corrected q-map space. It satisfies the following:
Corollary <ref> and <ref>: let $\widetilde{N}$ be defined by (<ref>) and take $c_{\ell}=\frac{\chi}{192\pi}$. Then the instanton corrected q-map metric $g_{\overline{N}}$ associated to (<ref>) and (<ref>) is defined and positive definite on $\widetilde{N}$. Furthermore, it carries an effective action by isometries by a group of the form $\mathrm{SL}(2,\mathbb{Z})\ltimes(\mathbb{R}\ltimes H_4)$.
Theorem <ref>: Let $(\widetilde{N},g_{\overline{N}})$ be as in Corollary <ref>. Then:
* There is a free and properly discontinuous action by isometries of a discrete group of the form $\mathrm{SL}(2,\mathbb{Z})'\ltimes \Lambda$, where $\Lambda \subset \mathbb{R}\ltimes H_4$ is a lattice, $\mathrm{SL}(2,\mathbb{Z})' \subset \mathrm{SL}(2,\mathbb{Z})$ is a finite index subgroup and the QK manifold $(\widetilde{N}/(\mathrm{SL}(2,\mathbb{Z})'\ltimes \Lambda),g_{\overline{N}})$ has finite volume.
* Furthermore, there is a submanifold with boundary $\hat{N}\subset \widetilde{N}$ where $\mathrm{SL}(2,\mathbb{Z})'\ltimes \Lambda$ acts and the quotient $(\hat{N}/(\mathrm{SL}(2,\mathbb{Z})'\ltimes \Lambda),g_{\overline{N}})$ gives a complete QK manifold with boundary[Recall that a Riemannian manifold with boundary is complete if it is complete as a metric space with the induced distance function.] and of finite volume. The manifold with boundary $\hat{N}$ is of the form $\hat{N}=\widetilde{N}'\cup \partial \hat{N}$, where $\widetilde{N}'$ is defined in the second point of Remark <ref>.
We note that the example $(\widetilde{N},g_{\overline{N}})$ of Corollary <ref> is incomplete (see Remark <ref>). We do not know if the metric and the $\mathrm{SL}(2,\mathbb{Z})$-action can be extended to a complete manifold (without boundary).
At the end of Remark <ref> we comment about some expectations of a related example associated to the resolved conifold.
Finally, we make a short comment related to the swampland program in physics. Among the geometric properties expected from the moduli space $\mathcal{M}$ of a low energy effective theory consistent with quantum gravity, are that $\mathcal{M}$ should be non-compact, have finite volume, and be geodesically complete (see [37] and <cit.>). In particular, applied to type IIA/B string theory, they imply that $\mathcal{M}_{\text{HM}}^{\text{IIA/B}}(X)$ must be a non-compact complete QK manifold of finite volume (after including all quantum corrections). On the other hand, the example from Theorem <ref> produces a non-compact QK manifold of finite volume, with “partial completeness" in the sense that it has a complete end, and a boundary where the metric is geodesically incomplete. It would be interesting to see if a suitable extension of the example of Theorem <ref> would produce a QK manifold with the required geometric properties expected by the swampland conjectures.
§.§ Organization of topics
The topics are organized as follows:
* In Section <ref> we review the construction of instanton corrected c-map metrics from [25], and also discuss their twistor description. In particular, the instanton corrected c-map spaces from [25] are in the image of the HK/QK correspondence, and we want to recall a description of the QK twistor space in terms of the HK data, as done in <cit.>.
* In Section <ref> we start by specifying the class of instanton corrected q-map spaces within the class of instanton corrected c-map metrics. Following closely the work in the physics literature of [15, 11], we study when an instanton corrected q-map space carries an $\mathrm{SL}(2,\mathbb{Z})$-action by isometries, or at least an action by some of its the subgroups, like $\langle S \rangle \subset \mathrm{SL}(2,\mathbb{Z})$.
* In Section <ref> we study certain universal isometries of instanton corrected q-map spaces and how they are related to the $\mathrm{SL}(2,\mathbb{Z})$ S-duality symmetry. We collect the main results from Section <ref> and Section <ref> in Theorem <ref>.
* In Section <ref>, we give an explicit example of an instanton corrected q-map space where the full $S$-duality acting by isometries is realized. Furthermore, we show that it admits a quotient of finite volume by a discrete group of isometries.
* Finally, in Appendix <ref> we collect some useful integral identities involving Bessel functions, and in Appendix <ref> we include a rather long computation that is not really needed for the main points of the paper, but we include for completeness.
Acknowledgements: this work was supported by the Deutsche Forschungsgemeinschaft (German Research Foundation) under Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306. As with our previous related joint works [25, 26], the idea for this work originated from discussions within our Swampland seminar, which is part of the Cluster of Excellence Quantum Universe. The authors would like to thank Murad Alim, Jörg Teschner and Timo Weigand for their contributions in the aforementioned discussions.
§ INSTANTON CORRECTED QK METRICS AND THEIR TWISTOR DESCRIPTION
The main aims of this section are the following:
* On one hand, we want to recall the main results of [25], concerning the construction of QK metrics associated to certain CASK manifolds with mutually local variation of BPS structures. In the setting of Calabi-Yau compactifications of string theory, these metrics are related to the type IIA hypermultiplet metric with mutually local D-instanton corrections, studied in the physics literature in [2].
* On the other hand, we want to recall certain general facts about the twistor space of QK metrics in the image of the HK/QK correspondence. This part will be mostly based on <cit.>. In particular, the QK metrics from the previous point lie in the image of the HK/QK correspondence, and we want to write down an explicit expression for the holomorphic contact structure of its twistor space in terms of the HK data (see (<ref>) and (<ref>)). These formulas will be used throughout the rest of this work, and in particular in Section <ref>, where we study the isometries of instanton corrected q-map spaces.
* Finally, we write down certain “type IIA" Darboux coordinates for the holomorphic contact structure, see (<ref>). These have been previously written down in the physics literature [12], under slightly different conventions. Using the explicit formula for the contact structure obtained in the previous point, we will give in the Appendix <ref> a direct proof of the fact that they are Darboux coordinates. This particular result is not needed for Section <ref>, where certain "type IIB" Darboux coordinates are found, but we include it for completeness.
§.§ QK metrics associated to CASK manifolds with mutually local variations of BPS structures
We briefly recall the main ingredients in the construction of [25].
An integral conical affine special Kähler (CASK) manifold is a tuple $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ where:
* $(M,g_M,\omega_M,\nabla)$ is an affine special Kähler (ASK) manifold. Namely, $(M,g_M,\omega_M)$ is pseudo-Kähler, with the complex structure $J$ determined by the metric $g_M$ and the Kähler form $\omega_M$ by $g_M(J-,-)=\omega_M(-,-)$; $\nabla$ is a torsion-free flat connection with $\nabla \omega_M=0$; and if $d_{\nabla}:\Omega^k(M,TM)\to \Omega^{k+1}(M,TM)$ denotes the extension of $\nabla:\Omega^0(M,TM)\to \Omega^{1}(M,TM)$ to higher degree forms, then $d_{\nabla}J=0$, where we think of $J$ as an element of $\Omega^1(M,TM)$.
* $\Gamma\subset TM$ is a sub-bundle of $\nabla$-flat lattices with $\Gamma\otimes_{\mathbb{Z}}\mathbb{R}=TM$. Around any $p\in M$, we can find a local trivialization $(\widetilde{\gamma}_i,\gamma^i)$ of $\Gamma$ of Darboux frames with respect to $\omega_M$. We denote $\langle -,- \rangle:=\omega_M(-,-)|_{\Gamma\times \Gamma}$, and our conventions are that $\langle \widetilde{\gamma}_i,\gamma^j\rangle=\delta_i^j$.
* $\xi$ is a vector field on $M$ such that $\nabla \xi=D\xi=\text{Id}_{TM}$, where $D$ denotes the Levi-Civita connection of $g_M$, and $\text{Id}_{TM}$ is the identity endomorphism of $TM$. Furthermore, we assume that $g_M$ is positive definite on $\mathcal{D}:=\text{span}\{\xi,J\xi\}$ and negative definite on $\mathcal{D}^{\perp}$.
On the other hand, the data corresponding to the mutually local instanton corrections in the string theory setting was specified in terms of the notion of a mutually local variation of BPS structures (see for example [17] for the more general notion of a variation of BPS structures).
A variation of mutually-local BPS structures over the complex manifold $M'$ is a tuple $(M',\Gamma',Z,\Omega)$ where
* $\Gamma' \to M'$ is a local system of lattices with a skew-pairing $\langle - , - \rangle:\Gamma' \times \Gamma' \to \mathbb{Z}$.
* $Z$ is a holomorphic section of $(\Gamma')^*\otimes \mathbb{C}\to M'$, where $(\Gamma')^*$ denotes the dual local system of $\Gamma'$. If $\gamma$ is a local section of $\Gamma'$, then we denote by $Z_{\gamma}:=Z(\gamma)$ the corresponding local holomorphic function on $M'$.
* $\Omega: \Gamma' -\{0\}\to \mathbb{Z}$ is a function of sets satisfying $\Omega(\gamma)=\Omega(-\gamma)$ and the following properties
* Mutual-locality: if we define $\text{Supp}(\Omega):=\{\gamma \in \Gamma' -\{0\}\;\; | \;\; \Omega(\gamma)\neq 0\}$, then $\gamma_1,\gamma_2\in \Gamma'_p\cap \text{Supp}(\Omega)$ implies that $\langle \gamma_1,\gamma_2\rangle=0$.
* Support property: for any compact set $K\subset M'$ and a choice of covariantly constant norm $|\cdot |$ on $\Gamma'|_{K}\otimes_{\mathbb{Z}}\mathbb{R}$, there is a constant $C>0$ such that for all $\gamma \in \Gamma'|_{K}\cap \text{Supp}(\Omega)$
\begin{equation}
\end{equation}
* Convergence property: for any $R>0$, the series
\begin{equation}\label{convpropBPS}
\sum_{\gamma\in \Gamma'|_p}\Omega(\gamma)e^{-R|Z_{\gamma}|}
\end{equation}
converges normally over compact subsets of $M'$.
* The numbers $\Omega(\gamma)$, called BPS indices, are monodromy invariant. Namely if $\gamma$ has monodromy $\gamma \to A\gamma$ around a loop, then $\Omega(\gamma)=\Omega(A\gamma)$.
Given an integral CASK manifold $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$, we will only consider mutually local variations of BPS structures $(M',\Gamma',Z,\Omega)$ where $(M,\Gamma)=(M',\Gamma')$, $\langle -, - \rangle=\omega_M(-,-)|_{\Gamma\times \Gamma}$, and where $Z$ is the canonical central charge associated to the integral CASK manifold <cit.>. The later is determined as follows: if $\xi^{1,0}=\frac{1}{2}(\xi-\I J\xi)$, then
\begin{equation}
\end{equation}
In particular, given a local Darboux frame $(\widetilde{\gamma}_i,\gamma^i)$ of $\Gamma$, the locally defined functions $\{Z_{\widetilde{\gamma}_i}\}_{i=0}^{n}$ and $\{Z_{\gamma^i}\}_{i=0}^{n}$ give conjugate systems of holomorphic special coordinates for the CASK geometry, where $n+1=\text{dim}_{\mathbb{C}}(M)$.
§.§.§ Associated instanton corrected HK manifold
To the data $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ and $(M,\Gamma,Z,\Omega)$ one can associate an “instanton-corrected" hyperkähler (HK) geometry <cit.>. This HK manifold can be thought as a deformation of the canonical HK manifold associated to $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ via the rigid c-map (also known as the associated “semi-flat" HK manifold) [22, 24, 31, 5]. In the physics literature, see [33], such instanton corrected HK metrics were studied in the context of $S^1$-compactifications of 4d $\mathcal{N}=2$ SUSY gauge theories. There the description of the HK geometry is in terms of its associated twistor space, which in turn in described in terms of a twistor family of holomorphic Darboux coordinates satisfying “TBA"-like integral equations.
In order to describe the instanton corrected HK manifold, we first let $N:=T^*M/\Gamma^*$. This can be canonically identified with
\begin{equation}
N\cong \{\zeta: \Gamma \to \mathbb{R}/ \mathbb{Z} \; | \; \zeta_{\gamma+\gamma'}=\zeta_{\gamma}+\zeta_{\gamma'}\}\,.
\end{equation}
In particular, slightly abusing notation and denoting by $\zeta$ the evaluation map on $N$, and given a local Darboux frame $(\widetilde{\gamma}_i,\gamma^i)$ of $\Gamma$, we obtain local coordinates on $N$ by $(Z_{\gamma^i},\zeta_{\widetilde{\gamma}_i},\zeta_{\gamma^i})$ (or $(Z_{\widetilde{\gamma}_i},\zeta_{\widetilde{\gamma}_i},\zeta_{\gamma^i})$). Note that $Z_{\gamma^i}$ and $Z_{\widetilde{\gamma_i}}$ are (pull-backs of local) holomorphic functions on the base manifold $M$ while $\zeta_{\widetilde{\gamma}_i}$ and $\zeta_{\gamma^i}$ are “fiber coordinates" taking values in the circle.
In the following, we will also denote by $\langle -, -\rangle$ the pairing on $\Gamma^*$ induced by the isomorphism $\gamma \mapsto \langle \gamma, - \rangle$. With this definition, the dual of a Darboux basis of $\Gamma$ is a Darboux basis of $\Gamma^*$. We will also denote by $\langle -, -\rangle$ the $\mathbb{C}$-linear extension of the pairing to $\Gamma^*\otimes \mathbb{C}$.
Finally, if $K_i: \mathbb{R}_{>0}\to \mathbb{R}$ denotes the $i$-th modified Bessel function of the second kind, and $\gamma$ is a local section of $\Gamma$ with $\gamma \in \text{Supp}(\Omega)$, we define the following local function (resp. 1-form) on $N$:
\begin{equation}\label{VAinst}
V_{\gamma}^{\text{inst}}:=\frac{1}{2\pi}\sum_{n>0}e^{2\pi\I n\zeta_{\gamma}}K_0(2\pi n|Z_{\gamma}|), \quad A_{\gamma}^{\text{inst}}:=-\frac{1}{4\pi}\sum_{n>0}e^{2\pi\I n\zeta_{\gamma}}|Z_{\gamma}|K_1(2\pi n|Z_{\gamma}|)\Big( \frac{\mathrm{d}Z_{\gamma}}{Z_{\gamma}}-\frac{\mathrm{d}\overline{Z}_{\gamma}}{\overline{Z}_{\gamma}}\Big)\,.
\end{equation}
Due to the convergence property and support property of variations of BPS structures, these expressions are well-defined local smooth functions (resp. $1$-forms) on $N$ (see <cit.>).
Finally, we will need the following compatibility condition between the data $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ and $(M,\Gamma,Z,\Omega)$:
Let $\pi:N\to M$ be the canonical projection. We will say that $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ and $(M,\Gamma,Z,\Omega)$ are compatible if the tensor
\begin{equation}\label{non-deg}
T:=\pi^*g_{M}- \sum_{\gamma}\Omega(\gamma)V_{\gamma}^{\text{inst}}\pi^*|\mathrm{d}Z_{\gamma}|^2
\end{equation}
on $N$ is horizontally non-degenerate.
We then have the following:
<cit.> Let $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ and $(M,\Gamma,Z,\Omega)$ be as before. Furthermore, let $\omega_i\in \Omega^2(N)$ for $i=1,2,3$ be defined by
\begin{equation} \label{holsym}
\omega_1+\mathrm{i}\omega_2:=-2\pi\left(\langle \mathrm{d}Z\wedge \mathrm{d}\zeta \rangle + \sum_{\gamma}\Omega(\gamma)\left(\mathrm{d}Z_{\gamma}\wedge A_{\gamma}^{\text{inst}} +\I V_{\gamma}^{\text{inst}}\mathrm{d}\zeta_{\gamma}\wedge \mathrm{d}Z_{\gamma}\right)\right)
\end{equation}
\begin{equation}\label{invKF}
\omega_{3}:=2\pi\left(\frac{1}{4}\langle \mathrm{d}Z\wedge \mathrm{d}\overline{Z}\rangle-\frac{1}{2} \langle \mathrm{d}\zeta\wedge \mathrm{d}\zeta \rangle -\sum_{\gamma}\Omega(\gamma)\left( \frac{\I}{2} V^{\text{inst}}_{\gamma}\mathrm{d}Z_{\gamma}\wedge \mathrm{d}\overline{Z}_{\gamma}+\mathrm{d}\zeta_{\gamma}\wedge A_{\gamma}^{\text{inst}}\right)\right)\,.
\end{equation}
Then the triple of real $2$-forms $(\omega_1,\omega_2, \omega_3)$ corresponds to the Kähler forms of a pseudo-HK structure[Our terminology is such that the signature of the metric is not assumed to be constant, in case $N$ has several components.] on $N$ if and only if $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ and $(M,\Gamma,Z,\Omega)$ are compatible.
We denote the resulting instanton corrected HK manifold from the previous theorem by $(N,g_N,\omega_1,\omega_2,\omega_3)$.
* Compared to <cit.>, we have rescaled the above 2-forms $\omega_i$ by a factor of $2\pi$, and rescaled $\zeta$ by $2\pi$ (i.e. $\zeta:\Gamma \to \mathbb{R}/\mathbb{Z}$ instead of $\zeta:\Gamma \to \mathbb{R}/2\pi\mathbb{Z}$). Furthermore, we have changed by a sign the convention of how the BPS indices $\Omega$ enter into the formulas (<ref>), (<ref>) (i.e. the above formulas would correspond in [25] to the HK metric associated to the mutually local variation of BPS structures $(M,\Gamma,Z,-\Omega)$). We do this change of conventions in order to simplify the formulas taken from <cit.> below and also to be able to compare more easily with the physics literature in Section <ref> below.
* In the expressions (<ref>) and (<ref>) we are combining the wedge $\wedge$ with the pairing $\langle -, - \rangle$ on $\Gamma^*\otimes \mathbb{C}$. For example, with respect to a Darboux frame $(\widetilde{\gamma}_i,\gamma^i)$ of $\Gamma$, we have $\langle \mathrm{d}Z\wedge \D\overline{Z}\rangle=\D Z_{\widetilde{\gamma}_i}\wedge \D \overline{Z}_{\gamma^i} -\D Z_{\gamma^i}\wedge \D \overline{Z}_{\widetilde{\gamma}_i}$ and $\langle \mathrm{d}\zeta \wedge \mathrm{d}\zeta \rangle=\mathrm{d}\zeta_{\widetilde{\gamma}_i}\wedge \mathrm{d}\zeta_{\gamma^i}-\mathrm{d}\zeta_{\gamma^i}\wedge\mathrm{d}\zeta_{\widetilde{\gamma}_i}=2\mathrm{d}\zeta_{\widetilde{\gamma}_i}\wedge \mathrm{d}\zeta_{\gamma^i}$. Furthermore, the expressions in (<ref>) and (<ref>) are actually global and well-defined due to the monodromy invariance of $\Omega(\gamma)$, and the support and convergence property of the variations of BPS structures.
* $(N,g_N,\omega_1,\omega_2,\omega_3)$ carries an infinitesimal rotating circle action <cit.>. Namely, there is a vector field $V$ on $N$ such that
\begin{equation}\label{rotvec}
\mathcal{L}_V(\omega_1+\I\omega_2)=2\I(\omega_1+\I\omega_2), \quad \mathcal{L}_V\omega_3=0\,.
\end{equation}
Note that due to the factor $2$ in (<ref>) the vector field $V$ is twice the vector field denoted $V$ in [25].
* Under the mild assumption on the flow of $\xi$ that it generates a free-action on $M$ of the (multiplicative) monoid $\mathbb{R}_{\geq 1}$, we can guarantee that $g_N$ has signature $(4,4n)$ where $n+1=\text{dim}_{\mathbb{C}}(M)$ <cit.>.
* If one sets $\Omega(\gamma)=0$ for all $\gamma \in \Gamma$, then $(N,g_N,\omega_1,\omega_2,\omega_3)$ reduces to the semi-flat HK manifold obtained via the rigid c-map.
§.§.§ Associated instanton corrected QK manifold via HK/QK correspondence
The instanton corrected QK manifold $(\overline{N},g_{\overline{N}})$ associated to the data of $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ and $(M,\Gamma,Z,\Omega)$ was constructed in [25] by applying the HK/QK correspondence to $(N,g_N,\omega_1,\omega_2,\omega_3)$. In order to do this, one needs the following additional data:
* A hyperholomorphic principal $S^1$-bundle $(\pi_N:P\to N,\eta)$, where $\pi_N:P\to N$ is constructed via <cit.>, and $\eta$ is a connection on $P$ having curvature
\begin{equation}
\D \eta=\pi_N^*(\omega_3 -\frac{1}{2}\D\iota_V g_N)\,.
\end{equation}
The connection $\eta$ is given by <cit.>
\begin{equation}\label{etadef}
\eta:=\Theta +\pi^*_N\Big(\frac{\pi \I}{2}\pi^*_M( \overline{\partial} r^2-\partial r^2) - \sum_{\gamma}2\pi \Omega(\gamma)\eta_{\gamma}^{\text{inst}}-\frac{1}{2}\iota_Vg_N\Big)
\end{equation}
where $r^2:=g_M(\xi,\xi)$; $V$ is the rotating vector field satisfying (<ref>); $\Theta$ is another connection on $P$ having curvature $-\pi\langle \D \zeta \wedge \D \zeta \rangle=2\pi \mathrm{d}\zeta_{\gamma^i}\wedge \mathrm{d}\zeta_{\widetilde{\gamma}_i}$,
\begin{equation}\label{etagammainst}
\eta_{\gamma}^{\text{inst}}:=\frac{\I}{8\pi^2}\sum_{n>0}\frac{e^{2\pi \I n\zeta_{\gamma}}}{n}|Z_{\gamma}|K_1(2\pi n|Z_{\gamma}|)\Big(\frac{\D Z_{\gamma}}{Z_{\gamma}}-\frac{\D \overline{Z}_{\gamma}}{\overline{Z}_{\gamma}}\Big)\,.
\end{equation}
If $\sigma$ denotes a local coordinate for the $S^1$-fiber, then one can write
\begin{equation}\label{Thetacon}
\Theta=\pi\left(\D\sigma -\pi_N^*\langle \zeta,\D\zeta \rangle\right)\,.
\end{equation}
* We need to furthermore specify a Hamiltonian for $\omega_3$ with respect to the rotating vector field $V$ satisfying (<ref>), which in this case is given by <cit.>
\begin{equation}\label{fdef}
f=2\pi (r^2- 8c_{\ell} - \sum_{\gamma}\Omega(\gamma)\iota_V\eta_{\gamma}^{\text{inst}}),\quad c_{\ell} \in \mathbb{R},
\end{equation}
together with the lift $V^P$ of $V$ to $P$ given by
\begin{equation}
V^P:=\widetilde{V}+f_3\partial_{\sigma}, \quad f_3:=f-\frac{1}{2}g_{N}(V,V)
\end{equation}
where $\widetilde{V}$ denotes the horizontal lift with respect to $\eta$ and $\partial_{\sigma}$ is the vertical vector field of $P$ generating the $S^1$-action.
* Finally, we consider the open subset $N'\subset N$ given by
\begin{equation}\label{n'def}
N'=\{p\in N \quad | \quad f(p)\neq 0, \quad f_3(p)\neq 0, \quad g_N(V_p,V_p)\neq 0\},
\end{equation}
and the $1$-forms on $P$ given by
\begin{equation}\label{thetadef}
\theta_0^P=-\frac{1}{2}\pi_N^*\mathrm{d}f, \quad \theta_3^P:=\eta +\frac{1}{2}\pi_N^*\iota_Vg_N, \quad
\theta_1^P:=\frac{1}{2}\pi_N^*\iota_V\omega_2,\quad
\theta_2^P:=-\frac{1}{2}\pi_N^*\iota_V\omega_1\,,
\end{equation}
We then have
Let $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ be an integral CASK manifold and $(M,\Gamma,Z,\Omega)$ a compatible mutually local variation of BPS structures. Furthermore, let $(N,g_N,\omega_1,\omega_2,\omega_3)$, $(\pi_N:P\to N,\eta)$, $f$, $f_3$, $\theta_i^P$, $V^P$ and $N'$ the associated data defined in the previous points. Given any submanifold $\overline{N}\subset P|_{N'}$ transverse to $V^P$, the symmetric $2$-tensor
\begin{equation}
\end{equation}
defines a pseudo-QK metric on $\overline{N}$. Furthermore, if $(N,g_N,\omega_1,\omega_2,\omega_3)$ has signature $(4,4n)$, then $g_{\overline{N}}$ is positive definite on $\overline{N}_{+}=\overline{N}\cap\{f>0,f_3<0\}$.
Recall that we can guarantee that $(N,g_N,\omega_1,\omega_2,\omega_3)$ has signature $(4,4n)$ if the flow of $\xi$ generates a free-action on $M$ of the monoid $\mathbb{R}_{\geq 1}$ (see Section <ref>).
§.§.§ The case of a CASK domain
We now specialize the previous construction to the case of a CASK domain. This will be the case of interest in the following Section <ref>.
A CASK domain is a tuple $(M,\mathfrak{F})$ where
* $M\subset \mathbb{C}^{n+1}-\{0\}$ is a $\mathbb{C}^{\times}$-invariant domain. We denote the canonical holomorphic coordinates by $Z^i$, $i=0,1,...,n$. To avoid inessential coordinate changes later on, we
will assume for simplicity that $Z^0$ does not vanish on $M$.
* $\mathfrak{F}:M\to \mathbb{C}$ is a holomorphic function, homogeneous of degree $2$ with respect to the natural $\mathbb{C}^{\times}$-action on $M$.
* The matrix
\begin{equation}
\text{Im}\left(\tau_{ij}\right), \quad \tau_{ij}:=\frac{\partial^2 \mathfrak{F}}{\partial Z^i\partial Z^j}
\end{equation}
has signature $(n,1)$, and $\text{Im}(\tau_{ij})Z^i\overline{Z}^j<0$.
A CASK domain $(M,\mathfrak{F})$ induces in the usual way a CASK manifold $(M,g_M,\omega_M,\nabla,\xi)$ [5]. With our conventions on the signature of the CASK manifold, if $Z_i:=\frac{\partial \mathfrak{F}}{\partial Z^i}$, then $\{Z^i\}$ and $\{-Z_i\}$ are a global system of conjugate conical special holomorphic coordinates. If $x^i=\text{Re}(Z^i)$ and $y_i:=\text{Re}(Z_i)$, then $\nabla$ is defined such that $\D x^i$ and $\D y_i$ are flat. Furthermore
\begin{equation}\label{CASKconvention}
g_M=-\text{Im}(\tau_{ij})\D Z^i \D\overline{Z}^j, \quad \omega_M=-\frac{\mathrm{i}}{2}\text{Im}(\tau_{ij})\D Z^i\wedge \D\overline{Z}^j=\D x^i\wedge \D y_i, \quad \xi=Z^i\partial_{Z^i} +\overline{Z}^i\partial_{\overline{Z}^i}\,.
\end{equation}
Given a CASK domain $(M,\mathfrak{F})$ we can induce a canonical integral structure on the CASK manifold by defining $\Gamma \to M$ to be $\Gamma=\text{span}_{\mathbb{Z}}\{\partial_{x^i},\partial_{y_i}\}$. In the following, we will assume that:
* $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ is an integral CASK manifold induced by a CASK domain $(M,\mathfrak{F})$ with the canonical integral structure. In this case, we will sometimes use the notation $(\partial_{x^i},\partial_{y_i})=(\widetilde{\gamma}_i,\gamma^i)$.
* Given a mutually local variation of BPS structures $(M,\Gamma,Z,\Omega)$ with $(M,\Gamma)$ as in the previous point, we assume that $Z$ is the canonical central charge, and that $\text{Supp}(\Omega)\subset \text{span}_{\mathbb{Z}}\{\partial_{y_i}\}$. In particular, the canonical central charge satisfies in this case
\begin{equation}
Z=Z_{\widetilde{\gamma}_i}\mathrm{d}x^i+Z_{\gamma^i}\mathrm{d}y_i=-Z_i\D x^i+Z^i\D y_i\,.
\end{equation}
In order to construct the associated QK metric, we need to choose $\overline{N}\subset P|_{N'}$ transverse to $V^P$. Denoting by $\pi_M:=\pi\circ \pi_N$ the composition of the projections $\pi_N:P\to N$, and $\pi:N\to M$, we have that $\pi_M(p)=(Z^0,...,Z^n)$. We define $\overline{N}$ by
\begin{equation}\label{QKCASKdomain}
\overline{N}:=\{p \in P|_{N'} \quad| \quad \text{Arg}(Z^0)=0,\quad \text{ where $\pi_M(p)=(Z^0,...,Z^n)$} \}\,.
\end{equation}
Throughout the paper, we will use coordinates $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)$ on (<ref>) defined as follows:
* $16\pi\rho:=2\pi r^2 -16\pi c_{\ell}$ where $r^2=g_M(\xi,\xi) = -\text{Im}(\tau_{ij}) Z^i \overline{Z}^j$ and $c_{\ell}\in \mathbb{R}$. Note that $r^2$ is a Kähler potential for the affine special Kähler metric $g_M$.
* $z^a:=Z^a/Z^0$ for $a=1,...,n$. These are global holomorphic coordinates on the induced projective special Kähler (PSK) manifold $(\overline{M},g_{\overline{M}})$ induced by the CASK domain. In particular, we have a projection $\pi_{\overline{M}}: M\to \overline{M}$.
* $(\zeta^i,\widetilde{\zeta}_i)$ are given by $\zeta^i:=-\zeta_{\partial_{y_i}}=-\zeta_{\gamma^i}$ and $\widetilde{\zeta}_i:=\zeta_{\partial_{x^i}}=\zeta_{\widetilde{\gamma}_i}$, where the latter are the evaluation map on $N$ contracted with $\partial_{x^i}$ and $\partial_{y_i}$.
* $\sigma$ is a local coordinate for the $S^1$-fiber of $\pi_N:P\to N$ satisfying (<ref>).
In the string theory setting, the coordinates $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)$ can be identified with certain fields from the type IIA hypermultiplet. Namely, $\rho$ is the 4d-dilaton, $z^a$ are coordinates for the complex moduli of the Calabi-Yau, $(\zeta^i,\widetilde{\zeta}_i)$
are the RR-axions, and $\sigma$ is the NS-axion. Furthermore, the constant $c_{\ell}$ appearing in the definition of $\rho$ is identified with the $1$-loop correction to the tree-level Ferrara-Sabharwal metric.
In <cit.> an explicit expression of the resulting QK metric $(\overline{N},g_{\overline{N}})$ in these coordinates is given (with slightly different conventions). In order to write down the formula we introduce the following notation:
* We denote by $\widetilde{Z}_{\gamma}:=Z_{\gamma}/Z^0$ the normalized central charge. In particular, we have $\widetilde{Z}_{\gamma^i}=z^i$ with $z^0=1$. If furthermore we let $\mathcal{K}=-\log(-2\text{Im}(\tau_{ij})z^i\overline{z}^j)$, then $\mathcal{K}$ is a global Kähler potential for the projective special Kähler manifold $(\overline{M},g_{\overline{M}})$ induced by the CASK domain. Note that
\begin{equation} \label{r2:eq}r^2 = |Z^0|^2\frac{e^{-\mathcal{K}}}{2}.\end{equation}
* We denote $N_{ij}=-2\text{Im}(\tau_{ij})$. If $\gamma \in \text{Supp}(\Omega)$ we write $\gamma=q_i(\gamma)\gamma^i$ (recall that we assume that $\text{Supp}(\Omega)\subset \text{span}_{\mathbb{Z}}\{\gamma^i\}$), and define
\begin{equation}
\end{equation}
* We let
\begin{equation}\label{instW}
W_i:=\D\zeta_{\widetilde{\gamma}_i}+\tau_{ij}\D\zeta_{\gamma^j}, \;\;\;\;\;\; W_i^{\text{inst}}:=-\sum_{\gamma}\Omega(\gamma)q_i(\gamma)( A_{\gamma}^{\text{inst}}-\I V_{\gamma}^{\text{inst}}\D\zeta_{\gamma}),
\end{equation}
\begin{equation}\label{etainstdef}
\eta^{\text{inst}}:=-\sum_{\gamma}\Omega(\gamma) \eta_{\gamma}^{\text{inst}}-\frac{1}{2}\iota_V\left(\frac{g_N}{2\pi}- \pi^*_Mg_{M}\right) \,.
\end{equation}
* We split $f=16\pi\rho+f^{\text{inst}}=16\pi(\rho+\rho^{\text{inst}})$ in (<ref>) using that $16\pi \rho=2\pi r^2-16\pi c_{\ell}$, and where $f^{\text{inst}}=16\pi \rho^{\text{inst}}$ contains the terms with the BPS indices $\Omega(\gamma)$, namely
\begin{equation}
\end{equation}
Finally, we denote $ f_3^{\text{inst}}=16\pi \rho_3^{\text{inst}}=2\pi \iota_V \eta^{\text{inst}}$. We remark that in the case where $\Omega(\gamma)=0$ for all $\gamma$ we have $f^{\text{inst}}=f_3^{\text{inst}}=0$, and similarly for all the other quantities with an $^\text{inst}$ superscript.
We then have
<cit.> Let $(M,\mathfrak{F})$ and $(M,\Gamma,Z,\Omega)$ be as before. By possibly restricting $M$, we assume that $M$ is the maximal open subset where $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ and $(M,\Gamma,Z,\Omega)$ are compatible. Furthermore, let $\overline{N}$ be as (<ref>). Then in the coordinates $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)$ from Definition <ref> the instanton corrected QK metric $(\overline{N},g_{\overline{N}})$ associated to $(M,g_M,\omega_M,\nabla,\xi,\Gamma)$ and $(M,\Gamma,Z,\Omega)$ has the form:
\begin{equation}\label{coordQKmetric2}
\begin{split}
g_{\overline{N}}=& \frac{\rho+c_{\ell}}{\rho+\rho^{\text{inst}}}\Big(g_{\overline{M}}+2e^{\mathcal{K}}\sum_{\gamma}\Omega(\gamma)V_{\gamma}^{\text{inst}}\Big|\mathrm{d}\widetilde{Z}_{\gamma} + \widetilde{Z}_{\gamma}\Big(\frac{\D\rho}{2(\rho+c_{\ell})}+\frac{\D\mathcal{K}}{2}\Big)\Big|^2\Big)\\
&+\frac{1}{2(\rho+\rho^{\text{inst}})^2}\Big(\frac{\rho +2c_{\ell} -\rho^{\text{inst}}}{2(\rho+c_{\ell})}\D\rho^2+2\D\rho \D \rho^{\text{inst}}|_{\overline{N}}+(\D \rho^{\text{inst}})^2|_{\overline{N}}\Big)\\
&+\frac{\rho+c_{\ell}+\rho_-^{\text{inst}}}{64(\rho+\rho^{\text{inst}})^2(\rho+2c_{\ell}-\rho_3^{\text{inst}})}\Big(\D \sigma -\langle \zeta,\D\zeta \rangle -4 c_{\ell}\D^c\mathcal{K}+\eta_+^{\text{inst}}|_{\overline{N}}+\frac{\rho_+^{\text{inst}}-c_{\ell}}{\rho+c_{\ell}+\rho_-^{\text{inst}}}\eta_-^{\text{inst}}|_{\overline{N}}\Big)^2\\
&-\frac{1}{4(\rho+\rho^{\text{inst}})}(W_i+W_i^{\text{inst}}|_{\overline{N}})(N+N^{\text{inst}})^{ij}(\overline{W}_j+\overline{W}_j^{\text{inst}}|_{\overline{N}}) \\
&+\frac{(\rho+c_{\ell})e^{\mathcal{K}}}{2(\rho+\rho^{\text{inst}})^2}\Big|z^i(W_i+W_i^{\text{inst}}|_{\overline{N}})- \frac{\mathrm{i}}{2}\sum_{\gamma}\Omega(\gamma) A_{\gamma}^{\text{inst}}(V)\Big(\D\widetilde{Z}_{\gamma} + \widetilde{Z}_{\gamma}\Big(\frac{\D\rho}{2(\rho+c_{\ell})}+\frac{\D \mathcal{K}}{2}\Big)\Big)\Big|^2\\
\end{split}
\end{equation}
where $\mathrm{d}^c:=\mathrm{i}(\overline{\partial}-\partial)$, $\eta^{\text{inst}}_{\pm}$ are given by
\begin{equation}
\eta_{\pm}^{\text{inst}}:=\Big( \eta^{\text{inst}}-4\rho_3^{\text{inst}}\widetilde{\eta}\Big)\pm \Big( \sum_{\gamma}\Omega(\gamma)\eta_{\gamma}^{\text{inst}}-4\rho^{\text{inst}}\widetilde{\eta}\Big)\,, \quad \widetilde{\eta}:=\mathrm{d}^c \log(r)\,,
\end{equation}
\begin{equation}
\rho_{\pm}^{\text{inst}}:=(\rho^{\text{inst}}\pm \rho_3^{\text{inst}})/2 \,.
\end{equation}
Furthermore, the open subset $\overline{N}_{+}=\overline{N}\cap\{f>0,f_3<0\}$ is non-empty, and $g_{\overline{N}}$ is positive-definite on $\overline{N}_{+}$.
* In Theorem <ref> we have relaxed a bit the possible restriction of $M$ compared to <cit.>. In [25] we assume that we can restrict to an $M$ invariant under the action of the monoid $\mathbb{R}_{\geq 1}\times S^1$ to make the CASK structure compatible with the BPS structure. This ensures that no matter the point of $\overline{M}$ the metric is defined for $\rho>K$ for some sufficiently big uniform $K$. Our weakened assumption makes it so that the constant $K$ might depend on the point $z^a\in \overline{M}$.
* When $\Omega(\gamma)=0$ for all $\gamma \in \Gamma$ the expression reduces to the 1-loop corrected Ferrara-Sabharwal metric:
\begin{equation}
\begin{split}
g_{\overline{N}}=& \frac{\rho+c_{\ell}}{\rho}g_{\overline{M}}+\frac{\rho +2c_{\ell}}{4\rho^2(\rho+c_{\ell})}\D\rho^2+\frac{\rho+c_{\ell}}{64\rho^2(\rho+2c_{\ell})}\Big(\D \sigma -\langle \zeta,\D\zeta \rangle -4 c_{\ell}\D^c\mathcal{K}\Big)^2\\
&-\frac{1}{4\rho}\left(N^{ij}-\frac{2(\rho+c_{\ell})e^{\mathcal{K}}}{\rho }z^i\overline{z}^j\right)W_i\overline{W}_j\,.
\end{split}
\end{equation}
In particular $(\overline{N},g_{\overline{N}})$ can be thought as a deformation of the $1$-loop corrected metric.
* Since the instanton corrections of the HK geometry are exponentially suppressed as $|Z_{\gamma}|\to \infty$ for $\gamma \in \text{Supp}(\Omega)$, it is easy to check that the possibly restricted $M$ from above satisfying the required conditions is never empty. Furthermore, on $\overline{N}$ the instanton corrections of the QK geometry are exponentially suppressed as $\rho \to \infty$ (due to the relation $|Z_{\gamma}|=|Z^0||\widetilde{Z}_{\gamma}|=4\sqrt{\rho+c_{\ell}}e^{\mathcal{K}/2}|\widetilde{Z}_{\gamma}|$), so we can ensure that $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)\in \overline{N}_{+}$ by taking $\rho$ sufficiently big.
* The function $f=16\pi\rho+f^{\text{inst}}$ can be thought in the string theory setting as the D-instanton corrected 4d dilaton (up to different conventions in the normalization) [9].
* When comparing (<ref>) to <cit.>, we note that here we are using different conventions for the normalization of $g_{N}$; the rotating vector field $V$; the functions $N_{ij}$, $N_{ij}^{\text{inst}}$, $e^{-\mathcal{K}}$, $\eta_{\pm}^{\text{inst}}$; the signature of $\text{Im}(\tau_{ij})$; and the coordinates $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)$. Compared to [25], $g_{N}$ is scaled by $2\pi$;
$V$, $N_{ij}$, $N_{ij}^{\text{inst}}$ and $e^{-\mathcal{K}}$ are scaled by $2$; $\eta_{\pm}^{\text{inst}}$ are scaled by $\pi^{-1}$; and the signature of $\text{Im}(\tau_{ij})$ is opposite. Furthermore, the coordinates and $1$-loop constant of [25] are related to the ones in Definition <ref> by performing the scaling
\begin{equation}\label{scaling}
\rho \to 16\pi\rho,\quad c_{\ell}\to 16\pi c_{\ell}, \quad \sigma \to \pi \sigma, \quad \zeta^i\to -2\pi\zeta^i, \quad \widetilde{\zeta}_i \to 2\pi \widetilde{\zeta}_i\,.
\end{equation}
Finally, as mentioned in Remark <ref>, the sign with which the $\Omega(\gamma)$ enter the formula in (<ref>) is opposite to <cit.>. This changes of convention will make the formulas from subsequent section look more simple and more easily comparable to the physics literature.
In what follows, it will be useful to consider the following lift of $(\overline{N},g_{\overline{N}})$:
We will denote by $(\widetilde{N},g_{\overline{N}})$ the QK manifold obtained by lifting the QK metric $(\overline{N},g_{\overline{N}})$ obtained in Theorem <ref> to the open subset $\widetilde{N}\subset \mathbb{R}_{>0}\times \overline{M}\times \mathbb{R}^{2n+2}\times \mathbb{R}$ obtained by considering $(\zeta^i,\widetilde{\zeta}_i,\sigma)\in \mathbb{R}^{2n+2}\times \mathbb{R}$ as (non-periodic) global coordinates of $\mathbb{R}^{2n+2}\times \mathbb{R}$. We will call such a space an instanton corrected c-map space.
§.§ Twistor space description and Darboux coordinates
Let $(\overline{N},g_{\overline{N}},Q)$ denote a QK manifold, where $Q\to \overline{N}$ denotes the associated quaternionic structure. Namely, a parallel subbundle $Q\subset \text{End}(T\overline{N})$ admiting local trivializations $(J_1,J_2,J_3)$ by skew endomorphisms satisfying the quaternion relations. The structure of a QK manifold $(\overline{N},g_{\overline{N}},Q)$ can be encoded in a holomorphic object, known as the twistor space $(\mathcal{Z},\mathcal{I},\lambda,\tau)$ [39, 35]. Here $\mathcal{Z}\to \overline{N}$ is a sphere subbundle of $Q$ defined by
\begin{equation}
\mathcal{Z}_p:=\{J\in Q_p \; |\; J^{2}=-1\}, \quad p \in \overline{N};
\end{equation}
$\mathcal{I}$ is a canonical holomorphic structure on $\mathcal{Z}$; $\lambda \in \Omega^1(\mathcal{Z},\mathcal{L})$ defines a holomorphic contact structure on $\mathcal{Z}$, where $\mathcal{L}\to \mathcal{Z}$ is a certain holomorphic line bundle; and $\tau$ is a real structure on $\mathcal{Z}$ (i.e. an antiholomorphic involution).
In what follows, we consider $(\overline{N},g_{\overline{N}})$ built in the previous Section <ref>, and its lift $(\widetilde{N},g_{\overline{N}})$. We start by recalling the following from the discussion in <cit.>:
Let $(\widetilde{N},g_{\overline{N}})$ be the lift of an instanton corrected QK manifold associated to a CASK domain $(M,\mathfrak{F})$ and mutually local variation of BPS structures $(M,\Gamma,Z,\Omega)$. Then $\mathcal{Z}\cong \widetilde{N}\times \mathbb{C}P^1$ (non-holomorphically), and there is a holomorphic coordinate $t$ on the $\mathbb{C}P^1$ factor and a holomorphic section $s$ of the holomorphic line bundle $\mathcal{L}\to \mathcal{Z}$ vanishing at $t=0,\infty$, such that
\begin{equation}\label{contactlocal}
\lambda=\left(f\frac{\D t}{t}+t^{-1}\theta_{+}^P|_{\overline{N}}-2\mathrm{i} \theta_3^P|_{\overline{N}} +t \theta_{-}^{P}|_{\overline{N}}\right)\cdot s
\end{equation}
where $\theta_{\pm}^P:=\theta_1^P\pm \mathrm{i}\theta_2^P$, and $\theta^P_i$ for $i=1,2,3$ are defined as in (<ref>), and $f$ is defined as in (<ref>).
This follows from the discussion in <cit.>, where the twistor space of a QK manifold $(\overline{N},g_{\overline{N}})$ obtained via HK/QK correspondence is described in terms of the “HK data" given by $(N,g_{N},\omega_1,\omega_2,\omega_3)$, $(\pi_N:P\to N,\eta)$, $f$, and the associated HK cone. In particular, whenever the QK manifold admits a global chart of coordinates, which is the case of $(\widetilde{N},g_{\overline{N}})$ obtained in Section <ref>, it follows that $\mathcal{Z}\cong \widetilde{N}\times \mathbb{C}P^1$ non-holomorphically. The formula (<ref>) follows from <cit.>.
The lift to $\widetilde{N}\rightarrow \overline{N}$ is understood.
In what follows, we will be concerned in describing Darboux coordinates for the contact structure $\lambda$ expressed as (<ref>), in the case where $(\widetilde{N},g_{\overline{N}})$ is the instanton corrected QK metric obtained in Section <ref>. For this it will be important to have an explicit expression for $f$, $\theta_{+}^P|_{\overline{N}}$ and $\theta_3^P|_{\overline{N}}$ ($\theta_-^P|_{\overline{N}}$ can be obtained from $\theta_-^P=\overline{\theta_+^P}$).
Consider the CASK domain $(M,\mathfrak{F})$ and mutually local variation of BPS structures $(M,\Gamma,Z,\Omega)$ as in Section <ref>. Then $f$, $\theta_{+}^P|_{\overline{N}}$ and $\theta_3^P|_{\overline{N}}$ from (<ref>) have the following formulas with respect to the coordinates $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)$:
\begin{align*}
f&=16\pi\rho + \frac{2R}{\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\mathrm{i}n\zeta_{\gamma}}}{n}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\\
\theta^{P}_{+}|_{\overline{N}}&=-4\pi R\langle \widetilde{Z},\D\zeta \rangle +2\mathrm{i}R\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\mathrm{i}n\zeta_{\gamma}}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\D\zeta_{\gamma}\\
&\quad \quad +2R^2\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi \mathrm{i}n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi R n|\widetilde{Z}_{\gamma}|)\left(\frac{\D\widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}+\frac{\D\overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}+\frac{\D\rho}{(\rho+c_{\ell})}+\D\mathcal{K}\right)\\
\theta_3^P|_{\overline{N}}&=\pi\D\sigma -\pi \langle\zeta, \D\zeta \rangle-4\pi(\rho+c_{\ell})\D^c\mathcal{K}-\frac{\mathrm{i}R}{2\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi \mathrm{i}n\zeta_{\gamma}}}{n}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D\widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}-\frac{\D\overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right) \numberthis \label{fthetaexpres}
\end{align*}
where $R:=2\sqrt{\rho+c_{\ell}}e^{\mathcal{K}/2}$, $\widetilde{Z}_{\gamma}=q_0+q_az^a$ and $\zeta_{\gamma}=-q_i\zeta^i$ for $\gamma=q_i\gamma^i$, and in the last formula we have used $\mathrm{d}^c=\mathrm{i}(\overline{\partial}-\partial)$.
To obtain the above identities we will use that
\begin{equation}\label{z0rrel}
\end{equation}
together with $\Omega(\gamma)=\Omega(-\gamma)$ and the fact that the rotating vector field is given globally in the case of a CASK domain by
\begin{equation}\label{vcaskd}
\end{equation}
To obtain the formula for $f$ we just use (<ref>) together with (<ref>) and a relabeling of the sum variable $\gamma \to -\gamma$. To obtain the formula for $\theta_{+}^P|_{\overline{N}}$ we use the formulas (<ref>) for $\omega_1+\mathrm{i}\omega_2$, the definitions for $\theta_1^P$ and $\theta_2^P$ in (<ref>), a relabeling $\gamma \to -\gamma$ of the sums over $\gamma$, the CASK relation $Z_{\widetilde{\gamma}_i}=-Z_i=-\tau_{ij}Z^j=-\tau_{ij}Z_{\gamma^j}$, and the fact that
\begin{equation}\label{restrictionrel}
\left(\frac{\D Z_{\gamma}}{Z_{\gamma}}-\frac{\D\overline{Z}_{\gamma}}{\overline{Z}_{\gamma}}\right)\Bigg|_{\overline{N}}=\left(\frac{\D\widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}-\frac{\D\overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right), \quad \D Z_{\gamma}|_{\overline{N}}=|Z^0|\widetilde{Z}_{\gamma}\left(\frac{\D\widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}} +\frac{\D \rho}{2(\rho+c_{\ell})}+\frac{\D\mathcal{K}}{2}\right)\,.
\end{equation}
Finally, the equality for $\theta_3^P|_{\overline{N}}$ follows from the formulas (<ref>) for $\eta$, (<ref>) for $\theta_3^P$, the first equation from (<ref>), and again a relabeling $\gamma \to -\gamma$ in the sum over $\gamma$. We also use the fact that
\begin{equation}
\D^c r^2|_{\overline{N}}=-r^2\D^c \mathcal{K}\,,
\end{equation}
which follows from the second relation in (<ref>).
We remark that if we denote $F_i:=\partial_{Z^i}\mathfrak{F}/Z^0=Z_i/Z^0$, then in the expression (<ref>) we have
\begin{equation}
\langle \widetilde{Z},\mathrm{d}\zeta\rangle= \widetilde{Z}_{\widetilde{\gamma}_i}\D\zeta_{\gamma^i}-\widetilde{Z}_{\gamma^i}\D\zeta_{\widetilde{\gamma}_i}=(-F_i)\D(-\zeta^i) - z^i\D\widetilde{\zeta}_i=F_i\D\zeta^i-z^i\mathrm{d}\widetilde\zeta_i\,.
\end{equation}
Similarly, we have
\begin{equation}
\langle \zeta,\mathrm{d}\zeta\rangle=\zeta_{\widetilde{\gamma}_i}\D\zeta_{\gamma^i}-\zeta_{\gamma^i}\D\zeta_{\widetilde{\gamma}_i}=-\widetilde{\zeta}_i\mathrm{d}\zeta^i+\zeta^i\mathrm{d}\widetilde{\zeta}_i\,.
\end{equation}
§.§.§ Darboux coordinates for c-map spaces associated to CASK domains
In this section we focus on the easier case of a c-map space associated to a CASK domain. The following coordinates have been been previously obtained in the physics literature in [36] in the $c_{\ell}=0$ case, and in the 1-loop corrected case in [14] via slightly different methods.
Consider the QK manifold $(\widetilde{N},g_{\overline{N}})$ obtained from a CASK domain $(M,\mathfrak{F})$ via the c-map (i.e. by taking $\Omega(\gamma)=0$ for all $\gamma$ in our previous constructions). If $F_i:=Z_i/Z^0=\partial_{Z^i} \mathfrak{F}/Z^0$, $t$ denotes the twistor fiber coordinate from Proposition <ref> and $R=2\sqrt{\rho+c_{\ell}}e^{\mathcal{K}/2}$, then the functions on the twistor space given by
\begin{equation}\label{Darbouxcoords1loop}
\begin{split}
\xi^{i}&=\zeta^{i}- \mathrm{i}R(t^{-1}z^i +t\overline{z}^i)\\
\widetilde{\xi}_i&=\widetilde{\zeta}_{i}- \mathrm{i}R(t^{-1}F_{i}+t\overline{F}_i)\\
\alpha&= \sigma -\mathrm{i}R(t^{-1}\langle \widetilde{Z},\zeta \rangle +t \langle \overline{\widetilde{Z}}, \zeta \rangle)-8\mathrm{i} c_{\ell}\log(t)\,,
\end{split}
\end{equation}
define Darboux coordinates for the holomorphic contact structure $\lambda$, in the sense that
\begin{equation}
\lambda=-2\pi\mathrm{i}(\D\alpha +\widetilde{\xi}_i\D\xi^i-\xi^i\D\widetilde{\xi}_i)\cdot s\,.
\end{equation}
By Proposition <ref>, it is enough to check that
\begin{equation}
-2\pi \I(\D\alpha +\widetilde{\xi}_i\D\xi^i-\xi^i\D\widetilde{\xi}_i)=f\frac{\D t}{t}+t^{-1}\theta_{+}^P|_{\overline{N}}-2\mathrm{i} \theta_3^P|_{\overline{N}} +t \theta_{-}^{P}|_{\overline{N}}
\end{equation}
where $f$, $\theta_{+}^P=\overline{\theta_{-}^P}$ and $\theta_3^P$ are obtained by setting $\Omega(\gamma)=0$ for all $\gamma$ in (<ref>). That is:
\begin{equation}
f=16\pi \rho, \quad\quad \theta_3^P|_{\overline{N}}=\pi\D\sigma -\pi\langle\zeta, \D\zeta \rangle-4\pi(\rho+c_{\ell})\D^c\mathcal{K}, \quad\quad
\theta_{+}^P|_{\overline{N}}=-4\pi R\langle \widetilde{Z},\D\zeta \rangle\,.
\end{equation}
We now compute
\begin{equation}
\begin{split}
\widetilde{\xi}_i\D\xi^i-\xi^i\D\widetilde{\xi}_i=&-\langle \zeta, \D\zeta \rangle +\I(t^{-1}\langle\widetilde{Z},\zeta \rangle +t \langle\overline{\widetilde{Z}}, \zeta \rangle)\D R +\I R(-t^{-2}\langle\widetilde{Z},\zeta\rangle \D t+\langle \overline{\widetilde{Z}},\zeta \rangle \D t)+ 8\I(\rho+c_{\ell})\frac{\D t}{t}\\
& +\I R(t^{-1}\langle \D \widetilde{Z},\zeta \rangle + t \langle \D\overline{\widetilde{Z}},\zeta \rangle) -\I R(t^{-1}\langle \widetilde{Z},\D\zeta \rangle + t \langle \overline{\widetilde{Z}},\D\zeta \rangle)-4(\rho+c_{\ell})\D^c\mathcal{K}\,,
\end{split}
\end{equation}
where for the terms $8\I (\rho+c_{\ell})\frac{\D t}{t}$ and $-4(\rho+c_{\ell})\D^c\mathcal{K}$ we have used that the CASK relation $F_i=\tau_{ij}z^j$ implies
\begin{equation}
\begin{split}
\Big(-8(\rho+c_{\ell})e^{\mathcal{K}}F_i\overline{z}^i+8(\rho+c_{\ell})e^{\mathcal{K}}\overline{F}_iz^i\Big)\frac{\D t}{t}&= \Big(-8(\rho+c_{\ell})e^{\mathcal{K}}\tau_{ij}z^j\overline{z}^i+8(\rho+c_{\ell})e^{\mathcal{K}}\overline{\tau}_{ij}\overline{z}^jz^i\Big)\frac{\D t}{t}\\
&=\Big(8\I(\rho+c_{\ell})e^{\mathcal{K}}(-2\text{Im}(\tau_{ij}))z^j\overline{z}^i\Big)\frac{\D t}{t}=8\I(\rho+c_{\ell})\frac{\D t}{t}\\
\end{split}
\end{equation}
and by using the relation $\D F_i=\tau_{ij}\D z^i$
\begin{equation}
\begin{split}
-4(\rho+c_{\ell})e^{\mathcal{K}}F_i\D \overline{z}^i&-4(\rho+c_{\ell})e^{\mathcal{K}}\overline{F}_i\D z^i +4(\rho+c_{\ell})e^{\mathcal{K}}\D \overline{F}_iz^i+4(\rho+c_{\ell})e^{\mathcal{K}}\D F_i\overline{z}^i\\
&=-4(\rho+c_{\ell})e^{\mathcal{K}}(-2\text{Im}(\tau_{ij}))\Big(i\overline{z}^j\D z^i -iz^i\D \overline{z}^j\Big)\\
&=-4(\rho+c_{\ell})\D ^c\mathcal{K}\,.
\end{split}
\end{equation}
(Recall that $\mathcal{K}=-\log K$ where $K= -2 \mathrm{Im}(\tau_{ij}) z^i\bar z^j$).
On the other hand, we find that
\begin{equation}
\begin{split}
\D \alpha=&\D \sigma -\I(t^{-1}\langle \widetilde{Z},\zeta \rangle +t \langle \overline{\widetilde{Z}},\zeta \rangle)\D R + \I R(t^{-2}\langle \widetilde{Z},\zeta \rangle - \langle \overline{\widetilde{Z}},\zeta \rangle)\D t\\
&-\I R(t^{-1}\langle \D \widetilde{Z},\zeta \rangle +t \langle \D \overline{\widetilde{Z}},\zeta \rangle)-\I R(t^{-1}\langle \widetilde{Z},\D \zeta \rangle +t \langle \overline{\widetilde{Z}},\D \zeta \rangle)-8\I c_{\ell}\frac{\D t}{t},
\end{split}
\end{equation}
so we conclude that
\begin{equation}
\begin{split}
\D \alpha +\widetilde{\xi}_i\D \xi^i-\xi^i\D \widetilde{\xi}_i&=8\I \rho\frac{\D t}{t}+\D \sigma -\langle \zeta, \D \zeta \rangle-2\I R(t^{-1}\langle \widetilde{Z},\D \zeta \rangle + t \langle \overline{\widetilde{Z}},\D \zeta \rangle) -4(\rho+c_{\ell})\D ^c\mathcal{K}\\
\end{split}
\end{equation}
so that
\begin{equation}
\begin{split}
-2\pi \mathrm{i}\left(\D \alpha +\widetilde{\xi}_i\D \xi^i-\xi^i\D \widetilde{\xi}_i\right)&=f\frac{\D t}{t} -4\pi R(t^{-1}\langle \widetilde{Z},\D \zeta \rangle + t \langle \overline{\widetilde{Z}},\D \zeta \rangle)-2\I \left(\pi\D\sigma -\pi\langle\zeta, \D\zeta \rangle-4\pi(\rho+c_{\ell})\D^c\mathcal{K}\right)\\
&=f\frac{\D t}{t}+t^{-1}\theta_+^P|_{\overline{N}} -2\mathrm{i}\theta_3^P|_{\overline{N}} +t\theta_-^P|_{\overline{N}}
\end{split}
\end{equation}
and the result follows.
§.§.§ The case with instanton corrections
We now consider the case where the BPS indices are not all equal to $0$. We want to write down the modifications of the coordinates (<ref>), such that
\begin{equation}
\lambda=-2\pi \mathrm{i}\left(\D \alpha + \widetilde{\xi}_i\D \xi^i -\xi^i\D \widetilde{\xi}_i\right)\cdot s=\left(f\frac{\D t}{t}+t^{-1}\theta_+^P|_{\overline{N}} -2\mathrm{i}\theta_3^P|_{\overline{N}} +t\theta_-^P|_{\overline{N}}\right)\cdot s
\end{equation}
where $f$ and $\theta_{\alpha}^{P}$ are as in (<ref>).
The expressions for the coordinates below have been previously found in the physics literature (using slightly different arguments and conventions), for example in [13, 12].
Consider the instanton corrected QK manifold $(\widetilde{N},g_{\overline{N}})$ associated to a CASK domain and $(M,\Gamma,Z,\Omega)$, as before. Then the functions on the twistor space $\mathcal{Z}$ given by
\begin{equation}\label{Darbouxcoordsinst}
\begin{split}
\xi^{i}&=\zeta^{i}- \mathrm{i}R(t^{-1}z^i +t\overline{z}^i)\\
\widetilde{\xi}_i&=\widetilde{\zeta}_{i}- \mathrm{i}R(t^{-1}F_{i}+t\overline{F}_i)-\frac{1}{8\pi^2 }\sum_{\gamma}\Omega(\gamma)n_i(\gamma) \int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}\log(1-\exp(2\pi\mathrm{i}\xi_{\gamma}(\zeta)))\\
\alpha&= \sigma -\mathrm{i}R(t^{-1}\langle \widetilde{Z},\zeta \rangle +t \langle \overline{\widetilde{Z}}, \zeta \rangle)-8\mathrm{i} c_{\ell}\log(t)+\frac{1}{4\pi^2\mathrm{i}}\left(t^{-1}\mathcal{W} +t\overline{\mathcal{W}}-\frac{1}{2\pi}\sum_{\gamma}\Omega(\gamma)\int_{l_{\gamma}}\frac{\D \zeta}{\zeta}\frac{t+\zeta}{t-\zeta}\mathrm{L}(\exp{2\pi\mathrm{i}\xi_{\gamma}(\zeta)})\right)\,,
\end{split}
\end{equation}
where the integration contours are given by $l_{\gamma}:=\mathbb{R}_{<0}\widetilde{Z}_{\gamma}$ oriented from $0$ to $\infty$, $\xi_\gamma(\zeta) = q_i \xi^i(\zeta)$ for $\gamma=q_i\gamma^i$ and where we abbreviate $\xi^i(\zeta)=\xi^i(R,z^i,\zeta^i,t=\zeta)$,
\begin{equation}\label{wdef}
\mathcal{W}:=R\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}\frac{e^{-2\pi\mathrm{i}n\zeta_{\gamma}}}{n}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\,,
\end{equation}
\begin{equation}\label{Rogersdilogarithm}
\mathrm{L}(\exp({2\pi\mathrm{i} \xi_{\gamma}}))=\mathrm{Li}_2(\exp(2\pi\mathrm{i}\xi_{\gamma}))+\pi\mathrm{i}\xi_{\gamma}\log(1-\exp(2\pi\mathrm{i}\xi_{\gamma}));
\end{equation}
define Darboux coordinates for the contact structure $\lambda$ in the sense that
\begin{equation}
\lambda=-2\pi \I(\D \alpha +\widetilde{\xi}_i\D \xi^i-\xi^i\D \widetilde{\xi}_i)\cdot s\,.
\end{equation}
A explicit proof is given in Appendix <ref>. This result is not needed for the following sections and is only included for completeness. See also the work in the physics literature [12].
* The function $\mathrm{Li}_2(x)$ appearing in (<ref>) is the dilogarithm function, while the function $\mathrm{L}(x)=\mathrm{Li}_2(x)+\frac{\log(x)}{2}\log(1-x)$ is the Rogers dilogarithm.
* We remark that along the chosen $l_{\gamma}$ contours, the integrands are exponentially decreasing near $0$ and $\infty$. In particular, one can deform the contour $l_{\gamma}$ within the half plane centered at $l_{\gamma}$ without changing the value of the integral, as long as it does not collide with another ray of the form $l_{\gamma}$ for $\gamma \in \text{Supp}(\Omega)$.
§ S-DUALITY ON INSTANTON CORRECTED Q-MAP SPACES
The structure of the section is as follows:
* In Section <ref> we define instanton corrected q-map spaces $(\overline{N},g_{\overline{N}})$ in terms of the construction of [25] reviewed in Section <ref>,
applied to certain special pairs of compatible initial data $(M,\mathfrak{F})$ and $(M,\Gamma,Z,\Omega)$. These spaces will be the main objects of study in the rest of the section.
* In Section <ref> we define the quantum corrected mirror map, previously defined in [15], and relating the type IIA variables $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)$ with the type IIB variables $(\tau_1+\I\tau_2,b^a+\I t^a, c^a,c_a,c_0,\psi)$. In terms of the type IIB variables, we define the corresponding $\mathrm{SL}(2,\mathbb{Z})$-action.
* In Section <ref> we show that the holomorphic contact structure of the twistor space associated to $(\overline{N},g_{\overline{N}})$ admits Darboux coordinates of the form studied in [15]. We show this directly by Poisson resumming the contact form (<ref>), which in turn is expressed in terms of the Bessel function expressions (<ref>) specialized to the particular form of the BPS indices below (<ref>).
* In Section <ref> we give conditions that guarantee that either the S-duality $\mathrm{SL}(2,\mathbb{Z})$-transformations or the subgroup $\langle S \rangle \subset \mathrm{SL}(2,\mathbb{Z})$ acts by isometries on an instanton corrected q-map space.
§.§ Setting
Let us first recall the notion of a PSK manifold in the image of the r-map, and the associated CASK domain.
A projective special real (PSR) manifold is a Riemannian manifold $(\mathcal{H},g_{\mathcal{H}})$ where $\mathcal{H}\subset \mathbb{R}^n$ is a hypersurface, for which there is a cubic polynomial $h:\mathbb{R}^n\to \mathbb{R}$ such that $\mathcal{H}\subset \{h(t)=1\}$ and $g_{\mathcal{H}}=-\partial^2h|_{T\mathcal{H}\times T\mathcal{H}}$.
If we denote the canonical coordinates of $\mathbb{R}^n$ by $t^a$, then we can write
\begin{equation}\label{cubicpsr}
\end{equation}
with $k_{abc}\in \mathbb{R}$ symmetric in the indices. Now let $U:=\mathbb{R}_{>0}\cdot \mathcal{H}\subset \mathbb{R}^{n}-\{0\}$ and $\overline{M}^{\text{cl}}:=\mathbb{R}^n+\I U\subset \mathbb{C}^n$ with canonical holomorphic coordinates $z^a:=b^a+\I t^a$. On $\overline{M}^{\text{cl}}$ we have a PSK metric by defining
\begin{equation}\label{8h:eq}
g_{\overline{M}^{\text{cl}}}:=\frac{\partial^2\mathcal{K}}{\partial z^a \partial \overline{z}^b}\mathrm{d}z^a\mathrm{d}\overline{z}^b\, \quad \; \mathcal{K}=-\log(8h(t))\,.
\end{equation}
The PSK manifold $(\overline{M}^{\text{cl}},g_{\overline{M}^{\text{cl}}})$ is associated to the CASK domain $(M^{\text{cl}},\mathfrak{F}^{\text{cl}})$ of signature $(n,1)$ given by
\begin{equation}
M^{\text{cl}}:=\{(Z^0,...,Z^n)=Z^0\cdot(1,z) \in \mathbb{C}^{n+1}\; | \; Z^0\in \mathbb{C}^{\times}, z\in \overline{M}^{\text{cl}}\},\, \quad \mathfrak{F}^{\text{cl}}:=-\frac{1}{6}k_{abc}\frac{Z^aZ^bZ^c}{Z^0}\,.
\end{equation}
Notice in particular the relation $Z^a/Z^0=z^a=b^a+\I t^a$, which we use repeatedly below.
The construction given above that associates the PSK manifold $(\overline{M}^{\text{cl}}, g_{\overline{M}^{\text{cl}}})$ to the PSR manifold $(\mathcal{H},g_{\mathcal{H}})$ is called the r-map. Furthermore, the QK metric obtained by applying the c-map (resp. tree-level c-map) to a PSK manifold in the image of the r-map is called a q-map space (resp. tree-level q-map space).
We now want to consider a tuple $(M,\mathfrak{F})$ of the following special type:
* We start with a PSR manifold $(\mathcal{H},g_{\mathcal{H}})$ and consider the associated CASK domain $(M^{\text{cl}},\mathfrak{F}^{\text{cl}})$. We further let $\Lambda^{+}:=\text{span}_{\mathbb{Z}_{\geq 0}}\{\gamma^a\}_{a=1}^{n}-\{0\}$ be a commutative semi-group generated by $\{\gamma^a\}_{a=1}^{n}$, where $n=\text{dim}(\mathcal{H})+1$. We want to consider a new CASK domain with a holomorphic prepotential of the form
\begin{equation}\label{prepotential}
\mathfrak{F}(Z^i)=\mathfrak{F}^{\text{cl}}(Z^i)+\mathfrak{F}^{\text{w.s.}}(Z^i)
\end{equation}
where $\mathfrak{F}^{\text{cl}}$ is as before, and
\begin{equation}\label{Fws:eq}
\mathfrak{F}^{\text{w.s.}}:=\chi\frac{(Z^0)^2\zeta(3)}{2(2\pi \I )^3}-\frac{(Z^0)^2}{(2\pi \I)^3}\sum_{\hat{\gamma}=q_a\gamma^a \in \Lambda^+}n_{\hat{\gamma}}\mathrm{Li}_3(e^{2\pi \I q_aZ^a/Z^0})\,\,.
\end{equation}
In the above expression $\chi \in \mathbb{Z}$, $n_{\hat{\gamma}}\in \mathbb{Z}$, $\mathrm{Li}_n(z)$ denotes the n-th polylogarithm, and $\zeta(x)$ denotes the Riemann zeta function.
* We assume that the $n_{\hat{\gamma}}$ are such that the $\mathbb{C}^{\times}$-invariant subset of $\mathbb{C}^{n+1}$ defined by
\begin{equation}\label{defM}
M^q:=M^{\text{cl}}\cap\{(Z^0,...,Z^n)\in \mathbb{C}^{n+1} \; | \; q_at^a>0 \; \text{for all $\hat{\gamma}=q_a\gamma^a\in \Lambda^+$ with $n_{\hat{\gamma}}\neq 0$}\}\;
\end{equation}
is open[This condition is automatic if the commutative semigroup generated by $\{\hat{\gamma}\in \Lambda^+ \; | \; n_{\hat{\gamma}}\neq 0\}$ is finitely generated.]. We further assume that the growth of the $n_{\hat{\gamma}}$ with $\hat{\gamma}$ is such that for any $\epsilon>0$ the series
\begin{equation}\label{convpropn}
\sum_{\hat{\gamma}=q_a\gamma^a\in \Lambda^+}|n_{\hat{\gamma}}|e^{-\epsilon q_at^a}
\end{equation}
has compact normal convergence on $M^q$, so that $\mathfrak{F}$ defines a holomorphic function on $M^q$.
In particular, the condition $q_at^a>0$ ensures that $\mathrm{Li}_3(e^{2\pi \I q_aZ^a/Z^0})$ is well defined and can be expressed as
\begin{equation}
\mathrm{Li}_3(e^{2\pi \I q_aZ^a/Z^0})=\sum_{k>0}\frac{e^{2\pi \I kq_aZ^a/Z^0}}{k^3}\,.
\end{equation}
* We denote by $M\subset M^q$ the maximal open subset of $M^q$ where $\text{Im}(\tau_{ij})=\text{Im}(\partial_i\partial_j\mathfrak{F})$ has signature $(n,1)$ and $\text{Im}(\tau_{ij})Z^i\overline{Z}^j<0$. These conditions are $\mathbb{C}^{\times}$-invariant, so that $(M,\mathfrak{F})$ defines a CASK domain on each connected component of $M$.
* To such a tuple $(M,\mathfrak{F})$ we associate the canonical lattice $\Gamma \to M$, together with the canonical central charge $Z$ (recall Section <ref>). Namely, if $Z_i:=\partial_{Z^i}\mathfrak{F}$ and $x^i=\text{Re}(Z^i)$, $y_i=\text{Re}(Z_i)$, then $(\partial_{x^i},\partial_{y_i})=(\widetilde{\gamma}_i,\gamma^i)$ defines a global Darboux frame for $\Gamma\to M$. For the canonical central charge we then have
\begin{equation}\label{Z:eq}
Z_{\gamma^i}=Z^i, \quad Z_{\widetilde{\gamma}_i}=-Z_i=-\frac{\partial \mathfrak{F}}{\partial Z^i}\,.
\end{equation}
We will identify the semigroups $\Lambda^+\cong \text{span}_{\mathbb{Z}_{\geq 0}}\{\partial_{y_a}\}_{a=1}^n-\{0\}$.
* We remark that given any $(Z^0,...,Z^n)=Z^0\cdot(1,b^a+\I t^a)\in M^q$, all points of the form $Z^0\cdot(1,b^a+\lambda\I t^a)$ with $\lambda>0$ must also be in $M^q$. In particular, for $\lambda>0$ sufficiently big we have $\text{Im}(\tau_{ij})\sim \text{Im}(\partial_i\partial_j\mathfrak{F}^{\text{cl}})$ due to the exponential decay of the terms with polylogarithms. Since $\text{Im}(\partial_i\partial_j\mathfrak{F}^{\text{cl}})$ has signature $(n,1)$ and $\text{Im}(\partial_i\partial_j\mathfrak{F}^{\text{cl}})Z^i\overline{Z}^j<0$, it follows that at the points $Z^0\cdot(1,b^a+\lambda\I t^a)$ for $\lambda>0$ sufficiently big we have that $\text{Im}(\tau_{ij})$ has signature $(n,1)$ and $\text{Im}(\tau_{ij})Z^i\overline{Z}^j<0$, so that the required $M$ is never empty, provided $M^q$ is not empty.
* We would also like to comment on the particular form of the prepotential (<ref>). In the setting of Calabi-Yau compactifications of type IIB string theory, the term $\mathfrak{F}^{\text{cl}}$ has $k_{abc}\in \mathbb{Z}$ equal to the triple interesection numbers of the Calabi-Yau threefold $X$, and corresponds to the holomorphic prepotential determining (via the c-map) the classical QK geometry of the hypermultiplet moduli space. On the other hand, $\mathfrak{F}^{\text{w.s.}}$
correspond to world-sheet instanton corrections. In such a setting, $\chi$ coincides with the Euler characteristic $\chi(X)$ of $X$, and the numbers $n_{\hat{\gamma}}$ are the so-called genus zero Gopakumar-Vafa invariants, and usually denoted by $n_{\hat{\gamma}}^{(0)}$. Since we are in a setting independent of string theory, we will use the simpler notation of $n_{\hat{\gamma}}$ for the coefficients appearing in (<ref>). Furthermore, it will be useful to define $n_{0}:=-\frac{\chi}{2}$. Then, using that $\mathrm{Li}_3(1)=\zeta(3)$ we can write
\begin{equation}\label{Fws}
\mathfrak{F}^{\text{w.s.}}=-\frac{(Z^0)^2}{(2\pi \I)^3}\sum_{\hat{\gamma} \in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}\mathrm{Li}_3(e^{2\pi \I q_aZ^a/Z^0})\,.
\end{equation}
* In the physics literature sometimes an additional term of the form
\begin{equation}
\frac{1}{2}A_{ij}Z^iZ^j, \quad A_{ij}=A_{ji}\in \mathbb{R}
\end{equation}
is added to $\mathfrak{F}$ in (<ref>). While the addition of such a term does not alter the PSK metric $g_{\overline{M}}$ or the CASK metric $g_{M}$ (since $A_{ij}$ is real), the inclusion of such a term turns out to be important for mirror symmetry between D-branes of type IIA and type IIB string theory (see for example the review <cit.>, and the references therein). However, below we will focus on a case analogous to including D(-1) and D1 instanton corrections on type IIB, and we can safely ignore the inclusion of such a term.
Finally, after possibly restricting $M$, we assume that $M$ is the maximal open subset such that $(M,\Gamma,Z,\Omega)$ with $\Gamma$ and $Z$ as before and with
\begin{equation}\label{varBPS}
\begin{cases}
\Omega(q_0\gamma^0)=-\chi=2n_0, \quad q_0\in \mathbb{Z}-\{0\}\\
\Omega(q_0\gamma^0\pm q_a\gamma^a)=\Omega(\pm q_a\gamma^a)=n_{q_a\gamma^a} \quad \text{for $q_a\gamma^a \in \Lambda^+$, $q_0\in \mathbb{Z}$}\\
\Omega(\gamma)=0 \quad \text{else},\\
\end{cases}
\end{equation}
is a mutually local variation of BPS structures compatible (in the sense of Definition <ref>) with the CASK manifold $(M,g_M,\omega_M,\nabla,\xi)$ associated to $(M,\mathfrak{F})$. To check that $(M,\Gamma,Z,\Omega)$ defines a mutually local variation of BPS structures, one only needs to check the support property, since the convergence property follows easily from (<ref>), while the mutual locality and invariance under monodromy is obvious.
We remark that the BPS indices in (<ref>) are determined by the holomorphic prepotential $\mathfrak{F}$, see (<ref>). Such a prescription of BPS indices has previously appeared in the physics literature in [15, 11] or more explicitly in <cit.>. See also [34] (in particular, <cit.>), where such a BPS spectrum is conjectured for a non-compact Calabi-Yau 3-folds without compact divisors).
In the following, we will consider the associated instanton corrected QK manifold $(\overline{N},g_{\overline{N}})$ associated to $(M,\mathfrak{F})$ and $(M,\Gamma,Z,\Omega)$ as in Theorem <ref>. It will be convenient to lift the QK metric $(\overline{N},g_{\overline{N}})$ to $(\widetilde{N},g_{\overline{N}})$ as in Definition <ref>, where the $(\zeta^i,\widetilde{\zeta}_i,\sigma)$ directions are no longer periodic. In particular, we recall that we have the following:
* $\widetilde{N}\subset \mathbb{R}_{>0}\times \overline{M}\times \mathbb{R}^{2n+2}\times \mathbb{R}$ is an open subset, where the splitting is such that $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)\in \mathbb{R}_{>0}\times \mathbb{C}^n\times \mathbb{R}^{2n+2}\times \mathbb{R}$, where $\overline{M}$ is the associated PSK manifold.
* The twistor space $\mathcal{Z}$ of $(\widetilde{N},g_{\overline{N}})$ smoothly decomposes as $\mathcal{Z}=\widetilde{N}\times \mathbb{C}P^1$. In particular, the expression (<ref>) for the contact structure holds for the global coordinates $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma,t)$ of $\mathcal{Z}$.
Here we are slightly abusing notation by considering $t$, the identity map on $\mathbb{C}P^1$, as a global coordinate.)
If $(M,\mathfrak{F})$ and $(M,\Gamma,Z,\Omega)$ are given as before, we will call the resulting QK manifold $(\overline{N},g_{\overline{N}})$ (or its lift $(\widetilde{N},g_{\overline{N}})$) an instanton corrected q-map space.
The reason for the calling $(\overline{N},g_{\overline{N}})$ (or $(\widetilde{N},g_{\overline{N}})$) an instanton corrected q-map space is the following:
* By setting the one-loop parameter to $c_{\ell}=0$, and $n_{\hat{\gamma}}=0$ for all $\hat{\gamma}\in \Lambda^+\cup\{0\}$ (and hence, also $\Omega(\gamma)=0$ for all $\gamma$), one recovers a q-map space. That is the QK metric obtained by applying the c-map to a PSK manifold in the image of the r-map:
\begin{equation}\label{QKtree}
g_{\overline{N}}=g_{\overline{M}}+\frac{\D\rho^2}{4\rho^2}-\frac{1}{4\rho}(N^{ij}-2e^{\mathcal{K}}z^i\overline{z}^j)W_i\overline{W}_j + \frac{1}{64\rho^2}(\D\sigma + \widetilde{\zeta}_{i}\D\zeta^{i}-\zeta^{i}\D\widetilde{\zeta}_{i})^2\,.
\end{equation}
where $g_{\overline{M}}$, $N_{ij}$, $\mathcal{K}$ and $W_i$ are constructed in terms of $\mathfrak{F}^{\text{cl}}$ (see Section <ref> for the definition of the above terms).
* In the setting of Calabi-Yau compactification of type IIB string theory, the terms due to the BPS structure $(M,\Gamma,Z,\Omega)$ are thought as D(-1), D1 instanton corrections, and those due to $\mathfrak{F}^{\text{w.s.}}$ are thought as world-sheet instantons corrections. Hence, the QK metric obtained by the above $(M,\mathfrak{F})$ and $(M,\Gamma,Z,\Omega)$ can be thought as a q-map with the inclusion of the analog of the above corrections.
§.§ The quantum corrected mirror map and the S-duality action
In the sections that follow we will rescale the twistor fiber coordinate $t\to -\I t$. The contact structure on the twistor space is then expressed by (compare with (<ref>))
\begin{equation}\label{contstr}
\lambda=\left(f\frac{\D t}{t}+t^{-1}\I \theta_+^P|_{\overline{N}} -2\I \theta_3^P|_{\overline{N}} -t\I \theta_-^P|_{\overline{N}}\right)\cdot s\,.
\end{equation}
In order to define the S-duality action, we consider the following diffeomorphism, first defined (under slightly different conventions) in [15]:
Let $\overline{M}:=\{z \in \mathbb{C}^n \; | \; (1,z)\in M\}$ where $M$ was given in the previous section, and
consider the manifold $\overline{\mathcal{N}}_{\text{IIA}}:=\mathbb{R}_{>-c_{\ell}}\times \overline{M}\times \mathbb{R}^{2n+2}\times \mathbb{R}$ with global coordinates $(\rho,z^a,\zeta^i,\widetilde{\zeta}_i,\sigma)$. We will call such coordinates type IIA coordinates. On the other hand, if $H\subset \mathbb{C}$ denotes the upper half-plane, we define $\overline{\mathcal{N}}_{\text{IIB}}:=H\times \overline{M}\times \mathbb{R}^{2n}\times \mathbb{R}^2$ with global coordinates $(\tau_1+\mathrm{i}\tau_2,b^a+\I t^a,c^a,c_a,c_0,\psi)$. We call the latter type IIB coordinates. The type IIB coordinates are related to the type IIA coordinates via the diffeomorphism (see Remark <ref> below) $\mathcal{M}: \overline{\mathcal{N}}_{\text{IIB}}\to \overline{\mathcal{N}}_{\text{IIA}}$ defined by
\begin{equation}\label{MM}
\begin{split}
z^a&=b^a+\I t^a, \;\;\;\;\; \rho=\frac{\tau_2^2}{16}e^{-\mathcal{K}}-c_{\ell}, \;\;\;\;\;\;\; \zeta^0=\tau_1, \;\;\;\;\; \zeta^a=-(c^a-\tau_1b^a),\\
\widetilde{\zeta}_a&=c_a +\frac{k_{abc}}{2}b^b(c^c-\tau_1b^c)+\widetilde{\zeta}_{a}^{\text{inst}}, \;\;\;\;\;\; \widetilde{\zeta}_0=c_0-\frac{k_{abc}}{6}b^ab^b(c^c-\tau_1b^c)+\widetilde{\zeta}_{0}^{\text{inst}}\\
\sigma&= -2(\psi + \frac{1}{2}\tau_1c_0) + c_a(c^a-\tau_1b^a) -\frac{k_{abc}}{6}b^ac^b(c^c-\tau_1b^c)+\sigma^{\text{inst}}\,,
\end{split}
\end{equation}
where $\mathcal{K}=-\log(-2\text{Im}(\tau_{ij})z^i\overline{z}^j)$ with $z^0=1$, $\tau_{ij}=\frac{\partial^2 \mathfrak{F}}{\partial Z^i\partial Z^j}$, $c_{\ell}\in \mathbb{R}$ is the 1-loop parameter, and
\begin{equation}\label{typeiiacoordcor}
\begin{split}
\widetilde{\zeta}^{\text{inst}}_{a}&:=\frac{1}{8\pi^2}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}q_a\sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\frac{m\tau_1+ n}{m|m\tau+ n|^2}e^{-S_{\hat{\gamma},m,n}}\\
\widetilde{\zeta}^{\text{inst}}_0&:=\frac{\I}{16\pi^3}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\left(\frac{(m\tau_1+ n)^2}{|m\tau + n|^3}+ 2\pi q_a\left(t^a+\I b^a\frac{m\tau_1+n}{|m\tau+ n|}\right)\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{m|m\tau+ n|}\\
\sigma^{\text{inst}}&:=\tau_1\widetilde{\zeta}_0^{\text{inst}}-(c^a-\tau_1b^a)\widetilde{\zeta}_a^{\text{inst}}-\frac{\I\tau_2^2}{8\pi^2}\sum_{\hat{\gamma}\in \Lambda_{+}}n_{\hat{\gamma}}q_at^a\sum_{n\in \mathbb{Z}-\{0\}}\frac{e^{-S_{\hat{\gamma},0,n}}}{n|n|}\\
&\quad\quad +\frac{\I}{8\pi^3}\sum_{\hat{\gamma}\in \Lambda^{+}}n_{\hat{\gamma}}\sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\left(2-\frac{(m\tau_1+n)^2}{|m\tau+ n|^2}\right)\frac{(m\tau_1+ n)e^{-S_{\hat{\gamma},m,n}}}{m^2|m\tau+ n|^2}
\end{split}
\end{equation}
\begin{equation}\label{mnaction}
S_{\hat{\gamma},m,n}:=2\pi q_a(|m\tau+ n|t^a +\I mc^a+ \I nb^a)\,.
\end{equation}
We will refer to the diffeomorphism $\mathcal{M}$ as the quantum corrected mirror map.
* Note that since $z=(z^a)\in \overline{M}$ and $M\subset M^{q}$, we have $\text{Re}(S_{\hat{\gamma},m,n})>0$ for all $\hat{\gamma}\in \Lambda^{+}$ with $n_{\hat{\gamma}}\neq 0$, so (<ref>) implies that the sums in (<ref>) have compact normal convergence on $\overline{\mathcal{N}}_{\text{IIB}}$. Furthermore, we remark that (<ref>) really defines a diffeomorphism. Indeed, since $-\text{Im}(\tau_{ij})z^i\overline{z}^j>0$ on $\overline{M}$ and $\tau_2>0$, we can always invert the relation involving $\rho$, while $\widetilde{\zeta}_i^{\text{inst}}$ and $\sigma^{\text{inst}}$ only depend on $\tau=\tau_1+\mathrm{i}\tau_2$, $z^a$ and $c^a$, so it is easy to invert all the other relations. Note, in particular, that the expressions for $\widetilde{\zeta}_i^{\text{inst}}$ and $\sigma^{\text{inst}}$ are real. Furthermore, if we set $c_{\ell}=0$, $n_{\hat{\gamma}}=\chi=0$ for all $\hat{\gamma}\in \Lambda^{+}$, then we recover the classical mirror map.
* While the expressions (<ref>) look complicated, they satisfy nice transformation properties with respect to the expected isometry groups of the metric. See Section <ref>, Lemma <ref> and Corollary <ref>.
Let $\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}:=H\times \overline{M}^{\text{cl}}\times \mathbb{R}^{2n}\times \mathbb{R}^2$ with global type IIB variables $(\tau=\tau_1+\I \tau_2,b^a+\I t^a,c^a,c_a,c_0,\psi)$. We define an $\mathrm{SL}(2,\mathbb{Z})$-action on $\overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}$ by
\begin{equation}\label{sl2can}
\begin{split}
\tau&\to \frac{a\tau +b}{c\tau +d},\;\;\;\;\; t^a \to |c\tau +d|t^a, \;\;\;\; c_a \to c_a, \\
\begin{pmatrix}c^a\\ b^a\end{pmatrix}&\to \begin{pmatrix} a & b \\ c & d\\ \end{pmatrix}\begin{pmatrix}c^a \\ b^a\end{pmatrix}, \;\;\;\; \begin{pmatrix}c_0\\ \psi\end{pmatrix}\to \begin{pmatrix} d & -c \\ -b & a\\ \end{pmatrix}\begin{pmatrix}c_0 \\ \psi \end{pmatrix}, \;\;\;\; \begin{pmatrix} a & b \\ c & d\\ \end{pmatrix}\in \mathrm{SL}(2,\mathbb{Z})
\end{split}
\end{equation}
We call this action the S-duality action. We note that there is the possibility that S-duality does not act on $\overline{\mathcal{N}}_{\text{IIB}}\subset \overline{\mathcal{N}}_{\text{IIB}}^{\text{cl}}$, since it may happen that $\overline{M}$ is not invariant under the scaling of the $t^a$ in (<ref>).
Recall from Section <ref> that the quaternionic Kähler manifold $\overline{N}$ was constructed in [25] via HK/QK correspondence (by specializing [6]) as the hypersurface
of the bundle $P\to N$ which is
defined by the equation $\mathrm{Arg}\, Z^0 =0$.
Using the IIB variables we can now simply match $|Z^0|=\tau_2$.
§.§ Type IIB Darboux coordinates
We now want to write a distinguished set of Darboux coordinates which will be useful for studying S-duality or the action by specific elements of $\mathrm{SL}(2,\mathbb{Z})$. The coordinates below have been previously written in [15]. Nevertheless, our approach will be different from [15] in the sense that we will start from the mathematical construction of the instanton QK metric obtained in [25], and then explicitly show that (<ref>) indeed define Darboux coordinates for the contact structure on its twistor space.
Recalling (<ref>) and (<ref>), we let
\begin{equation}
\widetilde{\zeta}_i^{\text{cl}}:=\widetilde{\zeta}_i-\widetilde{\zeta}_i^{\text{inst}}, \quad \sigma^{\text{cl}}:=\sigma - \sigma^{\text{inst}}\,,
\end{equation}
and define
\begin{equation}\label{dcclassical}
\begin{split}
\xi^{i,\text{cl}}&:=\zeta^{i}+ R(t^{-1}z^i -t\overline{z}^i)\\
\widetilde{\xi}_i^{\text{cl}}&:=\widetilde{\zeta}_{i}^{\text{cl}}+ R(t^{-1}F^{\text{cl}}_{i}-t\overline{F}^{\text{cl}}_i)\\
\alpha^{\text{cl}}&:= \sigma^{\text{cl}} +R(t^{-1}\langle \widetilde{Z}^{\text{cl}},\zeta^{\text{cl}} \rangle -t \langle \overline{\widetilde{Z}}^{\text{cl}}, \zeta^{\text{cl}} \rangle)\,,
\end{split}
\end{equation}
where $\widetilde{Z}^{\text{cl}}:= Z^{\text{cl}}/Z^0$ and $Z^{\text{cl}}$ is defined replacing
$Z$ and $\mathfrak F$ in equation (<ref>) by $Z^{\text{cl}}$ and $\mathfrak F^{\text{cl}}$, and $F_i^{\text{cl}}:=\partial_{Z^i}\mathfrak{F}^{\text{cl}}/Z^0$.
The expressions (<ref>) match the coordinates (<ref>) for the case $\mathfrak{F}=\mathfrak{F}^{\text{cl}}$ after the scaling $t \to -\mathrm{i}t$ done in (<ref>) and setting $c_{\ell}=0$. In particular, they define Darboux coordinates for the contact structure of twistor space of the tree-level q-map space defined by $\mathfrak{F}^{\text{cl}}$.
Finally, if $c\in \mathbb{R}-\{0\}$ and $d\in \mathbb{R}$, we denote by $t_{\pm}^{c,d}$ the roots of $t(c\xi^{0,\text{cl}}+d)=0$ in the variable $t$. Using that $\zeta^0=\tau_1$ and $R=2\sqrt{\rho+c_{\ell}}e^{\mathcal{K}/2}=\tau_2/2$ (recall (<ref>)), we find
\begin{equation}\label{troots}
t_{\pm}^{c,d}=\frac{c\tau_1+d \pm |c\tau +d|}{c\tau_2}\,.
\end{equation}
Consider an instanton corrected q-map space $(\widetilde{N},g_{\overline{N}})$ with $1$-loop parameter $c_{\ell}=\frac{\chi}{192\pi}$. Then the functions $(\xi^i,\widetilde{\xi}_{i},\alpha)$ of the associated twistor space $\mathcal{Z}\cong \widetilde{N}\times \mathbb{C}P^1$ given by
\begin{align*}
\xi^i&=\xi^{i,\text{cl}}\\
\widetilde{\xi}_{a}&=\widetilde{\xi}_a^{\text{cl}}+\frac{\tau_2}{8\pi^2}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}q_a\left(\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\frac{1+t_+^{m,n}t}{t-t_{+}^{m, n}}\right)\\
\widetilde{\xi}_0&=\widetilde{\xi}_0^{\text{cl}}+\frac{\I \tau_2}{16\pi^3}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(\frac{1}{m\xi^0+ n}+\frac{m\tau_1+ n}{|m\tau+ n|^2}\right)\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}\\
&\quad-\frac{\tau_2}{8\pi^2}\sum_{\hat{\gamma}\in \Lambda^{+}\cup \{0\}}n_{\hat{\gamma}} \sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(q_ab^a \frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}+\I q_at^a\frac{1-t_{+}^{m, n}t}{t-t_{+}^{m, n}}\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+ n|^2}\\
\alpha&=-\frac{1}{2}(\alpha^{\text{cl}}-\xi^{i}\widetilde{\xi}_i^{\text{cl}})+\frac{\I \tau_2^2}{32\pi^3}\sum_{\hat{\gamma}\in \Lambda^{+}\cup\{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left((m\tau_1+ n)(t^{-1}-t) -2m\tau_2\right)\frac{1+t_{+}^{m, n}t}{t-t_{+}^{m, n}}\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau+n|^4}\,, \numberthis \label{typeiibdc}
\end{align*}
define Darboux coordinates for the contact structure of $\mathcal{Z}$, in the sense that
\begin{equation}\label{contdarbiib}
\lambda=4\pi \I\left(\D\alpha -\widetilde{\xi}_i\D \xi^i\right)\cdot s\,.
\end{equation}
In the above expressions, for the sums in the case where $m=0$ we set
\begin{equation}\label{m=0def}
\frac{1+t_{+}^{0, n}t}{t-t_{+}^{0, n}}:=\begin{cases}
1/t, \quad n<0\\
-t, \quad n>0
\end{cases}, \quad \quad \quad \quad \frac{1-t_{+}^{0, n}t}{t-t_{+}^{0, n}}:=\begin{cases}
1/t, \quad n<0\\
t, \quad n>0\,.
\end{cases}
\end{equation}
This will be proven in Section <ref> below.
* The definitions (<ref>) above for the case $m=0$ (where $t_{\pm}^{m,n}$ is not defined) can be motivated as follows. If $n>0$ then it is easy to check that
\begin{equation}
\lim_{c\to 0}\frac{1+t_{+}^{c, n}t}{t-t_{+}^{c, n}}=-t\,, \quad \lim_{c\to 0}\frac{1-t_{+}^{c, n}t}{t-t_{+}^{c, n}}=t.
\end{equation}
On the other hand, by using that for $c\neq 0$ we have $t_{+}^{c,d}t_{-}^{c,d}=-1$, one similarly finds for $n<0$
\begin{equation}
\lim_{c\to 0}\frac{t_-^{c,n}}{t_-^{c,n}}\left(\frac{1+t_{+}^{c, n}t}{t-t_{+}^{c, n}}\right)= \lim_{c\to 0}\frac{t_-^{c,n}-t}{t_-^{c,n}t+1}=\frac{1}{t}\,, \quad\quad \lim_{c\to 0}\frac{t_-^{c,n}}{t_-^{c,n}}\left(\frac{1-t_{+}^{c, n}t}{t-t_{+}^{c, n}}\right)=\lim_{c\to 0}\frac{t_-^{c,n}+t}{t_-^{c,n}t+1}=\frac{1}{t}.
\end{equation}
* As previously mentioned in the introduction, one can relate the Darboux coordinates (<ref>) to the Darboux coordinates (<ref>) by Poisson resumming the later and applying a contact transformation, as done in [15]. We will follow a similar approach by Poisson resumming the contact structure directly, and then checking that (<ref>) give Darboux coordinates.
Using (<ref>) it is not hard to see that the coordinates are well defined on an open dense subset of $\mathcal{Z}\cong \widetilde{N}\times \mathbb{C}P^1$, namely the points
\begin{equation}
\{(\tau,b^a+\I t^a, c^a,c_a,c_0,\psi, t)\in \mathcal{Z} \; | \; t\neq 0, \infty, \;\; t\not\in \{t_{+}^{m,n}(\tau)\}_{m\in \mathbb{Z}-\{0\}, n \in \mathbb{Z}}\}\,,
\end{equation}
where we have used the quantum corrected mirror map (<ref>) to put type IIB coordinates on $\widetilde{N}\subset \overline{\mathcal{N}}_{\text{IIA}}$.
In subsequent sections, we will also frequently use the notation
\begin{equation}
\widetilde{\xi}_i^{\text{inst}}:=\widetilde{\xi}_i-\widetilde{\xi}_i^{\text{cl}}, \quad \alpha^{\text{inst}}:=\alpha + \frac{1}{2}(\alpha^{\text{cl}}-\xi^{i}\widetilde{\xi}_i^{\text{cl}})\,.
\end{equation}
When comparing (<ref>) and (<ref>) with the formulas appearing in [15], we remark that we are using different conventions for the definition of $t_{\pm}^{c,d}$ and $S_{\hat{\gamma},m,n}$. Namely, what we call $S_{\hat{\gamma},m,n}$ in their notation is $S_{\hat{\gamma},-m,-n}$ and $t_{\pm}^{c,d}$ in their notation is $t_{\mp}^{c,d}$.
As an immediate corollary of Theorem <ref>, we obtain the following
Consider an instanton corrected q-map space $(\widetilde{N},g_{\overline{N}})$ with one-loop parameter $c_{\ell}\in \mathbb{R}$. If $(\xi^i,\widetilde{\xi}_i,\alpha)$ are the functions on $\mathcal{Z}$ defined by (<ref>), then if $\alpha_{c_{\ell}}:=\alpha+4\I \left(c_{\ell}-\frac{\chi}{192\pi}\right)\log(t)$, the functions $(\xi^i,\widetilde{\xi}_{i},\alpha_{c_{\ell}})$ of the associated twistor space are Darboux coordinates for the contact structure $\lambda$ of $\mathcal{Z}$, in the sense that
\begin{equation}
\lambda=4\pi \I\left(\D\alpha_{c_{\ell}} -\widetilde{\xi}_i\D \xi^i\right)\cdot s\,.
\end{equation}
Notice that since the one loop parameter only enters in the contact structure via $f$ by (<ref>) and $\rho=2\pi r^2 -c_{\ell}$, by writing $f=(f+16\pi (c_{\ell}-\frac{\chi}{192\pi}))-16\pi (c_{\ell}-\frac{\chi}{192\pi}) $ we have by Theorem <ref> that
\begin{equation}
4\pi \I\left(\D\alpha -\widetilde{\xi}_i\D \xi^i\right)=\left(f+16\pi \left(c_{\ell}-\frac{\chi}{192\pi}\right)\right)\frac{\D t}{t}+t^{-1}\I \theta_+^P|_{\overline{N}} -2\I \theta_3^P|_{\overline{N}} -t\I \theta_-^P|_{\overline{N}}\,.
\end{equation}
It then follows immediately that
\begin{equation}
4\pi \I\left(\D\alpha_{c_{\ell}} -\widetilde{\xi}_i\D \xi^i\right)=f\frac{\D t}{t}+t^{-1}\I \theta_+^P|_{\overline{N}} -2\I \theta_3^P|_{\overline{N}} -t\I \theta_-^P|_{\overline{N}}\,,
\end{equation}
compare (<ref>).
§.§.§ Preliminary lemmas
In this section we give some preliminary lemmas and expressions that will be useful to prove Theorem <ref>.
We will divide each of the $1$-forms $\theta_{+}^{P}|_{\overline{N}}$, $\theta_3^P|_{\overline{N}}$ appearing in the contact structure (<ref>) as follows:
\begin{equation}
\theta_{+}^P|_{\overline{N}}=\theta_{+}^{P, \text{cl}}+\theta_{+}^{P, \text{w.s.}}+\theta_{+}^{P, \text{inst}}, \quad\quad \theta_{3}^P|_{\overline{N}}=\theta_{3}^{P, \text{cl}}+\theta_{3}^{P, \text{w.s.}}+\theta_{3}^{P, \text{inst}},
\end{equation}
where using that $R=2\sqrt{\rho+c_{\ell}}e^{\mathcal{K}/2}=\tau_2/2$, the decompositions $F_i=F_i^{\text{cl}}+F_i^{\text{ws}}$ where $F_i^{\text{cl}}:=\partial_{Z^i}\mathfrak{F}^{\text{cl}}/Z^0$ and $F_i^{\text{w.s.}}:=\partial_{Z^i}\mathfrak{F}^{\text{w.s.}}/Z^0$, $\widetilde{\zeta}_i=\widetilde{\zeta}_i^{\text{cl}}+\widetilde{\zeta}_i^{\text{inst}}$ and $\sigma=\sigma^{\text{cl}}+\sigma^{\text{inst}}$, and (<ref>) we have
\begin{equation}\label{theta+split}
\begin{split}
\theta_{+}^{P,\text{cl}}&:=-2\pi\tau_2\left(F_i^{\text{cl}}\D \zeta^i-z^i\D \widetilde{\zeta}_i^{\text{cl}}\right)\\
\theta_{+}^{P,\text{w.s.}}&:=-2\pi\tau_2\left(F_i^{\text{w.s.}}\D \zeta^i-z^i\D \widetilde{\zeta}_i^{\text{inst}}\right)\\
\theta_{+}^{P,\text{inst}}&=\I \tau_2\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\D \zeta_{\gamma}\\
&\quad+\frac{\tau_2^2}{2}\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi \I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}+\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}+\frac{2}{\tau_2}\D \tau_2\right).\\
\end{split}
\end{equation}
For $\theta_3^P|_{\overline{N}}$ we use the relation
\begin{equation}
\D^c\mathcal{K}=e^{\mathcal{K}}(-2\text{Im}(\tau_{ij}))\Big(\I \overline{z}^j\D z^i -\I z^i\D \overline{z}^j\Big),
\end{equation}
so that
\begin{equation}
4\pi(\rho+c_{\ell})\D ^c\mathcal{K}=\pi R^2(-2\text{Im}(\tau_{ij}))\Big(\I\overline{z}^j\D z^i -\I z^i\D \overline{z}^j\Big)=\pi\tau_2^2\mathrm{Im}(\tau_{0a})\D t^a +\pi\tau_2^2\mathrm{Im}(\tau_{ab})(b^a\D t^b-t^a\D b^b),
\end{equation}
and hence we can split $\theta_3^P|_{\overline{N}}$ as follows using (<ref>)
\begin{equation}\label{theta3split}
\begin{split}
\theta_{3}^{P,\text{cl}}&:=\pi\D \sigma^{\text{cl}} +\pi\left(\widetilde{\zeta}_i^{\text{cl}}\D \zeta^i-\zeta^i\D \widetilde{\zeta}_i^{\text{cl}}\right)-\pi\tau_2^2\mathrm{Im}(\tau_{0a}^{\text{cl}})\D t^a -\pi\tau_2^2\mathrm{Im}(\tau_{ab}^{\text{cl}})(b^a\D t^b-t^a\D b^b)\\
\theta_{3}^{P,\text{w.s.}}&:=\pi\D \sigma^{\text{inst}} +\pi\left(\widetilde{\zeta}_i^{\text{inst}}\D \zeta^i-\zeta^i\D \widetilde{\zeta}_i^{\text{inst}}\right)-\pi\tau_2^2\mathrm{Im}(\tau_{0a}^{\text{w.s.}})\D t^a -\pi\tau_2^2\mathrm{Im}(\tau_{ab}^{\text{w.s.}})(b^a\D t^b-t^a\D b^b)\\
\theta_3^{P,\text{inst}}&=-\frac{\I\tau_2}{4\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}-\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right)\,,
\end{split}
\end{equation}
where $\tau_{ij}^{\text{cl}}=\partial_{Z^i}\partial_{Z^j}\mathfrak{F}^{\text{cl}}$ and $\tau_{ij}^{\text{w.s.}}=\partial_{Z^i}\partial_{Z^j}\mathfrak{F}^{\text{w.s.}}$.
Similarly, using that $f=16\pi \rho+f^{\text{inst}}$ and (using the second relation in (<ref>))
\begin{equation}\label{rhodef}
\rho=\frac{\tau_2^2}{16}e^{-\mathcal{K}}-c_{\ell}= \frac{\tau_2^2}{16}K-c_{\ell}, \quad K=-2\text{Im}(\tau_{ij})z^i\overline{z}^j\,,
\end{equation}
we can decompose $f$ as
\begin{equation}\label{fsplitting2}
f=f^{\text{cl}}+f^{\text{w.s.}}-16\pi c_{\ell}+f^{\text{inst}},
\end{equation}
\begin{equation}\label{fsplitting}
\begin{split}
f^{\text{w.s.}}&:=-2\pi\tau_2^2\text{Im}(\tau_{ij}^{\text{w.s.}})z^i\overline{z}^j=\frac{\tau_2^2}{2\pi^2}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}\text{Re}\left(\mathrm{Li}_3(e^{2\pi \I q_az^a})+2\pi q_at^a\mathrm{Li}_2(e^{2\pi \I q_az^a})\right)\\
f^{\text{inst}}&=\frac{\tau_2}{\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\,.\\
\end{split}
\end{equation}
In order to prove Theorem <ref>, we start with the following lemma, which will allow us to rewrite the Bessel function expressions of the instanton terms $f^{\text{inst}}$, $\theta_{\pm}^{P,\text{inst}}$ and $\theta_3^{P,\text{inst}}$ in terms of expressions similar to those appearing in (<ref>).
Given $\hat{\gamma}\in \Lambda^{+}$ with $n_{\hat{\gamma}}\neq 0$, and $\nu \in \mathbb{N}$, we define
\begin{equation}\label{Idef}
\begin{split}
\mathcal{I}_{\hat{\gamma}}^{(\nu)}&:=2\sum_{q_0\in \mathbb{Z}}\sum_{s=\pm 1}\sum_{n=1}^{\infty}\frac{e^{-2\pi\I ns\zeta_{q_0\gamma^0+\hat{\gamma}}}}{(sn)^{\nu}}K_0(4\pi Rn|\widetilde{Z}_{q_0\gamma^0+\hat{\gamma}}|)\\
\mathcal{I}_0^{(\nu)}&:=2\sum_{q_0\in \mathbb{Z}-\{0\}}\sum_{s=\pm 1}\sum_{n=1}^{\infty}\frac{e^{-2\pi\I ns\zeta_{q_0\gamma^0}}}{(sn)^\nu}K_0(4\pi Rn|\widetilde{Z}_{q_0\gamma^0}|)\,.
\end{split}
\end{equation}
Notice that in $\mathcal{I}_{0}^{(\nu)}$ we exclude the term $q_0=0$ from the sum, since $K_0(x)$ diverges at $x=0$. Furthermore, since $\hat{\gamma}\in \Lambda^{+}$ with $n_{\hat{\gamma}}\neq 0$ implies that $\pm (q_0\gamma^0+\hat{\gamma})\in \text{Supp}(\Omega)$ for all $q_0\in \mathbb{Z}$, by the support property of variations of BPS structures one finds that $|Z_{q_0\gamma^0+\hat{\gamma}}|\neq 0$, so that $|\widetilde{Z}_{q_0\gamma^0+\hat{\gamma}}|\neq 0$ on $\widetilde{N}$ (or $\overline{N})$. By the exponential decay of $K_0(x)$ as $x\to \infty$, it is easy to check that the convergence of the series $\mathcal{I}_{\hat{\gamma}}^{(\nu)}$ and $\mathcal{I}_{0}^{(\nu)}$ is compact normal on $\widetilde{N}$. In particular, they define smooth functions on $\widetilde{N}$ and we can interchange sums with differentiation.
The function $\mathcal{I}_{\hat{\gamma}}^{(\nu)}$ can be expressed as
\begin{equation}\label{Presum1}
\mathcal{I}_{\hat{\gamma}}^{(\nu)}=\sum_{n\in \mathbb{Z}}\sum_{m\in \mathbb{Z}-\{0\}}\frac{e^{-S_{\hat{\gamma},m,n}}}{m^{\nu}|m\tau + n|},
\end{equation}
where $S_{\hat{\gamma},m,n}$ was defined in (<ref>). Furthermore, we have for $\nu \geq 2$
\begin{equation}\label{I0der}
\partial_{\tau_2}\mathcal{I}_{0}^{(\nu)}=\partial_{\tau_2}\mathcal{I}_{\hat{\gamma}}^{(\nu)}|_{\hat{\gamma}=0}+\frac{2}{\tau_2}\sum_{s=\pm 1}\sum_{n>0}\frac {1}{(sn)^{\nu}}, \quad \partial_{\tau_1}\partial_{\tau_2}\mathcal{I}_{0}^{(\nu)}=\partial_{\tau_1}\partial_{\tau_2}\mathcal{I}_{\hat{\gamma}}^{(\nu)}|_{\hat{\gamma}=0}, \quad \partial_{\tau_1}^2\mathcal{I}_{0}^{(\nu)}=\partial_{\tau_1}^2\mathcal{I}_{\hat{\gamma}}^{(\nu)}|_{\hat{\gamma}=0}\,,
\end{equation}
where $\partial_{\tau_2}\mathcal{I}_{\hat{\gamma}}^{(\nu)}|_{\hat{\gamma}=0}$ (and similarly for the other derivatives) means taking the derivative of (<ref>) and then evaluating at $\hat{\gamma}=0$.
We will use a similar notation to <cit.> and follow the same idea of Poisson resummation, but in an easier case than the one presented in <cit.>. We denote $x=q_ab^a$, $y=q_at^a$ and $\Theta:=q_a(\zeta^a-b^a\zeta^0)=-q_ac^a$, so that we can write
\begin{equation}\label{Poisson1}
\mathcal{I}_{\hat{\gamma}}^{(\nu)}=\sum_{q_0\in \mathbb{Z}}f(x+q_0,y,\zeta^i,R)
\end{equation}
\begin{equation}
f(x,y,\zeta^i,R):=2\sum_{s=\pm 1}\sum_{n=1}^{\infty}\frac{e^{2\pi\I ns(\Theta+x\zeta^0)}}{(sn)^{\nu}}K_0(4\pi Rn|x+\I y|)\,.
\end{equation}
We have used above that $\zeta_{q_i\gamma^i}=-q_i\zeta^i$ (recall Definition <ref>).
Equation (<ref>) makes it clear that $\mathcal{I}_{\hat{\gamma}}^{(\nu)}$ is a function that is invariant under integer shifts in the $x$ variable, and the idea is to now compute the Poisson resummation with respect to the $x$-variable. Namely,
\begin{equation}\label{Prformula}
\mathcal{I}_{\hat{\gamma}}^{(\nu)}=\sum_{q_0\in \mathbb{Z}}f(x+q_0,y,\zeta^i,R)=\sum_{q_0\in \mathbb{Z}}\hat{f}(q_0,y,\zeta^i,R)e^{2\pi \I q_0x},
\end{equation}
\begin{equation}
\hat{f}(w)=\int_{-\infty}^{\infty}\D x\; f(x)e^{-2\pi \I xw}
\end{equation}
denotes the Fourier transform (from now on, we omit the dependence of $f$ of the variables $y$, $\zeta^i$ and $R$ from the notation).
Using the integral representation (<ref>) of $K_0$ we have
\begin{equation}
\begin{split}
K_0(4\pi Rn|x+\I y|)&=\int_{0}^{\infty}\D t\exp(-4\pi Rn|x+\I y|\cosh(t))=\frac{1}{2}\int_{-\infty}^{\infty}\D t\exp(-4\pi Rn|x+\I y|\cosh(t))\\
&=\frac{1}{2}\int_{0}^{\infty}\frac{\D z}{z}\exp(-2\pi Rn|x+\I y|(z^{-1}+z))\,.
\end{split}
\end{equation}
Furthermore, since the integral exponentially
decays at both ends, provided $x\neq 0$, we can deform
the integration path by changing $z$ to $z\to z\frac{|x+\I y|}{x+\I y}\text{sign}(x)$, obtaining
\begin{equation}
K_0(4\pi Rn|x+\I y|)=\frac{1}{2}\int_{0}^{\infty}\frac{\D z}{z}\exp(-2\pi Rn\cdot\text{sign}(x)((x+\I y)z^{-1}+(x-\I y)z))\,.
\end{equation}
In particular, we have
\begin{align*}
\hat{f}(w)&=\int_{-\infty}^{\infty}\D x\sum_{s=\pm 1}\sum_{n=1}^{\infty}\frac{e^{2\pi \I ns\Theta}}{(sn)^{\nu}}\int_{0}^{\infty} \frac{\D z}{z}\exp(- 2\pi nR\cdot\text{sign}(x)((x+\I y)z^{-1}+(x-\I y)z))e^{2\pi\I x(- w+ns\zeta^0)}\\
&=\sum_{s=\pm 1}\sum_{n=1}^{\infty}\frac{e^{ 2\pi\I ns\Theta}}{(sn)^{\nu}}\int_{0}^{\infty} \frac{\D z}{z}\int_{-\infty}^{\infty}\D x\exp(- 2\pi nR\cdot\text{sign}(x)((x+\I y)z^{-1}+(x-\I y)z))e^{2\pi\I x(- w+ns\zeta^0)}\\
&=\sum_{s=\pm 1}\sum_{n=1}^{\infty}\frac{e^{ 2\pi\I ns\Theta}}{(sn)^{\nu}}\int_{0}^{\infty} \frac{\D z}{z}\int_{0}^{\infty}\D x\;e^{2\pi x(-nRz^{-1}-nRz -\I w+\I ns\zeta^0)+2\pi y(-\I nRz^{-1}+\I nRz)}\\
&\quad +\sum_{s=\pm 1}\sum_{n=1}^{\infty}\frac{e^{2\pi \I ns\Theta}}{(sn)^{\nu}}\int_{0}^{\infty} \frac{\D z}{z}\int_{0}^{\infty}\D x\;e^{2\pi x(-nRz^{-1}-nRz + \I w -\I ns\zeta^0) + 2\pi y(\I nRz^{-1}-\I nRz)}\\
&=\frac{1}{2\pi}\sum_{s=\pm 1}\sum_{n=1}^{\infty}\frac{e^{2\pi \I ns\Theta}}{(sn)^{\nu}}\left(\int_{0}^{\infty} \frac{\D z}{z}\frac{e^{2\pi y(-\I nRz^{-1}+\I nRz)}}{nRz^{-1}+nRz +\I w-\I ns\zeta^0}+\int_{0}^{\infty} \frac{\D z}{z}\frac{e^{2\pi y(\I nRz^{-1}-\I nRz)}}{nRz^{-1}+nRz -\I w+\I ns\zeta^0}\right). \numberthis \\
\end{align*}
We can combine the $s=1$ and $s=-1$ terms of the previous sum into an integral over $\mathbb{R}$ and a sum over $n\in \mathbb{Z}-\{0\}$ to obtain
\begin{equation}
\hat{f}(w)=\frac{1}{2\pi}\sum_{n\in \mathbb{Z}-\{0\}}\frac{e^{2\pi \I n\Theta}}{n^{\nu-1}|n|}\int_{-\infty}^{\infty}\frac{\D z}{z}\frac{e^{-2\pi\I nRy(z^{-1}-z)}}{nR(z^{-1}+z) + \I w-\I n\zeta^0}\,.
\end{equation}
The integrand of the $n$-th term of $\hat{f}(m)$ with $m\in \mathbb{Z}$ has poles at $z=\I t_{\pm}^{n,- m}$ where $t_{\pm}^{c,d}$ was defined before in (<ref>) as the roots in $t$ of $t(c\xi^0(t)+d)=0$. Since one of our defining conditions on the manifold $M$ is that $\text{sign}(y)=\text{sign}(q_at^a)>0$ for $n_{\hat{\gamma}}\neq 0$ (see (<ref>)), we can compute the previous integral by closing the contour in the upper half plane when $n>0$, and in the lower half plane if $n<0$. Independently of the sign of $n$, only the pole at $\mathrm{i}t_{+}^{n,-m}$ contributes to the $n$-th integral. In particular, we obtain
\begin{equation}
\begin{split}
\hat{f}(m)&= \sum_{n\in \mathbb{Z}-\{0\}}\frac{e^{2\pi \I n\Theta}}{n^{\nu}}\frac{e^{-2\pi nRy(1/t_{+}^{n,- m}+t_{+}^{n,- m})}}{nR(t_{+}^{n,- m}-t_{-}^{n,- m})}= \sum_{n\in \mathbb{Z}-\{0\}}\frac{e^{2\pi \I n\Theta}}{n^{\nu}}\frac{e^{-2\pi y|n\tau- m|}}{|n\tau- m|}\,,
\end{split}
\end{equation}
where we used that $t_{+}^{n,-m}t_{-}^{n,-m}=-1$ and the expressions (<ref>) of $t_{\pm}^{n,-m}$.
After plugging the previous result in (<ref>), using $2R=\tau_2$, (<ref>) and relabeling the summation indices we then obtain
\begin{equation}\label{resum1}
\begin{split}
\mathcal{I}_{\hat{\gamma}}^{(\nu)}&= \sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\frac{e^{2\pi \I m\Theta-2\pi \I nx}}{m^{\nu}}\frac{e^{-2\pi y|m\tau+ n|}}{|m\tau+ n|}= \sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\frac{e^{-S_{\hat{\gamma},m,n}}}{m^{\nu}|m\tau + n|}\,.\\
\end{split}
\end{equation}
Finally, we use the following identities to prove (<ref>):
\begin{equation}\label{Bessellimit}
\lim_{z\to 0}zK_1(2\pi\tau_2 nz)=\frac{1}{2\pi n\tau_2}, \quad \lim_{z\to 0}z^2K_1(2\pi\tau_2 n z)=0, \quad \lim_{z\to 0} z^2K_0(2\pi \tau_2 nz)=0\,.
\end{equation}
We show the first identity in (<ref>), with the others following similarly. By applying the first limit in (<ref>), the dominated convergence theorem (here we use that $\nu\geq 2)$ and $K_0'=-K_1$, we obtain:
\begin{equation}
\begin{split}
\partial_{\tau_2}\mathcal{I}_{0}^{(\nu)}&=\lim_{\zeta_{\hat{\gamma}}\to 0,\widetilde{Z}_{\hat{\gamma}}\to 0}\left(\partial_{\tau_2}\mathcal{I}_{\hat{\gamma}}^{(\nu)}+2\sum_{s=\pm 1}\sum_{n>0}\frac{e^{-2\pi \mathrm{i}ns \zeta_{\hat{\gamma}}}}{(sn)^{\nu}}2\pi n|\widetilde{Z}_{\hat{\gamma}}|K_1(2\pi n\tau_2|\widetilde{Z}_{\hat{\gamma}}|)\right)\\
&=\left(\partial_{\tau_2}\mathcal{I}_{\hat{\gamma}}^{(\nu)}\right)\Big|_{\hat{\gamma}=0}+\frac{2}{\tau_2}\sum_{s=\pm 1}\sum_{n>0}\frac{1}{(sn)^{\nu}}\,.
\end{split}
\end{equation}
§.§.§ Poisson resummation of the quantum corrections of the contact structure
In this section, we seek to rewrite the quantum corrections terms in $f$, $\theta_{+}^P$ and $\theta_3^P$ using the Poisson resummation of Lemma <ref>. This will help us show that the coordinates (<ref>) are Darboux coordinates for the contact structure describing the instanton corrected q-map metric associated to the data of Section <ref>. We start with the following proposition for $f$:
The following holds for $f^{\text{w.s.}}$ and $f^{\text{inst}}$ defined in (<ref>):
\begin{equation}\label{typeiibfquant}
f^{\text{w.s.}}+f^{\text{inst}}=\frac{\chi}{12}+\frac{\tau_2^2}{(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(\frac{1}{|m\tau+n|}+2\pi q_at^a\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}
\end{equation}
On one hand, we have
\begin{equation}
\begin{split}
f^{\text{inst}}&=\frac{2R}{\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi\I n\zeta_{\gamma}}}{n}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\\
&=\frac{2R}{\pi}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\sum_{s=\pm 1}\sum_{q_0\in \mathbb{Z}}\sum_{n>0}\frac{e^{-2\pi\I ns\zeta_{q_0\gamma^0+\hat{\gamma}}}}{n}|\widetilde{Z}_{q_0\gamma^0+\hat{\gamma}}|K_1(4\pi Rn|\widetilde{Z}_{q_0\gamma^0+\hat{\gamma}}|)\\
&\quad \quad-\frac{R}{\pi}\chi\sum_{s=\pm 1}\sum_{q_0\in \mathbb{Z}-\{0\}}\sum_{n>0}\frac{e^{-2\pi\I ns\zeta_{q_0\gamma^0}}}{n}|\widetilde{Z}_{q_0\gamma^0}|K_1(4\pi Rn|\widetilde{Z}_{q_0\gamma^0}|)\\
&=-\frac{R}{2\pi^2}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\partial_{\tau_2}\mathcal{I}_{\hat{\gamma}}^{(2)}+\frac{R}{(2\pi)^2}\chi \partial_{\tau_2}\mathcal{I}_{0}^{(2)}\,,\\
\end{split}
\end{equation}
where we have used the particular form of the BPS indices (<ref>), the expressions (<ref>) and the fact that $R=\tau_2/2$. Using Lemma <ref> we therefore obtain that
\begin{equation}\label{typeiibfinst}
\begin{split}
f^{\text{inst}}&=\frac{\tau_2^2}{(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\left(\frac{1}{|m\tau+ n|}+2\pi q_at^a\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}\\
&\quad\quad -\frac{\tau_2^2}{2(2\pi)^2}\chi \sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\frac{1}{|m\tau + n|^3}+\frac{\tau_2\chi}{2(2\pi)^2}\frac{4}{\tau_2}\sum_{n>0}\frac{1}{n^2}\\
&=\frac{\chi}{12}+\frac{\tau_2^2}{(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}\sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\left(\frac{1}{|m\tau+ n|}+2\pi q_at^a\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}\,,\\
\end{split}
\end{equation}
where in the last equality we have used that $\sum_{n>0}\frac{1}{n^2}=\frac{\pi^2}{6}$ and combined the other two sums by using the convention $n_{0}=-\frac{\chi}{2}$ from Section <ref>.
On the other hand, it is not hard to check using (<ref>) that one can write
\begin{equation}\label{typeiibfws}
f^{\text{w.s.}}=\frac{\tau_2^2}{(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}\sum_{n\in \mathbb{Z}-\{0\}}\left(\frac{1}{|n|}+2\pi q_at^a\right)\frac{e^{-S_{\hat{\gamma},0,n}}}{ n^2}\\
\end{equation}
By summing (<ref>) and (<ref>) we therefore obtain (<ref>).
Combining Proposition <ref> with (<ref>) we see that
\begin{equation}\label{Poissonresumedf}
f=8\pi \tau_2^2h(t) +16\pi\left(\frac{\chi}{192\pi}-c_{\ell}\right)+ \frac{\tau_2^2}{(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}\left(\frac{1}{|m\tau+n|}+2\pi q_at^a\right)\frac{e^{-S_{\hat{\gamma},m,n}}}{|m\tau + n|^2}\,,
\end{equation}
where $f^{\text{cl}}=-2\pi\tau_2^2\text{Im}(\tau_{ij}^{\text{cl}})z^i\overline{z}^j=8\pi\tau_2^2h(t)$ and $h(t)$ is given in (<ref>). In particular, we see that the quantum corrected $f$ has the same transformation rule as $f^{\text{cl}}$ under $S$-duality if and only if $c_{\ell}=\frac{\chi}{192\pi}$. Namely, for this value of $c_{\ell}$, $f$ transforms under $\begin{pmatrix}
a & b\\
c & d\\ \end{pmatrix}\in \mathrm{SL}(2,\mathbb{Z})
$ as
\begin{equation}
f\to \frac{f}{|c\tau+d|}\,.
\end{equation}
This suggests that this particular value of the 1-loop constant $c_{\ell}$ could play a special role in studying $S$-duality, and indeed, we will see in Section <ref> that for this value S-duality acts by isometries on the instanton corrected q-map space (provided $S$-duality acts on the domain of definition of the metric). Furthermore, in the setting of Calabi-Yau compactification of type IIB string theory, the quantity $f$ with $c_{\ell}=\frac{\chi}{192}$ is proportional to the 4d dilaton with perturbative, world-sheet instanton, D(-1) and D1 instanton corrections, and the corresponding transformation property has been previously remarked in the physics literature (see for example <cit.> and the paragraphs below the equation).
We now continue with the Poisson resummation of the other terms of the contact form.
The 1-form $-2\I (\theta_3^{P, \text{w.s.}}+\theta_3^{P, \text{inst}})$ can be rewritten in terms of the coordinates $(\tau_2,b^a,t^a,\tau_1=\zeta^0,\zeta^a, \widetilde{\zeta}_i,\sigma)$ as follows
\begin{equation}\label{typeiibtheta3quantum}
\begin{split}
&-2\I (\theta_3^{P, \text{w.s.}}+\theta_3^{P, \text{inst}})\\
&=\sum_{\hat{\gamma}\in \Lambda^{+}\cup\{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^{2}-\{0\}}e^{-S_{\hat{\gamma},m,n}}\Bigg[\frac{\I \tau_2^4}{2\pi}\frac{m^2}{|m\tau +n|^4}q_a\mathrm{d}b^a -\frac{\tau_2^2}{2\pi}\frac{m\tau_1+ n}{|m\tau+ n|^3}q_a\mathrm{d}t^a+\frac{\I \tau_2^2}{2\pi }\frac{m(m\tau_1+ n)}{|m\tau+ n|^4}q_a\mathrm{d}\zeta^a
\\
&\quad-\left(\frac{\tau_2^3}{\pi^2} \frac{m^2(m\tau_1+ n)}{|m\tau+ n|^6}+\frac{\tau_2}{2\pi}q_at^a\frac{(m\tau_1+ n)^3+2m^2\tau_2^2(m\tau_1+ n)}{|m\tau+ n|^5}\right)\mathrm{d}\tau_2\\
&\quad+\left(-\frac{\I \tau_2^2}{2\pi}q_ab^a \frac{m(m\tau_1+ n)}{|m\tau+ n|^4}+\frac{\tau_2^4}{2\pi}q_at^a \frac{m^3}{|m\tau+ n|^5}+\frac{\tau_2^2}{2\pi^2}\frac{m(m^2\tau_2^2-(m\tau_1+ n)^2)}{|m\tau+ n|^6}\right)\mathrm{d}\tau_1\Bigg].\\
\end{split}
\end{equation}
We follow the same idea as in the previous proposition. Namely, notice that using (<ref>) and (<ref>), $\theta_{3}^{P,\text{inst}}$ can be written as
\begin{equation}
\begin{split}
\theta_3^{P,\text{inst}}=&-\frac{\I R}{2\pi}\sum_{\gamma}\Omega(\gamma)\sum_{n>0}\frac{e^{-2\pi \I n\zeta_{\gamma}}}{n}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}-\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}\right)\\
&=-\frac{\I R}{2\pi}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\sum_{s=\pm 1}\sum_{q_0\in \mathbb{Z}}\sum_{n>0}\frac{e^{-2\pi \I ns\zeta_{q_0\gamma^0+\hat{\gamma}}}}{n}|\widetilde{Z}_{q_0\gamma^0+\hat{\gamma}}|K_1(4\pi Rn|\widetilde{Z}_{q_0\gamma^0+\hat{\gamma}}|)\left(\frac{\D \widetilde{Z}_{\hat{\gamma}}}{\widetilde{Z}_{q_0\gamma^0+\hat{\gamma}}}-\frac{\D \overline{\widetilde{Z}_{\hat{\gamma}}}}{\overline{\widetilde{Z}}_{q_0\gamma^0+\hat{\gamma}}}\right)\\
&=\frac{\I }{8\pi^2}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\partial_{z^a}\mathcal{I}_{\hat{\gamma}}^{(2)}\D z^a-\frac{\I }{8\pi^2}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\partial_{\overline{z}^a}\mathcal{I}_{\hat{\gamma}}^{(2)}\D \overline{z}^a,
\end{split}
\end{equation}
where in second equality we have used that no $\hat{\gamma}=0$ terms appear due to $\mathrm{d}\widetilde{Z}_{q_0\gamma^0}=\mathrm{d}q_0=0$.
Using Lemma <ref> we then obtain the following:
\begin{equation}\label{typeiibtheta3inst}
\begin{split}
-2\I &\theta_3^{P,\text{inst}}\\ &=\frac{\I }{2\pi}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\frac{e^{-S_{\hat{\gamma},m,n}}}{m^2}q_a\D b^a+\frac{1}{2\pi}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}e^{-S_{\hat{\gamma},m,n}}\frac{m\tau_1+ n}{m^2|m\tau+ n|}q_a\D t^a\,.\\
\end{split}
\end{equation}
Observe for the calculation that the partial derivatives $\partial_{z^a}$ and $\partial_{\overline{z}^a}$ are to be taken with respect to the mixed coordinates $(\tau_2,b^a,t^a,\tau_1=\zeta^0,\zeta^a, \widetilde{\zeta}_i,\sigma)$, which are neither the IIA nor the IIB coordinates.
On the other hand, using directly the formulas (<ref>) and (<ref>) for $\sigma^{\text{inst}}$, $\widetilde{\zeta}_i^{\text{inst}}$ and $\mathfrak{F}^{\text{w.s.}}$ one can compute $-2\I \theta_3^{P,\text{w.s.}}$ in (<ref>). The $\D \zeta^a$, $\D \tau_1$ and $\D \tau_2$ components of $-2\I \theta_3^{P,\text{w.s.}}$ can be seen to match the corresponding components in the right-hand side of (<ref>). On the other hand, the $\D t^a$ and $\D b^a$ components of $-2\I \theta_3^{P,\text{w.s.}}$ are as follows
\begin{align*}\label{typeiibtheta3ws}
-2\I \theta_3^{P,\text{w.s.}}|_{\D b^a}&= -\frac{\I }{2\pi}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}q_a\sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\left(2-\frac{(m\tau_1+ n)^2}{|m\tau+n|^2}\right)\frac{e^{-S_{\hat{\gamma},m,n}}(m\tau_1+ n)^2}{m^2|m\tau+ n|^2}\\
-2\I \theta_3^{P,\text{w.s.}}|_{\D t^a}&= -\frac{\tau_2^2}{2\pi}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}q_a\sum_{n\in \mathbb{Z}-\{0\}}\frac{e^{-S_{\hat{\gamma},0,n}}}{n|n|}\\
& \quad-\frac{1}{2\pi}\sum_{\hat{\gamma}\in \Lambda^+\cup\{0\}}n_{\hat{\gamma}}q_a\sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\left(2-\frac{(m\tau_1+ n)^2}{|m\tau+ n|^2}\right)\frac{e^{-S_{\hat{\gamma},m,n}}(m\tau_1+ n)}{m^2|m\tau+ n|}\,.\numberthis\\
\end{align*}
One can then directly check that the $\D b^a$ and $\D t^a$ components of (<ref>) and (<ref>) combine into the corresponding components of (<ref>). We therefore obtain the required expression.
Finally, we compute a similar expression for $\theta_{+}^P$:
The 1-form $\I (\theta_{+}^{P,\text{w.s.}}+\theta_{+}^{P,\text{inst}})$ can be rewritten in terms of the coordinates $(\tau_2,b^a,t^a,\tau_1=\zeta^0,\zeta^a, \widetilde{\zeta}_i,\sigma)$ as follows
\begin{equation}\label{typeiibthetaplusquant}
\begin{split}
&\I (\theta_{+}^{P,\text{w.s.}}+\theta_{+}^{P,\text{inst}})\\
=&\sum_{\hat{\gamma}\in \Lambda^+\cup \{0\}}n_{\hat{\gamma}}\sum_{(m,n)\in \mathbb{Z}^2-\{0\}}e^{-S_{\hat{\gamma},m,n}}\Bigg[\frac{\I \tau_2^3}{4\pi}\frac{m(|m\tau+n|-(m\tau_1+n))}{|m\tau+ n|^4}q_a\mathrm{d}b^a\\
&-\frac{\tau_2^3}{4\pi } \frac{m}{|m\tau+ n|^3}q_a\mathrm{d}t^a+\frac{\I \tau_2}{4\pi}\frac{(m\tau_1+ n)(|m\tau+ n|-(m\tau_1+ n))}{|m\tau+ n|^4}q_a\mathrm{d}\zeta^a \\
&-\frac{\tau_2}{8\pi^2}\frac{2(m\tau_1+ n)(m^2\tau_2^2-(m\tau_1+ n)^2) +|m\tau+ n|(2(m\tau_1+ n)^2-m^2\tau_2^2)}{|m\tau+ n|^6}\mathrm{d}\tau_1\\
\end{split}
\end{equation}
As before, we first notice that we can write the terms of $\theta_{+}^{P,\text{inst}}$ as follows. For the first sum in (<ref>) we have
\begin{equation}
\begin{split}
&2R\I\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_0(4\pi Rn|\widetilde{Z}_{\gamma}|)\D \zeta_{\gamma}\\
&=2R\I\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\sum_{q_0\in \mathbb{Z}}\sum_{s=\pm 1}\sum_{n>0}\widetilde{Z}_{q_0\gamma^0+\hat{\gamma}}e^{-2\pi \I ns\zeta_{q_0\gamma^0+\hat{\gamma}}}K_0(4\pi Rn|\widetilde{Z}_{q_0\gamma^0+\hat{\gamma}}|)\D \zeta_{q_0\gamma^0+\hat{\gamma}}\\
&\quad -R\I \chi\sum_{q_0\in \mathbb{Z}}\sum_{s=\pm 1}\sum_{n>0}\widetilde{Z}_{q_0\gamma^0}e^{-2\pi \I ns\zeta_{q_0\gamma^0}}K_0(4\pi Rn|\widetilde{Z}_{q_0\gamma^0}|)\D \zeta_{q_0\gamma^0}\\
&=-\frac{1}{(2\pi)^3}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\D \zeta^0\partial_{\zeta^0}\partial_{\tau_2}\partial_{\overline{z}_{\hat{\gamma}}}\mathcal{I}_{\hat{\gamma}}^{(3)}-\frac{\I }{(2\pi)^2}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\D \zeta^a\partial_{\tau_2}\partial_{\overline{z}^a}\mathcal{I}_{\hat{\gamma}}^{(2)}-\frac{R\I }{2(2\pi)^2}\chi \D \zeta^0\partial_{\zeta^0}^2\mathcal{I}_{0}^{(2)}\,,\\
\end{split}
\end{equation}
where $\partial_{z_{\hat{\gamma}}}:=\frac{1}{\#q}\frac{1}{q_a}\partial_{z^a}$ with the sum in the index $a$ only for the $a=1,...,n$ such that $q_a\neq0$, and $\#q$ is the number of non-zero $q_a$[It is natural to denote $z_{\hat{\gamma}}:=q_az^a$ for $\hat{\gamma}\in \Lambda^{+}$. With this notation, the differential operator $\partial_{\widetilde{z}_{\hat{\gamma}}}$ satisfies $\partial_{z_{\hat{\gamma}}}z_{\hat{\gamma}}=1$.]. By Lemma <ref>, we then find
\begin{align*}
&2R\I \sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_0(2Rn|\widetilde{Z}_{\gamma}|)\D \zeta_{\gamma}\\ &=\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}e^{-S_{\hat{\gamma},m,n}}\Bigg[-\frac{\I \tau_2}{8\pi^2}\left(\frac{1}{|m\tau+ n|^3}\left(1-3\frac{(m\tau_1+ n)^2}{|m\tau+ n|^2}\right)-2\pi\I \frac{q_ab^a(m\tau_1+ n)}{|m\tau+ n|^3}\right)\D \zeta^0\\
&\quad-\frac{\I \tau_2}{4\pi}\frac{q_at^a}{|m\tau+ n|^4}\left(m^2\tau_2^2-|m\tau+ n|(m\tau_1+ n)-2(m\tau_1+ n)^2\right)\D \zeta^0\\
&\quad+\frac{\I \tau_2}{2}q_at^aq_b\frac{m\tau_1+ n+|m\tau+ n|}{|m\tau+ n|^2}\left(\I b^b+\frac{(m\tau_1+ n)t^b}{|m\tau+ n|}\right)\D \zeta^0\\
&\quad+\frac{\tau_2}{4\pi}\frac{q_a}{|m\tau+ n|^2}\left(2\pi q_bt^b(m\tau_1+ n+|m\tau+ n|)+\frac{m\tau_1+ n}{|m\tau + n|}\right)\mathrm{\D }\zeta^a\Bigg]\\
&\quad +\frac{\I \tau_2}{(4\pi)^2}\chi \sum_{\substack{m\in \mathbb{Z}-\{0\}\\ n\in \mathbb{Z}}}\frac{1}{|m\tau+n|^3}\left(1-3\frac{(m\tau_1+ n)^2}{|m\tau+ n|^2}\right)\D \zeta^0\,. \numberthis \label{typeiibthetaplusinst1}\\
\end{align*}
Similarly, for the remaining term of $\theta_{+}^{P,\text{inst}}$ in (<ref>) we have that
\begin{equation}
\begin{split}
&2R^2\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi \I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\left(\frac{\D \widetilde{Z}_{\gamma}}{\widetilde{Z}_{\gamma}}+\frac{\D \overline{\widetilde{Z}}_{\gamma}}{\overline{\widetilde{Z}}_{\gamma}}+\frac{2}{\tau_2}\D \tau_2\right)\\
&=2R^2\sum_{\gamma}\Omega(\gamma)\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\D \widetilde{Z}_{\gamma}+2R^2\sum_{\gamma}\Omega(\gamma)\frac{\widetilde{Z}_{\gamma}^2}{|\widetilde{Z}_{\gamma}|}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\D \overline{\widetilde{Z}}_{\gamma}\\
&\quad +\tau_2\sum_{\gamma}\Omega(\gamma)\widetilde{Z}_{\gamma}\sum_{n>0}e^{-2\pi\I n\zeta_{\gamma}}|\widetilde{Z}_{\gamma}|K_1(4\pi Rn|\widetilde{Z}_{\gamma}|)\D \tau_2\\
&=\frac{\I \tau_2^2}{16\pi^2}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\partial_{\zeta^a}\partial_{\tau_2}\mathcal{I}_{\hat{\gamma}}^{(2)}\D z^a-\frac{1}{(2\pi)^3}\sum_{\hat{\gamma}\in \Lambda^+}n_{\hat{\gamma}}\partial_{\overline{z}^a}\partial_{\tau_2}\partial_{\overline{z}_{\hat{\gamma}}}\mathcal{I}_{\hat{\gamma}}^{(3)}\D \overline{z}^a\\
|
# Confronting the primordial black hole scenario with the gravitational-wave
events detected by LIGO-Virgo
Zu-Cheng Chen<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical
Physics, Institute of Theoretical Physics, Chinese Academy of Sciences,
Beijing 100190, China School of Physical Sciences, University of Chinese
Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China Chen Yuan
<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, No.
19A Yuquan Road, Beijing 100049, China Qing-Guo Huang Corresponding author:
<EMAIL_ADDRESS>CAS Key Laboratory of Theoretical Physics, Institute of
Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physical Sciences, University of Chinese Academy of Sciences, No.
19A Yuquan Road, Beijing 100049, China School of Fundamental Physics and
Mathematical Sciences Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China Center for Gravitation and Cosmology, College of Physical
Science and Technology, Yangzhou University, Yangzhou 225009, China
###### Abstract
Adopting a binned method, we model-independently reconstruct the mass function
of primordial black holes (PBHs) from GWTC-3 and find that such a PBH mass
function can be explained by a broad red-tilted power spectrum of curvature
perturbations. Even though GW190521 with component masses in upper mass gap
$(m>65M_{\odot})$ can be naturally interpreted in the PBH scenario, the events
(including GW190814, GW190425, GW200105, and GW200115) with component masses
in the light mass range $(m<3M_{\odot})$ are quite unlikely to be explained by
binary PBHs although there are no electromagnetic counterparts because the
corresponding PBH merger rates are much smaller than those given by LIGO-
Virgo. Furthermore, we predict that both the gravitational-wave (GW)
background generated by the binary PBHs and the scalar-induced GWs
accompanying the formation of PBHs should be detected by the ground-based and
space-borne GW detectors and pulsar timing arrays in the future.
Introduction. Primordial black holes (PBHs) Carr and Hawking (1974); Carr
(1975) are formed in the very early Universe due to the collapse of over-
densed regions which are generated by the enhanced curvature power spectrum on
small scales compared to those on the cosmic microwave background (CMB)
scales. PBHs can not only represent the dark matter (DM) in the Universe, but
also may potentially provide an explanation to the merger events detected by
LIGO-Virgo Collaboration (LVC) if the fraction of the stellar mass PBHs in
cold dark matter (CDM) is $f_{\mathrm{PBH}}\sim\mathrm{few}\times 10^{-3}$
Sasaki _et al._ (2016); Chen and Huang (2018); Raidal _et al._ (2019); De
Luca _et al._ (2020a); Hall _et al._ (2020); Bhagwat _et al._ (2021); Hütsi
_et al._ (2020); Wong _et al._ (2021); De Luca _et al._ (2021a); Franciolini
_et al._ (2021).
Among all the merger events detected by LVC, some of them are likely to have
ambiguities in the astrophysical scenario. Firstly, the primary component of
GW190521 has a high probability to be within the pair-instability supernovae
mass gap Abbott _et al._ (2020a), implying that the primary black hole (BH)
might not have a stellar origin. Secondly, even though the component masses of
GW190425 lie in the mass range of $[1.12M_{\odot},2.52M_{\odot}]$ and are
consistent with the individual binary component being neutron star (NS) Abbott
_et al._ (2020b), the source-frame chirp mass and total mass are significantly
larger than any known binary NS system. Thirdly, GW190814 is reported to have
a compact object with a mass of $2.5-2.67M_{\odot}$ Abbott _et al._ (2020c),
which falls in the “lower mass gap” where no NSs or BHs have been observed in
a double compact-object system. Finally, LVC recently reported GW200105 and
GW200115 Abbott _et al._ (2021a) in which the secondary component masses are
respectively $1.9_{-0.2}^{+0.3}M_{\odot}$ and $1.5_{-0.3}^{+0.7}M_{\odot}$,
indicating that both of them are well below the maximal mass of an NS. In
addition, there are no electromagnetic counterparts to confirm and PBHs are
speculated to provide an explanation to these five gravitational-wave (GW)
events Abbott _et al._ (2020b, c, 2021a); De Luca _et al._ (2021b).
In this letter, we will give a comprehensive investigation for the possibility
that if the GW events, in particular the five events mentioned above, can be
explained in the PBH scenario. We adopt a binned method to model-independently
reconstruct the PBH mass function from GWTC-3 Abbott _et al._ (2021b), and we
find that GW190521 can be explained by a binary PBH merger. But the other four
GW events (GW190814, GW190425, GW200105, and GW200115) are quite unlikely to
be interpreted as binary PBHs because the corresponding merger rates of binary
PBHs are much smaller than those given by LVC even though the PBH mass
function around $\sim 1M_{\odot}$ is significantly enhanced due to the
softening of the equation of state during the QCD phase transition Byrnes _et
al._ (2018).
In addition, we compute the gravitational-wave background (GWB) from both the
PBH binary coalescences Chen _et al._ (2019) and the so-called scalar-induced
gravitational waves (SIGWs) generated by the curvature perturbation during the
formation of PBHs Saito and Yokoyama (2009). We find that the GWB associated
with PBHs is compatible with the current limits of observations and should be
detected by the ground-based and space-borne GW detectors and pulsar timing
arrays in the future.
A model-independent reconstruction of PBH mass function. In this work, the PBH
mass is considered in the range of $[1M_{\odot},130M_{\odot}]$. To infer the
PBH mass function from the GWTC-3, we adopt a model-independent approach using
the following binned mass function
$P(m)=\begin{cases}P_{1},&1\,M_{\odot}\leq m<3\,M_{\odot}\\\
P_{2},&3\,M_{\odot}\leq m<10\,M_{\odot}\\\ P_{3},&10\,M_{\odot}\leq
m<40\,M_{\odot}\\\ P_{4},&40\,M_{\odot}\leq m<80\,M_{\odot}\\\
P_{5},&80\,M_{\odot}\leq m\leq 130\,M_{\odot}\end{cases}$ (1)
in which the mass function $P(m)$ is normalized by $\int P(m)dm=1$. Therefore,
only four of $P_{i}(i=1,\cdots,5)$ are independent, and
$\vec{\theta}=\\{P_{1},P_{2},P_{3},P_{4}\\}$ are chosen to be the free
parameters. The merger rate density in units of
$\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$ for a general mass function,
$P(m|\vec{\theta})$, takes the form of Chen and Huang (2018)
$\displaystyle\mathcal{R}_{12}$ $\displaystyle(t|\vec{\theta})\approx 2.8\cdot
10^{6}\times\left({\frac{t}{t_{0}}}\right)^{-\frac{34}{37}}f_{\mathrm{PBH}}^{2}(0.7f_{\mathrm{PBH}}^{2}+\sigma_{\mathrm{eq}}^{2})^{-{21\over
74}}$ (2)
$\displaystyle\times\min\left(\frac{P(m_{1}|\vec{\theta})}{m_{1}},\frac{P(m_{2}|\vec{\theta})}{m_{2}}\right)\left({P(m_{1}|\vec{\theta})\over
m_{1}}+{P(m_{2}|\vec{\theta})\over m_{2}}\right)$
$\displaystyle\times(m_{1}m_{2})^{{3\over 37}}(m_{1}+m_{2})^{36\over 37},$
where the component masses $m_{1}$ and $m_{2}$ are in units of $M_{\odot}$,
$f_{\mathrm{PBH}}\equiv\Omega_{\mathrm{PBH}}/\Omega_{\mathrm{CDM}}$ is the
energy density fraction of PBHs in CDM, and $\sigma_{\mathrm{eq}}\approx
0.005$ Ali-Haïmoud _et al._ (2017); Chen and Huang (2018) is the variance of
density perturbations of the rest CDM on scale of order
$\mathcal{O}(10^{0}\sim 10^{3})M_{\odot}$ at radiation-matter equality.
We perform the hierarchical Bayesian inference Abbott _et al._ (2016a, b, c);
Wysocki _et al._ (2019); Fishbach _et al._ (2018); Mandel _et al._ (2019);
Thrane and Talbot (2019) to extract the population parameters
$\\{\vec{\theta},f_{\mathrm{PBH}}\\}$ from observed BBHs. Given the data of
$N$ binary BH (BBH) detections, $\vec{d}=(d_{1},\dots,d_{N})$, the likelihood
for an inhomogeneous Poisson process is Wysocki _et al._ (2019); Fishbach
_et al._ (2018); Mandel _et al._ (2019); Thrane and Talbot (2019)
$p(\vec{d}|\vec{\theta},R)\propto
e^{-\beta(\vec{\theta})}\prod_{i}^{N}\int\mathrm{d}\vec{\lambda}\
p(\vec{\lambda}|d_{i})\ \mathcal{R}_{12}(\vec{\lambda}|\vec{\theta}),$ (3)
where $\vec{\lambda}\equiv\\{m_{1},m_{2}\\}$, $p(\vec{\lambda}|d_{i})$ is the
posterior of an individual event, and
$\beta(\vec{\theta})\equiv\int\mathrm{d}\vec{\lambda}\ VT(\vec{\lambda})\
\mathcal{R}_{12}(\vec{\lambda}|\vec{\theta})$ where $VT(\vec{\lambda})$ is the
spacetime sensitivity volume of LIGO-Virgo detectors. We use the GW events
from GWTC-3 by discarding events with false alarm rate larger than $1\
\mathrm{yr}^{-1}$, and events with the secondary component mass smaller than
$3M_{\odot}$ to avoid contamination from putative events involving neutron
stars De Luca _et al._ (2021a). On the other hand, although PBHs are expected
to have negligible spin at formation De Luca _et al._ (2019a); Mirbabayi _et
al._ (2020), they might become fast rotating through accretion effects De Luca
_et al._ (2020b, c). However, given that the accretion model is very sensitive
to the cut-off points of the red-shift and there is no evidence to support the
accretion effects on PBHs so far, we consider two cases where in case I we
discard the events with non-vanishing effective spin while we keep these
events in case II.
The median value and $90\%$ equal-tailed credible intervals for the parameters
$\\{\vec{\theta},f_{\mathrm{PBH}}\\}$ are represented by crosses in Fig. 1.
For the case I, the results are $P_{1}=1.5^{+0.6}_{-0.7}\times
10^{-1}M_{\odot}^{-1}$, $P_{2}=2.5^{+1.8}_{-1.1}\times 10^{-2}M_{\odot}^{-1}$,
$P_{3}=1.4^{+0.4}_{-0.3}\times 10^{-2}M_{\odot}^{-1}$,
$P_{4}=2.0^{+0.9}_{-0.7}\times 10^{-3}M_{\odot}^{-1}$, and
$f_{\mathrm{PBH}}=2.9^{+0.7}_{-0.6}\times 10^{-3}$. Therefore, the total local
merger rate is $154^{+186}_{-88}\,\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$
according to Eq. (2). For the case II, the results are
$P_{1}=1.1^{+0.8}_{-0.5}\times 10^{-1}M_{\odot}^{-1}$,
$P_{2}=4.2^{+1.2}_{-1.8}\times 10^{-2}M_{\odot}^{-1}$,
$P_{3}=1.3^{+0.3}_{-0.3}\times 10^{-2}M_{\odot}^{-1}$,
$P_{4}=2.0^{+0.7}_{-0.6}\times 10^{-3}M_{\odot}^{-1}$, and
$f_{\mathrm{PBH}}=3.4^{+0.7}_{-0.5}\times 10^{-3}$. Therefore, the total local
merger rate is $175^{+144}_{-68}\,\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}$.
Figure 1: The median values and the $90\%$ credible intervals of the
population parameters $f_{\mathrm{PBH}}P_{i}$ with $i=1,\cdots,5$. The blue
(red) curves correspond to the mass function of PBHs generated by a broad
tilted curvature power spectrum corresponding to discarding (keeping) non-
vanishing effective spin events.
From our binned reconstruction of PBH mass function, the inferred merger rates
of binary PBHs for the events with component masses in the upper mass gap
$(m>65M_{\odot})$ and the light mass range $(m<3M_{\odot})$ are listed in the
third column of Tab. 1. Compared to the merger rates given by LVC (the second
column of Tab. 1), we conclude that the upper mass gap event GW190521 can be
explained by the PBH scenario, while the events with at least one component
mass being smaller than $3M_{\odot}$ (GW190425, GW190814, GW200105, and
GW200115) are quite unlikely to be explained in the PBH scenario because the
merger rates of corresponding binary PBHs are at least one order of magnitude
smaller than those given by LVC.
Event | $R_{\mathrm{LVC}}[\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}]$ | $R_{\mathrm{PBH}}[\mathrm{Gpc}^{-3}\,\mathrm{yr}^{-1}]$
---|---|---
case I | case II
GW190521 | $0.13^{+0.30}_{-0.11}$ Abbott _et al._ (2020a) | $0.12^{+0.11}_{-0.07}$ | $0.16^{+0.11}_{-0.08}$
GW190425 | $460^{+1050}_{-390}$ Abbott _et al._ (2020b) | $3.6^{+6.8}_{-2.8}$ | $2.5^{+6.8}_{-1.8}$
GW190814 | $7^{+16}_{-6}$ Abbott _et al._ (2020c) | $0.13^{+0.09}_{-0.07}$ | $0.12^{+0.11}_{-0.05}$
GW200105 | $16^{+38}_{-14}$ Abbott _et al._ (2021a) | $1.9^{+1.8}_{-1.0}$ | $2.8^{+2.1}_{-1.4}$
GW200115 | $36^{+82}_{-30}$ Abbott _et al._ (2021a) | $6.3^{+7.4}_{-3.8}$ | $8.7^{+8.1}_{-4.5}$
Table 1: The local merger rate inferred by LIGO-Virgo and PBH.
It is also worthy figuring out a physical explanation for the PBH mass
function reconstucted from GWTC-3 in Fig. 1. Here, we take into account a
broad tilted power spectrum for the curvature perturbations, namely
$\mathcal{P}_{\mathcal{R}}(k)=A(k/k_{\min})^{\alpha}\Theta(k-k_{\min})\Theta(k_{\max}-k),$
(4)
where $\Theta$ is the heaviside theta function, $k_{\min}=3.2\times
10^{5}\mathrm{Mpc}^{-1}$ and $k_{\max}=9.5\times 10^{6}\mathrm{Mpc}^{-1}$.
After PBHs are formed, they evolve like dust-like matter during radiation
dominated era, and then
$f_{\mathrm{PBH}}P(m)=\frac{1}{\Omega_{\mathrm{CDM}}m}\left(\frac{M_{\mathrm{eq}}}{m}\right)^{1/2}\beta\left(m\right),$
(5)
where $M_{\mathrm{eq}}\approx 2.8\times 10^{17}M_{\odot}$ is the horizon mass
at matter-radiation equality and $\beta(m)$ describes the mass fraction of the
Universe that collapse to form PBHs. Using Press-Schechter formalism Press and
Schechter (1974), $\beta(m)$ can be evaluated by integrating the probability
distribution function (PDF) $P(\delta)$ of the density contrast $\delta$ that
is larger than the threshold value, namely
$\beta(m)=\int_{\delta_{c}}^{\infty}d\delta\frac{m}{M_{H}}P(\delta),$ (6)
where
$M_{H}\simeq
17\left(\frac{g}{10.75}\right)^{-1/6}\left(\frac{k}{10^{6}\mathrm{Mpc}^{-1}}\right)^{-2}M_{\odot}$
(7)
is the horizon mass. Here $g$ is the degress of freedom of relativistic
particles, $\delta_{c}$ is the threshold value of density contrast for the
formation of PBHs,
$P(\delta)=e^{-\delta^{2}/(2\sigma_{k}^{2})}/\sqrt{2\pi\sigma_{k}^{2}}$ with
variance $\sigma_{k}$ related to the curvature power spectrum by
$\sigma^{2}_{k}=\left(\frac{4}{9}\right)^{2}\int_{0}^{\infty}\frac{dq}{q}{W}^{2}(q,R_{H})\left(\frac{q}{k}\right)^{4}T^{2}(q,R_{H})\mathcal{P}_{\mathcal{R}}(q),$
(8)
where
$T(k,R_{H})=3\frac{\sin(kR_{H}/\sqrt{3})-(kR_{H}/\sqrt{3})\cos(kR_{H}/\sqrt{3})}{(kR_{H}/\sqrt{3})^{3}}$
(9)
is the transfer function during radiation dominated era. A window function
$W(k,R_{H})$ is needed to smooth out the density contrast on a comoving length
$R\sim k^{-1}$, for which we use a top-hap window function in real space,
namely
$W(k,R_{H})=3\frac{\sin(kR_{H})-(kR_{H})\cos(kR_{H})}{(kR_{H})^{3}}.$ (10)
The PBH mass $m$ is related to the density contrast by the critical collapse,
namely $m=M_{H}\kappa\left(\delta_{m}-\delta_{c}\right)^{\gamma}$ Choptuik
(1993); Evans and Coleman (1994); Niemeyer and Jedamzik (1998) with
$\kappa=3.3$ and $\gamma=0.36$ Koike _et al._ (1995). The nonlinear relation
between the density contrast and the curvature perturbation $\delta_{\zeta}$
leads to $\delta_{m}=\delta_{\zeta}-3/8\delta_{\zeta}^{2}$ Young _et al._
(2019); De Luca _et al._ (2019b); Kawasaki and Nakatsuka (2019). For the
equation of state $w=1/3$, numerical simulation indicates $\delta_{c}\approx
0.45$ Musco _et al._ (2005, 2009). The threshold value of density contrast
$\delta_{c}$ slightly decreases due to the softening of equation of state
Musco and Miller (2013); Saikawa and Shirai (2018) during the QCD phase
transition Borsanyi _et al._ (2016); Saikawa and Shirai (2018), and therefore
the PBH mass function around $m\sim 1M_{\odot}$ should be significantly
enhanced Byrnes _et al._ (2018). In this letter, the data of equation of
state and the sound speed are adopted from Saikawa and Shirai (2018).
Here, the best-fit values of $A$ and $\alpha$ are $A=0.004647$ and
$\alpha=-0.1033$ for case I, and $A=0.004640$ and $\alpha=-0.1003$ for case
II, and then the PBH mass function generated by such a broad tilted curvature
power spectrum is illustrated by the blue (case I) and red (case II) curves in
Fig. 1. In particular, the enhancement of PBH mass function around $m\sim
1M_{\odot}$ just attributes to the softening of the equation of state during
the QCD phase transition. In a word, our results indicate that such a broad
red-tilted curvature power spectrum provides a quite reasonable explanation
for the PBH mass function implied by GWTC-3.
GWB associated with PBHs. GWB is supposed to be a superposition of incoherent
GWs that are characterized statistically. The GWB associated with PBHs
includes two main contributions: one is the coalescences of binary PBHs Chen
_et al._ (2019) (see more discussion in Mukherjee and Silk (2021); Mukherjee
_et al._ (2021)) and the other is SIGW inevitably generated by the curvature
perturbations during the formation of PBHs Saito and Yokoyama (2009).
Figure 2: The GWB associated with PBHs. The blue (red) band corresponds to
the GWB from the coalescences of binary PBHs from Case I (Case II), and the
purple curve corresponds to the SIGWs generated by the curvature perturbation
during the formation of PBHs. The two dark shaded regions are ruled out by the
LIGO O3 Abbott _et al._ (2021c) and NANOGrav 12.5-year data Arzoumanian _et
al._ (2020), respectively. We also show the sensitivity curves of FAST Nan
_et al._ (2011), LISA Audley _et al._ (2017), TianQin Luo _et al._ (2016),
Taiji Hu and Wu (2017), LIGO Design, Einstein Telescope (ET) Punturo _et al._
(2010), Cosmic Explorer (CE) Abbott _et al._ (2017) and NEMO Ackley _et al._
(2020).
For binary systems, the energy-density spectrum of a GWB which defined as the
energy density of GWs per logarithm frequency, $f=k/(2\pi)$, can be calculated
as Allen and Romano (1999); Phinney (2001); Regimbau and Mandic (2008); Zhu
_et al._ (2011, 2013)
$\Omega_{\mathrm{GW}}(f)=\frac{f}{\rho_{c}H_{0}}\int\mathrm{d}z\mathrm{d}m_{1}\mathrm{d}m_{2}\frac{\mathcal{R}_{12}(z)}{(1+z)\,E(z)}\frac{\mathrm{d}E_{\mathrm{GW}}}{\mathrm{d}f_{s}},$
(11)
where $\rho_{c}=3H_{0}^{2}/(8\pi)$ is the critical energy density of our
Universe, $f_{s}$ is frequency of GWs in the source frame, $H_{0}$ is the
Hubble constant, and
$E(z)=\sqrt{\Omega_{r}\left(1+z\right)^{4}+\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}}$
accounts for the evolution of our Universe with $\Omega_{\mathrm{r}}$,
$\Omega_{\mathrm{m}}$ and $\Omega_{\mathrm{\Lambda}}$ the density parameters
for radiation, matter and dark energy. Here, we adopt the best-fit results
from Planck 2018 (Aghanim _et al._ , 2020) and approximate the energy
spectrum $\mathrm{d}E_{\mathrm{GW}}/\mathrm{d}f_{s}$ emitted by an individual
BBH using expressions from Cutler _et al._ (1993); Chernoff and Finn (1993);
Zhu _et al._ (2011). The corresponding GWBs for case I and case II are
respectively shown as the blue and red band in Fig. 2, indicating that both of
them are compatible with the current limits given by LIGO O3 Abbott _et al._
(2021c), and should be detected by the future ground-based and space-borne GW
detectors, such as Neutron Star Extreme Matter Observatory (NEMO) Ackley _et
al._ (2020), Cosmic Explorer (CE) Abbott _et al._ (2017), Einstein Telescope
(ET) Punturo _et al._ (2010), Taiji Hu and Wu (2017), TianQin Luo _et al._
(2016) and LISA Audley _et al._ (2017).
On the other hand, it is known that the GWs should be generated by the scalar
curvature perturbations at second order in perturbation theory Tomita (1967);
Matarrese _et al._ (1993, 1994, 1998); Noh and Hwang (2004); Carbone and
Matarrese (2005); Nakamura (2007); Yuan _et al._ (2020, 2019a, 2019b); Yuan
and Huang (2021). In this sense, SIGWs are inevitably produced during the
formation of PBHs. The perturbed metric in Newton gauge is given by
$\mathrm{d}s^{2}=a^{2}\left\\{-(1+2\Phi)\mathrm{d}\eta^{2}+\left[(1-2\Phi)\delta_{ij}+\frac{h_{ij}}{2}\right]\mathrm{d}x^{i}\mathrm{~{}d}x^{j}\right\\},$
(12)
where $\Phi=-2\mathcal{R}/3$ is the Bardeen potential, $h_{ij}$ is the second-
order transverse and traceless tensor mode and $\eta$ is the conformal time.
The equation of motion for $\Phi$ is govern by
$\Phi^{\prime\prime}+3\mathcal{H}(\eta)\left(1+c_{s}^{2}\right)\Phi^{\prime}+3\mathcal{H}^{2}\left(c_{s}^{2}-w\right)\Phi-
c_{s}^{2}\nabla^{2}\Phi=0,$ (13)
where $w$ and $c_{s}$ are the equation of state and the sound speed,
respectively. $\Phi_{k}(\eta)$ is related to its initial value
$\Phi_{k}\equiv\Phi_{k}(\eta\rightarrow 0)$ by
$\Phi_{k}(\eta)=\Phi_{k}T_{\Phi}(k\eta)$, where $T_{\Phi}(k\eta)$ is the
transfer function. The equation of motion for the second-order tensor modes,
$h_{ij}$, takes the form
$h_{ij}^{\prime\prime}+2\mathcal{H}h_{ij}^{\prime}-\nabla^{2}h_{ij}=-4\mathcal{T}_{ij}^{\ell
m}\mathcal{S}_{\ell m},$ (14)
with $\mathcal{T}_{ij}^{\ell
m}=e_{ij}^{(+)}(\bm{k})e^{(+)lm}(\bm{k})+e_{ij}^{(\times)}(\bm{k})e^{(\times)lm}(\bm{k})$
selects the transverse-traceless part of the source term, with $e_{ij}$ the
polarization tensor and $\mathcal{H}=a^{\prime}/a$. The source term is given
by
$S_{ij}=2\Phi\partial_{i}\partial_{j}\Phi-\frac{4}{3(1+w)}\left(\partial_{i}\Phi+\frac{\partial_{i}\Phi^{\prime}}{\mathcal{H}(\eta)}\right)\left(\partial_{j}\Phi+\frac{\partial_{j}\Phi^{\prime}}{\mathcal{H}(\eta)}\right).$
(15)
Here the prime denotes the derivative with respect to $\eta$. Following Kohri
and Terada (2018), Eq. (14) can be solved by the Green’s function and the
transfer function method, and the energy density parameter by today is given
by
$\displaystyle\Omega_{\mathrm{GW}}$
$\displaystyle=\frac{\Omega_{\mathrm{r}}}{6}\int_{0}^{\infty}\mathrm{d}u\int_{|1-u|}^{1+u}\mathrm{~{}d}v\frac{v^{2}}{u^{2}}\left[1-\left(\frac{1+v^{2}-u^{2}}{2v}\right)^{2}\right]^{2}$
(16)
$\displaystyle\qquad\qquad\times\mathcal{P}_{\mathcal{R}}(uk)\mathcal{P}_{\mathcal{R}}(vk)\overline{I^{2}(u,v)},$
The kernel function takes the form Kohri and Terada (2018)
$\displaystyle\overline{I^{2}(u,v)}=\frac{9(u^{2}+v^{2}-3)^{2}}{32u^{6}v^{6}}\Bigg{\\{}\Big{(}-4uv+(u^{2}+v^{2}-3)$
$\displaystyle\times\ln\Big{|}{3-(u+v)^{2}\over
3-(u-v)^{2}}\Big{|}\Big{)}^{2}+\pi^{2}\left(u^{2}+v^{2}-3\right)^{2}\Theta(u+v-\sqrt{3})\Bigg{\\}}.$
For the curvature power spectrum given in Eq. (4) with $A$ and $\alpha$ taking
their best-fit values, the SIGWs are illustrated as the purple curve
(coincides for case I and II) in Fig. 2. We find that the predicted SIGWs are
compatible with NANOGrav 12.5-yr data Arzoumanian _et al._ (2020), and should
be detected by FAST Nan _et al._ (2011) in the future. Note that recent
analysis implies there is no statistically significant evidence for the tensor
transverse polarization mode in the NANOGrav 12.5-yr data set Arzoumanian _et
al._ (2020); Chen _et al._ (2021a); Arzoumanian _et al._ (2021), PPTA second
data release Goncharov _et al._ (2021); Wu _et al._ (2021), IPTA second data
release Chen _et al._ (2021b), and EPTA second data release Chalumeau _et
al._ (2021).
Conclusion and Discussion. In this letter, we use a binned PBH mass function
to model-independently reconstruct the PBH mass function from GWTC-3, and find
that such a mass function can be naturally explained by a broad red-tilted
curvature power spectrum. By comparing the merger rates of binary PBHs with
those given by LVC, we conclude that GW190521 with the primary component being
within the pair-instability supernovae mass gap can be explained by the merger
of binary PBHs, but the light components (i.e. $m<3M_{\odot}$) in GW190814,
GW190425, GW200105, and GW200115 events should be NSs or other exotic compact
objects. In addition, the PBH scenario proposed in this letter can be tested
by searching for the GWB generated by the binary PBHs and the SIGW inevitably
produced by the curvature perturbations during the formation of PBHs.
Acknowledgments. We acknowledge the use of HPC Cluster of ITP-CAS and HPC
Cluster of Tianhe II in National Supercomputing Center in Guangzhou. This work
is supported by the National Key Research and Development Program of China
Grant No.2020YFC2201502, grants from NSFC (grant No. 11975019, 11991052,
12047503), Key Research Program of Frontier Sciences, CAS, Grant NO. ZDBS-
LY-7009, CAS Project for Young Scientists in Basic Research YSBR-006, the Key
Research Program of the Chinese Academy of Sciences (Grant NO. XDPB15), and
the science research grants from the China Manned Space Project with NO. CMS-
CSST-2021-B01.
## References
* Carr and Hawking (1974) Bernard J. Carr and S. W. Hawking, “Black holes in the early Universe,” Mon. Not. Roy. Astron. Soc. 168, 399–415 (1974).
* Carr (1975) Bernard J. Carr, “The Primordial black hole mass spectrum,” Astrophys. J. 201, 1–19 (1975).
* Sasaki _et al._ (2016) Misao Sasaki, Teruaki Suyama, Takahiro Tanaka, and Shuichiro Yokoyama, “Primordial Black Hole Scenario for the Gravitational-Wave Event GW150914,” Phys. Rev. Lett. 117, 061101 (2016), [erratum: Phys. Rev. Lett.121,no.5,059901(2018)], arXiv:1603.08338 [astro-ph.CO] .
* Chen and Huang (2018) Zu-Cheng Chen and Qing-Guo Huang, “Merger Rate Distribution of Primordial-Black-Hole Binaries,” Astrophys. J. 864, 61 (2018), arXiv:1801.10327 [astro-ph.CO] .
* Raidal _et al._ (2019) Martti Raidal, Christian Spethmann, Ville Vaskonen, and Hardi Veermäe, “Formation and Evolution of Primordial Black Hole Binaries in the Early Universe,” JCAP 1902, 018 (2019), arXiv:1812.01930 [astro-ph.CO] .
* De Luca _et al._ (2020a) V. De Luca, G. Franciolini, P. Pani, and A. Riotto, “Primordial Black Holes Confront LIGO/Virgo data: Current situation,” JCAP 06, 044 (2020a), arXiv:2005.05641 [astro-ph.CO] .
* Hall _et al._ (2020) Alex Hall, Andrew D. Gow, and Christian T. Byrnes, “Bayesian analysis of LIGO-Virgo mergers: Primordial vs. astrophysical black hole populations,” Phys. Rev. D 102, 123524 (2020), arXiv:2008.13704 [astro-ph.CO] .
* Bhagwat _et al._ (2021) S. Bhagwat, V. De Luca, G. Franciolini, P. Pani, and A. Riotto, “The importance of priors on LIGO-Virgo parameter estimation: the case of primordial black holes,” JCAP 01, 037 (2021), arXiv:2008.12320 [astro-ph.CO] .
* Hütsi _et al._ (2020) Gert Hütsi, Martti Raidal, Ville Vaskonen, and Hardi Veermäe, “Two populations of LIGO-Virgo black holes,” (2020), arXiv:2012.02786 [astro-ph.CO] .
* Wong _et al._ (2021) Kaze W. K. Wong, Gabriele Franciolini, Valerio De Luca, Vishal Baibhav, Emanuele Berti, Paolo Pani, and Antonio Riotto, “Constraining the primordial black hole scenario with Bayesian inference and machine learning: the GWTC-2 gravitational wave catalog,” Phys. Rev. D 103, 023026 (2021), arXiv:2011.01865 [gr-qc] .
* De Luca _et al._ (2021a) V. De Luca, G. Franciolini, P. Pani, and A. Riotto, “Bayesian Evidence for Both Astrophysical and Primordial Black Holes: Mapping the GWTC-2 Catalog to Third-Generation Detectors,” JCAP 05, 003 (2021a), arXiv:2102.03809 [astro-ph.CO] .
* Franciolini _et al._ (2021) Gabriele Franciolini, Vishal Baibhav, Valerio De Luca, Ken K. Y. Ng, Kaze W. K. Wong, Emanuele Berti, Paolo Pani, Antonio Riotto, and Salvatore Vitale, “Quantifying the evidence for primordial black holes in LIGO/Virgo gravitational-wave data,” (2021), arXiv:2105.03349 [gr-qc] .
* Abbott _et al._ (2020a) R. Abbott _et al._ (LIGO Scientific, Virgo), “GW190521: A Binary Black Hole Merger with a Total Mass of $150M_{\odot}$,” Phys. Rev. Lett. 125, 101102 (2020a), arXiv:2009.01075 [gr-qc] .
* Abbott _et al._ (2020b) B. P. Abbott _et al._ (LIGO Scientific, Virgo), “GW190425: Observation of a Compact Binary Coalescence with Total Mass $\sim 3.4M_{\odot}$,” Astrophys. J. Lett. 892, L3 (2020b), arXiv:2001.01761 [astro-ph.HE] .
* Abbott _et al._ (2020c) R. Abbott _et al._ (LIGO Scientific, Virgo), “GW190814: Gravitational Waves from the Coalescence of a 23 Solar Mass Black Hole with a 2.6 Solar Mass Compact Object,” Astrophys. J. Lett. 896, L44 (2020c), arXiv:2006.12611 [astro-ph.HE] .
* Abbott _et al._ (2021a) R. Abbott _et al._ (LIGO Scientific, KAGRA, VIRGO), “Observation of Gravitational Waves from Two Neutron Star–Black Hole Coalescences,” Astrophys. J. Lett. 915, L5 (2021a), arXiv:2106.15163 [astro-ph.HE] .
* De Luca _et al._ (2021b) V. De Luca, V. Desjacques, G. Franciolini, P. Pani, and A. Riotto, “GW190521 Mass Gap Event and the Primordial Black Hole Scenario,” Phys. Rev. Lett. 126, 051101 (2021b), arXiv:2009.01728 [astro-ph.CO] .
* Abbott _et al._ (2021b) R. Abbott _et al._ (LIGO Scientific, VIRGO, KAGRA), “GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run,” (2021b), arXiv:2111.03606 [gr-qc] .
* Byrnes _et al._ (2018) Christian T. Byrnes, Mark Hindmarsh, Sam Young, and Michael R. S. Hawkins, “Primordial black holes with an accurate QCD equation of state,” JCAP 08, 041 (2018), arXiv:1801.06138 [astro-ph.CO] .
* Chen _et al._ (2019) Zu-Cheng Chen, Fan Huang, and Qing-Guo Huang, “Stochastic Gravitational-wave Background from Binary Black Holes and Binary Neutron Stars and Implications for LISA,” Astrophys. J. 871, 97 (2019), arXiv:1809.10360 [gr-qc] .
* Saito and Yokoyama (2009) Ryo Saito and Jun’ichi Yokoyama, “Gravitational wave background as a probe of the primordial black hole abundance,” Phys. Rev. Lett. 102, 161101 (2009), [Erratum: Phys. Rev. Lett.107,069901(2011)], arXiv:0812.4339 [astro-ph] .
* Ali-Haïmoud _et al._ (2017) Yacine Ali-Haïmoud, Ely D. Kovetz, and Marc Kamionkowski, “Merger rate of primordial black-hole binaries,” Phys. Rev. D96, 123523 (2017), arXiv:1709.06576 [astro-ph.CO] .
* Abbott _et al._ (2016a) B. P. Abbott _et al._ (LIGO Scientific, Virgo), “The Rate of Binary Black Hole Mergers Inferred from Advanced LIGO Observations Surrounding GW150914,” Astrophys. J. Lett. 833, L1 (2016a), arXiv:1602.03842 [astro-ph.HE] .
* Abbott _et al._ (2016b) B. P. Abbott _et al._ (LIGO Scientific, Virgo), “Supplement: The Rate of Binary Black Hole Mergers Inferred from Advanced LIGO Observations Surrounding GW150914,” Astrophys. J. Suppl. 227, 14 (2016b), arXiv:1606.03939 [astro-ph.HE] .
* Abbott _et al._ (2016c) B. P. Abbott _et al._ (LIGO Scientific, Virgo), “Binary Black Hole Mergers in the first Advanced LIGO Observing Run,” Phys. Rev. X6, 041015 (2016c), [erratum: Phys. Rev.X8,no.3,039903(2018)], arXiv:1606.04856 [gr-qc] .
* Wysocki _et al._ (2019) Daniel Wysocki, Jacob Lange, and Richard O’Shaughnessy, “Reconstructing phenomenological distributions of compact binaries via gravitational wave observations,” Phys. Rev. D 100, 043012 (2019), arXiv:1805.06442 [gr-qc] .
* Fishbach _et al._ (2018) Maya Fishbach, Daniel E. Holz, and Will M. Farr, “Does the Black Hole Merger Rate Evolve with Redshift?” Astrophys. J. Lett. 863, L41 (2018), arXiv:1805.10270 [astro-ph.HE] .
* Mandel _et al._ (2019) Ilya Mandel, Will M. Farr, and Jonathan R. Gair, “Extracting distribution parameters from multiple uncertain observations with selection biases,” Mon. Not. Roy. Astron. Soc. 486, 1086–1093 (2019), arXiv:1809.02063 [physics.data-an] .
* Thrane and Talbot (2019) Eric Thrane and Colm Talbot, “An introduction to Bayesian inference in gravitational-wave astronomy: parameter estimation, model selection, and hierarchical models,” Publ. Astron. Soc. Austral. 36, e010 (2019), [Erratum: Publ.Astron.Soc.Austral. 37, e036 (2020)], arXiv:1809.02293 [astro-ph.IM] .
* De Luca _et al._ (2019a) V. De Luca, V. Desjacques, G. Franciolini, A. Malhotra, and A. Riotto, “The initial spin probability distribution of primordial black holes,” JCAP 05, 018 (2019a), arXiv:1903.01179 [astro-ph.CO] .
* Mirbabayi _et al._ (2020) Mehrdad Mirbabayi, Andrei Gruzinov, and Jorge Noreña, “Spin of Primordial Black Holes,” JCAP 03, 017 (2020), arXiv:1901.05963 [astro-ph.CO] .
* De Luca _et al._ (2020b) V. De Luca, G. Franciolini, P. Pani, and A. Riotto, “The evolution of primordial black holes and their final observable spins,” JCAP 04, 052 (2020b), arXiv:2003.02778 [astro-ph.CO] .
* De Luca _et al._ (2020c) V. De Luca, G. Franciolini, P. Pani, and A. Riotto, “Constraints on Primordial Black Holes: the Importance of Accretion,” Phys. Rev. D 102, 043505 (2020c), arXiv:2003.12589 [astro-ph.CO] .
* Press and Schechter (1974) William H. Press and Paul Schechter, “Formation of galaxies and clusters of galaxies by selfsimilar gravitational condensation,” Astrophys. J. 187, 425–438 (1974).
* Choptuik (1993) Matthew W. Choptuik, “Universality and scaling in gravitational collapse of a massless scalar field,” Phys. Rev. Lett. 70, 9–12 (1993).
* Evans and Coleman (1994) Charles R. Evans and Jason S. Coleman, “Observation of critical phenomena and selfsimilarity in the gravitational collapse of radiation fluid,” Phys. Rev. Lett. 72, 1782–1785 (1994), arXiv:gr-qc/9402041 .
* Niemeyer and Jedamzik (1998) Jens C. Niemeyer and K. Jedamzik, “Near-critical gravitational collapse and the initial mass function of primordial black holes,” Phys. Rev. Lett. 80, 5481–5484 (1998), arXiv:astro-ph/9709072 .
* Koike _et al._ (1995) Tatsuhiko Koike, Takashi Hara, and Satoshi Adachi, “Critical behavior in gravitational collapse of radiation fluid: A Renormalization group (linear perturbation) analysis,” Phys. Rev. Lett. 74, 5170–5173 (1995), arXiv:gr-qc/9503007 .
* Young _et al._ (2019) Sam Young, Ilia Musco, and Christian T. Byrnes, “Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation,” JCAP 11, 012 (2019), arXiv:1904.00984 [astro-ph.CO] .
* De Luca _et al._ (2019b) V. De Luca, G. Franciolini, A. Kehagias, M. Peloso, A. Riotto, and C. Ünal, “The Ineludible non-Gaussianity of the Primordial Black Hole Abundance,” JCAP 07, 048 (2019b), arXiv:1904.00970 [astro-ph.CO] .
* Kawasaki and Nakatsuka (2019) Masahiro Kawasaki and Hiromasa Nakatsuka, “Effect of nonlinearity between density and curvature perturbations on the primordial black hole formation,” Phys. Rev. D 99, 123501 (2019), arXiv:1903.02994 [astro-ph.CO] .
* Musco _et al._ (2005) Ilia Musco, John C. Miller, and Luciano Rezzolla, “Computations of primordial black hole formation,” Class. Quant. Grav. 22, 1405–1424 (2005), arXiv:gr-qc/0412063 .
* Musco _et al._ (2009) Ilia Musco, John C. Miller, and Alexander G. Polnarev, “Primordial black hole formation in the radiative era: Investigation of the critical nature of the collapse,” Class. Quant. Grav. 26, 235001 (2009), arXiv:0811.1452 [gr-qc] .
* Musco and Miller (2013) Ilia Musco and John C. Miller, “Primordial black hole formation in the early universe: critical behaviour and self-similarity,” Class. Quant. Grav. 30, 145009 (2013), arXiv:1201.2379 [gr-qc] .
* Saikawa and Shirai (2018) Ken’ichi Saikawa and Satoshi Shirai, “Primordial gravitational waves, precisely: The role of thermodynamics in the Standard Model,” JCAP 05, 035 (2018), arXiv:1803.01038 [hep-ph] .
* Borsanyi _et al._ (2016) Sz. Borsanyi _et al._ , “Calculation of the axion mass based on high-temperature lattice quantum chromodynamics,” Nature 539, 69–71 (2016), arXiv:1606.07494 [hep-lat] .
* Mukherjee and Silk (2021) Suvodip Mukherjee and Joseph Silk, “Can we distinguish astrophysical from primordial black holes via the stochastic gravitational wave background?” Mon. Not. Roy. Astron. Soc. 506, 3977–3985 (2021), arXiv:2105.11139 [gr-qc] .
* Mukherjee _et al._ (2021) Suvodip Mukherjee, Matthew S. P. Meinema, and Joseph Silk, “Prospects of discovering sub-solar primordial black holes using the stochastic gravitational wave background from third-generation detectors,” (2021), arXiv:2107.02181 [astro-ph.CO] .
* Abbott _et al._ (2021c) R. Abbott _et al._ (KAGRA, Virgo, LIGO Scientific), “Upper limits on the isotropic gravitational-wave background from Advanced LIGO and Advanced Virgo’s third observing run,” Phys. Rev. D 104, 022004 (2021c), arXiv:2101.12130 [gr-qc] .
* Arzoumanian _et al._ (2020) Zaven Arzoumanian _et al._ (NANOGrav), “The NANOGrav 12.5 yr Data Set: Search for an Isotropic Stochastic Gravitational-wave Background,” Astrophys. J. Lett. 905, L34 (2020), arXiv:2009.04496 [astro-ph.HE] .
* Nan _et al._ (2011) Rendong Nan, Di Li, Chengjin Jin, Qiming Wang, Lichun Zhu, Wenbai Zhu, Haiyan Zhang, Youling Yue, and Lei Qian, “The Five-Hundred-Meter Aperture Spherical Radio Telescope (FAST) Project,” Int. J. Mod. Phys. D 20, 989–1024 (2011), arXiv:1105.3794 [astro-ph.IM] .
* Audley _et al._ (2017) Heather Audley _et al._ (LISA), “Laser Interferometer Space Antenna,” (2017), arXiv:1702.00786 [astro-ph.IM] .
* Luo _et al._ (2016) Jun Luo _et al._ (TianQin), “TianQin: a space-borne gravitational wave detector,” Class. Quant. Grav. 33, 035010 (2016), arXiv:1512.02076 [astro-ph.IM] .
* Hu and Wu (2017) Wen-Rui Hu and Yue-Liang Wu, “The Taiji Program in Space for gravitational wave physics and the nature of gravity,” National Science Review 4, 685–686 (2017), https://academic.oup.com/nsr/article-pdf/4/5/685/31566708/nwx116.pdf .
* Punturo _et al._ (2010) M. Punturo _et al._ , “The Einstein Telescope: A third-generation gravitational wave observatory,” _Proceedings, 14th Workshop on Gravitational wave data analysis (GWDAW-14): Rome, Italy, January 26-29, 2010_ , Class. Quant. Grav. 27, 194002 (2010).
* Abbott _et al._ (2017) Benjamin P. Abbott _et al._ (LIGO Scientific), “Exploring the Sensitivity of Next Generation Gravitational Wave Detectors,” Class. Quant. Grav. 34, 044001 (2017), arXiv:1607.08697 [astro-ph.IM] .
* Ackley _et al._ (2020) K. Ackley _et al._ , “Neutron Star Extreme Matter Observatory: A kilohertz-band gravitational-wave detector in the global network,” Publ. Astron. Soc. Austral. 37, e047 (2020), arXiv:2007.03128 [astro-ph.HE] .
* Allen and Romano (1999) Bruce Allen and Joseph D. Romano, “Detecting a stochastic background of gravitational radiation: Signal processing strategies and sensitivities,” Phys. Rev. D 59, 102001 (1999), arXiv:gr-qc/9710117 .
* Phinney (2001) E. S. Phinney, “A Practical theorem on gravitational wave backgrounds,” astro-ph/0108028 (2001).
* Regimbau and Mandic (2008) T. Regimbau and V. Mandic, “Astrophysical Sources of Stochastic Gravitational-Wave Background,” _Proceedings, 12th Workshop on Gravitational wave data analysis (GWDAW-12): Cambridge, USA, December 13-16, 2007_ , Class. Quant. Grav. 25, 184018 (2008), arXiv:0806.2794 [astro-ph] .
* Zhu _et al._ (2011) Xing-Jiang Zhu, E. Howell, T. Regimbau, D. Blair, and Zong-Hong Zhu, “Stochastic Gravitational Wave Background from Coalescing Binary Black Holes,” Astrophys. J. 739, 86 (2011), arXiv:1104.3565 [gr-qc] .
* Zhu _et al._ (2013) Xing-Jiang Zhu, Eric J. Howell, David G. Blair, and Zong-Hong Zhu, “On the gravitational wave background from compact binary coalescences in the band of ground-based interferometers,” Mon. Not. Roy. Astron. Soc. 431, 882–899 (2013), arXiv:1209.0595 [gr-qc] .
* Aghanim _et al._ (2020) N. Aghanim _et al._ (Planck), “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641, A6 (2020), arXiv:1807.06209 [astro-ph.CO] .
* Cutler _et al._ (1993) C. Cutler, Eric Poisson, G. J. Sussman, and L. S. Finn, “Gravitational radiation from a particle in circular orbit around a black hole. 2: Numerical results for the nonrotating case,” Phys. Rev. D47, 1511–1518 (1993).
* Chernoff and Finn (1993) David F. Chernoff and Lee Samuel Finn, “Gravitational radiation, inspiraling binaries, and cosmology,” Astrophys. J. 411, L5–L8 (1993), arXiv:gr-qc/9304020 [gr-qc] .
* Tomita (1967) Kenji Tomita, “Non-linear theory of gravitational instability in the expanding universe,” Progress of Theoretical Physics 37, 831–846 (1967).
* Matarrese _et al._ (1993) Sabino Matarrese, Ornella Pantano, and Diego Saez, “A General relativistic approach to the nonlinear evolution of collisionless matter,” Phys. Rev. D47, 1311–1323 (1993).
* Matarrese _et al._ (1994) Sabino Matarrese, Ornella Pantano, and Diego Saez, “General relativistic dynamics of irrotational dust: Cosmological implications,” Phys. Rev. Lett. 72, 320–323 (1994), arXiv:astro-ph/9310036 [astro-ph] .
* Matarrese _et al._ (1998) Sabino Matarrese, Silvia Mollerach, and Marco Bruni, “Second order perturbations of the Einstein-de Sitter universe,” Phys. Rev. D58, 043504 (1998), arXiv:astro-ph/9707278 [astro-ph] .
* Noh and Hwang (2004) Hyerim Noh and Jai-chan Hwang, “Second-order perturbations of the Friedmann world model,” Phys. Rev. D69, 104011 (2004).
* Carbone and Matarrese (2005) Carmelita Carbone and Sabino Matarrese, “A Unified treatment of cosmological perturbations from super-horizon to small scales,” Phys. Rev. D71, 043508 (2005), arXiv:astro-ph/0407611 [astro-ph] .
* Nakamura (2007) Kouji Nakamura, “Second-order gauge invariant cosmological perturbation theory: Einstein equations in terms of gauge invariant variables,” Prog. Theor. Phys. 117, 17–74 (2007), arXiv:gr-qc/0605108 [gr-qc] .
* Yuan _et al._ (2020) Chen Yuan, Zu-Cheng Chen, and Qing-Guo Huang, “Scalar induced gravitational waves in different gauges,” Phys. Rev. D 101, 063018 (2020), arXiv:1912.00885 [astro-ph.CO] .
* Yuan _et al._ (2019a) Chen Yuan, Zu-Cheng Chen, and Qing-Guo Huang, “Probing Primordial-Black-Hole Dark Matter with Scalar Induced Gravitational Waves,” Phys. Rev. D100, 081301 (2019a), arXiv:1906.11549 [astro-ph.CO] .
* Yuan _et al._ (2019b) Chen Yuan, Zu-Cheng Chen, and Qing-Guo Huang, “Log-dependent slope of scalar induced gravitational waves in the infrared regions,” (2019b), arXiv:1910.09099 [astro-ph.CO] .
* Yuan and Huang (2021) Chen Yuan and Qing-Guo Huang, “A topic review on probing primordial black hole dark matter with scalar induced gravitational waves,” (2021), arXiv:2103.04739 [astro-ph.GA] .
* Kohri and Terada (2018) Kazunori Kohri and Takahiro Terada, “Semianalytic calculation of gravitational wave spectrum nonlinearly induced from primordial curvature perturbations,” Phys. Rev. D97, 123532 (2018), arXiv:1804.08577 [gr-qc] .
* Chen _et al._ (2021a) Zu-Cheng Chen, Chen Yuan, and Qing-Guo Huang, “Non-tensorial gravitational wave background in NANOGrav 12.5-year data set,” Sci. China Phys. Mech. Astron. 64, 120412 (2021a), arXiv:2101.06869 [astro-ph.CO] .
* Arzoumanian _et al._ (2021) Zaven Arzoumanian _et al._ (NANOGrav), “The NANOGrav 12.5-year Data Set: Search for Non-Einsteinian Polarization Modes in the Gravitational-wave Background,” Astrophys. J. Lett. 923, L22 (2021), arXiv:2109.14706 [gr-qc] .
* Goncharov _et al._ (2021) Boris Goncharov _et al._ , “On the evidence for a common-spectrum process in the search for the nanohertz gravitational-wave background with the Parkes Pulsar Timing Array,” (2021), 10.3847/2041-8213/ac17f4, arXiv:2107.12112 [astro-ph.HE] .
* Wu _et al._ (2021) Yu-Mei Wu, Zu-Cheng Chen, and Qing-Guo Huang, “Constraining the Polarization of Gravitational Waves with the Parkes Pulsar Timing Array Second Data Release,” (2021), arXiv:2108.10518 [astro-ph.CO] .
* Chen _et al._ (2021b) Zu-Cheng Chen, Yu-Mei Wu, and Qing-Guo Huang, “Searching for Isotropic Stochastic Gravitational-Wave Background in the International Pulsar Timing Array Second Data Release,” (2021b), arXiv:2109.00296 [astro-ph.CO] .
* Chalumeau _et al._ (2021) A. Chalumeau _et al._ , “Noise analysis in the European Pulsar Timing Array data release 2 and its implications on the gravitational-wave background search,” (2021), 10.1093/mnras/stab3283, arXiv:2111.05186 [astro-ph.HE] .
|
(7.22$\pm$14.25 km s-1) F38 Sinfoni 14/04/11 (-3.69$\pm$5.60 km s-1), 17/07/13
(-14.65$\pm$6.12 km s-1), 06/08/13 (-14.28$\pm$5.89 km s-1), 26/04/17
(2.17$\pm$5.99 km s-1), 30/04/17 (-7.13$\pm$5.11 km s-1), 07/05/18
(5.92$\pm$5.13 km s-1), 07/08/18 (6.23$\pm$5.40 km s-1), 12/08/18
(10.94$\pm$5.99 km s-1), 17/08/18 (low S/N), 18/08/18 (14.49$\pm$6.09 km s-1)
F40 Sinfoni 29/05/13 (-9.57$\pm$9.51 km s-1), 18/06/13 (-3.30$\pm$18.00 km
s-1), 17/07/13 (-20.06$\pm$11.19 km s-1), 03/08/13 (-2.71$\pm$10.88 km s-1),
04/08/13 (-0.87$\pm$10.08 km s-1), 06/08/13 (0.90$\pm$11.02 km s-1), 08/08/13
(-18.25$\pm$14.85 km s-1), 26/04/17 (18.17$\pm$11.69 km s-1), 30/04/17
(19.44$\pm$9.69 km s-1), 23/06/17 (-28.02$\pm$10.18 km s-1), 27/07/17
(19.52$\pm$9.07 km s-1), 07/05/18 (-5.06$\pm$10.54 km s-1), 07/08/18
(-4.57$\pm$14.84 km s-1), 12/08/18 (8.52$\pm$12.17 km s-1), 18/08/18
(25.86$\pm$11.06 km s-1)
## Appendix B The LBV phenomenon and secular variability within the Arches
As massive stars transition away from the Main Sequence they appear to
encounter an instability of unknown origin: the so-called luminous blue
variable (LBV) phase. Historically it has been thought that the extreme mass
loss that characterises LBVs helps facilitate the formation of H-depleted WRs,
although observations of Romano’s star (Clark et al. [2012], Polcaro et al.
[2016]) and the erupting component of the eclipsing hierarchical system HD5980
indicate this behaviour may also be present in WN-subtype WRs. The LBV
phenomenon appears to extend to very high masses, with the LBV component of
HD5980 having a dynamical mass of M${}_{\rm current}\sim 60M_{\odot}$,
implying M${}_{\rm initial}>80M_{\odot}$ (Koenigsberger et al. [2014]; Hillier
et al. [2019]). However, this extreme is exceeded by that of the primary in
the binary $\eta$ Carinae, which is generally supposed to be $>100M_{\odot}$
(Hillier et al. [2001]), with a mass of $\sim 40-60M_{\odot}$ suggested for
the unseen secondary (Mehner et al. [2010]). $\eta$ Carinae has been the
subject of intensive observation due to both its current exceptional
properties and its unprecedented eruption in the $19^{\rm th}$ Century, during
which a luminosity of $\gtrsim 10^{7}L_{\odot}$ and a mass loss rate of
$\gtrsim 0.5M_{\odot}$ yr-1 have been inferred (Smith et al. [2003]).
$\eta$ Carinae is located within the Trumpler 16 + Collinder 228 cluster
aggregate, for which Smith ([2006]) infers an age of 2-3Myr. As such -
although of significantly lower integrated mass - it appears directly
comparable to the Arches; indeed it hosts three WNLha stars, a single O4
supergiant and a number of mid-O giants and main sequence stars (cf. papers I
and III). Given this similarity one might ask whether the Arches hosts any
star exhibiting the LBV phenomenon? Trivially, despite the presence of
similarly massive binaries within the Arches, there appears to be no direct
analogue of $\eta$ Carinae; its K-band morphology is very different from any
cluster member (Hamann et al. [1994]), while any nebula comparable to the
Homunculus would be clearly visible (cf. the Pistol nebula; Figer et al.
[1999]).
Surprisingly, none of the stars within our sample showed any evidence of long
term changes in their spectral morphologies during this period. Simulations of
$60M_{\odot}$ stars by Groh et al. ([2014]) predict that the LBV phenomenon
occurs directly after a late-O/early-B hypergiant phase around $\gtrsim 3$Myr,
implying that the cohort of mid-O supergiants and, potentially, the mid- to
late-hypergiants have yet to reach this evolutionary juncture. Unfortunately
no predictions exist for when, or even if, more massive stars such as the
Arches WNLha cohort become LBVs, nor how long this episode endures. Given the
limited cadence of the observations, one could imagine that if the
outbursts/eruption duty cycle were long, or outbursts rapid, they could have
been missed. Indeed, while $\eta$ Carinae was highly (photometrically) active
throughout the 19th and 20th Centuries (Smith & Frew [2011]) and the
characteristic variability of LBVs occurs over months to years (e.g. Humphreys
& Davidson [1994]), observations of P Cygni reveal that LBVs may reside in a
quiescent state for centuries (cf Lamers & de Groot [1992]). Alternatively one
might suppose that WNLha stars have yet to reach the LBV phase, or that such
stars circumvent it entirely. However such a hypothesis would require the 19th
Century eruption of $\eta$ Carina to be driven by a mechanism other than the
nuclear evolution of the star, such as binary interaction/merger (Portegies
Zwart & van den Heuvel [2015]).
## Appendix C The X-ray and binary properties of WNLh stars
In order to better understand the Arches binaries it is instructive to
contrast their properties with those of other known binary populations. Young
massive clusters provide the perfect laboratory to undertake such a
comparison, since their ensemble properties allow for constraints to be placed
on their distances, extinction and ages, in turn aiding the determination of
properties such as luminosity for individual members. Moreover, selection
effects may be more easily understood and controlled in such an environment.
When assessing the binary fraction of clusters, multi-epoch spectroscopic
surveys clearly represent the most powerful methodology available, especially
if combined with photometric observations. However not all the clusters that
host WNLha stars have been subject to such an approach, and in such cases
X-ray observations - which are sensitive to the signature of shocked material
generated by wind collisions - are an excellent substitute.
In order to undertake such an assessment, the X-ray properties of both single
and binary WNLh stars must first be quantified. Clark et al. ([2009])
presented such a summary; however a reappraisal is timely, and we provide that
here. For the reasons described above, stars located within stellar aggregates
form a ‘gold standard’; moreover many more such regions have now been subject
to targeted X-ray investigation (Sect. C.1.1 and refs. therein). Nevertheless,
we supplement the resultant sample with the remaining, isolated examples for
which suitable data exist, an inclusion driven by the possibility of better
understanding the intrinsic multi-wavelength properties of single stars (Sect.
C.1.2). For the avoidance of bias, we emphasise that the isolated stellar
cohort are not included in the subsequent determination of binary fractions.
We also highlight that a large number of WNLha stars have been identified
within the the central molecular zone of the Galaxy (Clark et al. [2021]).
Unlike the cohorts listed above these were in large part identified via X-ray
observations; as a consequence of the obvious resultant bias towards detecting
colliding wind binaries we explicitly exclude them from the analysis presented
here.
### C.1 Data collation and presentation
In the following section we present the results of a literature review
undertaken to determine both the X-ray and binary properties of WNLha stars,
encompassing both isolated galactic examples and the larger population found
within star clusters and associations. In order to produce the mostly complete
dataset possible, so as to interpret their X-ray behaviour, we also present
the preliminary results of tailored model-atmosphere analysis (cf. Sect. 4) of
the X-ray bright WNLha stars found within the Galactic clusters Danks 1 and
Mercer 81 (Najarro et al. in prep.).
#### C.1.1 WNLh stars within stellar aggregates
Given their rich stellar populations the young massive cluster R136 and its
host star forming region 30 Doradus are obvious targets for such an approach.
Unfortunately, their location within the Large Magellanic Cloud leads to
significant blending in the crowded core regions of R136, and limits spectral
analysis to the brightest of the X-ray sources. As a consequence we also
review the contents of Galactic clusters, following the compilation of Rate et
al. ([2020b]). From an initial sample of 19 candidate clusters, a number were
rejected due to spectral re-classification of relevant cluster members on the
basis of published data101010 Cl 1813-178 (Messineo et al. [2011]) and VVV
CL009 (Chené et al. [2013]). and/or the cluster population suggesting an age
at which, by analogy with the Quintuplet and Westerlund 1 (Clark et al.
[2018b], [2020]), core-H burning WNLha stars should not be present111111The
Quartet (Messineo et al. [2009]), VVV CL036, VVV CL074, and VVV CL099 (Chené
et al. [2013]).. Four further clusters host WNLha stars but lack the multi-
epoch spectroscopic, X-ray and radio observations required to identify
binaries121212 [DBS2003]-179 (WR84-1, WR84-6, and WR84-7; Borissova et al.
[2012]), VVV CL041 (WR62-2; Chené et al. [2015]), VVV CL073 (WR75-26; Chené et
al. [2013]), and Mercer 23 (WR125-3; Hanson et al. [2010]). This left ten
clusters hosting WNLha stars for which the requisite observational data were
available, in addition to R136 and 30 Dor131313While 13 WNLha stars, including
the binary CXOU J1745-28, have currently been identified within the Central
Molecular Zone of the Galaxy (Mikles et al. [2006], Clark et al. [2009]) we do
not consider them here since a large number have been selected on the basis of
their X-ray properties, potentially biasing the sample towards colliding wind
binaries.. The binary and X-ray properties of the WNLh (and related) stars in
these clusters are summarised in Tables C.1 and C.2, while the thumbnail
sketches below present a synopsis of the physical properties in each
aggregate, including relevant references and conclusions derived from the
reported observations.
Table 8: The X-ray properties of WNha stars in 30 Doradus and galactic
clusters
Cluster | Dist. | ID | log | $L_{\rm x}$ | $kT_{1}$ | $kT_{2}$ | log
---|---|---|---|---|---|---|---
| (kpc) | | $(L_{\rm bol}/L_{\odot})$ | ($10^{32}$ergs-1) | (keV) | (keV) | $(L_{\rm X}/L_{\rm bol})$
R136 | $\sim 50$ | R136a1 | 6.94a | 100 | - | 1.2 | -
| | +a2 | 6.63a | | | |
| | R136a3 | 6.58a | 70 | - | 4.2 | -
| | +a6 | 6.52a | | | |
| | R136c∗ | 6.75a | 700 | - | 3.0 | -5.5
30 Dor | $\sim 50$ | Mk30∗ | 6.16a | $<10$ | - | - | $<-6.8$
| | Mk34∗ | 6.70b | 1000 | 1.2 | 4.5 | -5.3
| | Mk35 | 6.3a | 10 | - | - | -6.9
| | Mk37∗ | 6.48a | 100 | - | - | -6.1
| | Mk37Wb | 6.21a | $<10$ | - | - | $<-6.8$
| | Mk39 | 6.4a | 130 | - | 2.0 | -5.9
| | Mk51 | 6.2a | $<10$ | - | - | $<-6.8$
| | RMC134 | 6.2a | 7 | - | - | -7.0
| | RMC135∗ | 6.2a | $<6.3$ | - | - | $<-7.0$
| | RMC140b∗ | 6.4a | 30 | - | 2.5 | -6.5
| | RMC144∗ | 6.72c | 25 | - | 2.1 | -6.9
| | RMC145∗ | 6.5a | 19 | - | 1.6 | $-6.8$
| | RMC146 | 6.29a | $<10$ | - | - | $<-6.9$
| | RMC147 | 6.36a | $<10$ | - | - | $<-7.0$
| | VFTS682 | 6.51a | $<10$ | - | - | $<-7.1$
Arches | $\sim 8$ | F6∗ | 6.32d | 110 | - | 2.2 | -5.9
| | F7∗ | 6.27d | 72 | - | 1.8 | -6.0
| | F9 | 6.15e | 46 | - | 2.5 | -6.1
Bochum 7 | $\sim$6.0 | WR12∗ | 5.98f | $<0.4$ | - | - | $<-8.0$
Westerlund 2 | $\sim 4.2$ | WR20a∗ | 6.3g | 57 | 0.4 | 1.6 | -6.1
| | WR20b | - | 120 | 0.4 | 5.4 | -
| | [WR20aa | - | 24 | 0.9 | - | -]
| | [WR21a* | - | 194 | 0.8 | 3.0 | -]
Carina | $\sim 2.6$ | WR22∗ | 6.35f | 4.2 | 0.6 | 2.0 | -7.3
| | WR24 | 6.19f | 6.4 | 0.7 | 3.3 | -7.0
| | WR25∗ | 6.61f | 84 | 0.6 | 2.7 | -6.3
NGC 3603 | $\sim 7.6$ | WR43A∗ | 6.39h | 360 | 0.8 | 3.1 | -
| | +WR43B | 6.46h | | | |
| | WR43C∗ | 6.35h | 260 | 1.0 | 7.9 | -5.5
| | [WR42e | - | 2.3 | - | - | \- ]
| | [WR43-2∗ | - | 2.9 | - | - | \- ]
Danks 1 | $\sim 3.8$ | WR48-8 | 5.44e | 17 | - | - | -5.8
| | WR48-9 | 5.99e | 43 | - | - | -6.0
| | WR48-10 | 5.70e | 14 | - | - | -6.2
Mercer 81 | $\sim 10$ | WR76-3 | 6.17e | 14 | - | 2.0 | -6.6
| | WR76-7 | 6.39e | 35 | - | 2.6 | -6.4
Sco OB1 | $\sim$2 | WR78∗ | 5.8f | 6.9 | 0.6 | 2.3 | -6.6
| | WR79a∗ | 5.78i | 1.4 | 0.6 | 2.7 | -7.2
HM-1 | $\sim 3.3$ | WR87 | 6.21f | 1.6 | - | 2 | -7.6
| | WR89∗ | 6.33f | 154 | 0.6 | 2 | -5.7
W43 | $\sim 6$ | WR121a∗ | - | 154 | 1.0 | 3.6 | -
Sources in the vicinity of the clusters but for which a physical association
has yet to be proved are given in square brackets. Identifiers marked with an
asterisk are those which are identified as binaries via optical or IR
observations, non-thermal radio emission (WR78 and WR89) or periodically
modulated X-ray activity (WR121a). Variable X-ray sources have their
representative fluxes given in italics. Parameters for R136a1+a2, R136a3+a6
and WR43A+WR43B represent a blend of both sources (see text for further
details). References for the X-ray properties are given in the body of the
text, while those for the bolometric luminosities of the system primaries are
aCrowther et al. ([2016]), bTehrani et al. ([2019]), cShenar et al. ([2021]),
dthis work, eNajarro et al. (in prep.), fHamann et al. ([2019]), gRauw et al.
([2005]), hCrowther et al. ([2010]), and iCrowther & Bohannan ([1997]). We
highlight that Mk34, RMC144, and WR20a are composed of two ‘twins’ - as with
other systems the bolometric luminosities presented are those of the combined
binary; individual components in these systems will be 0.3dex fainter. Finally
we have scaled the bolometric luminosities for WR24, WR24, and WR25 (Hamann et
al. ([2019]) to a common distance of 2.6kpc for the Carina complex.
Table 9: The X-ray properties of isolated WNha stars
ID | Dist. | log | $L_{\rm x}$ | log
---|---|---|---|---
| (kpc) | $(L_{\rm bol}/L_{\odot})$ | ($10^{32}$ergs-1) | $(L_{\rm X}/L_{\rm bol})$
WR16 | 2.6 | 5.72a | 0.13 | -8.2
WR29${}^{b}*$ | 6.0 | 5.73b | 2 - 3 | -7.1 - -6.9
WR40 | 4.0 | 5.91a | $<0.7$ | $<-7.7$
WR105∗ | 1.8 | 5.89a | 4 - 12 | -6.9 - -6.4
WR148∗ | 8.3 | 6.3a,c | 2.1 | -7.6
Identifiers marked with an asterisk are those which are identified as binaries
via optical (WR29 and WR148) and radio (WR105) observations. References for
the X-ray properties are given in the body of the text, while those for the
bolometric luminosities of the system primaries are aHamann et al. ([2019]),
bNaze et al. ([2020]) and cZhekov ([2017]). Note the X-ray luminosities listed
for WR29 and WR105 represent the range allowed under the modelling assumptions
of Naze et al. ([2020]).
* •
R136: Schnurr et al. ([2009b]) undertook the most comprehensive RV survey of
the six WN5h stars within R136 to date, but report only a single potential
binary - R136c - with a tentative $\sim 8.4$d period (noting the survey was
only sensitive to periods $\lesssim 40$d). X-ray observations reveal that it
is one of the brightest objects within the LMC and demonstrates a hard
spectrum (Table 8); strongly suggestive of a colliding wind binary (see
below). Of the remaining WN5h stars within R136, Townsley et al. ([2006])
report a blend of emission from R136a1+a2 and R136a3+a6 (the latter being O2
If; Crowther et al. [2016]). Both composite sources are luminous and hard,
noting that only single temperature models were employed due to data quality
(Table 8). Finally, R136b is heavily blended with other core stars and so is
not discussed further, while no mention is made of R136a5 by these authors.
* •
30 Doradus: A further 15 WNLh and related stars are found within the wider
confines of 30 Dor (Crowther et al. [2016]). Of these, seven have been
confirmed as binaries via spectroscopic observations141414Mk30 (O2 If/WN5),
Mk34 (WN6(h)), Mk37 (O3.5 If/WN7), RMC135 (WN6h), RMC140b (WN5h), RMC144
(WN5-6h), and RMC145 (WN6h).; one is an RV variable but lacks an orbital
solution151515Mk35 (O2 If/WN5).; three appear to lack RV variability161616Mk51
(O3.5 If/WN7), RMC134 (WN6(h)), and VFTS682 (WN5h).; and appropriate datasets
are lacking for four171717Mk37Wb (O2 If/WN5), Mk39 (02.5 If/WN6), RMC146
(WN5ha), and RMC147 (WN5h). (Table 3, Schnurr et al. [2008b], Bestenlehner et
al. [2011], Tehrani et al. [2019]). X-ray data have been presented for a
number of these stars, resulting in eight detections. Of these six arise from
confirmed binaries/RV variables (Table 8). The most comprehensive analyses are
of Mk34 and RMC144. Pollock et al. ([2018]) show that the former is an
extremely luminous source (median $L_{\rm X}\sim 10^{35}$ergs-1) that is
modulated on the orbital period, with a spectrum comprising soft and hard
components. Despite having similarly luminous components RMC demonstrates a
much more modest X-ray flux, although it too displays orbital modulation and
an intrinsically hard spectrum (Townsley et al. [2006], Shenar et al. [2021]).
Two further stars - RMC140b and RMC145 - are also luminous and hard sources
($L_{\rm X}\sim 10^{33}-10^{34}$ergs-1; $kT\gtrsim 1.6$keV; noting that only
single temperature models were employed by Townsley et al. [2006]). Mk37, the
remaining binary detection, and the RV variable Mk35 are similarly luminous,
while RMC134 - which appears not to be RV variable - is comparatively faint;
unfortunately no spectral information is available for any of these three
stars (Table 8; Schnurr et al. ([2008b]). Finally, Mk39 has yet to be subject
to RV monitoring, but has X-ray properties directly comparable to confirmed
binaries (Table 8; Townsley et al. [2006]); Schnurr et al. ([2008b]) returned
individual upper limits for the last two binaries (Mk30 and RMC135; Table 8).
This leaves a total of five stars for which we adopt the generic detection
threshold of $10^{33}$ergs-1 reported by Townsley et al. ([2006]) as upper
limits, noting that in practice deviations from this are expected depending on
the local environment of individual stars.
* •
Bochum 7: Bo7 hosts a single WN8h star, WR12, for which Fahed & Moffat
([2012]) present a spectroscopic orbital solution. Pollock et al. ([1995])
report an X-ray detection, albeit of low detection significance ($(10\pm
5)\times 10^{32}$ergs-1 for a distance of 6kpc, subsequently quoted by Ignace
et al. [2000]). However more recent and sensitive observations by Nazé et al.
([2020]) failed to detect it (Table 8), the authors noting that it was
observed at a phase at which the WNLha primary and its dense stellar wind
would have (partially) obscured the secondary and any putative colliding wind
region (see also WR22 below). Clearly future observations will be required to
ascertain the nature of its (variable?) emission although we adopt the upper
limit of Nazé et al. ([2020]) in this work.
* •
Westerlund 2: Two WNLha stars have traditionally been associated with Wd2;
WR20a (WN6ha+WN6ha) and WR20b (WN6h). The former is a well known massive
eclipsing system (Rauw et al. [2005]; Table 3) and is also a luminous, hard
X-ray source that varies on the orbital period (Nazé et al. [2008]); in Table
8 we list the properties for the brightest of the epochs presented by the
authors. They further demonstrate that WR20b is a fainter, harder source
(representative $L_{\rm X}\sim 2\times 10^{33}$ergs-1, $kT\sim 3.6$keV)
although Skinner et al. ([2010]) prefer a two-temperature fit resulting in a
higher luminosity (see Table 8). As a consequence they propose that WR20b is
also a colliding wind system; however spectral monitoring by Rauw et al.
([2011]) apparently excludes short binary periods ($<50$d - subject to the
usual caveats regarding unfavourable orbital inclination). Three further WNLha
stars are found in the periphery of Wd2, for which a physical association is
currently uncertain: WR20aa (O2If∗/WN6), WR20c (O2IF∗/WN6ha), and WR21a
(O3If∗/WN6ha+O3V). The very massive short period binary WR21a (Table 3) is, as
may be anticipated, a highly luminous, hard and periodically variable X-ray
source (Gosset & Naze [2016]). WR20aa and WR20c are possible runaways
originating from Wd2 (Roman-Lopes et al. [2011]). Townsley et al. ([2019])
report that the former appears to be a moderately bright and soft X-ray source
(Table 8), while no X-ray data are available for the latter. In neither case
has RV spectral monitoring been attempted.
* •
Carina: The Carina star-forming complex, encompassing the young massive
clusters Trumpler 16 and Collinder 229, is host to three WNLha stars - WR22
(=HD92740A; WN7+abs +09), WR24 (=HD93131; WN6ha), and WR25 (=HD93162; O2.5
If∗/WN6 + O). X-ray observations have been made of all three systems (Gosset
et al. [2009], Skinner et al. [2010], Raassen et al. [2003], Pollock &
Corcoran [2006]) and are summarised in Table 8. WR22 and WR25 have already
been confirmed as binaries via RV studies, and both sources also appear to
show orbitally modulated X-ray fluxes (Gosset et al. [2009], Arora et al.
[2020]). WR22 is of particularly interest since it is observed to be
particularly faint at the point at which the WNLha and associated wind might
be expected to eclipse both the secondary and any associated wind interaction
zone ($L_{\rm X}\sim 6.3\times 10^{31}$erg-1). Finally, while spectroscopic
evidence for binarity is lacking for WR24, we highlight the close similarity
of its X-ray properties to those of WR22.
* •
NGC 3603: Three WNLha stars are found within the core of NGC3603 - WR43A (=
cl* NGC 3603 BLW A1; WN6ha+WN6ha), WR43B (= cl* NGC 3603 MDS B; WN6ha) and
WR43C (= cl* NGC3603 MDS C; WN6ha + ?), of which WR43A and WR43C are known
short period binaries (Table 3). X-ray observations are complicated by
blending due to the compact nature of NGC3603 but Moffat et al. ([2002]; their
Fig. 4) suggest fluxes of $L_{\rm x}\sim 10^{33}$ergs-1 for WR43B, $L_{\rm
x}\gtrsim 4\times 10^{34}$ergs-1 for WR43C and $L_{\rm x}\gtrsim
10^{34}$ergs-1 for the blend comprising WR43A, cl* NGC3603 BLW A2 (O3 V) and
cl* NGC3603 BLW A3 (O3 III(f∗)); unfortunately no spectral information was
presented for any source. Subsequently, Huenemoerder et al. ([2009]) analysed
the emission spectra deriving from a blend of WR43A+WR43B and from WR43C,
finding both to be highly luminous and demonstrating hard spectral components
(cf. Table 8). Of these Pollock et al. ([2018]) report WR43C as being
variable. Two further WNLha stars are located in the surrounds of NGC3603.
Displaced $\sim 6$pc from the cluster core, Roman-Lopes ([2012]) suggests that
the O2 If∗/WN6h stars WR42e could be a runaway (and hence single);
nevertheless they report a X-ray luminosity that is comparable with the binary
WR22 (Table 8) but no spectral information. Finally WR43-2 (= cl* NGC3603 MTT
58; O2-3.5 If∗/WN5-6ha + ?)) is located within a compact H ii region on the
periphery of the wind-blown bubble encompassing NGC3603, some $\sim$2pc
distant from the cluster core. Again lacking spectral information, it is of
comparable luminosity to WR42e (Roman-Lopes [2013]), but is unambiguously a
short-period binary on the basis of periodic photometric modulation (Jaque
Arancibia et al. [2015]).
* •
Mercer 30: de la Fuente et al. ([2016]) report two WNLh stars within the
cluster Mercer 30 - WN46-5 (= [DNB2016] Mc30-7; WN6h) and WN46-6 (= [DNB2016]
Mc30-8; WN7h), which modelling suggests are highly luminous, and hence massive
(log$L/L_{\odot}\gtrsim 6.0$ and $M_{\rm current}\gtrsim 50M_{\odot}$).
Despite a limited dataset comprising three epochs of spectroscopic
observations, the former is clearly a binary by virtue of $\Delta$RV$\sim
191$km s-1 between observations in 2009 and 2011; unfortunately no X-ray
observations exist for this cluster.
* •
Mercer 81: Davies et al. ([2012a]) and de la Fuente et al. ([2013]) report
five WN7-8ha stars within the distant and highly reddened cluster Mercer 81;
WR76-9 (=[DDN2012] 3), WN76-6 (=[DDN2012] 7), WR76-2 (=[DDN2012] 5), WR76-7
(=[DDN2012] 2), and WR76-3 (=[DDN2012] 8). No radial velocity surveys have
been undertaken but the X-ray survey of Norma Arm by Rahoui et al. ([2014])
report detections WR76-3 and WR76-7 (their sources #1278 and #1279
respectively), finding them to be hard, bright sources (Table 8; assuming a
single temperature fit). Bolometric luminosities for the two X-ray detections
derive from quantitative modelling to be presented in Najarro et al. (in
prep.); we find both stars to be intrinsically highly luminous (Table 8).
* •
Danks 1: In the absence of RV observations, Zhekov et al. ([2017]) presents a
preliminary analysis of X-ray data, quoting fluxes for the three WNLha stars
WR48-8 (= [DCT2012] D1-5; WN8-9ha), WR48-9 (= [DCT2012] D1-1; WN8-9ha), and
WR48-10 (= [DCT2012] D1-2; WN8-9ha), with Townsley et al. ([2019])
subsequently reporting that the first two sources are variable, with fluxes
decreasing by factors of $\sim 5$ and $\sim 10$ respectively, between two
epochs of observations separated by $\sim 4$yr. While neither author presents
spectral information for these detections both conclude that they are bona
fide binaries based on their X-ray properties (a claim we revisit below). As
with Mercer 81 we provide bolometric luminosities for the three X-ray
detections derived from modelling in Table 8 (Najarro et al. in prep.). We
highlight that while WR48-9 and 48-10 are broadly comparable to other Galactic
examples, WR48-8 is significantly fainter, while the emission lines in the
K-band spectrum appear anomalously broad and weak (Davies et al. [2012b]). As
a consequence we suspect its nature and/or evolutionary pathway differs from
other examples; we return to this topic in Najarro et al. (in prep.).
* •
Havlen Moffat 1; As with Danks 1, HM-1 as a whole has yet to be subject to a
systematic, modern RV survey, and so we are forced to fall back on
observations at other wavelengths; WR89 (WN7h) appears to be a non-thermal
radio source and hence colliding wind binary (Cappa et al. [2004]), although
inspection of archival spectra fails to reveal RV shifts (Nazé et al.
[2013])181818The lack of spectral variability is potentially explicable by
unfavourable inclination and/or a long or highly eccentric orbit..
Nevertheless, these authors reveal it to have an extreme (variable) X-ray
luminosity and a spectrum comprising both soft and hard thermal components.
Unfortunately, only X-ray observations are available to determine the nature
of WR87 (WN7h); while it is comparatively faint with respect to WR79a, it too
has a hard spectral component at least consistent with emission from a wind
collision zone (Table 8 and discussion below).
* •
Sco OB1: Hosting the young massive cluster NGC6321 at its core, the Sco OB1
association contains two WNLh stars - WR 78 (= NGC 6231 305; WN7ha) and WR79a
(= NGC6231 327; WN9ha) - that, in contrast to many other examples considered,
demonstrate relatively modest bolometric luminosities Table 8). Both stars are
detected at X-ray energies, and share similar properties, being rather faint
and hard sources (Skinner et al. [2010], [2012]). WR78 appears to demonstrate
aperiodic optical variability (Skinner et al. [2012] and refs. therein) while
Chini et al. ([2012]) lists WR79a as an SB2 system but provides no links to an
appropriate reference; as such, we refrain from assuming binarity for either
on the basis of these works. However de Becker & Ruacq ([2013]) and Cappa et
al. ([2004]) report that both stars are non-thermal radio sources and hence
colliding-wind binaries: a conclusion we adopt for the remainder of this
study.
* •
W43: The giant H ii region hosts a young, heavily embedded cluster at its
core, which in turn contains a single WN7ha star: WR121a. Arora & Pandey
([2020]) provide an analysis of time-resolved X-ray observations of this star,
finding it to be a hard and highly luminous source, which demonstrates an
apparent $\sim 4.1$d periodicity that they attribute to orbital modulation in
a massive compact binary. Examining near-IR and radio data, Luque-Escamilla et
al. ([2011]) further show that WR121a maybe resolved into two components
separated by a projected angular distance of $\sim 0.6$", suggesting the
possible presence of a further stellar companion.
* •
Quintuplet: Finally we turn to the Quintuplet, which Clark et al. ([2018b])
report contains two WN8-9ha (qF256 and qF274) and an hybrid O7-8 Ia+/WN8-9ha
star (LHO001). Given the remaining cluster population their presence is a
surprise, with the authors suggesting they could be the result of binary
interaction. Unfortunately only LHO001 has been the subject of multi-epoch
spectroscopy, which reveal it to be highly variable, although insufficient
data exists to attempt to identify periodic modulation. Wang et al. ([2006])
present X-ray observations of the Quintuplet; none of these sources are
detected and, as with the Arches (Sect. 3.2.1), a combination of potential
blending and the presence of spatial variable diffuse emission prevent the
derivation of upper limits. Given the limited observational data for these
stars as well as their uncertain provenance, we refrain from discussing them
further.
#### C.1.2 Isolated WNLh stars
We may supplement the above dataset with five isolated single and binary
stars. Of the former, Skinner et al. ([2012]) report a detection of the WN8h
star WR16; unfortunately it is too faint for spectral analysis (Table 9, with
the flux scaled to the distance adopted by Hamann et al. [2019]). Although
optically variable it currently cannot be unambiguously identified as a binary
(Antokhin et al. [1995]). Similar conclusions are also reported for WR40
(WN8h) by these authors; unfortunately Gosset et al. ([2009]) are only able to
provide a (conservative) upper limit to the X-ray flux which, suitably scaled,
is repeated in Table 9. Turning to the binaries, Nazé et al. ([2020]) find
WR29 (WN7h+O; Table 3) to be a faint X-ray source; assuming a canonical
$kT\sim 0.6+6.9$keV thermal spectrum it appears moderately luminous ($\sim
2-3\times 10^{32}$ergs-1; Table 9). WR105 (WN9h) is a non-thermal radio source
(Montes et al. [2009]) and hence a strong colliding wind binary candidate.
Although a weak X-ray detection (Nazé et al. [2020]), under the same modelling
assumptions adopted for WR29 it appears to be intrinsically luminous ($\sim
4-12\times 10^{32}$ergs-1; Table 9).
Finally we turn to the runaway binary WR148 (WN8h), the nature of which has
been the subject of some debate e.g. Drissen et al. [1986]. Recently Munoz et
al. ([2017]) have suggested an O5V classification for the secondary; however
the canonical mass of such an object - they quote $\sim 37M_{\odot}$ \- is in
tension with the inclination derived by Drissen et al. ([1986]). Upon
reappraisal of the relevant photometric and polarimetric data the former
authors suggest that the results are ‘inconclusive’. Allowing for a lower
inclination, the canonical mass of the O5V secondary implies a current mass of
$\sim 33M_{\odot}$ for the WN7ha primary - unexpectedly low in comparison to
other examples (Table 3) and for a luminosity of $\sim
10^{6.3}L_{\odot}$191919Obtained by scaling the luminosity of Hamann et al.
([2019]) to the consensus distance of 8.3kpc (Zhekov [2012]), noting that Rate
& Crowther ([2020a]) suggest $\sim 9.47^{+1.77}_{-1.49}$kpc.. Nevertheless,
Zhekov ([2012]) report a rather moderate X-ray luminosity ($\sim 2.1\times
10^{32}$ergs-1) and a thermal spectrum ($kT\sim 0.4+1.0$keV) for the system.
For completeness we briefly discuss the Galactic centre source CXO
J174536.1-285638 (WN9ha). Although excluded from the sample for the reasons
given above, we note that it is a long period binary ($\sim 189$ d; Mikles et
al. [2008]). Adopting a two-component spectral fit ($kT\sim 0.7+4.6$keV)
implies an exceptional X-ray luminosity ($L_{\rm X}\sim 10^{35}$ergs-1) that
is directly comparable to that of Mk34 (Table 8). This indicates that the
latter is not a pathological system; however, the lower luminosity of the
primary ($L_{\rm bol}\sim 10^{6.24}$; Clark et al. [2009]) leads to an even
more extreme log$(L_{\rm X}/L_{\rm bol})\sim-4.8$ for CXO J174536.1-285638
than was found for Mk34 (Table 8).
### C.2 Data interpretation
#### C.2.1 The X-ray properties of WNLh stars
Historically, and by analogy to O stars, WR binaries have been expected to be
hard, overluminous X-ray sources. However recent studies suggest that such a
picture may be overly simplistic. In particular, while a number of confirmed O
star and WR binaries within Wd1 conform to this expectation, others are found
to have comparatively soft spectra consistent with emission from single stars
(Clark et al. [2019c]; see also Oskinova [2015], [2016]). Considering a
dataset exclusively comprising WR stars allowed Skinner et al. ([2010],
[2012]) to report the ubiquity of a hard ($kT\gtrsim 2$keV) component in the
X-ray spectra of all WN stars with spectra sufficient to accommodate
modelling. Conversely, Oskinova ([2016]) state that, on average, single WN
stars display a softer ($kT\sim 0.4$keV) spectrum. Clearly, reconciling these
potentially conflicting findings requires an accurate determination of the
binary properties of the stars studied. Nevertheless, studies focusing on
individual binary systems via multi-epoch, phase-resolved observations
indicate that where present - and as with O stars - the high-temperature
spectral component originates in the wind collision zone (e.g. Gosset et al.
[2009], Montes et al. [2013], Parkin & Gosset [2011]).
Mindful of the fact that the cohorts considered by Skinner et al. ([2012]) and
Oskinova ([2015], [2016]) comprise in part H-free WNE stars - and so are not
directly comparable to our sample - we highlight that the spectral properties
of the WNLh stars considered here are uniformly characterised by the presence
of a hard thermal component (Table 8), potentially indicative of binarity.
However, while the possibility remains that single WR stars may also
demonstrate similarly hard X-ray spectra (Skinner et al. [2012]) it appears
premature to utilise this observational feature to unambiguously identify WR
binaries.
This leaves X-ray luminosity as a prospective binary diagnostic. Oskinova
([2015]) and Gagné et al. ([2012]) provide compilations of X-ray properties
for single and binary WRs, finding that the luminosities of both show an
exceptional variance, spanning $\sim$four orders of magnitude. Nevertheless,
they identify potential trends whereby (i) binaries appear to be more
prevalent amongst the brighter X-ray sources, apparently obeying the canonical
$L_{\rm X}/L_{\rm bol}\sim 10^{-7}$ relationship delineated by hot, massive
stars and (ii) the most luminous systems are biased towards the binaries with
the longest periods. However, interpretation is complicated by the
heterogenous nature of the samples appraised by these authors, comprising both
WNLh and the more evolved H-free ‘classical’ WNE stars (with the latter being
of systematically lower luminosity).
Motivated by their evolutionary uniformity we plot $L_{\rm bol}$ versus
$L_{\rm X}$ for our sample of WNLh stars in Fig. C.1 (c.f. Tables C.1 and C.2;
note that due to uncertainty regarding individual contributions, the blended
R136 and NGC3603 sources are not included). For consistency, in all instances
we plot the integrated bolometric luminosity of the stellar system in question
(with that of F9 obtained via an identical methodology to that described in
Sect. 4; Najarro et al. in prep.). It is expected that for the majority of
objects considered this will be equivalent to the output from the WNLh star
that dominates the emergent spectrum. However at least three systems - WR20a,
MK34 and RMC144 - are known binaries comprising twin WNLh stars; in such cases
the luminosity of individual components is $\sim 0.3$ dex lower than
represented in Fig. C.1. We find that X-ray luminosities of the stars range
over four orders of magnitude: from $\sim 10^{31}$ergs-1 (for the apparently
single WR16) to $\gtrsim 10^{35}$ergs-1 (the binaries Mk34 and CXO
J174536.1-285638), the latter stars being amongst the most luminous colliding-
wind systems known (Gagne et al. [2012]). Despite considerable scatter, a
positive trend between X-ray and bolometric luminosities is apparent. With
23/39 of the WNLh stars considered exceeding the empirical $L_{\rm X}/L_{\rm
bol}\sim 10^{-7}$ relationship found for O stars, it appears that the former
have a systematically greater X-ray luminosity than the latter. Moreover there
are hints that the correlation is also steeper for WNLh stars, with 12/15
examples with log$(L_{\rm bol}/L_{\odot})>6.3$ exceeding this threshold (Fig.
C.1).
Of the remaining stars, the constant RV sources WR24 and R134 essentially
straddle this division202020Based on the results presented in Fig. 4 of Moffat
et al. ([2002]), WR43B also lies on this threshold which, based on the values
in Table 8 would imply that WR43A significantly exceeds it., while the upper
limits to the X-ray flux of five stars - the binary Mk30 and apparently single
star Mk51 as well as MK37Wb, RMC146, and RMC147 (which are currently of
indeterminate nature) - render their placement uncertain. Indeed, only nine
stars are unambiguously found below this threshold; these comprise the
apparently single stars WR16, WR40 and VFTS682; the binaries WR12, WR22,
WR79a, WR148 and RMC135; and WR87, the nature of which is currently uncertain.
Of these the binaries WR12 and WR22 are of particular interest given the
stringent upper limits obtained for their X-ray fluxes ($L_{\rm X}\lesssim
10^{31.8}$ergs-1; Fig. C.1). These derive from observations of both systems
obtained at orbital phases when their WNLha primaries potentially obscure
their companions and any putative wind collision zones. Following the
arguments of Gosset et al. ([2009]) and Nazé et al. ([2020]), one might expect
the dense, metal-rich winds of the WNLh stars to efficiently absorb high
energy emission from the latter components, leaving only their own
intrinsically weak emission visible. Indeed, the observations of both WR16 and
WR40 support the hypothesis that single WNLh stars are particularly faint
X-ray sources.
A corollary of this hypothesis would be that all of the X-ray luminous WNLh
stars are colliding wind binaries. Indeed 16 of the 21 confirmed WNLh
binaries212121WR20a, WR25, WR29, WR43C, WR78, WR89, WR105, F6, F7, Mk34, Mk35,
Mk37, RMC136, RMC140b, RMC144, and RMC145 are found above the $L_{\rm
X}/L_{\rm bol}\sim 10^{-7}$ threshold; of the remaining seven co-located stars
six currently lack diagnostic observations222222WR48-8, WR48-9, WR48-10,
WR76-3, WR76-7, and Mk39 and only Arches F9 shows no evidence for binary-
induced RV variability at this time (although we note that its radio
properties are indicative of binarity; Sect. 3.2). Thus we might infer a
binary fraction of 15/16 ($\sim 94$%) amongst stars exceeding this limit at
this time (and a minimum of $\sim 70$% if the remainder are found to be
single).
Consequently, we adopt $L_{\rm X}/L_{\rm bol}\sim 10^{-7}$ as a conservative
and empirical lower limit for the identification of colliding wind binaries
with WNLh primaries via their X-ray emission. In doing so we reiterate that
five known binaries fall below this threshold; however lowering the limit to
encompass all these system at all times would potentially permit the inclusion
of genuinely single stars. This is evident upon consideration of the variable
X-ray emission from the binaries WR12 and WR24, which appear to manifest as
single stars at certain orbital phases. In addition to their intrinsic
variability, the large variance in the X-ray properties of binaries likely
results from the diversity of their physical properties - in particular the
nature of the secondary, orbital separation and eccentricity (Gagné et al.
[2012] and Table 3) - which may allow some systems to remain inherently faint.
That the absolute X-ray luminosity of binaries with WNLh primaries appears
greater than that of those containing O stars likely reflects the difference
in wind properties, with the former supporting more powerful, optically thick
winds in comparison to the optically thin outflows of the latter (e.g.
Gräfener ([2021]).
Figure 11: Plot of bolometric and X-ray luminosities for the WNLh stars given
in Tables C.1 and C.2 (black), with the addition of the Arches detections F6,
F7, and F9 (red symbols). Confirmed binaries (including the RV variable Mk35
for simplicity) are given by filled squares, stars with no indication of
binarity are depicted by open squares, and the subset for which the requisite
observations have yet to be undertaken represented by open circles. Stars
known to be X-ray variable are indicated by the nested symbols. The second,
open symbol for W22 represents the upper limit to the intrinsic emission from
the WN7h primary according to Gosset et al. ([2009]). The solid lines joining
the two points for WR29 and WR105 indicate the range of possible luminosities
for both stars (Nazé et al. [2020]). Luminosities are shifted slightly for
RMC140b (-0.003 dex), RMC144 (+0.003 dex), Mk51 (+0.02 dex) and Mk37Wb (+0.02
dex) to aid clarity. The canonical $L_{\rm X}=10^{-7}L_{\rm bol}$ relationship
that Oskinova ([2015]) suggests is followed by O and WN stars is given by the
dotted line; stars falling below this limit are individually labeled, as are
certain additional stars of interest (subject to figure clarity). Finally the
low luminosity star WR48-8 is not plotted, but lies significantly above the
threshold.
#### C.2.2 The binary fraction of WNLh stars within clusters
How may we interpret these data in terms of the incidence of binarity and the
X-ray properties of such systems? Regarding the frequency of occurrence, as a
first step we may simply consider those stars that appear unambiguously binary
via spectroscopic determination of an orbital period. Mindful of the
limitations of extant RV and X-ray surveys, we find 9/17 ($\sim 53$%)
confirmed binaries within R136/30 Dor (including the RV variable Mk35).
Likewise, recognising that clusters such as Danks 1, HM-1 and Mercer 81 have
yet to be subject to systematic RV surveys (and the limited nature of the
Mercer 30 study) we arrive at a total of 7/11 in five galactic aggregates
(Table 9; Bochum 7, Westerlund 2, Carina, NGC3603 and Mercer 30). To this
tally we may add the periodic X-ray source WR121 (W43) and the non-thermal
radio sources WR78 (Sco OB1) and WR89 (HM-1) where emission is though to arise
in shocked material in the wind collision zone, increasing the Galactic count
to 10/16. This implies a minimum binary fraction of 19/33 ($\gtrsim 58$%)
before allowance is made for the insensitivity of the extant RV surveys to
longer period systems and those seen at unfavourable orbital inclinations.
This brings us to interpretation of the X-ray properties of these stars.
Orbitally modulated X-ray emission is apparent in the spectroscopically
confirmed binaries Mk34, RMC144, WR20a, WR21a, WR22, and WR25; on this basis
the periodically variable X-ray source WR121a is also a compelling binary
candidate (see also CXO J1745-28; Table 3). However, many of the remaining
candidate and confirmed WNLh binary systems have an insufficient timeline of
X-ray observations to permit a search for periodicities. For such systems we
adopt $L_{\rm X}/L_{\rm bol}\gtrsim 10^{-7}$ as a conservative threshold for
binarity; in doing so we are mindful of (i) the relative paucity of
observational constraints for apparently bona fide single stars, (ii) that
genuine binaries may fall below this threshold (cf. Fig. C.1 and Sect. 3.2.2)
and (iii) that we may not assess binarity for a number of X-ray detected WNLh
stars that currently lack bolometric luminosity determinations232323WR20aa,
WR20b, WR21a, WR42e and WR43-2..
Nevertheless, adopting this discriminant means we may assert binarity for
Arches F9 (see Sect. 3.3.2) as well as Mk39 (within 30 Dor), WR48-8, WR48-9,
WR48-10 (within Danks 1), and WR76-3 and WR76-7 (Mercer 81). Critically, this
would mean recognising the possibility of a binary fraction approaching unity
for the most luminous subset of stars (log$(L_{\rm bol}/L_{\odot})>6.3$; Sect.
C.2.1). Such a conclusion, however, is subject to the caveat that the nature
of some members of the very luminous cohorts within R136 (a1, a2, a3, and a5)
and NGC3603 (WR43A and WR43B) remain difficult to ascertain due to blending,
although the properties of the respective composite X-ray sources are strongly
suggestive of binarity amongst one or more of the component stars (Table 8).
More provocatively, if we were to take the observations summarised in Fig. C.1
at face value, one would infer that (apparently) single WNLha stars are
intrinsically X-ray faint ($L_{\rm X}<10^{32}$ergs-1). This would lead to the
conclusion that all but one of the stars reported as X-ray detections in Table
8 \- which, with the exception of WR12, uniformly exceed this value at one or
more epochs - are binaries: a verdict consistent with their uniformly hard
X-ray spectra.
Nevertheless, under the former, more conservative X-ray cut, the binary count
for R136/30 Dor rises to 10/21 ($\sim 48$%) with the inclusion of Mk39 (noting
that the sample size has risen since the full cohort of WNLh stars has been
subject to X-ray observations). Similarly, consideration of the additional
X-ray diagnostic leads to the identification of 15/24 ($\sim 63$%) binaries
within the ten galactic aggregates considered above. Given the foregoing
discussion and the diverse and incomplete nature of the multi-wavelength
observations employed in the preceding assessment, we consider it highly
likely that the simple number counts presented above are lower limits to the
true binary fractions, but the heterogenous nature of the extant datasets
means we are unable to assess the degree of incompleteness at this time. We
discuss the Arches cluster in the context of these observations - and the
resultant conclusions - further in Sect. 4 and 5.
|
# Emergence of two-fold spectral topology through non-Abelian gauge
engineering
Ronika Sarkar<EMAIL_ADDRESS>Solid State and Structural Chemistry
Unit, Indian Institute of Science, Bangalore 560012, India Department of
Physics, Indian Institute of Science, Bangalore 560012, India Ayan Banerjee
Solid State and Structural Chemistry Unit, Indian Institute of Science,
Bangalore 560012, India Awadhesh Narayan<EMAIL_ADDRESS>Solid State and
Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012,
India
###### Abstract
Non-Abelian phenomena and non-Hermitian systems have both been widely explored
in recent years. As a bridge between the two, we introduce and develop non-
Abelian gauge engineering for realizing _multi-fold_ spectral topology. As an
example of our proposal, we engineer non-Hermiticity in the paradigmatic Su-
Schrieffer-Heeger (SSH) model by introducing a generalized non-Abelian gauge,
leading to an emergent two-fold spectral topology that governs the decoupled
behaviour of the corresponding non-Hermitian skin effect. As a consequence of
the non-Abelian gauge choice, our model exhibits a rich phase diagram
consisting of distinct topological phases, which we characterize by
introducing the notion of _paired_ winding numbers, which, in turn, predict
the direction of skin localization under open boundaries. We demonstrate that
the choice of gauge parameters enables control over the directionality of the
skin effect, allowing for it to be unilateral or bilateral. Furthermore, we
discover non-dispersive flat bands emerging within the inherent SSH model
framework, arising from the non-Abelian gauge. We also introduce a simplified
toy model to capture the underlying physics, thereby giving direct physical
insights. Our findings pave way for the exploration of unconventional spectral
topology through non-Abelian gauges.
Introduction.$-$ Non-Abelian phenomena are widespread in classical and quantum
systems – prominent examples range from the Yang-Mills theory of the standard
model Yang and Mills (1954) to the emergence of anyons obeying non-Abelian
statistics Nayak _et al._ (2008). In recent years, there has been remarkable
progress in the generation and control over non-Abelian features in a variety
of platforms Yang _et al._ (2023), leading to the unveiling of intriguing
physics Sugawa _et al._ (2018); Guo _et al._ (2021); Yang _et al._ (2019);
Brosco _et al._ (2021); Zhou (2023); Pang _et al._ (2023). The intricate
interplay between non-Abelian topology and quantum condensed matter has
yielded a rich tapestry of phenomena. This includes the discovery of non-
Abelian band topology in non-interacting metals Wu _et al._ (2019), the
realization of non-Abelian topological insulators Jiang _et al._ (2021), the
emergence of non-Abelian statistics in superconducting vortices Ivanov (2001);
Sato and Fujimoto (2009), and the exploration of non-Abelian braiding Noh _et
al._ (2020); Chen _et al._ (2022); Zhang _et al._ (2022a), to name just a
few.
In parallel, the synergy of non-Hermiticity and topology has ushered in a
remarkable wave of activity Bergholtz _et al._ (2021); Ashida _et al._
(2020); Ding _et al._ (2022); Banerjee _et al._ (2023a). This has led to a
variety of intriguing discoveries, including non-Hermitian spectral topology
Okuma and Sato (2023), non-Hermitian skin effect Yao and Wang (2018); Alvarez
_et al._ (2018); Zhang _et al._ (2022b); Lin _et al._ (2023), as well as
unexpected connections to topics in modern mathematics Hu and Zhao (2021);
Jaiswal _et al._ (2023); Wang _et al._ (2022); Banerjee _et al._ (2023b);
Hu (2023). The ramified symmetries in non-Hermitian systems give rise to
distinctive topological phases with the notion of “line” and “point” gap
topology Kawabata _et al._ (2019a). The role of non-Hermitian time dynamics
in spectral topology has revealed new insights into the interplay between non-
Hermiticity and topological features Lee _et al._ (2019), providing a deeper
understanding of phenomena such as spectral singularities, i.e., exceptional
points (EPs) Heiss (2004); Ding _et al._ (2016); Miri and Alu (2019); Chen
_et al._ (2020); Kawabata _et al._ (2019b), and the emergence of novel
topological phases with exotic properties Lee (2016); Shen _et al._ (2018);
Li _et al._ (2020); Liang _et al._ (2022); Kawabata _et al._ (2022); Longhi
(2019); Kunst _et al._ (2018); Weidemann _et al._ (2020); Yokomizo and
Murakami (2019); Song _et al._ (2019); Xue _et al._ (2022). Traversing
through EPs leads to phase transitions, and unveils unconventional features
such as intersecting Riemann sheet topology Dembowski _et al._ (2004);
Doppler _et al._ (2016), manifesting as Berry phases and fractional
topological charges Zhou _et al._ (2018), which have a range of potential
applications Wiersig (2014); Özdemir _et al._ (2019); Chen _et al._ (2017).
Figure 1: Non-Abelian gauge engineering in the Su-Schrieffer-Heeger model.
(a) Non-Hermiticity introduced by generalized gauge engineering in the SSH
model. (b) A simplified schematic of the four-band model for our choice of
gauge. Non-Abelian gauge fixing associates two degrees of freedom with each
unit cell site $A$ and $B$, effectively dissociating them to $A_{1},A_{2}$ and
$B_{1},B_{2}$, respectively. Each hopping label corresponds to the same
coloured arrow. Additionally, the rightward (leftward) red arrows denote
$-it_{2}\sin{\theta_{R}}$ ($it_{2}\cos{\theta_{L}}$) and the rightward
(leftward) green arrows denote $it_{2}\sin{\theta_{R}}$
($-it_{2}\cos{\theta_{L}}$). A similar approach is directly applicable to
other models.
In this work, we introduce and develop non-Abelian gauge engineering as a
promising approach to realize non-Hermitian multi-fold spectral topology.
Using the iconic Su-Schrieffer-Heeger (SSH) model as an example, we
demonstrate a remarkably rich phase diagram and the emergence of a two-fold
spectral topology, which are a direct consequence of the underlying non-
Abelian structure. We discover a variety of distinct topological phases, which
we characterize by introducing _paired_ winding numbers. These topological
invariants enable the prediction of the skin effect under open boundary
conditions, and the selection of gauge parameters allows exquisite control
over the direction and nature of the skin effect. We show that the spectral
topology, as well as the chiral dynamics, are robust to disorder. Notably, the
phase diagram features flat bands which are accessible in a wide parameter
regime. We further introduce a toy model comprising of coupled Hatano-Nelson
chains to understand the underlying principles of such a non-Abelian gauge
engineering.
Figure 2: Phase diagram and two-fold spectral topology. The phase diagram
shows five distinct topological phases as a function of the gauge parameters,
$\theta_{L}$ and $\theta_{R}$. The red and blue lines marking the phase
boundaries denote EP2 and EP4 transition lines, respectively. The spectral
topology associated with the $E_{1},E_{2}$ energy bands are shown in purple in
each phase, while $E_{3},E_{4}$ spectra are shown in green. The corresponding
arrows illustrate the direction of spectral winding. The associated coloured
circles denote the dynamically robust chiral modes associated with each
spectral loop. The paired topological invariant for each phase is presented in
the legend. The pink dashed lines illustrate parameter ranges which result in
two non-dispersive flat bands. Here we set $t_{1}=t_{2}=1$ and $\phi=0$.
Model.$-$ Our proposal is to introduce a non-Abelian gauge to a parent
Hermitian model, thus endowing it with additional degrees of freedom. A
suitable choice of the gauge parameters in turn allows generating non-
Hermiticity and the concomitant spectral topology [Fig. 1(a)]. Originally
introduced to study solitons in polyacetylene Su _et al._ (1979), the
paradigmatic SSH model provides a fundamental framework for studying
topological systems Asbóth _et al._ (2016). We consider the Hermitian
bipartite SSH model with different intra- and inter-unit cell hopping terms.
We engineer non-Hermiticity into the model by introducing a non-Abelian gauge
to the hopping potentials. A general method of engineering a non-Hermitian
gauge to the inter-unit cell hopping $t_{2}$ is schematically shown in Fig.
1(a). The Hamiltonian consists of the intra- and inter-unit cell couplings,
i.e., $H=H_{\textup{intra}}+H_{\textup{inter}}$. Each component can be
expressed as
$\displaystyle
H_{\textup{intra}}=\sum_{i}\>t_{1}\>e^{i\phi\>\sigma_{0}}\>c^{\dagger}_{A,i}c_{B,i}+t_{1}\>e^{-i\phi\>\sigma_{0}}\>c^{\dagger}_{B,i}c_{A,i},$
(1) $\displaystyle
H_{\textup{inter}}=\sum_{i}\>[\>t_{2}\>e^{i\theta_{R}\>\vec{\nu_{R}}.\vec{\sigma}}+\eta_{R}(\theta_{R})\>]\>c^{\dagger}_{B,i}c_{A,i+1}+$
$\displaystyle[\>t_{2}\>e^{i\theta_{L}\>\vec{\nu_{L}}.\vec{\sigma}}+\eta_{L}(\theta_{L})\>]\>c^{\dagger}_{A+1,i}c_{B,i}.$
Here, $L$ and $R$ denote leftward and rightward hopping terms, respectively,
and $\vec{\sigma}$ refers to the Pauli matrices (where $\sigma_{0}$ is the
identity matrix), which form the generators of the gauge transformations.
$\theta_{R}$ and $\theta_{L}$ are parameters of this invoked local
transformation. For a non-Abelian gauge transformation, the generators
($M^{i},M^{j},M^{k}$) must form a Lie group, i.e.,
$\commutator{M^{i}}{M^{j}}=C^{ijk}M^{k}$ and satisfy
$\Tr{M^{i}M^{j}}=C^{ij}\delta_{ij}$ Jacobson (1979). In our general model, the
gauge becomes non-Abelian when at least one of the following conditions is
fulfilled: (i)
$\commutator{\>\vec{\nu_{R}}.\vec{\sigma}}{\>\vec{\nu_{L}}.\vec{\sigma}\>}\neq
0$, (ii)
$\commutator{\>\vec{\nu_{R}}.\vec{\sigma}}{\>\eta_{L}(\theta_{L})\>}\neq 0$,
(iii) $\commutator{\>\eta_{R}(\theta_{R})}{\>\vec{\nu_{L}}.\vec{\sigma}\>}\neq
0$ or (iv) $\commutator{\>\eta_{R}(\theta_{R})}{\eta_{L}(\theta_{L})\>}\neq
0$. For simplicity, we choose $\vec{\nu_{L}}.\vec{\sigma}=-i\sigma_{y}$,
$\vec{\nu_{R}}.\vec{\sigma}=\sigma_{x}$,
$\eta_{L}(\theta_{L})=-\sqrt{2}t_{2}\cos({\frac{3\pi}{4}+\theta_{L}})\sigma_{0}$
and $\eta_{R}(\theta_{R})=0$. This satisfies
$\commutator{\vec{\nu_{L}}.\vec{\sigma}}{\vec{\nu_{R}}.\vec{\sigma}}\neq 0$,
following the aforementioned conditions, and conforms to an $SU(2)$ non-
Abelian gauge transformation Kasper _et al._ (2023). A schematic for our
proposed model is illustrated in Fig. 1(b). The $\sigma$ matrices associate a
synthetic spin to each lattice site modifying the SSH model to a
_quadripartite_ lattice. The Hamiltonian for the periodic system can be
reduced to the matrix form, $H(k)=\begin{pmatrix}0&M_{1}(k)\\\
M_{2}(k)&0\end{pmatrix}$, whose components are
$\begin{split}M_{1}(k)=\begin{bmatrix}t_{1}e^{i\phi}+t_{2}e^{-ik}\cos{\theta_{L}}&-it_{2}e^{-ik}\cos{\theta_{L}}\\\
it_{2}e^{-ik}\cos{\theta_{L}}&t_{1}e^{i\phi}+t_{2}e^{-ik}\cos{\theta_{L}}\\\
\end{bmatrix},\\\ \\\
M_{2}(k)=\begin{bmatrix}t_{1}e^{-i\phi}+t_{2}e^{ik}\cos{\theta_{R}}&-it_{2}e^{ik}\sin{\theta_{R}}\\\
it_{2}e^{ik}\sin{\theta_{R}}&t_{1}e^{-i\phi}+t_{2}e^{ik}\cos{\theta_{R}}\end{bmatrix}.\end{split}$
(2)
We compute the energy eigenvalues (detailed expressions for which are
presented in the supplementary information sup ), and derive the conditions
for the occurrence of EPs Heiss (2004). For simplicity, we set $\phi=0$. The
choice of our non-Abelian gauge results in $E_{1}$ and $E_{2}$ energy bands
being independent of $\theta_{L}$, which expedites our analytical
calculations. $E_{1}$ and $E_{2}$ gap closing points occur at
$\sin{\theta_{R}}-\cos{\theta_{R}}=\pm t_{1}/t_{2}$, whereas, $E_{3}$ and
$E_{4}$ gap closings occur at $\cos{\theta_{L}}=\pm t_{1}/2t_{2}$ and
$\sin{\theta_{R}}+\cos{\theta_{R}}=\pm t_{1}/t_{2}$. To obtain EPs, the
condition $\absolutevalue{t_{1}/t_{2}}\leq 2$ must be satisfied. We set
$t_{1}=t_{2}=1$ for subsequent calculations to illustrate our results. The
nature of the energy bands around different order EPs have been examined in
detail in the supplementary information sup .
Emergence of two-fold spectral topology and flat bands.$-$ The contours of EPs
reveal the phase transition lines, partitioning the phase space
$\theta_{L}-\theta_{R}$ into topologically distinct regions. In particular,
from the general conditions for obtaining EPs, we find that EP4 (fourth-order
EP) phase transitions occur for $\theta_{R}=0,\>\pm\pi/2,\>\pm\pi$ (shown by
dark blue vertical lines in Fig. 2); whereas, the EP2 (second-order EP)
transitions occur along $\theta_{L}=\pm\pi/3,\>\pm 2\pi/3,\>\pm 4\pi/3$ (shown
by red horizontal lines in Fig. 2). These exceptional contours divide the
parameter space into different phases, which can be classified uniquely
through the spectral topology and our newly defined topological invariant –
paired winding numbers.
A striking feature of our non-Hermitian gauge-engineered model is the
emergence of a two-fold spectral topology. Spectral topology refers to the
topological structure of the complex energy spectra, usually featuring
distinct point-gaps and line-gaps Kawabata _et al._ (2019a), unique for each
phase. The complex spectra are generally arc-like or loop-like, signifying the
occurrence of the non-Hermitian skin effect for the latter under open
boundaries Zhang _et al._ (2022c). In non-Hermitian models explored so far,
each phase has one distinctive spectral topology, which is contributed by all
the underlying energy bands of the system. In our model, each pair of energy
bands ($E_{1},E_{2}$) and ($E_{3},E_{4}$) individually contribute to a
different spectral topology which can be independently characterized. The
spectral topology is thus rendered two-fold, necessitating a two-fold
topological classification of the rich phase diagram obtained. The distinct
spectral topology uncovers five different topological phases (illustrated in
Fig. 2). The purple and green curves show the complex spectra contributed by
($E_{1},E_{2})$ and ($E_{3},E_{4})$, respectively. The filled circles on top
of each complex loop denote the dynamically stable chiral modes. These are the
energy modes which persist after a non-unitary long-time evolution of the
system with lifetimes $\sim\hbar/\textup{Im}(E_{\textrm{chiral}})$ Lee _et
al._ (2019). Under adiabatic deformations, owing to its topological origin,
each spectral loop survives. Consequently, the associated chiral mode also
survives, rendering them topologically robust attributes, using which we
define a topological invariant $(\nu,\gamma)$ for each pair of energy bands,
$\begin{split}\nu=\frac{1}{2}\sum_{\begin{subarray}{c}n=1,2\\\
\textup{Im}(E)>0\end{subarray}}\text{sgn}\>[\textup{Im}\>E_{n}(k_{n\alpha})]\>\text{sgn}\>[\partial_{k}\>[\textup{Re}\>E_{n}(k_{n\alpha})]],\\\
\gamma=\frac{1}{2}\sum_{\begin{subarray}{c}n=3,4\\\
\textup{Im}(E)>0\end{subarray}}\text{sgn}\>[\textup{Im}\>E_{n}(k_{n\alpha})]\>\text{sgn}\>[\partial_{k}\>[\textup{Re}\>E_{n}(k_{n\alpha})]].\end{split}$
(3)
Here ${k_{n\alpha}:\textup{Re}\>E_{n}(k_{n\alpha})=E_{b,\alpha}}$, where
$E_{b,\alpha}$ denotes the set of point gaps of each spectral loop, which
essentially locates the corresponding chiral mode. The first sign function
takes into account the sign of the imaginary energy of the chiral mode, while
the second sign function tracks the velocity of the chiral mode. The Nielsen-
Ninomiya theorem Bessho and Sato (2021) establishes that chiral modes always
come in pairs of opposite chirality. However, the effective chirality of long-
time dynamics is characterized by $\textup{Im}(E)>0$ modes Banerjee _et al._
(2022). Thereby, the paired winding numbers $(\nu,\gamma)$ defined above track
the chirality of the dynamically robust chiral modes of each spectral loop.
Each phase is uniquely classified by its characteristic paired winding number.
Additionally, for a topological transition between any two phases, phase 1:
$(\nu_{1},\gamma_{1})$ and phase 2: $(\nu_{2},\gamma_{2})$, we introduce
$\Lambda=\absolutevalue{\absolutevalue{\nu_{1}}-\absolutevalue{\nu_{2}}}+\absolutevalue{\absolutevalue{\gamma_{1}}-\absolutevalue{\gamma_{2}}}$.
If $\Lambda=1/2$, then the phase transition must cross an EP2 line, while
$\Lambda=1$ denotes a phase transition across an EP4 line. This is also true
for non-adjacent phases. For instance, the shortest path for transiting
between the blue $(1,0)$ and grey $(1/2,-1)$ phases requires crossing one EP2
as well as one EP4 line – here $\Lambda=3/2$. The distinct characteristic
features of each phase are summarized in Table 1.
Table 1: Characterization of different phases. The paired winding number, the number of dynamically stable chiral modes, the nature of line gap, and the skin effect characterize each phase uniquely. Here, all paired quantities refer to the spectral physics of $(E_{1,2},E_{3,4})$. Here $+(-)$ in the winding number refers to counter-clockwise (clockwise) spectral winding. The indicative color corresponds to the representation in the phase diagram in Fig. 2. Re/Im imply the presence of a real/imaginary line gap in the respective spectra, whereas ‘No’ refers to no line gap. The contents in the column ‘skin effect’ indicates the direction of the skin effect due to each band-pair. Paired winding no. | Indicative colour | Chiral modes | Line gap | Skin effect
---|---|---|---|---
$(1,-1)$ | Purple | (2,2) | (Re, Re) | (Left, Right)
$(\frac{1}{2},-\frac{1}{2})$ | Red | (1,1) | (No, No) | (Left, Right)
$(1,0)$ | Blue | (2,0) | (Re, Im) | (Left, Right)
$(1,\frac{1}{2})$ | Green | (2,1) | (Re, No) | (Left, Left)
$(\frac{1}{2},-1)$ | Grey | (1,2) | (No, Re) | (Left, Right)
Another remarkable consequence of our non-Abelian gauge engineering is that by
solely tuning the gauge parameter, $\theta_{R}$, we can generate two
dynamically robust flat bands at
$\theta_{R}=\pi/4,\>3\pi/4\>(\>\forall\>\theta_{L})$. $E_{1}$ and $E_{2}$
spectral loops shrink to points on the real energy axis, resulting in two flat
bands at $\textup{Re}(E)=\pm t_{1}$ and $\textup{Im}(E)=0$. These flat bands
have infinite lifetimes and are robust to non-unitary time dynamics, thereby
rendering them accessible to experiments. Later, we further elucidate the
origin of these flat bands by mapping our model to a toy model of coupled
Hatano-Nelson chains. Moreover, as a consequence of the point-like spectra
exhibited by $E_{1},E_{2}$ under the flat band condition, the complex spectral
area of the two left-chiral bands is lost. Consequently, the skin effect
vanishes at the left edge of the lattice model. Thus, exclusively under the
flat band condition, the occurrence of the skin effect is limited solely to
the right edge of the lattice. We present further details in the supplementary
information sup .
Figure 3: Spectral winding, generalized Brillouin zone, and tunable skin
effect. (a) The spectral topology in the ‘grey’ phase showing that the
spectral winding of $E_{1},E_{2}$ (in purple) is counter-clockwise while that
of $E_{3},E_{4}$ (in green) is clockwise. This results in (b) a non-Hermitian
skin effect at both edges depicted by the local density of states (LDOS). The
inset shows the GBZ (in corresponding colours) with $|\beta(E_{1},E_{2})|<1$
and $|\beta(E_{3},E_{4})|>1$. The black dashed lines mark the unit circle. In
the green phase, (c) the direction of spectral winding is counter-clockwise
for both pairs of bands, resulting in (d) skin effect only at the left edge.
Here both $|\beta(E_{1},E_{2})|,\>|\beta(E_{3},E_{4})|<1$. (e)-(f) The real
and imaginary spectra in the ‘green’ phase as a function of $k_{x}$. A base
energy $E_{0}$ intersects $\textup{Re}(E_{1})$ and $\textup{Re}(E_{3})$ at
$k_{\alpha}$ and $k_{\beta}$, respectively. The corresponding $\textup{Im}(E)$
are marked in filled (unfilled) circles for long-time surviving (evanescent)
modes. For $E_{1}$ ($E_{3}$) the mode at $k_{\alpha}$ ($k_{\beta}$) survives
owing to higher $\textup{Im}(E)$. At $k_{\alpha}$, $E_{1}$ has a left-mover
while at $k_{\beta}$, $E_{3}$ has a left-mover. Hence, skin effect in the
green phase occurs at the left edge. (g)-(h) Behaviour of the spectra and skin
effect under disorder ($\Delta=0.1$) in the grey phase, illustrating its
robustness. Here $t_{1}=t_{2}=1,\phi=0$. For grey phase:
$\theta_{R}=3\pi/4,\theta_{L}=2\pi/5$. For green phase:
$\theta_{R}=\pi/8,\theta_{L}=2\pi/5$.
Correspondence to skin effect and chiral dynamics.$-$ We now illustrate how
our paired winding numbers ($\nu,\gamma$) can predict the skin effect under
open boundary conditions. Notably, our two-fold spectral topology also renders
the resultant non-Hermitian skin effect two-fold. We find that in all phases
(except the ‘green’ phase), $\>\nu$ and $\gamma$ have opposite signs,
signifying that the winding direction and, thereby, the chirality of the
dynamically stable modes are opposite. In contrast, in the green phase with a
paired winding number (1,1/2), the chirality of the two pairs of energy bands
is in the same direction. This peculiarity is reflected in the skin effect. To
exemplify this, we contrast the bi-chiral ‘grey’ phase and the uni-chiral
‘green’ phase in Fig. 3(a)-(d). The purple curves denote the spectral topology
of energy bands ($E_{1},E_{2}$), while the green curves show ($E_{3},E_{4}$).
The opposite signs of the winding number in the grey phase [Fig. 3(a)-(b)]
manifests as a ‘bilateral’ skin effect. The counter-clockwise propagating
energy bands ($E_{1},E_{2}$) lead to skin effect at the left edge, whereas the
clockwise propagating bands ($E_{3},E_{4}$) induce skin effect at the right
edge. The generalized Brillouin zone (GBZ) denoted by $\beta$ [inset of Fig.
3(b)] shows that $\absolutevalue{\beta(E_{1},E_{2})}<1$ (in purple) whereas,
$\absolutevalue{\beta(E_{3},E_{4})}>1$ (in green) – this establishes the
directionally decoupled skin effect. In contrast, in the uni-chiral green
phase [see Fig. 3(c)-(d)], the winding direction of both pairs of energy bands
is counter-clockwise, resulting in a ‘unilateral’ skin effect at the left
edge. The GBZ shows that both $\absolutevalue{\beta(E_{1},E_{2})}<1$ and
$\absolutevalue{\beta(E_{3},E_{4})}<1$. Detailed calculations of the GBZ are
presented in the supplementary information sup .
We highlight the physical origin of the directionally-dependent skin effect in
our model in Fig. 3(e)-(f) for the green phase, which exhibits a unilateral
skin effect. The different colours correspond to the different energy bands
specified in the legend. We consider a real excitation energy $E_{0}$, which
intersects $\textup{Re}(E_{1})$ and $\textup{Re}(E_{3})$ at $k_{\alpha}$ and
$k_{\beta}$, respectively. At $k_{\alpha}$, $E_{1}$ has a left-moving mode
while $E_{3}$ has a right-mover. However, at $k_{\beta}$, $E_{1}$ has a right-
mover while $E_{2}$ has a left-mover. The net flow of eigenstates would
counterbalance in the Hermitian regime. However, the inherent non-Hermiticity
– a consequence of non-Abelian gauge choice – associates a finite lifetime
with every eigenmode, leading to a dynamically persistent current. Owing to
the non-unitary time evolution $\sim e^{-iEt}\rightarrow
e^{-i\mathrm{Re}(E)t}\>e^{\mathrm{Im}(E)t}$, the eigenmodes with negative
imaginary energy die out over a timescale $\sim\hbar/\textup{Im}(E)$. For
energy band $E_{1}$, the left-mover at $k_{\alpha}$ survives, and for $E_{3}$,
the left-mover at $k_{\beta}$ survives. The persistent modes are denoted by
filled red circles, and the corresponding evanescent modes are shown using
unfilled red circles. Thus, the cumulative flow of states in the green phase
is leftwards, leading to the unilateral skin effect at the left edge.
Our model facilitates control over the direction of the skin effect by
suitably tuning parameters $\theta_{L}$ and $\theta_{R}$. Due to the
topological origin of the skin effect and the point-gap spectra, we expect
them to be robust to moderate amounts of disorder, which would make the
observation of the bilateral skin effect experimentally feasible. To
illustrate this robustness, we introduce random on-site disorder in our model
as,
$H_{\textup{dis}}=\sum_{i}\Delta_{i}(c^{\dagger}_{A,i}c_{A,i}+c^{\dagger}_{B,i}c_{B,i})$,
where $\Delta_{i}=\Delta[-1,1]$ is chosen randomly and the disorder strength
is $\Delta$. Fig. 3(g)-(h) show the robust nature of the spectral topology and
bilateral skin effect in the exemplary grey phase at disorder strength
$\Delta=0.1\>t_{1}$, confirming our expectations.
Figure 4: Coupled Hatano-Nelson chains as toy model. (a) Two Hatano-Nelson
chains linked using coupling strength $t_{o}$. The hopping parameters are
denoted below the schematic, mapping it to our non-Abelian SSH model. (b)
Dynamically robust ($\textup{Im}(E)=0$) flat bands at $\textup{Re}(E)=\pm
t_{o}$, in the strong coupling limit ($t_{o}/t_{1}=0.01$). (c) Bilateral skin
effect and (d) unilateral skin effect attained by tuning parameters
$\theta_{L}$ and $\theta_{R}$ in the weak coupling limit ($t_{1}/t_{o}=0.1$).
Here $t_{2}=1,\phi=0$ for all plots. We set (b) $\theta_{R,L}=\pi/4,3\pi/4$ ,
(c) $\theta_{R,L}=\pi/2,\pi/2$ , and (d) $\theta_{R,L}=-\pi/5,3\pi/4$.
Mapping to a toy model: Coupled Hatano-Nelson chains.$-$ To elucidate the
controllable nature of our proposed direction-selective skin effect, we devise
a simple toy model. We consider coupled chains of the Hatano-Nelson model,
which is the simplest model with rich non-Hermitian features Hatano and Nelson
(1996). As illustrated in Fig. 4, we couple two Hatano-Nelson chains with non-
reciprocal hopping, with coupling strength $t_{o}$. The hopping parameters are
set as represented in the schematic, in order to map it to our original non-
Abelian SSH model. In the weak coupling limit, i.e., $t_{o}/t_{1}<<1$, we
discover that by tuning parameters $\theta_{L}$ and $\theta_{R}$, we can
control the direction of the skin effect. Remarkably, we find that it is
possible to obtain the bilateral skin effect [Fig. 4(c)], as well as the
unilateral skin effect [Fig. 4(d)], even for fixed values of $t_{1}$ and
$t_{2}$. Furthermore, in the strong coupling limit, i.e., $t_{o}/t_{1}>>1$, we
can tune the free parameters such that $\theta_{R}=\pi/4$ and
$\theta_{L}=3\pi/4$, leading to the emergence of two non-dispersive flat bands
at $\textup{Re}(E)=\pm t_{o}$. Similar to our original model, these flat bands
are also dynamically robust with $\textup{Im}(E)=0$ [shown in Fig. 4(b)]. In
the strong coupling limit, there is a dissociation process, resulting in unit
cells that consist of one atom each from the upper and lower chains. In this
regime, the wavefunction within each unit cell becomes localized – this leads
to the emergence of a flat band, with the wavefunction pinned to its
respective unit cell.
Conclusions and outlook.$-$ In summary, we proposed non-Abelian gauge
engineering to realize multi-fold spectral topology and demonstrated that by
utilizing a non-Abelian $SU(2)$ gauge, a two-band model can exhibit a two-fold
spectral topology which in turn leads to a multi-chiral non-Hermitian skin
effect. We introduced a paired winding number which characterizes the distinct
spectral topology of each phase and provides insights into the nature of phase
transitions through EPs, while also capturing the chiral dynamics which
determines the direction of skin effect. We note that applying a generalized
$SU(N)$ gauge can ramify the spectral topology into an $N$-fold structure,
which may have intriguing consequences for skin effects in higher dimensions.
This multi-fold spectral topology offers the possibility of directionally-
controllable skin effects and mode selection. Importantly, our proposed models
and their associated effects are experimentally accessible, providing a
promising avenue for future studies and potential applications. As an
illustration of the experimental feasibility, we have developed a topoelectric
circuit that can faithfully emulate the underlying non-Abelian SSH
Hamiltonian. This circuit, presented in the supplementary information sup ,
provides a promising platform for realizing our predicted effects. We hope
that our proposal motivates further synergy between non-Abelian and non-
Hermitian physics.
Acknowledgments.$-$ R.S. and A.B. are supported by the Prime Minister’s
Research Fellowship (PMRF). A.N. acknowledges support from the Indian
Institute of Science.
## References
* Yang and Mills (1954) C.-N. Yang and R. L. Mills, Physical review 96, 191 (1954).
* Nayak _et al._ (2008) C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. D. Sarma, Reviews of Modern Physics 80, 1083 (2008).
* Yang _et al._ (2023) Y. Yang, B. Yang, G. Ma, J. Li, S. Zhang, and C. Chan, arXiv preprint arXiv:2305.12206 (2023).
* Sugawa _et al._ (2018) S. Sugawa, F. Salces-Carcoba, A. R. Perry, Y. Yue, and I. Spielman, Science 360, 1429 (2018).
* Guo _et al._ (2021) Q. Guo, T. Jiang, R.-Y. Zhang, L. Zhang, Z.-Q. Zhang, B. Yang, S. Zhang, and C. T. Chan, Nature 594, 195 (2021).
* Yang _et al._ (2019) Y. Yang, C. Peng, D. Zhu, H. Buljan, J. D. Joannopoulos, B. Zhen, and M. Soljačić, Science 365, 1021 (2019).
* Brosco _et al._ (2021) V. Brosco, L. Pilozzi, R. Fazio, and C. Conti, Physical Review A 103, 063518 (2021).
* Zhou (2023) L. Zhou, arXiv preprint arXiv:2302.05710 (2023).
* Pang _et al._ (2023) Z. Pang, J. Hu, and Y. Yang, arXiv preprint arXiv:2304.01876 (2023).
* Wu _et al._ (2019) Q. Wu, A. A. Soluyanov, and T. Bzdušek, Science 365, 1273 (2019).
* Jiang _et al._ (2021) T. Jiang, Q. Guo, R.-Y. Zhang, Z.-Q. Zhang, B. Yang, and C. T. Chan, Nature Communications 12, 6471 (2021).
* Ivanov (2001) D. A. Ivanov, Physical review letters 86, 268 (2001).
* Sato and Fujimoto (2009) M. Sato and S. Fujimoto, Physical Review B 79, 094504 (2009).
* Noh _et al._ (2020) J. Noh, T. Schuster, T. Iadecola, S. Huang, M. Wang, K. P. Chen, C. Chamon, and M. C. Rechtsman, Nature Physics 16, 989 (2020).
* Chen _et al._ (2022) Z.-G. Chen, R.-Y. Zhang, C. T. Chan, and G. Ma, Nature Physics 18, 179 (2022).
* Zhang _et al._ (2022a) X.-L. Zhang, F. Yu, Z.-G. Chen, Z.-N. Tian, Q.-D. Chen, H.-B. Sun, and G. Ma, Nature Photonics 16, 390 (2022a).
* Bergholtz _et al._ (2021) E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Reviews of Modern Physics 93, 015005 (2021).
* Ashida _et al._ (2020) Y. Ashida, Z. Gong, and M. Ueda, Advances in Physics 69, 249 (2020).
* Ding _et al._ (2022) K. Ding, C. Fang, and G. Ma, Nature Reviews Physics 4, 745 (2022).
* Banerjee _et al._ (2023a) A. Banerjee, R. Sarkar, S. Dey, and A. Narayan, Journal of Physics: Condensed Matter (2023a).
* Okuma and Sato (2023) N. Okuma and M. Sato, Annual Review of Condensed Matter Physics 14, 83 (2023).
* Yao and Wang (2018) S. Yao and Z. Wang, Physical review letters 121, 086803 (2018).
* Alvarez _et al._ (2018) V. M. Alvarez, J. B. Vargas, and L. F. Torres, Physical Review B 97, 121401 (2018).
* Zhang _et al._ (2022b) X. Zhang, T. Zhang, M.-H. Lu, and Y.-F. Chen, Advances in Physics: X 7, 2109431 (2022b).
* Lin _et al._ (2023) R. Lin, T. Tai, L. Li, and C. H. Lee, Frontiers of Physics 18, 53605 (2023).
* Hu and Zhao (2021) H. Hu and E. Zhao, Physical Review Letters 126, 010401 (2021).
* Jaiswal _et al._ (2023) R. Jaiswal, A. Banerjee, and A. Narayan, New Journal of Physics 25, 033014 (2023).
* Wang _et al._ (2022) H.-Y. Wang, F. Song, and Z. Wang, arXiv preprint arXiv:2212.11743 (2022).
* Banerjee _et al._ (2023b) A. Banerjee, R. Jaiswal, M. Manjunath, and A. Narayan, Proceedings of the National Academy of Sciences 120, e2302572120 (2023b).
* Hu (2023) H. Hu, arXiv preprint arXiv:2306.12022 (2023).
* Kawabata _et al._ (2019a) K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Physical Review X 9, 041015 (2019a).
* Lee _et al._ (2019) J. Y. Lee, J. Ahn, H. Zhou, and A. Vishwanath, Physical review letters 123, 206404 (2019).
* Heiss (2004) W. Heiss, Journal of Physics A: Mathematical and General 37, 2455 (2004).
* Ding _et al._ (2016) K. Ding, G. Ma, M. Xiao, Z. Zhang, and C. T. Chan, Physical Review X 6, 021007 (2016).
* Miri and Alu (2019) M.-A. Miri and A. Alu, Science 363, eaar7709 (2019).
* Chen _et al._ (2020) H.-Z. Chen, T. Liu, H.-Y. Luan, R.-J. Liu, X.-Y. Wang, X.-F. Zhu, Y.-B. Li, Z.-M. Gu, S.-J. Liang, H. Gao, _et al._ , Nature Physics 16, 571 (2020).
* Kawabata _et al._ (2019b) K. Kawabata, T. Bessho, and M. Sato, Physical review letters 123, 066405 (2019b).
* Lee (2016) T. E. Lee, Physical review letters 116, 133903 (2016).
* Shen _et al._ (2018) H. Shen, B. Zhen, and L. Fu, Physical review letters 120, 146402 (2018).
* Li _et al._ (2020) L. Li, C. H. Lee, S. Mu, and J. Gong, Nature communications 11, 5491 (2020).
* Liang _et al._ (2022) Q. Liang, D. Xie, Z. Dong, H. Li, H. Li, B. Gadway, W. Yi, and B. Yan, Physical review letters 129, 070401 (2022).
* Kawabata _et al._ (2022) K. Kawabata, K. Shiozaki, and S. Ryu, Physical Review B 105, 165137 (2022).
* Longhi (2019) S. Longhi, Physical Review Research 1, 023013 (2019).
* Kunst _et al._ (2018) F. K. Kunst, E. Edvardsson, J. C. Budich, and E. J. Bergholtz, Physical review letters 121, 026808 (2018).
* Weidemann _et al._ (2020) S. Weidemann, M. Kremer, T. Helbig, T. Hofmann, A. Stegmaier, M. Greiter, R. Thomale, and A. Szameit, Science 368, 311 (2020).
* Yokomizo and Murakami (2019) K. Yokomizo and S. Murakami, Physical review letters 123, 066404 (2019).
* Song _et al._ (2019) F. Song, S. Yao, and Z. Wang, Physical review letters 123, 170401 (2019).
* Xue _et al._ (2022) W.-T. Xue, Y.-M. Hu, F. Song, and Z. Wang, Physical Review Letters 128, 120401 (2022).
* Dembowski _et al._ (2004) C. Dembowski, B. Dietz, H.-D. Gräf, H. Harney, A. Heine, W. Heiss, and A. Richter, Physical Review E 69, 056216 (2004).
* Doppler _et al._ (2016) J. Doppler, A. A. Mailybaev, J. Böhm, U. Kuhl, A. Girschik, F. Libisch, T. J. Milburn, P. Rabl, N. Moiseyev, and S. Rotter, Nature 537, 76 (2016).
* Zhou _et al._ (2018) H. Zhou, C. Peng, Y. Yoon, C. W. Hsu, K. A. Nelson, L. Fu, J. D. Joannopoulos, M. Soljačić, and B. Zhen, Science 359, 1009 (2018).
* Wiersig (2014) J. Wiersig, Physical review letters 112, 203901 (2014).
* Özdemir _et al._ (2019) Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Nature materials 18, 783 (2019).
* Chen _et al._ (2017) W. Chen, Ş. Kaya Özdemir, G. Zhao, J. Wiersig, and L. Yang, Nature 548, 192 (2017).
* Su _et al._ (1979) W.-P. Su, J. R. Schrieffer, and A. J. Heeger, Physical review letters 42, 1698 (1979).
* Asbóth _et al._ (2016) J. K. Asbóth, L. Oroszlány, A. Pályi, J. K. Asbóth, L. Oroszlány, and A. Pályi, A Short Course on Topological Insulators: Band Structure and Edge States in One and Two Dimensions , 1 (2016).
* Jacobson (1979) N. Jacobson, _Lie algebras_ , 10 (Courier Corporation, 1979).
* Kasper _et al._ (2023) V. Kasper, T. V. Zache, F. Jendrzejewski, M. Lewenstein, and E. Zohar, Physical Review D 107, 014506 (2023).
* (59) “See supplementary information for energy eigenvalue expressions, discussion of band dispersions, generalized brillouin zone calculations, detailed characterization of other phases, and details of the topoelectric circuit realization.” .
* Zhang _et al._ (2022c) K. Zhang, Z. Yang, and C. Fang, Nature communications 13, 2496 (2022c).
* Bessho and Sato (2021) T. Bessho and M. Sato, Physical Review Letters 127, 196404 (2021).
* Banerjee _et al._ (2022) A. Banerjee, S. S. Hegde, A. Agarwala, and A. Narayan, Physical Review B 105, 205403 (2022).
* Hatano and Nelson (1996) N. Hatano and D. R. Nelson, Physical review letters 77, 570 (1996).
|
# Optical fields spanning the 4D space of nonparaxial polarization
David Marco 1, 2 and Miguel A. Alonso 1, 3, 4, 5
1 Aix Marseille Univ., CNRS, Centrale Marseille, Institut Fresnel, UMR 7249,
13397 Marseille Cedex 20, France
2 Instituto de Bioingeniería, Universidad Miguel Hernández de Elche, 03202
Elche, Spain
3 The Institute of Optics, University of Rochester, Rochester, NY14627, USA
4 Center for Coherence and Quantum Optics, University of Rochester, Rochester,
NY14627, USA
5 Laboratory for Laser Energetics, University of Rochester, Rochester,
NY14627, USA
E-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
We present simple mathematical solutions for optical fields that span all
nonparaxial polarization ellipses. These fields, given by a superposition of
five plane waves with non-coplanar polarizations, are experimentally
realizable. By slowly varying in time the relative amplitude between two
groups of plane waves, the resulting fields take the form of a periodic
polarization lattice, where all possible nonparaxial polarization ellipses are
localized in a 4D spatio-temporal cell. The field distributions in these cells
can be regarded as 4D skyrmionic textures.
## 1 Introduction
For a monochromatic optical field, the electric field vector at any given
point traces in time an ellipse [1]. The parameters that describe the shape
and orientation of this ellipse (other than its size, which is related to the
field’s intensity) are what we call polarization. For a monochromatic paraxial
optical field, the ellipse is essentially constrained to the plane
perpendicular to the main direction of propagation, so only two parameters are
required to describe polarization: the ellipticity and the orientation angle
within the plane. These two quantities are well represented by the two
coordinates over the surface of a unit sphere known as the Poincaré sphere.
Polarization can change from point to point across (or along) a beam. Since
the Poincaré sphere is a 2D abstract space, it is possible for a beam to
contain many or even all polarization states at a given physical plane
transverse to its main direction of propagation. Simple solutions such as the
so-called full Poincaré beams [2] contain all polarization states at each
transverse plane. For these beams, the Poincaré sphere is mapped onto the
plane according to a stereographic projection. Since the sphere is covered
entirely without any singularities of the mapping (except at infinity), this
field can be regarded as a 2D skyrmionic structure. Skyrmions correspond to
topological structures in which a closed abstract parameter space (the
Poincaré sphere in this case) is mapped completely and monotonically onto the
flat physical space. They have been studied in several areas of physics,
especially in magnetic systems [3].
For monochromatic nonparaxial fields, the plane containing the polarization
ellipse at any given point is not fixed by the main direction of propagation.
Four parameters are then required to fully describe polarization, such as the
ellipticity and the three Euler angles determining the orientation of the
ellipse. These parameters define then a 4D space of nonparaxial polarization,
for which several geometric representations have been proposed in terms of two
points over the surface of a unit sphere [4, 5, 6]. For a nonparaxial field,
polarization typically changes from point to point. However, the 3D physical
space is not sufficient to fully explore the 4D polarization space, unless it
is supplemented with a four dimension, such as time.
Here we present what we refer to for brevity as full 3D polarization (F3DP)
fields, which are nonparaxial optical fields containing all possible
polarization ellipses. We describe a family of experimentally realizable
mathematical solutions of such fields in the form of optical lattices whose
polarization state changes not only in 3D space but also adiabatically in
time, so that all possible nonparaxial polarization ellipses exist inside a
cuboid in 3D space within a given temporal interval. The fields are
constructed as superpositions of five noncoplanar plane waves. The adiabatic
temporal variation of these quasimonochromatic fields is implemented by slowly
varying the relative amplitude between two sets of plane waves. Some of the
solutions present uniform intensity in space and time, and they all present a
rich polarization structure. These field can be experimentally implemented in
the focal region of a high numerical aperture lens.
Moreover, we identified 4D skyrmionic structures in these F3DP fields. Most
skyrmionic structures studied so far in vector waves are 2D skyrmions (also
known as baby skyrmions), which fully span at a plane or a section of a plane
a spherical space corresponding to an aspect of the polarization ellipse, such
as: the normalized Stokes vector for paraxial beams [2, 7, 8]; the direction
of the field vector at a fixed time for the electromagnetic evanescent waves
of surface plasmon polaritons [9], evanescent acoustic waves [10] and
traveling acoustic waves [11]; the direction of the spin vector (normal to the
ellipse) in evanescent optical fields [12] and at the focal plane of tightly
focused optical fields [13]; and the direction of the ellipse major axis for
tightly focused optical fields [13]. Recently, a 3D skyrmionic Hopfion was
found in the propagating volume of a paraxial field [14], which spans a
3-sphere where every 2D polarization state with any global phase is
represented. To study the 4D skyrmionic structures within F3DP fields, we
define a Skyrme density as the Jacobian of the mapping between the coordinates
over the unit sphere of two points representing nonparaxial polarization
states and the flat space-time. The integral of this density over a skyrmionic
spatiotemporal cell is shown to give a Skyrme number of unit size.
## 2 Constructing the optical field
In this section, we propose a temporally-variant quasi-monochromatic field
that contains all possible states of full polarization within a spatio-
temporal cell with spatial dimensions of the order of a wavelength. This field
consists of a simple combination of five plane waves. By changing the angles
between these plane waves the field can be made more directional, and the cell
containing all polarization states can be made larger.
### 2.1 The construction and laboratory reference frames
At any given point the complex electric vector $\mathbf{E}$ of a nonparaxial
(quasi)monochromatic field can be expressed as
$\mathbf{E}=A\exp(i\Phi)\left[\sin{\gamma_{1}}\left(\mathbf{x}\,\cos{\gamma_{2}}e^{-i\delta_{1}}+\mathbf{z}\,\sin{\gamma_{2}}e^{-i\delta_{2}}\right)+\mathbf{y}\,\cos{\gamma_{1}}\right],$
(1)
where $\mathbf{x}$, $\mathbf{y}$ and $\mathbf{z}$ are unit vectors in the $x$,
$y$ and $z$ directions, respectively, and the six real parameters
$A,\Phi,\gamma_{1},\gamma_{2},\delta_{1},\delta_{2}$ are functions of
position, and even time as long as their variation is negligible over the
scale of an optical cycle. The real electric field is obtained by taking the
real part of ${\bf E}\exp{(-i\omega t)}$, where $\omega$ is the carrier
temporal angular frequency of oscillation. Note that only four parameters are
required to describe a 3D polarization state, since the global amplitude $A$
and the global phase $\Phi$ do not affect the shape and orientation of the
ellipse traced by the electric field over a temporal period. In order for the
field to span all polarization states, these four parameters must span a range
of values, for example: $0\leq\gamma_{1},\gamma_{2}\leq\pi/2$, which account
for all relative amplitudes between the three field components, and
$0\leq\delta_{1},\delta_{2}<2\pi$, which account for all relative phases.
Therefore, in order to design a field that covers all states of polarization,
we must map the complete abstract four-parameter space
$(\gamma_{1},\gamma_{2},\delta_{1},\delta_{2})$ onto the physical space-time
$(x,y,z,t)$, with the constraint that the field must be a solution of
Maxwell’s equations in free space. Under the approximation that the temporal
dependence of the polarization parameters is much slower than the scale set by
$\omega$, these conditions reduce to requiring the form in Eq. (1) to satisfy
the Helmholtz equation and to have zero divergence.
It turns out that a simple solution can be found by making two assumptions: i)
the global amplitude $A$ is constant and hence can be set to unity, and ii)
each of the four polarization parameters depends linearly on one of the
spatio-temporal coordinates. The following combination, for example, achieves
the desired goals: $\gamma_{2}\rightarrow ky/\sqrt{2}$, $\delta_{1}\rightarrow
kx/\sqrt{2}$, $\delta_{2}\rightarrow kz/\sqrt{2}$, where
$k=\omega/c=2\pi/\lambda$ is the wavenumber with $\lambda$ being the
wavelength, and $\gamma_{1}\rightarrow\Omega t$, where $\Omega\ll\omega$. By
also letting $\Phi=k(x+z)/\sqrt{2}$ we arrive at the normalized field
$\displaystyle\mathbf{E}$ $\displaystyle=\left[\sin{\Omega
t}\left(\mathbf{x}\,e^{-i\frac{kx}{\sqrt{2}}}\cos{\frac{ky}{\sqrt{2}}}+\mathbf{z}\,e^{-i\frac{kz}{\sqrt{2}}}\sin{\frac{ky}{\sqrt{2}}}\right)+\mathbf{y}\,\cos{\Omega
t}\right]e^{ik\frac{x+z}{\sqrt{2}}}$
$\displaystyle=\mathbf{x}\,\frac{\sin{\Omega
t}}{2}\left(e^{i\mathbf{k}_{\mathbf{x}\mathrm{A}}\cdot{\bf
r}}+e^{i\mathbf{k}_{\mathbf{x}\mathrm{B}}\cdot{\bf
r}}\right)+\mathbf{z}\,\frac{\sin{\Omega
t}}{2i}\left(e^{i\mathbf{k}_{\mathbf{z}\mathrm{A}}\cdot{\bf
r}}-e^{i\mathbf{k}_{\mathbf{z}\mathrm{B}}\cdot{\bf
r}}\right)+\mathbf{y}\cos{\Omega t}\,e^{i\mathbf{k}_{\mathbf{y}}\cdot{\bf
r}},$ (2)
where in the second step the $y$-dependent trigonometric functions were
separated into complex exponentials to show that this field is a superposition
of five plane waves. The amplitudes of these plane waves have a slow temporal
dependence that adiabatically changes the relative weight of the first four
with respect to the fifth, as depicted in Fig. 1(a). The wavevectors of these
five plane waves are
$\displaystyle\mathbf{k}_{\mathbf{x}\mathrm{A}}$
$\displaystyle=k\left(0,1,1\right)/\sqrt{2},$ (3a)
$\displaystyle\mathbf{k}_{\mathbf{x}\mathrm{B}}$
$\displaystyle=k\left(0,-1,1\right)/\sqrt{2},$ (3b)
$\displaystyle\mathbf{k}_{\mathbf{z}\mathrm{A}}$
$\displaystyle=k\left(1,1,0\right)/\sqrt{2},$ (3c)
$\displaystyle\mathbf{k}_{\mathbf{z}\mathrm{B}}$
$\displaystyle=k\left(1,-1,0\right)/\sqrt{2},$ (3d)
$\displaystyle\mathbf{k}_{\mathbf{y}}$
$\displaystyle=k\left(1,0,1\right)/\sqrt{2},$ (3e)
each being orthogonal to the polarization of the corresponding plane wave
(indicated in the subindex). All nonparaxial polarization ellipses then exist
inside the cuboid defined by $x,z\in[0,\sqrt{2}\lambda)$ and
$y\in[0,\sqrt{2}\lambda/4]$ within a time interval $\Omega t\in[0,\pi/2]$,
referred to here as a full 3D polarization (F3DP) cell. This spatio-temporal
cell is depicted in Fig. 1(b). As mentioned earlier, while all polarization
states are explored by this field, its intensity $|{\bf E}|^{2}$ is constant
in space and time.
Figure 1: (a,c) Wavevectors and their polarization for five plane waves that
compose a nonparaxial optical field containing all possible polarization
states, in (a) the construction frame and (c) the laboratory frame. The field
is constructed by adiabatically harmonically varying in time the relative
amplitude between two sets of plane waves (green, and red and blue). The
temporal evolution of the normalized intensity of each group of plane waves is
shown. (b,d) Spatial cell where every possible nonparaxial polarization
ellipse is contained within a temporal interval (F3DP cell) $\Omega
t\in[0,\pi/2]$. The cells are aligned with the reference frame coordinates in
the (b) construction and (d) laboratory frames.
The reference frame just used (termed here the construction frame) is
convenient for understanding why the field spans all possible polarization
states. However, we now change to a reference frame that is better suited for
visualizing the polarization structure and for describing its experimental
generation. We call this new frame the laboratory frame, for which the $z$
direction is chosen to coincide with the wavevector ${\bf k}_{\bf y}$, which
bisects the other four wavevectors and can therefore be regarded as the main
propagation direction. The laboratory frame results from a rotation by
$-\pi/4$ radians around the $y$ axis of the solution in the construction
frame, as shown in Fig. 1(c). Following this rotation both for the spatial
coordinates and the polarization components, we arrive at the expression
$\displaystyle\mathbf{E}$ $\displaystyle=\left[e^{-ik(z+x)/2}\sin{\Omega
t}\left(\mathbf{p}\cos{\frac{ky}{\sqrt{2}}}+\mathbf{m}\,e^{ikx}\sin{\frac{ky}{\sqrt{2}}}\right)+\mathbf{y}\cos{\Omega
t}\right]e^{ikz}$ $\displaystyle=\mathbf{p}\,\frac{\sin{\Omega
t}}{2}\left(e^{i\mathbf{k}_{\mathbf{p}\mathrm{A}}\cdot{\bf
r}}+e^{i\mathbf{k}_{\mathbf{p}\mathrm{B}}\cdot{\bf
r}}\right)+\mathbf{m}\,\frac{\sin{\Omega
t}}{2i}\left(e^{i\mathbf{k}_{\mathbf{m}\mathrm{A}}\cdot{\bf
r}}-e^{i\mathbf{k}_{\mathbf{m}\mathrm{B}}\cdot{\bf
r}}\right)+\mathbf{y}\cos{\Omega t}\,e^{i\mathbf{k}_{\mathbf{y}}\cdot{\bf
r}},$ (4)
where the polarization components $\mathbf{x}$ and $\mathbf{z}$ turned into
the components $\mathbf{p}$ and $\mathbf{m}$, respectively, defined as
$\mathbf{p},\mathbf{m}=\frac{\pm\mathbf{x}+\mathbf{z}}{\sqrt{2}},$ (5)
and the corresponding wavevectors are given by
$\displaystyle\mathbf{k}_{\mathbf{p}\mathrm{A}}$
$\displaystyle=k\left(-1,\sqrt{2},1\right)/2,$ (6a)
$\displaystyle\mathbf{k}_{\mathbf{p}\mathrm{B}}$
$\displaystyle=k\left(-1,-\sqrt{2},1\right)/2,$ (6b)
$\displaystyle\mathbf{k}_{\mathbf{m}\mathrm{A}}$
$\displaystyle=k\left(1,\sqrt{2},1\right)/2,$ (6c)
$\displaystyle\mathbf{k}_{\mathbf{m}\mathrm{B}}$
$\displaystyle=k\left(1,-\sqrt{2},1\right)/2,$ (6d)
$\displaystyle\mathbf{k}_{\mathbf{y}}$ $\displaystyle=k\left(0,0,1\right).$
(6e)
Note that the F3DP cell in Fig. 1(b) now has sides that in the laboratory
frame are at $\pm\pi/4$ with respect to the $x$ and $z$ axes. We then choose
new F3DP cells whose boundaries are aligned with the coordinates of the
laboratory frame, by considering the range that the coordinates must sweep to
span all possible relative amplitudes and phases between the coefficients of
$\mathbf{p}$, $\mathbf{m}$ and $\mathbf{y}$. One such cell is given by
$x\in[-\lambda/2,\lambda/2)$, $y\in[0,\sqrt{2}\lambda/4]$, and
$z\in[0,2\lambda)$, for the same time interval as before. The spatial volume
of this cell is the same as for the previous one: $\lambda^{3}/\sqrt{2}$. This
cell is depicted in Fig. 1(d).
### 2.2 Increasing the size of the F3DP cell
The spatial dimensions of this F3DP cell can be increased by making the field
more directional, that is, by bringing the wavevectors of the four plane waves
with polarization states $\mathbf{p}$ and $\mathbf{m}$ (red and blue in Fig.
1(c)) closer to the main propagation direction ($z$). It is useful to regard
this transformation as a composition of the two transformations depicted in
Fig. 2(a): First, a decrease of the angle $2\alpha$ subtended by each pair of
wavevectors with the same polarization (blue-blue and red-red pairs in Fig.
2(a)). Second, a decrease in the angle $\beta$ between the projection onto the
$xz$ plane of each of these four wavevectors and the $z$ axis (so that
$2\beta$ is the angle between the projections of the blue-red pairs in Fig.
2(a)). The wavevectors then take the form
$\displaystyle\mathbf{k}_{\mathbf{p}\mathrm{A}}$
$\displaystyle=k\left(-\cos\alpha\sin\beta,\sin\alpha,\cos\alpha\cos\beta\right),$
(7a) $\displaystyle\mathbf{k}_{\mathbf{p}\mathrm{B}}$
$\displaystyle=k\left(-\cos\alpha\sin\beta,-\sin\alpha,\cos\alpha\cos\beta\right),$
(7b) $\displaystyle\mathbf{k}_{\mathbf{m}\mathrm{A}}$
$\displaystyle=k\left(\cos\alpha\sin\beta,\sin\alpha,\cos\alpha\cos\beta\right),$
(7c) $\displaystyle\mathbf{k}_{\mathbf{m}\mathrm{B}}$
$\displaystyle=k\left(\cos\alpha\sin\beta,-\sin\alpha,\cos\alpha\cos\beta\right),$
(7d) $\displaystyle\mathbf{k}_{\mathbf{y}}$
$\displaystyle=k\left(0,0,1\right),$ (7e)
where the particular case in Eq. (6) corresponds to $\alpha=\beta=\pi/4$. Note
that if $\beta=\pi/4$ these wavevectors remain perpendicular to the
corresponding polarization vectors (${\bf p}$, ${\bf m}$, or ${\bf y}$)
regardless of the value of $\alpha$, and in fact the intensity of the field
remains constant. However, changing to $\beta\neq\pi/4$ does require the
rotation of the polarization vectors for the first four plane waves according
to
$\mathbf{p},\mathbf{m}=\pm\mathbf{x}\cos{\beta}+\mathbf{z}\sin{\beta},$ (8)
in order to ensure plane-wave transversality. Since for $\beta\neq\pi/4$ these
two polarization components are not mutually orthogonal, the intensity of the
field is no longer spatio-temporally uniform. The field can now be written as
$\displaystyle\mathbf{E}$
$\displaystyle=\left[e^{-i(\kappa_{z}z+\kappa_{x}x)}\sin{\Omega
t}\left(\mathbf{p}\cos{\kappa_{y}y}+\mathbf{m}\,e^{i2\kappa_{x}x}\sin{\kappa_{y}y}\right)+\mathbf{y}\cos{\Omega
t}\right]e^{ikz}$ $\displaystyle=\mathbf{p}\,\frac{\sin{\Omega
t}}{2}\left(e^{i\mathbf{k}_{\mathbf{p}\mathrm{A}}\cdot{\bf
r}}+e^{i\mathbf{k}_{\mathbf{p}\mathrm{B}}\cdot{\bf
r}}\right)+\mathbf{m}\,\frac{\sin{\Omega
t}}{2i}\left(e^{i\mathbf{k}_{\mathbf{m}\mathrm{A}}\cdot{\bf
r}}-e^{i\mathbf{k}_{\mathbf{m}\mathrm{B}}\cdot{\bf
r}}\right)+\mathbf{y}\cos{\Omega t}\,e^{i\mathbf{k}_{\mathbf{y}}\cdot{\bf
r}},$ (9)
where the spatial frequencies between contributions are given by
$\displaystyle\kappa_{x}$ $\displaystyle=k\cos{\alpha}\sin{\beta},$ (10a)
$\displaystyle\kappa_{y}$ $\displaystyle=k\sin{\alpha},$ (10b)
$\displaystyle\kappa_{z}$ $\displaystyle=k(1-\cos{\alpha}\cos{\beta}).$ (10c)
The dimensions of the F3DP cell are inversely proportional to these spatial
frequencies, as shown in Fig. 2(b), and can be made in principle arbitrarily
large by making the angles $\alpha$ and $\beta$ smaller. Note that for
$\beta\neq\pi/4$ the polarization vectors $\mathbf{p}$, $\mathbf{m}$ and
$\mathbf{y}$ no longer define an orthogonal set, but they are still linearly
independent, and all possible relative amplitudes and phases between them are
swept in the cell, giving rise to all polarization states. However, for small
$\beta$ some polarization states are present only in regions of relatively low
intensity, and their coverage becomes less uniform.
Figure 2: (a) The dimensions of the F3DP cell can be controlled by changing
the angles $\alpha$ and $\beta$ for the wavevectors and the corresponding
polarization vectors. The angle between two wavevectors with the same
polarization state (blue-blue or red-red pairs) is $2\alpha$, and the angle
between the projections onto the $xz$ plane of these two pairs of wavevectors
(blue-red pairs) is $2\beta$. (b) The size of the F3DP cell increases as
$\alpha$ and $\beta$ decrease: the smaller cell in the figure corresponds to
$\alpha=\beta=\pi/4$, while the larger one (whose dimensions in $x$ and $y$
are twice those of the smaller one) corresponds to
$\alpha=\tilde{\alpha}=\arcsin[1/(2\sqrt{2})]=0.3614$ and
$\beta=\tilde{\beta}=\arcsin(1/\sqrt{14})=0.2706$.
This field can be implemented experimentally by focusing with a microscope
objective the light emerging from five points at the back focal plane, where
the polarization and phase of each is appropriately prepared. The separation
of the points sets the value of the angles $\alpha$ and $\beta$. The minimum
numerical aperture (in air) required to produce this field is
$(1-\cos^{2}\alpha\cos^{2}\beta)^{1/2}$.
## 3 The polarization structure of the field
We now describe the polarization structure of the field. We start by
describing the global field distribution by finding its spatial unit cell. We
then study the spatio-temporal structure of polarization singularities inside
a F3DP cell.
### 3.1 The field’s global structure
Figure 3(a) depicts a unit cell of the field, which consists of two stacked
F3DP cells (red and green cuboids). The cell dimensions are expressed in units
of $\kappa_{j}j$ ($j=x$, $y$, $z$), so the representation is valid for any
value of $\alpha$ and $\beta$. The cell A (red) is the F3DP cell described in
the previous section, which can be written as $\kappa_{x}x\in[-\pi/2,\pi/2)$,
$\kappa_{y}y\in[0,\pi/2]$ and $\kappa_{z}z\in[0,2\pi)$. The cell B (green) is
an F3DP cell displaced by $\pi/2$ in each dimensionless coordinate
$\kappa_{j}j$ with respect to A. It turns out that the polarization
distributions in A and B are identical except for a mirroring in $y$, as one
can see from applying a shift of $\pi/2$ to $\kappa_{x}x$, $\kappa_{y}y$ and
$\kappa_{z}z$ in Eq. (9). The entire field can then be constructed by stacking
these unit cells, as shown in Figs. 3(b,c). The positions where the unit cells
must be stacked are deduced from the changes in $\kappa_{j}j$ that leave the
field in Eq. (9) invariant: i) a displacement of $n\pi$ ($n\in\mathbb{Z}$) in
$\kappa_{x}x$ and $\kappa_{z}z$ (which results in the brick structure in Fig.
3(b) for each type of cell), and ii) a displacement of $n\pi$ in $\kappa_{y}y$
and a shift of $n\pi$ in $\kappa_{x}x$ or $\kappa_{z}z$ (depicted in Fig.
3(c)).
Figure 3: (a) The field unit cell consists of two F3DP cells (red and green
boxes) of dimensions $\Delta x=\pi/\kappa_{x}=\lambda/(2\cos\alpha\sin\beta)$,
$\Delta y=\pi/(2\kappa_{y})=\lambda/(4\sin\alpha)$ and $\Delta
z=2\pi/\kappa_{z}=\lambda/(1-\cos\alpha\cos\beta)$. The cell dimensions are
expressed in units of $\kappa_{j}j$ ($j=x$, $y$, $z$). Projections onto the
$xy$, $xz$ and $yz$ planes are shown in order to aid with 3D visualization.
The (green) cell B is displaced by $\pi/2$ rad in $\kappa_{x}x$, $\kappa_{y}y$
and $\kappa_{z}z$ with respect to the (red) cell A, and its polarization state
distribution is the same as in the red cell but mirrored in $y$. The global
field structure is reproduced by stacking unit cells following two rules:
stacking unit cells displaced (b) by $n\pi$ (where $n\in\mathbb{Z}$) in
$\kappa_{x}x$ and $\kappa_{z}z$ (brick structure), and (c) by $n\pi$ in
$\kappa_{y}y$ and displaced by $n\pi$ in $\kappa_{x}x$ or in $\kappa_{z}z$.
### 3.2 The skeleton of the polarization distribution: L-lines and C-lines
In order to gain intuition about the spatiotemporal distribution of the
polarization states, we now study the L-lines (lines of linear polarization)
and C-lines (lines of circular polarization) inside the F3DP cell A. Figure 4
shows the evolution of these lines for $\alpha$, $\beta=\pi/4$. (The
corresponding polarization structure for $\alpha=\tilde{\alpha}$ and
$\beta=\tilde{\beta}$ is shown in the supplementary material in section A.1.)
Once we describe the polarization state distribution inside cell A, the
corresponding distribution in cell B can be obtained by just mirroring in $y$,
as mentioned earlier.
Figure 4: Temporal evolution of the L-lines and C-lines inside the F3DP cell A
for $\alpha$, $\beta=\pi/4$.
At different times this idealized field presents L-lines, L-planes and even an
L-volume (the latter two being of course unstable under field perturbations).
For $\Omega t=0$, the polarization state is $\mathbf{y}$, so the entire 3D
space is an L-volume. On the other hand, for $\Omega t=\pi/2$, the zones of
linear polarization are two sets of L-planes, which are constant in $x$ and in
$y$, respectively. Within the interval $\Omega t\in(0,\pi/2)$, the field
presents L-lines whose positions remain constant in time. A detailed
explanation of the calculations of the L-lines is provided in the
supplementary material in subsection A.1.1.
Figure 5: Temporal evolution on the unit sphere in real space of (a,b) the
electric field vector direction for linearly polarized states, and (c,d) the
normalized spin density vector for circularly polarized states, for (a,c)
$\alpha=\beta=\pi/4$, and (b,d) $\alpha=\tilde{\alpha}$ and
$\beta=\tilde{\beta}$. Each curve correspond to one temporal instant, and the
spacing between temporal intervals is constant. The C-lines are plotted
starting from the time $t_{C}$ when they appear: $\Omega t_{C}=\pi/4$ for
$\beta=\pi/4$ and $\Omega t_{C}=0.6331$ for $\beta=\tilde{\beta}$.
Figures 5(a,b) show that, for $\Omega t\in[0,\pi/2]$, the electric field
orientation at the regions of linear polarization covers the complete unit
sphere. Note that antipodal points correspond to the same linear polarization
state. Each curve (or more correctly, each pair of curves) represents all the
values of the orientation of the linearly polarized states at a given instant.
These curves are independent of $\alpha$, and, for $\beta=\pi/4$ (Fig. 5(a)),
they are parallels. Decreasing $\beta$ (Fig. 5(b) for $\beta=\tilde{\beta}$)
distorts the curves while maintaining full directional coverage. It is clear
from Eq. (9) that, for any value of $\beta$, the $\mathbf{y}$ polarization
component of the field decreases while the $\mathbf{p}$ and $\mathbf{m}$
components grow. Eventually, all the polarization ellipses lie on the $xz$
plane at $\Omega t=\pi/2$. This fact explains why, in the unit sphere in Figs.
5(a,b), the curves move from the poles towards the equator as time increases.
The C-lines are found from the constraint $\mathbf{E}\cdot\mathbf{E}=0$ [15].
Since $\mathbf{E}\cdot\mathbf{E}$ is a complex quantity, this constraint
imposes two conditions, each defining a surface, and the intersections of
these surfaces are the C-lines. Figure 4 shows the normalized spin density
vector
$\mathbf{s}=\mathrm{Im}\>(\mathbf{E}^{*}\times\mathbf{E})/|\mathbf{E}|^{2}$
along the C-lines. This vector is normal to the plane containing the ellipse,
and its magnitude is proportional to the area of the normalized ellipse,
ranging from 0 (linear polarization) to 1 (circular polarization). The
equations describing the C-lines and $\mathbf{s}$ are inlcuded in the
supplementary material in subsection A.1.2.
There are no C-lines at $t=0$; they appear at a time $t_{C}$, which depends
only on $\beta$, and then undergo two topological transitions within the
interval $\Omega t\in[0,\pi/2]$. The first transition always occurs at the
time $\Omega t_{T1}=\pi/4$, and the other one occurs at $\Omega t_{T2}$, which
depends on $\beta$. The dependence of $t_{C}$ and $t_{T2}$ with $\beta$ is
discussed in the supplementary material. For $\beta=\pi/4$ the two topological
transitions coincide with the appearance of the C-lines, $\Omega t_{C}=\Omega
t_{T1}=\Omega t_{T2}=\pi/4$, while choosing a smaller $\beta$ causes $t_{C}$
to decrease and $t_{T1}$ to increase. Conversely, if $\alpha$ is reduced, the
C-lines get stretched out along $y$ and $z$ and shrink slightly along $x$, but
$t_{C}$ and $t_{T2}$ remain unaltered, i.e., $\alpha$ does not alter the
topological properties of the C-lines. Studying the case when $\beta=\pi/4$
and one case when $\beta<\pi/4$ (in subsection A.1.2 in the supplementary
material) is sufficient then to illustrate the topological behavior of the
C-lines. Figure 4 shows the L-lines and C-lines for $\alpha=\beta=\pi/4$.
Right after appearing at $\Omega t=\pi/4$ as interconnected straight lines,
the C-lines split into two helices within the cell. As time increases, these
helices straighten until they become parallel straight lines for $\Omega
t=\pi/2$. At this time the $\mathbf{y}$ field component vanishes, so the spin
at every point is in the $\mathbf{y}$ direction (normal to the main
propagation direction), leading to “photonic wheels” with transverse spin [16,
17].
We plot the spin directions $\mathbf{s}$ along the C-lines for a given
temporal instant as a curve on the unit sphere in Figs. 5(c,d). For $\Omega
t=\pi/4$, $\mathbf{s}$ spans all directions within the $xz$ plane for
$\beta=\pi/4$ (equator of the sphere in Fig. 5(c)). After $\Omega t=\pi/4$,
each of the two C-lines traces identical parallels, each on a different
hemisphere: the vector $\mathbf{s}$ at the C-line in the region
$-0.5\lambda<x<0$ ($0<x<0.5\lambda$) evolves in time towards the direction
$+y$ ($-y$). As was the case in Figs. 5(a,b), the curves do not depend on
$\alpha$ but they are no longer parallels for $\beta<\pi/4$ (Fig. 5(d)), while
still covering all possible spin directions along the C-lines within the time
interval $\Omega t\in[0,\pi/2]$. Note from Fig. 5(d) that the topological
transition at $t_{T2}$ ($\Omega t_{T2}=1.2094$ for $\beta=\tilde{\beta}$)
corresponds to topological transitions of the spin curves at $\pm{\bf x}$ over
the unit sphere: the curves that cycle around the $z$ axis correspond to
$t<t_{T2}$, while those that cycle around the $y$ axis correspond to
$t>t_{T2}$.
The temporal evolution of the field skeleton for $\alpha=\beta=\pi/4$ (Fig. 4)
and for $\alpha=\tilde{\alpha}$, $\beta=\tilde{\beta}$ (Fig. 8) are included
in the supplementary video 1 and video 2, respectively. The evolution of the
electric field orientation at the linearly polarized regions (Figs. 5 (a,c))
and of the vector $\mathbf{s}$ at the circularly polarized regions (Figs. 5
(b,d)) are also included in each video.
## 4 4D optical skyrmionic textures
As pointed out in the introduction, a 3D polarization ellipse can be fully
characterized by the positions of two indistinguishable points,
$\mathbf{p}_{1}$ and $\mathbf{p}_{2}$, over the unit sphere
($|\mathbf{p}_{1}|=|\mathbf{p}_{2}|=1$). Different variants of this type of
two-point construction exist: one proposed by Hannay [4] based on Majorana’s
representation of spin systems; one referred to as the Poincarana
representation that is naturally linked to geometric phase for nonparaxial
fields [5]; and a more recent one [6] defined to satisfy certain statistical
conditions. For all of them, the bisector of $\mathbf{p}_{1}$ and
$\mathbf{p}_{2}$ is parallel to $\mathbf{s}$, the line joining
$\mathbf{p}_{1}$ and $\mathbf{p}_{2}$ is parallel to the major axis of the
ellipse, and the angular separation between $\mathbf{p}_{1}$ and
$\mathbf{p}_{2}$ is a monotonic function of the ellipticity: the two points
coincide for circular polarization, and they are antipodal for linear
polarization. However, each construction encodes ellipticity in the angular
separation of the points in a different manner.
For a field to span all 3D polarization states, each of the two points in any
of these representations must span the sphere independently, so that all
combinations of the two point positions are represented. We then define a
Skyrme number for this 4D space as
$n_{S}=\iiiint\Sigma_{S}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t,$
(11)
where
$\Sigma_{S}=\frac{1}{2(4\pi)^{2}}\sum^{4}_{i,j,k,l=1}\epsilon_{ijkl}\;\left[\mathbf{p}_{1}\cdot\left(\partial_{i}\mathbf{p}_{1}\times\partial_{j}\mathbf{p}_{1}\right)\right]\;\left[\mathbf{p}_{2}\cdot\left(\partial_{k}\mathbf{p}_{2}\times\partial_{l}\mathbf{p}_{2}\right)\right]$
(12)
is the Skyrme density, which corresponds to the Jacobian determinant between
the coordinates in the abstract closed space (in this case the coordinates of
the two points over the sphere) and the physical space (in this case space-
time). Here $\epsilon_{ijkl}$ is the fourth-rank Levi-Civita tensor (which
differs from zero only for the 24 terms for which all indices are different,
out of the 64 in the sum), and $\partial_{i}$ for $i=1,2,3,4$ represents a
derivative in $x,y,z,t$, respectively. The factor of $1/(4\pi)^{2}$ is for
normalizing the area of the sphere for each point, and the factor $1/2$
accounts for the indistinguishability of the two points. Due to the properties
of the cross product, the 24 nonzero terms in the above expression reduce to
the 6 terms corresponding to
$(i,j,k,l)\rightarrow(x,y,z,t),(z,t,x,y),(x,z,y,t),(x,t,z,y),(z,y,x,t),(t,y,z,x)$,
multiplied by a factor of 4.
This Skyrme number can be used for any of the three constructions mentioned
earlier [4, 5, 6]. Here, we choose to use the Poincarana construction [5]
(Fig. 6), where the midpoint between the points matches exactly $\mathbf{s}$.
The Poincarana points are given by
$\displaystyle\mathbf{p}_{1,2}=\mathbf{s}\pm\sqrt{1-|\mathbf{s}|^{2}}\frac{\mathbf{a}}{|\mathbf{a}|},$
(13)
where $\mathbf{a}$ is the major semi-axis of the polarization ellipse, which
is obtained as $\mathbf{a}=\mathrm{Re}(e^{-i\phi}\mathbf{E}/|\mathbf{E}|)$
with $\phi=\mathrm{Arg}(\mathbf{E}\cdot\mathbf{E})/2$ [6]. We used this
construction because it seems to lead to the simplest expressions for the
Skyrme density in some cases.
Figure 6: Poincarana representation construction: A fully polarized 3D
polarization state is described by two indistinguishable points
$\mathbf{p}_{1,2}$ on the unit sphere. The normalized spin density vector
$\mathbf{s}$ is the centroid of the two points, and the major semi-axis vector
$\mathbf{a}$ is parallel to the point separation.
For any $\alpha$ and $\beta$, the sign of $\Sigma_{S}$ is uniform within each
cell, A or B, but it is opposite between the two types of cell (due to the
reversal in $y$ of the polarization distribution). This sign also switches
temporally every time $\Omega t$ is an integer multiple of $\pi/2$. We found
that $\Sigma_{S}$ is independent of $z$ for any $\alpha$ and $\beta$, and it
has periodic dependence on $x$ for $\beta<\pi/4$ without changing sign (the
sign depending only on $y$ and $t$). Further, for $\beta=\pi/4$, we observed
through numeric evaluation that $\Sigma_{S}$ actually does not depend on $x$,
and the long analytic expression for $\Sigma_{S}$ greatly simplifies by
setting $x=z=0$, giving
$\Sigma_{S}=\frac{k^{3}\Omega}{\pi^{2}}\cos\alpha(\sqrt{2}-\cos\alpha)\cos{\Omega
t}\sin^{3}{\Omega t}\sin{2\kappa_{y}y}.$ (14)
This expression for $\Sigma_{S}$ is plotted in Fig. 7(a) as a function of $y$
and $t$ for $\alpha=\beta=\pi/4$. For $\beta=\pi/4$, the Skyrme number inside
a cell of type A or B can then be easily calculated analytically by
integrating the Skyrme density in Eq. (14), giving the results $n_{S}=-1$ for
cell A and $n_{S}=+1$ for cell B within the interval $\Omega t\in[0,\pi/2]$.
We also verified numerically that $n_{S}=-(+)1$ in cell A(B) for several
values of $\alpha$ and $\beta$. We then refer to the F3DP cells where
$|n_{S}|=1$ as skyrmionic F3DP cells (Fig. 7(b)). Other versions of A and B
displaced in $x$ and/or $z$ are also skyrmionic F3DP cells, since these
displacements simply shift the relative phases between the components.
Note that, since $\Sigma_{S}$ does not depend on $z$ for any $\beta$, the
Skyrme number accumulates linearly and indefinitely as the integration range
in $z$ increases. As a consequence, one can construct cells where $n_{S}=\pm
1$ that are not F3DP cells. There are also F3DP cells (containing each
polarization state) whose volume is the same as that for A or B, but for which
$n_{S}=0$; we call these nonskyrmionic F3DP cells. An example corresponding to
$\kappa_{x}x\in[0,\pi]$, $\kappa_{y}y\in[0,2\pi]$ and
$\kappa_{z}z\in[0,\pi/2]$ is shown in Fig. 7(c), which is composed of segments
of cells of both types leading to cancellation of the Skyrme density
contributions. More details about the construction of F3DP nonskyrmionic cells
can be found in Section A.2 of the supplementary material.
Figure 7: (a) Skyrme density for $\alpha=\beta=\pi/4$ in units of $\Omega
k^{3}$. The Skyrme density beyond the plotted region is repeated. (b) F3DP
skyrmionic cells. The Skyrme number is $-(+)1$ in cell A (B). (c) A F3DP
nonskyrmionic cell that contains all 3D polarization ellipses and has the same
volume as the F3DP skyrmionic cells, but has Skyrme number equal to 0.
## 5 Conclusions
We proposed a simple optical field that covers all possible states of
nonparaxial polarization. This field is given by a superposition of five
linearly-polarized plane waves whose amplitudes vary slowly in time. The
polarization texture is periodic, and is composed of 4D skyrmionic spatio-
temporal cells. The polarization distribution in each cell is described in
terms of its “skeleton” given by the spatiotemporal manifolds of purely linear
or circular polarization.
While the description in this work is theoretical, the experimental
implementation of the field proposed here is probably not difficult to
achieve, and this is currently underway in our group. The main challenge is
most likely not the implementation of the field but its measurement. Current
techniques for measuring nonparaxial polarization distributions rely on
scanning a nanoparticle in order to measure one point at a time. Measuring
this field would require scanning over four dimensions. We are currently
working on an efficient measurement scheme that will allow sampling the 4D
space more efficiently.
The simple forms proposed here are most likely not the only F3DP
electromagnetic fields possible, although they might be the simplest ones. It
would be interesting to investigate whether F3DP fields exist for which the
Skyrme density does not change sign in space or time (making the skyrmionic
structure more resilient to field perturbations). These fields might be
periodic arrays of cells like the F3DP field here, or consist of a single cell
occupying a theoretically infinite volume, in analogy to the 2D skyrmionic
beams in [2]. Nevertheless, the constraints imposed by the Helmholtz equation
and the divergence condition are stronger in the nonparaxial case than they
are for paraxial beams.
## Acknowledgement
This research received funding from the Agence Nationale de Recherche (ANR)
through the project 3DPol, ANR-21-CE24-0014-01. D. M. acknowledges Ministerio
de Universidades, Spain, Universidad Miguel Hernández and the European Union
(Next generation EU fund) for a Margarita Salas grant from the program Ayudas
para la Recualificación del Sistema Universitario Español.
## References
* [1] M. Born and E. Wolf, Principles of Optics: electromagnetic theory of propagation, interference and diffraction of light (Cambridge University Press, 2001), pp. 25-33.
* [2] A.M. Beckley, T.G. Brown, and M.A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777-10785 (2010).
* [3] N. Nagaosa and Y. Tokura, “Topological properties and dynamics of magnetic skyrmions,” Nat. Nanotechnol. 8, 899-911 (2013).
* [4] J.H. Hannay, “The Majorana representation of polarization, and the Berry phase of light,” J. Mod. Opt. 45, 1001-1008 (1998).
* [5] K.Y. Bliokh, M.A. Alonso and M.R. Dennis, “Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects,” Rep. Prog. Phys. 82, 122401 (2019).
* [6] M. A. Alonso, “Geometric descriptions for the polarization of nonparaxial light: a tutorial,” arXiv:2008.02720v2 (2022).
* [7] S. Donati, L. Dominici, G. Dagvadorj, D. Ballarini, M. De Giorgi, A. Bramati, G. Gigli, Y. G. Rubo, M. H. Szymańska and D. Sanvitto, “Twist of generalized skyrmions and spin vortices in a polariton superfluid,” Proc. Natl. Acad. Sci. U.S.A. 113, 14926–14931 (2016).
* [8] S. Gao, F. C. Speirits, F. Castellucci, S. Franke-Arnold, S.M. Barnett and J.B. Götte, “Paraxial skyrmionic beams,” Phys. Rev. A 102, 053513 (2020).
* [9] S. Tsesses, E. Ostrovsky, K. Cohen, B. Gjonaj, N.H. Lindner, and G. Bartal, “Optical skyrmion lattice in evanescent electromagnetic fields,” Science 361, 993-996 (2018).
* [10] H. Ge, X.-Y. Xu, L. Liu, R. Xu, Z.-K. Lin, S.-Y. Yu, M. Bao, J.-H. Jiang, M.-H. Lu, and Y.-F. Chen, “Observation of Acoustic Skyrmions,” Phys. Rev. Lett. 127, 144502 (2021).
* [11] R. D. Muelas-Hurtado, K. Volke-Sepulveda, J. L. Ealo, F. Nori, M. A. Alonso, K. Y. Bliokh, E. Brasselet, “Observation of polarization singularities and topological textures in sound waves,” Phys. Rev. Lett. 129, 204301 (2022).
* [12] L. Du, A. Yang, A. V. Zayats, and X. Yuan, “Deep-subwavelength features of photonic skyrmions in a confined electromagnetic field with orbital angular momentum,” Nat. Phys. 15, 650-654 (2019).
* [13] R. Gutiérrez-Cuevas and E. Pisanty, “Optical polarization skyrmionic fields in free space,” J. Opt. 23, 024004 (2021).
* [14] D. Sugic, R. Droop, E. Otte, D. Ehrmanntraut, F. Nori, J. Ruostekoski, C. Denz, and M.R. Dennis, “Particle-like topologies in light,” Nat. Commun. 12, 6785 (2021).
* [15] M.V. Berry and M.R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. A 457, 141-55 (2001).
* [16] K.Y. Bliokh and F. Nori, “Transverse and longitudinal angular momenta of light,” Phys. Rep. 592, 1-38 (2015).
* [17] A. Aiello, P. Banzer, M. Neugebauer, and G. Leuchs, “From transverse angular momentum to photonic wheels,” Nat. Photon. 9, 789-795 (2015).
## Appendix A Supplementary material
We include in the supplementary material details about the L-lines, C-lines,
and of the nonskyrmionic F3DP cells.
### A.1 Skeleton
In this section, we provide details of the derivation of the L-lines, and
equations for the C-lines, the normalized spin denity vector $\mathbf{s}$, and
the topological transitions that the C-lines undergo. We also study the
evolution of the L-lines and C-lines for $\beta<\pi/4$, in particular for the
case $\alpha=\tilde{\alpha}$ and $\beta=\tilde{\beta}$, shown in Fig. 8.
#### A.1.1 The L-lines within cell A
Here, we describe how to obtain the L-lines for the field defined in Eq. (9).
As mentioned in the main text, for $\Omega t=0$, the field has a polarization
state $\mathbf{y}$, which means that the 3D space itself is an L-volume. For
$\Omega t=\pi/2$, on the other hand, only the components $\mathbf{p}$ and
$\mathbf{m}$ survive. If, in addition, $\kappa_{y}y=n\pi/2$
($n\in\mathbb{Z}$), then only one linear polarization component remains, so
this condition defines planes of constant $y$ with $\mathbf{p}$ (for
$\kappa_{y}y=n\pi$) or $\mathbf{m}$ (for $\kappa_{y}y=(2n+1)\pi/2$)
polarization state. Conversely, for $\kappa_{y}y\neq n\pi/2$, the phase
between the $\mathbf{p}$ and $\mathbf{m}$ components must be $n\pi$ for the
state to be linear, which defines a set of L-planes $x=n\pi/(2\kappa_{x})$.
Within the intermediate interval $\Omega t\in(0,\pi/2)$ we find L-lines. The
first subset of L-lines is deduced for $\kappa_{y}y=n\pi/2$. Under this
spatial constraint, only two polarization components survive ($\mathbf{y}$ and
$\mathbf{p}$ or $\mathbf{y}$ and $\mathbf{m}$), and in order to have linearly
polarized light, the relative phase between these components must be $n\pi$,
which, together with the condition $\kappa_{y}y=n\pi/2$, defines a set of
L-lines that lie on the planes $y=0$ and $y=\lambda/(4\sin\alpha)$ in the F3DP
cell A. The second set of L-lines is deduced for $\kappa_{y}y\neq n\pi/2$
(i.e., when the three linear polarization components survive), imposing that
the two relative phases between the three linear polarization components must
be integer multiples of $\pi$.
#### A.1.2 The C-lines within cell A
The equations defining the C-lines for the field defined in Eq. (9) can be
found from the condition $\mathbf{E}\cdot\mathbf{E}=0$, whose real and
imaginary parts lead to
$\displaystyle\cos^{2}{\Omega t}\cos{2\kappa_{z}z}+\sin^{2}{\Omega
t}\left(\cos{2\kappa_{x}x}-\cos{2\beta}\sin{2\kappa_{y}y}\right)$
$\displaystyle=0,$ (15a) $\displaystyle\cos^{2}{\Omega
t}\sin{2\kappa_{z}z}-\sin^{2}{\Omega t}\cos{2\kappa_{y}y}\sin{2\kappa_{x}x}$
$\displaystyle=0.$ (15b)
Using
$\mathbf{s}=\mathrm{Im}\>(\mathbf{E}^{*}\times\mathbf{E})/|\mathbf{E}|^{2}$,
we obtain the normalized spin density vector $\mathbf{s}$ for the field
defined in Eq. (9):
$\mathbf{s}=\frac{1}{|\mathbf{E}|^{2}}\begin{pmatrix}-\sin{2\Omega
t}\sin{\beta}\left[\cos{\kappa_{y}y}\sin{(\kappa_{z}z+\kappa_{x}x)}+\sin{\kappa_{y}y}\sin{(\kappa_{z}z-\kappa_{x}x)}\right]\\\
-\sin^{2}{\Omega t}\sin{2\beta}\sin{2\kappa_{x}x}\sin{2\kappa_{y}y}\\\
\sin{2\Omega
t}\cos{\beta}\left[\cos{\kappa_{y}y}\sin{(\kappa_{z}z+\kappa_{x}x)}-\sin{\kappa_{y}y}\sin{(\kappa_{z}z-\kappa_{x}x)}\right]\end{pmatrix},$
(16)
where
$|\mathbf{E}|^{2}=1-\sin^{2}{\Omega
t}\cos{2\beta}\cos{(2\kappa_{x}x)\sin{(2\kappa_{y}y)}}$ (17)
is the field intensity.
For $\beta<\pi/4$, we observed for multiple values of $\beta$ that the C-lines
in the F3DP cell A always appear as points at $x=\pi/(2\kappa_{x})$,
$y=\pi/(4\kappa_{y})$, $z=\pi/\kappa_{z}$, and $x=\pi/(2\kappa_{x})$,
$y=\pi/(4\kappa_{y})$, $z=0$. (It is only for $\beta=\pi/4$ that they appear
as lines and they also appear at other regions.) Evaluating $\mathbf{s}$ at
these spatial coordinates reveals that $\mathbf{s}$ points in the $\pm z$
direction at these points, as can be seen for $\alpha=\tilde{\alpha}$ and
$\beta=\tilde{\beta}$ in Fig. 8 and on the sphere in Fig. 5(d). Setting
$|\mathbf{s}|=1$ (which implies circular polarization) at these points, we
arrive at the expression
$\Omega
t_{C}=\arccos\left(\frac{\cos\beta}{\sqrt{1+\frac{\cos{2\beta}}{2}}}\right)$
(18)
for the time $t_{C}$ when the C-lines appear: $\Omega t_{C}=\pi/4$ for
$\beta=\pi/4$ (Fig. 4) and $\Omega t_{C}=0.6331$ for $\beta=\tilde{\beta}$
(Fig. 8). As mentioned in the main text, we observed a topological transition
at $\Omega t_{T1}=\pi/4$ for any $\beta$. At a given time $t_{T2}$, there is
another topological transition for the C-lines. We observed that there are two
helical C-lines inside the F3DP cell A meeting at the points $x=0$,
$y=\pi/(4\kappa_{y})$ and $z=\pi/(2\kappa_{z})$, and $x=0$,
$y=\pi/(4\kappa_{y})$ and $z=\pi/\kappa_{z}$ for any $\beta$. Evaluating the
vector $\mathbf{s}$ in these coordinates we found that the vector $\mathbf{s}$
points along $+x$ (-$x$) for any $\beta$ at these points. Imposing that
$|\mathbf{s}|=1$, we arrive at
$\Omega
t_{T2}=\arccos\left(\frac{\sin\beta}{\sqrt{1-\frac{\cos{2\beta}}{2}}}\right).$
(19)
The topological transition is noticeable from the unit sphere in Fig. 5(d). It
occurs when $\mathbf{s}$ points along $x$ and $-x$ and it is a very fast
transition. For $\beta=\tilde{\beta}$, $\Omega t_{T2}=1.2094$, while for
$\beta=\pi/4$, $\Omega t_{T2}=\pi/4$. As pointed out in the main text, for
$\beta=\pi/4$, the topological transitions and the appearance of the C-lines
occur at the same instant $\Omega t_{C}=\Omega t_{T1}=\Omega t_{T2}=\pi/4$.
However, as $\beta$ decreases, the instants $t_{C}$ and $t_{T2}$ move away
from each other in time while $\Omega t_{T1}$ remains at $\pi/4$ as shown in
the plot in Fig. 9.
Figure 8: Temporal evolution of the L-lines and C-lines inside the F3DP cell A
for $\alpha=\tilde{\alpha}$ and $\beta=\tilde{\beta}$ (for which the
dimensions of the F3DP cell in $x$ and $y$ are twice as large as for the cell
corresponding to $\alpha=\beta=\pi/4$). The polarization ellipses are plotted
for the C-lines and L-lines, and the normalized spin density vector is plotted
for the C-lines. At $\Omega t_{C}=0.6331$, the C-lines appear as points. There
are two topological transitions, occurring at $\Omega t_{T1}=\pi/4$ and at
$\Omega t_{T2}=1.2094$.
Figure 8 shows the L-lines, the C-lines and the $\mathbf{s}$ vector inside the
F3DP cell A for $\alpha=\tilde{\alpha}$ and $\beta=\tilde{\beta}$. Here, the
C-lines appear as points at $\Omega t_{C}=0.6331$. The plot for $\Omega
t=\pi/4-\pi/50$ in Fig. 8 shows that shortly after $t_{C}$, the C-lines become
small closed loops that spill into neighboring F3DP cells (as can be deduced
from the brick-like structure in Fig. 3(b)). These loops grow until, right
before $\Omega t=\Omega t_{T1}=\pi/4$, segments of them almost lie on the
$y=0$ plane while others almost lie on the $y=\lambda/(4\sin\alpha)$ plane. At
the topological transition at $\Omega t_{T1}=\pi/4$, pairs of C-lines segments
lying on the same plane of constant $y$ get linked by straight C-lines. These
straight lines come from closed C-lines loops in the upper and lower F3DP
cells. Right after $\Omega t=\pi/4$, the segments almost lying on the $y=0$
and $y=\lambda/(4\sin\alpha)$ surfaces before $\Omega t=\pi/4$ disappear from
the cell; they move to the F3DP cells above and below to form the same
structure they form in A but displaced and reflected along $y$. This structure
consists of two tangled helical C-lines winding around each other. At the
second topological transition at $\Omega t_{T2}$, the two helical lines meet
at two points, and right after $t_{T2}$, the C-line segments within $x>0$
combine to form a new C-line, and the same for $x<0$, so there are now two
untangled helical C-lines. After $t_{T2}$, the C-lines behave as they did for
$\beta=\pi/4$: as $\Omega t$ increases the lines straighten gradually until
they become straight for $\Omega t=\pi/2$, when the ellipses are constrained
to the $xz$ plane, and pure transverse spin appears.
Figure 9: Dependence on $\beta$ of the time $t_{C}$ when the C-lines appear,
and of the topological transition times $t_{T1}$ and $t_{T2}$.
### A.2 F3DP nonskyrmionic cells
In section 4, we reported the existence of F3DP cells where $n_{S}=0$. We show
in Fig. 10 examples of regions containing all 3D polarization ellipses where
$n_{S}=0$. The regions in (a) and (b) are constructed as follows: First, we
take one or more regions in A that sum up to half of the A volume by slicing A
along $z$. Then, we take the regions in B containing all the polarization
states that are not in the chosen A regions (i.e., the regions in B that are
not a displacement of the A regions by $\pi/2$ in $\kappa_{x}x$, $\kappa_{y}y$
and $\kappa_{z}z$). Since $\Sigma_{S}$ does not depend on $z$, $n_{S}$ grows
linearly with $z$, and thus the total $n_{S}$ is $-1/2$ ($+1/2$) in the A (B)
sub-regions, so their Skyrme numbers always cancel. A simple example is
depicted in Fig. 10(a), where we took the second half of cell A. We can choose
the A volumes to be split into two parts as shown in Fig. 10(b). Following the
stacking rules to reproduce the field in subsection 3.1, one realizes that the
cell in Fig. 7(c) ($\kappa_{x}x\in[0,\pi]$, $\kappa_{y}y\in[0,2\pi]$ and
$\kappa_{z}z\in[0,\pi/2]$), which is also shown in Fig. 10(c), can be built
from Fig. 10(b), but now all the parts of the cell are connected forming a
cuboidal F3DP nonskyrmionic cell.
Figure 10: Nonskyrmionic F3DP regions. The unit cell of the polarization
structure of the field has been depicted as two dashed cuboids to better
visualize the nonskyrmionic F3DP regions.
### A.3 Videos
We show the evolution of the L-lines, C-lines and normalized spin density
vector $\mathbf{s}$ inside the F3DP cell A for $\alpha=\beta=\pi/4$ (video 1)
and for $\alpha=\tilde{\alpha}$ and $\beta=\tilde{\beta}$ (video 2). The
evolution of the electric field orientation at the regions of linear
polarization and of the $\mathbf{s}$ vector at the C-lines over the unit
sphere is included.
|
# Faster Inference of Integer SWIN Transformer by Removing the GELU Activation
Mohammadreza Tayaranian111Correspondence to
<EMAIL_ADDRESS>Seyyed Hasan Mozafari,
James J. Clark, Brett Meyer, Warren Gross
###### Abstract
SWIN transformer is a prominent vision transformer model that has state-of-
the-art accuracy in image classification tasks. Despite this success, its
unique architecture causes slower inference compared with similar deep neural
networks. Integer quantization of the model is one of the methods used to
improve its inference latency. However, state-of-the-art has not been able to
fully quantize the model. In this work, we improve upon the inference latency
of the state-of-the-art methods by removing the floating-point operations,
which are associated with the GELU activation in Swin Transformer. While
previous work proposed to replace the non-integer operations with linear
approximation functions, we propose to replace GELU with ReLU activation. The
advantage of ReLU over previous methods is its low memory and computation
complexity. We use iterative knowledge distillation to compensate for the lost
accuracy due to replacing GELU with ReLU. We quantize our GELU-less SWIN
transformer and show that on an RTX 4090 NVIDIA GPU we can improve the
inference latency of the quantized SWIN transformer by at least $11\%$ while
maintaining an accuracy drop of under $0.5\%$ on the ImageNet evaluation
dataset.
## Introduction
The attention mechanism has gained popularity in recent years after its
successful debut in transformer architecture (Vaswani et al. 2017). While the
transformer architecture has been initially used for natural language
processing (NLP) tasks, it was also brought to the computer vision domain with
the introduction of vision transformer models (Dosovitskiy et al. 2021). SWIN
transformer (Liu et al. 2021a) is a well-known vision transformer which
improves on the original design by using shifted windows in the input. It
shows state-of-the-art performance in a variety of computer vision tasks.
However, SWIN transformer’s inference latency is negatively affected due to
its use of windowed attention. The windowed attention relies on shifting of
the input activations, and the shift operation is highly memory intensive,
thus having a high impact on the inference latency. For instance, running
inference on an NVIDIA V100 GPU, SWINSMALL is shown to be $55\%$ slower
compared to ViTSMALL (Liu et al. 2022). For mobile devices, Wang et al. (2022)
demonstrated a more pronounced gap in the inference latency where SWINSMALL is
$2.2$ times slower than ViTSMALL.
Layer-normWindowShifting$Q$Fused Op #1QKVGemm$dQ$QKVBias$Q$Fused Op #2$Q\cdot
K$Gemm$dQ$Softmax &Pos. Bias$Q$Fused Op
#3$V\cdot\textrm{att}$GemmProjGemm$dQ$ProjBiasAdd(Residual)$Q$Fused Op
#4FC1Gemm$dQ$FC1 Bias & GELU activation$Q$Fused Op #5FC2Gemm$dQ$FC2 BiasAdd
&Layer-normFused Op #6 Figure 1: High-level depiction of the components of a
transformer block in the quantized SWIN-transformer. $Q$ and $dQ$ denote the
quantization and de-quantization operations, respectively.
Quantization is one of the techniques used for the improvement of the
inference latency of deep neural networks. It involves representing the values
of the neural network using data types with lower bit-widths.
Despite the theoretical possibility of using arbitrary data types and bit-
widths for quantization, the achieved inference speedup depends on hardware on
which the model is being deployed (Sun et al. 2022). For instance, consider a
quantization method which uses the 4-bit integer data type to represent
weights and activations of a deep neural network. Running this method on
hardware that doesn’t support 4-bit arithmetic operations results in a speedup
lower than the expected theoretical speedup. Besides, the overhead of
converting the quantized values to a data type which is supported by the GPU
will further reduce the speedup of the method.
Another important factor in the speedup of partially quantization models is
their non-integer components (Li and Gu 2022). Such component is the non-
linear function, e.g. Softmax, that due to its non-linearity is not easily
quantizable. Given a non-linear function $f$, its quantized input $\hat{x}$,
and the quantization scale $s$, we have $f(s\hat{x})\neq sf(\hat{x})$. As a
result, some integer implementations opt to use the floating-point data type
for these components. This enforces the inclusion of memory intensive
quantization and de-quantization functions in the inference pipeline which
results in notable overhead.
With the goal of avoiding non-integer components, previous work focused on
quantizing the non-linear operations of transformer-based models (Li and Gu
2022; Kim et al. 2021; Lin et al. 2021). The main theme of these works is to
substitute the non-linear components with a linear or piece-wise linear
version without losing accuracy. In this work, we propose to replace the GELU
activation with the piece-wise linear ReLU function (Fukushima 1975). Compared
with which needs to compute the maximum of the input tensor to approximate
GELU (Li and Gu 2022), ReLU can be simply applied with the help of a
comparator. The advantage of ReLU is its low complexity and simple logic
whereas previous work’s shift-based GELU required (Li and Gu 2022). We apply
these changes to the SWIN transformer model in a layer-by-layer fashion and
use knowledge distillation in the process to maintain the model’s accuracy.
The weights and input activations of the resulting model, which we call GELU-
less SWIN, are then quantized using post-training quantization.
The results of this comparison show that our model has a maximum accuracy drop
of $0.5\%$ while achieving more than $11\%$ inference latency reduction
compared to the FasterTransformer framework.
## Previous Work
### Quantization of Linear Components
The majority of previous work focuses on the quantization of linear components
of vision transformers, i.e. fully connected and convolutional layers. In the
quantized version of these components, either the weight, the input
activation, or both matrices are quantized. The quantization scale is obtained
by either quantization aware training or via a calibration phase in a post-
training quantization fashion.
Liu et al. (2021b) propose a mixed-precision quantization in which each linear
layer in different transformer blocks has a different bit-width. Yuan et al.
(2021) uses a Hessian-guided similarity measure for finding quantization
scales. To keep the precision in the layers that are more sensitive to the
quantization noise, they use two scale factors for each fully connected layer
with each of the scales responsible for only a part of the tensor. Li et al.
(2022a) quantizes the linear layers of vision transformers down to 2 bits.
They do so by adding trainable parameters that help the quantized weights
follow the distribution of floating-point weights. Li et al. (2022b) uses
different bit-widths and scales for each attention head. Li et al. (2022a) and
Li et al. (2022b) both show promising results in terms of accuracy when using
4-bit and 3-bit weights. Despite being able to maintain the model’s test
accuracy, all of these works lack studies of hardware performance metrics,
e.g. latency, of their quantized model and only discuss the model size, which
is not a reliable proxy for latency. In the present work, in addition to
accuracy, we measure the inference latency of our quantization method and its
speedup compared to the baseline.
### Quantization of Non-Linear Components
Another line of work tries to quantize the non-linear components of the model.
Softmax, LayerNorm, and GELU activation are the three main non-linear
components of vision transformers that are not straightforward to quantize.
Lin et al. (2020) and Kim et al. (2021) use polynomial approximations to
provide quantizable versions of the non-linear components. Although their
methods were proven successful for transformer-based language models, Li and
Gu (2022) have shown that these methods cannot be used to vision transformers.
Lin et al. (2021) propose a $\log_{2}$ quantization method which adapts the
methodology of (Kim et al. 2021) for fully integer vision transformers.
Although the authors show their method’s ability to maintain accuracy, the
performance of their method in terms of hardware metrics like latency is not
discussed.
The closest work to our work is I-ViT, which provides shift-based replacements
for the non-linear components (Li and Gu 2022). Their integer-friendly
replacement functions use the power of two approximations of the $e^{x}$
function (Stevens et al. 2021). They also improve on the integer LayerNorm
proposed by Lin et al. (2020) by using a shift-based iterative function to
compute the square root of the variance. Their experimental results show that
their quantized model has a $5.8\%$ shorter inference latency compared with
implementing quantized SWIN transformer based on NVIDIA’s FasterTransformer
framework. Despite its high accuracy, their proposed shift-based GELU
approximation has a high memory and computation complexity as it needs to
compute the maximum value of the input tensor to approximate GELU. In
comparison, our proposed method of replacing the GELU activation with ReLU has
the advantage of lower memory and computation complexity given the ReLU’s
simple logic.
## Background and Motivation
SWIN transformer addresses an architectural problem with the original vision
transformer model. It uses a windowed attention mechanism to avoid global
attention and its considerable computation. The window attention divides each
input activation into smaller windows and computes the attention over the
image patches in each window. This enables the use of SWIN transformer as a
backbone model for tasks such as semantic segmentation that have larger input
images (Liu et al. 2021a).
Despite its state-of-the-art accuracy, SWIN transformer’s use of window
shifting operations has negative effects on its hardware performance. This
negative effect is revealed when comparing the inference latency of SWIN
transformer with the original vision transformer. For instance, SWINSMALL is
$55\%$ slower in terms of inference latency when compared to DeiTSMALL which
has the same architecture as the original vision transformer (Liu et al.
2022). In the case of mobile GPU, Mehta and Rastegari (2022) demonstrated that
the window shifting operations are not supported by iPhone GPUs, making it
impossible to implement SWIN on this hardware. Nevertheless, on mobile GPUs
where SWIN can be implemented, SWINSMALL is $2.2$ times slower than DeiTSMALL
(Wang et al. 2022).
| Fused Op #
---|---
Model | 1 | 2 | 3 | 4 | 5 | 6
SWINTINY | $0.32$ | $1.51$ | $2.55$ | $0.88$ | $2.03$ | $1.16$
SWINSMALL | $0.3$ | $2.01$ | $4.1$ | $1.17$ | $2.48$ | $1.36$
SWINBASE | $0.57$ | $2.53$ | $5.44$ | $1.57$ | $3.89$ | $2.69$
SWINLARGE | $1.22$ | $3.93$ | $8.18$ | $2.92$ | $8.13$ | $3.66$
Average | $0.6$ | $2.5$ | $5.07$ | $1.64$ | $4.13$ | $2.22$
Table 1: Latency (ms) of the fused operations of the quantized SWIN
transformer, depicted in Figure 1. The latency value are measured on an NVIDIA
RTX 4090 and are averaged over 1000 inference runs.
Inspired by the gap in the inference latency, we use integer quantization to
speed up the inference of the SWIN transformer model. Since our target
hardware is the NVIDIA GPU, we start with the FasterTransformer framework’s
proposed quantized SWIN. Figure 1 depicts a high-level overview of the
components of this quantized SWIN transformer. $Q$ and $dQ$ are the
quantization and de-quantization functions that are used to convert between
the integer and floating-point data types. The quantized SWIN uses 8-bit
integer for the weights and input activations of the linear layers. It also
uses GPU’s integer tensor cores to accelerate the integer matrix
multiplication operations.
In this quantized SWIN architecture, components like biases, Softmax, layer-
norms, and the residual connections use the floating-point data type. The
quantized SWIN uses fused operations, shown in Figure 1 with dashed
rectangles, to spread the overhead of quantization and de-quantization of the
integer values over multiple functions. The functions inside each fused
operation use the shared memory of the GPU to pass values between each other.
This design minimizes the accesses to the slow global memory and thus keeps
the latency of the fused operation, and the entire model, at a minimum.
## GELU-less SWIN Transformer
The quantized SWIN transformer is depicted in Figure 1. The components that
are using the 8-bit integer GEMM, shown in light green, are already quantized
to the minimum bit-width supported by the GPU. Thus, we turn our attention to
the fused operations and measure their latencies.
The latency of each fused operation, calculated as the drop in inference
latency resulted from removing it, is provided in Table 1. Based on these
results, Softmax and GELU activation are the two non-integer components that
have the highest latency in the quantized SWIN transformer pipeline.
Furthermore, our experiments reveal that the latency of all the fused
operations are dominated by the global memory accesses of the required
quantization and de-quantization functions in a not fully quantized
implementation. As a result, instead of modifying only parts of a fused
operation, we need to remove it entirely to achieve higher inference speedup.
Considering these observations, we propose to remove the fused operation
associated with the GELU activation and substitute it with an integer
activation function. We propose to replace GELU with the piece-wise linear
ReLU activation. While the Shift-based GELU proposed by Li and Gu (2022) needs
to compute the maximum value of the input tensor, the ReLU function is a
simple activation function and thus has a lower memory complexity than its
Shift-based alternative. In order to completely remove the fused operation of
GELU, we also need to eliminate the FC1 Bias as well. We avoid changing the
Softmax fused operation as it also contains the relative position bias which
is essential to converting input image to tokens.
Algorithm 1 Our proposed GELU replacement method with knowledge distillation.
Input: SWIN, dataset
Parameter: $N$: Number of Transformer Blocks
Output: GELU-less SWIN
1: $\textrm{student}\leftarrow\operatorname{clone}(\textrm{SWIN})$
2: for $i\leftarrow 1$ to $N$ do
3: $\textrm{block}\leftarrow\textrm{student}.\textrm{blocks}[i]$
4: $\textrm{block}.\textrm{activation}\leftarrow\operatorname{ReLU}$
5: $\textrm{block}.\textrm{bias}\leftarrow 0$
6: $\operatorname{disable\\_gradient}(\textrm{block}.\textrm{bias})$
7: $\operatorname{kd\\_epoch}(\textrm{student},\textrm{SWIN})$
8: end for
9: $\textrm{GELU-less SWIN}\leftarrow\textrm{student}$
10: return GELU-less SWIN
Model | Method | Datatype | Top-1 Acc. | Latency | Speedup
---|---|---|---|---|---
(%) | (ms)
SWINTINY | Baseline | FP32 | $81.2$ | $60.27$ | $\times 1$
Half-precision | FP16 | $81.2$ | $24.96$ | $\times 2.41$
FasterTransformer | int8 | $80.1$ | $17.04$ | $\times 3.54$
Ours | int8 | $80.0$ | $15.01$ | $\times\mathbf{4.02}$
SWINSMALL | Baseline | FP32 | $83.2$ | $103.21$ | $\times 1$
Half-precision | FP16 | $83.2$ | $40.26$ | $\times 2.56$
FasterTransformer | int8 | $83.0$ | $25.05$ | $\times 4.12$
Ours | int8 | $82.5$ | $22.57$ | $\times\mathbf{4.57}$
SWINBASE | Baseline | FP32 | $83.4$ | $157.31$ | $\times 1$
Half-precision | FP16 | $83.4$ | $58.39$ | $\times 2.69$
FasterTransformer | int8 | $83.3$ | $35.55$ | $\times 4.43$
Ours | int8 | $84.6$ | $31.66$ | $\times\mathbf{4.97}$
SWINLARGE | Baseline | FP32 | $86.2$ | $284.41$ | $\times 1$
Half-precision | FP16 | $86.2$ | $104.76$ | $\times 2.71$
FasterTransformer | int8 | $85.8$ | $61.35$ | $\times 4.64$
Ours | int8 | $85.5$ | $53.22$ | $\times\mathbf{5.34}$
Table 2: Comparison of our proposed method with the floating-point baselines
and FasterTransformer’s quantizated model. The accuracy are from evaluating a
pre-trained SWIN transformer on the ImageNet evaluation dataset. The reported
inference latency is for a batch size of 128, averaged over 1000 runs.
Algorithm 1 describes our proposed method for replacing the GELU activation
with ReLU. To avoid risking model divergence, we gradually apply our changes
to the model and modify the transformer blocks one at a time. After each block
modification, we use knowledge distillation to distill the soft labels of the
fully GELU SWIN to our partially GELU and partially ReLU model. The result of
the algorithm, which we call the GELU-less SWIN, does not have a floating-
point fused operation for its activation function. As ReLU is easily
quantizable, it can be fused, as an integer operation, to the previous GEMM.
This way, ReLU’s latency will be completely masked as it will be directly
applied to the output of the GEMM. Finally, we apply the post-training
quantization method of the FasterTransformer framework to this GELU-less SWIN
transformer and quantize its weights and input activations.
## Experiments
### Experimental Setup
We study the evaluation accuracy and latency of our method by applying it to
variuos configurations of the SWIN transformer. We use a pre-trained SWIN
model which is pre-trained on the ImageNet training dataset and evaluate it on
the ImageNet evaluation dataset (Deng et al. 2009). As our methodology mainly
targets NVIDIA GPU hardware, we implement our method on top of NVIDIA’s
FasterTransformer222https://github.com/NVIDIA/FasterTransformer framework. We
compare the performance of our quantized SWIN with FasterTransformer’s
quantized SWIN and also the 32-bit and 16-bit floating-point models.
For knowledge distillation, we use the SGD optimizer with a constant learning
rate of $10^{-2}$ and a momentum of $0.9$. As described in Algorithm 1, the
number of knowledge distillation epochs is the same as the number of
transformer blocks of the SWIN model. So, SWINTINY undergoes 12 epochs of the
knowledge distillation while for other SWIN configurations this number is 24
epochs. We use $10\%$ of the ImageNet training dataset for our knowledge
distillation, with a batch size of $32$. The wall clock time for the knowledge
distillation is $74$, $203$, $213$, and $337$ minutes, respectively, for
models from SWINTINY to SWINLARGE.
For our proposed method, we apply the same post-training quantization that is
used in the FasterTransformer framework on our GELU-less SWIN. The inference
latency is an average of 1000 runs and is measured on a quantized model which
had its GELU fused operation removed. Our batch size for the evaluation
experiments is $128$. All our experiments are performed on an NVIDIA RTX 4090
GPU.
### Experimental Results
Table 2 provides the evaluation top-1 accuracy and latency of SWIN transformer
configurations using different methods. As the Table demonstrates, our
proposed GELU-less quantized SWIN has the smallest inference latency across
the SWIN configurations. Our method is able to improve the latency of the
FasterTransformer by $12\%$, $11\%$, $11\%$, and $13\%$, respectively, for
models from SWINTINY to SWINLARGE.
As an ablation study, we also apply our method without the use of knowledge
distillation which results in an evaluation accuracy of less than $0.9\%$
which shows the knowledge distillation is essential to the proposed algorithm.
## Conclusion
In this work we proposed a method to reduce the inference latency of int-8
quantized SWIN transformer model. We analyzed the latency of the operations in
the existing int-8 quantized SWIN piepline. Based on the analysis, we proposed
to replace the floating-point GELU activation with the ReLU activation. ReLU
is a piece-wise linear function which is easily quantizable and has a very low
complexity. Our proposed method replaces GELU with ReLU and removes the bias
that is fused to it. We also use knowledge distillation to maintain the
accuracy. Our experiments show that quantizing our proposed GELU-less SWIN
results in at least $11\%$ reduction of inference latency compared to the
original quantized SWIN transformer model.
## References
* Deng et al. (2009) Deng, J.; Dong, W.; Socher, R.; Li, L.-J.; Li, K.; and Fei-Fei, L. 2009. Imagenet: A large-scale hierarchical image database. In _2009 IEEE conference on computer vision and pattern recognition_ , 248–255. Ieee.
* Dosovitskiy et al. (2021) Dosovitskiy, A.; Beyer, L.; Kolesnikov, A.; Weissenborn, D.; Zhai, X.; Unterthiner, T.; Dehghani, M.; Minderer, M.; Heigold, G.; Gelly, S.; Uszkoreit, J.; and Houlsby, N. 2021. An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale. In _International Conference on Learning Representations_.
* Fukushima (1975) Fukushima, K. 1975. Cognitron: A self-organizing multilayered neural network. _Biological cybernetics_ , 20(3-4): 121–136.
* Kim et al. (2021) Kim, S.; Gholami, A.; Yao, Z.; Mahoney, M. W.; and Keutzer, K. 2021. I-BERT: Integer-only BERT Quantization. In Meila, M.; and Zhang, T., eds., _Proceedings of the 38th International Conference on Machine Learning_ , volume 139 of _Proceedings of Machine Learning Research_ , 5506–5518. PMLR.
* Li et al. (2022a) Li, Y.; Xu, S.; Zhang, B.; Cao, X.; Gao, P.; and Guo, G. 2022a. Q-ViT: Accurate and Fully Quantized Low-bit Vision Transformer. _arXiv preprint arXiv:2210.06707_.
* Li and Gu (2022) Li, Z.; and Gu, Q. 2022. I-ViT: integer-only quantization for efficient vision transformer inference. _arXiv preprint arXiv:2207.01405_.
* Li et al. (2022b) Li, Z.; Yang, T.; Wang, P.; and Cheng, J. 2022b. Q-vit: Fully differentiable quantization for vision transformer. _arXiv preprint arXiv:2201.07703_.
* Lin et al. (2020) Lin, Y.; Li, Y.; Liu, T.; Xiao, T.; Liu, T.; and Zhu, J. 2020. Towards fully 8-bit integer inference for the transformer model. _arXiv preprint arXiv:2009.08034_.
* Lin et al. (2021) Lin, Y.; Zhang, T.; Sun, P.; Li, Z.; and Zhou, S. 2021. Fq-vit: Post-training quantization for fully quantized vision transformer. _arXiv preprint arXiv:2111.13824_.
* Liu et al. (2021a) Liu, Z.; Lin, Y.; Cao, Y.; Hu, H.; Wei, Y.; Zhang, Z.; Lin, S.; and Guo, B. 2021a. Swin transformer: Hierarchical vision transformer using shifted windows. In _Proceedings of the IEEE/CVF international conference on computer vision_ , 10012–10022.
* Liu et al. (2022) Liu, Z.; Mao, H.; Wu, C.-Y.; Feichtenhofer, C.; Darrell, T.; and Xie, S. 2022. A convnet for the 2020s. In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , 11976–11986.
* Liu et al. (2021b) Liu, Z.; Wang, Y.; Han, K.; Zhang, W.; Ma, S.; and Gao, W. 2021b. Post-training quantization for vision transformer. _Advances in Neural Information Processing Systems_ , 34: 28092–28103.
* Mehta and Rastegari (2022) Mehta, S.; and Rastegari, M. 2022. MobileViT: Light-weight, General-purpose, and Mobile-friendly Vision Transformer. In _International Conference on Learning Representations_.
* Stevens et al. (2021) Stevens, J. R.; Venkatesan, R.; Dai, S.; Khailany, B.; and Raghunathan, A. 2021. Softermax: Hardware/software co-design of an efficient softmax for transformers. In _2021 58th ACM/IEEE Design Automation Conference (DAC)_ , 469–474. IEEE.
* Sun et al. (2022) Sun, M.; Ma, H.; Kang, G.; Jiang, Y.; Chen, T.; Ma, X.; Wang, Z.; and Wang, Y. 2022. VAQF: fully automatic software-hardware co-design framework for low-bit vision transformer. _arXiv preprint arXiv:2201.06618_.
* Vaswani et al. (2017) Vaswani, A.; Shazeer, N.; Parmar, N.; Uszkoreit, J.; Jones, L.; Gomez, A. N.; Kaiser, Ł.; and Polosukhin, I. 2017. Attention is all you need. _Advances in neural information processing systems_ , 30.
* Wang et al. (2022) Wang, X.; Zhang, L. L.; Wang, Y.; and Yang, M. 2022. Towards efficient vision transformer inference: A first study of transformers on mobile devices. In _Proceedings of the 23rd Annual International Workshop on Mobile Computing Systems and Applications_ , 1–7.
* Yuan et al. (2021) Yuan, Z.; Xue, C.; Chen, Y.; Wu, Q.; and Sun, G. 2021. Ptq4vit: Post-training quantization framework for vision transformers. _arXiv preprint arXiv:2111.12293_.
|
# Parametric Shape Holomorphy of Boundary Integral Operators with Applications
Jürgen Dölz Institute for Numerical Simulation, University of Bonn, Friedrich-
Hirzebruch-Allee 7, 53115 Bonn, Germany<EMAIL_ADDRESS>Fernando
Henríquez Chair of Computational Mathematics and Simulation Science, École
Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland ()
<EMAIL_ADDRESS>
###### Abstract
We consider a family of boundary integral operators supported on a collection
of parametrically defined bounded Lipschitz boundaries. Thus, the boundary
integral operators themselves also depend on the parametric variables, leading
to a parameter-to-operator map. One of the main results of this article is to
establish the analytic or holomorphic dependence of said boundary integral
operators upon the parametric variables, i.e., of the parameter-to-operator
map. As a direct consequence we also establish holomorphic dependence of
solutions to boundary integral equations, i.e., holomorphy of the parameter-
to-solution map. To this end, we construct a holomorphic extension to complex-
valued boundary deformations and investigate the _complex_ Fréchet
differentiability of boundary integral operators with respect to each
parametric variable. The established parametric holomorphy results have been
identified as a key property to derive best $N$-term approximation rates to
overcome the so-called curse of dimensionality in the approximation of
parametric maps with distributed, high-dimensional inputs.
To demonstrate the applicability of the derived results, we consider as a
concrete example the sound-soft Helmholtz acoustic scattering problem and its
frequency-robust boundary integral formulations. For this particular
application, we explore the consequences of our results in reduced order
modelling, Bayesian shape inversion, and the construction of efficient
surrogates using artificial neural networks.
## 1 Introduction
In this article, we focus on shape-parametric boundary integral operators
(BIO) of the form
(1.1) $\mathsf{A}_{\bm{{y}}}\colon L^{2}(\Gamma_{\bm{{y}}})\to
L^{2}(\Gamma_{\bm{{y}}}),$
where $\Gamma_{\bm{{y}}}\subset\mathbb{R}^{3}$ is assumed to be a bounded
Lipschitz boundary depending on the parametric input
${\bm{{y}}}\in\mathbb{U}\coloneqq[-1,1]^{\mathbb{N}}$. The boundary integral
operator in Eq. 1.1 is assumed to be of the form
(1.2) $(\mathsf{A}_{\bm{{y}}}\varphi)({\bf
x})\mathrel{\mathrel{\mathop{:}}=}\int\limits_{\Gamma_{\bm{{y}}}}{\sf a}({\bf
y},{\bf x}-{\bf y})\varphi({\bf y})\mathrm{ds}_{\bf y},\quad{\bf
x}\in\Gamma_{\bm{{y}}},$
for any $\varphi\in L^{2}(\Gamma_{\bm{{y}}})$, where the kernel function ${\sf
a}:\Gamma_{\bm{{y}}}\times\mathbb{R}\backslash\\{{\bf
0}\\}\rightarrow\mathbb{C}$ is bounded in the first argument, and possibly
_singular_ when ${\bf x}={\bf y}$ in the second one. A typical example
application is the $L^{2}(\Gamma_{\bm{{y}}})$-based frequency-stable
formulation for the acoustic scattering by random surfaces, wherein the
boundary’s randomness is encoded through a Karhunen-Loève-type expansion of a
given random field. This is, for example, the setting considered in [21, 5,
23, 4].
In the following, we are interested in the regularity of the _parameter-to-
operator_ map
(1.3)
$\mathbb{U}\ni{\bm{{y}}}\mapsto\mathsf{A}_{\bm{{y}}}\in\mathscr{L}\left(L^{2}(\Gamma_{\bm{{y}}}),L^{2}(\Gamma_{\bm{{y}}})\right)$
and, assuming $\mathsf{A}_{\bm{{y}}}$ to be boundedly invertible for each
${\bm{{y}}}\in\mathbb{U}$, of the _parameter-to-solution_ map
(1.4) $\mathbb{U}\ni{\bm{{y}}}\mapsto
u_{\bm{{y}}}\mathrel{\mathrel{\mathop{:}}=}\mathsf{A}_{\bm{{y}}}^{-1}f_{\bm{{y}}}\in
L^{2}(\Gamma_{\bm{{y}}}),$
where $f_{\bm{{y}}}\in L^{2}(\Gamma_{\bm{{y}}})$ for each
${\bm{{y}}}\in\mathbb{U}$. For these maps, constructing efficient surrogates
is a central task in computational UQ and reduced order modelling of
parametric partial differential and boundary integral equations. However, it
is well-known that the so-called curse of dimensionality in the parameter
space renders most naïve approximation methods computationally unfeasible. To
overcome this challenge and effectively design dimension-robust algorithms for
the approximation of the maps in Eq. 1.3 and Eq. 1.4 one requires a better
understanding of the _parametric regularity_ of the BIOs upon the shape-
parametric representations.
In [10], it is proposed to construct a holomorphic extension over a tensor
product of ellipses in the complex domain as a key step to finally break the
curse of dimensionality in the sparse polynomial approximation of high-
dimensional parametric problems. Indeed, by varying the size of these ellipses
on each parameter, i.e. by exploiting the _anisotropic_ parameter dependence,
this _$(\bm{b},p,\varepsilon)$ -holomorphy_ allows to state provable
convergence rates that do not depend on the input’s parametric dimension. From
a computational point of view, $(\bm{b},p,\varepsilon)$-holomorphy constitutes
a stepping stone in the analysis and implementation of a plethora of
approximation techniques such as sparse grid interpolation and quadrature [56,
49, 41, 24], higher-order Quasi-Monte Carlo [19, 20, 17, 18] integration
(HoQMC), construction of artificial neural network (ANNs) surrogates [53, 30,
42, 29, 1], and to establish sparsity of parametric posterior densities in
Bayesian inverse problems [9, 49, 51].
The main objective of this work is to derive shape-parametric regularity
results for BIOs of the form Eq. 1.1. In addition, we aim to demonstrate the
usefulness of this property by studying its implications in reduced order
modelling, expression rates of artificial neural network (ANN) surrogates, and
Bayesian shape inversion.
### 1.1 Previous Work
The analytic dependence of the parameter-to-operator and parameter-to-solution
map is the primary object of study in many recent works tackling parametric
shape deformations. In the context of volume-based variational formulations,
this property has been established for subsurface flows [6, 25, 26], time-
harmonic electromagnetic wave scattering by perfectly conducting and
dielectric obstacles [35, 3], stationary Stokes and Navier-Stokes equation
[11], volume formulations for acoustic wave scattering by a penetrable
obstacle in a low-frequency regime [33], and high-frequency Helmholtz
formulations [54, 22].
As pointed out previously, to establish the sought parametric holomorphy
property one verifies _complex_ Fréchet differentiability of the parameter-to-
operator or parameter-to-solution map. This approach is strongly connected to
the notion of shape differentiability of the domain/boundary-to-solution map.
In [44, 46] and [8], the shape differentiability in the Fréchet sense of BIOs
for acoustic and elastic problems is studied. This result is extended in [45,
13, 14] to the BIOs that arise in electromagnetic wave scattering. The same
approach used to obtain these results is also employed in [13, 14, 55] to
establish the infinite differentiability of a collection of BIOs characterized
by a class of pseudo-homogeneous kernels. However, regardless of the specific
setting considered on the previous works, none of the imply an analytic or
holomorphic dependance upon shape perturbations. Another drawback of the
previously discussed results is that none of them are capable of
systematically account for polygons, let alone Lipschitz boundaries.
Recently, in [28, 27] the holomorphic dependence of the Calderón projector
upon boundaries of class $\mathscr{C}^{2}$ has been established. Therein, the
two-dimensional analysis for both the Laplace and Helmholtz BIOs is thoroghly
analyzed. In [43] this work has been extended to address the case of multiple
open arcs in two-dimensions. In [16], based on [36, 37], a analytic shape
differentiability result is established, and this approach is extended in [15]
to the layer potentials for the heat equation. Yet useful in the three
dimensional case, these works still rule out the possibility of polygonal
boundaries in the analysis.
### 1.2 Contributions
We establish parametric holomorphy by verifying _complex_ Fréchet
differentiability of suitable complex extensions of the parameter-to-operator
map Eq. 1.3 and paramter-to-solution map Eq. 1.4. These maps are viewed as
elements of the complex Banach space of bounded linear operators or functions,
with respect to a collection of an affinely parametrized bounded Lipschitz
boundaries.
The contribution of this article is three-fold:
1. 1.
By considering a collection of affine shape-parametric boundaries and suitable
assumptions on the integral kernel $\mathsf{a}$, we prove that the parameter-
to-operator map Eq. 1.3 is $(\bm{b},p,\varepsilon)$-holomorphic in the sense
of [10, Definition 2.1] in a $L^{2}$-based setting. We conclude that the
parameter-to-solution map Eq. 1.4 is also
$(\bm{b},p,\varepsilon)$-holomorphic. Even though we do not explore the entire
Calderón projector in a scale of fractional Sobolev spaces as in [28, 16], we
let the boundary to be globally Lipschitz, yet possibly be patchwise more
regular. This allows the possibility of polygonal domain in the analysis.
2. 2.
We demonstrate the practical applicability of our
$(\bm{b},p,\varepsilon)$-holomorphy results by exemplarily applying our result
to frequency-stable integral equations used in the boundary reduction of
acoustic wave propagation in unbounded domains. This, in turn, serves a
theoretical foundation for the analysis of the computational technique
introduced in [21].
3. 3.
We demonstrate the theoretical usability of our
$(\bm{b},p,\varepsilon)$-holomorphy results by using results from the
literature to obtain _best $n$-term_ approximation rates for Eq. 1.3 and Eq.
1.4. In addition, we explore the consequence of our result in reduced order
modelling, Bayesian shape inversion, and in the construction of efficient
surrogates using ANNs.
### 1.3 Outline
The remainder of the article is structured as follows. First, in Section 2, we
recall some preliminary facts on holomorphy in Banach spaces, parametrized
surfaces, and boundary integral operators. $(\bm{b},p,\varepsilon)$-holomorphy
of the boundary integral operators is established in Section 3, whereas the
illustration to boundary integral equations from acoustic scattering is done
in Section 4. Section 5 demonstrates how the established
$(\bm{b},p,\varepsilon)$-holomorphy leads to various _best- $n$-term_
approximation rates and is followed by concluding remarks in Section 6.
## 2 Preliminaries
Let ${\text{D}}\subset\mathbb{R}^{2}$ be a bounded Lispchitz domain [34,
Definition 3.3.1]. We denote by $L^{p}({\text{D}})$, $p\in[1,\infty)$ the set
of $p$-integrable functions over D, by $H^{s}({\text{D}})$, $s\geq 0$,
stantard Sobolev spaces in D. Let
$\mathscr{C}^{k,\lambda}(\partial{\text{D}})$ be the space of $k$ times
continuously differentiable functions in $\partial{\text{D}}$ with Hölder
continuous partial derivatives of order $\lambda$, and, in particular, by
$\mathscr{C}^{0,1}(\partial{\text{D}})$ the space of Lipschitz continuous
functions on $\partial{\text{D}}$. In addition, let
$H^{s}(\partial{\text{D}})$, for $s\in[0,1]$, be the Sobolev space of traces
on $\partial{\text{D}}$ ([48, Section 2.4], [34, Section 4.2]). As is
customary, we identify $H^{0}(\partial{\text{D}})$ with
$L^{2}({\partial{\text{D}}})$ and, for $s\in[0,1]$,
$H^{-s}(\partial{\text{D}})$ with the dual space of
$H^{s}(\partial{\text{D}})$.
For (complex) Banach spaces $X$ and $Y$, we denote by $\mathscr{L}(X,Y)$ the
space of bounded linear operators from $X$ into $Y$ and by
$\mathscr{L}_{\text{iso}}(X,Y)$ the (open) subset of isomorphisms, i.e.
bounded linear operators with a bounded inverse. Recall that
$\mathscr{L}(X,Y)$ is a (complex) Banach space equipped with the standard
operator norm [47, Theorem III.2].
Finally, we equipp $\mathbb{U}$ with the product topology, and, according to
Tychonoff’s theorem, this renders $\mathbb{U}$ compact with this topology.
### 2.1 Holomorphy in Banach Spaces
For the convenience of the reader, we recall the basic facts of holmorphy in
Banach spaces. To this end, let $E$ and $F$ be _complex_ Banach spaces
equipped with the norms $\left\lVert\cdot\right\rVert_{E}$ and
$\left\lVert\cdot\right\rVert_{F}$, respectively.
###### Definition 2.1 ([39, Definition 13.1]).
Let $U$ be an open, nonempty subset of $E$. A map $\mathcal{M}:U\rightarrow F$
is said to be _complex Fréchet differentiable_ if for each $r\in U$ there
exists a map $(\frac{d}{dr}\mathcal{M})(r,\cdot)\in\mathscr{L}(E,F)$ such that
(2.1)
$\displaystyle\left\lVert{\mathcal{M}(r+\xi)}-\mathcal{M}(r)-\left(\frac{d}{dr}\mathcal{M}\right)(r,\xi)\right\rVert_{F}=o\left(\left\lVert\xi\right\rVert_{E}\right).$
We say that $\left(\frac{d}{dr}\mathcal{M}\right)(r,\xi)$ is the _Fréchet
derivative_ of the map $\mathcal{M}:U\rightarrow F$ at $r\in U$ in the
direction $\xi\in E$.
The following results are extensions of some classical results from complex
analysis to Banach spaces.
###### Theorem 2.2 ([39, Theorem 14.7]).
Let $U$ be an open, nonempty subset of $E$. For the map
$\mathcal{M}:U\rightarrow F$ the following conditions are equivalent:
1. 1.
$\mathcal{M}$ is complex Fréchet differentiable.
2. 2.
$\mathcal{M}$ is infinitely complex Fréchet differentiable.
3. 3.
$\mathcal{M}$ is holomorphic.
###### Remark 2.3.
In view of Theorem 2.2 it sufficies to show _complex_ Fréchet
differentiability to establish an holomorphic dependence.
###### Lemma 2.4 ([31, Theorem 3.1.5, item c)]).
Let $U$ be an open, nonempty subset of $E$, and let
$\left(\mathcal{M}_{n}\right)_{n\in\mathbb{N}}$ be a sequence of holomorphic
maps from $U\subset E$ to $F$. Assume that $\mathcal{M}_{n}$ converges
uniformly to $\mathcal{M}$ in $U$. Then $\mathcal{M}:U\subset E\rightarrow F$
is holomorphic.
###### Proposition 2.5 ([28, Proposition 4.20]).
Let $X$, $Y$ be complex Banach spaces and denote by
$\mathscr{L}_{\text{iso}}(X,Y)$ the (open) subset of operators with bounded
inverse in $\mathscr{L}(X,Y)$.
1. 1.
Let $\mathsf{M}\in\mathscr{L}_{\text{iso}}(X,Y)$. Then with
(2.2)
$\displaystyle\mathcal{C}_{\mathsf{M}}\coloneqq\left\\{\mathsf{T}\in\mathscr{L}(X,Y):\;\left\lVert\mathsf{M}-\mathsf{T}\right\rVert_{\mathscr{L}(X,Y)}<\left\lVert\mathsf{M}^{-1}\right\rVert^{-1}_{\mathscr{L}(Y,X)}\right\\}$
$\mathcal{C}_{\mathsf{M}}\subset\mathscr{L}_{\text{iso}}(X,Y)$ and for all
$\mathsf{T}\in\mathcal{C}_{\mathsf{M}}$ it holds
(2.3)
$\displaystyle\left\lVert\mathsf{T}^{-1}\right\rVert_{\mathscr{L}(Y,X)}\leq\frac{\left\lVert\mathsf{M}^{-1}\right\rVert_{\mathscr{L}(Y,X)}}{1-\left\lVert\mathsf{M}-\mathsf{T}\right\rVert_{\mathscr{L}(X,Y)}\left\lVert\mathsf{M}^{-1}\right\rVert_{\mathscr{L}(Y,X)}}.$
2. 2.
The inversion map
(2.4)
$\displaystyle\text{{inv}}:\;\mathscr{L}_{\text{{iso}}}(X,Y)\mapsto\mathscr{L}_{\text{{iso}}}(Y,X):\mathsf{M}\mapsto\mathsf{M}^{-1}$
is holomorphic.
3. 3.
The application map
(2.5) $\displaystyle\text{{app}}:\;\left(\mathscr{L}(X,Y),X\right)\rightarrow
Y:\;(\mathsf{M},\mathsf{g})\mapsto\mathsf{M}\,\mathsf{g}$
is holomorphic.
### 2.2 Parametric Holomorphy
We proceed to define precisely the notion parametric holomorphy to be used
throughout this work. To this end, for $s>1$ we define the so-called Bernstein
ellipse as
(2.6)
$\displaystyle\mathcal{E}_{s}\coloneqq\left\\{\frac{z+z^{-1}}{2}:\;1\leq\left\lvert
z\right\rvert\leq s\right\\}\subset\mathbb{C}.$
This ellipse has foci at $z=\pm 1$ and semi-axes of length
$a\coloneqq(s+s^{-1})/2$ and $b\coloneqq(s-s^{-1})/2$. In addition, we define
the tensorized poly-ellipse
(2.7) $\displaystyle\mathcal{E}_{\bm{\rho}}\coloneqq\bigotimes_{j\geq
1}\mathcal{E}_{\rho_{j}}\subset\mathbb{C}^{\mathbb{N}},$
where $\bm{\rho}\coloneqq(\rho_{j})_{j\geq 1}$ is such that $\rho_{j}>1$, for
$j\in\mathbb{N}$. We adopt the convention $\mathcal{E}_{1}\coloneqq[-1,1]$ to
include the limit case $\rho_{j}=1$.
The next definition, originally introduced in [10], constitutes the precise
the definition of parametric holomorphy. It has been considered in a variety
of previous works encompassing the approximation of high-dimensional maps, and
it is, consequently, the one to be used in this work.
###### Definition 2.6 ([10, Definition 2.1]).
Let $X$ be a complex Banach space equipped with the norm
$\left\lVert\cdot\right\rVert_{X}$. For $\varepsilon>0$ and $p\in(0,1)$, we
say that map
$\mathbb{U}\ni{\bm{{y}}}\mapsto u_{\bm{{y}}}\in X$
is $(\bm{b},p,\varepsilon)$-holomorphic if and only if
1. 1.
The map $\mathbb{U}\ni{{\bm{{y}}}}\mapsto u_{\bm{{y}}}\in X$ is uniformly
bounded, i.e.
(2.8) $\displaystyle\sup_{{\bm{{y}}}\in\mathbb{U}}\left\lVert
u_{\bm{{y}}}\right\rVert_{X}\leq C_{0},$
for some finite constant $C_{0}>0$.
2. 2.
There exists a positive sequence $\bm{b}\coloneqq(b_{j})_{j\geq
1}\in\ell^{p}(\mathbb{N})$ and a constant $C_{\varepsilon}>0$ such that for
any sequence $\bm{\rho}\coloneqq(\rho_{j})_{j\geq 1}$ of numbers strictly
larger than one that is $(\bm{b},\varepsilon)$-admissible, i.e. satisyfing
(2.9) $\displaystyle\sum_{j\geq 1}(\rho_{j}-1)b_{j}\leq\varepsilon,$
the map ${\bm{{y}}}\mapsto u_{{\bm{{y}}}}$ admits a complex extension
${\bm{z}}\mapsto u_{{\bm{z}}}$ that is holomorphic with respect to each
variable $z_{j}$ on a set of the form
(2.10)
$\displaystyle\mathcal{O}_{\bm{\rho}}\coloneqq\displaystyle{\bigotimes_{j\geq
1}}\,\mathcal{O}_{\rho_{j}},$
where
$\mathcal{O}_{\rho_{j}}=\\{z\in\mathbb{C}\colon\operatorname{dist}(z,[-1,1])\leq\rho_{j}-1\\}.$
3. 3.
This extension is bounded on $\mathcal{E}_{\bm{\rho}}$ according to
(2.11) $\displaystyle\sup_{{\bm{z}}\in\mathcal{E}_{\bm{\rho}}}\left\lVert
u_{\bm{z}}\right\rVert_{X}\leq C_{\varepsilon}.$
According to [10, Lemma 4.4], for any $s>1$, it holds
$\mathcal{E}_{s}\subset\mathcal{O}_{s}$. This fact will be used extensively
throughout this work.
###### Remark 2.7.
Let $\bm{\rho}\coloneqq(\rho_{j})_{j\geq 1}$ be any sequence of numbers
strictly larger than one that is $(\bm{b},\varepsilon)$-admissible. In view of
Definition 2.1 and Theorem 2.2, Item 2 in Definition 2.6 is equivalent to
stating that for each $j\in\mathbb{N}$ and
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$ there exists
$\partial_{z_{j}}u_{{\bm{z}}}\in X$, referred to as the derivative in the
direction $z_{j}$, such that
(2.12) $\left\lVert
u_{{\bm{z}}+h\bm{e}_{j}}-u_{{\bm{z}}}-\partial_{z_{j}}u_{{\bm{z}}}h\right\rVert_{X}=o\left(\left\lvert
h\right\rvert\right),\quad\text{as }\left\lvert h\right\rvert\rightarrow
0^{+},\;h\in\mathbb{C},$
where $\bm{e}_{j}$ signifies the $j$-th unit vector.
### 2.3 Affine-Parametric Boundary Transformations
Let $\hat{\Gamma}\subset\mathbb{R}^{3}$, in the following referred to as the
reference boundary, be a bounded Lipschitz boundary. We refer to
$\hat{\Gamma}$ as the reference boundary. To define parametrized families of
bounded Lipschitz boundaries in $\mathbb{R}^{3}$, we consider affine-
parametric vector fields of Karhunen-Loève type. To this end, we set
${{\bm{r}}}_{\bm{{y}}}\colon\hat{\Gamma}\rightarrow\mathbb{R}^{3}$, depending
on the parametric input ${\bm{{y}}}\in\mathbb{U}$, as
(2.13) ${{\bm{r}}}_{\bm{{y}}}(\hat{{\bf x}})={\bm{\varphi}}_{0}(\hat{{\bf
x}})+\sum_{j\geq 1}y_{j}{\bm{\varphi}}_{j}(\hat{{\bf x}}),\quad\hat{{\bf
x}}\in\hat{\Gamma},\quad{\bm{{y}}}=(y_{j})_{j\geq 1}\in\mathbb{U},$
with ${\bm{\varphi}}_{j}\colon\hat{\Gamma}\to\mathbb{R}^{3}$ for
$j\in\mathbb{N}$. This gives rise to a collection of parametric boundaries
$(\Gamma_{\bm{{y}}})_{{\bm{{y}}}\in\mathbb{U}}$ of the form
(2.14) $\Gamma_{{\bm{{y}}}}\coloneqq\\{{\bf x}\in\mathbb{R}^{3}:\;{\bf
x}={{\bm{r}}}_{\bm{{y}}}(\hat{{\bf x}}),\quad\hat{{\bf x}}\in\hat{\Gamma}\\}.$
This family of affinely parametrized boundaries have recently been
investigated in the context of forward and inverse UQ. We refer to [21] for a
detailed discussion on their construction using NURBS.
We make the following assumptions on ${{\bm{r}}}_{\bm{{y}}}$:
###### Assumption 2.8.
Let $\hat{\Gamma}$ be the reference Lipschitz boundary.
1. 1.
The functions
$({\bm{\varphi}}_{i})_{i\in\mathbb{N}}\subset\mathscr{C}^{0,1}(\hat{\Gamma};\mathbb{R}^{3})$
are such that for each ${\bm{{y}}}\in\mathbb{U}$ one has that
${{\bm{r}}}_{\bm{{y}}}\colon\hat{\Gamma}\to\Gamma_{\bm{{y}}}$ is bijective and
bi-Lipschitz.
2. 2.
There exists $p\in(0,1)$ such that
$\bm{b}\coloneqq\left(\left\lVert{\bm{\varphi}}_{j}\right\rVert_{\mathscr{C}^{0,1}(\hat{\Gamma},\mathbb{R}^{3})}\right)_{j\in\mathbb{N}}\in\ell^{p}(\mathbb{N}).$
###### Remark 2.9.
2.8 has the following consequences:
1. 1.
Item 1 implies that $\Gamma_{\bm{{y}}}$ is indeed a bounded Lipschitz boundary
itself.
2. 2.
Item 2 yields absolute uniform convergence of the series Eq. 2.13 as an
element of $\mathscr{C}^{0,1}(\hat{\Gamma},\mathbb{R}^{3})$.
We are interested in extending the set of affine-parametric boundary
parametrizations to a complex set of the form $\mathcal{O}_{\bm{\rho}}$, and
establish its holomorphic dependence upon the parametric variables, in the
sense of Definition 2.6. To this end, we set
(2.15)
$\displaystyle\left(\hat{\Gamma}\times\hat{\Gamma}\right)^{\star}\coloneqq\left\\{\hat{{\bf
x}},\hat{{\bf y}}\in\hat{\Gamma}:\hat{{\bf x}}\neq\hat{{\bf y}}\right\\}.$
###### Lemma 2.10.
Let 2.8 be satisfied with $\bm{b}\in\ell^{p}(\mathbb{N})$ and $p\in(0,1)$.
There exists $\eta>0$ and $\varepsilon>0$ such that for any
$(\bm{b},\varepsilon)$–admisible sequence $\bm{\rho}\coloneqq(\rho_{j})_{j\geq
1}$ of numbers strictly larger than one it holds
(2.16) $\inf_{{\bm{z}}\in\mathcal{O}_{{\bm{\rho}}}}\inf_{(\hat{\bf
x}\times\hat{\bf
y})\in(\hat{\Gamma}\times\hat{\Gamma})^{\star}}\Re\left\\{\frac{\left(\bm{r}_{{\bm{z}}}(\hat{\bf
x})-\bm{r}_{{\bm{z}}}(\hat{\bf y})\right)\cdot\left(\bm{r}_{{\bm{z}}}(\hat{\bf
x})-\bm{r}_{{\bm{z}}}(\hat{\bf y})\right)}{\left\lVert\hat{\bf x}-\hat{\bf
y}\right\rVert^{2}}\right\\}\geq\eta>0,$
###### Proof 2.11.
The proof of this result follows the exact same steps as in the proof of [27,
Proposition 3.2]. For the sake of brevity, we skip it.
###### Lemma 2.12.
Let 2.8 be satisfied. Then, there exists $\varepsilon>0$ such that the map
$\mathbb{U}\ni{\bm{{y}}}\mapsto\bm{r}_{\bm{{y}}}\in\mathscr{C}^{0,1}(\hat{\Gamma},\mathbb{R}^{3})$
defined in Eq. 2.13 is $(\bm{b},p,\varepsilon)$-holomorphic and continuous.
###### Proof 2.13.
The proof of this result follows the exact same steps as in the proof of [11,
Lemma 5.8]. For the sake of brevity, we skip it.
### 2.4 Parametric Holomorphy of Boundary Functions
We study the parametric holomorphy property for a variety of functions
appearing in the parametric holomorphy analysis of BIOs to be carried out in
Section 3.
#### 2.4.1 Parametric Holomorphy of the Gramian
Let ${\bm{{y}}}\in\mathbb{U}$, $f\colon\Gamma_{\bm{{y}}}\rightarrow\mathbb{C}$
a measurable function. Since we assume $\hat{\Gamma}$ to be Lipschitz, there
exists a decomposition $\mathcal{G}$ of $\hat{\Gamma}$ into finitely many open
and disjoint patches $\tau$, i.e.,
(2.17)
$\displaystyle\hat{\Gamma}=\bigcup_{\tau\in\mathcal{G}}\overline{\tau},$
such that each $\tau\in\mathcal{G}$ is the image of a Lipschitz diffeomorphism
$\chi_{\tau}\colon\tilde{\tau}\to\tau$ with
$\tilde{\tau}\subset\mathbb{R}^{2}$. For each $\tau\in\mathcal{G}$, we set
$\chi_{\tau,{\bm{{y}}}}={{\bm{r}}}_{\bm{{y}}}\circ\chi_{\tau}$ and
$\tilde{f}_{\tau,{\bm{{y}}}}=f\circ{{\bm{r}}}_{\bm{{y}}}\circ\chi_{\tau}$.
Then, the surface integral of $f$ over $\Gamma_{\bm{{y}}}$ can be written as
$\displaystyle\int\limits_{\Gamma_{\bm{{y}}}}f(\mathbf{x})\mathrm{d}s_{\mathbf{x}}=\sum_{\tau\in\mathcal{G}}\int\limits_{\tilde{\tau}}\tilde{f}_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}})\sqrt{g_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}})}\mathrm{d}\tilde{\mathbf{u}},$
with the Gramian determinant
$g_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}})=\operatorname{det}G_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}}),\qquad\tilde{\mathbf{u}}\in\tilde{\tau},$
of the Gramian matrix
$G_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}})=\big{(}D_{\tilde{\mathbf{u}}}\chi_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}})\big{)}^{\top}\big{(}D_{\tilde{\mathbf{u}}}\chi_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}})\big{)},\qquad\tilde{\mathbf{u}}\in\tilde{\tau},$
where
$D_{\tilde{\mathbf{u}}}\chi_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}})\in\mathbb{R}^{3\times
2}$ signifies the Jacobian of $\chi_{\tau,{\bm{{y}}}}$.
###### Lemma 2.14.
Let 2.8 be satisfied. Then there exists $\varepsilon>0$ such that for each
$\tau\in\mathcal{G}$ the maps
(2.18) $\displaystyle\mathbb{U}\ni{\bm{{y}}}\mapsto g_{\tau,{\bm{{y}}}}\in
L^{\infty}(\widetilde{\tau})$
and
(2.19)
$\displaystyle\mathbb{U}\ni{\bm{{y}}}\mapsto\sqrt{g_{\tau,{\bm{{y}}}}}\in
L^{\infty}(\widetilde{\tau})$
are $(\bm{b},p,\varepsilon)$-holomorphic and continuous. In particular, for
the complex extension
(2.20) $\displaystyle\mathcal{O}_{\bm{\rho}}\ni{\bm{z}}\mapsto
g_{\tau,{\bm{z}}}\in L^{\infty}(\tilde{\tau})$
there exist $0<\underline{\zeta}\leq\overline{\zeta}<\infty$ such that
(2.21) $\displaystyle
0<\underline{\zeta}\leq\inf_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\inf_{\tilde{\mathbf{u}}\in\tilde{\tau}}\Re(g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}}))\leq\sup_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\sup_{\tilde{\mathbf{u}}\in\tilde{\tau}}\Re(g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}}))\leq\overline{\zeta}<\infty.$
###### Proof 2.15.
It follows from Item 2 in 2.8 that for each $\tau\in\mathcal{G}$ we have that
$\mathbb{U}\ni{\bm{{y}}}\mapsto D_{\tilde{{\bf
x}}}\chi_{\tau,{\bm{{y}}}}\in[L^{\infty}(\tilde{\tau})]^{3\times 2}$
and, thus, also
(2.22) $\mathbb{U}\ni{\bm{{y}}}\mapsto G_{\tau,{\bf
y}}\in\left[L^{\infty}(\tilde{\tau})\right]^{2\times 2}$
is $(\bm{b},p,\varepsilon^{\prime})$-holomorphic and continuous with
$(\bm{b},p,\varepsilon^{\prime})$ as in Lemma 2.12. Since the determinant is a
polynomial, this yields $(\bm{b},p,\varepsilon^{\prime})$-holomorphy of
$g_{\tau,{\bm{z}}}$. Moreover, for each ${\bm{{y}}}\in\mathbb{U}$, the Gramian
matrix $G_{\tau,{\bm{{y}}}}$ is positive definite and symmetric. Thus, it has
real and positive eigenvalues whose product coincides with
$g_{\tau,{\bm{{y}}}}$. Since the eigenvalues depend continuosly on
$G_{\tau,{\bm{{y}}}}$, this yields that $g_{\tau,{\bm{{y}}}}$ is bounded
uniformly from below and above. Thus, there is $\varepsilon>0$ such that Eq.
2.21 holds, which implies the $(\bm{b},p,\varepsilon)$-holomorphy and
continuity of Eq. 2.19.
#### 2.4.2 The Pullback Operator
For each ${\bm{{y}}}\in\mathbb{U}$ and $\varphi\in L^{2}(\Gamma_{\bm{{y}}})$,
we set
(2.23) $\tau_{\bm{{y}}}\varphi\coloneqq\varphi\circ\bm{r}_{\bm{{y}}}\in
L^{2}(\hat{\Gamma}).$
###### Lemma 2.16.
Let 2.8 be satisfied. Then, for each ${\bm{{y}}}\in\mathbb{U}$ the map
(2.24)
$L^{2}(\Gamma_{\bm{{y}}})\ni\varphi\mapsto\hat{\varphi}\coloneqq\tau_{\bm{{y}}}\varphi\in
L^{2}(\hat{\Gamma})$
defines an isomorphism, i.e.
$\tau_{\bm{{y}}}\in\mathscr{L}_{{\text{iso}}}(L^{2}_{\bm{{y}}}(\Gamma),L^{2}(\hat{\Gamma}))$
for each ${\bm{{y}}}\in\mathbb{U}$.
###### Proof 2.17.
As a consequence of Assumptions 2.8 for each ${\bm{{y}}}\in\mathbb{U}$ we
obtain
(2.25)
$\int\limits_{\Gamma_{\bm{{y}}}}\left\lvert\varphi(\mathbf{x})\right\rvert^{2}\text{d}s_{\mathbf{x}}=\sum_{\tau\in\mathcal{G}}\int\limits_{\tilde{\tau}}\left\lvert\left(\tau_{\bm{{y}}}\varphi\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\right\rvert^{2}\sqrt{g_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}})}\text{d}\tilde{\mathbf{u}},$
and
(2.26)
$\int\limits_{\hat{\Gamma}}\left\lvert\left(\tau_{\bm{{y}}}\varphi\right)(\hat{\mathbf{x}})\right\rvert^{2}\text{d}s_{\hat{\mathbf{x}}}=\sum_{\tau\in\mathcal{G}}\int\limits_{\tilde{\tau}}\left\lvert\left(\tau_{\bm{{y}}}\varphi\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\right\rvert^{2}\sqrt{g_{\tau}(\tilde{\mathbf{u}})}\text{d}\tilde{\mathbf{u}},$
where $g_{\tau}$ is the Gram determinant of $\chi_{\tau}$. Using Eq. 2.21 on
$\mathbb{U}\subset\mathcal{O}_{\bm{\rho}}$ yields the assertion.
#### 2.4.3 Parametric Holomorphy of the Normal Vector
Certain integral operators, such as the double layer operator in potential
layer theory, depend on the outward pointing normal. To this end, we establish
parametric holomorphy of the outward pointing normal vector.
###### Lemma 2.18.
Let 2.8 be fulfilled and $(\bm{b},p,\varepsilon)$ as in Lemma 2.14. Then,
pullback of the outward pointing normal
(2.27)
$\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{\mathbf{n}}_{\bm{{y}}}\mathrel{\mathrel{\mathop{:}}=}\tau_{\bm{{y}}}\mathbf{n}_{\Gamma_{\bm{{y}}}}\in
L^{\infty}(\hat{\Gamma})$
is $(\bm{b},p,\varepsilon)$-holomorphic and continuous.
###### Proof 2.19.
Let $\tau\in\mathcal{G}$,
$\chi_{\tau,{\bm{{y}}}}={{\bm{r}}}_{\bm{{y}}}\circ\chi_{\tau}$, and $\hat{{\bf
x}}=\chi_{\tau}(\tilde{\mathbf{u}})$. Then, the non-normalized outward-
pointing normal
$\hat{\mathbf{m}}_{\bm{{y}}}(\hat{{\bf
x}})=\big{(}\partial_{1}\chi_{\tau,{\bm{{y}}}}\times\partial_{2}\chi_{\tau,{\bm{{y}}}}\big{)}(\hat{\mathbf{u}})$
is $(\bm{b},p,\varepsilon)$-holomorphic due to Lemma 2.12. Moreover, due to
$\left\lVert\hat{\mathbf{m}}_{{\bm{{y}}}}(\hat{\mathbf{x}})\right\rVert=\sqrt{g_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}})}$
for all ${\bm{{y}}}\in\mathbb{U}$, Lemma 2.14 implies that also
$\left\lVert\hat{\mathbf{m}}_{{\bm{{y}}}}\right\rVert$ is
$(\bm{b},p,\varepsilon)$-holomorphic and continuous. Thus, since
$\hat{\mathbf{n}}_{\bm{{y}}}(\hat{{\bf
x}})=\frac{\hat{\mathbf{m}}_{\bm{{y}}}(\hat{{\bf
x}})}{\left\lVert\hat{\mathbf{m}}_{\bm{{y}}}(\hat{{\bf
x}})\right\rVert},\quad\hat{{\bf x}}\in\hat{\Gamma},$
LABEL:{eq:gramianbound} implies the assertion.
### 2.5 Riesz-Thorin Interpolation Theorem
We briefly recall a result concerning the boundedness of BIOs in a
$L^{2}$-based setting, to be used in the parametric holomorphy analysis in
Section 3. To this end, let $\Gamma$ be a bounded Lipschitz boundary in
$\mathbb{R}^{3}$ and
(2.28) $(\mathsf{T}_{\Gamma}\varphi)({\bf
x})\mathrel{\mathrel{\mathop{:}}=}\int\limits_{\Gamma}{\sf t}({\bf y},{\bf
x}-{\bf y})\varphi({\bf y})\mathrm{ds}_{\bf y},\quad{\bf x}\in\Gamma,$
an integral operator with kernel
$\mathsf{t}:\Gamma\times\mathbb{R}^{3}\backslash\\{{\bf
0}\\}\rightarrow\mathbb{C}$. In the following, to show the mapping property
$\mathsf{T}_{\Gamma}\colon L^{2}(\Gamma)\to L^{2}(\Gamma)$, we rely on the
_Riesz-Thorin interpolation theorem_ [38, Theorem 2.b.14], which states that
the operator norm of $\mathsf{T}_{\Gamma}:L^{2}(\Gamma)\rightarrow
L^{2}(\Gamma)$ is bounded according to
(2.29)
$\displaystyle\left\lVert\mathsf{T}_{{\Gamma}}\right\rVert_{\mathscr{L}\left(L^{2}(\Gamma),L^{2}(\Gamma)\right)}\leq\left\lVert\mathsf{T}_{{\Gamma}}\right\rVert_{\mathscr{L}\left(L^{1}(\Gamma),L^{1}(\Gamma)\right)}^{\frac{1}{2}}\left\lVert\mathsf{T}_{{\Gamma}}\right\rVert_{\mathscr{L}\left(L^{\infty}(\Gamma),L^{\infty}(\Gamma)\right)}^{\frac{1}{2}}.$
To estimate the operator norms in the upper bound of Eq. 2.29 we use their
explicit characterizations
(2.30)
$\displaystyle\left\lVert\mathsf{T}_{{\Gamma}}\right\rVert_{\mathscr{L}\left(L^{1}(\Gamma),L^{1}(\Gamma)\right)}$
$\displaystyle=\underset{{\bf
y}\in{\Gamma}}{\operatorname{es}\sup}\int\limits_{\Gamma}|\mathsf{t}({\bf
y},{\bf x}-{\bf y})|\mathrm{d}s_{\bf x},$ (2.31)
$\displaystyle\left\lVert\mathsf{T}_{{\Gamma}}\right\rVert_{\mathscr{L}\left(L^{\infty}(\Gamma),L^{\infty}(\Gamma)\right)}$
$\displaystyle=\underset{{\bf
x}\in{\Gamma}}{\operatorname{es\sup}}\int\limits_{\Gamma}|\mathsf{t}({\bf
y},{\bf x}-{\bf y})|\mathrm{d}s_{\bf y}.$
## 3 Parametric Shape Holomorphy of Boundary Integral Operators
In this section, we present and prove the parametric holomorphy results of the
parameter-to-operator map Eq. 1.3 and parameter-to-solution map Eq. 1.4 in an
$L^{2}$-based setting. To this end, for each ${\bm{{y}}}\in\mathbb{U}$ and by
using the pullback operator
$\tau_{\bm{{y}}}:L^{2}(\Gamma_{\bm{{y}}})\rightarrow L^{2}(\hat{\Gamma})$, we
can transport the BIO
$\mathsf{A}_{{\bm{{y}}}}:L^{2}(\Gamma_{\bm{{y}}})\rightarrow
L^{2}(\Gamma_{\bm{{y}}})$ introduced in Eq. 1.2 to the reference boundary
$\hat{\Gamma}$ as through
(3.1)
$\displaystyle\hat{\mathsf{A}}_{\bm{{y}}}\coloneqq\tau_{\bm{{y}}}\,\mathsf{A}_{{\bm{{y}}}}\,\tau^{-1}_{\bm{{y}}}.$
As a consequence of Lemma 2.16, for each ${\bm{{y}}}\in\mathbb{U}$, one has
that ${\mathsf{A}}_{{\bm{{y}}}}:L^{2}(\Gamma_{\bm{{y}}})\rightarrow
L^{2}(\Gamma_{\bm{{y}}})$ is a bounded linear operator if and only if
$\hat{\mathsf{A}}_{\bm{{y}}}\colon L^{2}(\hat{\Gamma})\to L^{2}(\hat{\Gamma})$
is so as well.
The remainder of the section is structured as follows. In Section 3.1,
assuming that the kernel function ${\sf
a}:\Gamma_{\bm{{y}}}\times\mathbb{R}\rightarrow\mathbb{C}$ is bounded, we
prove parametric holomorphy the the map
${\bm{{y}}}\mapsto\hat{\mathsf{A}}_{\bm{{y}}}$, in the sense of Definition
2.6. Next, in Section 3.2, we no longer assume ${\sf a}(\cdot,\cdot)$ to be
bounded, but only integrable, and extend the parametric holomorphy result to
this scenario. Finally, in Section 3.3, we establish parametric holomorphy of
the parameter-to-solution map.
### 3.1 Parametric Shape Holomorphy for BIOs with Bounded Kernel
We proceed to establish parametric shape holomorphy of a class of BIOs
characterized by a bounded kernel function. More precisely, upon assuming an
affinely parametrized boundary as the one introduced in Section 2.3, we
establish $(\bm{b},p,\varepsilon)$-holomorphy for the said BIO.
###### Theorem 3.1.
Let 2.8 be fulfilled. For each $\tau\in\mathcal{G}$ we define
(3.2) $\displaystyle\mathsf{k}_{\tau,{\bm{{y}}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\coloneqq\mathsf{a}\left(\bm{r}_{\bm{{y}}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right),\bm{r}_{\bm{{y}}}(\hat{\mathbf{x}})-\bm{r}_{\bm{{y}}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right),\quad(\hat{{\bf
x}},\tilde{\mathbf{u}})\in\hat{\Gamma}\times\tilde{\tau}.$
In addition, for each $\tau\in\mathcal{G}$ we assume that the map
(3.3) $\mathbb{U}\ni{\bm{{y}}}\mapsto\mathsf{k}_{\tau,{\bm{{y}}}}\in
L^{\infty}\left(\hat{\Gamma}\times\tilde{\tau}\right)$
is $(\bm{b},p,\varepsilon)$-holomorphic and continuous for some
$\varepsilon>0$. Then, the map
(3.4)
$\mathcal{A}:\mathbb{U}\rightarrow\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right):{\bm{{y}}}\mapsto\hat{\mathsf{A}}_{{\bm{{y}}}}$
with $\hat{\mathsf{A}}_{\bm{{y}}}$ as in Eq. 3.1 is
$(\bm{b},p,\varepsilon)$-holomorphic and continuous.
###### Proof 3.2.
Throughout, let $\bm{\rho}\coloneqq(\rho_{j})_{j\geq 1}$ be any
$(\bm{b},\varepsilon)$-admissible sequence of numbers of numbers strictly
larger than one. For the sake of readability, we divide the proof into three
steps.
Step A: Localization of $\hat{\mathsf{A}}_{\bm{{y}}}$ to a panel. Using the
decomposition into panels $\mathcal{G}$ of the reference boundary
$\hat{\Gamma}$ as introduced in Eq. 2.17, for any $\hat{\varphi}\in
L^{2}(\hat{\Gamma})$ and ${\bm{{y}}}\in\mathbb{U}$ one has
(3.5)
$\displaystyle\left(\hat{\mathsf{A}}_{\bm{{y}}}\,\hat{\varphi}\right)\left(\hat{{\bf
x}}\right)=\sum_{\tau\in\mathcal{G}}\left(\hat{\mathsf{A}}_{\tau,{\bm{{y}}}}\,\hat{\varphi}\right)\left(\hat{{\bf
x}}\right),\quad\hat{\bf x}\in\hat{\Gamma}.$
For each $\tau\in\mathcal{G}$, let $\mathsf{k}_{\tau,{\bm{z}}}$ denote the
holomorphic extension of $\mathsf{k}_{\tau,{\bm{{y}}}}$ to
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$ and define
(3.6) $\mathsf{b}_{\tau,{\bm{z}}}(\hat{\bf x},\hat{{\bf
u}})\coloneqq\mathsf{k}_{\tau,{\bm{z}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})},\quad(\hat{\bf
x},\hat{\bf u})\in\hat{\Gamma}\times\tilde{\tau}.$
For $\hat{\varphi}\in L^{2}(\hat{\Gamma})$, set
(3.7)
$\left(\hat{\mathsf{A}}_{\tau,{\bm{z}}}\,\hat{\varphi}\right)\left(\hat{{\bf
x}}\right)\coloneqq\int\limits_{\tilde{\tau}}\mathsf{b}_{\tau,{\bm{z}}}(\hat{\bf
x},\hat{{\bf
u}})\left(\hat{\varphi}\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\,\text{d}\tilde{\mathbf{u}}.$
The map
(3.8) $\mathbb{U}\ni{\bm{{y}}}\mapsto\mathsf{b}_{\tau,{\bm{{y}}}}\in
L^{\infty}(\hat{\Gamma}\times\tilde{\tau})$
is $(\bm{b},p,\varepsilon)$-holomorphic and continuous as a consequence of
Lemma 2.14 and the properties of the map
${\bm{{y}}}\mapsto\mathsf{k}_{\tau,{\bm{{y}}}}$, stated as an assumption. For
$j,m\in\mathbb{N}$, $\tau\in\mathcal{G}$, Cauchy’s integral formula yields
(3.9)
$\displaystyle\partial^{m}_{z_{j}}\mathsf{b}_{\tau,{\bm{z}}}=\frac{m!}{2\pi\imath}\int\limits_{z^{\prime}_{j}\in\partial
B(z_{j},\zeta)}\frac{\mathsf{b}_{\tau,{\bm{z}}^{\prime}}}{(z^{\prime}_{j}-z_{j})^{m+1}}{{\text{d}}}z^{\prime}_{j}$
with ${\bm{z}}^{\prime}=(z_{1},\dots,{z}^{\prime}_{j},\dots)$. Hence, for
$\zeta>0$ such that $B(z_{j},\zeta)\subset\mathcal{O}_{\rho_{j}}$ we obtain
(3.10)
$\left\lVert\partial^{m}_{z_{j}}\mathsf{b}_{\tau,{\bm{z}}}\right\rVert_{L^{\infty}\left(\hat{\Gamma}\times\tilde{\tau}\right)}\leq\frac{m!}{\zeta^{m}}\sup_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\left\lVert\mathsf{b}_{\tau,{\bm{z}}}\right\rVert_{L^{\infty}\left(\hat{\Gamma}\times\tilde{\tau}\right)}.$
Setting
(3.11)
$\displaystyle\left(\hat{\mathsf{A}}^{(j,m)}_{\tau,{\bm{z}}}\hat{\varphi}\right)\left(\hat{{\bf
x}}\right)\coloneqq\int\limits_{\tilde{\tau}}\left(\partial^{m}_{z_{j}}\mathsf{b}_{\tau,{\bm{z}}}\right)(\hat{\bf
x},\hat{\bf
u})\left(\hat{\varphi}\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\,\text{d}\tilde{\mathbf{u}},\quad\hat{\bf
x}\in\hat{\Gamma}$
for $\hat{\varphi}\in L^{2}(\hat{\Gamma})$, yields as a consequence of the
Riesz-Thorin interpolation theorem from Section 2.5
(3.12)
$\left\lVert\hat{\mathsf{A}}^{(j,m)}_{\tau,{\bm{z}}}\right\rVert_{\mathscr{L}(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma}))}\leq
C_{\tau}\left\lVert\partial^{m}_{z_{j}}\mathsf{b}_{\tau,{\bm{z}}}\right\rVert_{L^{\infty}\left(\hat{\Gamma}\times\tilde{\tau}\right)},$
where $C_{\tau}>0$ depends only on $\tau\in\mathcal{G}$, i.e.,
$\hat{\mathsf{A}}^{(j,m)}_{\tau,{\bm{z}}}:L^{2}(\hat{\Gamma})\rightarrow
L^{2}(\hat{\Gamma})$ is linear and bounded for each
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$. As a special case, we obtain
(3.13)
$\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}\right\rVert_{\mathscr{L}(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma}))}\leq
C_{\tau}\left\lVert\mathsf{b}_{\tau,{\bm{z}}}\right\rVert_{L^{\infty}\left(\hat{\Gamma}\times\tilde{\tau}\right)},$
which, together with the $(\bm{b},p,\varepsilon)$-holomorphy of the map
${\bm{{y}}}\mapsto\mathsf{b}_{\tau,{\bm{{y}}}}$ from Eq. 3.8, allows to verify
Item 1 and Item 3 in the Definition 2.6 of $(\bm{b},p,\varepsilon)$-holomorphy
for the map
(3.14)
$\mathcal{A}_{\tau}:\mathbb{U}\rightarrow\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right):{\bm{{y}}}\mapsto\hat{\mathsf{A}}_{\tau,{\bm{{y}}}}.$
It thus only remains to verify Item 2.
Step B: Differentiability of $\mathcal{A}_{\tau}$. Let
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$, $j\in\mathbb{N}$ and consider
$h\in\mathbb{C}$ small enough such that ${\bm{z}}+\eta
h\bm{e}_{j}\in\mathcal{O}_{\bm{\rho}}$ for all $\eta\in[0,1]$. Taylor’s
expansion of the map ${\bm{z}}\mapsto\mathsf{b}_{\tau,{\bm{z}}}$ in the
direction $z_{j}$ yields
(3.15)
$\mathsf{b}_{\tau,{\bm{z}}+h\bm{e}_{j}}=\mathsf{b}_{\tau,{\bm{z}}}+h\,\partial_{{z_{j}}}\mathsf{b}_{\tau,{\bm{z}}}+h^{2}\int\limits_{0}^{1}(1-\eta)\left(\partial^{2}_{z_{j}}\mathsf{b}_{\tau,{\bm{z}}+\eta
h\bm{e}_{j}}\right){{\text{d}}}\eta.$
Let $\widetilde{\rho}_{j}<{\rho}_{j}$. For each
${\bm{z}}\in\mathcal{O}_{\widetilde{\bm{\rho}}}$ with
$\widetilde{\bm{\rho}}=(\rho_{1},\rho_{2},\dots,\widetilde{\rho}_{j},\dots)$
there exists ${\bm{{y}}}\in\mathbb{U}$ such that $\left\lvert
z_{j}-y_{j}\right\rvert<\widetilde{\rho}_{j}-1$. Consider $h\in\mathbb{C}$
satisfying $0<\left\lvert h\right\rvert<{\rho}_{j}-\widetilde{\rho}_{j}$. Set
(3.16) $\zeta_{h}={\rho}_{j}-\widetilde{\rho}_{j}-\left\lvert h\right\rvert$
We need to ensure $B(z_{j}+\eta h,\zeta_{h})\subset\mathcal{O}_{\rho_{j}}$.
For $\widetilde{z}_{j}\in B(z_{j}+\eta h,\zeta_{h})$ we have
(3.17) $\displaystyle\left\lvert\widetilde{z}_{j}-y_{j}\right\rvert$
$\displaystyle\leq\left\lvert\widetilde{z}_{j}-z_{j}-\eta
h\right\rvert+\left\lvert h\right\rvert+\left\lvert z_{j}-y_{j}\right\rvert$
$\displaystyle\leq\zeta_{h}+\left\lvert
h\right\rvert+\widetilde{\rho}_{j}-1\leq\rho_{j}-1.$
Thus, for $\tau\in\mathcal{G}$, $\hat{\varphi}\in L^{2}(\hat{\Gamma})$ and
${\bm{z}}\in\mathcal{O}_{\widetilde{\bm{\rho}}}$ one has
(3.18)
$\hat{\mathsf{A}}_{\tau,{\bm{z}}+h\bm{e}_{j}}\hat{\varphi}=\hat{\mathsf{A}}_{\tau,{\bm{z}}}\hat{\varphi}+h\,\hat{\mathsf{A}}^{(j,1)}_{\tau,{\bm{z}}}\hat{\varphi}+h^{2}\int\limits_{0}^{1}(1-\eta)\left(\hat{\mathsf{A}}^{(j,2)}_{\tau,{\bm{z}}+\eta
h\bm{e}_{j}}\hat{\varphi}\right){{\text{d}}}\eta.$
Next, we have
(3.19)
$\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}+h\bm{e}_{j}}\hat{\varphi}-\hat{\mathsf{A}}_{\tau,{\bm{z}}}\hat{\varphi}-h\,\hat{\mathsf{A}}^{(j,1)}_{\tau,{\bm{z}}}\hat{\varphi}\right\rVert_{L^{2}\left(\hat{\Gamma}\right)}\leq\left\lvert
h\right\rvert^{2}\sup_{\eta\in[0,1]}\left\lVert\hat{\mathsf{A}}^{(j,2)}_{\tau,{\bm{z}}+\eta
h\bm{e}_{j}}\,\hat{\varphi}\right\rVert_{L^{2}\left(\hat{\Gamma}\right)}.$
Thus, recalling Eq. 3.12 and Eq. 3.10 one obtains
(3.20)
$\sup_{\eta\in[0,1]}\left\lVert\hat{\mathsf{A}}^{(j,2)}_{\tau,{\bm{z}}+\eta
h\bm{e}_{j}}\,\hat{\varphi}\right\rVert_{L^{2}\left(\hat{\Gamma}\right)}\leq\frac{2}{\zeta^{2}_{h}}\sup_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\left\lVert\mathsf{b}_{\tau,{\bm{z}}}\right\rVert_{L^{\infty}\left(\hat{\Gamma}\times\tilde{\tau}\right)}.$
Hence, it holds
(3.21)
$\displaystyle\left\lVert\frac{\hat{\mathsf{A}}_{\tau,{\bm{z}}+h\bm{e}_{j}}-\hat{\mathsf{A}}_{\tau,{\bm{z}}}}{h}-\hat{\mathsf{A}}^{(j,1)}_{\tau,{\bm{z}}}\right\rVert_{{\mathscr{L}(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma}))}}\leq\left\lvert
h\right\rvert\frac{2}{\zeta^{2}_{h}}\sup_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\left\lVert\mathsf{b}_{\tau,{\bm{z}}}\right\rVert_{L^{\infty}\left(\hat{\Gamma}\times\tilde{\tau}\right)},$
which shows that $\hat{\mathsf{A}}^{(j,1)}_{\tau,{\bm{z}}}$ is the derivative
of $\hat{\mathsf{A}}_{\tau,{\bm{z}}}$. Hence, the map
(3.22)
$\mathcal{A}_{\tau}:\mathcal{O}_{\widetilde{\bm{\rho}}}\rightarrow\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right):{\bm{z}}\mapsto\hat{\mathsf{A}}_{\tau,{\bm{z}}}.$
is holomorphic at each ${\bm{z}}\in\mathcal{O}_{\widetilde{\bm{\rho}}}$ in
each direction $z_{j}$.
To show that $\mathcal{A}_{\tau}$ is $(\bm{b},p,\varepsilon)$-holomorphic, we
observe that for any $\rho>1$ one has
(3.23)
$\mathcal{O}_{{{\rho}}}=\bigcup_{\widetilde{\rho}:\widetilde{\rho}<\rho}\mathcal{O}_{\widetilde{{\rho}}}$
and one can extend the aforementioned property to
${\bm{z}}\in\mathcal{O}_{{\bm{\rho}}}$. Therefore, we have verified Item 2 in
Definition 2.6 and Eq. 3.14 is $(\bm{b},p,\varepsilon)$-holomorphic for each
$\tau\in\mathcal{G}$.
Step C: $(\bm{b},p,\varepsilon)$-holomorphy of $\mathcal{A}$. It remains to
conclude that $\mathcal{A}$ is $(\bm{b},p,\varepsilon)$-holomorphic. It
follows from Eq. 3.5 that
(3.24) $\mathcal{A}=\sum_{\tau\in\mathcal{G}}\mathcal{A}_{\tau}$
and since the addition of a finite number of holomorphic maps preserves
holomorphy, this yields the desired result.
### 3.2 Parametric Shape Holomorphy of BIOs with Singular Kernel
We now address the case in which the kernel function ${\sf
a}:\Gamma_{\bm{{y}}}\times\mathbb{R}^{3}\rightarrow\mathbb{C}$ of Eq. 1.2 is
allowed to have a singularity in the second variable, however we assume that
is integrable. To this end, we define
(3.25) $\displaystyle\mathfrak{U}\coloneqq\\{{\bf
z}\in\mathbb{C}^{d}:\Re\\{{\bf z}\cdot{\bf z}\\}>0\\},$
where $d\in\\{2,3\\}$.
###### Theorem 3.3.
Let 2.8 be fulfilled. In addition, assume that
1. 1.
There exists $\varepsilon>0$ such that for each $\tau\in\mathcal{G}$,
$\tilde{\mathbf{u}}\in\tilde{\tau}$ and for all ${\bf z}\in\mathfrak{U}$ the
map
(3.26)
$\mathbb{U}\ni{\bm{{y}}}\mapsto\mathsf{a}\left(\bm{r}_{\bm{{y}}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right),{\bf
z}\right)\in\mathbb{C}$
is $(\bm{b},p,\varepsilon)$-holomorphic and continuous.
2. 2.
Let $\bm{\rho}\coloneqq(\rho_{j})_{j\geq 1}$ be any
$(\bm{b},\varepsilon)$-admissible sequence of numbers strictly larger than
one. For any ${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$ the map
(3.27) $\mathfrak{U}\ni{\bf
z}\mapsto\mathsf{a}\left(\bm{r}_{\bm{z}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right),{\bf
z}\right)\in\mathbb{C}$
is entire.
3. 3.
Again, let $\bm{\rho}\coloneqq(\rho_{j})_{j\geq 1}$ be any
$(\bm{b},\varepsilon)$-admissible sequence of numbers strictly larger than
one. There exists $C_{\mathsf{a}}>0$ and $\nu\in[0,2)$ such that for any
$\tau\in\mathcal{G}$, $\tilde{\mathbf{u}}\in\tilde{\tau}$, and
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$ one has
(3.28)
$\left\lvert\mathsf{a}\left(\bm{r}_{\bm{z}}\circ\chi_{\tau}\left(\hat{{\bf
y}}\right),\bm{r}_{\bm{z}}(\hat{{\bf x}})-\bm{r}_{\bm{z}}(\hat{{\bf
y}})\right)\right\rvert\leq\frac{C_{\mathsf{a}}}{\left\lVert\bm{r}_{\bm{z}}(\hat{{\bf
x}})-\bm{r}_{\bm{z}}(\hat{{\bf y}})\right\rVert^{\nu}},\quad\left(\hat{{\bf
x}},\hat{{\bf
y}}\right)\in\left(\hat{\Gamma}\times\hat{\Gamma}\right)^{\star},$
Then, the map
(3.29)
$\mathcal{A}:\mathbb{U}\rightarrow\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right):{\bm{{y}}}\mapsto\hat{\mathsf{A}}_{{\bm{{y}}}}$
with $\hat{\mathsf{A}}_{\bm{{y}}}$ as in Eq. 3.1 is
$(\bm{b},p,\varepsilon)$-holomorphic and continuous.
For the proof of the theorem, we follow the general strategy of the proof of
Theorem 3.1 and make the necessary modifications for singular kernel
functions.
###### Proof 3.4 (Proof of Theorem 3.3).
As per usual, throughout this proof let $\bm{\rho}\coloneqq(\rho_{j})_{j\geq
1}$ be any sequence of numbers strictly larger than one that is
$(\bm{b},\varepsilon)$-admissible for some $\varepsilon>0$.
Step A: Localization of $\hat{\mathsf{A}}_{\bm{{y}}}$ to a patch. In analogy
to the proof of Theorem 3.1 we set
(3.30) $\displaystyle\mathsf{k}_{\tau,{\bm{{y}}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\coloneqq\mathsf{a}\left(\bm{r}_{\bm{{y}}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right),\bm{r}_{\bm{{y}}}(\hat{\mathbf{x}})-\bm{r}_{\bm{{y}}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right)$
and
(3.31) $\mathsf{b}_{\tau,{\bm{z}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\coloneqq\mathsf{k}_{\tau,{\bm{{y}}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})}$
for each $(\hat{{\bf x}},\tilde{\mathbf{u}})\in\hat{\Gamma}\times\tilde{\tau}$
with $\hat{{\bf x}}\neq\chi_{\tau}(\tilde{\mathbf{u}})$, $\tau\in\mathcal{G}$,
and ${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$ and define
(3.32)
$\left(\hat{\mathsf{A}}_{\tau,{\bm{z}}}\,\hat{\varphi}\right)\left(\hat{{\bf
x}}\right)\coloneqq\int\limits_{\tilde{\tau}}\mathsf{b}_{\tau,{\bm{z}}}(\hat{\bf
x},\hat{{\bf
u}})\left(\hat{\varphi}\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\,\text{d}\tilde{\mathbf{u}},$
for $\hat{\varphi}\in L^{2}(\hat{\Gamma})$. Using the Riesz-Thorin
interpolation introduced in Section 2.5 we obtain for each
$\tau\in\mathcal{G}$
(3.33)
$\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}\right\rVert_{\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right)}\leq\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}\right\rVert_{\mathscr{L}\left(L^{1}(\hat{\Gamma}),L^{1}(\hat{\Gamma})\right)}^{\frac{1}{2}}\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}\right\rVert_{\mathscr{L}\left(L^{\infty}(\hat{\Gamma}),L^{\infty}(\hat{\Gamma})\right)}^{\frac{1}{2}}.$
Lemma 2.10 yields for each $\tau\in\mathcal{G}$ and for any
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$
(3.34) $\displaystyle\left\lVert\bm{r}_{{\bm{z}}}(\hat{\bf
x})-\bm{r}_{\bm{z}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{2}$
$\displaystyle\geq\Re\\{\left(\bm{r}_{{\bm{z}}}(\hat{\bf
x})-\bm{r}_{\bm{z}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right)\cdot\left(\bm{r}_{{\bm{z}}}(\hat{\bf
x})-\bm{r}_{{\bm{z}}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right)\\}$
$\displaystyle\geq\eta\left\lVert\hat{\bf
x}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{2}$
$\displaystyle>0.$
This, together with Item 3 from the assumptions, yields
(3.35) $\left\lvert\mathsf{b}_{\tau,{\bm{z}}}(\hat{\bf x},\hat{\bf
u})\right\rvert\leq\frac{C_{\mathsf{a}}}{\left\lVert\bm{r}_{\bm{z}}(\hat{\bf
x})-\bm{r}_{\bm{z}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu}}\left\lvert\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})}\right\rvert\leq\frac{\widetilde{C}_{\mathsf{a}}}{\left\lVert\hat{\bf
x}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu}}\left\lvert\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})}\right\rvert,$
with $\widetilde{C}_{\mathsf{a}}=C_{\mathsf{a}}\eta^{-\nu/2}$ and for any
$(\hat{\bf x},\hat{\bf u})\in\hat{\Gamma}\times\tilde{\tau}$ with $\hat{\bf
x}\neq\chi_{\tau}\left(\tilde{\mathbf{u}}\right)$. Thus, for each
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$, it holds
(3.36)
$\displaystyle\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}\right\rVert_{\mathscr{L}\left(L^{1}(\hat{\Gamma}),L^{1}(\hat{\Gamma})\right)}$
$\displaystyle=\underset{\tilde{\mathbf{u}}\in\tilde{\tau}}{\operatorname{ess}\sup}\int\limits_{\hat{\Gamma}}\left\lvert\mathsf{b}_{\tau,{\bm{z}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\right\rvert\text{d}\text{s}_{\hat{\mathbf{x}}}$
$\displaystyle\leq\widetilde{C}_{\mathsf{a}}\,\underset{\tilde{\mathbf{u}}\in\tilde{\tau}}{\operatorname{ess}\sup}\left\lvert\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})}\right\rvert\,\underset{\tilde{\mathbf{u}}\in\tilde{\tau}}{\operatorname{ess}\sup}\int\limits_{\hat{\Gamma}}\frac{1}{\left\lVert\hat{\bf
x}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu}}\text{d}\text{s}_{\hat{\mathbf{x}}},$
where the latter integral is finite for any
$\tilde{\mathbf{u}}\in\tilde{\tau}$ since $\nu\in[0,2)$.
Furthermore, for each ${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$, it holds
$\displaystyle\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}\right\rVert_{\mathscr{L}\left(L^{\infty}(\hat{\Gamma}),L^{\infty}(\hat{\Gamma})\right)}$
$\displaystyle=\underset{\hat{\mathbf{x}}\in\hat{\Gamma}}{\operatorname{ess}\sup}\int\limits_{\tilde{\tau}}\left\lvert\mathsf{b}_{\tau,{\bm{z}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\right\rvert\text{d}\tilde{\mathbf{u}}$
$\displaystyle\leq\widetilde{C}_{\mathsf{a}}\underset{\hat{\bf
u}\in\tilde{\tau}}{\operatorname{ess}\sup}\left\lvert\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})}\right\rvert\,\underset{\hat{\mathbf{x}}\in\hat{\Gamma}}{\operatorname{ess}\sup}\int\limits_{\tilde{\tau}}\frac{1}{\left\lVert\hat{\bf
x}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu}}\text{d}\tilde{\mathbf{u}}$
$\displaystyle\leq\widetilde{C}_{\mathsf{a}}\underset{\hat{\bf
u}\in\tilde{\tau}}{\operatorname{ess}\sup}\left\lvert\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})}\right\rvert\,\underset{\hat{\mathbf{x}}\in\hat{\Gamma}}{\operatorname{ess}\sup}\int\limits_{\hat{\Gamma}}\frac{1}{\left\lVert\hat{\bf
x}-\hat{{\bf y}}\right\rVert^{\nu}}\text{d}\text{s}_{\hat{{\bf y}}},$
where the assumptions on $\nu$ imply again that latter integral is uniformly
bounded for all $\hat{{\bf x}}\in\hat{\Gamma}$. This shows that
$\hat{\mathsf{A}}_{\tau,{\bm{z}}}\colon L^{2}(\hat{\Gamma})\to
L^{2}(\hat{\Gamma})$ is a bounded linear operator and allows us to define for
each ${\tau}\in\mathcal{G}$
(3.37)
$\mathcal{A}_{\tau}:\mathcal{O}_{\bm{\rho}}\rightarrow\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right):{\bm{z}}\mapsto\hat{\mathsf{A}}_{\tau,{\bm{z}}}.$
Step A’: Regularization of localized operators
$\hat{\mathsf{A}}_{\tau,{\bm{z}}}$. Let $\phi:[0,\infty)\rightarrow\mathbb{R}$
be a $\mathscr{C}^{1}([0,\infty))$ function with
$\phi(t)\begin{cases}=0&t\in[0,\frac{1}{2}],\\\
\in[0,1]&t\in(\frac{1}{2},1),\\\ =1&t\in[1,\infty).\end{cases}$
For each $n\in\mathbb{N}$, $\tau\in\mathcal{G}$, and
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$ define
(3.38) $\mathsf{b}^{(n)}_{\tau,{\bm{z}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\coloneqq\mathsf{b}_{\tau,{\bm{z}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})$
and
(3.39)
$\left(\hat{\mathsf{A}}^{(n)}_{\tau,{\bm{z}}}\,\hat{\varphi}\right)\left(\hat{{\bf
x}}\right)\coloneqq\int\limits_{\tilde{\tau}}\mathsf{b}^{(n)}_{\tau,{\bm{z}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\left(\hat{\varphi}\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\,\text{d}\tilde{\mathbf{u}},\quad\hat{\bf
x}\in\hat{\Gamma},$
for all $\hat{\varphi}\in L^{2}(\hat{\Gamma})$. Using the Riesz-Thorin
interpolation theorem Eq. 2.29 we obtain
(3.40)
$\displaystyle\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}-\hat{\mathsf{A}}_{\tau,{\bm{z}}}^{(n)}\right\rVert_{\mathscr{L}(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma}))}$
$\displaystyle\hskip
42.67912pt\leq\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}-\hat{\mathsf{A}}_{\tau,{\bm{z}}}^{(n)}\right\rVert_{\mathscr{L}(L^{1}(\hat{\Gamma}),L^{1}(\hat{\Gamma}))}^{\frac{1}{2}}\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}-\hat{\mathsf{A}}_{\tau,{\bm{z}}}^{(n)}\right\rVert_{\mathscr{L}(L^{\infty}(\hat{\Gamma}),L^{\infty}(\hat{\Gamma}))}^{\frac{1}{2}}.$
To estimate this quantity, we note that for $\hat{\bf x}\in\hat{\Gamma}$ it
holds
$\displaystyle\left(\hat{\mathsf{A}}_{\tau,{\bm{z}}}\,\hat{\varphi}\right)\left(\hat{{\bf
x}}\right)-\left(\hat{\mathsf{A}}^{(n)}_{\tau,{\bm{z}}}\,\hat{\varphi}\right)\left(\hat{{\bf
x}}\right)$
$\displaystyle\qquad\qquad\qquad=\int\limits_{\tilde{\tau}}\mathsf{b}_{\tau,{\bm{z}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\left(1-\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})\right)\left(\hat{\varphi}\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\,\text{d}\tilde{\mathbf{u}}.$
Therein, along the lines of step A, the integrand can be estimated as
(3.41) $\displaystyle\left\lvert\mathsf{b}_{\tau,{\bm{z}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\left(1-\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})\right)\right\rvert\leq\widetilde{C}_{\mathsf{a}}\frac{\left(1-\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})\right)}{\left\lVert\hat{\bf
x}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu}}\left\lvert\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})}\right\rvert,$
for $(\hat{\bf x},\hat{\bf u})\in\hat{\Gamma}\times\tilde{\tau}$ with
$\hat{\bf x}\neq\chi_{\tau}\left(\tilde{\mathbf{u}}\right)$. Thus, it holds
(3.42)
$\displaystyle\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}-\hat{\mathsf{A}}_{\tau,{\bm{z}}}^{(n)}\right\rVert_{\mathscr{L}\left(L^{1}(\hat{\Gamma}),L^{1}(\hat{\Gamma})\right)}$
$\displaystyle\qquad\qquad=\underset{\tilde{\mathbf{u}}\in\tilde{\tau}}{\operatorname{ess}\sup}\int\limits_{\hat{\Gamma}}\left\lvert\mathsf{b}_{\tau,{\bm{z}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\right\rvert\left(1-\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})\right)\text{ds}_{\hat{\mathbf{x}}}$
$\displaystyle\qquad\qquad\leq\widetilde{C}_{\mathsf{a}}\underset{\tilde{\mathbf{u}}\in\tilde{\tau}}{\operatorname{ess}\sup}\left\lvert\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})}\right\rvert\,\underset{\hat{{\bf
y}}\in\hat{\Gamma}}{\operatorname{ess}\sup}\int\limits_{\hat{\Gamma}}\frac{1-\phi(n\left\lVert\hat{\mathbf{x}}-\hat{\mathbf{y}}\right\rVert^{\nu})}{\left\lVert\hat{\mathbf{x}}-\hat{\mathbf{y}}\right\rVert^{\nu}}\text{ds}_{\hat{\mathbf{x}}}.$
On the other hand, it holds
(3.43)
$\displaystyle\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}-\hat{\mathsf{A}}_{\tau,{\bm{z}}}^{(n)}\right\rVert_{\mathscr{L}\left(L^{\infty}(\hat{\Gamma}),L^{\infty}(\hat{\Gamma})\right)}$
$\displaystyle\qquad\qquad\leq\underset{\hat{{\bf
x}}\in\hat{\Gamma}}{\operatorname{ess}\sup}\int\limits_{\tilde{\tau}}\left\lvert\mathsf{b}_{\tau,{\bm{z}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\left(1-\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})\right)\right\rvert\text{d}\tilde{\mathbf{u}}$
$\displaystyle\qquad\qquad\leq\widetilde{C}_{\mathsf{a}}\underset{\tilde{\mathbf{u}}\in\tilde{\tau}}{\operatorname{ess}\sup}\left\lvert\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})}\right\rvert\,\underset{\hat{{\bf
x}}\in\hat{\Gamma}}{\operatorname{ess}\sup}\int\limits_{\tilde{\tau}}\frac{1-\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})}{\left\lVert\hat{\bf
x}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu}}\text{d}\tilde{\mathbf{u}}$
$\displaystyle\qquad\qquad\leq\widetilde{C}_{\mathsf{a}}\underset{\tilde{\mathbf{u}}\in\tilde{\tau}}{\operatorname{ess}\sup}\left\lvert\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})}\right\rvert\underset{\hat{{\bf
x}}\in\hat{\Gamma}}{\operatorname{ess}\sup}\int\limits_{\hat{\Gamma}}\frac{1-\phi(n\left\lVert\hat{\mathbf{x}}-\hat{{\bf
y}}\right\rVert^{\nu})}{\left\lVert\hat{\bf x}-\hat{{\bf
y}}\right\rVert^{\nu}}\text{d}\text{s}_{\hat{{\bf y}}}.$
Upon noting that the integrals in Eq. 3.42 and Eq. 3.43 are equal, it remains
to estimate the essential supremum of this integral. Let us calculate
(3.44) $\displaystyle\underset{\hat{{\bf
x}}\in\hat{\Gamma}}{\operatorname{ess}\sup}\int\limits_{\hat{\Gamma}}\frac{1-\phi(n\left\lVert\hat{\mathbf{x}}-\hat{{\bf
y}}\right\rVert^{\nu})}{\left\lVert\hat{\bf x}-\hat{{\bf
y}}\right\rVert^{\nu}}\text{d}\text{s}_{\hat{{\bf y}}}$
$\displaystyle\leq\underset{\hat{{\bf
x}}\in\hat{\Gamma}}{\operatorname{ess}\sup}\int\limits_{\hat{\Gamma}\cap
B(\hat{{\bf x}},1/n)}\frac{1}{\left\lVert\hat{\bf x}-\hat{{\bf
y}}\right\rVert^{\nu}}\text{d}\text{s}_{\hat{{\bf y}}}$ $\displaystyle\leq
C\int_{B(0,1/n)}\frac{1}{\|\tilde{\mathbf{u}}\|^{\nu}}\text{d}\tilde{\mathbf{u}}$
$\displaystyle=\int\limits_{0}^{1/n}\int\limits_{0}^{2\pi}r^{1-\nu}\text{d}r\text{d}t$
$\displaystyle=\frac{2\pi}{2-\nu}n^{\nu-2}.$
Next, recalling Eq. 3.40 we get
(3.45)
$\sup_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}-\hat{\mathsf{A}}_{\tau,{\bm{z}}}^{(n)}\right\rVert_{\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right)}\leq\widetilde{C}_{\mathsf{a}}\frac{2\pi}{2-\nu}\underbrace{\sup_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\underset{\tilde{\mathbf{u}}\in\tilde{\tau}}{\operatorname{ess}\sup}\left\lvert\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})}\right\rvert}_{(\spadesuit)}n^{\nu-2}.$
As a consequence of Lemma 2.14, the term $(\spadesuit)$ in (3.45) is bounded.
Recalling that we have assumed $\nu\in[0,2)$ implies
(3.46)
$\lim_{n\rightarrow\infty}\sup_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\left\lVert\hat{\mathsf{A}}_{\tau,{\bm{z}}}-\hat{\mathsf{A}}_{\tau,{\bm{z}}}^{(n)}\right\rVert_{\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right)}=0.$
This implies that for each $\tau\in\mathcal{G}$
(3.47)
$\mathcal{A}_{\tau}^{(n)}:\mathcal{O}_{\bm{\rho}}\rightarrow\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right):{\bm{z}}\mapsto\hat{\mathsf{A}}_{\tau,{\bm{z}}}^{(n)}$
converges uniformly in $\mathcal{O}_{\bm{\rho}}$ to $\mathcal{A}_{\tau}$ from
Eq. 3.37.
Step B: $(\bm{b},p,\varepsilon)$-holomorphy of localized, regularized
operators $\hat{\mathsf{A}}_{\tau,{\bm{{y}}}}^{(n)}$. We aim to verify the
assumptions of Theorem 3.1 to show that
(3.48)
$\mathcal{A}_{\tau}^{(n)}:\mathbb{U}\rightarrow\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right):{\bm{{y}}}\mapsto\hat{\mathsf{A}}_{\tau,{\bm{{y}}}}^{(n)}$
is $(\bm{b},p,\varepsilon)$-holomorphic with $\varepsilon>0$ _independent_ of
$n\in\mathbb{N}$. Thus, by setting
(3.49) $\mathsf{k}_{\tau,{\bm{{y}}}}^{(n)}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\coloneqq\mathsf{k}_{\tau,{\bm{{y}}}}\left(\hat{{\bf
x}},\tilde{\mathbf{u}}\right)\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})$
with $\mathsf{k}_{\tau,{\bm{{y}}}}$ from Eq. 3.30 we need to show that
(3.50) $\mathbb{U}\ni{\bm{{y}}}\mapsto\mathsf{k}_{\tau,{\bm{{y}}}}^{(n)}\in
L^{\infty}\left(\hat{\Gamma}\times\tilde{\tau}\right)$
is $(\bm{b},p,\varepsilon)$-holomorphic with $\varepsilon>0$, again,
independent of $n$ and continuous. To this end, due to Eq. 3.34, for each
fixed $(\hat{\bf x},\hat{\bf u})\in\hat{\Gamma}\times\tilde{\tau}$ such that
${\bf x}\neq\chi_{\tau}\left(\tilde{\mathbf{u}}\right)$ and any
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$ it holds
(3.51) $\left(\bm{r}_{{\bm{z}}}(\hat{\bf
x})-\bm{r}_{\bm{z}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right)\cdot\left(\bm{r}_{{\bm{z}}}(\hat{\bf
x})-\bm{r}_{{\bm{z}}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right)\in\mathfrak{U}.$
Thus, due to Item 3 in the assumptions and Eq. 3.34, it follows that
(3.52)
$\displaystyle\sup_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\left\lvert\mathsf{k}^{(n)}_{\tau,{\bm{z}}}(\hat{\bf
x},\hat{\bf u})\right\rvert$ $\displaystyle\leq
C_{\mathsf{a}}\sup_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\frac{\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})}{\left\lVert\bm{r}_{\bm{z}}(\hat{\bf
x})-\bm{r}_{\bm{{y}}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu}}$
$\displaystyle\leq\widetilde{C}_{\mathsf{a}}\frac{\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})}{\left\lVert\hat{{\bf
x}}-\chi_{\tau}(\tilde{\mathbf{u}})\right\rVert^{\nu}}$
$\displaystyle\leq\widetilde{C}_{\mathsf{a}}n\sup_{t\in[0,\infty)}\left\lvert\phi^{\prime}(t)\right\rvert.$
Thus,
(3.53)
$\sup_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\left\lVert\mathsf{k}^{(n)}_{\tau,{\bm{z}}}\right\rVert_{L^{\infty}(\hat{\Gamma}\times\tilde{\tau})}\leq\widetilde{C}_{\mathsf{a}}n\sup_{t\in[0,\infty)}\left\lvert\phi^{\prime}(t)\right\rvert.$
This allows to verify Item 1 and Item 3 in the Definition 2.6 of
$(\bm{b},p,\varepsilon)$-holomorphy and it only remains to verify Item 2.
To this end, we observe that the map
$\mathbb{U}\ni{\bm{{y}}}\mapsto\bm{r}_{\bm{{y}}}(\hat{\mathbf{x}})-\bm{r}_{\bm{{y}}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\in\mathbb{C}$
is $(\bm{b},p,\varepsilon)$-holomorphic itself as a consequence of Lemma 2.12,
with $\varepsilon>0$ independent of $(\hat{\bf x},\hat{\bf
u})\in\hat{\Gamma}\times\tilde{\tau}$. Consequently, recalling Item 1 and Item
2 in the assumptions, we may conclude that for each fixed $(\hat{\bf
x},\hat{\bf u})\in\hat{\Gamma}\times\tilde{\tau}$ such $\hat{\bf
x}\neq\chi_{\tau}\left(\tilde{\mathbf{u}}\right)$ that the map
(3.54)
$\displaystyle\mathbb{U}\ni{\bm{{y}}}\mapsto\mathsf{k}_{\tau,{\bm{{y}}}}(\hat{\bf
x},\hat{\bf u})\in\mathbb{C}$
is $(\bm{b},p,\varepsilon)$-holomorphic for some $\varepsilon>0$ that does not
depend on $(\hat{\bf x},\hat{\bf u})\in\hat{\Gamma}\times\tilde{\tau}$. To
show holomorphy on $\hat{\Gamma}\times\tilde{\tau}$, let
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$, $j\in\mathbb{N}$ and consider
$h\in\mathbb{C}$ small enough such that
${\bm{z}}+h\bm{e}_{j}\in\mathcal{O}_{\bm{\rho}}$. Using Taylor’s expansion in
the direction $z_{j}$ we get
(3.55) $\displaystyle\mathsf{k}^{(n)}_{\tau,{\bm{z}}+h\bm{e}_{j}}(\hat{\bf
x},\hat{\bf u})$ $\displaystyle=\mathsf{k}^{(n)}_{\tau,{\bm{z}}}(\hat{\bf
x},\hat{\bf
u})+h\left(\partial_{z_{j}}\mathsf{k}_{\tau,{\bm{z}}}\right)\\!(\hat{\bf
x},\hat{\bf
u})\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})$
$\displaystyle\qquad+h^{2}\int\limits_{0}^{1}(1-\eta)\left(\partial^{2}_{z_{j}}\mathsf{k}_{\tau,{\bm{z}}+\eta
h\bm{e}_{j}}\right)\\!(\hat{\bf x},\hat{\bf
u})\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu}){{\text{d}}}\eta.$
Thus, for $\zeta_{h}$ such that
$B(z_{j}+h\eta,\zeta_{h})\subset\mathcal{O}_{\rho_{j}}$, Cauchy’s integral
formula yields
(3.56) $\displaystyle\left(\partial^{2}_{z_{j}}\mathsf{k}_{\tau,{\bm{z}}+\eta
h\bm{e}_{j}}\right)(\hat{\bf x},\hat{\bf
u})\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})$
$\displaystyle\qquad\qquad=\frac{1}{\pi\imath}\int\limits_{z^{\prime}_{j}\in\partial
B(z_{j}+h\eta,\zeta_{{\bm{z}},h})}\frac{\mathsf{k}_{\tau,{\bm{z}}^{\prime}}(\hat{\bf
x},\hat{\bf u})}{(z^{\prime}_{j}-z_{j}-\eta
h)^{3}}\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu}){{\text{d}}}z^{\prime}_{j},$
with ${\bm{z}}^{\prime}=(z_{1},\dots,z_{j}^{\prime},\dots)$, and
(3.57)
$\displaystyle\left\lvert\left(\partial^{2}_{z_{j}}\mathsf{k}_{\tau,{\bm{z}}+\eta
h\bm{e}_{j}}\right)(\hat{\bf x},\hat{\bf
u})\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu})\right\rvert\leq\frac{2}{\zeta_{h}^{2}}\sup_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\left\lvert\mathsf{k}_{{\bm{z}}}^{(n)}(\hat{\bf
x},\hat{\bf u})\right\rvert.$
Hence, from Eq. 3.55,
(3.58)
$\left\lVert\frac{\mathsf{k}^{(n)}_{{\bm{z}}+h\bm{e}_{j}}-\mathsf{k}^{(n)}_{{\bm{z}}}}{h}-\mathsf{g}^{(n)}_{\tau,{\bf
z}}\right\rVert_{L^{\infty}\left(\hat{\Gamma}\times\tilde{\tau}\right)}\leq\frac{\left\lvert
h\right\rvert}{\zeta_{h}^{2}}\sup_{{\bm{z}}\in\mathcal{O}_{\bm{\rho}}}\left\lVert\mathsf{k}_{{\bm{z}}}^{(n)}\right\rVert_{L^{\infty}(\hat{\Gamma}\times\tilde{\tau})},$
where $\mathsf{g}^{(n)}_{\tau,{\bf z}}\in
L^{\infty}\left(\hat{\Gamma}\times\tilde{\tau}\right)$ is defined for each
$n\in\mathbb{N}$ as
(3.59) $\mathsf{g}^{(n)}_{\tau,{\bm{z}}}(\hat{\bf x},\hat{\bf
u})\coloneqq\left(\partial_{z_{j}}\mathsf{k}_{\tau,{\bm{z}}}\right)\\!(\hat{\bf
x},\hat{\bf
u})\phi(n\left\lVert\hat{\mathbf{x}}-\chi_{\tau}\left(\tilde{\mathbf{u}}\right)\right\rVert^{\nu}),$
and using Eq. 3.53 we obtain
(3.60)
$\left\lVert\frac{\mathsf{k}^{(n)}_{{\bm{z}}+h\bm{e}_{j}}-\mathsf{k}^{(n)}_{{\bm{z}}}}{h}-\mathsf{g}^{(n)}_{\tau,{\bf
z}}\right\rVert_{L^{\infty}\left(\hat{\Gamma}\times\tilde{\tau}\right)}\leq\widetilde{C}_{\mathsf{a}}\frac{\left\lvert
h\right\rvert}{\zeta^{2}_{h}}n\sup_{t\in[0,\infty)}\left\lvert\phi^{\prime}(t)\right\rvert.$
Thus, upon noting that $\zeta_{h}$ is bounded away from zero for $h\to 0$,
this shows that for each $n\in\mathbb{N}$ the map Eq. 3.50 and, due to Theorem
3.1 also Eq. 3.48, are $(\bm{b},p,\varepsilon)$-holomorphic with
$\varepsilon>0$ _independent_ of $n$.
Step C: $(\bm{b},p,\varepsilon)$-holomorphy of $\mathcal{A}$. Now, fix
$\tau\in\mathcal{G}$, and consider
$\mathcal{A}_{\tau}^{(n)}\colon\mathcal{O}_{\bm{\rho}}\to\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right)$
with $\mathcal{A}_{\tau}^{(n)}$ as defined in Eq. 3.47. Then, A and A’ show
that $\mathcal{A}_{\tau}^{(n)}$ converges uniformly to $\mathcal{A}_{\tau}$ on
$\mathcal{O}_{\bm{\rho}}$ with $\mathcal{A}_{\tau}$ as in Eq. 3.37. Step B
shows that $\mathcal{A}_{\tau}^{(n)}$ is $(\bm{b},p,\varepsilon)$-holomorphic
with $\varepsilon$ independent of $n$. Thus, Lemma 2.4 yields that
$\mathcal{A}_{\tau}$ is $(\bm{b},p,\varepsilon)$-holomorphic.
In the same way as in Step C in the proof of Theorem 3.1, we now conclude that
(3.61) $\mathcal{A}=\sum_{\tau\in\mathcal{G}}\mathcal{A}_{\tau}$
with $\mathcal{A}$ as in Eq. 3.29 is $(\bm{b},p,\varepsilon)$-holomorphic for
some $\varepsilon>0$, which is the assertion.
### 3.3 Parametric Holomorphy of the Parameter–to–solution Map
A key consequence of the presently obtained parametric holomorphy result of
the parameter-to-operator map Eq. 1.3 is that of the parameter-to-solution map
Eq. 1.4.
###### Corollary 3.5.
Let 2.8 be satisfied with $\bm{b}\in\ell^{p}(\mathbb{N})$ and $p\in(0,1)$, and
let the assumptions of Theorem 3.3 be fulfilled as well. In addition, assume
that
$\hat{\mathsf{A}}_{\bm{{y}}}\in\mathscr{L}_{{\text{iso}}}(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma}))$
for each ${\bm{{y}}}\in\mathbb{U}$ and that the map
$\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{f}_{{\bm{{y}}}}\in L^{2}(\hat{\Gamma})$ is
$(\bm{b},p,\varepsilon)$-holomorphic and continuous for some $\varepsilon>0$.
Then there exists $\varepsilon^{\prime}>0$ such that the parameter-to-solution
map
(3.62) $\mathcal{S}:\mathbb{U}\to L^{2}(\hat{\Gamma}):{\bm{{y}}}\mapsto
u_{\bm{{y}}}\mathrel{\mathrel{\mathop{:}}=}\hat{\mathsf{A}}_{{\bm{{y}}}}^{-1}\hat{f}_{\bm{{y}}},$
is $(\bm{b},p,\varepsilon^{\prime})$-holomorphic and continuous.
###### Proof 3.6.
The assertion then follows as a direct consequence from Item 2 and Item 3 in
Proposition 2.5.
## 4 Applications to Boundary Integral Equations on Parametric Boundaries
In this section, we explore the applicability of the main result of Section 3,
i.e. Theorem 3.3, to the boundary integral operators obtained from the
boundary reduction of the sound-soft Helmholtz scattering problem.
### 4.1 Model Problem: Sound-Soft Acoustic Scattering
Let ${\text{D}}\subset\mathbb{R}^{3}$ be a bounded Lipschitz domain with
boundary $\Gamma:=\partial{\text{D}}$ and denote by
${\text{D}}^{{\text{c}}}:=\mathbb{R}^{3}\backslash\overline{{\text{D}}}$ the
corresponding exterior domain. Given $\kappa>0$ and
$\bm{\hat{d}}_{\text{inc}}\in\mathbb{S}^{2}\coloneqq\\{{\bf
x}\in\mathbb{R}^{3}:\left\lVert{\bf x}\right\rVert=1\\}$, we define an
incident plane wave $u^{\text{inc}}({\bf x})\coloneqq\exp(\imath\kappa{\bf
x}\cdot\bm{\hat{d}}_{\text{inc}})$ and aim to find the scattered wave
$u^{\text{scat}}\in H^{1}_{{\text{loc}}}({\text{D}}^{{\text{c}}})$ such that
the total field $u\coloneqq u^{\text{inc}}+u^{\text{scat}}$ satisfies
(4.1a) $\displaystyle\Delta u+\kappa^{2}u$ $\displaystyle=0,\quad\text{in
}{\text{D}}^{{\text{c}}},$ (4.1b) $\displaystyle u$
$\displaystyle=0,\quad\text{on }\Gamma,$
and the scattered field additionally satisfies the Sommerfeld radiation
conditions
(4.2) $\displaystyle\frac{\partial u^{\text{scat}}}{\partial r}({\bf
x})-\imath\kappa u^{\text{scat}}({\bf x})=o\left(r^{-1}\right)$
as $r\coloneqq\left\lVert{\bf x}\right\rVert\rightarrow\infty$, uniformly in
$\hat{{\bf x}}:={\bf x}/r$. Thus, since $u^{\text{inc}}$ satisfies Eq. 4.1a on
its own, we seek $u^{\text{scat}}\in
H_{{\text{loc}}}^{1}({\text{D}}^{{\text{c}}})$ such that
(4.3a) $\displaystyle\Delta u^{\text{scat}}+\kappa^{2}u^{\text{scat}}$
$\displaystyle=0,\quad\text{in }{\text{D}}^{{\text{c}}},$ (4.3b)
$\displaystyle u^{\text{scat}}$ $\displaystyle=-u^{\text{inc}},\quad\text{on
}\Gamma,$
and Eq. 4.2 holds.
### 4.2 Boundary Integral Operators
We proceed to perform a boundary reduction of Eq. 4.2 and Eq. 4.3 by means of
boundary potentials in order to obtain a equivalent boundary integral
formulation. To this end, we first introduce the Green’s function of the
Helmholtz operator with leading frequency $\kappa\in\mathbb{R}$ in
$\mathbb{R}^{3}$
(4.4) $\text{G}^{(\kappa)}({\bf
z})\coloneqq\frac{\exp(\imath\kappa\left\lVert{\bf
z}\right\rVert)}{4\pi\left\lVert{\bf z}\right\rVert},\quad{\bf
z}\in\mathbb{R}^{3}\backslash\\{{\bf 0}\\},$
with $\imath^{2}=-1$ being the imaginary complex unit. Next, we define the
acoustic single layer potential
$\mathcal{S}_{\Gamma}^{(\kappa)}:H^{-\frac{1}{2}}(\Gamma)\rightarrow
H^{1}_{{\text{loc}}}(\Delta,{\text{D}}^{c})$
$\left(\mathcal{S}^{(\kappa)}_{\Gamma}\varphi\right)({\bf
x})\mathrel{\mathrel{\mathop{:}}=}\int\limits_{\Gamma}\text{G}^{(\kappa)}({\bf
x}-{\bf y})\varphi({\bf y})\text{d}\text{s}_{\bf y},\quad{\bf
x}\in{\text{D}}^{{\text{c}}},$
and the double layer potential
$\mathcal{D}_{\Gamma}^{(\kappa)}:H^{\frac{1}{2}}(\Gamma)\rightarrow
H^{1}_{{\text{loc}}}(\Delta,{\text{D}}^{{\text{c}}})$
$\left(\mathcal{D}_{\Gamma}^{(\kappa)}\phi\right)({\bf
x})\mathrel{\mathrel{\mathop{:}}=}\int\limits_{\Gamma}\frac{\partial\text{G}^{(\kappa)}({\bf
x}-{\bf y})}{\partial\mathbf{n}_{\Gamma}({\bf y})}\phi({\bf
y})\text{d}\text{s}_{\bf y},\quad{\bf x}\in{\text{D}}^{{\text{c}}}.$
By $\mathbf{n}_{\Gamma}({\bf y})$ we denote the outward pointing normal on
${\bf y}\in\Gamma$. Both operators are continuous and satisfy Eq. 4.3a and the
Sommerfeld radiation condition Eq. 4.2, see, e.g., [48].
By taking exterior Dirichlet and Neumann traces
$\gamma_{\Gamma}^{{\text{c}}}\colon
H^{1}_{{\text{loc}}}({\text{D}}^{{\text{c}}})\to
H^{\frac{1}{2}}(\Gamma)\quad\text{and}\quad\frac{\partial}{\partial\mathbf{n}_{\Gamma}}\colon
H^{1}(\Delta,{\text{D}}^{{\text{c}}})\to H^{-\frac{1}{2}}(\Gamma)$
we obtain the single layer operator
(4.5)
$\displaystyle\mathsf{V}^{(\kappa)}_{\Gamma}\mathrel{\mathrel{\mathop{:}}=}\gamma_{\Gamma}^{{\text{c}}}\mathcal{S}^{(\kappa)}_{\Gamma}\colon
H^{-\frac{1}{2}}(\Gamma)\to H^{\frac{1}{2}}(\Gamma),$
the double layer operator
(4.6)
$\displaystyle\frac{1}{2}\mathsf{Id}+\mathsf{K}^{(\kappa)}_{\Gamma}\mathrel{\mathrel{\mathop{:}}=}\gamma_{\Gamma}^{{\text{c}}}\mathcal{D}^{(\kappa)}_{\Gamma}\colon
H^{\frac{1}{2}}(\Gamma)\to H^{\frac{1}{2}}(\Gamma),$
and the adjoint double layer operator
(4.7)
$\displaystyle\frac{1}{2}\mathsf{Id}-\mathsf{K}^{(\kappa)^{\prime}}_{\Gamma}\mathrel{\mathrel{\mathop{:}}=}\frac{\partial}{\partial\mathbf{n}_{\Gamma}}\mathcal{S}^{(\kappa)}_{\Gamma}\colon
H^{-\frac{1}{2}}(\Gamma)\to H^{-\frac{1}{2}}(\Gamma).$
The double layer operator and the adjoint double layer operator are indeed
adjoint operators in the bilinear $L^{2}(\Gamma)$-duality product [48].
### 4.3 Combined Boundary Integral Formulation
By means of the introduced potentials and boundary integral operators, the
sound-soft scattering problem from Section 4.1 can be cast into an equivalent
boundary integral formulation. Here, we explore two widely well-know
approaches.
#### 4.3.1 Direct Boundary Integral Formulation
According to Green’s representation formula, the total field
$u:{\text{D}}^{{\text{c}}}\rightarrow\mathbb{C}$ admits the expression
(4.8) $\displaystyle u({\bf x})=u^{\text{inc}}({\bf
x})-\mathcal{S}^{(\kappa)}_{\Gamma}\left(\frac{\partial
u}{\partial\mathbf{n}_{\Gamma}}\right)({\bf x}),\quad{\bf
x}\in{\text{D}}^{{\text{c}}}.$
The unknown boundary data $\frac{\partial u}{\partial\mathbf{n}_{\Gamma}}$
can, for example, be obtained by applying Dirichlet and Neumann traces to Eq.
4.8 and solving
(4.9a) $\displaystyle\mathsf{V}^{(\kappa)}_{\Gamma}\frac{\partial
u}{\partial\mathbf{n}_{\Gamma}}$
$\displaystyle=\gamma^{{\text{c}}}_{\Gamma}u^{\text{inc}},\quad\text{on}\;\Gamma,\quad\text{and},$
(4.9b)
$\displaystyle\left(\frac{1}{2}\mathsf{Id}+\mathsf{K}^{(\kappa)^{\prime}}_{\Gamma}\right)\frac{\partial
u}{\partial\mathbf{n}_{\Gamma}}$ $\displaystyle=\frac{\partial
u^{\text{inc}}}{\partial\mathbf{n}_{\Gamma}},\quad\text{on}\;\Gamma.$
However, well–posedness of Eq. 4.9a and Eq. 4.9b collapses when $\kappa^{2}$
corresponds to a Dirichlet or Neumann eigenvalue of the Laplacian in D,
respectively. A fix to this issue consists in taking a suitable linear
combination of Eq. 4.9a and Eq. 4.9b. Provided
$\eta\in\mathbb{R}\backslash\\{0\\}$, the so–called coupling parameter, we
define
(4.10)
$\displaystyle\mathsf{A}^{(\kappa,\eta)^{\prime}}_{\Gamma}\coloneqq\frac{1}{2}\mathsf{Id}+\mathsf{K}^{(\kappa)^{\prime}}_{\Gamma}-\imath\eta\mathsf{V}^{(\kappa)}_{\Gamma}.$
Exploiting that $\frac{\partial u}{\partial\mathbf{n}_{\Gamma}}\in
L^{2}(\Gamma)$, see, e.g., [40], a new boundary integral formulation of Eq.
4.1a using the direct approach reads as follows.
###### Problem 4.1.
Seek $\phi\coloneqq\frac{\partial u}{\partial\mathbf{n}_{\Gamma}}\in
L^{2}(\Gamma)$ such that
(4.11)
$\displaystyle\mathsf{A}^{(\kappa,\eta)^{\prime}}_{\Gamma}\phi=\frac{\partial
u^{\text{inc}}}{\partial\mathbf{n}_{\Gamma}}-\imath\eta\gamma^{{\text{c}}}_{\Gamma}u^{\text{inc}}\in
L^{2}(\Gamma).$
Thus, one may establish that for each $\kappa\in\mathbb{R}_{+}$ it holds
$\mathsf{A}^{(\kappa,\eta)^{\prime}}_{\Gamma}\in\mathscr{L}_{{\text{iso}}}(L^{2}(\Gamma),L^{2}(\Gamma)),$
thereby implying that 4.1 is well–posed as an operator equation set in
$L^{2}(\Gamma)$.
#### 4.3.2 Indirect Boundary Integral Formulation
Another approach to obtain a boundary integral formulation for the sound-soft
scattering problem consists in expressing $u$ by means of a combined single
and double layer potential, namely we use the following _ansatz_ to represent
$u\in H^{1}_{{{\text{loc}}}}({\text{D}}^{{\text{c}}})$:
(4.12) $\displaystyle u({\bf x})=u^{\text{inc}}({\bf
x})+\left(\mathcal{D}^{(\kappa)}_{\Gamma}\varphi\right)({\bf
x})-\imath\eta\,\left(\mathcal{S}^{(\kappa)}_{\Gamma}\varphi\right)({\bf
x}),\quad{\bf x}\in{\text{D}}^{{\text{c}}},$
where $\varphi\in L^{2}(\Gamma)$ is an unknown boundary density and
$\eta\in\mathbb{R}\backslash\\{0\\}$ is the coupling parameter. Application of
the exterior Dirichlet trace to Eq. 4.12 leads us to the following equivalent
boundary integral formulation for Eq. 4.1a:
###### Problem 4.2.
We seek $\varphi\in L^{2}(\Gamma)$ such that
(4.13)
$\displaystyle\mathsf{A}^{(\kappa,\eta)}_{\Gamma}\varphi=-\gamma_{\Gamma}^{{\text{c}}}u^{\text{inc}}\in
L^{2}(\Gamma),$
where
(4.14)
$\displaystyle\mathsf{A}^{(\kappa,\eta)}_{\Gamma}\coloneqq\frac{1}{2}\mathsf{Id}+\mathsf{K}^{(\kappa)}_{\Gamma}-\imath\eta\mathsf{V}^{(\kappa)}_{\Gamma}.$
We readily remark that the operators Eq. 4.10 and Eq. 4.14 are adjoint to each
other in the bilinear $L^{2}(\Gamma)$-duality pairing. Thus, using similar
tools to the ones as for the direct approach in 4.1, one can show that for
each $\kappa\in\mathbb{R}_{+}$ and $\eta\in\mathbb{R}\backslash\\{0\\}$ it
holds
$\mathsf{A}^{(\kappa,\eta)}_{\Gamma}\in\mathscr{L}_{{\text{iso}}}(L^{2}(\Gamma),L^{2}(\Gamma)).$
In the following, to show shape holomorphy of the boundary combined boundary
integral operators Eq. 4.10 and Eq. 4.14 from 4.1 and 4.2. To this end, we
first show shape holomorphy for each of the involved operators Eqs. 4.5, 4.6
and 4.7.
### 4.4 Shape Parametric Holomorphy of the Single Layer Operator
For each ${\bm{{y}}}\in\mathbb{U}$ we set
(4.15)
$\displaystyle\hat{\mathsf{V}}^{(\kappa)}_{\bm{{y}}}\coloneqq\tau_{\bm{{y}}}\,\mathsf{V}^{(\kappa)}_{\Gamma_{\bm{{y}}}}\,\tau^{-1}_{\bm{{y}}}\in\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right).$
The mapping property stated in Eq. 4.15 follows straightforwardly from Lemma
2.16 and the fact that
$\mathsf{V}^{(\kappa)}_{\Gamma_{\bm{{y}}}}:L^{2}({\Gamma}_{\bm{{y}}})\rightarrow
L^{2}({\Gamma}_{\bm{{y}}})$ defines a bounded linear operator, see e.g. [7,
Theorem 3]. The following results establishes the parametric holomorphy of the
single layer BIO.
###### Theorem 4.3.
Let 2.8 be satisfied. Then the map
(4.16)
$\displaystyle\mathcal{V}^{(\kappa)}:\mathbb{U}\rightarrow\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right):{\bm{{y}}}\mapsto\hat{\mathsf{V}}^{(\kappa)}_{\bm{{y}}}$
is $(\bm{b},p,\varepsilon)$-holomorphic and continuous.
###### Proof 4.4.
We first construct a complex extension of
$\hat{\mathsf{V}}^{(\kappa)}_{\bm{{y}}}$ and then show that this extension
satisfies all assumptions of Theorem 3.3. To this end, we note that the single
layer operator admits the representation Eq. 1.2 with
(4.17) $\displaystyle\mathsf{a}({\bf y},{\bf
z})=\frac{\exp(\imath\kappa\left\lVert{\bf
z}\right\rVert)}{4\pi\left\lVert{\bf z}\right\rVert},\quad{\bf
z}\in\mathbb{R}^{3}\backslash\\{{\bf 0}\\}.$
Thus, setting $\left\lVert{\bf z}\right\rVert_{\mathbb{C}}\coloneqq\sqrt{{\bf
z}\cdot{\bf z}}$ with $\sqrt{\cdot}$ signifiying the principal branch of the
square root yields a complex extension
(4.18) $\displaystyle\mathsf{a}({\bf y},{\bf
z})=\frac{\exp(\imath\kappa\left\lVert{\bf
z}\right\rVert_{\mathbb{C}})}{4\pi\left\lVert{\bf
z}\right\rVert_{\mathbb{C}}}$
for all ${\bm{z}}\in\mathfrak{U}$. Thus, it remains to verify each item of the
assumptions of Theorem 3.3. As per usual, let
$\bm{\rho}\coloneqq(\rho_{j})_{j\geq 1}$ be any sequence of numbers strictly
larger than one that is $(\bm{b},\varepsilon)$-admissible.
Item 1:
Our first observation is that $\mathsf{a}(\cdot,\cdot)$ as in Eq. 4.18 is
independent of the first argument. Consequently, in this particular case, for
each fixed ${\bf z}\in\mathfrak{U}$ the map
$\mathbb{U}\ni{\bm{{y}}}\mapsto\mathsf{a}\left(\bm{r}_{\bm{{y}}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right),{\bf
z}\right)\in\mathbb{C}$ is trivially $(\bm{b},p,\varepsilon)$-holomorphic in
each direction, as it is constant.
Item 2:
We first note that the principal branch of the square root is holomorphic in
$\mathbb{C}\backslash(-\infty,0]$ and that the map ${\bf z}\mapsto{\bf
z}\cdot{\bf z}$ is a multivariate polynomial, and thus holomorphic. Thus, the
map $\mathfrak{U}\ni{\bf z}\mapsto\left\lVert{\bf
z}\right\rVert_{\mathbb{C}}\in\mathbb{C}$ is holomorphic. Moreover, the
function $t\mapsto\exp(\imath\kappa t)$ is entire and thus holomorphic and
$t\mapsto\frac{1}{t}$ is holmorphic as well, except at $t=0$. Consequently,
for any ${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$ the map
(4.19) $\mathfrak{U}\ni{\bf
z}\mapsto\mathsf{a}\left(\bm{r}_{\bm{z}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right),{\bf
z}\right)\in\mathbb{C}$
is holomorphic.
Item 3:
For each $\tau\in\mathcal{G}$, $\tilde{\mathbf{u}}\in\tilde{\tau}$,
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$, and $\hat{{\bf x}}\in\hat{\Gamma}$ and
abbreviating ${\bf z}=\bm{r}_{\bm{z}}(\hat{{\bf
x}})-\bm{r}_{\bm{z}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right)$ one has
(4.20)
$\mathsf{a}\left(\bm{r}_{\bm{z}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right),{\bf
z}\right)=\frac{\exp(\imath\kappa\Re\\{\left\lVert{\bf
z}\right\rVert_{\mathbb{C}}\\}-\kappa\Im\\{\left\lVert{\bf
z}\right\rVert_{\mathbb{C}}\\})}{4\pi\left\lVert{\bf
z}\right\rVert_{\mathbb{C}}}.$
Observing that $\left\lvert\left\lVert{\bf
z}\right\rVert_{\mathbb{C}}\right\rvert=\left\lVert{\bf z}\right\rVert$ yields
(4.21) $\left\lvert\mathsf{a}({\bf y},{\bf
z})\right\rvert\leq\frac{\exp(\kappa\left\lvert\Im\\{\left\lVert{\bf
z}\right\rVert_{\mathbb{C}}\\}\right\rvert)}{4\pi\left\lvert\left\lVert{\bf
z}\right\rVert_{\mathbb{C}}\right\rvert}\leq\frac{\exp(\kappa\left\lVert{\bf
z}\right\rVert)}{4\pi\left\lVert{\bf z}\right\rVert}.$
We remark that at this point the bound in Eq. 4.21 does not completely fulfil
Item 3 in the assumptions of Theorem 3.3. It follows from the definition of
$\mathcal{O}_{\bm{\rho}}$ that for each ${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$
there exists ${\bm{{y}}}\in\mathbb{U}$ such that $\left\lvert
z_{j}-y_{j}\right\rvert\leq\rho_{j}-1$, for all $j\in\mathbb{N}$ Next, we
observe that for any $\hat{{\bf x}},\hat{{\bf y}}\in\hat{\Gamma}$ one has
(4.22) $\displaystyle\left\lVert\bm{r}_{\bm{z}}(\hat{{\bf
x}})-\bm{r}_{\bm{z}}(\hat{{\bf y}})\right\rVert\leq$
$\displaystyle\left\lVert\bm{r}_{\bm{z}}(\hat{{\bf
x}})-\bm{r}_{\bm{{y}}}(\hat{{\bf
x}})\right\rVert+\left\lVert\bm{r}_{\bm{z}}(\hat{{\bf
y}})-\bm{r}_{\bm{{y}}}(\hat{{\bf y}})\right\rVert$
$\displaystyle+\left\lVert\bm{r}_{\bm{{y}}}(\hat{{\bf
x}})-\bm{r}_{\bm{{y}}}(\hat{{\bf y}})\right\rVert.$
It follows from Eq. 2.13 that for any $\hat{{\bf x}}\in\hat{\Gamma}$
(4.23) $\displaystyle\left\lVert\bm{r}_{\bm{z}}(\hat{{\bf
x}})-\bm{r}_{\bm{{y}}}(\hat{{\bf x}})\right\rVert$
$\displaystyle\leq\sum_{j\geq 1}\left\lvert
z_{j}-y_{j}\right\rvert\left\lVert{\bm{\varphi}}_{j}\right\rVert_{L^{\infty}(\hat{\Gamma};\mathbb{R}^{3})}$
$\displaystyle\leq\sum_{j\geq
1}\left(\rho_{j}-1\right)\left\lVert{\bm{\varphi}}_{j}\right\rVert_{\mathscr{C}^{0,1}(\hat{\Gamma};\mathbb{R}^{3})}$
$\displaystyle\leq\varepsilon,$
where the last inequality comes from the fact that the sequence
$\bm{\rho}\coloneqq(\rho_{j})_{j\geq 1}$ is $(\bm{b},\varepsilon)$-admissible.
In addition, for any $\hat{{\bf x}},\hat{{\bf y}}\in\hat{\Gamma}$ and each
${\bm{{y}}}\in\mathbb{U}$ one has
(4.24) $\left\lVert\bm{r}_{\bm{{y}}}(\hat{{\bf
x}})-\bm{r}_{\bm{{y}}}(\hat{{\bf y}})\right\rVert\leq\left\lVert\hat{{\bf
x}}-\hat{{\bf
y}}\right\rVert\left\lVert\bm{r}_{\bm{{y}}}\right\rVert_{\mathscr{C}^{0,1}(\hat{\Gamma};\mathbb{R}^{3})}.$
The map
$\mathbb{U}\ni{\bm{{y}}}\mapsto\bm{r}_{\bm{{y}}}\in\mathscr{C}^{0,1}(\hat{\Gamma};\mathbb{R}^{3})$
is continuous according to Lemma 2.12, thus rendering the map
$\mathbb{U}\ni{\bm{{y}}}\mapsto\left\lVert\bm{r}_{\bm{{y}}}\right\rVert_{\mathscr{C}^{0,1}(\hat{\Gamma};\mathbb{R}^{3})}$
continuous as well, as any norm depends continuoulsy on its argument. As
pointed out previously, $\mathbb{U}$ defines a compact subset when equipped
with the product topology. Therefore, the map
${\bm{{y}}}\mapsto\left\lVert\bm{r}_{\bm{{y}}}\right\rVert_{\mathscr{C}^{0,1}(\hat{\Gamma};\mathbb{R}^{3})}$
attains a maximum. Consequently, by collection Eq. 4.22 Eq. 4.23, and Eq.
4.22, one has that for any $\hat{{\bf x}},\hat{{\bf y}}\in\hat{\Gamma}$ and
each ${\bm{{y}}}\in\mathbb{U}$ it holds
(4.25) $\displaystyle\left\lVert\bm{r}_{\bm{z}}(\hat{{\bf
x}})-\bm{r}_{\bm{z}}(\hat{{\bf y}})\right\rVert\leq
2\varepsilon+\text{diam}(\hat{\Gamma})\,\max_{{\bm{{y}}}\in\mathbb{U}}\left\lVert\bm{r}_{\bm{{y}}}\right\rVert_{\mathscr{C}^{0,1}(\hat{\Gamma};\mathbb{R}^{3})}.$
Hence, with
(4.26)
$C_{\mathsf{a}}\coloneqq\exp\left(\kappa\left(2\varepsilon+\text{diam}(\hat{\Gamma})\,\max_{{\bm{{y}}}\in\mathbb{U}}\left\lVert\bm{r}_{\bm{{y}}}\right\rVert_{\mathscr{C}^{0,1}(\hat{\Gamma};\mathbb{R}^{3})}\right)\right)$
$\mathsf{a}(\cdot,\cdot)$ as in Eq. 4.20 satisfies Item 3 in the assumptions
of Theorem 3.3.
Therefore, it follows from Theorem 3.3 that the map in Eq. 4.16 is
$(\bm{b},p,\varepsilon)$-holomorphic and continuous.
### 4.5 Shape Holomorphy of the Double Layer Operator
For each ${\bm{{y}}}\in\mathbb{U}$ we define
(4.27)
$\displaystyle\hat{\mathsf{K}}_{\bm{{y}}}^{(\kappa)}\coloneqq\tau_{\bm{{y}}}\;\mathsf{K}_{\Gamma_{\bm{{y}}}}^{(\kappa)}\tau^{-1}_{\bm{{y}}}\in\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right).$
Again, as argued previously for the single layer operator, the mapping
property stated in Eq. 4.27 follows straightforwardly from Lemma 2.16 and the
fact that for each ${\bm{{y}}}\in\mathbb{U}$ one has
$\mathsf{K}_{\Gamma_{\bm{{y}}}}^{(\kappa)}:L^{2}(\Gamma_{\bm{{y}}})\rightarrow
L^{2}(\Gamma_{\bm{{y}}})$ defines a bounded linear operator, see e.g. [7,
Theorem 3.5]
However, unlike in the analysis for the single layer operator, we can not
directly argue using solely 2.8 and Theorem 4.3 as the term $\frac{\partial
G^{(\kappa)}}{\partial\mathbf{n}}$ does not yield an integrable kernel when
considering the double layer operator set on a Lipschitz boundary. The latter
difficulty can be overcome by assuming that all surfaces are piecewise
$\mathscr{C}^{1,1}$ as in the following assumption.
###### Assumption 4.5.
There is a decomposition $\mathcal{G}$ of $\hat{\Gamma}$ such that for each
${\bm{{y}}}\in\mathbb{U}$ and each ${\tau}\in\mathcal{G}$ one has that
${{\bm{r}}}_{\bm{{y}}}\circ\chi_{\tau}\in\mathscr{C}^{1,1}(\tilde{\tau};\mathbb{R}^{3})$.
Next, we proceed to prove the parametric holomorphy result for the double
layer BIO.
###### Theorem 4.6.
Let 2.8 and 4.5 be fulfilled with $\bm{b}\in\ell^{p}(\mathbb{N})$ and
$p\in(0,1)$. Then there exists $\varepsilon>0$ such that the map
(4.28)
$\displaystyle\mathcal{K}^{(\kappa)}:\mathbb{U}\rightarrow\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right):{\bm{{y}}}\mapsto\hat{\mathsf{K}}_{\bm{{y}}}^{(\kappa)}$
is $(\bm{b},p,\varepsilon)$-holomorphic and continuous.
###### Proof 4.7.
As in the analysis of the parametric holomorphy result for single layer
operator, we start by constructing a complex extension of
$\hat{\mathsf{K}}_{{\bm{{y}}}}$. To this end, we start by remarking that the
double layer operator has the representation Eq. 1.2 with
(4.29) $\displaystyle\mathsf{a}({\bf y},{\bf z})=\big{(}{\bf n}({\bf
y})\cdot{\bf z}\big{)}\frac{\exp(\imath\kappa\left\lVert{\bf
z}\right\rVert)}{4\pi\left\lVert{\bf
z}\right\rVert^{3}}\left(\imath\kappa\left\lVert{\bf
z}\right\rVert-1\right),\quad{\bf y}\in\hat{\Gamma},\quad{\bf
z}\in\mathbb{R}^{3}\backslash\\{{\bf 0}\\}.$
Thus, for each $\tau\in\mathcal{G}$, $\tilde{\mathbf{u}}\in\tilde{\tau}$,
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$, and $\hat{{\bf x}}\in\hat{\Gamma}$ and
abbreviating ${\bf z}=\hat{{\bf
x}}-\bm{r}_{\bm{z}}\circ\chi_{\tau}\left(\tilde{\mathbf{u}}\right)$ we obtain
the complex extension
(4.30) $\displaystyle\mathsf{a}(\bm{r}_{\bm{z}}(\hat{\mathbf{x}}),{\bf
z})=\big{(}\hat{\mathbf{n}}_{\bm{z}}(\hat{\mathbf{x}})\cdot{\bf
z}\big{)}\frac{\exp(\imath\kappa\left\lVert{\bf
z}\right\rVert_{\mathbb{C}})}{4\pi\left\lVert{\bf
z}\right\rVert^{3}_{\mathbb{C}}}\left(\imath\kappa\left\lVert{\bf
z}\right\rVert_{\mathbb{C}}-1\right),\quad{\bf z}\in\mathfrak{U}.$
As per usual, let $\bm{\rho}\coloneqq(\rho_{j})_{j\geq 1}$ be any sequence of
numbers strictly larger than one that is $(\bm{b},\varepsilon)$-admissible. We
verify each item of the assumptions of Theorem 3.3.
Item 1:
Follows directly from Lemma 2.18.
Item 2:
As in the proof of Theorem 4.3, the maps $\mathfrak{U}\ni{\bf
z}\mapsto\left\lVert{\bf z}\right\rVert_{\mathbb{C}}\in\mathbb{C}$ and
$\mathbb{C}\setminus\\{0\\}\ni t\mapsto\frac{1}{t}\in\mathbb{C}$ are
holomorphic. Consequently, for any ${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$ the
map $\mathfrak{U}\ni{\bf
z}\mapsto\mathsf{a}\left(\bm{r}_{\bm{z}}(\hat{\mathbf{x}}),{\bf
z}\right)\in\mathbb{C}$ is holomorphic as well.
Item 3:
We first need to address the non-integrability of $\frac{\partial
G^{(\kappa)}}{\partial\mathbf{n}}$. To this end, it is well known that for
each $\tau\in\mathcal{G}$, $\hat{\bf x},\hat{\bf y}\in\tau$, and
${\bm{{y}}}\in\mathbb{U}$ it holds
(4.31) $\displaystyle\underline{c}\left\lvert\hat{\bf n}_{\bm{{y}}}(\hat{\bf
y})\cdot\left(\bm{r}_{\bm{{y}}}(\hat{\bf x})-\bm{r}_{\bm{{y}}}(\hat{\bf
y})\right)\right\rvert\leq\left\lVert\hat{\bf x}-\hat{\bf
y}\right\rVert^{2}\leq\overline{c}\left\lVert\bm{r}_{\bm{{y}}}(\hat{\bf
x})-\bm{r}_{\bm{{y}}}(\hat{\bf y})\right\rVert^{2},$
for some $0<\underline{c}\leq\overline{c}<\infty$, see, e.g., [48, Lemma
2.2.14]. We note particularly that the proof together with our assumptions
guarantee that the constant can be chosen independently of ${\bm{{y}}}$. By a
perturbation argument we can conclude that there is $\varepsilon>0$ and two
possibly different constants $0<\underline{c}\leq\overline{c}<\infty$ such
that
(4.32) $\displaystyle\underline{c}\left\lvert\hat{\bf n}_{\bm{z}}(\hat{\bf
y})\cdot\left(\bm{r}_{\bm{z}}(\hat{\bf x})-\bm{r}_{\bm{z}}(\hat{\bf
y})\right)\right\rvert\leq\left\lVert\hat{\bf x}-\hat{\bf
y}\right\rVert^{2}\leq\overline{c}\left\lVert\bm{r}_{\bm{z}}(\hat{\bf
x})-\bm{r}_{\bm{z}}(\hat{\bf y})\right\rVert^{2},$
for all ${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$. Abbreviating ${\bf
z}=\bm{r}_{\bm{z}}(\hat{{\bf x}})-\bm{r}_{\bm{z}}(\hat{\bf y})$ and proceeding
as in the proof of Theorem 4.3 implies
$|\mathsf{a}\left(\bm{r}_{\bm{z}}(\hat{\bf y}),{\bf
z}\right)|\leq\frac{\overline{c}}{\underline{c}}\frac{\exp(\kappa\left\lVert{\bf
z}\right\rVert)}{4\pi\left\lVert{\bf
z}\right\rVert}\left(\kappa\left\lVert{\bf z}\right\rVert+1\right),$
i.e., the singularity of the complex extension of the double layer operator
behaves similarly to the single layer operator case. The conclusion follows in
complete analogy to the proof of Theorem 4.3.
Finally, recalling Theorem 3.3 one may conclude that the map introduced in Eq.
4.28 is $(\bm{b},p,\varepsilon)$-holomorphic and continuous.
### 4.6 Shape Holomorphy of the Adjoint Double Layer Operator
In order to establish the parametric holomorphy result for the adjoint double
layer operator from the double layer operator itself, we consider for each
${\bm{{y}}}\in\mathbb{U}$ the bilinear $L^{2}(\Gamma_{\bm{{y}}})$-inner
product $(\cdot,\cdot)_{L^{2}(\Gamma_{\bm{{y}}})}$. For all $\varphi,\phi\in
L^{2}(\Gamma_{\bm{{y}}})$ it holds that
(4.33)
$\displaystyle\left({\mathsf{K}}^{(\kappa)}_{\Gamma_{\bm{{y}}}}\varphi,\phi\right)_{L^{2}(\Gamma_{\bm{{y}}})}=\left(\varphi,{\mathsf{K}}^{(\kappa)^{\prime}}_{\Gamma_{\bm{{y}}}}\phi\right)_{L^{2}(\Gamma_{\bm{{y}}})}.$
For each ${\bm{{y}}}\in\mathbb{U}$ we may define
(4.34)
$\displaystyle\hat{\mathsf{K}}^{(\kappa)^{\prime}}_{{\bm{{y}}}}\coloneqq\tau_{\bm{{y}}}\;{\mathsf{K}}^{(\kappa)^{\prime}}_{\Gamma_{\bm{{y}}}}\;\tau^{-1}_{\bm{{y}}}\in\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right),$
where the mapping properties are immediate. In fact, defining a modified,
real-valued inner product on $L^{2}(\hat{\Gamma})$ by
(4.35) $\displaystyle\left(\varphi,\phi\right)_{L^{2}(\Gamma_{\bm{{y}}})}$
$\displaystyle=\int\limits_{\Gamma_{\bm{{y}}}}\varphi({\bf y})\phi({\bf
y})\text{d}s_{{\bf y}}$
$\displaystyle=\sum_{\tau\in\mathcal{G}}\int\limits_{\tilde{\tau}}\left(\tau_{\bm{{y}}}\varphi\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\left(\tau_{\bm{{y}}}\phi\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\sqrt{g_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}})}\text{d}\tilde{\mathbf{u}}$
$\displaystyle\eqqcolon\left(\hat{\varphi},\hat{\phi}\right)_{L^{2}(\hat{\Gamma}),{\bm{{y}}}},$
with $\hat{\varphi}=\tau_{\bm{{y}}}\varphi,\hat{\phi}=\tau_{\bm{{y}}}\phi\in
L^{2}(\hat{\Gamma})$ and ${\bm{{y}}}\in\mathbb{U}$, one readily verifies that
$\hat{\mathsf{K}}^{(\kappa)^{\prime}}_{{\bm{{y}}}}$ is indeed the adjoint
operator of $\hat{\mathsf{K}}^{(\kappa)}_{{\bm{{y}}}}$ with respect to
$\left(\cdot,\cdot\right)_{L^{2}(\hat{\Gamma}),{\bm{{y}}}}$. I.e., for each
${\bm{{y}}}\in\mathbb{U}$ and any $\hat{\varphi},\hat{\phi}\in
L^{2}(\hat{\Gamma})$ it holds
(4.36)
$\displaystyle\left(\hat{\mathsf{K}}^{(\kappa)}_{{\bm{{y}}}}\hat{\varphi},\hat{\phi}\right)_{L^{2}(\hat{\Gamma}),{\bm{{y}}}}=\left(\hat{\varphi},\hat{\mathsf{K}}^{(\kappa)^{\prime}}_{{\bm{{y}}}}\hat{\phi}\right)_{L^{2}(\hat{\Gamma}),{\bm{{y}}}}.$
Using Eq. 4.33, parametric holomorphy of the adjoint double layer operator can
be derived from the corresponding result for the double layer operator as
throroughly described in the following result.
###### Theorem 4.8.
Let 2.8 and 4.5 be fulfilled. Then, there exists $\varepsilon>0$ such that the
map
(4.37)
$\displaystyle\mathcal{K}^{(\kappa)^{\prime}}:\mathbb{U}\rightarrow\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right):{\bm{{y}}}\mapsto\hat{\mathsf{K}}^{(\kappa)^{\prime}}_{\bm{{y}}}$
is $(\bm{b},p,\varepsilon)$-holomorphic and continuous.
###### Proof 4.9.
The complex extension of the modified inner product Eq. 4.35 to
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$ reads
(4.38)
$\displaystyle\left(\hat{\varphi},\hat{\phi}\right)_{L^{2}(\hat{\Gamma}),{\bm{z}}}=\sum_{\tau\in\mathcal{G}}\int\limits_{\tilde{\tau}}\left(\hat{\varphi}\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\left(\hat{\phi}\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\sqrt{g_{\tau,{\bm{z}}}(\tilde{\mathbf{u}})}\text{d}\tilde{\mathbf{u}}$
and takes arguments $\hat{\varphi},\hat{\phi}\in L^{2}(\hat{\Gamma})$. Thus,
in accordance with Eq. 4.36, we can extend
$\hat{\mathsf{K}}^{(\kappa)^{\prime}}_{{\bm{{y}}}}$ into
$\mathcal{O}_{\bm{\rho}}$ as the (unique) bounded linear operator satisfying
$\hat{\mathsf{K}}^{(\kappa)^{\prime}}_{{\bm{z}}}:L^{2}(\hat{\Gamma})\rightarrow
L^{2}(\hat{\Gamma})$
(4.39)
$\displaystyle\left(\hat{\mathsf{K}}^{(\kappa)}_{{\bm{z}}}\hat{\varphi},\hat{\phi}\right)_{L^{2}(\hat{\Gamma}),{\bm{z}}}=\left(\hat{\varphi},\hat{\mathsf{K}}^{(\kappa)^{\prime}}_{{\bm{z}}}\hat{\phi}\right)_{L^{2}(\hat{\Gamma}),{\bm{z}}},$
for all $\hat{\varphi},\hat{\phi}\in L^{2}(\hat{\Gamma})$ and
${\bm{z}}\in\mathcal{O}_{\bm{\rho}}$. The map
$\mathbb{U}\ni{\bm{{y}}}\mapsto\left(\cdot,\cdot\right)_{{\bm{{y}}},L^{2}(\hat{\Gamma})}$
into the Banach space of complex-valued bilinear forms on
$L^{2}(\hat{\Gamma})\times L^{2}(\hat{\Gamma})$ is
$(\bm{b},p,\varepsilon)$-holomorphic and continuous as a consequence of Lemma
2.14, and so is
$\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{\mathsf{K}}^{(\kappa)}_{\bm{{y}}}\in\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right)$
as stated in Theorem 4.6.
### 4.7 Parametric Shape holomorphy of Combined Field Integral Equations
With the shape holomorphy results of the single, double, and adjoint double
layer operator available, we can now conclude that Eq. 4.10 and Eq. 4.14,
i.e.,
$\mathsf{A}^{(\kappa,\eta)}_{\Gamma}:L^{2}(\Gamma)\rightarrow
L^{2}(\Gamma)\quad\text{and}\quad\mathsf{A}^{(\kappa,\eta)^{\prime}}_{\Gamma}:L^{2}(\Gamma)\rightarrow
L^{2}(\Gamma),$
are shape holomorphic as well.
###### Corollary 4.10.
Let 2.8 and 4.5 be satisfied with $\bm{b}\in\ell^{p}(\mathbb{N})$ and
$p\in(0,1)$. Then there exists $\varepsilon>0$ such that
(4.40a)
$\displaystyle\mathcal{A}^{(\kappa,\eta)}\colon\mathbb{U}\rightarrow\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right)$
$\displaystyle\colon{\bm{{y}}}\mapsto\hat{\mathsf{A}}_{\bm{{y}}}^{(\kappa,\eta)}\coloneqq\tau_{\bm{{y}}}\;\mathsf{A}_{\Gamma_{\bm{{y}}}}^{(\kappa,\eta)}\tau^{-1}_{\bm{{y}}},$
(4.40b)
$\displaystyle\mathcal{A}^{(\kappa,\eta)^{\prime}}\colon\mathbb{U}\rightarrow\mathscr{L}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right)$
$\displaystyle\colon{\bm{{y}}}\mapsto\hat{\mathsf{A}}_{\bm{{y}}}^{(\kappa,\eta)^{\prime}}\coloneqq\tau_{\bm{{y}}}\;\mathsf{A}_{\Gamma_{\bm{{y}}}}^{(\kappa,\eta)^{\prime}}\tau^{-1}_{\bm{{y}}},$
are $(\bm{b},p,\varepsilon)$-holomorphic and continuous.
###### Proof 4.11.
This result is a consequence of Theorem 4.3, Theorem 4.6, and Theorem 4.8.
### 4.8 Parametric Holomorphy of the Parameter-to-Solution Map
We are interested in the depedence of the solution to 4.1 and 4.2 upon the
parametric boundary shape. More precisely, we are interested in establishing
the holomorphic _parameter-to-solution_ maps
(4.41)
$\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{\phi}_{\bm{{y}}}\coloneqq\left(\hat{\mathsf{A}}^{(\kappa,\eta)^{\prime}}_{{\bm{{y}}}}\right)^{-1}\hat{f}_{\bm{{y}}}\in
L^{2}(\hat{\Gamma}),$
and
(4.42)
$\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{\varphi}_{\bm{{y}}}\coloneqq\left(\hat{\mathsf{A}}^{(\kappa,\eta)}_{{\bm{{y}}}}\right)^{-1}\hat{g}_{\bm{{y}}}\in
L^{2}(\hat{\Gamma}),$
where
(4.43)
$\displaystyle\hat{f}_{\bm{{y}}}\coloneqq\tau_{\bm{{y}}}{f}_{\Gamma_{\bm{{y}}}}\in
L^{2}(\hat{\Gamma})\quad\text{and}\quad\hat{g}_{\bm{{y}}}\coloneqq\tau_{\bm{{y}}}{g}_{\Gamma_{\bm{{y}}}}\in
L^{2}(\hat{\Gamma}).$
With the results stated in Theorem 4.3, Theorem 4.6, and Theorem 4.8, together
with Proposition 2.5, one can establish the following result.
###### Corollary 4.12.
Let 2.8 and 4.5 be satisfied with $\bm{b}\in\ell^{p}(\mathbb{N})$ and
$p\in(0,1)$. Then there exists $\varepsilon>0$ such that the parameter-to-
solution maps introduced in Eq. 4.41 and Eq. 4.42 are
$(\bm{b},p,\varepsilon)$-holomorphic and continuous.
###### Proof 4.13.
The proof is in complete analogy to the proof of Corollary 3.5. To this end,
we remark that Corollary 4.10 implies that there exists an $\varepsilon>0$
such that the maps $\mathcal{A}^{(\kappa,\eta)^{\prime}}$ and
$\mathcal{A}^{(\kappa,\eta)}$ are $(\bm{b},p,\varepsilon)$-holomorphic and
continuous. Moreover, it holds
$\mathsf{A}^{(\kappa,\eta)^{\prime}}_{\Gamma},\mathsf{A}^{(\kappa,\eta)}_{\Gamma}\in\mathscr{L}_{{\text{iso}}}(\Gamma_{\bm{{y}}},\Gamma_{\bm{{y}}})$
for all ${\bm{{y}}}\in\mathbb{U}$. Thus, Lemma 2.16 implies that
$\hat{\mathsf{A}}_{\bm{{y}}}^{(\kappa,\eta)},\;\hat{\mathsf{A}}_{\bm{{y}}}^{(\kappa,\eta)^{\prime}}\in\mathscr{L}_{{\text{iso}}}\left(L^{2}(\hat{\Gamma}),L^{2}(\hat{\Gamma})\right)$
for each ${\bm{{y}}}\in\mathbb{U}$.
It only remains to show the parametric holomorphy of
${\bm{{y}}}\mapsto\hat{f}_{\bm{{y}}}$ and
${\bm{{y}}}\mapsto\hat{g}_{\bm{{y}}}$. To this end, in 4.1, the right-hand
side is given through a plane incident wave $u^{\text{i}}({\bf
x}):=\exp(\imath\kappa{\bf x}\cdot\bm{\hat{d}}_{\text{inc}})$, yielding
(4.44) $\hat{f}_{\bm{{y}}}(\hat{{\bf
x}})=\kappa\hat{\mathbf{n}}_{\bm{{y}}}(\hat{{\bf
x}})\cdot\bm{\hat{d}}_{\text{inc}}\exp\left(\imath\kappa\bm{r}_{\bm{{y}}}(\hat{{\bf
x}})\cdot\bm{\hat{d}}_{\text{inc}}\right)-\imath\eta\exp\left(\imath\kappa\bm{r}_{\bm{{y}}}(\hat{{\bf
x}})\cdot\bm{\hat{d}}_{\text{inc}}\right).$
Lemmas 2.12 and 2.18 imply that that
$\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{f}_{\bm{{y}}}\in L^{2}(\hat{\Gamma})$ is
indeed $(\bm{b},p,\varepsilon)$-holomorphic and continuous for some
$\varepsilon>0$. Similarly, for 4.2, we have
(4.45) $\hat{g}_{\bm{{y}}}(\hat{{\bf
x}})=-\exp\left(\imath\kappa\bm{r}_{\bm{{y}}}(\hat{{\bf
x}})\cdot\bm{\hat{d}}_{\text{inc}}\right),\quad\hat{{\bf x}}\in\hat{\Gamma},$
for which it follows with Lemma 2.12 that
$\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{g}_{\bm{{y}}}\in L^{2}(\hat{\Gamma})$ is
$(\bm{b},p,\varepsilon)$-holomorphic and continuous for some $\varepsilon>0$.
### 4.9 Parametric Holomorphy of the Far-Field Pattern
For later reference, following [12, Chapter 5], we also consider the far-field
patterns of the single and double layer potentials
$\mathcal{S}^{(\kappa)}_{\Gamma}$ and $\mathcal{D}^{(\kappa)}_{\Gamma}$
introduced in Section 4.2. For $\lambda\in L^{2}(\Gamma)$ and an _observation
direction_ $\bm{d}_{o}\in\mathbb{S}^{2}$ they are defined as
(4.46) $\displaystyle\text{FF}^{(\kappa)}_{\mathcal{S},\Gamma}(\lambda)$
$\displaystyle\coloneqq-\frac{\exp(\imath\pi/4)}{\sqrt{2\pi\kappa}}\int\limits_{\Gamma}\exp(-\imath\kappa\,\bm{d}_{o}\cdot{\bf
y})\lambda({\bf y})\text{d}\text{s}_{{{\bf y}}},$ (4.47)
$\displaystyle\text{FF}^{(\kappa)}_{\mathcal{D},\Gamma}(\lambda)$
$\displaystyle\coloneqq\frac{\exp(\imath\pi/4)\sqrt{\kappa}}{\sqrt{2\pi}}\int\limits_{\Gamma}\left(\bm{d}_{o}\cdot\mathbf{n}_{\Gamma}\right)\exp(-\imath\kappa\,\bm{d}_{o}\cdot{\bf
y})\lambda({\bf y})\text{d}\text{s}_{{{\bf y}}},$
respectively. In the following, we show that these far-field patterns are
shape holomorphic as well. To this end, for each ${\bm{{y}}}\in\mathbb{U}$, we
set
(4.48)
$\displaystyle\hat{\text{FF}}^{(\kappa)}_{\mathcal{S},{\bm{{y}}}}\coloneqq\tau_{\bm{{y}}}\,{\text{FF}}^{(\kappa)}_{\mathcal{S},\Gamma_{\bm{{y}}}}\,\tau^{-1}_{\bm{{y}}}\quad\text{and}\quad\hat{\text{FF}}^{(\kappa)}_{\mathcal{D},{\bm{{y}}}}\coloneqq\tau_{\bm{{y}}}\,{\text{FF}}^{(\kappa)}_{\mathcal{D},\Gamma_{\bm{{y}}}}\,\tau^{-1}_{\bm{{y}}}.$
###### Lemma 4.14.
Let 2.8 and 4.5 be satisfied with $\bm{b}\in\ell^{p}(\mathbb{N})$ and
$p\in(0,1)$. Assume that there exists $\varepsilon^{\prime}>0$ such that the
map $\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{\lambda}_{\bm{{y}}}\in
L^{2}(\hat{\Gamma})$ is $(\bm{b},p,\varepsilon^{\prime})$-holomorphic and
continuous. Then there exists $\varepsilon>0$ such that
(4.49)
$\displaystyle\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{\text{FF}}^{(\kappa)}_{\mathcal{S},{\bm{{y}}}}\left(\hat{\lambda}_{\bm{{y}}}\right)\in\mathbb{C}\quad\text{and}\quad\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{\text{FF}}^{(\kappa)}_{\mathcal{D},{\bm{{y}}}}\left(\hat{\lambda}_{\bm{{y}}}\right)\in\mathbb{C}$
are $(\bm{b},p,\varepsilon)$-holomorphic and continuous.
###### Proof 4.15.
For each ${\bm{{y}}}\in\mathbb{U}$, the explicit representation of Eq. 4.48 is
$\displaystyle\hat{\text{FF}}^{(\kappa)}_{\mathcal{S},{\bm{{y}}}}(\hat{\lambda}_{\bm{{y}}})$
$\displaystyle=-\frac{\exp(\imath\pi/4)}{\sqrt{2\pi\kappa}}\sum_{\tau\in\mathcal{G}}\int\limits_{\tilde{\tau}}\hat{k}^{(\kappa)}_{\mathcal{S},{\bm{{y}}}}(\tilde{\mathbf{u}})\left(\hat{\lambda}_{\bm{{y}}}\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\mathrm{d}\tilde{\mathbf{u}},$
$\displaystyle\hat{\text{FF}}^{(\kappa)}_{\mathcal{D},{\bm{{y}}}}(\hat{\lambda}_{\bm{{y}}})$
$\displaystyle=\frac{\exp(\imath\pi/4)\sqrt{\kappa}}{\sqrt{2\pi}}\sum_{\tau\in\mathcal{G}}\int\limits_{\tilde{\tau}}\hat{k}^{(\kappa)}_{\mathcal{D},{\bm{{y}}}}(\tilde{\mathbf{u}})\left(\hat{\lambda}_{\bm{{y}}}\circ\chi_{\tau}\right)(\tilde{\mathbf{u}})\mathrm{d}\tilde{\mathbf{u}},$
where $\hat{\lambda}_{\bm{{y}}}\coloneqq\tau_{\bm{{y}}}\lambda\in
L^{2}(\hat{\Gamma})$ and
$\displaystyle\hat{k}^{(\kappa)}_{\mathcal{S},{\bm{{y}}}}(\tilde{\mathbf{u}})$
$\displaystyle=\exp(-\imath\kappa\,\bm{d}_{o}\cdot\bm{r}_{\bm{{y}}}\circ\chi_{\tau}(\tilde{\mathbf{u}}))\sqrt{g_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}})}$
$\displaystyle\hat{k}^{(\kappa)}_{\mathcal{D},{\bm{{y}}}}(\tilde{\mathbf{u}})$
$\displaystyle=\left(\bm{d}_{o}\cdot\hat{\mathbf{n}}_{{\bm{{y}}}}\right)(\chi_{\tau}(\tilde{\mathbf{u}}))\exp(-\imath\kappa\,\bm{d}_{o}\cdot\bm{r}_{\bm{{y}}}\circ\chi_{\tau}(\tilde{\mathbf{u}}))\sqrt{g_{\tau,{\bm{{y}}}}(\tilde{\mathbf{u}})}.$
Therein, according to Lemma 2.18 and Lemma 2.14 the maps
$\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{\mathbf{n}}_{{\bm{{y}}}}\in
L^{\infty}(\hat{\Gamma},\mathbb{R}^{3})$ and
$\mathbb{U}\ni{\bm{{y}}}\mapsto\sqrt{g_{\tau,{\bm{{y}}}}}\in
L^{\infty}(\tilde{\tau})$ are $(\bm{b},p,\varepsilon)$-holomorphic and
continuous, respectively. Consequently, for each $\tau\in\mathcal{G}$ the maps
(4.50)
$\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{k}^{(\kappa)}_{\mathcal{S},{\bm{{y}}}}(\tilde{\mathbf{u}})\in
L^{2}(\widetilde{\tau})\quad\text{and}\quad\mathbb{U}\ni{\bm{{y}}}\mapsto\hat{k}^{(\kappa)}_{\mathcal{D},{\bm{{y}}}}(\tilde{\mathbf{u}})\in
L^{2}(\widetilde{\tau})$
are $(\bm{b},p,\varepsilon)$-holomorphic and continuous, thus rendering the
maps in Eq. 4.49 as well.
## 5 Implications in High-dimensional Approximation
In this section, we explore the implications of the presently established
parametric holomorphy result on examples in sparse polynomial approximation
and reduced order modelling, in Bayesian shape inversion, and in expression
rates of artificial neural networks.
### 5.1 Sparse Polynomial Approximation and Reduced Order Modelling
Our aim in the following is to derive best $n$-term polynomial approximation
results to the parameter-to-solution maps Eq. 4.41 and Eq. 4.42 of 4.1 and 4.2
and discuss its consequences. That is, we aim to approximate
(5.1) $\Phi:\mathbb{U}\rightarrow
L^{2}(\hat{\Gamma}):{\bm{{y}}}\mapsto\hat{\phi}_{\bm{{y}}},\hat{\varphi}_{\bm{{y}}},$
by polynomials of finite degree by truncating the infinite expansion
(5.2)
$\displaystyle\Phi({\bm{{y}}})=\sum_{\bm{\nu}\in\mathcal{F}}c_{\bm{\nu}}P_{\bm{\nu}}({\bm{{y}}})$
where
(5.3)
$\displaystyle\mathcal{F}\coloneqq\left\\{\bm{\nu}\in\mathbb{N}_{0}^{\mathbb{N}}:|\bm{\nu}|<\infty\right\\}$
and $P_{\bm{\nu}}(\bm{y})\coloneqq\prod_{j\geq
1}P_{\nu_{j}}\left(y_{j}\right)$, with $P_{n}$ denoting univariate polynomials
of degree $n$. Aiming to recall a result from [10], we consider Legendre
polynomials $P_{n}=P_{n}^{(q)}$ on the interval $[-1,1]$ with normalization
constraints $\|P_{n}^{(q)}\|_{L^{q}([-1,1])}=1$, $q\in\\{2,\infty\\}$. The
coefficients in Eq. 5.3 are denoted by
$(c_{\bm{\nu}}^{(q)})_{\bm{\nu}\in\mathcal{F}}$ accordingly.
We say that a set $\Lambda\subset\mathcal{F}$ is downward closed if
$\bm{\nu}\in\Lambda$ and $\bm{\mu}\leq\bm{\nu}$ implies $\bm{\mu}\in\Lambda$.
To state the sought result, we define for
$c:=\left(c_{\bm{\nu}}\right)_{\bm{\nu}\in\mathcal{F}}\subset\mathbb{R}$ its
downward closed envelope as
$\mathbf{c}_{\bm{\nu}}\coloneqq\sup_{\bm{\mu}\geq\bm{\nu}}\left\lVert
c_{\bm{\mu}}\right\rVert_{L^{2}(\hat{\Gamma})}$ for $\bm{\nu}\in\mathcal{F}$,
where $\bm{\mu}\geq\bm{\nu}$ signifies that the inequality is satisfied
component wise. It is the smallest monotone decreasing sequence which bounds
$\left(\left\lVert
c_{\bm{\nu}}\right\rVert_{L^{2}(\hat{\Gamma})}\right)_{\bm{\nu}\in\mathcal{F}}$
element wise. In addition, for $p>0$ we define $\ell_{m}^{p}(\mathcal{F})$ as
the space of sequences with downward closed envelope in
$\ell^{p}(\mathcal{F})$.
For $1\leq p\leq\infty$, we denote by $L^{p}(\mathbb{U};V)$ the Lebesgue-
Bochner space consisting of strongly $\varrho$-measurable functions
$f:\mathbb{U}\rightarrow{V}$ satisfying $\|f\|_{L^{p}(\mathbb{U};V)}<\infty$,
with
$\|f\|_{L^{p}(\mathbb{U};V)}:=\begin{cases}\left(\displaystyle\int_{\mathbb{U}}\|f(\bm{y})\|_{{V}}^{p}\mathrm{~{}d}\varrho(\bm{y})\right)^{1/p}&1\leq
p<\infty\\\
\underset{{\bm{{y}}}\in\mathbb{U}}{\operatorname{ess}\sup}\|f(\bm{y})\|_{V}&p=\infty,\end{cases}$
where $\mathrm{d}\varrho(\bm{y})$ signifies the tensor product of the uniform
measure in $[-1,1]$, which in turn defines a measure in $\mathbb{U}$. In
particular, we set $L^{p}(\mathbb{U})=L^{p}(\mathbb{U};\mathbb{R})$, for
$1\leq p\leq\infty$.
The next result, which is an application of the main result presented in [10],
addresses the best $n$-term polynomial approximation of the map in Eq. 5.1.
###### Theorem 5.1 ([10, Theorem 2.2, ff.]).
Let 2.8 and 4.5 be satisfied with $\bm{b}\in\ell^{p}(\mathbb{N})$ and
$p\in(0,1)$. Then, $\left(\left\lVert
c_{\bm{\nu}}^{(q)}\right\rVert_{L^{2}(\hat{\Gamma})}\right)_{\bm{\nu}\in\mathcal{F}}$,
$q\in\\{2,\infty\\}$, from Eq. 5.2 belong to $\ell^{p}_{m}(\mathcal{F})$ and
there exists $C>0$ such that for each $n\in\mathbb{N}$ there are finite,
downward closed sets $\Lambda^{2}_{n},\Lambda^{\infty}_{n}\in\mathcal{F}$ of
cardinality $n$ such that
(5.4)
$\left\lVert\Phi-\sum_{\bm{\nu}\in\Lambda^{\infty}_{n}}c_{\bm{\nu}}^{(\infty)}P_{\bm{\nu}}^{(\infty)}\right\rVert_{L^{\infty}(\mathbb{U};L^{2}(\hat{\Gamma}))}\leq
C(n+1)^{-\left(\frac{1}{p}-1\right)},$
and
(5.5)
$\left\lVert\Phi-\sum_{{\bm{\nu}}\in\Lambda^{2}_{n}}c_{\bm{\nu}}^{(2)}P_{\bm{\nu}}^{(2)}\right\rVert_{L^{2}(\mathbb{U};L^{2}(\hat{\Gamma}))}\leq
C(n+1)^{-\left(\frac{1}{p}-\frac{1}{2}\right)}.$
We explore the implications of our parametric holomorphy result in the context
of reduced order modelling. There, the goal is to find a low-dimensional
affine space approximating the solution manifold
$\displaystyle\mathcal{M}\coloneqq\left\\{\Phi({\bm{{y}}}):\;{\bm{{y}}}\in\mathbb{U}\right\\}$
to Eq. 5.1 up to a given accuracy. A commonly used concept in nonlinear
approximation to quantify the approximation error to such solution manifolds
is the so-called Kolmogorov’s $n$-width. For a compact subset $\mathcal{K}$ of
a Banach space $X$ it is defined for $n\in\mathbb{N}$ as
$\displaystyle
d_{n}(\mathcal{K})_{X}\coloneqq\inf_{\begin{subarray}{c}X_{n}\subset X\\\
\operatorname{dim}\left(X_{n}\right)\leq
n\end{subarray}}\sup_{v\in\mathcal{K}}\inf_{w\in X_{n}}\left\lVert
v-w\right\rVert_{X}.$
As a direct consequence of Theorem 5.1 we obtain the following bound on
Kolmogorov’s $n$-width, giving an upper bound on the minimal number of
dimensions to reach a certain approximation accuracy to the solution manifold.
###### Corollary 5.2.
Let 2.8 and 4.5 be satisfied with $\bm{b}\in\ell^{p}(\mathbb{N})$ and
$p\in(0,1)$. Then
$d_{n}(\mathcal{M})_{L^{2}(\hat{\Gamma})}\leq
C(n+1)^{-\left(\frac{1}{p}-1\right)}.$
### 5.2 Bayesian Shape Inversion
We proceed to discuss the significance of the parametric holomorphy results in
the context of computational Bayesian shape inversion.
#### 5.2.1 Bayesian Inverse Problems
Provided a _parameter–to–solution map_ $G:\mathbb{U}\rightarrow X$ with $X$
being a separable Banach space over $\mathbb{C}$ and a _prior_ probability
measure $\mu_{0}$ on $\mathbb{U}$, one seeks to update the prior measure from
observational data. In the following we will assume that $\mu_{0}$ is the
uniform probability measure in the sense of [52, Section 2]. To obtain
observations from solutions in $X$, an _observation operator_
$O:X\rightarrow\mathbb{R}^{K}$ is assumed to be given, allowing to introduce
the _uncertainty–to–observation_ map as
$\mathcal{J}\coloneqq O\circ G:\mathbb{U}\rightarrow\mathbb{R}^{K}.$
Further, the observations $\Upsilon\in\mathbb{R}^{K}$ to update the prior
measure are assumed to be distorted by additive Gaussian noise,
(5.6) $\displaystyle\Upsilon=\mathcal{J}({{\bm{{y}}}^{\star}})+\eta,$
where $\eta\sim\mathcal{N}(0,\Sigma)$, $\Sigma\in\mathbb{R}^{K\times
K}_{\text{sym}}$ is a symmetric, positive definite covariance matrix and
${\bm{{y}}}^{\star}$ is the _ground truth_. The updated _posterior probability
measure_ $\mu^{\Upsilon}$ can then be expressed with respect to the prior
measure as follows.
###### Theorem 5.3 ([52, Theorem 2.1]).
Assume that $\mathcal{J}:\mathbb{U}\rightarrow\mathbb{R}^{K}$ is bounded and
continuous. Then $\mu^{\Upsilon}({{\text{d}}}{\bm{{y}}})$, the distribution of
${\bm{{y}}}\in\mathbb{U}$ given the data $\Upsilon\in\mathbb{R}^{K}$, is
absolutely continuous with respect to $\mu_{0}({{\text{d}}}{\bm{{y}}})$, i.e.
there exists a parametric density $\Theta({\bm{{y}}})$ such that for each
${\bm{{y}}}\in\mathbb{U}$ the _Radon-Nikodym_ derivative is given by
(5.7)
$\displaystyle\frac{{{\text{d}}}\mu^{\Upsilon}}{{{\text{d}}}\mu_{0}}({\bm{{y}}})=\frac{1}{Z}\Theta({\bm{{y}}})$
with the posterior density
(5.8)
$\displaystyle\Theta({\bm{{y}}})\coloneqq\exp\left(-\Phi_{\Sigma}(\Upsilon,\bm{y})\right)\quad\text{and}\quad
Z\coloneqq\int\limits_{\mathbb{U}}\Theta({\bm{{y}}})\mu_{0}({{\text{d}}}\bm{y})>0,$
with
$\displaystyle\Phi_{\Sigma}(\Upsilon,\bm{y})=\frac{1}{2}\left(\Upsilon-\mathcal{J}({\bm{y}})\right)^{\top}\Sigma^{-1}\left(\Upsilon-\mathcal{J}({\bm{y}})\right).$
Given a quantity of interest $\phi\colon\mathbb{U}\to Y$, being $Y$ a
separable Banach space over $\mathbb{C}$, Theorem 5.3 allows us to compute
statistical moments of a quantity of interest with respect to the posterior
measure. For example, one can compute the mean with respect to
$\mu^{\Upsilon}({{\text{d}}}{\bm{{y}}})$ as follows
(5.9)
$\displaystyle\mathbb{E}^{\mu^{\Upsilon}}[\phi]=\int_{\mathbb{U}}\phi({\bm{{y}}})\mu^{\Upsilon}({{\text{d}}}{\bm{{y}}})=\frac{1}{Z}\int\limits_{\mathbb{U}}\phi({\bm{{y}}})\Theta({\bm{{y}}})\mu_{0}({{\text{d}}}\bm{y}),$
with $Z$ as in Eq. 5.8.
#### 5.2.2 Bayesian Shape Inversion in Acoustic Scattering
Equipped with the frequency-robust boundary integral formulations for the
sound-soft acoustic scattering problem introduced in Section 4, the
uncertainty–to–solution and observation operators of the Bayesian shape
inversion problem are
(5.10) $G\colon\mathbb{U}\to
L^{2}(\hat{\Gamma}):{\bm{{y}}}\mapsto\begin{cases}\hat{\phi}_{{\bm{{y}}}}=\left(\hat{\mathsf{A}}^{(\kappa,\eta)^{\prime}}_{{\bm{{y}}}}\right)^{-1}\hat{f}_{\bm{{y}}}&\text{for
\lx@cref{creftype~refnum}{prb:problem_Aprime_k_n}},\\\
\hat{\varphi}_{{\bm{{y}}}}=\left(\hat{\mathsf{A}}^{(\kappa,\eta)}_{{\bm{{y}}}}\right)^{-1}\hat{g}_{\bm{{y}}}&\text{for
\lx@cref{creftype~refnum}{prbm:BIE_sound_soft_A}},\end{cases}$
with $\hat{\mathsf{A}}^{(\kappa,\eta)}_{{\bm{{y}}}}$ and
$\hat{\mathsf{A}}^{(\kappa,\eta)^{\prime}}_{{\bm{{y}}}}$ as in Eq. 4.40 and
$\hat{f}_{\bm{{y}}}$ and $\hat{g}_{\bm{{y}}}$ as in Eq. 4.43 and
(5.11)
$O\colon\mathbb{U}\to\mathbb{R}^{2},\quad{\bm{{y}}}\mapsto\begin{pmatrix}\Re(u_{\bm{{y}}}^{\infty})\\\
\Im(u_{\bm{{y}}}^{\infty})\end{pmatrix}$
with
(5.12)
$u_{\bm{{y}}}^{\infty}=\begin{cases}\hat{\text{FF}}^{(\kappa)}_{\mathcal{S},\Gamma_{\bm{{y}}}}\left(\hat{\phi}_{{\bm{{y}}}}\right)&\text{for
\lx@cref{creftype~refnum}{prb:problem_Aprime_k_n}},\\\
\hat{\text{FF}}^{(\kappa)}_{\mathcal{S},\Gamma_{\bm{{y}}}}\left(\hat{\varphi}_{{\bm{{y}}}}\right)-\iota\eta\hat{\text{FF}}^{(\kappa)}_{\mathcal{D},\Gamma_{\bm{{y}}}}\left(\hat{\varphi}_{{\bm{{y}}}}\right)&\text{for
\lx@cref{creftype~refnum}{prbm:BIE_sound_soft_A}},\end{cases}$
and $\hat{\text{FF}}^{(\kappa)}_{\mathcal{S},\Gamma_{\bm{{y}}}}$ and
$\hat{\text{FF}}^{(\kappa)}_{\mathcal{D},\Gamma_{\bm{{y}}}}$ as in Eq. 4.48.
Multiple observation directions or directions of incident waves could also be
considered with the straightforward adaptions. The quantity of interest for
our Bayesian shape inversion problem is
(5.13)
$\phi\colon\mathbb{U}\to\mathscr{C}^{0,1}(\hat{\Gamma};\mathbb{R}^{3})\colon{\bm{{y}}}\mapsto\bm{r}_{\bm{{y}}},$
with $\bm{r}_{\bm{{y}}}$ is as in Eq. 2.13. Other quantities of interest could
also be considered.
#### 5.2.3 Parametric Regularity of the Posterior Distribution
In this section, we present the parametric regularity results for the
posterior density as defined in Eq. 5.8. Prior research conducted in [23] and
[50] has also investigated this topic and arrived at similar conclusions.
Specifically, in [23, Theorem 4 and Corollary 5], the authors established
parametric regularity estimates for the posterior distribution using a real-
valued differentiation approach.
In contrast, our approach centers on the sparsity of the posterior
distribution. We observe that it can be expressed as the composition of an
entire function and a $(\bm{b},p,\varepsilon)$-holomorphic map, which in turn
implies that the posterior density is also
$(\bm{b},p,\varepsilon)$-holomorphic. This finding provides a useful way to
derive the sparsity of the posterior density.
###### Corollary 5.4.
Let 2.8 and 4.5 be fulfilled with $\bm{b}\in\ell^{p}(\mathbb{N})$ and
$p\in(0,1)$. In addition, let $G:\mathbb{U}\rightarrow L^{2}(\hat{\Gamma})$,
$O:\mathbb{U}\rightarrow\mathbb{R}^{2}$, and
$\phi:\mathbb{U}\rightarrow\mathscr{C}^{0,1}(\hat{\Gamma},\mathbb{R}^{3})$ be
as in Eq. 5.10, Eq. 5.11, and Eq. 5.13, respectively. Then the maps
(5.14)
$\displaystyle\mathbb{U}\ni{\bm{{y}}}\mapsto\Theta({\bm{{y}}})\in\mathbb{R}\quad\text{and}\quad\mathbb{U}\ni{\bm{{y}}}\mapsto\phi({\bm{{y}}})\Theta({\bm{{y}}})\in\mathscr{C}^{0,1}(\hat{\Gamma},\mathbb{R}^{3})$
with $\Theta:\mathbb{U}\rightarrow\mathbb{C}$ as in Eq. 5.8 are
$(\bm{b},p,\varepsilon)$-holomorphic and continuous.
###### Proof 5.5.
We show that $O$ is indeed $(\bm{b},p,\varepsilon)$-holomorphic, which yields
$(\bm{b},p,\varepsilon)$-holomorphy of $\mathcal{J}=O\circ G$. The remainder
of the proof is in complete analogy to [50, Theorem 4.1]. Using Lemma 4.14,
the $(\bm{b},p,\varepsilon)$-holomorphy of $O$ is immediate, once we show that
taking the real and the imaginary part of a complex function is holomorphic.
Indeed, we provide a proof of this statement in Lemma A.1.
This result justifies the sparse polynomial interpolation results also for the
posterior distribution of the acoustic scattering problem. Moreover, along the
lines of the following section, it also allows to construct surrogates of the
posterior distribution using artificial neural networks.
### 5.3 Approximation of the Far-Field using ANNs
Let $L\in\mathbb{N}$, $\ell_{0},\dots,\ell_{L}\in\mathbb{N}$ and let
$\sigma:\mathbb{R}\rightarrow\mathbb{R}$ be a non-linear function, referred to
in the following as the _activation function_. Given $({\bf W}_{k},{\bf
b}_{k})_{k=1}^{L}$, ${\bf W}_{k}\in\mathbb{R}^{\ell_{k}\times\ell_{k-1}}$,
${\bf b}_{k}\in\mathbb{R}^{\ell_{k}}$, define the affine transformation ${\bf
A}_{k}:\mathbb{R}^{\ell_{k-1}}\rightarrow\mathbb{R}^{\ell_{k}}:{\bf
x}\mapsto{\bf W}_{k}{\bf x}+{\bf b}_{k}$ for $k\in\\{1,\ldots,L\\}$. We define
an _artificial neural network (ANN)_ with activation function $\sigma$ as a
map
$\Psi_{\mathcal{NN}}:\mathbb{R}^{\ell_{0}}\rightarrow\mathbb{R}^{\ell_{L}}$
with
(5.15) $\displaystyle\Psi_{\mathcal{NN}}({\bf x})\coloneqq\begin{cases}{\bf
A}_{1}(x),&L=1,\\\ \left({\bf A}_{L}\circ\sigma\circ{\bf
A}_{L-1}\circ\sigma\cdots\circ\sigma\circ{\bf A}_{1}\right)(x),&L\geq
2,\end{cases}$
where the activation function $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ is
applied componentwise. We define the depth and the width of an ANN as
$\text{width}(\Psi_{\mathcal{NN}})=\max\\{\ell_{1},\ldots,\ell_{L}\\}\quad\text{and}\quad\text{depth}(\Psi_{\mathcal{NN}})=L.$
In the present work, we consider as activation function the hyperbolic tangent
(5.16)
$\displaystyle\sigma(x)=\text{tanh}(x)=\frac{\exp(x)-\exp(-x)}{\exp(x)+\exp(-x)}.$
When this particular function is used, we refer to Eq. 5.15 as a tanh ANN.
However we remark that other choices exist, such as the Rectified Linear Unit
(ReLU), Rectified Power Unit (RePU), and the _swish_ activation function.
We are interested in the tanh ANN approximation of the far-field at a specific
observation direction $\bm{d}_{\text{obs}}\in\mathbb{S}^{2}$ using the
mathematical tools of [1]. To this end, we define
(5.17) $\Phi:\mathbb{U}\rightarrow\mathbb{R}:{\bm{{y}}}\mapsto\left\lvert
u_{\bm{{y}}}^{\infty}\right\rvert^{2},$
where, for each ${\bm{{y}}}\in\mathbb{U}$, $u_{\bm{{y}}}^{\infty}$ is given as
in Eq. 5.12. In the following, we present a streamlined and tailored-to-our-
purposes version of the results presented in [1] to this particular
application, which allows for a straightforward generalization to more
involved examples. To this end, we first recall the approximation of Legendre
polynomials by means of tanh ANNs.
###### Lemma 5.6 ([1, Theorem 7.4]).
Let $\Lambda\subset\mathcal{F}$ with $\mathcal{F}$ as in Eq. 5.3 be a finite
multi-index set, $m(\Lambda)=\max_{\bm{\nu}\in\Lambda}\|\bm{\nu}\|_{1}$, and
$\vartheta\subset\mathbb{N}$, $\left\lvert\vartheta\right\rvert=n$ satisfying
(5.18) $\bigcup_{\bm{\nu}\in\Lambda}\text{supp}(\bm{\nu})\subseteq\vartheta.$
Then, for any $\delta\in(0,1)$ there exists a tanh ANN
$\Psi_{\Lambda,\delta}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{\left\lvert\Lambda\right\rvert}$
with
$\Psi_{\Lambda,\delta}=\left(\Psi_{\bm{\nu},\delta}\right)_{\bm{\nu}\in\Lambda}$
such that
(5.19) $\left\lVert
P_{\bm{\nu}}^{(2)}-\Psi_{\bm{\nu},\delta}\circ\mathcal{T}_{\vartheta}\right\rVert_{L^{\infty}(\mathbb{U})}\leq\delta,\quad\bm{\nu}\in\Lambda,$
with
$\text{width}\left(\Psi_{\bm{\nu},\delta}\right)=\mathcal{O}\left(\left\lvert\Lambda\right\rvert
m(\Lambda)\right)$ and
$\text{depth}\left(\Psi_{\bm{\nu},\delta}\right)=\mathcal{O}\left(\log_{2}(m(\Lambda))\right)$,
where
$\mathcal{T}_{\vartheta}:\mathbb{R}^{\mathbb{N}}\rightarrow\mathbb{R}^{\left\lvert\vartheta\right\rvert}:{\bm{{y}}}=(y_{j})_{j\in\mathbb{N}}\mapsto(y_{j})_{j\in\vartheta}.$
In combination with the sparse polynomial approximation results from Section
5.1 this result allows to derive expression rates of tanh ANNs.
###### Lemma 5.7.
Let 2.8 and 4.5 be fulfilled with $\bm{b}\in\ell^{p}(\mathbb{N})$ and
$p\in(0,1)$. Let $\Phi:\mathbb{U}\rightarrow\mathbb{R}$ be as in Eq. 5.17.
Assume that $\bm{b}$ has a minimal monotone majorante in
$\ell^{p}(\mathbb{N})$. There exists a sequence of tanh neural network
$(\Psi_{\bm{\nu},\delta}^{(n)})_{n\in\mathbb{N}}$ and $C>0$ such that
(5.20)
$\left\lVert\Phi-\Psi_{\bm{\nu},\delta}^{(n)}\circ\mathcal{T}_{\vartheta}\right\rVert_{L^{2}(\mathbb{U})}\leq
Cn^{-\left(\frac{1}{p}-\frac{1}{2}\right)}$
with
$\text{width}\left(\Psi_{\bm{\nu},\delta}^{(n)}\right)=\mathcal{O}(n^{2})$,
$\text{depth}\left(\Psi_{\bm{\nu},\delta}^{(n)}\right)=\mathcal{O}\left(\log_{2}(n)\right)$,
and $\vartheta=\\{1,\dots,n\\}$.
###### Proof 5.8.
In analogy to Corollary 5.4, it follows from 2.8, 4.5, and Lemma A.1 that the
map $\Phi:\mathbb{U}\mapsto\mathbb{R}$ from Eq. 5.17 is
$(\bm{b},p,\varepsilon)$-holomorphic and continuous. Let $\widetilde{\bm{b}}$
be the monotone majorant of $\bm{b}$. Then, as discussed in the proof of [1,
Corollary 8.2], $\Phi$ as in is also
$(\widetilde{\bm{b}},p,\varepsilon)$-holomorphic and continuous. In analogy to
Eq. 5.2, we set
$\Phi=\sum_{\bm{\nu}\in\mathcal{F}}c_{\bm{\nu}}^{(2)}P_{\bm{\nu}}^{(2)}$, with
$\left(c_{\bm{\nu}}^{(2)}\right)_{\bm{\nu}\in\mathcal{F}}$ being the Legendre
coefficients of Eq. 5.17. Set
$\widetilde{\Lambda}_{n}\coloneqq\left\\{\bm{\nu}=(\nu_{k})_{k\in\mathbb{N}}\in\mathcal{F}:\>\prod_{k:\nu_{k}\neq
0}\left(\nu_{k}+1\right)\leq n,\;\nu_{k}=0,\;k>n\right\\}.$
This set contains all _anchored_ sets of a size at most $n$ as pointed out in
[2, Proposition 2.18]. By anchored set we mean a downward closed set
satisfying for each $j\in\mathbb{N}$
(5.21)
$\bm{e}_{j}\in\widetilde{\Lambda}_{n}\Rightarrow\\{\bm{e}_{1},\dots,\bm{e}_{j}\\}\in\widetilde{\Lambda}_{n}.$
Thus, recalling that Theorem 5.1 implies that $\left(\left\lVert
c_{\bm{\nu}}^{(2)}\right\rVert_{L^{2}(\hat{\Gamma})}\right)_{\bm{\nu}\in\mathcal{F}}$
belong to $\ell^{p}_{m}(\mathcal{F})$ and according to [2, Theorem 3.33] it
holds
(5.22) $\sqrt{\sum_{\bm{\nu}\notin\widetilde{\Lambda}_{n}}\left\lvert
c_{\bm{\nu}}^{(2)}\right\rvert^{2}}\leq
C(n+1)^{-\left(\frac{1}{p}-\frac{1}{2}\right)}.$
Moreover, as a consequence of Theorem 5.1, for each $n\in\mathbb{N}$ there
exists a downward closed set $\Lambda_{n}$ of cardinality $n$ such that
(5.23)
$\left\lVert\Phi-\sum_{\bm{\nu}\in\Lambda_{n}}c_{\bm{\nu}}^{(2)}P_{\bm{\nu}}^{(2)}\right\rVert_{L^{2}(\mathbb{U})}=\sqrt{\sum_{\bm{\nu}\notin\Lambda_{n}}\left\lvert
c_{\bm{\nu}}^{(2)}\right\rvert^{2}}\leq
C(n+1)^{-\left(\frac{1}{p}-\frac{1}{2}\right)},$
and
$\left(c_{\bm{\nu}}^{(2)}\right)_{\bm{\nu}\in\mathcal{F}}\in\ell^{p}_{m}(\mathcal{F})$.
Being $\widetilde{\bm{b}}$ monotonically increasing implies that the set
$\Lambda_{n}$ is anchored as well. Set
$\Lambda=\Lambda_{n}\cap\widetilde{\Lambda}_{n}$. Then,
(5.24)
$\left\lVert\Phi-\sum_{\bm{\nu}\in\Lambda}c_{\bm{\nu}}^{(2)}\Phi_{\bm{\nu},\delta}\circ\mathcal{T}_{\vartheta}\right\rVert_{L^{2}(\mathbb{U})}\leq(\spadesuit)+(\clubsuit),$
where
(5.25)
$\displaystyle(\spadesuit)^{2}=\left\lVert\Phi-\sum_{\bm{\nu}\in\Lambda}c_{\bm{\nu}}^{(2)}P_{\bm{\nu}}^{(2)}\right\rVert^{2}_{L^{2}(\mathbb{U})}$
$\displaystyle=\sum_{\bm{\nu}\notin\Lambda}\left\lvert
c_{\bm{\nu}}^{(2)}\right\rvert^{2}$
$\displaystyle<\sum_{\bm{\nu}\notin\Lambda_{n}}\left\lvert
c_{\bm{\nu}}^{(2)}\right\rvert^{2}+\sum_{\bm{\nu}\notin\widetilde{\Lambda}_{n}}\left\lvert
c_{\bm{\nu}}^{(2)}\right\rvert^{2}\leq
C(n+1)^{-2\left(\frac{1}{p}-\frac{1}{2}\right)}$
and
(5.26)
$\displaystyle(\clubsuit)=\left\lVert\sum_{{\bm{\nu}}\in\Lambda}c_{\bm{\nu}}^{(2)}P_{\bm{\nu}}^{(2)}-\sum_{\bm{\nu}\in\Lambda}c_{\bm{\nu}}^{(2)}\Phi_{\bm{\nu},\delta}\circ\mathcal{T}_{\vartheta}\right\rVert_{L^{2}(\mathbb{U})}\leq\sum_{\bm{\nu}\in\Lambda}\left\lvert
c_{\bm{\nu}}^{(2)}\right\rvert\left\lVert
P_{\bm{\nu}}^{(2)}-\Phi_{\bm{\nu},\delta}\circ\mathcal{T}_{\vartheta}\right\rVert_{L^{\infty}(\mathbb{U})}.$
To conclude, we observe that for any
$\bm{\nu}\in\Lambda\subset\widetilde{\Lambda}_{n}$, one has
$\text{supp}(\bm{\nu})=\\{1,\dots,n\\}$. Hence, observing that
$\left(\left\lvert
c_{\bm{\nu}}^{(2)}\right\rvert\right)_{\bm{\nu}\in\mathcal{F}}$ belongs to
$\ell^{p}_{m}(\mathcal{F})\subset\ell^{1}_{m}(\mathcal{F})$, Lemma 5.6 with
$\delta=(n+1)^{-\left(\frac{1}{p}-\frac{1}{2}\right)}$ and
$\vartheta=\\{1,\dots,n\\}$ yields the final result. Also, since any
$\bm{\nu}\in\Lambda$ implies $\bm{\nu}\in\widetilde{\Lambda}_{n}$, we have
(5.27) $\displaystyle m(\Lambda)=\max_{\bm{\nu}\in\Lambda}\|\bm{\nu}\|_{1}\leq
n,$
thus yielding the width and depth estimates stated in this result.
## 6 Conclusion and Outlook
In this work, we consider a class of boundary integral operators set on a
collection of parametrically defined boundaries. In this setting, we establish
the holomorphic dependence of said operators–as an element of a Banach spaces
of bounded linear operators–upon the parametric variables, i.e., we show that
the parameter-to-operator map is holomorphic. This result allows us to
establish parametric holomorphy results for the solution of boundary integral
equations involving the aforementioned operators, provided the invertibility
of the operator for each parametric input, i.e., we show that the parameter-
to-solution map is holomorphic. Though we only consider BIOs in a
$L^{2}$-based setting, this results enables parametric holomorphy analysis in
three dimensional domains defined on affinely parametrized polygonal domains.
To show the applicability of our parametric holomorphy results, we explore the
implications on the boundary integral formulation of the sound-soft acoustic
wave scattering. In particular, herein we discussed the implications in sparse
polynomial approximation of the parameter-to-solution map, reduced order
modelling, in establishing parametric regularity of the posterior distribution
in Bayesian shape inversion, and in the construction of efficient surrogates
with artificial neural networks for the approximation of the far-field
pattern. Even though not thoroughly discussed, using the parametric regularity
and examples established in this article allows to prove dimension-independent
convergence rates for multilevel algorithms in forward and inverse acoustic
domain uncertainty quantification such as in [21], where the domain
deformations were implemented using NURBS.
Further work comprises exploring the implications of the parametric holomorphy
result for BIOs. A canonical example is the analysis and implementation of the
Galerkin POD-ANN as in [32] for the sound-soft acoustic scattering problem.
## Acknowledgement
|
]https://github.com/NThakkar-IDM/covid_and_stat_mech
# A generating function perspective on the transmission forest
Niket Thakkar<EMAIL_ADDRESS>[ Mike Famulare The
Institute for Disease Modeling
Global Health — Bill $\&$ Melinda Gates Foundation
Seattle, Washington 98109
###### Abstract
In a previous paper, we showed that a compartmental stochastic process model
of SARS-CoV-2 transmission could be fit to time series data and then
reinterpreted as a collection of interacting branching processes drawn from a
dynamic degree distribution. We called this reinterpretation a transmission
forest. This paper builds on that idea. Specifically, leveraging generating
function methods from analytic combinatorics, we develop a theory describing
the transmission forest’s properties, allowing us to show for example that
transmission tree interactions fade with increasing disease prevalence. We
then validate the theory by computing forest statistics, like the tree
survival function, which we compare to estimates based on the sampling method
developed previously. The accuracy and flexibility of the analytic approach is
clear, and it allows us to comment on multi-scale features of more general
transmission processes.
disease modeling, probability, generating function, transmission tree,
COVID-19
## I. Epidemiology and graph theory
Transmission trees are the basic graphical unit of an epidemiology. Or, said
differently, if we knew the full transmission tree, it would shed light on a
range of questions – characteristics assigned to nodes would inform our
understanding of risk, features of edges would teach us about transmission
mechanisms, and geometric changes would help us estimate the effects of
interventions.
But of course the basic epidemiological problem is that we can’t measure
transmission trees directly. Even in perfectly observed, closed populations,
assigning edges between nodes is only possible inferentially when transmission
events are sufficiently staggered. In that situation, the gold standard is
outbreak investigations conducted by specialists, where interviews and follow
ups are used to construct a plausible subtree of the full transmission tree
[1]. As a result, even in the best case, this approach cannot scale to a full
epidemiological system.
Another, modern, alternative path forward comes from work on viral
phylogenetics. In that case, genetic sequences of sampled viruses can be
arranged into a phylogenetic tree, and features of that tree can be used to
infer features of the associated transmission trees [2]. This is easier in
theory than in practice. The phylogentic tree is a product of epidemiological
and evolutionary processes, the latter of which can be very complex and system
specific, and separating signals in general is a challenge.
Within this context, in a paper last year [3], we explored a third
perspective. We demonstrated that volatility in Washington’s COVID-19
epidemiological curves contains information on the underlying transmission
trees’ degree distribution. Once specified, that time-varying distribution
could be used to grow a set of interacting trees, which we called a
transmission forest, and while those trees lack the individual-level
resolution of conventionally estimated transmission trees, they were shown to
be predictive of observations from outbreak investigations and phylogenetics.
This step towards harmony between often discordant views of a cryptic epidemic
[4], that is the time series, phylogeny, and outbreak investigations, speaks
to the transmission tree as a fundamental structure for organizing
epidemiological information. Transmission tree inference is an ambitious goal,
but we stand to learn a lot working towards it.
Along those lines, this paper contributes to the broader project of
statistically characterizing transmission trees using more readily available
epidemiological data. Our concrete goal is to more completely and rigorously
define the transmission forest, and in some sense this paper is a mathematical
supplement to the original [3]. That said, more than clarifying the work from
last year, the formalism we develop here offers a tractable connection between
individual-level behavior, the resulting tree structures, and emergent
forests, leading to broadly applicable multi-scale insights into transmission
processes.
## II. The transmission forest model
Figure 1: Sampling the forest. (a) For a given $N_{t}$ trajectory (black
curve) a transmission forest is a random graph (edges in grey) constrained to
sum to $N_{t}$. A given day’s transmission (inset) between infectious parents
(red dots) and their newly infected children (blue circles) demonstrates how
trees compete for nodes. (b) We can sample under the sum constraint by
creating transmission constellations from a fixed number of edges. (c) The
sampling approach (10k samples, mean in purple, 95$\%$ CI shaded) is able to
efficiently explore tail-events in the skewed degree distributions (black
bars) underlying transmission.
We start by defining the transmission forest mathematically and in that way
encapsulating many of the previous paper’s main ideas. Consider a discrete
time, stochastic disease transmission process between susceptible individuals,
$S_{t}$, and infectious individuals, $I_{t}$. In a closed population, we have
$\displaystyle S_{t}$ $\displaystyle=S_{t-1}-N_{t-1}$ (1) $\displaystyle
E_{t}$ $\displaystyle=\left(1-\frac{1}{d_{E}}\right)E_{t-1}+N_{t-1}$ (2)
$\displaystyle I_{t}$
$\displaystyle=\left(1-\frac{1}{d_{I}}\right)I_{t-1}+\frac{1}{d_{E}}E_{t-1},$
(3)
where newly infected individuals, $N_{t}$, become exposed but not infectious,
$E_{t}$, for $d_{E}$ days on average before becoming infectious for $d_{I}$
days on average. On the one hand, following classical models [5], we can write
$N_{t}$ as
$N_{t}=(\beta_{t}\varepsilon_{t})S_{t}I_{t},$ (4)
with transmission rate $\beta_{t}$ and log-normally distributed volatility
$\varepsilon_{t}$, capturing the idea that new infections come from a random
fraction of interacting pairs. Equivalently, inspired by branching processes
[6], we also have
$N_{t}=\sum_{i=1}^{I_{t}}T_{it},$ (5)
capturing the idea that new infections are the sum total of realized daily
transmission events, $T_{it}$, across infectious individuals indexed by $i$.
We model transmission events as independent and identically distributed with
$T_{t}\sim\text{NegBin}\left(\mu_{t},k_{t}\right),$ (6)
which is approximately entropy maximizing [7] when the mean, $\mu_{t}$, and
over-dispersion, $k_{t}$, are set to maintain Eq. 5 up to second order moments
[3].
The transmission forest is a random graph drawn from this model. We visualize
a sample in Fig. 1a. For a given $N_{t}$ trajectory (black curve), nodes are
placed in daily columns and directed edges are drawn between nodes on day $t$
(blue circles) and their infectious parents between days $t-d_{E}$ and
$t-d_{E}-d_{I}$ (red dots) according to Eq. 6. The process is repeated for all
$t$ to fill the graph (grey lines, where we’ve suppressed the nodes for visual
clarity).111This definition makes it clear that we’ve taken $d_{E}$ and
$d_{I}$ to be deterministic across nodes. Strictly speaking, that
approximation is optional. In both the sampling approach and the theory,
latent and infectious durations can be arbitrarily distributed, but this
simplifying assumption is in keeping with the previous paper, and it helps us
focus the work. In the early part of the trajectory, where $t<d_{E}+d_{I}$,
nodes have no parents, making the graph a necessarily disjoint collection of
transmission trees, inspiring the name.
Model fitting to observed time series is discussed in detail in our previous
paper – here we assume the parameters $\beta_{t}$ and $\varepsilon_{t}$ as
well as the process’s initial conditions are known. Throughout this paper, we
use the same model fit to COVID-19 data from Washington from January 2020 to
March 2021 to illustrate the theory’s application.
## III. Sampling a forest
To generate sample forests as in Fig. 1a, we draw daily transmission graphs,
shown in the inset, and stitch them together over time. The sampling approach
gives scaffolding to the theory below, and we use it to validate results
throughout, so it’s worth a brief aside.
$I_{t}$ and $N_{t}$ trajectories can be drawn from Eqs. 1–4 using standard
normal samples. Then, for a given trajectory (rounded to the nearest
integers), the number of nodes in the graph is fixed, and as a result edges
drawn from Eq. 6 are constrained to satisfy Eq. 5 and are no longer
independent. In other words, that trees compete for nodes gives rise to their
interactions in the forest.
We might consider a rejection sampler in this case, essentially drawing
$I_{t}$ negative binomial samples repeatedly until they sum to $N_{t}$ for
every $t$, but that approach is unusably inefficient. With a fixed trajectory,
the right-hand-side of Eq. 5 is asymptotically Gaussian by the central limit
theorem, implying that on average
$\mathcal{O}(I_{t}\sqrt{I_{t}\text{V}[T_{t}]})$ samples have to be drawn per
time-step to create a single forest sample. For the Washington COVID-19 model
we’re working with, with a roughly 400 day time series, that translates to
over 1 billion negative binomial samples to draw one forest.
Fig. 1b illustrates a way forward. The daily transmission graph is a
collection of star graphs, which we call a constellation, and can be
represented as a length $I_{t}$ tuple of infectious node degrees constrained
to sum to $N_{t}$. Initializing a constellation as
$\displaystyle c=(1,1,1,...,1,0,0,0,...,0),$
a tuple of $N_{t}$ ones and $I_{t}-N_{t}$ zeros, we start at $c$’s first entry
($i=0$, representing edges assigned to the first red dot) by defining the
rolling sum $C=\sum_{j=0}^{i-1}c_{j}$ and drawing a sample, $\ell$, from
$\displaystyle
p(T_{t}=\ell|N_{t},C)=\frac{p(T_{t}=\ell)}{1-\sum_{m>N_{t}-C}p(T_{t}=m)},$
the conditionally renormalized degree distribution. We then set $c_{i}=\ell$,
set the next $\ell-1$ entries to 0 (taking edges from next red dots), and then
repeat the process with $i\rightarrow i+\ell$ until $C=N_{t}$. In Fig. 1b,
this is illustrated for $\ell=3$, and in Fig. 1c, we verify that the approach
can maintain the skewed target distributions we have in mind. Finally,
completing the forest requires us to link stars over time. We do this in a
simple way, shuffling $c$ and assigning the resulting events to infectious
nodes in order, without regard to past assignments.
This edge rewiring approach requires $\mathcal{O}(N_{t})$ negative binomial
samples and is, as a result, efficient enough to be performed every day for a
given model trajectory. In the Washington example, it leads to a 3 orders of
magnitude speed up over the rejection method, allowing us to sample large sets
of forests and compute statistics empirically.
## IV. Tree interactions are weak
Setting aside the empirical approach for now, our goal is to calculate
probabilities, like the distribution of tree sizes or their chance of
extinction over time. The next sections are the core of this paper, developing
a generating function [8] approach for modeling forest growth.
The sampling algorithm above motivates an overarching strategy, first
characterizing the possible daily transmission constellations (Fig. 1a inset)
and then considering the process of assigning stars to trees. With that in
mind, the negative binomial distribution in Eq. 6 can be represented as a
probability generating function [9],
$\displaystyle f_{T_{t}}(z)$
$\displaystyle=p(T_{t}=0)+p(T_{t}=1)z+p(T_{t}=2)z^{2}+...$
$\displaystyle=\sum_{\ell\geq
0}\binom{k_{t}+\ell-1}{\ell}(p_{t}z)^{\ell}(1-p_{t})^{k_{t}}$
$\displaystyle=\left(\frac{1-p_{t}}{1-p_{t}z}\right)^{k_{t}}$
where $p_{t}\equiv\mu_{t}/(\mu_{t}+k_{t})$ and $z$ is an arbitrary complex
number. Then, for a given day’s $I_{t}$, constellations drawn from independent
samples can be organized by the total number of edges in a generating function
product,
$\displaystyle C_{t}(z)$
$\displaystyle=f_{T_{t}}(z)^{I_{t}}=\left(\frac{1-p_{t}}{1-p_{t}z}\right)^{I_{t}k_{t}}$
$\displaystyle=\sum_{n\geq 0}p(n\text{ edges})z^{n}.$
As a first illustration of a probability calculation, using $[z^{n}]$ to
signify extracting the $z^{n}$’s coefficient from a formal power series,
Newton’s binomial theorem implies
$[z^{N_{t}}]C_{t}(z)=\binom{I_{t}k_{t}+N_{t}-1}{N_{t}}p_{t}^{N_{t}}(1-p_{t})^{I_{t}k_{t}}$
(7)
is the probability of satisfying Eq. 5 for a fixed $N_{t}$ over all possible
constellations with $I_{t}$ infectious nodes.
Eq. 7 represents the subset of graphs consistent with the trajectory
constraints, that is the $I_{t}$ and $N_{t}$ node populations. To get a more
detailed view of constellation structure, we can calculate the probability of
an infectious individual infecting $m$ new people across constrained graphs.
Introducing arbitrary complex number $u$ to highlight [8] an $m$-event,
$\displaystyle C_{t,m}(z,u)=\left(p(T_{t}=m)uz^{m}+\sum_{\ell\neq
m}p(T_{t}=\ell)z^{\ell}\right)^{I_{t}}$ $\displaystyle=(1-p_{t})^{k_{t}I_{t}}$
$\displaystyle\quad\times\left[\binom{k_{t}+m-1}{m}(u-1)(p_{t}z)^{m}+(1-p_{t}z)^{-k_{t}}\right]^{I_{t}},$
gives a path towards calculating the analog of Eq. 6 with tree interactions.
Note that $C_{t,m}(z,1)=C_{t}(z)$ for any $m$, as required. Then, the expected
number of $m$-events in day $t$’s constellation, $n_{m}$, is
$\text{E}[n_{m}|N_{t},I_{t}]=\frac{[z^{N_{t}}]\frac{\partial}{\partial
u}C_{t,m}(z,u)|_{u=1}}{[z^{N_{t}}]C_{t}(z)},$
and, making use of Eq. 7, we find
$\displaystyle\frac{\text{E}[n_{m}|N_{t},I_{t}]}{I_{t}}=\frac{\binom{k_{t}+m-1}{m}\binom{k_{t}(I_{t}-1)+N_{t}-m-1}{N_{t}-m}}{\binom{k_{t}I_{t}+N_{t}-1}{N_{t}}}$
(8)
$\displaystyle\approx\binom{k_{t}+m-1}{m}p_{t}^{m}(1-p_{t})^{k_{t}}\left[1+\mathcal{O}\left(\frac{m}{I_{t}}\right)\right],$
(9)
where the first line is exact for a given trajectory, and the second line uses
Sterling’s approximation to the Gamma function in the form
$\Gamma(x+\alpha)\sim\Gamma(x)x^{\alpha}[1+(\alpha^{2}+\alpha/2+1/12)/x]$ for
$\alpha<<x$. Eq. 9 shows that the expected degree distribution of infectious
nodes is asymptotically aligned with Eq. 6 but perturbed by tree interactions
that decay with $I_{t}$.
We can prove further that the degree distribution converges to the expected
distribution as graph size grows. A similar calculation, this time leveraging
$C_{t,m}(z,u)$ to compute $\text{E}[n_{m}(n_{m}-1)|N_{t},I_{t}]$, leads to an
intuitive asymptotic variance estimate
$\displaystyle\text{V}[n_{m}|N_{t},I_{t}]\approx
I_{t}p(T_{t}=m)\left[1-p(T_{t}=m)\right],$
implying that deviations around the expected $n_{m}$ are
$\mathcal{O}(1/\sqrt{I_{t}})$, vanishing as $I_{t}$ grows. Thus, the
infectious node degree distribution converges to Eq. 8 at a square-root rate,
and $P(\ell|N_{t},I_{t})\approx\text{E}[n_{\ell}|N_{t},I_{t}]/I_{t}$ is an
approximation that improves with constellation size. In other words, as
disease prevalence increases, we expect tree interactions to fade and
constellation geometry to stabilize.
Figure 2: Interactions and stability. Empirical degree distributions from 1000
sampled constellations (black dots) have interpretable dependence on the
infectious node population. Generating function calculations (blue lines, mean
and 2 standard deviations around it) show that the degree distribution
converges to its expected value with negligible tree interactions (red dashed
line) as disease prevalence increases.
We can validate these results by comparing $P(\ell|N_{t},I_{t})$ to sampled
constellations of various size. This is shown in Fig. 2 for time-averaged
values of $\mu_{t}$ and $k_{t}$ from the Washington COVID-19 model [3]. In the
figure, dots come from the degree distribution across 1000 sampled graphs
while the blue lines show the interacting generating function results (the
mean and 2 standard deviations around it). At low $I_{t}$, tree interactions
are significant and deviations from Eq. 6 (red dashed line) are clear in both
the samples and the generating function estimates. Meanwhile, volatility
around the mean, both in the samples and in the asymptotic distribution decay
with $I_{t}$.
Fig. 2 illustrates the regime where tree interactions warp the epidemiology.
This is a key structural feature of the transmission forest, particularly
relevant in near elimination contexts, and it’s something we intend to study
further. But in the Washington COVID-19 model, from March 2020 to March 2021,
the minimum $I_{t}$ is roughly 1500, implying that for the bulk of the model
time period trees grow in functional isolation.
## V. Trees as recursive functions
Moving from constellations to transmission trees requires us to link
$P(\ell|N_{t},I_{t})$-distributed events over time, being careful to track
when nodes start and stop being infectious. As we’ll see, this process lends
itself to a family of recursively defined generating functions that can be
efficiently evaluated and analysed in reverse time.
The previous section’s results can be written concisely in terms of the
generating function for stars,
$\displaystyle s_{t}(z)$ $\displaystyle=\sum_{\ell\geq
0}P(\ell|N_{t},I_{t})z^{\ell}$ $\displaystyle\approx\sum_{\ell\geq
0}\frac{\text{E}[n_{\ell}|N_{t},I_{t}]}{I_{t}}z^{\ell}$
$\displaystyle\approx\left(\frac{1-p_{t}}{1-p_{t}z}\right)^{k_{t}},$
where the first approximation comes from neglecting volatility around the
average constellation, and the second approximation comes from neglecting tree
interactions – both valid above low prevalence. It’s notable that this likely
would’ve been our naive choice based on Eq. 6 alone, but it’s nice to have
sound theoretical footing and a deeper understanding.
In any case, for a given trajectory, we can formally define the set of all
possible transmission trees rooted at time $r$, $\mathbb{T}_{r}$, by
associating with a tree a monomial $\tau=z_{1}^{n_{1}}z_{2}^{n_{2}}\dotsm
z_{\mathcal{T}}^{n_{\mathcal{T}}}$ where $n_{t}$ is the number of nodes at
time $t$ up to final time $\mathcal{T}$. The set is finite since $0\leq
n_{t}\leq N_{t}$ for all $t$, and we can consider the generating function of
complex vector $\mathbf{z}=(z_{1},...,z_{\mathcal{T}})$
$T_{r}(\mathbf{z})=\sum_{\tau\in\mathbb{T}_{r}}p(\tau|N_{t},I_{t})z_{r}z_{r+1}^{n_{r+1}}\dotsm
z_{\mathcal{T}}^{n_{\mathcal{T}}},$ (10)
which encapsulates tree structures rooted at time $r$. This expression is more
of a formal statement than anything else since it’s not clear how sums over
$\mathbb{T}_{r}$ are executed, and $\mathbf{z}$ is $\mathcal{T}$-dimensional.
We still need a more practical method for function evaluation. Towards that
end, we define $\rho(t)$ as the probability of being infectious $t$ days after
infection, which in the deterministic case is
$\rho(t)=\begin{cases}1&d_{E}<t\leq d_{E}+d_{I}\\\
0&\text{otherwise},\end{cases}$
but clearly could be defined based on a more sophisticated pathogenesis model.
Then, for a binary process, the binomial generating function
$b_{t}(z)=1-\rho(t)+\rho(t)z$ represents infectious status.
With this book-keeping machinery in hand, we can make progress by recognizing
that all trees are star graphs where the internal node is the root and the
external nodes are replaced by trees. This is a well-known recursive idea [8],
and it implies in our case that
$T_{r}(\mathbf{z})=z_{r}\prod_{t\geq{1}}^{\mathcal{T}}\left[1-\rho(t-r)+\rho(t-r)s_{t}(T_{t}(\mathbf{z}))\right],$
(11)
linking root $z_{r}$ to new trees at the appropriate infectious times through
nested compositions of $s_{t}(z)$ and $b_{t}(z)$.222Note that an additional
function composition can be used to incorporate surveillance. If cases are
reported with probability $\pi_{t}$, the binomial generating function
$o_{t}(z)=1-\pi_{t}+\pi_{t}z$ determines if a node is observed, and we can
replace $z_{r}$ with $o_{r}(z_{r})$ directly. The product in Eq. 11 is
tractable given the boundary condition
$T_{\mathcal{T}}(\mathbf{z})=z_{\mathcal{T}}$, which captures the idea that a
tree rooted at $\mathcal{T}$ has no time to grow (that is
$\rho(t-\mathcal{T})=0$ for all $t$ under consideration).
Eq. 11 defines a family of transmission tree generating functions indexed by
their root times, but it doesn’t give us a closed form expression for
$T_{r}(\mathbf{z})$. In practice, Eq. 10 reminds us of $T_{r}(\mathbf{z})$’s
power series structure, and it motivates statistically meaningful function
evaluations. Those evaluations can then be carried out in bespoke recursive
programs based on Eq. 11, starting at time $\mathcal{T}$ and working
backwards.
## VI. Calculating forest statistics
To illustrate the theory’s application and to simultaneously validate the
sampling and recursive approaches, we take up 3 example calculations in this
section: the size distribution of trees in the forest, the tree survival
function, and the relationship between size and lifetime.
Tree size. Evaluating $T_{r}(\mathbf{z})$ at $\mathbf{z}=(z,z,...,z)$, that is
dropping all delineation between $z_{t}$ over time, organizes trees by the
number of nodes. Specifically, Eq. 10 becomes
$\displaystyle T_{r}(z)$
$\displaystyle=\sum_{\tau\in\mathbb{T}_{r}}p(\tau|N_{t},I_{t})z^{n_{r}+n_{r+1}+\dotsm+n_{\mathcal{T}}}$
$\displaystyle=\sum_{n\geq 0}p(n|N_{t},I_{t})z^{n},$
where $n=\sum_{i=1}^{\mathcal{T}}n_{i}$, and in the second line we’ve grouped
terms in the sum. Our goal is to calculate the coefficients in this now one-
dimensional power series.
Complex analysis offers an efficient and well-known path forward [8]. For any
analytic generating function $f(z)$, Cauchy’s integral theorem implies that
$[z^{n}]f(z)=\frac{1}{2\pi i}\oint\frac{f(\xi)}{\xi^{n+1}}d\xi,$
over any closed contour in the complex plane. Choosing $|\xi|=1$ as the
contour leads to
$\displaystyle[z^{n}]f(z)$
$\displaystyle=\frac{1}{2\pi}\int_{0}^{2\pi}f\left(e^{i\theta}\right)e^{in\theta}d\theta$
$\displaystyle\approx\frac{1}{M}\sum_{m=1}^{M}f\left(e^{2\pi
i\frac{m}{M}}\right)e^{2\pi i\frac{nm}{M}},$
for angle $\theta$, which we’ve discretized into $M$ points along the unit
circle in the second line. The sum above is the discrete Fourier transform of
$f(z)$, which implies that a set of coefficients can be extracted by passing
the unit circle through $f(z)$ and transforming. Eq. 11 can be used to
evaluate $T_{r}(z)$ on the unit circle directly, building function evaluations
from $\mathcal{T}$ backwards.
Figure 3: Calculating forest statistics. (a) Complex-analysis based methods
can be used to compute the size distribution of trees (yellow) which compares
favorably to samples (black) (b) Structured evaluations can be used to compute
the tree survival function (orange) which captures key features of a Kaplan-
Meier estimator applied to samples (black) and can be used to compute the
daily hazard function (grey). (c) Expectations (pink curve) and intervals
(pink region) can be computed via Eq. 11 and its derivatives, shown here for
the relationship between tree size and lifetime.
The results of this process are compared to 250k sample trees rooted in early
March 2020 in Fig. 3a, using the parameters from the Washington COVID-19 model
[3]. Both methods yield consistent estimates, but with the generating function
approach (yellow) remaining stable at very low probabilities. As a consequence
of the skewed, individual-level transmission distribution (Eq. 6), tree size
is heavy-tailed. Most trees are simply roots with little growth, but a small
set of trees capture an out-sized fraction of nodes. In other words,
intuitively, super spreaders imply the existence of super trees.
Tree survival. Spontaneous transmission tree extinction implies, in terms of
monomials $\tau=z_{1}^{n_{1}}z_{2}^{n_{2}}\dotsm
z_{\mathcal{T}}^{n_{\mathcal{T}}}$, that $n_{t}=0$ for all $t$ above
extinction time $t^{*}$. This observation motivates a systematic evaluation
method for organizing transmission trees by their lifetimes. Consider
$\mathbf{z}^{*}=(z_{1},z_{2},...,z_{t^{*}},0,0,...)$. Then,
$T_{r}(\mathbf{z}^{*})=\sum_{\tau\in\mathbb{T}_{r}}p(\tau,n_{t>t^{*}}=0|N_{t},I_{t})z_{r}z_{r+1}^{n_{r+1}}\dotsm
z_{t^{*}}^{n_{t^{*}}},$
which, upon setting all remaining $z_{t}=1$, gives
$\displaystyle T_{r}(\Theta(t^{*}))$
$\displaystyle=\sum_{\tau\in\mathbb{T}_{r}}p(\tau,n_{t>t^{*}}=0|N_{t},I_{t})$
$\displaystyle=p(L\leq t^{*}-r)$
where $\Theta(t)$ is the Heaviside step function evaluated along $\mathbf{z}$
and $L$ is the tree lifetime. The survival function,
$S_{r}(t^{*})=P(L>t^{*}-r)$ is then
$\displaystyle S_{r}(t^{*})=1-T_{r}(\Theta(t^{*}))$ (12)
which can be evaluated for all $t^{*}$ again using Eq. 11.
In Fig. 3b, we compare Eq. 12 to a Kaplan-Meier estimator applied to the same
tree samples as above. The theory (orange) is consistent with the empirical
approach (black), even in finer details. Small drops in survival correspond to
times of more concerted transmission suppression in Washington, which are
further highlighted in the hazard function (grey). We refer to past work [10,
11, 12] for epidemiological details, but these survival analysis concepts
applied to transmission trees give perspective on intervention efficacy that
complements more conventional measures like the effective reproductive number
[5].
Size vs. lifetime. Finally, to illustrate how some statistics can be related,
consider $\mathbf{z}\rightarrow u\mathbf{z}$ for complex $u$. Then,
$T_{r}(\mathbf{z})=\sum_{\tau\in\mathbb{T}_{r}}p(\tau|N_{t},I_{t})z_{r}z_{r+1}^{n_{r+1}}\dotsm
z_{\mathcal{T}}^{n_{\mathcal{T}}}u^{n}$
consolidates tree size and structure. Based on the approach above, evaluations
of the form
$\displaystyle T_{r}(u\Theta(t^{*}))-T_{r}(u\Theta(t^{*}-1))$
$\displaystyle=\sum_{\tau\in\mathbb{T}_{r}}p(\tau|L\leq
t^{*}-r)u^{n}-\sum_{\tau\in\mathbb{T}_{r}}p(\tau|L\leq t^{*}-1-r)u^{n}$
$\displaystyle=\sum_{n\geq 0}p(n|L=t^{*}-r)u^{n},$
can be used to relate tree size to the extinction time, and we can compute
statistics. For example,
$\text{E}[n|L=t^{*}-r]=\frac{\frac{\partial}{\partial
u}[T_{r}(u\Theta(t^{*}))-T_{r}(u\Theta(t^{*}-1))]_{u=1}}{T_{r}(\Theta(t^{*}))-T_{r}(\Theta(t^{*}-1))},$
is the expected size at a fixed lifetime. To evaluate the numerator, we can
differentiate Eq. 11 to derive a recursion relation for $T_{r}(\mathbf{z})$’s
partial derivatives. We find
$\frac{\partial}{\partial
u}T_{r}(u\mathbf{z})=T_{r}(u\mathbf{z})\left(\frac{1}{u}+\sum_{t=1}^{\mathcal{T}}\frac{\rho(t-r)s_{t}^{\prime}(T_{t}(u\mathbf{z}))}{1-\rho(t-r)+\rho(t-r)s_{t}(T_{t}(u\mathbf{z}))}\frac{\partial}{\partial
u}T_{t}(u\mathbf{z})\right)$ (13)
where primes denote total derivatives and the evaluation depends on the full
family of rooted generating functions $T_{r}(\mathbf{z})$. Practically, for
some specific $\mathbf{z}$, we can evaluate all $T_{r}(\mathbf{z})$ first,
which then specifies the details in Eq. 13. A similar calculation can be used
to compute higher order derivatives and corresponding higher order statistics.
We compare the generating function approach above to samples with non-zero
lifetime in Fig. 3c. Statistics based on Eq. 11 (pink) gracefully capture
discretization effects, like the necessary spacing between trees surviving one
and two generations, as well as the transition to continuous lifetimes.
Features of the relationship, like local slope changes in keeping with the
mitigation environment in Washington at that time, are clearly reflected in
both the model and the samples. That this distribution can be calculated
efficiently based on time series data alone is striking, and it illustrates
how accessible transmission tree statistics might be with the right approach.
## VII. Conclusion
The key idea in this paper is that conventional stochastic process models of
disease transmission, based on Eq. 4, can be used to estimate transmission
tree properties without additional data. The bridge between the population-
scale of trajectories and the individual-scale of trees comes from Eq. 5,
which led us to a generating function family that can be used to efficiently
compute a variety of epidemiologically relevant statistics.
The theory developed here offers new perspective on individual-level data,
like distributions across outbreak investigations or associations among
collections of genetic sequences. Those types of comparisons were made
empirically in our previous paper [3], but the analytic approach gives
additional paths towards more quantitative comparisons and potentially joint
inferences. Speaking broadly, tree structures have a long mathematical
history. Eq. 11 represents a promising and intuitive connection between that
body of work and the classical compartmental models used to describe disease
transmission.
## Acknowledgements
This work was done with many people’s support. In particular, we want to thank
Kevin McCarthy for his attention and detailed comments. His input made the
writing much more clear and it brought to light implications we hadn’t
considered.
## References
* Arunkumar _et al._ [2019] G. Arunkumar, R. Chandni, D. T. Mourya, S. K. Singh, R. Sadanandan, P. Sudan, and B. Bhargava, Outbreak investigation of nipah virus disease in Kerala, India, 2018, The Journal of Infectious Diseases 219, 1867 (2019).
* Grenfell _et al._ [2004] B. T. Grenfell, O. G. Pybus, J. R. Gog, J. L. Wood, J. M. Daly, J. A. Mumford, and E. C. Holmes, Unifying the epidemiological and evolutionary dynamics of pathogens, Science 303, 327 (2004).
* Thakkar and Famulare [2022] N. Thakkar and M. Famulare, Covid-19 epidemiology as emergent behavior on a dynamic transmission forest, arXiv preprint arXiv:2205.02150 (2022).
* Bedford _et al._ [2020] T. Bedford, A. L. Greninger, P. Roychoudhury, L. M. Starita, M. Famulare, M.-L. Huang, A. Nalla, G. Pepper, A. Reinhardt, H. Xie, _et al._ , Cryptic transmission of SARS-CoV-2 in Washington state, Science 370, 571 (2020).
* Anderson and May [1992] R. M. Anderson and R. M. May, _Infectious diseases of humans: dynamics and control_ (Oxford University Press, 1992).
* Fewster [2014] R. Fewster, Stats 325, Chapter 6: Branching Processes: The Theory of Reproduction (Lecture notes, 2014).
* Jaynes [1957] E. T. Jaynes, Information theory and statistical mechanics, Physical Review 106, 620 (1957).
* Flajolet and Sedgewick [2009] P. Flajolet and R. Sedgewick, _Analytic combinatorics_ (Cambridge University Press, 2009).
* Miller [2018] J. C. Miller, A primer on the use of probability generating functions in infectious disease modeling, Infectious Disease Modelling 3, 192 (2018).
* Thakkar _et al._ [2020a] N. Thakkar, R. Burstein, H. Hu, P. Selvaraj, and D. Klein, Social distancing and mobility reductions have reduced COVID-19 transmission in King County, WA, Institute for Disease Modeling (March 29, 2020a).
* Thakkar and Famulare [2020] N. Thakkar and M. Famulare, One state, many outbreaks: a transmission modeling perspective on current COVID-19 trends in King, Pierce, and Yakima counties, Institute for Disease Modeling (August 17, 2020).
* Thakkar _et al._ [2020b] N. Thakkar, R. Burstein, and M. Famulare, Towards robust, real-time, high-resolution COVID-19 prevalence and incidence estimation, Institute for Disease Modeling (December 13, 2020b).
|
# Multi-core parallel tempering Bayeslands for basin and landscape evolution
Rohitash Chandra<EMAIL_ADDRESS>R. Dietmar Müller Ratneel
Deo Nathaniel Butterworth Tristan Salles Sally Cripps EarthByte Group,
School of Geosciences, University of Sydney, NSW 2006, Sydney, Australia
Centre for Translational Data Science, University of Sydney, NSW 2006, Sydney,
Australia Sydney Informatics Hub, University of Sydney, NSW 2006, Sydney,
Australia School of Mathematics and Statistics, University of Sydney, NSW
2006
Sydney, Australia
###### Abstract
In recent years, Bayesian inference has become a popular methodology for the
estimation and uncertainty quantification of parameters in geological and
geophysical forward models. Badlands is a basin and landscape evolution
forward model for simulating topography evolution at a large range of spatial
and time scales. Previously, Bayesian inference has been used for parameter
estimation and uncertainty quantification in Badlands, an extension known as
Bayeslands. It was demonstrated that the posterior surface of these parameters
could exhibit highly irregular features such as multi-modality and
discontinuities making standard Markov Chain Monte Carlo (MCMC) sampling
difficult. Parallel tempering (PT) is an advanced MCMC method suited for
irregular and multi-modal distributions. Moreover, PT is more suitable for
multi-core implementations that can speed up computationally expensive
geophysical models. In this paper, we present a multi-core PT algorithm
implemented in a high performance computing architecture for enhancing
Bayeslands. The results show that PT in Bayeslands not only reduces the
computation time over a multi-core architecture, but also provides a means to
improve the sampling process in a multi-modal landscape. This motivates its
usage in larger-scale problems in basin and landscape evolution models.
###### keywords:
Landscape evolution , Bayesian inference, Parallel tempering , Badlands ,
Bayeslands
††journal: IEEE Tran. Geo. Remote Sens.
## 1 Introduction
Recent developments in Earth evolution models have the capability to link
models for dynamic and isostatic topography through time [1] with landscape
evolution models [2]. This has given the opportunity to model landscape
evolution in response to surface uplift and subsidence over a large range of
spatial scales, and track sediments from source to sink [3, 4]. Geophysical
forward models depend on uncertain initial and boundary conditions [5], but a
large variety of observations are available to optimise and ground-truth these
models. Uncertain input parameters for these models include time-dependent
topography, relative sea level change, time-varying rainfall and rock
lithology and erodibility [6, 7] . Observations useful for model ground-
truthing include digital elevation models and river geometries, sediment
thickness and stratigraphy in sedimentary basins, and other geological proxies
for landscape evolution.
Bayesian inference provides a rigorous approach for uncertainty quantification
[8, 9, 10] and emerged as a popular tool for fitting complex parameter models
to data [11]. This provides advantages over optimization methods that provide
single point solutions and need multiple experimental runs for uncertainty
quantification [12, 13]. Estimation of parameters that fit observational data
in geophysical models have shifted from optimization [14] to inference that
addresses uncertainty quantification given that the models provide an
approximation of geophysical processes [15]. Markov Chain Monte Carlo (MCMC)
sampling methods implement Bayesian inference where a sample of the desired
distribution is obtained by observing the chain after a number of steps[16,
17, 18, 19]. Bayesian methods have become popular in geophysics in the past
few decades [20, 21, 22, 23]. The convergence in canonical MCMC sampling
methods face further challenges when the size of the problem increases in
terms of the number of parameters and computational time of running the model.
This is a challenge in geophysics that use forward models such as Badlands [3,
4]. Although advanced MCMC methods such as Hamiltonian based samplers have
been proposed [24, 25, 26], they require gradient information from the models
to form proposals. Acquiring gradient information from complex geo-physical
models such as Badlands is difficult, therefore, Bayesian inference via MCMC
random-walk proposals was proposed in a framework called Bayeslands [27].
Bayeslands demonstrated that the posterior surface of selected parameters
could exhibit highly irregular features, such as multi-modality and
discontinuities which made it difficult for sampling. Hence, there is a need
to explore efficient sampling methods that feature gradient free proposals and
addresses multi-modality.
Parallel tempering [28, 29] is a MCMC sampling scheme designed for sampling
multi-modal distributions. Another feature is their feasibility of
implementation in multi-core or parallel computing architectures. Due to
massive computational requirements of geo-physical models, parallel computing
architectures have been widely utilized in geosciences [30, 31, 32]. This can
therefore help us to extend and improve our previous approach called
Bayeslands that featured Bayesian inferences via MCMC random-walk sampler for
Badlands [27]. Hence, the major limitations such as computational time and the
need for efficient multi-modal distribution given by the MCMC random-walk
sampler in Bayeslands can be overcome by multi-core parallel tempering.
In this paper, we present a multi-core parallel tempering approach for
Bayeslands that features uncertainty quantification and estimation of selected
parameters for basin and landscape evolution. We use selected problems that
feature synthetic and real-world applications for selected parameters such as
rainfall and erodibility that affect the topography development over thousands
of years. We evaluate the performance of multi-core parallel tempering
Bayeslands (PT-Bayeslands) for selected number of multiprocessing cores and
report the overall computation time. The results report the successive
predicted topographies, successive sediments given by erosion-deposition,
their prediction accuracy, posterior distribution of selected parameters for
selected problem, and also the overall computation time.
The rest of the paper is organised as follows. Section 2 provides a background
on Badlands and Bayeslands. Section 3 presents the methodology and Section 4
presents the experiments and results. Section 5 provides a discussion and
Section 6 concludes the paper with directions for future research.
## 2 Background and Related Work
### 2.1 Badlands
Over the last decades, many numerical models have been proposed to simulate
how the Earth surface has evolved over geological time scales in response to
different driving forces such as tectonics or climatic variability [33, 34,
35, 36, 37]. These models combine empirical data and conceptual methods into a
set of mathematical equations that can be used to reconstruct landscape
evolution and associated sediment fluxes [38, 39]. They are currently used in
many research fields such as hydrology, soil erosion, hillslope stability and
general landscape studies.
In this paper, we use Badlands [3, 40, 41]. This framework is intended to
simulate regional to continental sediment deposition and associated
sedimentary basin architecture [4, 42]. In its most simple formulation, the
Earth surface elevation change is related to the interaction of three types of
processes, one driven by tectonic, another describing the smoothing effects
associated to diffusive processes, and a last one representing the erosive
power of water flow:
$\frac{\partial z}{\partial t}=-\nabla\cdot\mathbf{q_{s}}+u$
where $u$ in $m\cdot yr^{-1}$ is a source term that represents tectonic
uplift. The total downhill sediment flux $\mathbf{q_{s}}$ is defined by:
$\mathbf{q_{s}}=\mathbf{q_{r}}+\mathbf{q_{d}}$
$\mathbf{q_{s}}$ is the depth-integrated, bulk volumetric soil flux per width
($m^{2}\cdot yr^{-1}$). $\mathbf{q_{r}}$ represents transport by fluvial
system and $\mathbf{q_{d}}$ hillslope processes.
Here the fluvial incision is based on the classical detachment-limited stream
power law (SPL), in which erosion rate $\dot{\epsilon}$ depends on drainage
area $A$, net precipitation $P$ and local slope $S$ and takes the form:
$\dot{\epsilon}=\kappa(PA)^{m}S^{n}$
$\kappa$ is an erodibility coefficient while $m$ and $n$ are positive
exponents characterising the contribution of rain and slope respectively on
erosion processes. Despite its simplicity, the SPL reproduces many of the
characteristic features of natural systems where detachment-limited erosion
regime dominates [33]. With this formulation, sediment deposition occurs
solely in topographically closed depression and marine offshore regions.
Constraining the exact values of $\kappa$, $m$ and $n$ from natural landscapes
is relatively complex and not easily measured from direct field observations.
Their values depend on the geomorphological, climatic and tectonic context and
their ranges are commonly admitted to be:
* 1.
$0\,<\,m\,<\,2$
* 2.
$0\,<\,n\,<\,4$
* 3.
$\kappa$ varies by several orders of magnitude not only based on lithology,
climate, sedimentary flux or river channel width but also with the chosen
values of $m$ and $n$.
In addition to overland flow, semi-continuous processes of soil displacement
are accounted for using a linear diffusion law commonly referred to as soil
creep [34]:
$\frac{\partial z}{\partial t}=\kappa_{d}\nabla^{2}z$
in which $z$ is the elevation and $\kappa_{d}$ is the diffusion coefficient.
This transport rate depends linearly on topographic gradient and encapsulates
in a simple formulation the processes operating on superficial sedimentary
layers.
### 2.2 Bayeslands
Bayelands [27] illustrates how Bayesian inference, via the posterior
distribution, can be used to estimate and quantify the uncertainty surrounding
initial conditions of factors such as rainfall and erodibility. Formally,
given some data, $\mathbf{D}$, we make inference regarding unknown parameters,
denoted by $\theta$, via the posterior distribution $p(\theta|\mathbf{D})$,
given by Bayes rule,
$p(\theta|\mathbf{D})=\frac{p(\mathbf{D}|\theta)p(\theta)}{P(\mathbf{D})}$
where $p(\mathbf{D}|\theta)$ is the likelihood of the data given the
parameters, $p(\theta)$ is the prior, and $p(\mathbf{D})$ is a normalizing
constant and equal to $\int p(\mathbf{D}|\theta)p(\theta)d\theta$. Frequently
the posterior distribution does not have a closed form solution and it is
approximated by a sampling based estimate.
Bayeslands used MCMC to obtain samples of $\theta$ from the posterior
distribution by proposing values from some known distribution $q(\theta)$. The
value of $\theta$ is then set equal to these proposed values with a
probability which ensures the detailed balance condition is met, otherwise the
chain remains in its current position, see [43]. The transition kernel which
moves the Markov chain from one point to another in the parameter space is the
product of the proposal distribution $q$ and the acceptance probability. Under
certain conditions, the draws from this transition kernel converge to draws
from the stationary distribution, $p(\theta|\mathbf{D})$. This algorithm is
known as the Metropolis-Hastings (MH) algorithm [16] which is given in
Algorithm 1
Algorithm 1 Metropolis-Hastings (MH)
Result: Drawing from $p(\theta|\mathbf{D})$
Set maximum number of iterations, $K$, and initialize $\theta=\theta^{[0]}$
Set the current value of $\theta$, $\theta^{c}=\theta^{[0]}$
for _$k=1,\ldots,K$_ do
1. 1.
Propose a new value of $\theta$, $\theta^{p}\sim q(\theta|\theta^{c})$
2. 2.
Compute acceptance probability
$\alpha=\min\left(1,\frac{p(\theta^{p}|\mathbf{D})q(\theta^{c}|\theta^{p})}{p(\theta^{c}|\mathbf{D})q(\theta^{p}|\theta^{c})}\right)$
3. 3.
Draw $u\sim U[0,1]$
if _$u <\alpha$_ then
$\theta^{[k]}=\theta^{p}$
else
$\theta^{[k]}=\theta^{c}$
end
end
The development of transitions kernels which efficiently explore posterior
distributions has been the subject of much research in sampling methods [24].
Bayeslands has major limitations since MCMC random-walk requires thousands of
samples to accurately explore the posterior distribution. Executing it in a
sequential manner that employs only one processing unit becomes a
computational challenge taking into account the time needed to run Badlands
when the application problem covers a larger region featuring millions of
years. We will address these computational challenges using multi-core
parallel tempering in this paper.
### 2.3 Parallel tempering
Posterior distributions of parameters in geophysical inversions problems are
notoriously difficult to explore. They are high dimensional, multi-modal and
very irregular, by which we mean the derivative of posterior is often
difficult to compute and/or it has many discontinuities. An example is shown
in Figure 1 that features the likelihood surface of the Continental Margin
problem used in Bayeslands [27]. While proposal distributions based on
canonical random-walk MCMC methods are gradient free, in sampling multi-modal
posterior distributions, they become very inefficient [44]. As evident, the
likelihood surface can be highly multi-modal and irregular which makes it
natural for these algorithms to be trapped in a local minimum. In such
situations, proposal distributions that feature gradients such as Hamiltonian
methods will also face difficulties since there are local convergence issues
related to gradient based methods [45]. In complex geophysical inversion
problems, proposal distributions usually need to gradient free and explore
most, if not all the posterior modes. Methods such as gradient-free
Hamiltonian Monte-Carlo [46] do not address highly irregular and complex
multi-modal distributions.
Figure 1: Likelihood surfaces of the Continental Margin topography for the
rainfall and erodibility parameters only[27].
Parallel tempering has been motivated by thermodynamics of physical systems
where the temperature of a system determines the ability to change [47].
Parallel tempering is also known as replica exchange or the Multi-Markov Chain
method [48, 49] suitable for multi-modal distributions by running multiple
MCMC chains at different temperatures in parallel. Typically, gradient free
proposals within chains are used for proposals for exploring multi-modal and
discontinuous posteriors [50, 51] .
The Markov chains in the parallel replicas have stationary distributions which
are equal to (up to a proportionality constant)
$p(\theta|\mathbf{D})^{\beta}$, where $\beta\in[0,1]$, with $\beta=0$
corresponds to a stationary distribution which is uniform and $\beta=1$
corresponds to a stationary distribution which is the posterior. The replica
with smaller values of $\beta$ are able to explore a larger regions of
distribution, while those with higher values of $\beta$ typically explore
local regions.
Communication between the parallel replicas is essential for the efficient
exploration of the posterior distribution. This is done by considering the
chain and the parameters as part of the space to be explored as highlighted in
Algorithm 2. Suppose there are $M$ chains, indexed by $m$, with corresponding
stationary distributions,
$p_{m}(\theta|\mathbf{D})=p(\theta|\mathbf{D})^{\beta_{m}}$, for,
$m=1,\ldots,M$, with $\beta_{1}=1$ and
$\beta_{M}<\beta_{M-1}<\ldots,\beta_{1}$, then the pair $(m,\theta)$ are
jointly proposed and accepted/rejected according to the MH algorithm. The
stationary distribution of this sampler is proportional to
$p(\theta|\mathbf{D})^{\beta_{m}}p(m)$. The quantity $p(m)$ must be chosen by
the user and is referred to as a pseudo-prior. For efficient mixing of the
chain we use $q(m^{p}=m^{c}\pm 1|m^{c})=1/2$, for $1<m^{c}<M$, and
$q(m^{p}=m^{c}+1|m^{c}=1)=q(m^{p}=m^{c}-1|m^{c}=M)=1$.
Algorithm 2 Metropolis-Hastings Parallel Tempering (MHPT)
Result: Drawing from $p(\theta|\mathbf{D})$
Set maximum number of iterations, $K$, and initialize $\theta=\theta^{[0]}$,
and $m=m^{[0]}$
Set the current value of $\theta$, to $\theta^{c}=\theta^{[0]}$ and $m$ to
$m^{c}=m^{[0]}$
for _$k=1,\ldots,M$_ do
1. 1.
Update $\theta^{[k]}$, from the chain with
$p(\theta|\mathbf{D})^{\beta_{m^{c}}}$ as its invariant distribution according
to Alg 1.
2. 2.
Propose a new value of $m^{p}$, from $q(m^{p}|m^{c})$.
3. 3.
Compute acceptance probability
$\alpha=\min\left(1,\frac{p(\theta^{[k]}|\mathbf{D})^{\beta_{m^{p}}}p(m^{p})q(m^{c}|m^{p})}{p(\theta^{[k]}|\mathbf{D})^{\beta_{m^{c}}}p(m^{c})q(m^{p}|m^{c})}\right)$
4. 4.
Draw $u\sim U[0,1]$
if _$u <\alpha$_ then
$m^{[k]}=m^{p}$
else
$m^{[k]}=m^{c}$
end
end
A number of attempts have been made to improve the parallel tempering
algorithm, which includes the efficient way of finding $\beta_{m}$ for
$m=1,\ldots,M$, [52, 47], exchange of solutions among the replicas [53], and
joint adaption $\beta_{m}$ and the proposal distributions for
$\mathbf{\theta}$ in the corresponding chain, [54]. The potential for parallel
tempering in geoscience problems has been demonstrated [55] with examples that
better convergence than conventional approaches for complex multi-modal
optimization problems [50, 51, 14]. A number of factors need to be considered
in multi-core implementation which specifically takes into account operating
system concepts such as interprocess communications when considering exchange
of solutions between the chains [56]. In order to address this, a
decentralized implementation of parallel tempering was presented that
eliminates global synchronization and reduces the overhead caused by
interprocess communication in exchange of solutions between the chains that
run in parallel cores [57]. Parallel tempering has also been implemented in a
distributed volunteer computing network where computers belonging to the
general public are used with help of multi-threading and graphic processing
units (GPUs) [58]. Furthermore, parallel tempering has been implemented with
Field Programmable Gate Arrays (FPGAs) that has massive parallelizing
capabilities which showed much better performance than multi-core and GPU
implementations [59]. Furthermore, in terms of applications, other studies
have efficiently implemented parallel tempering via multi-core architectures
for exploration of Earth’s resources [60].
## 3 Methodology: Multi-Core Parallel Tempering Bayeslands
The task of sampling or running geophysical models is computationally
intensive, it is worthwhile to employ parallel computing approach to parallel
tempering since thousands of samples are required for efficiently sampling a
posterior distribution. Massive parallelizing in multi-core architectures that
implements the various chains of the parallel tempering sampling method can
help in speeding up the approach while exploring multi-modal posterior
distributions. Furthermore, Parallel tempering being a method for implementing
Bayesian inference naturally accounts for uncertainty quantification.
Therefore, we present a multi-core parallel tempering approach for addressing
the limitations of Bayeslands. The overall problem remains the same as
Bayeslands, where the task is in sampling from the posterior distribution of
selected parameters such as rainfall and erodibility that contribute to the
topography development over time with Badlands. In doing so, it is essential
to define the likelihood function that will be used to determine where the
proposals are accepted.
The remainder of the section defines the model that features the likelihood
and priors for parallel tempering Bayeslands. Furthermore, the generation of
systematic topographies for the selected problems and the details of
implementation of multi-core parallel tempering for Bayeslands is also given.
### 3.1 Model and Priors
The likelihood function essentially captures the quality to the proposals by
considering the difference between the simulated final topography and the
actual one. In the problems where sediment deposition occurs, the likelihood
function also considers the difference between the simulated and real
successive topographies at selected time intervals.
Let the initial topography be denoted by $\boldsymbol{D}_{0}$, with
$\boldsymbol{D}_{0}=(D_{0,s_{1}}\ldots,D_{0,s_{n}})$, where $s_{i}$
corresponds to site $s_{i}$, with co-ordinates latitude, $u_{i}$, and
longitude, $v_{i}$. Suppose that we are interested in the topography $t$ years
into the future, we will denote this by $\boldsymbol{D}_{t}$, with
$\boldsymbol{D}_{T}$ defined to be the current topography. Our model for the
process which generates the topography is
$D_{t,s_{i}}=f_{t,s_{i}}(\boldsymbol{\theta})+\epsilon_{t,s_{i}}\;\mbox{with}\;\epsilon_{t,s_{i}}\sim(0,\tau^{2})$
(1)
for $t=0,1,\ldots,T$ and $i=1,\ldots,n$, where $\boldsymbol{\theta}$ are the
parameters of the Badlands model and $f_{t,s_{i}}(\boldsymbol{\theta})$ is the
output of the Badlands forward model. This model states that the topography is
function of the Badlands forward model given parameters $\boldsymbol{\theta}$,
plus some Gaussian noise with zero mean and variance $\tau^{2}$. The
likelihood function $L_{l}(\boldsymbol{\theta})$,is given by
$L_{l}(\boldsymbol{\theta})=\frac{1}{(2\pi\tau^{2})^{n/2}}\exp\left\\{-\frac{1}{2}\frac{\sum_{t=1}^{T}\sum_{i=1}^{n}(D_{t,s_{i}}-f_{t,s_{i}}(\boldsymbol{\theta}))^{2}}{\tau^{2}}\right\\}$
where the subscript $l$, in $L_{l}(\theta)$, denotes that it is the landscape
likelihood.
Although it may be possible to observe previous topologies, it is not likely
and therefore we assume that only the current topology $\mathbf{D}_{T}$ is
available. However it may be possible to observe other landscape features of
the past, such as sediment deposits, and we wish to incorporate the
information contained in this data to constrain landscape evolution. We allow
for this possibility by defining another random variable
$\mathbf{z}_{t}=(z_{t,s_{1}}\ldots,z_{t,s_{m}})$ which are the sediment
deposits at sites $s_{1},\ldots,s_{m}$. We assume that observed values of
$\mathbf{z}_{t}$ are a function of the Badlands forward model, with parameter
$\boldsymbol{\theta}$ and some Gaussian noise
$z_{t,s_{i}}=g_{t,s_{i}}(\boldsymbol{\theta})+\eta_{t,s_{i}}\;\mbox{with}\;\eta_{t,s_{i}}\sim(0,\sigma^{2})$
(2)
then the sediment likelihood, $L_{s}(\theta)$ is
$L_{s}(\boldsymbol{\theta})=\frac{1}{(2\pi\sigma^{2})^{mT/2}}\exp\left\\{-\frac{1}{2}\sum_{t=1}^{T}\sum_{j=1}^{m}\frac{(Z_{t,s_{j}}-g_{t,s_{j}}(\boldsymbol{\theta}))^{2}}{\sigma^{2}}\right\\}$
giving a total likelihood as a function of theta $L(\boldsymbol{\theta})$ to
be
$L(\boldsymbol{\theta})=L_{s}(\boldsymbol{\theta})\times
L_{l}(\boldsymbol{\theta}).$
The prior distribution, also known as prior, refers to one’s belief in the
distribution of the parameter without taking into account the evidence or data
[61]. The prior distribution is adjusted by sampling from the posterior with
given likelihood function that takes into account the data and the model. In
the case of the priors for Bayeslands, we focus on a subset of the parameters
of the Badlands model depending upon the problem, see Table tab:truevalues.
The complete set of unknowns in our model is $\boldsymbol{\theta}$, $\tau^{2}$
and $\sigma^{2}$. We follow the setup in [27] and place uniform priors on the
unknown parameters with limits given in Table 3.
### 3.2 Synthetic topography data
Topography | Evo.(years) | Length [km, pts] | Width [km, pts] | Res. factor | Run-time (s)
---|---|---|---|---|---
Crater [27] | 15 000 | [0.24, 123] | [0.24, 123] | 0.002 | 0.5
Crater-extended | 50 000 | [0.24, 123] | [0.24, 123] | 0.002 | 2
CM [27] | 500 000 | [136.5, 91] | [123.0, 82] | 1.5 | 0.5
CM-extended | 1 000 000 | [136.0, 136] | [123.0, 123] | 1 | 2.5
Mountain | 1 000 000 | [80,202] | [40,102] | 1 | 10
Table 1: Landscape evolution problems presented in this paper. The run-time represents the approximate length of time for one model to run. Topography | Rainfall (m/a) | Erod. | n-value | m-value | c-marine | c-surface | Uplift (mm/a)
---|---|---|---|---|---|---|---
Crater [27] | 1.5 | 5.0-e05 | 1.0 | 0.5 | - | - | -
Crater-extended | 1.5 | 5.0-e05 | 1.0 | 0.5 | - | - | -
CM [27] | 1.5 | 5.0-e06 | 1.0 | 0.5 | 0.5 | 0.8 | -
CM-extended | 1.5 | 5.0-e06 | 1.0 | 0.5 | 0.5 | 0.8 | -
Mountain | 1.5 | 5.0-e06 | 1.0 | 0.5 | - | - | 1.0
Table 2: True values of parameters Topography | Rainfall (m/a) | Erod. | n-value | m-value | c-marine | c-surface | uplift
---|---|---|---|---|---|---|---
Crater [27] | [0,3.0 ] | [3.0-e05, 7.0-e05] | - | - | - | - |
Crater-ext | [0,3.0 ] | [3.0-e05, 7.0-e05] | [0, 2.0] | [0, 2.0] | - | - |
CM [27] | [0,3.0 ] | [3.0-e06, 7.0-e06] | - | - | - | - | -
CM-ext. | [0,3.0 ] | [3.0-e06, 7.0-e06] | [0, 2.0] | [0, 2.0] | [0.3, 0.7] | [0.6, 1.0] | -
Mountain | [0,3.0 ] | [3.0-e06, 7.0-e06] | [0, 2.0] | [0, 2.0] | - | - | [0.1, 1.7]
Table 3: Prior distribution range of model parameters
Badlands is a forward stratigraphic model that requires an initial topography
and values for $\boldsymbol{\theta}$ to simulate an expected final (ground-
truth) topography at time $T$, given by $\mathbf{f}_{T}(\boldsymbol{\theta})$
in equation 1, and expected sediment deposits at several points in time, given
by $g_{t,s_{i}}(\boldsymbol{\theta})$ using Equation 2.
We consider five landscape problems from which two are taken from Bayeslands
which consist of a crater and continental margin (CM) example, where only two
parameters (rainfall and erodibility) are sampled. In this work, we present
two variants of these problems, such as crater-extended and CM-extended. The
extended problems consider additional parameters and simulation time for
topography development. Furthermore, we also use a mountain building case that
features uplift rates as a parameter which is not present in the other
problems. The initial topographies for the crater-extended and CM-extended
cases appear in Figure 2 and Figure 3, respectively. The final topography
after 50,000 years and 1,000,000 years along with erosion-deposition is also
given. The Crater and CM cases have an initial and ground-truth topography
identical to [27]. Figure 4 shows the initial and ground-truth final
topography of the Mountain landscape problem after 1,000,000 years. Note that
there is no erosion-deposition catered for this case in the likelihood
function to demonstrate that this data is unavailable. Figure 9 shows the
evolution of the PT-Bayeslands model. Tables2 and 1 shows values of parameters
used to created the synthetic ground-truth topography. The parameters for the
two crater problems are, rainfall (m/a), erodibility, m-value, and n-value.
The CM-extended problem features six free parameters, which include marine
diffusion coefficient (c-marine given by $m^{2}$/a) and surface diffusion
coefficient (c-surface given by $m^{2}$/a) as shown in Table 2. The mountain
problem features 5 free parameters, rainfall (m/a), erodibility, m-value, and
n-value, and uplift (mm/a). The priors were drawn from a uniform distribution
with lower and upper limit given in Table 3.
---
(a) Crater-extended initial topography
(b) Crater-extended synthetic ground-truth topography
Figure 2: Crater-extended: Initial and eroded ground-truth topography and
sediment deposition after 50 000 years.
---
(a) CM-extended initial topography
(b) CM-extended synthetic ground-truth topography
Figure 3: Continental Margin (CM)-extended: Initial and eroded ground-truth
topography and sediment after 1 000 000 years.
---
(a) Mountain initial topography
(b) Mountain synthetic ground truth topography
Figure 4: Mountain: Initial and eroded ground-truth topography after 1 000 000
years evolution.
### 3.3 Multi-core parallel tempering Bayeslands
Parallel tempering can be regarded as an ensemble of replicas of a MCMC
sampler. Given $N$ replicas in a ensemble defined by multiple temperature
levels, the state of the ensemble is specified by
$X={\theta_{1},\theta_{2},...,\theta_{N}}$, where $\theta_{i}$ is the replica
at temperature level $T_{i}$. A Markov chain is constructed to sample
$\theta_{i}$ at each temperature level $T_{i}$ which is used to rescale the
likelihood value for the replica in order to give variates in exploration and
exploitations during sampling. In other words, . In principle, at every
iteration, the Markov chains can feature two types of transitions that include
1) the Metropolis transition and 2) a replica transition. In the Metropolis
transition, each replica is sampled independently to perform local Monte Carlo
moves defined by the temperature which is implemented by a change in the
likelihood function for each temperature level $T_{i}$.
The replica transition considers the exchange of two neighboring replicas that
are defined by two neighboring temperature levels, $i$ and $i+1$,
$\theta_{i}\leftrightarrow\theta_{i+1}$. The exchange is accepted by the
Metropolis-Hastings criterion with probability given in Algorithm 2. The
exchange enables a replica with a low temperature level to have a chance to
reach a high temperature level which gives enhanced exploration features that
can enable to escape a local minimum. In this way, the replica exchange can
shorten the sampling time required for convergence. The temperature level is
user-defined, which needs to be determined from trial experiments, and is
highly dependent on the nature of the problem in terms of likelihood surface
and multi-modality.
Figure 5: An overview of the different replicas that are executed on a multi-
processing architecture. Note that the main process controls the given
replicas and enables them to exchange the neighboring replicass given the swap
time and probability of exchange is satisfied.
Figure 5 gives an overview of the different replicas that are executed on a
multi-processing architecture where each replica runs on a separate core with
inter-process communication for exchanging neighboring replica. The Badlands
model is executed in the same processing core where the replicas local
sampling is executed. The main process is run on a separate core which
controls the replica and enables them to exchange the neighboring replicas
given the swap time and probability of exchange is satisfied. In order to
minimize interprocess communication between processes for reducing
computational costs, we only consider replica transition at fixed intervals
that are determined by user defined number of samples used for Metropolis
samples. Each replica is allocated a fixed sampling time. The main process
waits for all samplers to complete their required sampling after which the
samplers attempt configuration exchange. The main process waits for each
replica until the swap-interval sampling limit is fulfilled. Once the replica
reaches this junction, the main process proposes configuration swaps between
adjacent replications depending on the replica exchange probability. The main
process notifies the replicas post swapping to resume sampling with latest
configurations in the chain for each replica. The process continues with the
sampling and proposing swaps until the the maximum sampling time is reached.
In the multi-core implementation, we need to consider multi-processing
software development packages so that the resources have efficient inter-
process communication [56] considering the exchange of the solutions between
neighboring replica’s.
## 4 Experiments and Results
In this section, we present the experimental design and results for five
landscape evolution problems for evaluating the performance of PT-Bayeslands.
We use the landscape problems from Table 1, that involve synthetic and real-
world topographies. We implemented multi-core parallel tempering using the
Python multiprocessing library [62] and open-source software package
111Parallel tempering Bayeslands: https://github.com/badlands-
model/paralleltemp_Bayeslands .
### 4.1 Design of Experiments
PT-Bayeslands employs a random-walk sampler for each replica chain which is
able to run on separate cores (central processing units). The random-walk was
implemented by perturbing the chain in the respective replica with a small
amount of Gaussian noise with a parameter specific step-size or standard
deviation. The step-size $\beta_{i}$ for parameter $i$ is chosen to be a
combination of a fixed step size $\phi=0.02$, common to all parameters,
multiplied by the range of possible values for parameter $i$ so that
$\beta_{i}=(a_{i}-b_{i})*\phi$ (3)
where, $a_{i}$ and $b_{i}$ represent the maximum and minimum limits of the
priors for parameter and are given in Table 2. Note that PT-Bayeslands samples
the parameters with prior distribution highlighted for the respective problems
in Table 3. The step-size ratio $\phi$ was determined by trial experimental
runs that was used by all the problems. Similarly, the temperature change for
each of the replica’s $r$ was $\delta_{r}=2.5$. In trial experiments, the
selection of these parameters depended on the prediction accuracy. We used a
fix swap ratio, $\eta=0.1$, that determines when to swap with the neighboring
replica which is incorporated into the algorithm 2 and further illustrated in
Figure 5. Therefore, the number of samples $s$ required to check when to swap
for a given replica $r$ is given as follows.
$s_{r}=\eta*(\Phi/\Omega)$ (4)
where $\Phi$ is the total number of samples and $\Omega$ is the number of
replica’s for the problem. The experiments are evaluated in terms of total
simulation time, acceptance rate of proposals, and root-mean-squared-error
(RMSE) of elevation and sediment (Pred. RMSE), which have been defined in the
methodology section.
In parallel tempering, the number of replicas needs to be tailored for the
problem. They also determine the total simulation time for the problem. It is
generally expected that increasing the number of replica’s in a multi-core
architecture will shorten the computation time; however, additional time is
taken with interprocess communication through swapping of neighboring replicas
which is also dependent on the swap interval and number of replica’s that can
contribute to the wait-time by the main process. As as shown in Figure 5, the
main process runs on a separate core. Therefore given $N$ number of replicas,
the total number of processes running will be $N+1$. The task of the main
process is to manage the ensemble of replicas. It will hold and resume the
replica sampling process during the swap period. Moreover, we need to ensure
that the quality of the topography and sediment predictions by Badlands is
retained when number of cores are increased. Hence, we need to investigate the
effect on performance accuracy and time of increasing number of replicas for
PT-Bayeslands. The experiments are designed as follows.
* 1.
Step 1: Investigate the effects on computational time and accuracy when
increasing the number of replicas;
* 2.
Step 2: Evaluate the number of samples required for convergence defined in
terms of prediction accuracy;
* 3.
Step 3: Using knowledge from above experiments, apply PT-Bayeslands to all the
given problems and report the posterior distributions, computational time,
uncertainty and accuracy in topography and sediment predictions.
We present the crater problem for Step 1 and the CM problem for Step 2. Since
the CM problem has demonstrated an irregular multi-modal surface, as shown in
Figure 1 [27], it is important to evaluate the effect of sampling time on the
posterior distribution and the accuracy of prediction.
### 4.2 Results
We evaluate the effect of the number of replicas, executed in different cores,
has on the total time for the CM-extended topography. Table 4 presents a
summary of the results that employs 100,000 samples in PT-Bayeslands for the
CM-extended experiment. We observe that increasing the number of replicas
reduces the overall computation time. However the number of replicas does not
appear to effect the RMSE. Figure 6 shows the effect of the number of cores on
the time, panel (a) and on the RMSE, panel (b).
Figure 6: The effect of the number of cores on time, panel (a) and on the
RMSE, panel (b). Note that a 95% confidence interval for the slope of the
regression line in panel (b), is [-14.2,2.6], showing that the slope is not
significantly different from zero.
Next, we evaluate the effect of the prediction accuracy given an increase in
the number of samples. Table 5 presents summary of the results on the effect
of number of samples for the CM-extended topography with 24 replicas. We
observe that increasing the number of samples increases the overall time
taken, and reduces the acceptance rate of proposals. Moreover, the interval
between (10,000 - 100,000) samples does not have a major difference in the
prediction accuracy. Hence, even with 10,000 samples, PT-Bayeslands can
provide estimation and uncertainty quantification. Due to the stochastic
nature of the method and the difficulty of the likelihood surface shown in
Figure 1, we don’t see a clear trend in convergence when the number of cores
(replicas) or samples are increased. The performance depends on initial
conditions that is given by the initial points of the chains; depending on how
far they are from one of the modes in the likelihood surface, and the tendency
to be trapped in a local minimum.
In the next set of experiments, we used 24 cores with 100,000 samples for all
the problems. Table 6 shows a summary of results for the given problems. Note
that the total prediction accuracy is given by (Pred. RMSE) which consists of
sediment prediction accuracy (Sed. RMSE) and elevation prediction accuracy
(Elev. RMSE). Pred. RMSE considers the difference between the ground-truth and
final topography given by Badlands (after maximum number of evolution time in
years). However, the Sed. RMSE considers the successive predicted sediments
taken equally over 4 different timescales given in years. A major observation
here is the overall computation time for the five problems, which is due to
the time taken by Badlands to execute each proposal. The crater and crater-
extended cases are simpler problems (with shorter evolution time and fewer
parameters), hence more proposals are accepted when compared to the CM cases.
Moreover, the overall prediction performance (Pred. RMSE) has been
significantly improved by PT-Bayeslands. A cross-section of the crater with
the uncertainty of prediction is shown in Figure 7.
We select the results for the CM-extended problem from Table 6 and present
further details which include; 1.) successive predicted topographies and
erosion-deposition at four stages of evolution (Figure 8), 2.) cross-section
that gives uncertainty of topography prediction (Figure 10), and 3.) selected
posterior distributions, and trace-plots during sampling (Figure 11). Note
that erosion is represented as positive while deposition is represented as
negative (height in meters) (Figure 8). Posterior distributions, trace plots,
and other results for all parameters and all models are available in the on-
line supplementary material. In comparison with the true values given in Table
2 that was used to generate the synthetic ground-truth topography, we notice
that the true values were not recovered in some of the posterior distributions
(e. g. Figure 11). However, as shown in the likelihood surface (Figure 1), a
number of sub-optimal values exist. Due to multi-modality, although the true
values are not recovered for certain parameters, the sub-optimal modes give
accurate predictions. Note that the likelihood surface shown in Figure 1) is
for two parameters only (rainfall and erodibility).
Replica (cores) | Time (minutes) | Pred. RMSE | Accepted %
---|---|---|---
2 | 3540 | 78.8 | 0.1
4 | 1384 | 62.7 | 0.1
8 | 614 | 71.5 | 0.4
12 | 455 | 87.0 | 0.5
16 | 375 | 72.0 | 0.6
20 | 362 | 110.6 | 0.3
24 | 270 | 65.9 | 0.4
28 | 215 | 89.7 | 0.5
32 | 310 | 205.1 | 0.4
36 | 213 | 128.0 | 0.6
40 | 187 | 107.0 | 0.6
44 | 162 | 70.0 | 0.6
48 | 142 | 80.5 | 0.8
Table 4: Effect of number replicas/cores for the CM-extended topography with 100 000 samples. Num. Samples | Time (mins.) | Pred. RMSE | Accepted %
---|---|---|---
1000 | 3 | 120.6 | 14.0
2000 | 6 | 222.5 | 15.3
4000 | 10 | 200.8 | 9.4
6000 | 18 | 98.3 | 3.7
8000 | 18 | 213.1 | 8.3
10000 | 27 | 77.3 | 2.8
50000 | 131 | 77.7 | 1.3
100000 | 258 | 95.1 | 1.0
150000 | 436 | 69.3 | 0.2
Table 5: Effect of the number of samples for the CM-extended topography running with 24 replicas on 24 cpus. Topography | Time (minutes) | Sed. RMSE | Elev. RMSE | Pred. RMSE | Accepted %
---|---|---|---|---|---
Crater | 80 | 4.2 | 1.1 | 5.3 | 12.2
Crater-extended | 229 | 0.2 | 1.0 | 1.2 | 1.9
CM | 50 | 2.6 | 19.9 | 22.5 | 1.0
CM-extended | 257 | 50.2 | 49.0 | 99.2 | 0.6
Mountain | 375 | - | 617.0 | 617.0 | 0.63
Table 6: Typical results for respective problems (24 replicas/cpus and 100 000
samples). Note that ’-’ in case of Mountain Sed. RMSE indicates that it was
not part of the likelihood function Figure 7: Crater-extended: Cross-section
of the elevation along the x-axis comparing the ground-truth of the Badlands-
model with the PT-Bayelands predictions.
---
(a) CM-extended predicted topography after 50 % evolution time
(b) CM-extended erosion-deposition after 50 % evolution time
(c) CM-extended predicted topography after 100 % evolution time
(d) CM-extended erosion-deposition after 100 % evolution time
Figure 8: CM-extended: Topography and erosion-deposition development for
selected time frames. Note that erosion (positive) and deposition (negative)
values given by the height in meters
---
(a) Mountain predicted topography after 25 % evolution time
(b) Mountain predicted topography after 50 % evolution time
(c) Mountain predicted topography after 75 % evolution time
(d) Mountain predicted topography after 100 % evolution time
Figure 9: Mountain evolution over 1 000 000 years.
---
Figure 10: CM-extended: Cross-section of the elevation along the x-axis
comparing the ground-truth of the Badlands-model with the PT-Bayelands
predictions.
---
Figure 11: Typical posterior distribution (top panel) and replica traces
(bottom panel) exploring the marine diffusion parameter-space for the CM-
extended topography. In this model run 24 replicas are represented by the
coloured lines. Parameter, c-marine, units are $m^{2}a^{-1}$.
Comparison with literature (MCMC Bayeslands [27]) is given in Figure 12. The
trace-plot in these figures show that the PT-Bayeslands have advantages in
exploration with multiple replicas. The comparison of the computational time
is given in Table 7. These results show that PT-Bayeslands has a significant
improvement in computational time and accuracy in prediction performance.
---
(a) Rainfall distribution (MCMC- Bayeslands)
(b) Rainfall distribution (PT- Bayeslands)
Figure 12: Comparison of PT-Bayeslands with MCMC-Bayeslands for the CM problem. Note that the vertical dashed lines in different colours shows sub-optimal true values or modes that correspond to the likelihood surface in Figure 1. Topography | Samples | Time(mins) | Sed RMSE | Elev RMSE | Pred RMSE | Accepted%
---|---|---|---|---|---|---
MCMC Bayeslands results from [27]
Crater | 10,000 | 136 | 8.24 | 1.06 | 9.30 | 2.35
Crater | 100,000 | 1023 | 8.24 | 1.06 | 9.30 | 2.57
CM | 10,000 | 101 | 459.85 | 67.37 | 527.22 | 0.47
CM | 100,000 | 729 | 387.92 | 10.60 | 398.53 | 0.02
PT-Bayeslands using 24 replicas (this study)
Crater | 10,000 | 8 | 4.2 | 1.0 | 5.2 | 15.0
Crater | 100,000 | 78 | 4.2 | 1.1 | 5.3 | 12.2
CM | 10,000 | 7 | 7.4 | 25.3 | 32.7 | 4.9
CM | 100,000 | 50 | 2.6 | 19.9 | 22.5 | 1.0
Table 7: Comparison of results of PT-Bayeslands with single-threaded MCMC
Bayeslands [27] shows significant difference in performance given
computational time and accuracy in prediction.
---
Figure 13: Posterior distribution (top panel) and replica traces (bottom
panel) exploring the uplift parameter-space for the Mountain topography. The
24 replicas trace distribution are represented by the coloured lines.
Parameter, uplift, units are $mm\,a^{-1}$.
---
(a) Cross-section along the x-axis.
(b) Cross-section along the y-axis
Figure 14: Typical topography for the evolved Mountain problem, comparing PT-
Bayeslands predictions and the Badlands ground-truth. Result is for 24
replicas, 100,000 samples. Figure 15: Likelihood surface of the Mountain
topography for the rainfall and uplift parameters only.
In this paper, we only show details of the posterior distribution, successive
topography and successive erosion-deposition predictions for a sample of model
runs. Detailed results for all problems and model runs are provided in
supplementary material online 222Supplementary results:
https://github.com/badlands-
model/paralleltemp_Bayeslands/tree/master/supplementary_results.
## 5 Discussion
The results have shown that computation time for a given number of iterations
does not scale linearly with the number of replicas. There is a trade-off
between proposing to swap chains and the rate of convergence of the PT-
Bayeslands. Swapping proposals between neighboring replicas, in principle,
gives better mixing but introduces computational cost because each chain needs
to wait for all the replicas to complete in order to compute the neighbor swap
probability. This computational cost can be high if the number of replicas is
large. We note that the synthetic problems considered in this paper, only
required a few seconds of simulation time for Badlands, whereas, in real-world
problems, each simulation of Badlands could take several minutes to hours. In
such cases, the trade-off between the frequency of swapping and prediction
accuracy would need to be evaluated. In this study, we observed that
increasing the number of cores does not necessarily mean that the prediction
accuracy will get better. In particular it is advantageous to use a larger
number of cores for large scale problems where the Badlands model takes hours
to evaluate a single proposal. Moreover, the replica temperature is also an
important feature that can be evaluated in future works. Since evaluation of
such parameters is computationally expensive for large scale problems,
heuristics such as geometric temperature intervals can be used [52, 47].
Figures 12 show the improvement of parallel tempering over single chain MCMC.
Panel (a) in both figures represents output from the single MCMC-Bayeslands
while panel (b), represents the output from the PT-Bayeslands. These figures
show that PT-Bayelands mixes more and discovers more of the multiple nodes
than the single chain. The degree of improvement of using PT-Bayeslands over
single core based MCMC-Bayeslands depends upon the type of topography and the
parameter. For example, the estimated posteriors distributions for all
parameters in the Crater topography contained the true value of that
parameter, while in the CM-extended topography, only some of the posterior
distributions contained the true parameter. This is not surprising when one
considers the highly irregular shape of the likelihood surface, as given in
Figure 1[27]. Despite this variability in results the predictions were
accurate which could be attributable to a multi-modal posterior. Multi-modal
distributions are well known in Earth science problems in the literature (e.g.
[63] and [64]). Multi-modality implies that there are several optimal values
of the parameters where multiple combinations of the given parameters can
plausibly predict or simulate the topography that closely resembles the
synthetic or ground-truth topography [65].
Figure 13 shows the posterior distribution of the uplift parameter for the
Mountain topography. In this case the the chains found the true value early
and did not stray far in their exploration, likely because the uplift
parameter is not sensitive to the solution at the scales explored here (0.1 –
1.7 mm/a). The likelihood surface shown in Figure 15 suggests this, as
rainfall increases and uplift decreases (with other parameters set to their
true values) the likelihood remains high. Adjusting the temperature of the
chains and swap ratios can help influence the exploration of the chains to fit
the problem more appropriately. Yet here Figure 14 shows that the ground-truth
and the PT-Bayeslands predictions fit well, indicating that the explored space
collectively for all parameters taken together was sufficient to match the
data.
Furthermore, the experiments considered a fixed value for some parameters,
such as rainfall, that can have varying effects given different regions and
geological timescales. It is well known that climate change may affect
rainfall with varying implications to the environment, and in this case the
topography in terms of elevation via affecting erosion and sedimentation.
Therefore, it would be reasonable to implement region- and time-dependent
constraints for some parameters to fully take into account the effects of
climate changes.
## 6 Conclusions and Future Work
We presented a computationally efficient implementation of multi-core parallel
tempering for improving the computational time require to estimate parameters
of the Bayeslands model. The proposed methodology provides a general
systematic approach for uncertainty quantification of free parameters in basin
and landscape dynamics models (Badlands). The results showed that the method
not only reduces the computation time, but also provides a means to explore
the parameter space in highly irregular multi-modal landscapes. This has been
demonstrated by the results that show better posterior distributions of the
parameters along with improvement in prediction accuracy of topography and
sediment deposition.
Future work will extend the method for a larger number of parameters that
includes spatio-temporal variations in rainfall and uplift for different
timescales and regions. Moreover, for large-scale or continental-scale
problems, it would be reasonable to implement further enhancements to the
method for lowering the overall computational time. This could be done through
surrogate-assisted models where at times the surrogate of Badlands implemented
via machine learning evaluates the proposals. There is scope for real-world
applications, that feature landscape evolution for geological timescales.
Further, efficient gradient free proposals need to be constructed as the
number of parameters and the complexity of the model increases.
## Acknowledgements
We would like to thank Danial Azam from School of Geoscience, the University
of Sydney for providing technical assistance. We would also like to thank the
Strategic Research Excellence Initiative (SREI) Grant from the University of
Sydney. Furthermore, we would like to acknowledge the Artemis high performance
computing infrastructure provided by Sydney Informatics Hub of the University
of Sydney.
## References
## References
* [1] N. Flament, M. Gurnis, and R. D. Müller, “A review of observations and models of dynamic topography,” _Lithosphere_ , vol. 5, no. 2, pp. 189–210, 2013.
* [2] T. Coulthard, “Landscape evolution models: a software review,” _Hydrological processes_ , vol. 15, no. 1, pp. 165–173, 2001.
* [3] T. Salles and L. Hardiman, “Badlands: An open-source, flexible and parallel framework to study landscape dynamics,” _Computers & Geosciences_, vol. 91, pp. 77–89, 2016.
* [4] T. Salles, N. Flament, and D. Müller, “Influence of mantle flow on the drainage of eastern australia since the jurassic period,” _Geochemistry, Geophysics, Geosystems_ , vol. 18, no. 1, pp. 280–305, 2017\.
* [5] L. R. Scott and S. Zhang, “Finite element interpolation of nonsmooth functions satisfying boundary conditions,” _Mathematics of Computation_ , vol. 54, no. 190, pp. 483–493, 1990.
* [6] V. Godard, J. Lavé, and R. Cattin, “Numerical modelling of erosion processes in the himalayas of nepal: Effects of spatial variations of rock strength and precipitation,” _Geological Society, London, Special Publications_ , vol. 253, no. 1, pp. 341–358, 2006.
* [7] J. Rejman, R. Turski, and J. Paluszek, “Spatial and temporal variations in erodibility of loess soil,” _Soil and Tillage Research_ , vol. 46, no. 1-2, pp. 61–68, 1998.
* [8] G. Shafer, “Belief functions and parametric models,” _Journal of the Royal Statistical Society. Series B (Methodological)_ , pp. 322–352, 1982.
* [9] S. E. Fienberg _et al._ , “When did bayesian inference become” bayesian”?” _Bayesian analysis_ , vol. 1, no. 1, pp. 1–40, 2006.
* [10] N. L. Hjort, C. Holmes, P. Müller, and S. G. Walker, _Bayesian nonparametrics_. Cambridge University Press, 2010, vol. 28.
* [11] C. Robert and G. Casella, “A short history of markov chain monte carlo: Subjective recollections from incomplete data,” _Statistical Science_ , pp. 102–115, 2011.
* [12] K. Mosegaard and P. D. Vestergaard, “A simulated annealing approach to seismic model optimization with sparse prior information,” _Geophysical Prospecting_ , vol. 39, no. 5, pp. 599–611, 1991.
* [13] P. Rocca, M. Benedetti, M. Donelli, D. Franceschini, and A. Massa, “Evolutionary optimization as applied to inverse scattering problems,” _Inverse Problems_ , vol. 25, no. 12, p. 123003, 2009.
* [14] M. K. Sen and P. L. Stoffa, _Global optimization methods in geophysical inversion_. Cambridge University Press, 2013.
* [15] K. Gallagher, K. Charvin, S. Nielsen, M. Sambridge, and J. Stephenson, “Markov chain monte carlo (mcmc) sampling methods to determine optimal models, model resolution and model choice for earth science problems,” _Marine and Petroleum Geology_ , vol. 26, no. 4, pp. 525–535, 2009.
* [16] W. K. Hastings, “Monte carlo sampling methods using markov chains and their applications,” _Biometrika_ , vol. 57, no. 1, pp. 97–109, 1970.
* [17] N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” _The journal of chemical physics_ , vol. 21, no. 6, pp. 1087–1092, 1953.
* [18] A. E. Raftery and S. M. Lewis, “Implementing mcmc,” _Markov chain Monte Carlo in practice_ , pp. 115–130, 1996.
* [19] D. van Ravenzwaaij, P. Cassey, and S. D. Brown, “A simple introduction to markov chain monte–carlo sampling,” _Psychonomic bulletin & review_, pp. 1–12, 2016.
* [20] A. Malinverno, “Parsimonious bayesian markov chain monte carlo inversion in a nonlinear geophysical problem,” _Geophysical Journal International_ , vol. 151, no. 3, pp. 675–688, 2002.
* [21] K. Mosegaard and A. Tarantola, “Monte carlo sampling of solutions to inverse problems,” _Journal of Geophysical Research: Solid Earth_ , vol. 100, no. B7, pp. 12 431–12 447, 1995.
* [22] M. Sambridge and K. Mosegaard, “Monte carlo methods in geophysical inverse problems,” _Reviews of Geophysics_ , vol. 40, no. 3, 2002.
* [23] M. Sambridge, “Geophysical inversion with a neighbourhood algorithm—ii. appraising the ensemble,” _Geophysical Journal International_ , vol. 138, no. 3, pp. 727–746, 1999.
* [24] R. M. Neal _et al._ , “Mcmc using hamiltonian dynamics,” _Handbook of Markov Chain Monte Carlo_ , vol. 2, no. 11, 2011.
* [25] M. D. Hoffman and A. Gelman, “The no-u-turn sampler: adaptively setting path lengths in hamiltonian monte carlo.” _Journal of Machine Learning Research_ , vol. 15, no. 1, pp. 1593–1623, 2014.
* [26] M. Girolami and B. Calderhead, “Riemann manifold langevin and hamiltonian monte carlo methods,” _Journal of the Royal Statistical Society: Series B (Statistical Methodology)_ , vol. 73, no. 2, pp. 123–214, 2011.
* [27] R. Chandra, D. Azam, D. Muller, T. Salles, and S. Cripps, “Bayeslands: A bayesian inference approach for parameter tuning for modelling basin and landscape dynamics via badlands,” _Computers and Geoscience)_ , pp. In–Review, 2018. [Online]. Available: https://github.com/rohitash-chandra/research/blob/master/2018/BayesLands.pdf
* [28] E. Marinari and G. Parisi, “Simulated tempering: a new monte carlo scheme,” _EPL (Europhysics Letters)_ , vol. 19, no. 6, p. 451, 1992.
* [29] C. J. Geyer and E. A. Thompson, “Annealing markov chain monte carlo with applications to ancestral inference,” _Journal of the American Statistical Association_ , vol. 90, no. 431, pp. 909–920, 1995.
* [30] H. Zhang, M. Liu, Y. Shi, D. A. Yuen, Z. Yan, and G. Liang, “Toward an automated parallel computing environment for geosciences,” _Physics of the Earth and Planetary Interiors_ , vol. 163, no. 1-4, pp. 2–22, 2007.
* [31] J. A. Vrugt, B. O. Nualláin, B. A. Robinson, W. Bouten, S. C. Dekker, and P. M. Sloot, “Application of parallel computing to stochastic parameter estimation in environmental models,” _Computers & Geosciences_, vol. 32, no. 8, pp. 1139–1155, 2006.
* [32] K. Mills, G. Fox, and R. Heimbach, “Implementing an intervisibility analysis model on a parallel computing system,” _Computers & Geosciences_, vol. 18, no. 8, pp. 1047–1054, 1992.
* [33] K. X. Whipple and G. E. Tucker, “Implications of sediment-flux-dependent river incision models for landscape evolution,” _Journal of Geophysical Research: Solid Earth_ , vol. 107, no. B2, pp. 1–20, 2002.
* [34] G. E. Tucker and G. R. Hancock, “Modelling landscape evolution,” _Earth Surface Processes and Landforms_ , vol. 35, no. 1, pp. 28–50, 2010.
* [35] T. Salles and G. Duclaux, “Combined hillslope diffusion and sediment transport simulation applied to landscape dynamics modelling.” _Earth Surf. Process Landf._ , vol. 40, no. 6, p. 823–39, 2015.
* [36] B. Campforts, W. Schwanghart, and G. Govers, “Accurate simulation of transient landscape evolution by eliminating numerical diffusion: the ttlem 1.0 model,” _Earth Surface Dynamics_ , vol. 5, no. 1, pp. 47–66, 2017.
* [37] J. M. Adams, N. M. Gasparini, D. E. J. Hobley, G. E. Tucker, E. W. H. Hutton, S. S. Nudurupati, and E. Istanbulluoglu, “The landlab v1.0 overlandflow component: a python tool for computing shallow-water flow across watersheds,” _Geoscientific Model Development_ , vol. 10, no. 4, pp. 1645–1663, 2017.
* [38] A. D. Howard, W. E. Dietrich, and M. A. Seidl, “Modeling fluvial erosion on regional to continental scales,” _Journal of Geophysical Research: Solid Earth_ , vol. 99, no. B7, pp. 13 971–13 986, 1994.
* [39] D. E. J. Hobley, H. D. Sinclair, S. M. Mudd, and P. A. Cowie, “Field calibration of sediment flux dependent river incision,” _Journal of Geophysical Research: Earth Surface_ , vol. 116, no. F4, 2011.
* [40] T. Salles, “Badlands: A parallel basin and landscape dynamics model,” _SoftwareX_ , vol. 5, pp. 195–202, 2016.
* [41] T. Salles, X. Ding, and G. Brocard, “pybadlands: A framework to simulate sediment transport, landscape dynamics and basin stratigraphic evolution through space and time,” _PLOS ONE_ , vol. 13, pp. 1–24, 04 2018.
* [42] T. Salles, X. Ding, J. M. Webster, A. Vila-Concejo, G. Brocard, and J. Pall, “A unified framework for modelling sediment fate from source to sink and its interactions with reef systems over geological times,” _Scientific Reports_ , vol. 8, no. 1, p. 5252, 2018.
* [43] S. Chib and E. Greenberg, “Understanding the metropolis-hastings algorithm,” _The American Statistician_ , vol. 49, no. 4, pp. 327–335, 1995.
* [44] R. M. Neal, “Sampling from multimodal distributions using tempered transitions,” _Statistics and computing_ , vol. 6, no. 4, pp. 353–366, 1996\.
* [45] P. Baldi, “Gradient descent learning algorithm overview: A general dynamical systems perspective,” _IEEE Transactions on neural networks_ , vol. 6, no. 1, pp. 182–195, 1995.
* [46] H. Strathmann, D. Sejdinovic, S. Livingstone, Z. Szabo, and A. Gretton, “Gradient-free hamiltonian monte carlo with efficient kernel exponential families,” in _Advances in Neural Information Processing Systems_ , 2015, pp. 955–963.
* [47] A. Patriksson and D. van der Spoel, “A temperature predictor for parallel tempering simulations,” _Physical Chemistry Chemical Physics_ , vol. 10, no. 15, pp. 2073–2077, 2008.
* [48] K. Hukushima and K. Nemoto, “Exchange monte carlo method and application to spin glass simulations,” _Journal of the Physical Society of Japan_ , vol. 65, no. 6, pp. 1604–1608, 1996.
* [49] U. H. Hansmann, “Parallel tempering algorithm for conformational studies of biological molecules,” _Chemical Physics Letters_ , vol. 281, no. 1-3, pp. 140–150, 1997.
* [50] M. K. Sen and P. L. Stoffa, “Bayesian inference, gibbs’ sampler and uncertainty estimation in geophysical inversion,” _Geophysical Prospecting_ , vol. 44, no. 2, pp. 313–350, 1996.
* [51] M. Maraschini and S. Foti, “A monte carlo multimodal inversion of surface waves,” _Geophysical Journal International_ , vol. 182, no. 3, pp. 1557–1566, 2010.
* [52] A. Kone and D. A. Kofke, “Selection of temperature intervals for parallel-tempering simulations,” _The Journal of chemical physics_ , vol. 122, no. 20, p. 206101, 2005.
* [53] F. Calvo, “All-exchanges parallel tempering,” _The Journal of chemical physics_ , vol. 123, no. 12, p. 124106, 2005.
* [54] B. Miasojedow, E. Moulines, and M. Vihola, “An adaptive parallel tempering algorithm,” _Journal of Computational and Graphical Statistics_ , vol. 22, no. 3, pp. 649–664, 2013.
* [55] M. Sambridge, “A parallel tempering algorithm for probabilistic sampling and multimodal optimization,” _Geophysical Journal International_ , vol. 196, no. 1, pp. 357–374, 2013.
* [56] L. Lamport, “On interprocess communication,” _Distributed computing_ , vol. 1, no. 2, pp. 86–101, 1986.
* [57] Y. Li, M. Mascagni, and A. Gorin, “A decentralized parallel implementation for parallel tempering algorithm,” _Parallel Computing_ , vol. 35, no. 5, pp. 269 – 283, 2009.
* [58] K. Karimi, N. Dickson, and F. Hamze, “High-performance physics simulations using multi-core cpus and gpgpus in a volunteer computing context,” _The International Journal of High Performance Computing Applications_ , vol. 25, no. 1, pp. 61–69, 2011.
* [59] G. Mingas, L. Bottolo, and C.-S. Bouganis, “Particle mcmc algorithms and architectures for accelerating inference in state-space models,” _International Journal of Approximate Reasoning_ , vol. 83, pp. 413–433, 2017\.
* [60] A. Reid, E. V. Bonilla, L. McCalman, T. Rawling, and F. Ramos, “Bayesian joint inversions for the exploration of Earth resources,” in _IJCAI_ , 2013, pp. 2877–2884.
* [61] H. Jeffreys, “An invariant form for the prior probability in estimation problems,” _Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences_ , pp. 453–461, 1946.
* [62] N. Singh, L.-M. Browne, and R. Butler, “Parallel astronomical data processing with python: Recipes for multicore machines,” _Astronomy and Computing_ , vol. 2, pp. 1–10, 2013.
* [63] M. Dalla Mura, S. Prasad, F. Pacifici, P. Gamba, J. Chanussot, and J. A. Benediktsson, “Challenges and opportunities of multimodality and data fusion in remote sensing,” _Proceedings of the IEEE_ , vol. 103, no. 9, pp. 1585–1601, 2015.
* [64] A. Beskos, M. Girolami, S. Lan, P. E. Farrell, and A. M. Stuart, “Geometric mcmc for infinite-dimensional inverse problems,” _Journal of Computational Physics_ , vol. 335, pp. 327–351, 2017.
* [65] T. A. Cross and M. A. Lessenger, “Construction and application of a stratigraphic inverse model,” in _Numerical Experiments in Stratigraphy: Recent Advances in Stratigraphic and Sedimentologic Computer Simulations_ , J. Harbaugh, W. Watney, E. Rankey, R. Slingerland, R. Goldstein, and E. Franseen, Eds. Special publications of Society for Sedimentary Geology, 1999, ch. 62, pp. 69–83.
|
capbtabboxtable[][] [table]capposition=top
# Rule Covering for Interpretation and Boosting
Ş. İlker Birbil
Econometric Institute
Erasmus University Rotterdam
3000 DR Rotterdam, The Netherlands
<EMAIL_ADDRESS>
Mert Edali
Department of Medicine
University of Chicago
Chicago, IL 60637, USA
<EMAIL_ADDRESS>
Birol Yüceoğlu
Migros T.A.Ş
Ataşehir, 34758, İstanbul, Turkey
<EMAIL_ADDRESS>
Corresponding author.
###### Abstract
We propose two algorithms for interpretation and boosting of tree-based
ensemble methods. Both algorithms make use of mathematical programming models
that are constructed with a set of rules extracted from an ensemble of
decision trees. The objective is to obtain the minimum total impurity with the
least number of rules that cover all the samples. The first algorithm uses the
collection of decision trees obtained from a trained random forest model. Our
numerical results show that the proposed rule covering approach selects only a
few rules that could be used for interpreting the random forest model.
Moreover, the resulting set of rules closely matches the accuracy level of the
random forest model. Inspired by the column generation algorithm in linear
programming, our second algorithm uses a rule generation scheme for boosting
decision trees. We use the dual optimal solutions of the linear programming
models as sample weights to obtain only those rules that would improve the
accuracy. With a computational study, we observe that our second algorithm
performs competitively with the other well-known boosting methods. Our
implementations also demonstrate that both algorithms can be trivially coupled
with the existing random forest and decision tree packages.
## 1 Introduction
Although their prediction performances are remarkable, tree-based ensemble
methods are difficult to interpret due to large number of trees in the trained
model. Among these methods, Random Forest algorithm is probably the most
frequently used alternative for solving classification and regression problems
arising in different domains [2]. This performance versus interpretability
trade-off leads to the following two main questions in our work: Given a
trained random forest, can we obtain an interpretable representation with
mathematical programming that shows a performance close-enough to the overall
forest? Using a similar mathematical programming model, can we come up with a
learning algorithm, which generates only those rules that improve the
performance of a base tree?
To answer these questions, we focus on the rules constituting the trees in the
forest and propose two algorithms based on mathematical programming models.
The model in the first algorithm extracts the rules corresponding to the
leaves of the forest that result in the minimum total impurity while covering
all the samples. With this algorithm, we are able to mimick the performance of
the original forest with a significantly few number of rules (most of the time
even less than the number of rules in a regular decision tree). Being
encouraged with these results, we then propose our second algorithm based on
selective generation of rules. This algorithm improves the total impurity
gradually by solving a series of linear programming problems. This approach is
a variant of the column generation algorithm in mathematical optimization. In
our scheme, we use the dual optimal solutions of the linear programs as the
sample weights. We also observe that this scheme has close ties to well-known
ideas in boosting methods [13, 14, 17]. Thus, rule extraction and boosting are
the two keywords that help us to review the related literature.
Friedman and Popescu [9] propose RuleFit method to extract rules from
ensembles. The authors use decision trees as base learners, where each node of
the decision tree corresponds to a rule. They present an ensemble generation
algorithm to obtain a rule set. Then, they solve a linear regression problem
that minimizes the loss with a lasso penalty for selecting rules. Meinshausen
[15] presents a similar approach, called Node Harvest, which relies on rules
generated by a tree ensemble. Node Harvest minimizes the quadratic loss
function by partitioning the samples and relaxing the integrality constraints
on the weights. Even though the algorithm does not constrain the number of
rules, the author observes that only a small number of rules are used.
Mashayekhi and Gras [11] propose coverage of the samples in the training set
by the rules generated by Random Forest algorithm. They create a score for
each rule by considering the percentage of correctly and incorrectly
classified samples as well as the rule length. By using a hill climbing
algorithm, they are able to obtain slightly lower accuracy with much fewer
rules than random forest. Extending this work, Mashayekhi and Gras [12]
propose three different algorithms to extract if-then rules from decision tree
ensembles. The authors then compare their algorithms with both RuleFit and
Node Harvest. All these approaches use accuracy values to construct the
objective functions. Interpretability in terms of reducing the number of rules
is achieved by using lasso penalties. Our first algorithm, on the other hand,
reduces the number of rules directly through the objective function.
Gradient boosting algorithms use a sequence of weak learners and build a model
iteratively [7, 13, 8]. At each iteration the boosting algorithm focuses on
the misclassified samples by adjusting their weights with respect to the
errors of the previous iterations. A boosting method that uses linear
programming is called LPBoost [5]. This algorithm uses weak learners to
maximize the margins separating the different classes. Like us, the authors
also use column generation to create the weak learners iteratively. The final
ensemble then becomes a weighted combination of the weak learners. The
approach can accept decision stumps or decision trees as weak learners.
However, decision trees can perform badly in terms of convergence and solution
accuracy, if the dual variables associated with margin classification are not
constrained. Unlike LPBoost, we use the duals as sample weights in our column
generation routine. Another boosting method based on mathematical programming
is called IPBoost [16]. This approach uses binary decision variables
corresponding to misclassification by a predetermined margin. Due to binary
decision variables, the authors resort to an integer programming solution
technique called branch-and-bound-and-price. Dash et al. [4] also use column
generation to learn Boolean decision rules in either disjunctive normal form
or conjunctive normal form. They solve a sequence of integer programming
problems for pricing the columns. They choose to restrict the number of
selected clauses as a user-defined hyperparameter for improving
interpretability. Bertsimas and Dunn [1] propose an integer programming
formulation to create optimal decision trees for multi-class classification.
They present two approaches with respect to node splitting. The first approach
is more general but less interpretable. The second more interpretable model
considers the values of one feature for node splitting. One advantage of their
model is that it does not rely on an assumption for the nature of the values
of the features. Günlük et al. [10] propose a binary classification tree
formulation for only categorical features. The idea can be extended to
numerical variables by using thresholding. The approach is less general than
the approach presented by Bertsimas and Dunn [1], but has some advantages like
the number of integer variables being independent of the training dataset
size. Firat et al. [6] propose a column generation based heuristic for
learning decision tree. The approach is based on generating decision paths and
can be used for solving instances with tens of thousands of observations.
In the light of this review, we make the following contributions to the
literature: We propose a new mathematical programming approach for
interpretation and boosting of random forests and decision trees,
respectively. Unlike other work in the literature, the objective function in
our models aims at minimizing both the total impurity and the number of
selected rules. When applied to a trained random forest for interpretation,
our first algorithm obtains significantly few number of rules with an accuracy
level close to the random forest model. Our second algorithm is a new boosting
method based on rule generation and linear programming. As a novel approach,
the algorithm uses dual information to weigh samples in order to increase the
coverage with less impure rules. In our computational study, we demonstrate
that both algorithms are remarkably easy to implement within a widely-used
machine learning package.111(GitHub page) –
https://github.com/sibirbil/RuleCovering Without any fine tuning, we obtain
quite promising numerical results with both algorithms showing the potential
of the rule covering for interpretation and boosting.
## 2 Minimum rule cover: interpretation of forests
Let $(\bm{x}_{i},y_{i})$, $i\in\mathcal{I}$ be the set of samples, where the
vector $\bm{x}_{i}\in\mathcal{X}$ and the scalar $y_{i}\in\mathcal{K}$ denote
the input sample and the output class, respectively. We shall assume in the
subsequent part that the learning problem at hand is a classification problem.
However, our discussion here can be extended to regression problems as well
(we elaborate on this point in Section S.3.222All cross references starting
with letter “S” refer to the supplementary document.).
Suppose that a Random Forest algorithm is trained on this dataset and a
collection of trees is grown. Given one of these trees, we can easily generate
the path $j$ from the root node to a leaf node that results in a subset of
samples $\bm{x}_{i}$, $i\in\mathcal{I}(j)\subseteq\mathcal{I}$. Actually,
these paths constitute the rules that are later used for classification with
majority voting. Each rule corresponds to a sequence of if-then clauses and it
is likely that some of these sequences may appear multiple times. We denote
all such rules in all trees by set $\mathcal{J}$. Clearly, the size of
$\mathcal{J}$ is quite large. Therefore, we next construct a mathematical
programming model that aims at selecting the minimum number of rules from the
trained forest while preserving the performance. We also make sure that all
samples are covered with the selected rules.
As rule $j\in\mathcal{J}$ corresponds to a leaf node, we can also evaluate the
node impurity, $w_{j}$. Two most common measures for evaluating the node
impurity are known as Gini and Entropy criteria. We further introduce the
binary decision variables $z_{j}$ that mark whether the corresponding rules
$j\in\mathcal{J}$ are selected or not. The cost of $w_{j}$ is incurred, when
$z_{j}=1$. Note that if we use only these impurity evaluations as the
objective function coefficients (costs), then the model would tend to select
many rules that cover only a few samples, which are likely to be _pure nodes_.
However, we want to advocate the use of fewer rules in the model so that the
resulting set of rules is easy to interpret. In other words, we also aim at
minimizing the number of rules along with the impurity cost. Therefore, we
replace the cost coefficient so that selecting excessive number of rules is
penalized. Our mathematical programming model then becomes:
$\begin{array}[]{lll}\mbox{minimize\hskip
11.38109pt}&\sum_{j\in\mathcal{J}}(1+w_{j})z_{j}&\\\\[5.69054pt] \mbox{subject
to\hskip 11.38109pt}&\sum_{j\in\mathcal{J}(i)}z_{j}\geq
1,&i\in\mathcal{I},\\\\[5.69054pt]
&z_{j}\in\\{0,1\\},&j\in\mathcal{J},\end{array}$ (1)
where $\mathcal{J}(i)$, $i\in\mathcal{I}$ is the set of rules prescribing all
the leaves involving sample $\bm{x}_{i}$. In other words, sample $i$ is
covered with the rules in subset $\mathcal{J}(i)\subseteq\mathcal{J}$.
The mathematical programming model (1) is a standard weighted set covering
formulation, where sets and items correspond to rules and samples,
respectively. Although set covering problem is NP-hard, current off-the-shelf
solvers are capable of providing optimal solutions to moderately large
instances. In case the integer programming formulation becomes difficult to
solve, there are also powerful heuristics that return approximate solutions
very quickly. One such approach is the well-known greedy algorithm proposed by
Chvatál [3]. Algorithm 3 shows the steps of this heuristic using our notation.
Recently, Chvatál’s greedy algorithm, as we implement here, has been shown to
perform quite well on a set of test problems when compared against several
more recent heuristics [18]. Thus, we have reported our results also with this
algorithm in Section 4. Clearly, this particular heuristic can be replaced
with any other approach solving the set covering problem. Since these
heuristics provide an approximate solution, we denote the resulting set of
rules by $\hat{\mathcal{J}}$, and by definition, the cardinality of set
$\hat{\mathcal{J}}$ is larger than the cardinality of the optimal set of rules
obtained by solving (1).
Algorithm 1 gives the steps of our minimum rule cover algorithm (MIRCO). The
algorithm takes as input the trained random forest model and the training
dataset. Note that after running Algorithm 1, we obtain a subset of rules
$\hat{\mathcal{J}}$ that can also be used for classifying out-of-sample points
(test set) with majority voting. If we denote the predicted class of test
sample $\bm{x}_{0}$ with $C(\bm{x}_{0},\hat{\mathcal{J}})$, then
$C(\bm{x}_{0},\hat{\mathcal{J}})=\arg\max_{k\in\mathcal{K}}\big{\\{}\sum_{j\in\hat{\mathcal{J}}}n_{jk}\mathds{L}_{j}(\bm{x}_{0})\big{\\}},$
(2)
where $n_{jk}$ stands for the number of samples from class $k$ in leaf $j$,
and $\mathds{L}_{j}(\bm{x}_{0})$ is an operator showing whether $\bm{x}_{0}$
satisfies the sequence of clauses corresponding to leaf $j$. In fact, testing
in this manner would be a natural approach for evaluating the performance of
MIRCO. Since our main objective is to interpret the underlying model obtained
with Random Forest algorithm, we hope that this subset of rules would obtain a
classification performance closer to the performance obtained with all the
rules in set $\mathcal{J}$. We show with our computational study in Section 4
that this is indeed the case.
Input : Random forest model RF; training dataset $(\bm{x}_{i},y_{i})$,
$i\in\mathcal{I}$
1 $\mathcal{J}=\emptyset$;
2 for _Tree in RF_ do
3 $\mathcal{J}\leftarrow\mathcal{J}\cup\text{(rules in {Tree})}$;
4
5 end for
6
7Evaluate impurities $w_{j}$, $j\in\mathcal{J}$ using $(\bm{x}_{i},y_{i})$,
$i\in\mathcal{I}$;
8
9Construct sets $\mathcal{J}(i)$, $i\in\mathcal{I}$ ;
10
11Solve model (1) with Algorithm 3;
12
13Return $\hat{\mathcal{J}}$;
Algorithm 1 MIRCO
Figure 1 shows an example set of of rules obtained with MIRCO on a small data
set. Clearly, the resulting set of rules does not correspond to a decision
tree. On one hand, this implies that we cannot simply select one of the trees
grown by the Random Forest algorithm and replace its leaves with the set of
rules obtained with MIRCO. Consequently, multiple rules may cover the same
test sample, in which case we can again use majority voting to determine its
class. Region A in Figure 1 shows that any test sample in that region will be
classified with two overlapping rules. On the other hand, there is also the
risk that the entire feature space is not covered by the subset of rules. In
Figure 1 such a region is marked as B, where none of the rules cover this
region. Therefore, if one decides to use only the rules in $\hat{\mathcal{J}}$
as a classifier, then some of the samples in the test set may not be
classified at all. This is an anticipated behaviour, since MIRCO guarantees to
cover only those samples that are used to train the Random Forest algorithm.
Our numerical results show that the percentages of test samples that MIRCO
fails to classify are quite low.
Figure 1: A toy example illustrating that MIRCO does not necessarily
correspond to a tree. Rectangles denote the regions obtained with the rules in
$\hat{\mathcal{J}}$. The numbers in the rectangles depict the classification
results corresponding to three classes shown with three different markers.
Notice that the entire set of rules used by MIRCO does not have to be
constructed after training a Random Forest algorithm. If there is an oracle
that provides many rules to construct the set $\mathcal{J}$, then we can still
(approximately) solve the mathematical programming model (1) and obtain the
resulting set of rules. In fact, the development of this oracle decouples our
discussion from training a forest. In the next section, we propose such an
oracle that generates the necessary rules on the fly.
## 3 RCBoost: rule cover boosting
The main difficulty with the set covering problem is the integrality condition
on the decision variables. When this condition is ignored, the resulting
problem is a simple linear programming problem that can be solved quite
efficiently. For the same reason, many approximation methods devised for the
set covering problem are based on considering the linear programming
relaxation of (1). Formally, the relaxed problem simply becomes:
$\begin{array}[]{lllc}\mbox{minimize\hskip
11.38109pt}&\sum_{j\in\mathcal{J}}(1+w_{j})z_{j}&&\\\\[5.69054pt]
\mbox{subject to\hskip 11.38109pt}&\sum_{j\in\mathcal{J}(i)}z_{j}\geq
1,&i\in\mathcal{I},\hskip 34.1433pt(\lambda_{i})\\\\[5.69054pt] &z_{j}\geq
0,&j\in\mathcal{J},&\end{array}$ (3)
where $\lambda_{i}$, $i\in\mathcal{I}$ denote the dual variables corresponding
to the constraints. After solving this problem, we obtain both the primal and
the dual optimal solutions. As we will discuss shortly, the optimal dual
variables bear important information about the sample points. Since we relax
the integrality restrictions, one suggestion could be rewriting the objective
as solely the minimization of total impurity. We refrain from such an approach
because when there are many pure leaves, then the corresponding rule does not
contribute to the objective function value ($w_{j}=0$) and the corresponding
$z_{j}$ variable becomes free to attain any value. This implies that the
primal problem has multiple optimal solutions, and hence causing degeneracy in
the dual.
We have seen in the previous section that problem (3) may easily have quite
many variables (columns). One standard approach to solve this type of linear
programming problems is based on iterative generation of only necessary
variables that would lead to an improvement in the objective function value.
This approach is known as column generation. The main idea is to start with a
set of columns (column pool) and form a linear programming problem, called the
restricted master problem (RMP). After solving the restricted master problem,
the dual optimal solution is obtained. Then using this dual solution, a
pricing subproblem is solved to find the columns with negative reduced costs.
These columns are the only candidates for improving the objective function
value when they are added to the column pool. The next iteration continues
with obtaining the optimal dual solution of the new restricted master problem
with the extended column pool. Again, the pricing subproblem is solved to find
the columns with negative reduced costs. If there is no such column, then the
column generation algorithm terminates.
Suppose that we denote the column pool in the restricted master problem at
iteration $t$ of column generation algorithm by $\mathcal{J}_{t}$. Then, we
solve the following linear programming problem:
$\begin{array}[]{lllc}\mbox{minimize\hskip
11.38109pt}&\sum_{j\in\mathcal{J}_{t}}(1+w_{j})z_{j}&&\\\\[5.69054pt]
\mbox{subject to\hskip 11.38109pt}&\sum_{j\in\mathcal{J}_{t}(i)}z_{j}\geq
1,&i\in\mathcal{I},&\hskip 11.38109pt(\lambda^{t}_{i})\\\\[5.69054pt]
&z_{j}\geq 0,&j\in\mathcal{J}_{t},&\end{array}$ (RMP$(\mathcal{J}_{t})$)
where $\lambda^{t}_{i}$ denotes the dual variable corresponding to constraint
$i\in\mathcal{I}$ at iteration $t$, and $\mathcal{J}_{t}(i)$ stands for the
rules in $\mathcal{J}_{t}$ that cover sample $i$. The reduced cost of rule
$j\in\mathcal{J}_{t}$ is simply given by
$\bar{w}_{j}=(1+w_{j})-\sum_{i\in\mathcal{I}(j)}\lambda^{t}_{i}.$ (4)
The necessary rules that would improve the objective function value in the
next iteration are the ones with negative reduced costs. In all this
discussion, the key point is, in fact, the pricing subproblem, which would
generate the rules with negative reduced costs. Recall that the columns in
this approach are constructed by the rules in our setting. Thus, in the
remaining part of this section, we shall use the term “rule pool” instead of
“column pool.”
Algorithm 2 shows the training process of the proposed rule cover boosting
(RCBoost) algorithm. The steps of the algorithm is also given in Figure 2.
Here, DecisionTree routine plays the role of the pricing subproblem. The
parameter of this routine is a vector, which is used to assign different
weights to the samples. To obtain rules with negative reduced costs, the dual
optimal solution of the restricted master problems is passed as the sample
weight vector (line 2). If there are rules with negative reduced costs (line
2), then these are added to the rule pool $\mathcal{J}^{*}$ in line 2.
Otherwise, the algorithm terminates (line 2). Note that a bound on the maximum
number of RMP calls can also be considered as a hyperparameter. The resulting
set or rules in the final pool constitute the classifier (line 2). Like in
Random Forest algorithm, a test point is classified using the majority voting
system among all the rules (if-then clauses) that are satisfied. Note that
there is no danger of failing to classify a test sample like MIRCO, since the
initial rule pool is constructed with a decision tree (line 2). The output set
of rules $\mathcal{J}^{*}$ is then used for classifying out-of-sample point
with majority voting as in (2). That is, $C(\bm{x}_{0},\mathcal{J}^{*})$ is
the predicted class for $\bm{x}_{0}$.
Input : Training data $(\bm{x}_{i},y_{i})$, $i\in\mathcal{I}$
1 $\bar{\bm{\lambda}}=(1,1,\dots,1)$;
2 $\mathcal{J}^{*}\leftarrow\text{{DecisionTree}}(\bar{\bm{\lambda}})$;
3 for _$t=1,2,\dots$_ do
4 $\bm{\lambda}^{t}\leftarrow\text{{RMP}}(\mathcal{J}^{*})$;
5 $\bar{\bm{\lambda}}\leftarrow\bar{\bm{\lambda}}+\bm{\lambda}^{t}$;
6 $\bar{\mathcal{J}}\leftarrow\text{{DecisionTree}}(\bar{\bm{\lambda}})$;
7 $\bar{\mathcal{J}}^{-}=\\{j\in\bar{\mathcal{J}}~{}:~{}\bar{w_{j}}<0\\}$;
8 if _$\bar{\mathcal{J}}^{-}=\emptyset$_ then
9 break;
10
11 end if
12 $\mathcal{J}^{*}\leftarrow\mathcal{J}^{*}\cup\bar{\mathcal{J}}^{-}$;
13
14 end for
15Return $\mathcal{J}^{*}$;
Algorithm 2 RCBoost Figure 2: Flowchart of RCBoost
As line 2 shows, we accumulate the dual vectors at every iteration. This is
because many of the dual variables quickly become zero as either the
corresponding primal constraints are inactive or the dual optimal solution
becomes degenerate. So, we make sure with the accumulation that all the sample
weights remain nonzero. We should also note that calling the decision tree
algorithm with (cumulative) sample weights is a proxy to the pricing
subproblem. Because, the actual pricing subproblem is supposed to search for
the rules with negative reduced costs. However, we argue that using such a
proxy is still sensible. We appeal to the theory of sensitivity analysis in
linear programming and the role of sample weights in boosting algorithms. An
optimal dual variable shows the change in the optimal objective function, when
the right-hand side of this constraint (all ones in our case) is slightly
adjusted. Since each constraint corresponds to the samples in (1), the
corresponding optimal dual variables show us the role of these samples for
changing the objective function value. In fact the reduced cost calculation in
(4) also bears the same intuition and promotes having a rule with low impurity
and large dual variables. Thus, if the samples with large dual values
(weights) belong to the same class, then they are more likely to be classified
within the same leaf (rule) to decrease the impurity. On the other hand, if
these samples belong to different classes, then putting them into separate
leaves also yields a decrease in impurity. This is precisely what the sample
weights try to achieve for boosting methods, where the weights for the
incorrectly classified samples are increased. For instance, let
$w^{\lambda}_{j}$ denote the weighted Gini impurity for leaf $j\in\mathcal{J}$
given by
$w^{\lambda}_{j}=1-\sum_{k\in\mathcal{K}}\left(\frac{\sum_{i\in\mathcal{I}(j)}\mathds{1}_{k}(i)+\sum_{i\in\mathcal{I}(j)}\mathds{1}_{k}(i)\sum^{t}_{s=1}\lambda_{i}^{s})}{|\mathcal{I}(j)|+\sum_{i\in\mathcal{I}(j)}\sum^{t}_{s=1}\lambda_{i}^{s}}\right)^{2}.$
where $|\mathcal{I}(j)|$ denotes the cardinality of set $\mathcal{I}(j)$ and
the indicator operator $\mathds{1}_{k}(i)$ is one when sample $i$ belongs to
class $k$; otherwise, it is zero. This weighted impurity is minimized whenever
the samples with large accumulated duals from the same class are classified
within the same leaf. This choice is likely to trigger generation of leaves
with negative reduced costs, since the reduced cost value (4) is also
minimized when the same class of samples (lower Gini impurity) with large
duals are covered by rule $j$.
Using sample weights with decision trees also has a clear advantage from
implementation point of view. Almost all existing tools available for
efficiently training decision trees admit weights for the samples so that
boosting algorithms, like the well-known AdaBoost [7], can make use of this
feature. Those algorithms increase the weights of misclassified samples in an
iterative manner and obtain a sequence of classifiers. Then, these classifiers
are further weighed according to their classification performances. RCBoost,
however, only produces a sequence of rules by increasing weights and each rule
is evaluated by its impurity contribution to the objective function through
its reduced cost. Thus, the reduced costs play the role of the partial
derivatives like in gradient boosting algorithms. These observations naturally
beg for a discussion about the relationship between our algorithm RCBoost and
the other boosting algorithms. We reserve Section S.2 to elaborate on this
point.
## 4 Numerical results
We next present our computational study with MIRCO and RCBoost algorithms on
15 datasets with varying characteristics. We have implemented both algorithms
in Python 333(GitHub page) – https://github.com/sibirbil/RuleCovering. The
details of these datasets are given in Table S.1. While most of these datasets
correspond to binary classification problems, there are also problems with
three classes (seeds and wine), six classes (glass) and eight classes (ecoli).
In the literature, mammography, diabetes, oilspill, phoneme, glass and ecoli
are considered as imbalanced datasets.
MIRCO is compared against Decision Tree (DT) and Random Forest (RF) algorithms
in terms of accuracy and the number of rules (interpretability). We also
compare results of RCBoost (RCB) against RF, AdaBoost (ADA) and Gradient
Boosting (GB). For the latter two boosting methods, we use decision trees as
base estimators. We apply $10\times 4$ nested stratified cross validation in
all experiments with the following options for hyperparameter tuning:
$\mathtt{maxdepth}\in\left\\{5,10,20\right\\}$,
$\mathtt{numberoftrees}\in\left\\{10,50,100\right\\}$. We also use
$\mathtt{numberofRMPcalls}\in\left\\{5,10,50,100,200\right\\}$ option to set a
bound on the number of RMP calls.
We first report our results with DT, RF and MIRCO algorithms in Figure 3. As
the accuracy values (average over 10 replications) in Figure 3a show, MIRCO
performs on par with RF on most of the datasets (see Table S.2 for the
details). Recall that MIRCO is not guaranteed to classify all the test
samples. Thus, we also provide the fraction of the test samples missed by
MIRCO (Figure 3b). However, we still classify such a sample by applying the
rules that have the largest fraction of accepted clauses. Therefore, the
reported comparisons against RF and DT in Figure 3a are carried on all
samples. We observe that the fraction of the missed test samples is below 6%
for all datasets but one. This shows that MIRCO does not only approximate the
RF in terms of accuracy, but also covers a very high fraction of the feature
space. Figure 3c depicts the numbers of rules in log-scale to give an
indication about the interpretability. In all but one dataset, MIRCO has
generated less number of rules than DT. Furthermore, the standard deviation
figures in Table S.2 show that MIRCO shows less variability than DT in all but
two of the datasets.
(a) (b) (c)
Figure 3: (a) Accuracy values for RF, DT, and MIRCO. (b) Average percentage of
missed samples by MIRCO. (c) Average number of rules (in logarithmic scale)
generated by RF, DT, and MIRCO.
To compare RCB, we present the average accuracy values of RF, ADA, GB in
Figure 4 (see Table S.3 for the details). We also report the results with the
initial Decision Tree (iniDT) used in RCB algorithm (see line 2 in Algorithm
2) so that we can observe how rule generation improves performance. The number
in parentheses above each dataset shows the average number of RMP calls by
RCB. These results demosntrate that RCB exhibits a quite competitive
performance. In particular, RCB outperforms both boosting algorithms, ADA and
GB in four problems. One critical point here is that RCB makes only a small
number of RMP calls. In all but one datasets, RCB improves the performance of
the base decision tree, iniDT by using the proposed rule generation.
Figure 4: RCBoost results
## 5 Conclusion
We have proposed two algorithms for interpretation and boosting. Both
algorithms are based on mathematical programming approaches for covering the
rules in decision trees. The first algorithm is used for interpreting a model
trained with the Random Forest algorithm. The second algorithm focuses on the
idea of generating necessary rules to boost an initial model trained with the
Decision Tree algorithm. Both algorithms are quite straightforward to
implement within the existing packages. Even with our first implementation
attempt, we have obtained promising results. The first algorithm has
successfully extracted much less number of rules than the random forest model
leading to better interpretability. Moreover, we have observed that the
accuracy levels obtained with the extracted rules of our first algorithm
closely follow the accuracy levels of the trained random forest model. When
compared against other boosting algorithms, our second algorithm has shown a
competitive performance. As a new boosting algorithm, we have also provided a
discussion about the relationship between our approach and other boosting
algorithms. The idea of selecting rules for interpretation or boosting leads
to a plethora of extensions that could be further studied. Therefore, we have
prepared a separate supplementary discussion on extensions in Section S.3.
This section complements our discussion as well as lists several future
research paths.
## References
* Bertsimas and Dunn [2017] D. Bertsimas and J. Dunn. Optimal classification trees. _Machine Learning_ , 106(7):1039–1082, 2017.
* Breiman [2001] L. Breiman. Random forests. _Machine Learning_ , 45(1):5–32, 2001.
* Chvatal [1979] V. Chvatal. A greedy heuristic for the set-covering problem. _Mathematics of Operations Research_ , 4(3):233–235, 1979.
* Dash et al. [2018] S. Dash, O. Gunluk, and D. Wei. Boolean decision rules via column generation. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, _Advances in Neural Information Processing Systems_ , pages 4655–4665. 2018.
* Demiriz et al. [2002] A. Demiriz, K. P. Bennett, and J. Shawe-Taylor. Linear programming boosting via column generation. _Machine Learning_ , 46(1-3):225–254, 2002.
* Firat et al. [2020] M. Firat, G. Crognier, A. F. Gabor, C. Hurkens, and Y. Zhang. Column generation based heuristic for learning classification trees. _Computers & Operations Research_, 116:104866, 2020.
* Freund and Schapire [1997] Y. Freund and R. E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. _Journal of Computer and System Sciences_ , 55(1):119 – 139, 1997.
* Friedman [2001] J. H. Friedman. Greedy function approximation: A gradient boosting machine. _Annals of Statistics_ , 29(5):1189–1232, 2001\.
* Friedman and Popescu [2008] J. H. Friedman and B. E. Popescu. Predictive learning via rule ensembles. _The Annals of Applied Statistics_ , 2(3):916–954, 2008.
* Günlük et al. [2018] O. Günlük, J. Kalagnanam, M. Menickelly, and K. Scheinberg. Optimal decision trees for categorical data via integer programming. _arXiv preprint arXiv:1612.03225_ , 2018.
* Mashayekhi and Gras [2015] M. Mashayekhi and R. Gras. Rule extraction from random forest: the rf+hc methods. In D. Barbosa and E. Milios, editors, _Advances in Artificial Intelligence_ , pages 223–237, Cham, 2015. Springer International Publishing.
* Mashayekhi and Gras [2017] M. Mashayekhi and R. Gras. Rule extraction from decision trees ensembles: New algorithms based on heuristic search and sparse group lasso methods. _International Journal of Information Technology & Decision Making_, 16(06):1707–1727, 2017.
* Mason et al. [1999] L. Mason, J. Baxter, P. Bartlett, and M. Frean. Boosting algorithms as gradient descent. In S. Solla, T. Leen, and K. Müller, editors, _Advances in Neural Information Processing Systems_ , pages 512–518. 1999.
* Mason et al. [2000] L. Mason, , P. L. Bartlett, and J. Baxter. Improved generalization through explicit optimization of margins. _Machine Learning_ , 38:243–255, 2000.
* Meinshausen [2010] N. Meinshausen. Node harvest. _The Annals of Applied Statistics_ , 4(4):2049–2072, 2010.
* Pfetsch and Pokutta [2020] M. E. Pfetsch and S. Pokutta. IPBoost–non-convex boosting via integer programming. _arXiv preprint arXiv:2002.04679_ , 2020.
* Schapire et al. [1998] R. E. Schapire, Y. Freund, P. Bartlett, and W. S. Lee. Boosting the margin: A new explanation for the effectiveness of voting methods. _The Annals of Statistics_ , 26(5):1651–1686, 1998.
* Vasko et al. [2016] F. J. Vasko, Y. Lu, and K. Zyma. What is the best greedy-like heuristic for the weighted set covering problem? _Operations Research Letters_ , 44:366–369, 2016.
Supplementary Material for
“Rule Covering for Interpretation and Boosting”
Note that most of the cross references in this supplementary file refer to the
original manuscript. Therefore, clicking on those references may not take you
to the desired location.
## S.1 Chvatál’s greedy algorithm
We restate the well-known heuristic of Chvatál in our own notation. Recall
that the sets $\mathcal{I}$ and $\mathcal{J}$ stand for the set of samples and
the set of rules, respectively. Algorithm 3 gives the steps of the greedy
heuristic, where $\mathcal{I}_{\mathcal{R}}(j)$ represents those samples that
are in $\mathcal{R}$ covered by rule $j\in\mathcal{J}$. In line 3, the cost of
selecting rule $j$ is $1+w_{j}$, and the operation
$|\mathcal{I}_{\mathcal{R}}(j)|$ denotes the cardinality of the set
$\mathcal{I}_{\mathcal{R}}(j)$. Removal of the redundant rules in line 3 is
done in two steps: First, the rules in $\bar{\mathcal{J}}$ are sorted in
ascending order of their costs in a list. Then, the rules at the end of this
list are removed as long as the remaining rules cover all the samples.
Result: Set of rules $\hat{\mathcal{J}}$
1
2$\mathcal{R}=\mathcal{I}$, $\hat{\mathcal{J}}=\emptyset$;
3
4while _$\mathcal{R}\neq\emptyset$_ do
5
6
$j^{*}=\arg\underset{j\in\mathcal{J}\backslash\hat{\mathcal{J}}}{\min}\left\\{\frac{1+w_{j}}{|\mathcal{I}_{\mathcal{R}}(j)|}:\mathcal{I}_{\mathcal{R}}(j)\neq\emptyset\right\\}$;
7
8
$\mathcal{R}\leftarrow\mathcal{R}\backslash\mathcal{I}_{\mathcal{R}}(j^{*})$;
9
10 $\hat{\mathcal{J}}\leftarrow\hat{\mathcal{J}}\cup j^{*}$;
11
12 end while
13
14Remove redundant columns from $\hat{\mathcal{J}}$;
15
16Return $\hat{\mathcal{J}}$;
Algorithm 3 Greedy
## S.2 Relation to other boosting algorithms
The use of sample weights and reduced costs shows a similar trait among
RCBoost and other boosting algorithms. The reduced cost of rule $j$ can be
considered as the partial derivative of the objective function with respect to
the corresponding variable $z_{j}$. Thus, rules that have negative reduced
costs are added to the rule pool. This resembles a descent iteration. To
establish the relationship between RCBoost and other boosting methods from the
gradient descent point of view [13], we first define the Lagrangean function
$\mathcal{L}(\mathbf{z},\bm{\lambda})=\sum_{j\in\mathcal{J}}\big{(}1+w_{j}\big{)}z_{j}+\sum_{i\in\mathcal{I}}\lambda_{i}\big{(}1-\sum_{j\in\mathcal{J}(i)}z_{j}\big{)},$
where $\mathbf{z}$ and $\bm{\lambda}$ denote the primal and dual vectors.
Suppose that the optimal solution pair for (RMP$(\mathcal{J}_{t})$) is given
by $\mathbf{z}^{t}$ and $\bm{\lambda}^{t}$. Then, we have
$\begin{array}[]{rl}\mathcal{L}((\mathbf{z}^{t},\mathbf{z}^{c}),\bm{\lambda}^{t})&=\sum_{i\in\mathcal{I}}\lambda^{t}_{i}+\sum_{j\in\mathcal{J}_{t}}\bar{w}_{j}z^{t}_{j}+\sum_{j\in\mathcal{J}/\mathcal{J}_{t}}\bar{w}_{j}z^{t}_{j}\\\\[5.69054pt]
&=\sum_{i\in\mathcal{I}}\lambda^{t}_{i}+\sum_{j\in\mathcal{J}/\mathcal{J}_{t}}\bar{w}_{j}z^{c}_{j},\end{array}$
where $\bar{w}_{j}$ is the reduced cost of rule $j$ as in (4) and the tuple
$(\mathbf{z}^{t},\mathbf{z}^{c})$ shows the variables in $\mathcal{J}_{t}$ and
$\mathcal{J}/\mathcal{J}_{t}$, respectively. The last equality is a direct
consequence of optimality conditions, since $\mathbf{z}^{t}$ and
$\bm{\lambda}^{t}$ are complementary solution vectors. Thus, taking the
steepest descent step for $j\in\mathcal{J}/\mathcal{J}_{t}$ simply implies
involving those rules with negative reduced costs in the next iteration. From
this perspective, RCBoost is similar to gradient boosting algorithms.
Nonetheless, unlike those algorithms RCBoost does not fit a weak-classifier
but instead obtains a larger set $\mathcal{J}_{t+1}$ by adding a subset of
rules with negative reduced costs (line 2 of Algorithm 2). Then, solving the
linear programming problem over $\mathcal{J}_{t+1}$ yields the parameters
$\mathbf{z}^{t+1}$ and $\bm{\lambda}^{t+1}$. Suppose we denote the optimal
objective function of (RMP$(\mathcal{J}_{t})$) by $\Phi(\mathcal{J}_{t})$,
which is the total impurity that we use to evaluate the training loss of our
classifier. With each iteration we improve this loss, since
$\Phi(\mathcal{J}_{t})\geq\Phi(\mathcal{J}_{t+1})$ due to our linear
programming formulation. If we further define the subset
$\partial\mathcal{J}_{t}\subseteq\\{j\in\mathcal{J}:\bar{w}_{j}<0\\},$
then
$\Phi(\mathcal{J}_{t+1})=\Phi(\mathcal{J}_{t}\cup\partial\mathcal{J}_{t})$.
There has also been a focus on the margin maximization idea to discuss the
generalization performance of boosting methods [14, 17]. Here, one of the
fundamental points is to observe that a sample with a large margin is likely
to be classified correctly. Thus, margin maximization is about assigning
larger weights to those samples with small margins. RCBoost also implicitly
aims at obtaining large margins through selection of rules with less
impurities (classification errors decrease). In other words, minimizing the
objective function of the restricted master problem promotes large margins.
Formally, we can rewrite the optimal objective function value of
(RMP$(\mathcal{J}_{t})$) as
$\sum_{j\in\mathcal{J}_{t}}(1+w_{j})z^{t}_{j}=\sum_{i\in\mathcal{I}}\sum_{j\in\mathcal{J}_{t}(i)}\mu_{j}z^{t}_{j}=\sum_{i\in\mathcal{I}}\lambda_{i}^{t},$
where $\mu_{j}=\frac{1+w_{j}}{|\mathcal{I}(j)|}$, $j\in\mathcal{J}$. Using
complementary slackness condition, we know that $\lambda_{i}^{t}>0$ only if
$\sum_{j\in\mathcal{J}_{t}(i)}z_{j}^{t}=1$. Thus,
$\sum_{j\in\mathcal{J}_{t}(i)}\mu_{j}z^{t}_{j}$ is the convex combination of
margins obtained with different rules for sample $i$. The larger the obtained
impurity, the smaller the corresponding margin. Like other boosting methods,
RCBoost also assigns larger weights to those samples with smaller margins.
## S.3 Some extensions
We have used classification for our presentation. Nonetheless, both proposed
algorithms can also be adjusted to solve regression problems. To interpret
random forest regressors with MIRCO, the first step is to use a proper
criterion, like mean squared error, for evaluating $w_{j}$, $j\in\mathcal{J}$
values. Then, MIRCO can be used as it is given in Algorithm 1. The same
criteria can also be used in RCBoost to construct the restricted master
problem. When a regression problem is solved, one needs to pay attention to
two potential caveats: (i) The two components of the objective function
coefficients in optimization model (1) or (3) may not be balanced well, since
the criteria values $w_{j}$, $j\in\mathcal{J}$ are not bounded above by one.
This could be overcome by scaling the criteria values. If scaling is done with
a multiplier, then this multiplier becomes a new hyperparameter which may
require tuning. (ii) Regression trees tend to have too many leaves. Even after
applying MIRCO, the cardinality of the resulting set of rules
$\hat{\mathcal{J}}$ may be quite large, and hence, interpreting the random
forest regressor may be difficult. Similarly, RCBoost may make too many calls
to the restricted master problem and end up with a large number of rules in
the final rule pool $\mathcal{J}^{*}$.
Many boosting algorithms use decision trees as their weak estimators. Thus, it
seems that MIRCO can also be used with those algorithms to interpret their
results. However, one needs to pay attention to the fact that boosting
algorithms assign different weights to their weak estimators and MIRCO does
not take these weights into account. Therefore, if rule covering is used for
interpretation of boosting algorithms, Algorithm 1 should be adjusted to
incorporate the estimator weights.
Recall that we use a decision tree with sample weights in RCBoost as a proxy
to an actual pricing subproblem (line 2, Algorithm 2). An alternative could be
using the impurity and the negative reduced cost together while constructing
the decision tree. In that case, splitting can be done from non-dominated bi-
criteria values; i.e., impurity and negative reduced cost values.
Clearly, we can also use a random forest or any ensemble of trees with sample
weights as a proxy to the pricing subproblem. Even further, one may even model
a separate combinatorial problem to generate many different rules. At this
point, it is important to keep in mind that considering all possible rules
with negative reduced costs could prove to be a daunting job.
In its current stage, RCBoost adds all the rules with negative reduced costs
to the rule pool. An alternative approach could be to include only the rule
with the most negative reduced cost. Along the same vein, we may also remove
those rules that have high (positive) reduced costs from the column pool.
Actually, these are typical approaches in column generation to keep the size
of the column pool manageable. As the size of the rule pool becomes large,
RCBoost also becomes less interpretable. Fortunately, one can easily apply
MIRCO or other rule extraction approaches from the literature such as the
RuleFit algorithm [9] to a trained RCBoost. In that case, Algorithm 1 can be
applied starting from line 1 with $\mathcal{J}\leftarrow\mathcal{J}^{*}$. As a
final remark, we note that RCBoost treats the rules equally when a test point
is classified with the satisfied rules in $\mathcal{J}^{*}$. An alternative
could be giving different weights to different rules. These rule weights could
be evaluated by making use of the optimal values of $z_{j}$,
$j\in\mathcal{J}^{*}$ obtained from the final restricted master problem.
However, all these changes should be carefully contemplated, since the
resulting rule pool after shrinking may not necessarily cover the feature
space as we have discussed in Section 2 (see also Figure 1).
## S.4 Details of numerical results
Table S.1: Characteristics of the datasets | Number of | Number of | Number of
---|---|---|---
Dataset | Features | Classes | Samples
banknote | 4 | 2 | 1372
ILDP | 10 | 2 | 583
ionosphere | 34 | 2 | 351
transfusion | 4 | 2 | 748
liver | 6 | 2 | 345
tic-tac-toe | 9 | 2 | 958
WDBC | 30 | 2 | 569
mammography | 6 | 2 | 11183
diabetes | 8 | 2 | 768
oilspill | 48 | 2 | 937
phoneme | 5 | 2 | 5427
seeds | 7 | 3 | 210
wine | 13 | 3 | 178
glass | 10 | 6 | 214
ecoli | 7 | 8 | 336
Table S.2: MIRCO results | DT | RF | MIRCO
---|---|---|---
Dataset | Accuracy | Accuracy | Accuracy | % of Missed Pts
banknote | 0.99 (0.02) | 0.99 (0.01) | 0.98 (0.01) | 0.01 (0.01)
ILDP | 0.66 (0.07) | 0.71 (0.06) | 0.69 (0.03) | 0.06 (0.04)
ionosphere | 0.87 (0.07) | 0.92 (0.03) | 0.91 (0.04) | 0.02 (0.03)
transfusion | 0.79 (0.04) | 0.77 (0.04) | 0.76 (0.01) | 0.01 (0.01)
liver | 0.65 (0.09) | 0.73 (0.06) | 0.66 (0.06) | 0.06 (0.05)
tic-tac-toe | 0.94 (0.02) | 0.99 (0.01) | 0.97 (0.02) | 0.02 (0.02)
WDBC | 0.93 (0.03) | 0.96 (0.02) | 0.95 (0.03) | 0.01 (0.02)
mammography | 0.98 (0.00) | 0.99 (0.00) | 0.98 (0.00) | 0.00 (0.00)
diabetes | 0.74 (0.04) | 0.77 (0.04) | 0.73 (0.04) | 0.05 (0.04)
oilspill | 0.96 (0.02) | 0.96 (0.01) | 0.96 (0.01) | 0.01 (0.01)
phoneme | 0.88 (0.01) | 0.91 (0.01) | 0.88 (0.02) | 0.03 (0.01)
seeds | 0.91 (0.06) | 0.91 (0.09) | 0.91 (0.05) | 0.02 (0.03)
wine | 0.91 (0.07) | 0.98 (0.03) | 0.91 (0.07) | 0.04 (0.04)
glass | 0.67 (0.05) | 0.79 (0.07) | 0.71 (0.07) | 0.09 (0.06)
ecoli | 0.81 (0.09) | 0.88 (0.06) | 0.76 (0.08) | 0.03 (0.03)
∗ Numbers in parantheses show the standard deviations |
Table S.3: RCBoost results | RF | ADA | GB | iniDT | RCB
---|---|---|---|---|---
Dataset | Accuracy | Accuracy | Accuracy | Accuracy | Accuracy
banknote | 0.99 (0.01) | 1.00 (0.00) | 0.99 (0.01) | 0.99 (0.01) | 0.98 (0.01)
ILDP | 0.71 (0.06) | 0.70 (0.06) | 0.69 (0.03) | 0.66 (0.07) | 0.70 (0.04)
ionosphere | 0.92 (0.03) | 0.94 (0.05) | 0.92 (0.04) | 0.88 (0.03) | 0.89 (0.05)
transfusion | 0.77 (0.04) | 0.74 (0.04) | 0.77 (0.03) | 0.78 (0.03) | 0.78 (0.04)
liver | 0.73 (0.06) | 0.68 (0.07) | 0.68 (0.06) | 0.64 (0.10) | 0.71 (0.05)
tic-tac-toe | 0.99 (0.01) | 1.00 (0.01) | 0.99 (0.01) | 0.95 (0.02) | 0.99 (0.01)
WDBC | 0.96 (0.02) | 0.97 (0.03) | 0.96 (0.03) | 0.92 (0.03) | 0.94 (0.02)
mammography | 0.99 (0.00) | 0.99 (0.00) | 0.99 (0.00) | 0.98 (0.00) | 0.99 (0.00)
diabetes | 0.77 (0.04) | 0.74 (0.04) | 0.76 (0.04) | 0.70 (0.06) | 0.75 (0.03)
oilspill | 0.96 (0.01) | 0.97 (0.01) | 0.96 (0.02) | 0.95 (0.02) | 0.96 (0.02)
phoneme | 0.91 (0.01) | 0.92 (0.01) | 0.91 (0.01) | 0.88 (0.01) | 0.90 (0.01)
seeds | 0.91 (0.09) | 0.92 (0.07) | 0.92 (0.07) | 0.91 (0.09) | 0.94 (0.06)
wine | 0.98 (0.03) | 0.92 (0.08) | 0.93 (0.05) | 0.89 (0.07) | 0.94 (0.04)
glass | 0.79 (0.07) | 0.77 (0.08) | 0.73 (0.10) | 0.67 (0.11) | 0.71 (0.09)
ecoli | 0.88 (0.06) | 0.84 (0.07) | 0.83 (0.06) | 0.82 (0.09) | 0.83 (0.05)
∗ Numbers in parantheses show the standard deviations | |
|
# Weak completions of paratopological groups
Taras Banakh T.Banakh: Ivan Franko National University of Lviv (Ukraine) and
Jan Kochanowski University in Kielce (Poland<EMAIL_ADDRESS>and
Mikhail Tkachenko M.Tkachenko: Department of Mathematics, Universidad
Autónoma Metropolitana, Av. San Rafael Atlixco 186, Col. Vicentina, C.P.
09340, Iztapalapa, Mexico City, Mexico<EMAIL_ADDRESS>
###### Abstract.
Given a $T_{0}$ paratopological group $G$ and a class $\mathcal{C}$ of
continuous homomorphisms of paratopological groups, we define the
_$\mathcal{C}$ -semicompletion_ $\mathcal{C}[G)$ and _$\mathcal{C}$
-completion_ $\mathcal{C}[G]$ of the group $G$ that contain $G$ as a dense
subgroup, satisfy the $T_{0}$-separation axiom and have certain universality
properties. For special classes $\mathcal{C}$, we present some necessary and
sufficient conditions on $G$ in order that the (semi)completions
$\mathcal{C}[G)$ and $\mathcal{C}[G]$ be Hausdorff. Also, we give an example
of a Hausdorff paratopological abelian group $G$ whose
$\mathcal{C}$-semicompletion $\mathcal{C}[G)$ fails to be a $T_{1}$-space,
where $\mathcal{C}$ is the class of continuous homomorphisms of sequentially
compact topological groups to paratopological groups. In particular, the group
$G$ contains an $\omega$-bounded sequentially compact subgroup $H$ such that
$H$ is a topological group but its closure in $G$ fails to be a subgroup.
###### Key words and phrases:
Paratopological group, pseudocompact, precompact, Raĭkov completion,
Bicompletion
###### 2010 Mathematics Subject Classification:
22A15, 54D35, 54H11
The second author was supported by grant number CAR-64356 of the Program
“Ciencia de Frontera 2019” of the CONACyT, Mexico
## 1\. Introduction
It is well known [1, §3.6] that each topological group $G$ has the Raĭkov
completion, $\varrho{G}$, which coincides with the completion of $G$ with
respect to the two-sided group uniformity of the group. The Raĭkov completion
has a nice categorial characterization. It turns out that $\varrho{G}$ is a
unique topological group, up to topological isomorphism, that contains $G$ as
a dense subgroup and has the following extension property: Every continuous
homomorphism $h\colon X\to G$ defined on a dense subgroup $X$ of a topological
group $\widetilde{X}$ admits a unique extension to a continuous homomorphism
$\tilde{h}\colon\widetilde{X}\to\varrho{G}$ (see [1, Proposition 3.6.12]). A
topological group $G$ is Raĭkov complete if and only if it is complete in its
two-sided uniformity if and only if $G=\varrho{G}$.
In [4] and [3], it is shown that a kind of the Raĭkov completion can also be
defined for paratopological groups. By a _paratopological group_ we understand
a group $G$ endowed with a topology $\tau$ making the group multiplication
$G\times G\to G$, $\langle x,y\rangle\to xy$ continuous. If, in addition, the
inversion $G\to G$, $x\mapsto x^{-1}$, is continuous, then $\langle
G,\tau\rangle$ is a topological group. A topology $\tau$ on a group $G$ is
called a topological group topology on $G$ if $\langle G,\tau\rangle$ is a
topological group.
Let us recall that a topology $\tau$ on a set $X$ satisfies the $T_{0}$
separation axiom or, equivalently, $\tau$ is a $T_{0}$-topology if for any
distinct points $x,y\in G$ there exists an open set $U\in\tau$ such that
$U\cap\\{x,y\\}$ is a singleton. It is well known, on the one hand, that each
topological group satisfying the $T_{0}$ separation axiom is automatically
Tychonoff. On the other hand, the topology
$\tau=\\{\emptyset,\mathbb{R}\\}\cup\\{(a,\infty):a\in\mathbb{R}\\}$ turns the
additive group of the reals into a $T_{0}$ paratopological group which does
not satisfy the $T_{1}$ separation axiom. So ‘$T_{0}$ does not imply $T_{1}$’
in paratopological groups. From now on we assume that all paratopological
groups considered here satisfy the $T_{0}$ separation axiom.
Each paratopological group $\langle G,\tau\rangle$ admits a stronger
topological group topology $\tau^{\sharp}$, whose neighborhood base at the
identity $e$ consists of the set $U\cap U^{-1}$ where $e\in U\in\tau$. So,
$\langle G,\tau^{\sharp}\rangle$ is topologically isomorphic to the subgroup
$\\{\langle x,x^{-1}\rangle:x\in G\\}$ of the product group $G\times G$. The
topological group $G^{\sharp}=\langle G,\tau^{\sharp}\rangle$ is called the
group coreflection of the paratopological group $G=\langle G,\tau\rangle$. If
the topology $\tau$ satisfies the $T_{0}$ separation axiom, then the
topological group topology $\tau^{\sharp}$ is Hausdorff and hence Tychonoff. A
paratopological group $\langle G,\tau\rangle$ is a topological group if and
only if $\tau=\tau^{\sharp}$. A paratopological group $G$ is called
$\sharp$-complete if the topological group $G^{\sharp}$ is Raĭkov complete.
The Sorgenfrey line $\mathbb{S}$ is an easy example of a $\sharp$-complete
paratopological group since $\mathbb{S}^{\sharp}$ is the discrete group of the
reals.
A subset $D$ of a paratopological group $\langle G,\tau\rangle$ is called
$\sharp$-dense in $G$ if $D$ is dense in $\langle G,\tau^{\sharp}\rangle$.
Since $\tau\subseteq\tau^{\sharp}$, each $\sharp$-dense subset is dense (but
not vice versa).
According to [4] or [3], each paratopological group $G$ is a $\sharp$-dense
subgroup of a $\sharp$-complete paratopological group $\breve{G}$ that has the
following extension property: any continuous homomorphism $h\colon X\to G$
defined on a $\sharp$-dense subgroup $X$ of a paratopological group
$\widetilde{X}$ admits a unique extension to a continuous homomorphism
$\tilde{h}\colon\widetilde{X}\to\breve{G}$. This extension property of
$\breve{G}$ implies that the $\sharp$-complete paratopological group
$\breve{G}$ containing $G$ as a $\sharp$-dense subgroup is unique up to a
topological isomorphism. In [3] this unique paratopological group $\breve{G}$
is called the Raĭkov completion of $G$ and in [4] it is called the
bicompletion of $G$. By [3], a neighborhood base at the identity $e$ of the
paratopological group $\breve{G}$ consists of the sets
$\overline{U}^{\sharp}$, where $U$ is a neighborhood of $e$ in $G$ and
$\overline{U}^{\sharp}$ is the closure of $U$ in the Raĭkov completion
$\varrho{G}^{\sharp}$ of the topological group $G^{\sharp}$. So as a group,
$\breve{G}$ coincides with the Raĭkov completion of $G^{\sharp}$. If $G$ is a
topological group, then the Raĭkov completion $\breve{G}$ of $G$ as a
paratopological group coincides with the usual Raĭkov completion $\varrho{G}$
of $G$.
By [3], _for any regular paratopological group $G$, its Raĭkov completion
$\breve{G}$ is regular and hence Tychonoff_ (for the latter implication, see
[2]). However, the Raĭkov completion of a Hausdorff paratopological group $G$
is not necessarily Hausdorff (see [3, Example 2.9]). To bypass this pathology
of $\breve{G}$, in this article we consider some weaker notions of completion
of paratopological groups which preserve Hausdorffness in some special cases.
Our weaker notion of completion depends on a class $\mathcal{C}$ of continuous
homomorphisms between paratopological groups. For example, $\mathcal{C}$ can
be the class of all continuous homomorphisms of topological groups (possessing
some property like precompactness, pseudocompactness, countable compactness,
sequential compactness, etc.) to paratopological groups. So, we fix a class
$\mathcal{C}$ of continuous homomorphisms between paratopological groups.
A paratopological group $\widetilde{G}$ is called a
$\mathcal{C}$-semicompletion of a paratopological group $G$ if
$G\subseteq\widetilde{G}$ and any homomorphism $h\colon X\to G$ in the class
$\mathcal{C}$ has a unique extension
$\breve{h}\colon\breve{X}\to\widetilde{G}$ to the Raĭkov completion
$\breve{X}$ of the paratopological group $X$. The extension property of the
Raĭkov completion $\breve{G}$ ensures that $\breve{G}$ is a
$\mathcal{C}$-semicompletion of $G$. So every paratopological group has at
least one $\mathcal{C}$-semicompletion.
A $\mathcal{C}$-semicompletion $\widetilde{G}$ of a paratopological group $G$
is called minimal if $\widetilde{G}=H$ for any subgroup
$H\subseteq\widetilde{G}$ which is a $\mathcal{C}$-semicompletion of $G$.
Every $\mathcal{C}$-semicompletion $\tilde{G}$ of a paratopological group $G$
contains a unique minimal $\mathcal{C}$-semicompletion of $G$, which is equal
to the intersection $\bigcap\mathcal{S}$ of the family $\mathcal{S}$ of
subgroups $H\subseteq\widetilde{G}$ such that $H$ is a
$\mathcal{C}$-semicompletion of $G$. The Hausdorffness of the group
coreflections implies that $\bigcap\mathcal{S}$ is indeed a
$\mathcal{C}$-semicompletion of $G$.
The $\mathcal{C}$-semicompletion $\mathcal{C}[G)$ of a paratopological group
$G$ is defined to be the smallest $\mathcal{C}$-semicompletion of $G$ in its
Raĭkov completion $\breve{G}$. It is equal to the intersection
$\bigcap\mathcal{S}$ of the family $\mathcal{S}$ of all groups $H$ with
$G\subseteq H\subseteq\breve{G}$ such that $H$ is a
$\mathcal{C}$-semicompletion of $G$. Let us note that the
$\mathcal{C}$-semicompletion $\mathcal{C}[G)$ of $G$ is topologically
isomorphic to any minimal $\mathcal{C}$-semicompletion of $G$. Indeed, given a
minimal $\mathcal{C}$-semicompletion $H$ of $G$, consider the Raĭkov
completion $\breve{H}$ of the paratopological group $H$. Since $G\subseteq
H\subseteq\breve{H}$, the Raĭkov completion $\breve{G}$ of $G$ can be
identified with the $\sharp$-closure of $G$ in $\breve{H}$. Now the minimality
of the $\mathcal{C}$-semicompletion $H$ implies that
$H=H\cap\breve{G}\subseteq\breve{G}$ and, finally, $H=\mathcal{C}[G)$.
Having defined the $\mathcal{C}$-semicompletions of paratopological groups, we
can pose the following two general problems.
###### Problem 1.1.
Explore the categorial properties of $\mathcal{C}$-semicompletions.
###### Problem 1.2.
Characterize the paratopological groups $G$ for which the
$\mathcal{C}$-semicompletion $\mathcal{C}[G)$ of $G$ is Hausdorff.
Some partial answers to these problems will be given in Sections 2 and 3.
Again, let $\mathcal{C}$ be a class of continuous homomorphisms of
paratopological groups. Now we introduce the notion of a
$\mathcal{C}$-completion of a paratopological group. A paratopological group
$G$ is $\mathcal{C}$-complete if $G$ is a $\mathcal{C}$-semicompletion of $G$.
Equivalently, a paratopological group $G$ is $\mathcal{C}$-complete if each
homomorphism $h\colon X\to G$ in the class $\mathcal{C}$ extends to a
continuous homomorphism $\breve{h}\colon\breve{X}\to G$ defined on the Raĭkov
completion of $X$. By the extension property of Raĭkov completions, each
$\sharp$-complete paratopological group $G$ (hence, $\breve{G}$) is
$\mathcal{C}$-complete. In particular, the Sorgenfrey line $\mathbb{S}$ is
$\mathcal{C}$-complete.
By the $\mathcal{C}$-completion, $\mathcal{C}[G]$, of a paratopological group
$G$ we understand the intersection $\bigcap\mathcal{S}$ of the family
$\mathcal{S}$ of all $\mathcal{C}$-complete subgroups $H\subseteq\breve{G}$
that contain $G$. The Hausdorffness of the groups coreflections of
paratopological groups implies that the $\mathcal{C}$-completion
$\mathcal{C}[G]$ of any paratopological group $G$ is $\mathcal{C}$-complete
and, hence, $\mathcal{C}[G)\subseteq\mathcal{C}[G]$.
It is clear from the above definitions that if $\mathcal{C}$ is the empty
class (or, more generally, every homomorphism $h\colon X\to G$ with
$h\in\mathcal{C}$, if exists, is trivial), then $G$ is $\mathcal{C}$-complete
and, therefore, $G=\mathcal{C}[G)=\mathcal{C}[G]$.
###### Problem 1.3.
Explore the categorial properties of the operation of $\mathcal{C}$-completion
in the category of paratopological groups and their continuous homomorphisms.
###### Problem 1.4.
Characterize the paratopological groups $G$ such that the
$\mathcal{C}$-completion $\mathcal{C}[G]$ of $G$ is Hausdorff.
###### Problem 1.5.
For which paratopological groups (and classes $\mathcal{C}$ of homomorphisms)
do their $\mathcal{C}$-semicompletions coincide with their
$\mathcal{C}$-completions?
## 2\. Categorial properties of $\mathcal{C}$-completions and
$\mathcal{C}$-semicompletions
A class $\mathcal{C}$ of continuous homomorphisms between paratopological
groups is called composable if for any homomorphism $f\colon X\to Y$ in the
class $\mathcal{C}$ and any continuous homomorphism $g\colon Y\to Z$ of
paratopological groups, the composition $g\circ f$ is in $\mathcal{C}$. For
example, for any class $\mathcal{P}$ of paratopological groups, the class
$\mathcal{C}$ of continuous homomorphisms $h\colon X\to Y$ between
paratopological groups with $X\in\mathcal{P}$ is composable.
The following proposition shows that for a composable class $\mathcal{C}$, the
constructions of $\mathcal{C}$-semicompletion and $\mathcal{C}$-completion are
functorial in the category of paratopological groups and their continuous
homomorphisms.
###### Proposition 2.1.
Let $\mathcal{C}$ be a composable class of continuous homomorphisms of
paratopological groups. For any continuous homomorphism $h\colon X\to Y$ of
paratopological groups, its continuous extension
$\breve{h}\colon\breve{X}\to\breve{Y}$ satisfies
$\breve{h}[\mathcal{C}[X)]\subseteq\mathcal{C}[Y)$ and
$\breve{h}[\mathcal{C}[X]]\subseteq\mathcal{C}[Y]$.
###### Proof.
To see that $\breve{h}[\mathcal{C}[X)]\subseteq\mathcal{C}[Y)$, it suffices to
check that the preimage $\breve{h}^{-1}[\mathcal{C}[Y)]$ is a
$\mathcal{C}$-semicompletion of $X$. Given any homomorphism $f\colon Z\to X$
in the class $\mathcal{C}$, consider its continuous homomorphic extension
$\breve{f}\colon\breve{Z}\to\breve{X}$. Then
$\breve{h}\circ\breve{f}\colon\breve{X}\to\breve{Y}$ is a continuous extension
of the homomorphism $h\circ f\colon Z\to Y$. Taking into account that
$\mathcal{C}[Y)$ is a $\mathcal{C}$-semicompletion of $Y$ and the topology of
the group reflection of $\breve{Y}$ is Hausdorff, we conclude that
$(\breve{h}\circ\breve{f})[\breve{Z}]\subseteq\mathcal{C}[Y)$ and hence
$\breve{f}[\breve{Z}]\subseteq\breve{h}^{-1}[\mathcal{C}[Y)]$. This implies
that $\breve{h}^{-1}[\mathcal{C}[Y)]$ is a $\mathcal{C}$-semicompletion of $X$
and $\mathcal{C}[X)\subseteq\breve{h}^{-1}[\mathcal{C}[Y)]$, by the minimality
of $\mathcal{C}[X)$.
Next, we show that the preimage $\breve{h}^{-1}[\mathcal{C}[Y]]$ is a
$\mathcal{C}$-complete paratopological group. Given any homomorphism $f\colon
Z\to\breve{h}^{-1}[\mathcal{C}[Y]]$ in the class $\mathcal{C}$, consider its
unique continuous extension $\breve{f}\colon\breve{Z}\to\breve{X}$. Then
$\breve{h}\circ\breve{f}\colon\breve{Z}\to\breve{Y}$ is a unique continuous
extension of the homomorphism $h\circ f\colon Z\to\mathcal{C}[Y]$. Taking into
account that the paratopological group $\mathcal{C}[Y]$ is
$\mathcal{C}$-complete and the topology of the group reflection of $\breve{Y}$
is Hausdorff, we conclude that
$(\breve{h}\circ\breve{f})[\breve{Z}]\subseteq\mathcal{C}[Y]$, which implies
the inclusion $\breve{f}[\breve{Z}]\subseteq\breve{h}^{-1}[\mathcal{C}[Y]]$.
Therefore, the paratopological group $\breve{h}^{-1}[\mathcal{C}[Y]]$ is
$\mathcal{C}$-complete and
$\mathcal{C}[X]\subseteq\breve{h}^{-1}[\mathcal{C}[Y]]$, by the minimality of
$\mathcal{C}[X]$. ∎
## 3\. The Hausdorff property of $\mathcal{C}$-completions of paratopological
groups
In this section, we present some necessary and some sufficient conditions on a
paratopological group $G$ in order that the $\mathcal{C}$-semicompletion
$\mathcal{C}[G)$ or $\mathcal{C}$-completion $\mathcal{C}[G]$ of $G$ be
Hausdorff. Let us recall that for a regular paratopological group $G$, its
Raĭkov completion $\breve{G}$ is regular and so are the paratopological groups
$\mathcal{C}[G)$ and $\mathcal{C}[G]$.
###### Proposition 3.1.
The $\mathcal{C}$-completion $\mathcal{C}[G]$ of a paratopological group $G$
is Hausdorff if and only if $G$ is a subgroup of a Hausdorff
$\mathcal{C}$-complete paratopological group.
###### Proof.
The “only if” part is trivial. To prove the “if” part, assume that $G$ is a
subgroup of a Hausdorff $\mathcal{C}$-complete paratopological group $H$.
Taking into account that $G\subseteq H$ and $G^{\sharp}\subseteq H^{\sharp}$,
we conclude that $\breve{G}\subseteq\breve{H}$. The $\mathcal{C}$-completeness
of the paratopological groups $\breve{G}$ and $H$ implies that the
paratopological group $\breve{G}\cap H$ is $\mathcal{C}$-complete and hence
$\mathcal{C}[G]\subseteq\breve{G}\cap H$ is Hausdorff, being a subgroup of the
Hausdorff paratopological group $H$. ∎
Let $\mathcal{P}$ be a property. A paratopological group $H$ is said to be
_projectively $\mathcal{P}$_ if every neighborhood of the identity element in
$H$ contains a closed invariant (equivalently, “normal” in the algebraic
sense) subgroup $N$ such that the quotient paratopological group $H/N$ has
$\mathcal{P}$. Similarly, a paratopological group $X$ is said to be
projectively $\Psi$ if for every neighborhood $U$ of the identity in $X$,
there exists a continuous homomorphism $h\colon X\to Y$ to a Hausdorff
paratopological group of countable pseudocharacter such that
$h^{-1}(e_{Y})\subseteq U$; here $e_{Y}$ denotes the identity of the group
$Y$.
It is easy to see that the projectively $\Psi$ paratopological groups are
exactly the projectively $\mathcal{P}$ groups, where $\mathcal{P}$ is the
property of being a Hausdorff space of countable pseudocharacter. Indeed, let
$G$ be a projectively $\Psi$ paratopological group and $U$ be a neighborhood
of the identity in $G$. By our assumption, there exists a continuous
homomorphism $h\colon G\to H$ onto a Hausdorff paratopological group
$H_{\alpha}$ of countable pseudocharacter such that $h^{-1}(e_{H})\subseteq
U$, where $e_{H}$ is the identity of $H$. Let $N=h^{-1}(e_{H})$ be the kernel
of $h$ and $p\colon G\to G/N$ be the quotient homomorphism. Clearly there
exists a continuous one-to-one homomorphism $j\colon G/N\to H$ satisfying
$h=j\circ p$. Thus $j$ is a continuous bijection of $G/N$ onto $H$ and, hence,
the quotient group $G/N$ is Hausdorff and has countable pseudocharacter.
Therefore, $G$ is projectively $\mathcal{P}$. The inverse implication is
evident.
According to [6, Proposition 2] every projectively Hausdorff paratopological
group is Hausdorff. In particular, all projectively $\Psi$ paratopological
groups are Hausdorff. This fact also follows from the characterization of
projectively $\Psi$ groups presented in the next lemma.
###### Lemma 3.2.
A paratopological group $G$ is projectively $\Psi$ if and only if $G$ is
topologically isomorphic to a subgroup of a product of Hausdorff
paratopological groups of countable pseudocharacter.
###### Proof.
The sufficiency is evident, so we verify only the necessity. Assume that $G$
is a projectively $\Psi$ paratopological group. Applying [6, Proposition 2] we
conclude that $G$ is Hausdorff. Let $\\{U_{\alpha}:\alpha\in A\\}$ be a
neighborhood base at the identity $e$ of $G$. For every $\alpha\in A$, take an
open neighborhood $V_{\alpha}$ of $e$ in $G$ such that $V_{\alpha}^{2}\subset
U_{\alpha}$. By our assumption, there exists a continuous homomorphism
$f_{\alpha}$ of $G$ onto a Hausdorff paratopological group $H_{\alpha}$ of
countable pseudocharacter such that $f_{\alpha}^{-1}(e_{\alpha})\subseteq
V_{\alpha}$, where $e_{\alpha}$ is the identity of $H_{\alpha}$. Let
$N_{\alpha}$ be the kernel of $f_{\alpha}$ and $p_{\alpha}\colon G\to
G/N_{\alpha}$ be the quotient homomorphism. Clearly there exists a continuous
one-to-one homomorphism $j_{\alpha}\colon G/N_{\alpha}\to H_{\alpha}$
satisfying $f_{\alpha}=j_{\alpha}\circ p_{\alpha}$. Thus $j_{\alpha}$ is a
continuous bijection of $G/N_{\alpha}$ onto $H_{\alpha}$ and, hence, the
quotient group $G/N_{\alpha}$ is Hausdorff and has countable pseudocharacter.
Denote by $p$ the diagonal product of the family $\\{p_{\alpha}:\alpha\in
A\\}$. Then $p$ is a continuous homomorphism of $G$ to $P=\prod_{\alpha\in
A}G/N_{\alpha}$. We claim that $p$ is an isomorphic topological embedding.
First, the kernel of $p$ is trivial. Indeed, if $x\in G$ and $x\neq e_{G}$,
take a neighborhood $U$ of $e_{G}$ such that $x\notin U$. There exists
$\alpha\in A$ such that $p_{\alpha}^{-1}(e_{\alpha})\subseteq
U_{\alpha}\subseteq U$, whence it follows that $p_{\alpha}(x)\neq e_{\alpha}$.
Hence $p(x)\neq e_{H}$ and $p$ is injective.
Further, let $U$ be an arbitrary neighborhood of $e$ in $G$. Take $\beta\in A$
with $U_{\beta}\subseteq U$. Then
$f_{\beta}^{-1}f_{\beta}(V_{\beta})=V_{\beta}N_{\beta}\subseteq
V_{\beta}^{2}\subseteq U_{\beta}\subseteq U$. Since $j_{\beta}$ is one-to-one,
it follows from $f_{\beta}=j_{\beta}\circ p_{\beta}$ that
$p_{\beta}^{-1}p_{\beta}(V_{\beta})=f_{\beta}^{-1}f_{\beta}(V_{\beta})\subseteq
U$. Thus the open neighborhood $p_{\beta}(V_{\beta})$ of the identity in
$H_{\beta}$ satisfies $p_{\beta}^{-1}p_{\beta}(V_{\beta})\subseteq U$. Denote
by $\pi_{\beta}$ the projection of $\prod_{\alpha\in A}G/N_{\alpha}$ to the
factor $G/N_{\beta}$. Applying the equality $p_{\beta}=\pi_{\beta}\circ p$ we
deduce that the open neighborhood
$W=p(G)\cap\pi_{\beta}^{-1}p_{\beta}(V_{\beta})$ of the identity in $p(G)$
satisfies $p^{-1}(W)=p_{\beta}^{-1}(p_{\beta}(V_{\beta}))\subseteq U$. We have
thus proved that for every neighborhood $U$ of the identity in $G$, there
exists a neighborhood $W$ of the identity in $p(G)$ such that
$p^{-1}(W)\subseteq U$. This property of the continuous monomorphism $p$
implies that $p$ is an isomorphic topological embedding of $G$ into
$\prod_{\alpha\in A}G/N_{\alpha}$ (see also [1, Section 3.4]). ∎
A paratopological group $G$ is called _$\omega$ -balanced_ if for any
neighborhood $U$ of the identity in $G$, there exists a countable family
$\mathcal{V}$ of neighborhoods of the identity in $G$ such that for each $x\in
G$, one can find $V\in\mathcal{V}$ satisfying $xVx^{-1}\subseteq U$.
The _Hausdorff number_ $Hs(G)$ of a Hausdorff paratopological group $G$ is the
smallest cardinal $\kappa\geq 1$ such that for every neighborhood $U\subseteq
G$ of the identity in $G$, there exists a family
$\\{V_{\alpha}\\}_{\alpha\in\kappa}$ of neighborhoods of the identity in $G$
such that $\bigcap_{\alpha\in\kappa}V_{\alpha}V_{\alpha}^{-1}\subseteq U$. The
Hausdorff number was introduced and studied in [7].
According to [7, Theorem 2.7], every $\omega$-balanced paratopological group
$G$ with $Hs(G)\leq\omega$ is a subgroup of a Tychonoff product of first-
countable Hausdorff paratopological groups. By Lemma 3.2, the latter implies
that the $\omega$-balanced paratopological groups with countable Hausdorff
number are projectively $\Psi$.
###### Theorem 3.3.
Let $\mathcal{C}$ be a subclass of the class of continuous homomorphisms from
pseudocompact topological groups to paratopological groups. If a
paratopological group $G$ is projectively $\Psi$, then its
$\mathcal{C}$-completion $\mathcal{C}[G]$ is Hausdorff. If $G$ is
$\omega$-balanced and satisfies $Hs(G)\leq\omega$, then the paratopological
group $\mathcal{C}[G]$ is $\omega$-balanced, Hausdorff, and satisfies
$Hs(\mathcal{C}[G])\leq\omega$.
###### Proof.
Let $G$ be a projectively $\Psi$ paratopological group. By Lemma 3.2, $G$ is
topologically isomorphic to a subgroup of a Tychonoff product
$Y=\prod_{\alpha\in A}Y_{\alpha}$ of Hausdorff paratopological groups of
countable pseudocharacter. We claim that the paratopological group $Y$ is
$\mathcal{C}$-complete. Consider any homomorphism $f\colon Z\to Y$ in the
class $\mathcal{C}$. By the choice of $\mathcal{C}$, $Z$ is a pseudocompact
topological group. For every $\alpha\in A$, denote by $\pi_{\alpha}\colon Y\to
Y_{\alpha}$ the natural projection and consider the continuous homomorphism
$f_{\alpha}=\pi_{\alpha}\circ f\colon Z\to Y_{\alpha}$. Let $e_{\alpha}$ be
the identity of the group $Y_{\alpha}$. Clearly
$Z_{\alpha}=f_{\alpha}^{-1}(e_{\alpha})$ is a closed invariant subgroup of $Z$
and the quotient topological group $Z/Z_{\alpha}$ admits a continuous
injective homomorphism to the group $Y_{\alpha}$ of countable pseudocharacter.
Hence $Z/Z_{\alpha}$ has countable pseudocharacter as well and we can apply
[8, Proposition 2.3.12] to conclude that $Z/Z_{\alpha}$ is a compact
metrizable group. Let $q_{\alpha}\colon Z\to Z/Z_{\alpha}$ be the quotient
homomorphism and $g_{\alpha}\colon Z/Z_{\alpha}\to Y_{\alpha}$ be a unique
continuous injective homomorphism such that $f_{\alpha}=g_{\alpha}\circ
q_{\alpha}$. Since the compact topological group $Z/Z_{\alpha}$ is Raĭkov
complete, the quotient homomorphism $q_{\alpha}\colon Z\to Z/Z_{\alpha}$
admits a continuous homomorphic extension
$\breve{q}_{\alpha}\colon\breve{Z}\to Z/Z_{\alpha}$. Then
$\breve{f}_{\alpha}=g_{\alpha}\circ\breve{q}_{\alpha}\colon\breve{Z}\to
Y_{\alpha}$ is a continuous extension of the homomorphism $f_{\alpha}$.
The diagonal product of the homomorphisms
$\breve{f}_{\alpha}\colon\breve{Z}\to Y_{\alpha}$ with $\alpha\in A$ is a
continuous homomorphism $\breve{f}\colon\breve{Z}\to Y=\prod_{\alpha\in
A}Y_{\alpha}$ extending the homomorphism $f$ and witnessing that the
paratopological group $Y$ is $\mathcal{C}$-complete. By Proposition 3.1, the
$\mathcal{C}$-completion $\mathcal{C}[X]$ of $X$ is Hausdorff.
Now assume that the paratopological group $G$ is $\omega$-balanced and
satisfies $Hs(G)\leq\omega$. By [7, Theorem 2.7], $G$ is a subgroup of a
Tychonoff product $Y=\prod_{\alpha\in A}Y_{\alpha}$ of first-countable
Hausdorff paratopological groups. Hence $G$ is projectively $\Psi$, by Lemma
3.2. By the above argument, the paratopological group $Y$ is
$\mathcal{C}$-complete and, hence, $\mathcal{C}[G]$ can be identified with a
subgroup of $Y$. By Propositions 2.1–2.3 in [7], the subgroup $\mathcal{C}[G]$
of the Tychonoff product $Y$ of first-countable Hausdorff paratopological
groups is $\omega$-balanced and satisfies $Hs(\mathcal{C}[G])\leq\omega$. ∎
A topological group $G$ is called precompact if its Raĭkov completion
$\rho{G}$ is compact. This happens if and only if for any neighborhood $U$ of
the identity in $G$, there exists a finite set $F\subseteq G$ such that
$G=UF=FU$.
###### Proposition 3.4.
Let $\mathcal{C}$ be a subclass of the class of continuous homomorphisms from
precompact topological groups to paratopological groups. A Hausdorff
paratopological group $G$ is $\mathcal{C}$-complete if and only if for any
homomorphism $h\colon X\to G$ in the class $\mathcal{C}$, the image $h[X]$ has
compact closure in $G$.
###### Proof.
_Sufficiency._ Take any homomorphism $h\colon X\to G$ in the class
$\mathcal{C}$ and assume that the image $h[X]$ has compact closure
$\overline{h[X]}$ in $G$. It follows from [1, Proposition 1.4.10] that
$H=\overline{h[X]}$ is a Hausdorff compact topological semigroup and $H$,
being a subsemigroup of a group, is cancellative. By Numakura’s theorem (see
[5] or [1, Theorem 2.5.2]), $H$ is a compact topological group. Since $H$ is
Raĭkov complete, the homomorphism $h\colon X\to H$ has a continuous
homomorphic extension $\breve{h}\colon\breve{X}\to H\subseteq G$, witnessing
that the paratopological group $G$ is $\mathcal{C}$-complete.
_Necessity._ Assume that the paratopological group $G$ is
$\mathcal{C}$-complete. Then every homomorphism $h\colon X\to G$ in the class
$\mathcal{C}$ has a continuous homomorphic extension
$\breve{h}\colon\breve{X}\to G$ to the Raĭkov completion $\breve{X}=\rho{X}$
of the topological group $X$. The precompactness of $X$ guarantees that the
topological group $\breve{X}$ is compact and so is its image
$\breve{h}[\breve{X}]$ in $G$. Since the space $G$ is Hausdorff, the compact
subspace $\breve{h}[\breve{X}]$ is closed in $G$ and the closure
$\overline{h[X]}$ of $h[X]$ in $G$, being a closed subset of the compact space
$\breve{h}[\breve{X}]$, is compact. ∎
A subset $F$ of a topological space $X$ is called _functionally closed_ (or
else a _zero-set_) if $F=f^{-1}(0)$ for some continuous function $f\colon
X\to\mathbb{R}$. A paratopological group $G$ is said to be _simply
$sm$-factorizable_ if for every functionally closed set $A$ in $G$, there
exists a continuous homomorphism $h\colon G\to H$ onto a separable metrizable
paratopological group $H$ such that $A=h^{-1}[B]$, for some closed set
$B\subseteq H$ (see [1, Definition 5.6]).
A subspace $X$ of a topological space $Y$ is called _$C$ -embedded_ in $Y$ if
each continuous real-valued function on $X$ has a continuous extension over
$Y$.
###### Proposition 3.5.
Every regular simply $sm$-factorizable paratopological group $G$ is
$C$-embedded in its $\mathcal{C}$-completion $\mathcal{C}[G]$ provided
$\mathcal{C}$ is a subclass of the class of continuous homomorphisms of
pseudocompact topological groups to paratopological groups.
###### Proof.
The space $G$ is Tychonoff, by [2, Corollary 5]. We apply [9, Theorem 4.3]
according to which the realcompactification $\upsilon{G}$ of the space $G$
admits the structure of paratopological group containing $G$ as a dense
paratopological subgroup. Since the closure of every pseudocompact subspace in
$\upsilon{G}$ is compact and every pseudocompact topological group is
precompact, the paratopological group $\upsilon{G}$ is $\mathcal{C}$-complete
according to Proposition 3.4. Then
$G\subseteq\mathcal{C}[G]\subseteq\upsilon{G}$, which implies that $G$ is
$C$-embedded in $\mathcal{C}[G]$, being $C$-embedded in $\upsilon{G}$. ∎
Now we present a necessary condition on a paratopological group $G$ for the
Hausdorffness of its $\mathcal{C}$-semicompletion $\mathcal{C}[G)$. In view of
the inclusion $\mathcal{C}[G)\subseteq\mathcal{C}[G]$, the same condition is
necessary for the Hausdorffness of $\mathcal{C}[G]$.
###### Proposition 3.6.
Let $\mathcal{C}$ be a subclass of the class of continuous homomorphisms from
precompact topological groups to paratopological groups. If the
$\mathcal{C}$-semicompletion $\mathcal{C}[G)$ of $G$ is Hausdorff, then for
any homomorphism $h\colon X\to G$ in $\mathcal{C}$, the closure of $h[X]$ in
$G$ is a precompact topological group.
###### Proof.
Let $h\colon X\to G$ be any homomorphism in the class $\mathcal{C}$. By the
definition of a $\mathcal{C}$-semicompletion, the homomorphism $h$ has a
continuous extension $\breve{h}\colon\breve{X}\to\mathcal{C}[G)$. The
precompactness of $X$ ensures that $\breve{X}=\rho{X}$ is a compact
topological group. Assume that the paratopological group $\mathcal{C}[G)$ is
Hausdorff. Then the compact subspace $\breve{h}[\breve{X}]$ of
$\mathcal{C}[G)$ is closed. It follows that $\breve{h}[\breve{X}]$ is a
compact topological group that is topologically isomorphic to a quotient group
of the compact topological group $\breve{X}$. Then the closure of $h[X]$ in
$G$ is a precompact topological group, which is equal to the intersection
$G\cap\breve{h}[\breve{X}]$ of the group $G$ with the compact topological
group $\breve{h}[\breve{X}]\subseteq\mathcal{C}[G)$. ∎
Finally we present an example of a Hausdorff paratopological group $G$ with a
non-Hausdorff $\mathcal{C}$-semicompletion, where $\mathcal{C}$ is the class
of all continuous homomorphisms from sequentially compact topological groups
to paratopological groups. As usual, a space $X$ is _sequentially compact_ if
each sequence in $X$ contains a convergent subsequence. Also, $X$ is said to
be _$\omega$ -bounded_ if the closure in $X$ of every countable set is
compact.
###### Example 3.7.
Let $\mathcal{C}$ be the class of continuous homomorphisms from sequentially
compact topological groups to paratopological groups. There exists a Hausdorff
paratopological abelian group $G$ whose $\mathcal{C}$-semicompletion
$\mathcal{C}[G)$ fails to be a $T_{1}$-space. In addition, $G$ contains a
subgroup $H$ such that $H$ is a sequentially compact $\omega$-bounded
topological group but its closure in $G$ is not a group.
###### Proof.
Let $\mathbb{T}=\\{z\in\mathbb{C}:|z|=1\\}$ be the torus group with its usual
topology and multiplication inherited from the complex plane. So the identity
of $\mathbb{T}$ is $1$. Denote by $\Sigma$ the subgroup of the Tychonoff
product $\mathbb{T}^{\omega_{1}}$ defined as follows:
$\Sigma=\\{x\in\mathbb{T}^{\omega_{1}}:|\\{\alpha\in\omega_{1}:x_{\alpha}\neq
1\\}|\leq\omega\\}.$
According to Corollaries 1.6.33 and 1.6.34 of [1], the space $\Sigma$ is
Fréchet-Urysohn and $\omega$-bounded. Hence $\Sigma$ is sequentially compact.
Take an element $c\in\mathbb{T}^{\omega_{1}}$ of infinite order such that
$\langle c\rangle\cap\Sigma=\\{e\\}$, where $e$ is the identity element of
$\mathbb{T}^{\omega_{1}}$ and $\langle c\rangle$ is the cyclic group generated
by $c$. For every open neighborhood $U$ of $e$ in $\mathbb{T}^{\omega_{1}}$,
we define a subset $O_{U}$ of $G=\Sigma\times\mathbb{Z}$ by letting
$O_{U}=\bigcup_{n\in\omega}\big{(}c^{n}U\cap\Sigma\big{)}\times\\{n\\}.$
Here $\omega$ is identified with the subset $\\{0,1,2,\ldots\\}$ of
$\mathbb{Z}$. A routine verification shows that the sets $O_{U}$ with $U$ as
above, constitute a base at the identity $(e,0)\in G$ for a Hausdorff
paratopological group topology on $G$.
It turns out that the subgroup $H=\Sigma\times\\{0\\}$ of $G$ has the required
properties. Let us show that the closure of $H$ of $G$ is the set
$\Sigma\times(-\omega)$, where $-\omega=\\{0,-1,-2,\ldots\\}$, so this closure
is not a subgroup of $G$. First, we claim that any element $g=(x,k)\in G$ with
$k>0$ is not in the closure of $H$. Indeed, let $U$ be an arbitrary open
neighborhood of $e$ in $\mathbb{T}^{\omega_{1}}$. Then the set $gO_{U}$ is an
open neighborhood of $g$ in $G$ disjoint from $H$.
Further, consider an element $g=(x,-k)\in G$, where $x\in\Sigma$ and
$k\in\omega$. If $k=0$, then $g\in\Sigma\times\\{0\\}=H$. Assume that $k>0$
and take a basic open neighborhood $O_{U}$ of the identity in $G$, where $U$
is an open neighborhood of $e$ in $\mathbb{T}^{\omega_{1}}$. Then
$g\cdot
O_{U}=g\cdot\bigcup_{n\in\omega}\big{(}(c^{n}U\cap\Sigma)\times\\{n\\}\big{)}=\bigcup_{n\in\omega}(xc^{n}U\cap\Sigma)\times\\{n-k\\}.$
Therefore, $g\cdot O_{U}\cap H=g\cdot
O_{U}\cap(\Sigma\times\\{0\\})=(xc^{k}U\cap\Sigma)\times\\{k\\}\neq\emptyset$.
Hence $g$ is in the closure of $H$. We have thus shown that the closure of $H$
in $G$ is the asymmetric subset $\Sigma\times(-\omega)$ of $G$, which implies
the second claim of the example.
The definition of the topology of the paratopological group $G$ implies that
the topology of the group coreflection $G^{\sharp}$ coincides with the product
topology on $\Sigma\times\mathbb{Z}$ and then the Raĭkov completion
$\rho{G}^{\sharp}$ can be identified with the product
$\mathbb{T}^{\omega_{1}}\times\mathbb{Z}$, where $\mathbb{Z}$ is endowed with
the discrete topology. By [3, §2.2], a neighborhood base for the topology of
the Raĭkov completion $\breve{G}$ at its identity consists of the closures
$\overline{O_{U}}$ of the basic neighborhoods $O_{U}$ in the product space
$\mathbb{T}^{\omega_{1}}\times\mathbb{Z}$. It is easy to see that for any
neighborhood $U$ of $e$ in $\mathbb{T}^{\omega_{1}}$, the closure
$\overline{O_{U}}$ of
$O_{U}=\bigcup_{n\in\omega}(c^{n}U\cap\Sigma)\times\\{n\\}$ coincides with the
set $\bigcup_{n\in\omega}(c^{n}\overline{U})\times\\{n\\}$. We see, therefore,
that each neighborhood of the identity in $\breve{G}$ contains the point
$(c,1)$, which means that the Raĭkov completion $\breve{G}$ of $G$ does not
satisfy the $T_{1}$ separation axiom.
It remains to prove that $\breve{G}$ is the $\mathcal{C}$-semicompletion of
$G$. Since the subgroup $\Sigma$ of $\mathbb{T}^{\omega_{1}}$ is sequentially
compact, the continuous homomorphism
$h\colon\Sigma\to\Sigma\times\\{0\\}\subseteq G,\quad h:x\mapsto\langle
x,0\rangle,$
extends to a continuous homomorphism
$\bar{h}\colon\mathbb{T}^{\omega_{1}}\to\mathcal{C}[G)\subseteq\breve{G}$.
Since $\mathbb{T}^{\omega_{1}}$ is a topological group, the homomorphism
$\bar{h}$ remains continuous with respect to the topology of the Raĭkov
completion $\rho{G}^{\sharp}=\mathbb{T}^{\omega_{1}}\times\mathbb{Z}$ of the
topological group $G^{\sharp}$. Now the compactness of
$\mathbb{T}^{\omega_{1}}$ implies that
$\mathbb{T}^{\omega_{1}}\times\\{0\\}=\bar{h}[\mathbb{T}^{\omega_{1}}]\subseteq\mathcal{C}[G)$
and, hence, $\mathcal{C}[G)=\breve{G}$. ∎
## References
* [1] Alexander Arhangel’skii, Mikhail Tkachenko, Topological groups and related structures, Atlantis Press, Paris; World Sci. Publ., NJ, 2008.
* [2] Taras Banakh, Alex Ravsky, Each regular paratopological group is completely regular, Proc. Amer. Math. Soc. 145:3 (2017), 1373–1382.
* [3] Taras Banakh, Alex Ravsky, On feebly compact paratopological groups, Topol. Appl. 284 (2020) 107363.
* [4] Josefa Marín, Salvador Romaguera, A bitopological view of quasi-topological groups, Indian J. Pure Appl. Math. 27:4 (1996), 393–405.
* [5] Katsumi Numakura, On bicompact semigroups, Math. J. Okayama Univ. 1 (1952), 99–108.
* [6] Manuel Sanchis, Mikhail Tkachenko, Completions of paratopological groups and bounded sets, Monatsh. Math. 183 (2017), 699–721.
* [7] Mikhail Tkachenko, Embedding paratopological groups into topological products, Topol. Appl. 156 (2009), 1298–1305.
* [8] Mikhail Tkachenko, Pseudocompact Topological groups, Section 2 in: M. Hrušák, Á. Tamariz-Mascarúa, M. Tkachenko (Eds.), Pseudocompact Topological Spaces, Developments in Mathematics, vol. 55. Springer, Cham, 2018.
* [9] Li-Hong Xie, Mikhail Tkachenko, Simply sm-factorizable (para)topological groups and their completions, Monatsh. Math. 193 (2020), 507–529.
|
# Deriving the intrinsic properties of M51 with radiative transfer models
Christopher J. Inman,1 Cristina C. Popescu,1,2 Mark Rushton 3 and David
Murphy1
1University of Central Lancashire, Jeremiah Horrocks Institute, Preston, PR1
2HE, UK
2 Max Planck Institut für Kernphysik, Saupfercheckweg 1, 69117 Heidelberg,
Germany
3The Astronomical Institute of the Romanian Academy, Str. Cutitul de Argint 5,
Bucharest, Romania E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
A quantitative derivation of the intrinsic properties of galaxies related to
their fundamental building blocks, gas, dust and stars is essential for our
understanding of galaxy evolution. A fully self-consistent derivation of these
properties can be achieved with radiative transfer (RT) methods that are
constrained by panchromatic imaging observations. Here we present an axi-
symmetric RT model of the UV-optical-FIR/submm spectral and spatial energy
distribution of the face-on spiral galaxy M51. The model reproduces reasonably
well the azimuthally averaged radial profiles derived from the imaging data
available for this galaxy, from GALEX, SDSS, 2MASS, Spitzer and Herschel. We
model the galaxy with three distinct morphological components: a bulge, an
inner disc and a main disc. We derive the length parameters of the stellar
emissivity and of the dust distribution. We also derive the intrinsic global
and spatially resolved parameters of M51. We find a faint “outer disc”
bridging M51 with its companion galaxy M51b. Finally, we present and discuss
an alternative model, with dust properties that change within the galaxy.
###### keywords:
radiative transfer – galaxies: disc – galaxies: stellar content – galaxies:
structure – ISM: dust, extinction – galaxies: spiral
††pubyear: 2023††pagerange: Deriving the intrinsic properties of M51 with
radiative transfer models–C
## 1 Introduction
Dust is prevalent within the interstellar medium (ISM) of star-forming
galaxies (Trumpler 1930), and, although it constitutes only $1\%$ of the mass
of this ISM (Greenberg 1963), it plays an important role in the thermodynamic
balance of galaxies (Montier & Giard 2004; Giard et al. 2008; Popescu & Tuffs
2010; Natale et al. 2010), since it regulates the cooling and heating
mechanisms of the interstellar gas (Dwek & Werner 1981).
The interstellar dust also affects our view of galaxies. This is because
stellar photons, primarily in the ultraviolet (UV)/optical range, are
continuously absorbed and scattered by the interstellar dust. Photons that are
scattered can travel long distances within the ISM before they are scattered
again, absorbed or leave the galaxy. Due to scattering, what would be
otherwise an isotropic process related to the emission of stellar light in
galaxies, becomes highly anisotropic. As such, along any line of sight through
a galaxy, the stellar light may be diminished due to extinction, but also
amplified by stellar photons being scattered into the line of sight. Thus, for
a extended distribution of emitters and absorbers, the complex effect of
absorption and scattering processes is what we call attenuation.
The attenuation is dependent not only on the optical properties of the dust
grains (as is the case for the extinction processes), but also on the relative
distribution of stars and dust, and on the orientation of a galaxy with
respect to the observer (Byun et al. 1994; Bianchi et al. 1996; Kuchinski et
al. 1998; Ferrara et al. 1999; Baes & Dejonghe 2001; Tuffs et al. 2004;
Pierini et al. 2004; Fischera & Dopita 2004; Natale et al. 2015). Dust
attenuation affects the observed spatially integrated stellar luminosity of a
galaxy, but also its surface brightness distribution and the geometric
parameters derived from surface-brightness photometry, like scalelength of
discs, effective radii of bulges and Sérsic indexes: Byun et al. (1994); Evans
(1994); Cunow (2001); Möllenhoff et al. (2006); Gadotti et al. (2010); Pastrav
et al. (2013a, b); Savchenko et al. (2023).
Due to dust attenuation the observed UV/optical images of star forming
galaxies are far from providing a direct tracer of the constituent stellar
populations distributed throughout the galaxy. In order to derive the
intrinsic distribution of stars of all ages, and thus to understand how stars
form and evolve in galaxies, one also needs to jointly derive the distribution
of dust in galaxies (Popescu & Tuffs 2010). This can be done by considering
both the absorption and emission properties of interstellar dust. Thus, the
stellar light that is absorbed by dust is re-emitted in the infrared (IR) and
the energy balance between direct and re-radiated stellar light is an
important constraint on the modelling of transfer of radiation in galaxies. In
particular dust emission contains unique information on both the spatial and
spectral energy distribution (SED) of the energy density of the radiation
fields that heat the dust grains, and on the dust properties and dust opacity.
Accounting for both dust attenuation and dust emission is thus needed to
provide a self-consistent model of a galaxy.
Galaxy models that are based on radiative transfer (RT) methods (Kylafis &
Xilouris 2005; Steinacker et al. 2013) and constrained by panchromatic imaging
observations are the most powerful tools in quantifying the attenuation of
stellar light in galaxies and in deriving the intrinsic parameters of galaxies
(Popescu 2021). This type of modelling was initially performed on edge-on
galaxies, since in this orientation it is possible to directly constrain the
vertical distribution of stars and dust from imaging observations. The first
RT modelling of an edge-on galaxy was performed in the optical range, for NGC
891, by Kylafis & Bahcall (1987). This was the first time a RT model was
applied in conjunction with realistic distributions of stars and dust (finite
exponential discs as opposed to slabs or sandwich geometries - Disney et al.
1989 or spherical configurations - Witt et al. 1992; Witt & Gordon 1996,
2000). Fundamental work on characterising the relative distributions of stars
and dust and their statistical trends were later established in a series of
papers dedicated to the modelling of edge-on galaxies in the optical and NIR
by Xilouris et al. (1997, 1998, 1999).
The first panchromatic (UV/optical/FIR/submm) RT modelling of an edge-on
galaxy was achieved for the same edge-on spiral NGC 891 (see above for the
first RT modelling) by Popescu et al. (2000). Further work on modelling edge-
on spirals was performed by Misiriotis et al. (2001); Popescu et al. (2004);
Bianchi (2008); Baes et al. (2010); Popescu et al. (2011); MacLachlan et al.
(2011); Robitaille et al. (2012); De Geyter et al. (2015); Mosenkov et al.
(2016, 2018); Popescu et al. (2017); Natale et al. (2021).
More recently, non edge-on galaxies have also started to be modelled with RT
methods. There are two approaches to this type of modelling. The first
approach is to use non axi-symmetric RT models: De Looze et al. (2014); Viaene
et al. (2017); Williams et al. (2019); Verstocken et al. (2020); Nersesian et
al. (2020a, b). Non axi-symmetric models are, in principle, ideal to capture
the detailed geometry seen face-on. However, due to computational limitations,
they are implemented to only fit the integrated SED, with typically only four
free parameters that describe the total luminosity output for stars and the
dust mass, with the geometry being fixed. The second approach is to use axi-
symmetric RT models (Thirlwall et al. 2020; Rushton et al. 2022) and fit not
only the integrated SED, but also the geometry of stars and dust, albeit with
the assumption of axi-symmetry. So far only non interacting face-on spiral
galaxies have been modelled with axi-symmetric models (M33; Thirlwall et al.
2020, NGC 628; Rushton et al. 2022). The goal of this paper is to extend the
applicability of the model to a non-isolated spiral. We will apply the model
to M51, since, despite its interaction with a close companion, most of the
extent of the galaxy has regular spiral structure, and is not perturbed by the
interaction, thus making it suitable for the axi-symmetry approximation.
First discovered by Charles Messier in 1773, M51 is a system of interacting
galaxies consisting of M51a (=NGC 5194), the ‘Whirlpool Galaxy’ , a grand-
design spiral of Hubble type Sbc, and the smaller lenticular galaxy M51b (=NGC
5195), of Hubble type SB0. While interaction between the two galaxies is
evident from the “bridge” linking them, there is still debate of whether the
galaxies have interacted on a single fly-by collision (Toomre & Toomre 1972;
Durrell et al. 2003) or by multiple close passes (Salo & Laurikainen 2000;
Dobbs et al. 2010). Despite the interaction, as mentioned before, M51a
exhibits a good deal of azimuthal symmetry. This, in conjunction with the
close proximity ($\text{D}=8.58\,\text{Mpc}$; McQuinn et al. 2016) and the
wealth of detailed observational data, makes M51a ideal for studying the
energy balance between absorption by dust and re-emission in the far-infrared
for a non-isolated spiral galaxy that is seen not far from face-on. In the
following we will refer to M51a as M51.
In this work we will utilise the axisymmetric RT models developed in Popescu
et al. (2000), Tuffs et al. (2004) and Popescu et al. (2011), which have been
successfully used to account for the properties of star forming galaxies
derived from statistical samples (e.g. van der Giessen et al. 2022), as well
as for the detailed spatial and spectral energy distribution of individual
nearby galaxies (Popescu et al. 2000; Thirlwall et al. 2020; Rushton et al.
2022) and of the Milky Way (Popescu et al. 2017; Niederwanger et al. 2019;
Natale et al. 2021). These models describe the geometries of the stellar
emissivity and dust opacity from the UV to submm using parameterised analytic
functions. The models consider the absorption and anisotropic scattering of
stellar photons with dust grains of various sizes and chemical compositions,
with the optical constants taken from Draine & Lee (1984) and Draine & Li
(2007). The models also consider the heating of the dust grains and the re-
emission in the infrared, including explicit calculations for stochastic
heating.
Previous work on modelling M51 was done by De Looze et al. (2014) and
Nersesian et al. (2020b) using non-axisymmetric RT models. As discussed above,
these models have the inherent limitation that they cannot self-consistently
derive the geometry of emitters and absorbers. In this study we aim to model
M51, by self-consistently deriving the geometry of stars and dust through
fitting the detailed azimuthally averaged UV/optical/FIR/submm surface
brightness profiles of M51. As we will show here, our axi-symmetric model
produces a good fit to the imaging data within a radial distance of 7 kpc from
the centre of M51, within which the galaxy preserves azimuthal symmetry.
The paper is organised as follows: In Sect. 2 we present the imaging
observations used for constraining the model, and the data reduction steps
undertaken in this study. In Sect. 3 we briefly describe the model and its
geometrical components, with the fitting procedure detailed in Sect. 4. In
Sect. 5 we present the results of the fits for the surface-brightness
distributions and for the intrinsic parameters of M51, including the SFR, dust
optical depth, dust mass, the fractional contribution of the different stellar
populations in heating the dust and the attenuation curve. We also present the
results for the derived morphological components of M51. In Sect. 6 we compare
our model for M51 with previous RT modelling of this galaxy and discuss an
alternative to the standard model presented in this paper. We summarise our
main results in Sect. 7.
## 2 Observational Data
In this section we discuss the data set of M51 used in this study. To start
with, basic parameters like inclination and position angle were difficult to
determine with standard methods, and, because of this, there has been a lot of
uncertainty in the value of these parameters. Determinations of the
inclination angle vary from $15^{\circ}$ to $42^{\circ}$ (Danver 1942; Toomre
& Toomre 1972; Tully 1974; Bersier et al. 1994; Monnet et al. 1981; Considere
& Athanassoula 1982; Shetty et al. 2007; Hu et al. 2013), while determinations
for the PA vary from $-10^{\circ}$ to $36^{\circ}$ (Burbidge & Burbidge 1964;
Tully 1974; Monnet et al. 1981; Considere & Athanassoula 1982; Shetty et al.
2007; Tamburro et al. 2008; Walter et al. 2008; Hu et al. 2013). We adopted
the most recent values from Hu et al. (2013). The distance, inclination,
position angle and central coordinates used in this paper are listed in Table
1. We obtained panchromatic images from various missions, as summarised in
Table 2.
Table 1: Distance, inclination, position angle and coordinates adopted for the modelling of M51 (M51/NGC 5194), together with the relevant reference. Distance | $8.58\pm 0.1$ Mpc | McQuinn et al. (2016)
---|---|---
Inclination | $20.3^{\circ}\pm 2.8^{\circ}$ | Hu et al. (2013)
Position Angle | $12.0^{\circ}\pm 2.5^{\circ}$ | Hu et al. (2013)
Right Ascension | $13^{\text{h}}\,29^{\text{m}}\,57.11^{\text{s}}$ | Turner & Ho (1994)
Declination | $47^{\circ}\,11^{\prime}\,\,\,42.62^{\prime\prime}$ | Turner & Ho (1994)
All data was found on NASA/IPAC Extragalactic Database
(NED)111https://ned.ipac.caltech.edu, except for the GALEX data, which was
obtained from the Mikulski Archive for Space Telescopes
(MAST)222https://archive.stsci.edu/prepds/gcat/. Before analysis, the units of
the surface brightness maps were converted into [MJy sr-1].
Table 2: A summary of the observational data used in this study. The following references were used: (1) Gil de Paz et al. (2007), (2) Brown et al. (2014), (3) Bendo et al. (2012). Telescope | Filter/ | $\lambda_{0}$ [$\mu$m] | Pixel scale | $\dfrac{\epsilon_{\text{cal}}}{F_{\nu}}$[%] | $\sigma_{\text{bg}}$ $\left[\rm{kJy\,sr^{-1}}\right]$ | Band name
---|---|---|---|---|---|---
| Instrument | | [′′] | | |
GALEX1 | FUV | 0.1542 | 1.5 | 10 | 0.0019 | FUV
GALEX1 | NUV | 0.2274 | 1.5 | 10 | 0.0024 | NUV
SDSS2 | u | 0.3562 | 0.396 | 2 | 0.049 | u
SDSS2 | g | 0.4719 | 0.396 | 2 | 0.017 | g
SDSS2 | r | 0.6185 | 0.396 | 2 | 0.028 | r
SDSS2 | i | 0.7500 | 0.396 | 2 | 0.049 | i
2MASS2 | J | 1.2000 | 1.0 | 3 | 0 | J
2MASS2 | K${}_{\text{s}}$ | 2.2000 | 1.0 | 3 | 0 | K${}_{\text{s}}$
Spitzer2 | IRAC | 3.5070 | 0.75 | 3 | 0 | I1
Spitzer2 | IRAC | 4.4370 | 0.75 | 3 | 0 | I2
Spitzer2 | IRAC | 5.7390 | 0.75 | 3 | 0.27 | I3
Spitzer2 | IRAC | 7.9270 | 0.75 | 3 | 0 | I4
Spitzer2 | MIPS | 24.000 | 1.5 | 1 | 0 | MIPS24
Herschel3 | PACS | 70.000 | 1.4 | 10 | 20.0 | PACS70
Herschel3 | PACS | 160.00 | 2.85 | 20 | 0 | PACS160
Herschel3 | SPIRE | 250.00 | 6.0 | 15 | 0 | SPIRE250
Herschel3 | SPIRE | 350.00 | 8.0 | 15 | 0 | SPIRE350
Herschel3 | SPIRE | 500.00 | 12.0 | 15 | 0 | SPIRE500
### 2.1 Background removal
We performed a curve of growth analysis to determine the radius from the
centre of M51 at which the emission of the galaxy falls below the background
level. This was found to be 25 kpc. We also visually inspect each map to
ensure that the emission associated with M51 does not extend beyond this
point. The outer sampled radius (30 kpc) was chosen to ensure a large area for
measuring the background. This sampling region extended beyond the limit of
some maps, thus the full sampling area could not be used in those cases.
However, even in the most severe cases, this was a negligible portion of the
total sampling area. We then calculate the average background flux by taking
the mean surface brightness through 6 annuli of equal width contained in the
region $25\,\text{kpc}\leq R\leq 30\,\text{kpc}$. Although M51 is nearly face-
on with an inclination of 20.3∘, the interaction with M51b gives rise to some
stripped material in the outer disc, making the outer isophotes more elongated
than the inner ones. This change in isophotal shape was taken into account
when defining the background. Thus, it was found that the best fitting annuli
for the background had an inclination of 40∘ (and the same position angle as
the galaxy of 12∘).
### 2.2 Foreground star removal
Foreground stars are visible in the UV/optical/NIR wavelengths and can
influence the measured values of emission from M51. It is therefore important
to mask these stars. For this we first produce a median map where each pixel
of the median map is equal to the median of the $3\times 3$ pixels around it.
A bias map is also produced, that is principally the same as the median map,
but takes the median of the $21\times 21$ pixels. This bias map is used as an
additional check to mitigate against false positives when masking noisy data.
Pixels with values that exceed both the median and bias maps are then masked.
After this initial pass, we then manually inspect each mask for regions that
may have been missed, or unmask false positives, such as HII regions. HII
regions appear as point sources (similarly to foreground stars) and thus are
often falsely masked by this automated process. These HII regions were
identified by eye and the catalogue of Lee et al. (2011). We also masked the
companion galaxy M51b and it’s diffuse emission.
### 2.3 Convolutions
The SDSS, 2MASS and Spitzer data have a higher resolution than that of our
model, which would correspond to 50 pc at the distance of M51. We therefore
degrade the resolution of these data to match the resolution of our model
using a two dimensional Gaussian kernel. The data at all the other wavelengths
used in this study have lower resolutions, therefore, for these wavelengths,
we degrade the resolution of our model to match the corresponding ones from
the data. For this we used the kernels of Aniano et al. (2011) to convolve the
model with the PSF of the observational instrument.
### 2.4 GALEX
The Galaxy Evolution Explorer (GALEX; Martin et al. 2005) was a NASA Small
Explorer mission that performed an all-sky survey in the far-ultraviolet (FUV
$\sim 0.15\,\mu$m) and near-ultraviolet (NUV $\sim 0.22\,\mu$m). Both FUV and
NUV data are used in this study and were found on the Mikulski Archive for
Space Telescopes (MAST) 333https://galex.stsci.edu/GR6/ and made available by
Gil de Paz et al. (2007). The NUV and FUV data has a pixel scale of 1.5′′ (62
pc) per pixel and a FWHM of 4.2′′ and 5.3′′, respectively. The flux
calibration error is $\frac{\epsilon_{\text{cal}}}{F_{\nu}}$ = 10% following
Morrissey et al. (2007).
### 2.5 SDSS
The Sloan Digital Sky Survey (SDSS; York et al. 2000) has observed M51 with
two sets of CCD arrays. Each array has an integration time of 51 s (York et
al. 2000) capturing images through u, g, r, i and z filters. In this study we
use the u, g, r, and i SDSS bands, with images obtained from NED/IPAC and
processed by Brown et al. (2014). We do not use the z image as this wavelength
is not included in our model fitting. Each band has a pixel scale of 0.396′′
(16 pc) and FWHM of 1′′ (Gunn et al. 2006). Each band has a 2% rms photometric
calibration in all bands (Aihara et al. 2011). The u band required a
correction of -0.04 mag (Brown et al. 2014).
### 2.6 2MASS
The Two Micron All Sky Survey (2MASS; Jarrett et al. 2003) was performed in
the NIR bands, J, H and K${}_{\text{s}}$, and observed M51 as part of the
Large Galaxy Atlas Survey. We use the 2MASS data in the J and K${}_{\text{s}}$
bands provided on NED/IPAC, and processed by Brown et al. (2014). We do not
use the H band data as this wavelength is not included in our model. Both J
and K${}_{\text{s}}$ bands have a pixel scale of 1′′ (41 pc) and a FWHM of
2.5′′ (Skrutskie et al. 2006). The calibration errors for J and
K${}_{\text{s}}$ are 2%-3% (Jarrett et al. 2003).
### 2.7 Spitzer
#### 2.7.1 IRAC
The Spitzer Space Telescope contains the Infrared Array Camera (IRAC). IRAC is
a four channel camera that images the 3.6 $\mu$m, 4.5 $\mu$m, 5.8 $\mu$m and
8.0 $\mu$m bands simultaneously. In this study we use all four bands. The
images were found on NED/IPAC and were processed by Brown et al. (2014). Each
instrument has a pixel size of 0.75′′ (31 pc). The 3.6 $\mu$m, 4.5 $\mu$m,
5.8$\,\mu$m, and 8.0 $\mu$m data has a FWHM of 1.66′′, 1.72′′, 1.88′′, and
1.98′′, respectively (Fazio et al. 2004). The flux calibration error for the
IRAC data is 3% (Fazio et al. 2004).
#### 2.7.2 MIPS
The Multiband Imaging Photometer for Spitzer (MIPS; Rieke et al. 2004)
conducted imaging in three wavebands: 24 $\mu$m, 70 $\mu$m and 160 $\mu$m. In
this study we only use the 24 $\mu$m data, since at 70 $\mu$m and 160 $\mu$m
we use the higher quality data from Herschel. The data was processed by Brown
et al. (2014) and made available at NED/IPAC. MIPS24 has a pixel scale of
1.5′′ (62 pc) and a FWHM of 6′′ (Rieke et al. 2004). The calibration error is
1% (Rieke et al. 2004).
### 2.8 Herschel
#### 2.8.1 PACS
M51 is observed by the Photo Array Camera & Spectrometer (PACS; Poglitsch et
al. 2010) as part of the Very Nearby Galaxy Survey (VNGS) at the 70 $\mu$m and
160 $\mu$m bands. Both bands are utilised in this study and images were
obtained from NED/IPAC. The pixel scale of the 70 $\mu$m and 160 $\mu$m data
is 1.4′′ (58 pc) and 2.85′′ (118 pc), respectively. The FWHM of the 70 $\mu$m
and 160 $\mu$m data are 5.6′′ and 11.4′′, respectively (Bendo et al. 2012).
#### 2.8.2 SPIRE
The Spectral and Photometric Imaging Receiver (SPIRE; Griffin et al. 2010)
observed M51 as part of the Very Nearby Galaxy Survey (VNGS) at the 250
$\mu$m, 350 $\mu$m and 500 $\mu$m bands. All three bands were used in this
study and images were obtained from NED/IPAC and processed by Bendo et al.
(2012). The pixel scale of the 250 $\mu$m, 350 $\mu$m and 500 $\mu$m data is
6′′ (250 pc), 8′′ (333 pc), and 12′′ (500 pc), respectively. The FWHM of SPIRE
data for 250 $\mu$m, 350 $\mu$m, 500 $\mu$m are 18.2′′, 24.5′′, and 36′′,
respectively (Bendo et al. 2012). Flux calibration error is accurate to $\pm
15\%$(Griffin et al. 2010).
Examples of imaging data used in this analysis are given in the left panels of
Fig. 1. They show 15x15 kpc images of M51 in the GALEX NUV, 2MASS K band, and
the SPIRE 500 ${\mu}$m.
Figure 1: The observed (left), azimuthally averaged observed (middle) and
model (right) images of M51 of various wavebands. The emission from M51b is
masked out from the figure. We indicate on each image elliptical apertures
(green) with radial distances of 0.8 and 2.7 kpc, which correspond to the
inner radius of the inner and main disc of our model, respectively. We also
plot in white the elliptical aperture corresponding to $R_{l}=7$ kpc, the
radius within which the assumption of axi-symmetry holds. The masked
foreground stars are indicated by the white dots.
### 2.9 Photometry
The observed images were corrected for Galactic extinction using a value of
$E(B-V)=0.0308$ (Schlafly & Finkbeiner 2011) and $R_{V}=3.1$ (Fitzpatrick
1999). Azimuthally averaged surface brightness profiles and spatially
integrated flux densities $F_{\nu}$ were derived using curve of growth (CoG)
photometry. Each map was segmented into 250 annuli of a fixed width in which
the average brightness was calculated, with the corresponding radius taken as
the midpoint of the annulus. The elliptical annuli were defined by the
inclination and position angle of M51, as listed in Table 1. Examples of
azimuthally averaged images (NUV, K, and 500 ${\mu}$m) produced in this way
are given in the middle panels of Fig. 1. Comparison between azimuthally
averaged and corresponding original observed images shows that the assumption
of axi-symmetry seems to hold up to $\sim 7$ kpc, which we take to represent
the limit of applicability of our axi-symmetric model. We will refer to this
limit as $R_{l}$.
The photometric errors due to background fluctuations, calibration and
configuration noise were derived using the procedure from Thirlwall et al.
(2020).
## 3 The Model
Our model for M51 follows the axisymmetric RT models of Popescu et al. (2011),
hereafter PT11, which describe the dust opacity and stellar emissivity for
spiral galaxies from the UV to submm wavelengths using parameterised analytic
functions. The model considers the absorption and anisotropic scattering of
stellar photons with dust grains of various size and chemical composition. A
dust composition consisting of silicates, graphites and PAH molecules was
used, with a grain size distribution from Weingartner & Draine (2001), and
optical constants from Draine & Lee (1984) and Draine & Li (2007). As in our
previous work on modelling spiral galaxies, we consider a Milky Way-type dust,
since this has proven to provide consistent results for all modelled galaxies.
The parameters describing the grain size distribution and grain mixture are
taken from Table 1, line 7 of Weingartner & Draine (2001). The RT model of M51
also incorporates explicit calculations for the stochastic heating of the dust
grains.
The RT codes used for the modelling of M51 were the PT11 code and the DART-RAY
code (Natale et al. 2014, 2015; Natale et al. 2017). The PT11 code is a
modified version of the code of Kylafis & Bahcall (1987). Both the PT11 and
DART-RAY codes use ray-tracing algorithms.
Following Thirlwall et al. (2020), we adapt the model of PT11 to allow for
additional morphological components. In the case of M51 it was found that, in
addition to the bulge, an inner and a main disc component were present out to
$R_{l}$. Each morphological component comprises the generic stellar and dust
components from PT11: the stellar disc, the thin stellar disc, the dust disc
and thin dust disc. The stellar disc is made up of all the stars that are old
enough to have had time to migrate and form a thicker configuration (a few
hundred parsec scale height), while the thin stellar disc is made up by young
stars, that are still spatially closely related to the molecular layer from
which they formed (50-100 parsec scale-height). A clumpy component associated
with the star forming clouds is also incorporated in the model. All these are
described in detail in PT11. Following Thirlwall et al. (2020) we also found
the need to alter the analytic functions describing the exponential discs, to
include an inner truncation radius. Thus, we use Eq. (1) and (2) from
Thirlwall et al. (2020) to describe the stellar volume emissivity and the dust
density distribution for each disc component j:
$w_{\text{j}}(R,z)=\begin{cases}\\!0,&\text{if }R<R_{\text{tin,j}}\\\ \\\
\begin{aligned}
A_{0,\text{j}}\left[\frac{R}{R_{\text{in,}\text{j}}}\left(1-\chi_{\text{j}}\right)+\chi_{\text{j}}\right]\\\
\times\text{exp}\left(-\frac{R_{\text{in},\text{j}}}{h_{\text{j}}}\right)\text{exp}\left(-\frac{z}{z_{\text{j}}}\right),\end{aligned}&\text{if}R_{\text{tin,j}}\leq
R<R_{\text{in,j}}\\\ \\\
A_{0,\text{j}}\times\text{exp}\left(-\frac{R_{\text{in},\text{j}}}{h_{\text{j}}}\right)\text{exp}\left(-\frac{z}{z_{\text{j}}}\right),&R_{\text{in},\text{j}}\leq
R\leq R_{\text{t},\text{j}}\end{cases}$ (1)
with:
$\displaystyle\chi_{\text{j}}=\frac{w_{\text{j}}(0,z)}{w_{\text{j}}(R_{\text{in},\text{j}},z)}$
(2)
where $R$ and $z$ are the radial and vertical coordinates, $h_{\text{j}}$ and
$z_{\text{j}}$ are the scale-length and scale-height respectively,
$A_{0,\text{j}}$ is a constant scaling the amplitude, $\chi_{\text{j}}$
describes a linear slope of the radial distributions from the inner radius,
$R_{\rm{in}}$, to the centre of the galaxy. $R_{\text{tin}}$ and
$R_{\text{t}}$ are the inner and outer truncation radii.
Eqs. 1 and 2 are wavelength dependent. The spatial integration of these
formulae (Eqs. B1-B3 from Thirlwall et al. 2020) between $R_{\text{tin}}$ and
$R_{\text{t}}$ provides the intrinsic stellar luminosity $L_{\nu}$ if j
represents a stellar disc component, or the dust mass $M_{\rm d}$ if j
represents a dust disc component.
The emissivity of the bulge is described by the Sérsic distribution
$w_{\text{bulge}}(\lambda,R,z)=w(\lambda,0,0)\,\sqrt{\frac{b_{\rm
s}}{2\pi}}\,\frac{(a/b)}{R_{\rm e}}\,\eta^{(1/2n_{\rm s})-1}\exp{(-b_{\rm
s}\,\eta^{1/n_{\rm s}})}$ (3)
with:
$\eta(\lambda,R,z)=\frac{\sqrt{R^{2}+z^{2}(a/b)^{2}}}{R_{\rm e}}$ (4)
$\eta^{\text{bulge}}(\lambda,0,0)$ is the stellar emissivity at the centre of
the bulge, $R_{\text{e}}$ is the effective radius of the bulge, $a$ and $b$
are the semi-major and semi-minor axes of the bulge, respectively, and
$b_{\text{s}}$ is a constant which depends on the value of the Sérsic index
$n_{\text{s}}$ (see Eq. 8 from Natale et al. 2021). Following Thirlwall et al.
(2020), the parameters of the different components are as follows:
The stellar disc:
The stellar disc is made up of the old stellar population, defined as a
population of stars that had time to migrate from their molecular layer into a
thicker configuration (a few hundred parsec scale height). Following from
Popescu et al. (2000) and PT11, we make the approximation that the stellar
disc only emits in the optical/NIR (no UV counterpart). It is described by the
four geometrical parameters $h_{\rm{s}}^{\text{disc}}$,
$z_{\rm{s}}^{\text{disc}}$, $R_{\rm{in,s}}^{\text{disc}}$ and
$\chi_{\rm{s}}^{\text{disc}}$, and one amplitude parameter $L^{\text{disc}}$.
The free parameters are $h_{\rm{s}}^{\text{disc}}$,
$\chi_{\rm{s}}^{\text{disc}}$ and $L^{\text{disc}}(\lambda)$.
$z_{\rm{s}}^{\text{disc}}$ is fixed from the model, and
$R_{\rm{in,s}}^{\text{disc}}$ is fixed from the data. We allow
$L^{\text{disc}}$ to be a free parameter for each wavelength corresponding to
u, g, r, i, J, K, I1, I2, I3. We also allow $h_{\rm{s}}$ to be wavelength
dependent, hence we consider the scale-lengths for the g, r, i, J, K, I1, I2,
I3 bands to be free parameters. The scale-length in the u-band is taken to be
the same as in the g-band. The model of M51 includes an inner and a main
stellar disc.
The thin stellar disc
The thin stellar disc is made up of the young stellar population of stars that
did not have time to migrate from their molecular layer (50-100 pc). This disc
emits predominantly in the UV, although we assume a smooth decrease of its
emission in the optical/NIR. It is described by the four geometrical
parameters $h_{\rm{s}}^{\text{tdisc}}$, $z_{\rm{s}}^{\text{tdisc}}$,
$R_{\rm{in},s}^{\text{tdisc}}$ and $\chi_{\rm{s}}^{\text{tdisc}}$, and one
amplitude parameter $L^{\text{tdisc}}$. The free parameters are
$h_{\rm{s}}^{\text{tdisc}}$, $\chi_{\rm{s}}^{\text{tdisc}}$ and
$L^{\text{tdisc}}$. $z_{\rm{s}}^{\text{tdisc}}$ is fixed from the model and
$R_{\rm{in},s}^{\text{tdisc}}$ is fixed from the data. The scale-length of
this disc is assumed to be wavelength independent. The wavelength dependence
of $L^{\text{tdisc}}$ is fixed from the model. The model of M51 includes an
inner and a main thin stellar disc.
Keeping in line with previous modelling, we express the spectral integrated
luminosity of the young stellar disc $L^{\rm tdisc}$ in terms of a star
formation rate ${\rm SFR}$, using equations (16), (17), and (18) from PT11.
The dust disc
It is described by the geometrical parameters $h^{\rm{disc}}_{\rm{d}}$,
$z_{\rm{d}}^{\rm{disc}}$, $R_{\rm{in,d}}^{\rm{disc}}$ and
$\chi_{\rm{d}}^{\rm{disc}}$ and the amplitude parameter, the face-on optical
depth of the B-band at the inner radius
$\tau_{\rm{B}}^{\rm{f,disc}}\left(R_{\rm{in,d}}^{\rm{disc}}\right)$. The free
parameters are $h^{\rm{disc}}_{\rm{d}}$, $\chi_{\rm{d}}^{\rm{disc}}$ and
$\tau_{\rm{B}}^{\rm{f,disc}}\left(R_{\rm{in,d}}^{\rm{disc}}\right)$.
$z_{\rm{d}}^{\rm{disc}}$ is fixed from the model and
$R_{\rm{in,d}}^{\rm{disc}}$ is fixed from the data. Our model of M51 contains
an inner, and main dust disc.
The thin dust disc
The thin dust disc is described by the geometrical parameters
$h^{\rm{tdisc}}_{\rm{d}}$, $z_{\rm{d}}^{\rm{tdisc}}$,
$R_{\rm{in,d}}^{\rm{tdisc}}$ and $\chi_{\rm{d}}^{\rm{tdisc}}$ and the
amplitude parameter, the face-on optical depth of the B-band at the inner
radius $\tau_{\rm{B}}^{\rm{f,tdisc}}\left(R_{\rm{in,d}}^{\rm{tdisc}}\right)$.
Following PT11 we fix the geometric parameters of the thin dust disc to that
of the thin stellar disc. The model of M51 contains an inner, and main thin
dust disc.
The clumpy component
The clumpy component represents the dense dusty clouds associated with star
forming regions that are illuminated by the strong, UV-dominated radiation
fields produced by the photons emitted from the young stars inside the clouds.
The dust in these regions reaches thermal equilibrium with the intense
radiation fields and emits dominantly in the 24 $\mu$m to 70 $\mu$m region,
often surpassing the emission due to the diffuse component at these
wavelengths. Following PT11 we define a “clumpiness factor” $F$, which is the
amplitude parameter of the clumpy component, and represents the fraction of
the total luminosity of massive stars that is locally absorbed by the dust in
these star-forming clouds. The diffuse component is then illuminated by the
escape fraction $1-F$. The $F$ factor has a wavelength dependence that is not
determined by the optical properties of the dust grains, but by the
evolutionary stage of the birth clouds. This is because in our model the dust
cloud is very opaque, thus completely blocking any radiation from inside the
cloud. The only way photons escape is through holes in the cloud, produced by
the fragmentation in the cloud. The wavelength dependence of the escaping
radiation thus arises because lower mass and less blue stars spend a higher
proportion of their lifetime radiating when they are further away from their
birth-clouds, and because of the progressive fragmentation of the clouds. The
formalism is described in Tuffs et al. (2004). The dust emission SED of the
birth clouds is taken to follow the prescription from PT11.
## 4 Fitting the model
To directly compare the model to the data we produce azimuthally averaged
surface brightness profiles of both. The observed surface-brightness profiles
were obtained by carrying out curve of growth photometry on the maps listed in
Table 2, as described in Sect. 2. For the model profiles we carried out an
identical procedure. Examples of these profiles can be seen in Figs. 2-3.
Examples of model images are given in the right panels of Fig. 1.
From an initial inspection of the surface brightness profiles of the
observational data it was clear that a single disc described by Eqs. 1-2 would
provide an inadequate fit. Following Thirlwall et al. (2020), we segment the
galaxy into distinct morphological components to fully capture the detailed
morphology of M51. As mentioned in Sect. 3, we found that, within $R_{l}$, in
addition to the bulge, two distinct morphological components were needed to
model this galaxy: an inner and a main disc. We allow for each morphological
component to be made up of two stellar and two dust components.
We followed the same fitting procedure as in Thirlwall et al. (2020). In
brief, this involves an intelligent searching algorithm of the parameter
space, that takes into account the fact that different parameters do not
equally affect the emission at all wavelengths. This allows one to identify
pairs of parameters that predominantly shape emission at key wavelengths and
use this to fit one pair (usually made of a geometric parameter and an
amplitude parameter) of parameters at a time.
Thus, we begin modelling with some initial guess parameters. These parameters
are based on notable features of the surface brightness profiles, as well as
informed by previous modelling (Thirlwall et al. 2020). We start by fitting
the NUV profile. This is equivalent to constraining the spatial distribution
of the young stellar population residing in the thin stellar disc. Although
this stellar disc is highly attenuated in the UV, we can still derive an
estimate of the corresponding geometrical and amplitude parameters for an
initial guess of dust opacity. We thus derive an exponential scale length
$h^{\rm{tdisc}}_{\rm{s}}$ and a luminosity density $L^{\rm{tdisc}}$ in the NUV
filter, for the inner and main disc, assuming a known wavelength dependence of
the luminosity density at the other sampled wavelengths. There is also the
free parameter $\chi_{\rm s}^{\rm tdisc}$, which is fitted for the inner disc
in order to account for the fall of emissivity towards the galaxian centre.
For the main disc $\chi_{\rm s}^{\rm tdisc}$ is only used to provide a smooth
overlap between the inner and the main disc, resembling the observed averaged
profiles, instead of simply using delta functions. $\chi_{\rm s}^{\rm
m-tdisc}$ is therefore not a critical parameter in the fit. Inner radii
$R_{\rm in}$ and inner and outer truncation radii $R_{\rm tin}$ and $R_{\rm
t}$ are fixed from data and are not subject to iterations. Once we obtain a
good fit in the NUV (for the trial dust opacity), we then attempt to constrain
the dust distribution . To do this we re-run the RT calculation with the
parameters derived from fitting the NUV profile and for an initial guess for
the optical emission, and predict the dust emission at all infrared
wavelengths, including the 500 $\mu$m. We then constrain the spatial
distribution of the dust by comparing the predicted 500 $\mu$m model profiles
with the corresponding observed ones, since at this wavelength the dust
emission primarily traces dust column density rather than heating sources. At
this step small changes in the dust parameters may be needed for a best fit to
the 500 $\mu$m data. This results in a fit for the scale-length of the dust
disc $h^{\rm{disc}}_{\rm{d}}$ and a dust opacity (plus the parameter
$\chi_{\rm d}^{\rm tdisc}$). A few iterations are needed between NUV and the
500 $\mu$m, until a good fit at both NUV and the 500 $\mu$m is achieved.
Towards the end of the iteration the FUV luminosity density is also slightly
adjusted until the best fit to the FUV profile is also obtained. Once the
parameters related to the thin stellar disc and dust disc are constrained, we
run again the RT code to derive the distribution of the stellar disc and bulge
harbouring the old stellar population, by fitting the optical and NIR profiles
at each wavelength for which observational data is available. Since in a face-
on galaxy the optical and NIR emission is fairly optically thin to dust, the
final fits at these wavelengths do not alter the solution for for the dust
disc and the thin stellar disc (in the UV) obtained in previous iterations. A
clumpy component is finally fitted to account for any remaining 24-70 ${\mu}$m
observed emission not accounted for by the diffuse dust emission. Overall,
within a small number of iterations a solution is derived for the whole
spectral range. Notably, the 70-350 ${\mu}$m emission is not directly used in
the fit, but instead is predicted. To conclude, the fit mainly consists in
fitting only a pair of parameters (a scale-length and an amplitude) at any
given time, for each morphological component at a given wavelength. More
details on the fitting procedure can be found in Thirlwall et al. (2020).
Figure 2: Comparison between the azimuthally averaged surface brightness
profiles of the observations (solid black line) at selected UV/optical/NIR
wavelengths and of the corresponding model for dust attenuated stellar light
(solid red line). The blue shaded region around the observed profile
represents the corresponding errors in the averaged surface brightness, as
described in Sect. 2. The contribution from each morphological component is
plotted with dashed lines and colour-coded as follows: bulge in purple,
stellar disc in green and thin stellar disc in blue. The lower panels show the
residuals between the observations and our model, with the dashed blue lines
showing the $\pm 20\%$ residuals to guide the eye. Figure 3: Same as Fig. 2
but for selected wavelengths in the NIR/MIR/FIR/submm, where the emission from
the dust discs is plotted with the grey dotted lines and the dust emission
from the HII regions is shown by the dashed cyan lines.
The main (free) geometric parameters constrained by the procedure outlined
above are listed in Table 3. The free parameters related to stellar luminosity
densities are listed in Table 6. The free parameters related to dust opacity
are listed in Table 7. Other parameters of the model are listed in Table 8.
$h_{\text{s}}^{\text{i-disc}}$(g, r, i J, K, I1, I2, I3) | (0.74, 0.73, 0.72, 0.69, 0.66, 0.70, 0.72, 1.0)$\pm$7%
---|---
$h_{\text{s}}^{\text{m-disc}}$(g, r, i J, K, I1, I2, I3) | (3.8, 3.5, 3.7, 3.5, 3.5, 4.3, 4.9, 7.0)$\pm$13%
$h_{\text{s}}^{\text{i-tdisc}}$ | 0.6$\pm$0.15
$h_{\text{s}}^{\text{m-tdisc}}$ | 4.3$\pm$1.0
$h_{\text{d}}^{\text{i-disc}}$ | 5.0${}^{+3.5}_{-1.5}$
$h_{\text{d}}^{\text{m-disc}}$ | 6.0${}^{+3.0}_{-0.9}$
$\chi_{\text{s}}^{\text{i-tdisc}}$ | 0.1$\pm$0.02
$\chi_{\text{s}}^{\text{i-disc}}$ | 0
$\chi_{\text{d}}^{\text{i-disc}}$ | 0.45$\pm 0.05$
$R_{\text{eff}}$ | 0.35$\pm$0.1
Table 3: Main geometrical (free) parameters of the inner and main discs, and
of the bulge, derived for the standard model.
The results of the fits are presented in Figs. 2-3, showing an overall
reasonable agreement with the data. To help the comparison, the residuals $R$
between the surface brightness profiles of the observations and of the model
are also plotted on the bottom panels of Figs. 2-3. The residuals are defined
as:
$R=\frac{\text{model}-\text{observation}}{\text{observation}}$ (5)
In addition, the goodness of fit is also considered with the reduced chi-
squared $\text{chi}^{2}_{r}$ statistics444We use the notation $\text{chi}^{2}$
rather than $\chi^{2}$, as the later notation is reserved in this work for the
geometrical parameter describing the inner linear slope., which becomes more
relevant when finalising the model solution. We calculated
$\text{chi}^{2}_{r}$ for each wavelength by the following formulae:
$\displaystyle\text{chi}^{2}_{\lambda}=\sum_{n=1}^{N}\frac{\left(M_{n}-O_{n}\right)^{2}}{\epsilon^{2}_{\text{SB,}n}}$
(6)
$\displaystyle\text{chi}^{2}_{\text{r,}\lambda}=\frac{\text{chi}^{2}_{\lambda}}{N-m}\approx\frac{\text{chi}^{2}_{\lambda}}{N}$
(7)
where $N$ is the total number of annuli $n$, $m$ is the number of free
parameters fitted at wavelength ${\lambda}$, $O_{\text{n}}$ is the azimuthally
averaged flux through annuli $n$, $M_{\text{n}}$ is the model flux through
annuli $n$, $\epsilon^{2}_{\text{SB,}n}$ is the total error of the azimuthally
averaged surface brightness profile corresponding to annuli $n$. The total
number of annuli is 70 while $m$ varies between 2 in the FUV, to 5 (in the NUV
and 500 ${\mu}$m), to 8 (in the optical and NIR) for a fitted profile out to 7
kpc radius. At most other wavelengths that are predicted, we can assume zero
free parameters. Because $N$ is much larger than $m$ we neglect $m$ in Eq. 7.
Values of the $\text{chi}^{2}_{\lambda}$, for selected wavelengths, are listed
in Table 4. The total reduced chi-squared statistic over all wavelengths is:
$\text{chi}_{\rm{r}}^{2}=\frac{\sum_{l=1}^{L}\text{\text{chi}}^{2}_{\lambda,l}}{\sum_{l=1}^{L}N_{l}}$
(8)
Band | $\text{chi}^{2}_{r}$
---|---
NUV | 1.09
I2 | 0.74
PACS70 | 3.62
SPIRE500 | 1.33
Global | 2.44
Table 4: The $\text{chi}^{2}_{r}$ values at selected wavelengths, as well as
the global $\text{chi}^{2}_{r}$ (over all wavelengths), for the standard
model.
The $\text{chi}^{2}_{r}$ values from Table 4 confirm the trends seen in Figs.
2-3, proving that the axi-symmetric model provides a reasonable fit to the
data.
## 5 Results
### 5.1 The surface brightness distributions
Taking the azimuthally averaged surface brightness of the observations
produces relatively smooth exponential profiles, as seen in Figs. 2-3. Despite
the slight imperfections in the smoothness, the average profiles are still
suitable for description with analytic axi-symmetric functions. The most
significant deviations are in the UV and 24$\,\mu$m bands, where strong
asymmetries and clumpiness are present. The bumps in these profiles generally
coincide with the spiral arms as well as star forming regions. For example, a
significant bump in emission at 1.3 kpc, which can be seen most notably in the
FUV profile, is due to bright star forming regions. There is also a bump in
emission between 5 kpc and 6.5 kpc, which coincides with an overlap in the
spiral arms. Predictably, the 24 $\mu$m profiles exhibit the least smooth
curves, as at this wavelength the emission due to the diffuse dust and the
dust heated by localised SF regions is roughly equal, hence strong asymmetries
arise from these localised sources of emission. This is also seen in the IRAC
bands, where there is significant PAH emission.
Overall the model has reasonable agreement with the data with residuals
generally within 20%.
The scale-length of the observed profiles in the optical regime generally
decreases with increasing wavelength for each morphological component. For
wavelengths $\geq 3.6\,\mu$m this trend reverses for the main disc. This is
similar with trends found from previous modelling (M33; Thirlwall et al.
2020).
The observed profiles in the IRAC bands are modelled with emission from the
old stellar populations as well as dust emission, with a negligible
contribution from the young stellar population. At $3.6\,\mu$m and $4.5\,\mu$m
the emission is dominated by the old stellar population, but at $5.8\,\mu$m
the dust emission is almost equal in amplitude to that of the old stellar
population. At the $8.0\,\mu$m band the emission is dominated by the dust,
primarily due to the PAH emission.
### 5.2 The global SED of M 51
Figure 4: The global SED of the model (spatially integrated out to $R_{\rm
l}$) plotted together with the observed flux densities used to optimise our
model (shown with black squares). The error bars of the data are also shown,
although most are contained within the square symbols. The model SED is
represented by the solid black line, and covers both the dust attenuated
stellar SED in the UV/optical/NIR as well as the dust emission SED in the
NIR/MIR/submm. The contributions from each component of the model are plotted
with dashed lines as follows: thin stellar disc in dark blue, stellar disc in
green, bulge in purple, diffuse dust in grey and clumpy component in cyan. The
intrinsic stellar SED (as would be seen in the absence of dust) is plotted
with the solid brown line. We plot the relative residuals in the lower panel,
with the $\pm 20\%$ residuals indicated with the dotted blue line to guide the
eye. Figure 5: The predicted global SED of M 51a (solid black line) for the
intrinsic (dust de-attenuated stellar emission) and for the dust emission,
plotted together with the contribution from the individual morphological
components: the inner (dashed black line) and the main (dotted black line)
discs, and the bulge (dashed purple line).
The global SED of M51 was produced by spatially integrating the emission at
each wavelength out to ${\rm R}_{l}$. This was done both for observations and
for the model and the results are plotted in Fig. 4. The model SED closely
resembles the observed SED, with a maximum residual of $|R|=26.2\%$ and an
average residual of $\langle|R|\rangle=4.33\%$. The model performs worst at
$160\,\mu$m, where the emission is underestimated by our model.
We also show the contributions from the different morphological components to
the global SED in Fig. 5. It is important to note that the dust emission for a
given morphological component is not entirely powered by the stellar emission
of the same component, as for example, some fraction of the stellar photons
emitted from the inner disc will contribute to heating the dust in the main
disc. Nonetheless, we still show the separate contributions of the inner and
main discs to the dust emission, within the accepted understanding that there
is not a one-to-one spatial correspondence between the stellar and the dust
emission of these components. As expected due to its extent, the largest
contribution to the global SED is from the main disc responsible for 67.4% of
the total stellar emission, and 63.7% of the dust emission. The inner disc
contributes 30.7% to the total stellar emission and 36.3% to total dust
emission. The bulge contributes 1.89% to the total stellar emission. Here we
note that the separation between the inner disc and the bulge is rather
degenerate. We found no strong constraints for this separation. We therefore
consider the bulge versus inner disc luminosity to be subject to large
uncertainties.
The dust emission SED is dominated by the diffuse component. The HII component
has a smaller contribution, of $12.2\%$. We find a clumpiness factor of
$F=0.125$. The global dust emission SED peaks at $130\,\mu$m, which
corresponds to a dust temperature of 22 K. The hottest dust resides within the
inner disc, where the SED peaks at $102\,\mu$m, corresponding to a dust
temperature of 28 K. The main disc has the greatest contribution to the global
SED, therefore, as expected, it peaks at a similar wavelength, at around
$133\,\mu$m, corresponding to a dust temperature of 22 K.
We find that 38.3$\pm 4$% of the stellar light is absorbed and re-radiated by
the dust. This is a typical value for late-type spiral galaxies (Popescu &
Tuffs 2002). M33 was found to have a similar value of 35$\pm$3% (Thirlwall et
al. 2020). We also find that the young stellar population contributes 42.8% to
the total intrinsic stellar light, with 55.4% being due to the old stellar
population in the disc and the remaining 1.89% due to the bulge.
### 5.3 Star-formation rates in M51
We obtain a
$\text{SFR}=4.11^{+0.41}_{-0.39}\,$$\text{M}_{\odot}\text{yr}^{-1}$ which is
within the broad range of literature values. De Looze et al. (2014) obtained a
$\text{SFR}\sim 3.13\,$$\text{M}_{\odot}\text{yr}^{-1}$(555Value adjusted for
the distance of $D=8.58$ Mpc, as used in this study.) and Nersesian et al.
(2020b) found a $\text{SFR}=4.09\pm
0.35\,$$\text{M}_{\odot}\text{yr}^{-1}$(666Adjusted value.). Both these
studies use SKIRT, a Monte-Carlo RT model. Calzetti et al. (2005) obtain the
estimates $\text{SFR}(24\,\mu\text{m})\sim
3.7\,$$\text{M}_{\odot}\text{yr}^{-1}$(777Adjusted value.) and
$\text{SFR(FUV)}\sim 4.7\,$$\text{M}_{\odot}\text{yr}^{-1}$(888Adjusted
value.).
About two thirds of the star formation takes place in the main disc with
$\text{SFR}^{\text{m}}=2.73^{+0.28}_{-0.26}\,$$\text{M}_{\odot}\text{yr}^{-1}$.
This is expected due to both the size and bolometric output of the main disc.
For the inner disc we find
$\text{SFR}^{\text{i}}=1.38^{+0.15}_{-0.14}\,$$\text{M}_{\odot}\text{yr}^{-1}$.
We also calculate the surface density of star formation rate,
$\Sigma_{\text{SFR}}$, for each morphological component, by considering the
physical extent of each disc, which is shown in Tables 9-10. The global
surface density in SFR is $\Sigma_{\text{SFR}}=2.83^{+0.28}_{-0.27}\times
10^{-2}\,$$\text{M}_{\odot}\text{yr}^{-1}\text{kpc}^{-2}$.
### 5.4 Dust optical depth and dust mass
We find that the face-on optical depth in the B-band has a maximum value of
$\tau_{\text{B}}^{\text{f}}\left(R_{\text{in}}\right)=5.33\pm 0.33$ at the
inner radius of the inner disc. The main disc has a face-on optical depth at
the inner radius of $2.62\pm 0.08$. These values are listed in Table 7. We
find that M51 has a global dust mass (out to $R_{l}$) of
M${}_{\text{d}}=3.74^{+0.34}_{-0.30}\times 10^{7}\,\text{M}_{\odot}$. The dust
masses of the individual morphological components are listed in Table 11.
Using a gas mass of M${}_{\text{G}}=1.1\pm 0.03\times
10^{10}\,\text{M}_{\odot}$ (Mentuch Cooper et al. 2012) we obtain a gas-to-
dust ratio of $\text{GDR}=294\pm 4$.
### 5.5 Morphological components of M51
In modelling M51 we found three distinct morphological components out to
$R_{l}$, each requiring different geometrical parameters for the distribution
of stars and dust. Although we present the model for the entire galaxy, it is
also interesting to study the properties of these individual components. The
geometrical parameters of the three morphological components, as constrained
by our model are listed in Table 3.
#### 5.5.1 The inner disc
The inner disc extends from 0 kpc to 1.9 kpc and exists as a ring-like
structure for the young stars and dust, with an inner radius of 700 pc and 800
pc respectively. The distribution of the old stars, however, exponentially
increases towards the centre. The distribution of the dust is notably flat
with a scale length of 5 kpc, compared to a scale length of 600 pc for the
young stellar distribution. The inner disc has a
$\text{SFR}^{\text{i}}=1.38^{+0.15}_{-0.14}\,$$\text{M}_{\odot}\text{yr}^{-1}$,
contributing around 30% to the global SFR, and has the greatest surface
density of star formation rate
$\Sigma_{\text{SFR}}^{\text{i}}=1.21^{+0.28}_{-0.11}\times
10^{-1}\,$$\text{M}_{\odot}\text{yr}^{-1}\text{kpc}^{-2}$. The dust mass
within the inner disc is M${}_{\text{d}}^{\text{i}}=6.39^{+0.37}_{-0.21}\times
10^{6}\,\text{M}_{\odot}$. The temperature of the dust contained within the
inner disc is the hottest in the galaxy, at around 28 K.
#### 5.5.2 The bulge
The bulge was found to have an effective radius of 350 pc and was modelled
with a Sérsic index of 4. As mentioned in Sect. 5.2, the bulge parameters are
subject to uncertainty, in the sense that there are different configurations
in the bulge/inner disc decomposition that could result in a good fit to the
data. We based our choice on covering a small dip in the profile, that would
otherwise appear in other configurations, and as such we believe that our
choice is the most reliable result. Nonetheless, we caution the reader of the
subjectivity of the approach. The error estimates on the bulge/inner disc
parameters do not take into account the uncertainty due to the other possible
combinations.
#### 5.5.3 The main disc
The main disc resides in the region between 1.8 kpc and 6.8 kpc. The interior
of the main disc slightly overlaps with the outermost limits of the inner
disc. Both the stellar and dust distribution are relatively flat in the main
disc, with a scale length of 4.3 kpc for the young stellar disc and 6 kpc for
the dust disc. The main disc contributes most significantly to the global SFR,
with a
SFR${}^{\text{m}}=2.73^{+0.28}_{-0.26}\,$$\text{M}_{\odot}\text{yr}^{-1}$. The
main disc has a surface density SFR of
$\Sigma_{\text{SFR}}^{\text{m}}=2.00^{+0.18}_{-0.17}\times
10^{-2}\,$$\text{M}_{\odot}\text{yr}^{-1}\text{kpc}^{-2}$. The main disc has a
dust mass of M${}_{\text{d}}^{\text{m}}=3.10^{+0.26}_{-0.18}\times
10^{7}\,\text{M}_{\odot}$, with a dust temperature of 21.8 K.
### 5.6 The contribution of young and old stellar populations in heating the
dust
The fractional contribution of the different stellar populations in heating
the dust is an important quantity, since it allows a quantitative
understanding of the physical mechanisms related to the interaction between
stellar light and interstellar dust. In addition, this is key to the
understanding of key correlations, like for example the tight and universal
correlation between far-infrared and radio continuum emission of spiral
galaxies (de Jong et al. 1985; Helou et al. 1985; Wunderlich et al. 1987). The
standard interpretation of the so-called FIR-radio correlation is in terms of
young and massive star formation activity (Condon 1992). This picture assumes
that the dust emission is mainly powered by young stars which are also
responsible for the radio emission: the ionising radiation from the young
stars powers the thermal radio emission, and the remnants of the supernova
explosions which occur at the end of their lives accelerate the cosmic ray
electrons. Nonetheless both the effect of dust opacity and the contribution of
the old stellar populations play a role in shaping the slope of the
correlation (Pierini et al. 2003), and as such, knowledge of the fraction of
dust heating powered by the young and old stellar populations is a pre-
requisite for the analysis of the FIR/radio correlation. Furthermore,
knowledge of the different stellar populations in heating the dust is
equivalent to understanding the dust attenuation of the different stellar
populations, which, in turn, allows population synthesis models to be used for
deriving star formation histories in galaxies.
For the global emission of M51 we found that the fractional contribution of
the young stellar population in powering the dust emission, $F^{\rm dust}_{\rm
young}$, is $69\%$, the rest being attributed to the old stellar populations
in the disc and bulge. Previous modelling of M51 with radiative transfer
methods derived $F^{\rm dust}_{\rm young}=0.63$ (De Looze et al. 2014) and
$F^{\rm dust}_{\rm young}=0.72$ (Nersesian et al. 2020b). The latter value is
more in range with our determination.
Our value of $F^{\rm dust}_{\rm young}=0.69$ is similar to that derived with
the same models for other nearby spirals: $F^{\rm dust}_{\rm young}=0.71$ for
the Milky Way (Natale et al. 2021) and NGC 628 (Rushton et al. 2022), but
slightly lower than $F^{\rm dust}_{\rm young}=0.8$ for M33 (Thirlwall et al.
2020).
Following the formalism from Natale et al. (2015) and Popescu et al. (2000),
we use the radiative transfer calculations to not only derive $F^{\rm
dust}_{\rm young}$ and $F^{\rm dust}_{\rm old}$ for the global emission, but
also to understand the spatial variation of this quantity. In Fig. 6 we plot
$F^{\rm dust}_{\rm young}$ and $F^{\rm dust}_{\rm old}$ as a function of
radial distance. Within the inner 0.5 kpc dust emission is dominated by the
old stellar populations, mainly from the bulge. Beyond 0.5 kpc it is the young
stellar population that contributes a significant fraction of the dust
heating. At around 0.7-0.8 kpc (the inner radius of the inner disc) there is a
peak in $F^{\rm dust}_{\rm young}$, reaching above $70\%$. Within the extent
of the main disc, between 2.7 and 7 kpc, $F^{\rm dust}_{\rm young}$ is slowly
increasing with radial distance, from $60$ to $85\%$. At around 2 kpc,
corresponding to the transition region between the inner and the main disc,
$F^{\rm dust}_{\rm young}\approx F^{\rm dust}_{\rm old}$.
Figure 6: Radial profile of the fraction of dust heating $F^{\rm dust}$
powered by the young (blue) and old (red) stellar populations.
The overall trends are similar to those found for the Milky Way (Natale et al.
2021) and for NGC 628 (Rushton et al. 2022), in the sense that dust heating in
the inner regions has a significant contribution from old stellar populations,
while the rest of the dust in the disc is mainly heated by young stellar
populations, with $F^{\rm dust}_{\rm young}$ rather constant or slowly
increasing towards the outer disc. M51 has larger radial variations in $F^{\rm
dust}$ than the Milky Way or NGC 628.
### 5.7 The radial variation of $\Sigma_{\text{SFR}}$,
$\Sigma_{\text{M}_{\star}}$nd sSFR
The global SFR of M51 was found to be $4.11\,\text{M}_{\odot}\,{\rm yr}^{-1}$.
The rate of star formation is not constant throughout the extent of M51, but
has a strong radial dependence. To illustrate this we plot in the top panel of
Fig. 7 the surface density of SFR, $\Sigma_{\text{SFR}}$, versus radial
distance. One can see that there is a peak in $\Sigma_{\text{SFR}}$ at the
inner radius of the inner disc, followed by a steep decrease out to around 2
kpc. Beyond 3 kpc, within the extent of the main disc, the
$\Sigma_{\text{SFR}}$ follows a shallow decrease. Overall,
$\Sigma_{\text{SFR}}$ shows a marked difference between the inner and the main
disc. By contrast, the surface density of stellar mass,
$\Sigma_{\text{M}_{\star}}$, has a smoother radial variation, (middle panel,
Fig. 7), showing a shallow decline with increasing radial distance. The
stellar mass was calculated using the 3.6 and 4.5 ${\mu}$m flux densities and
the calibration from Eskew et al. (2012).
The specific SFR, sSFR is rather constant throughout the disc (bottom panel,
Fig. 7), except for the very central region. These trends are similar to the
trends found for M33, NGC 628 and the Milky Way.
Figure 7: Radial profiles of the SFR surface density. $\Sigma_{\text{SFR}}$
(top), stellar mass surface density, $\Sigma_{\text{M}_{\star}}$ (middle), and
specific star formation rate, sSFR (bottom).
### 5.8 The attenuation curve of M51
The attenuation curve of a galaxy depends not only on the optical constants of
the dust grains and their grain size distribution (incorporated in the dust
extinction), but also on the relative distribution of the stars and dust.
Neither the extinction nor the geometry is usually known, making predictions
for attenuation rather difficult. It is however the strength of radiative-
transfer modelling to accurately derive geometries of stars and dust, within
the framework of a given dust model.
In Fig. 8 we plot the predicted attenuation curve of M51, as derived from our
radiative transfer model. For comparison we also plot the extinction curve of
the MW from Fitzpatrick (1999). We fit the attenuation curve of our model with
a third-order polynomial plus a Drude profile, which is the functional form
presented in Salim et al. (2018). Using this form, we found that the best fit
is given by:
$\displaystyle k_{\lambda}$
$\displaystyle=-4.95+2.00\lambda^{-1}-0.29\lambda^{-2}+0.02\lambda^{-3}+D_{\lambda}+4.00$
(9) $\displaystyle D_{\lambda}$
$\displaystyle=\frac{3.00\,\lambda^{2}(0.035\,\mu\text{m})^{2}}{[\lambda^{2}-(0.2175\,\mu\text{m})^{2}]^{2}+\lambda^{2}(0.035\,\mu\text{m})^{2}}$
(10)
Figure 8: Comparison between the attenuation curve of M51 as seen at an
inclination of 20.3∘ (solid red line) with the average extinction curve of the
Milky Way (solid black line) (Fitzpatrick 1999). Both curves are normalised to
the corresponding values in the B-band.
The attenuation curve of M51 is slightly steeper than the MW extinction curve,
becoming increasingly disparate at shorter wavelengths. M51 also exhibits a
much greater 2200 Å bump compared to the MW extinction. We find that the
difference in width of the bumps to be negligible.
## 6 Discussion
Figure 9: The azimuthally averaged surface-brightness observed profiles
(black) plotted against our model (red) and the model of Nersesian et al.
(2020b) (blue), for a selection of wavebands. The lower panels indicate the
residuals for our model (red) and the model of Nersesian et al. (2020b)
(blue), with the $\pm 50$ per cent shown with the dashed blue line.
### 6.1 Comparison with the non axi-symmetric model of Nersesian et al. 2020b
As mentioned before, M51 was also modelled with non axi-symmetric RT codes by
De Looze et al. (2014) using the SKIRT code (Baes et al. 2011; Camps & Baes
2020), and more recently by Nersesian et al. (2020b), using a more up to date
version of the same formalism and code. Here we compare our model to that of
Nersesian et al. (2020b). Publicly available model images of Nersesian et al.
(2020b) were extracted from
DustPedia999https://sciences.ugent.be/skirtextdat/DustPediaData/M51/mock_images.zip
and processed in the same way as our model images, to produce azimuthally-
averaged radial profiles. We also checked that the observed fluxes used in
Nersesian et al. (2020b) were consistent with the values used in our work,
and, when small differences existed (at percent level), we adjusted for this.
In Fig. 9 we show examples of this comparison for a few selected wavelengths.
Overall our model performs better. This is to be expected, as the model of
Nersesian et al. (2020b) does not fit the geometry of the system, but only the
spatially integrated SED.
### 6.2 An outer disc
All the results presented so far were confined to the main extent of the
galaxy, out to the radial distance $R_{\rm l}=7$ kpc. Note however that
$R_{\rm l}$ is not the actual truncation radius of M51. There is very faint
and asymmetric emission beyond $R_{\rm l}$, also containing the bridge between
M51 and its companion M51b. This faint emission does not obey axi-symmetry,
therefore it is not a prime target for an axi-symmetric model. Nonetheless,
the emission is so faint that it is difficult to trace and model, and one
effective way to reveal it is through azimuthally averaging. By averaging, a
strong signal becomes apparent, in a form of an exponential decay. We model
this outer emission with another morphological component that we call “outer
disc” . While we acknowledge that an axi-symmetric model is a poor
approximation for this emission, we nevertheless attempt to at least give some
characterisation of the global properties of this emission.
The model for the outer disc morphology contains the same geometrical
components as for the main disc: a stellar disc, a thin stellar disc, a dust
disc and a thin dust disc. The analytic functions we fit are the same, and the
whole optimisation follows the same procedure as for the main body of the
galaxy.
Figure 10: Fits to the surface brightness profiles of the outer disc (7-20
kpc) at selected wavelengths, chosen to show examples of dust attenuated
stellar light in the NUV and 3.6 ${\mu}$m and dust emission in the 24 and 500
${\mu}$m. The black line is for the observed profiles and the red line is for
the corresponding model. The bottom panels show residuals between observations
and model.
Examples of fits to the outer disc are given in Fig. 10. We find the outer
disc to extend from 6.7 kpc to 20 kpc, with a scale length of 3.6 kpc and 1.45
kpc for the dust disc and young stellar disc, respectively. The main
geometrical parameters of the outer disc are listed in Table 5.
The outer disc is the morphological component with the smallest SFR, with a
SFR${}^{\text{o}}=0.57\pm 0.06\,$$\text{M}_{\odot}\text{yr}^{-1}$. The surface
density of SFR is the least dense of the components, with
$\Sigma_{\text{SFR}}^{\text{o}}=5.11\pm 0.56\times
10^{-4}\,$$\text{M}_{\odot}\text{yr}^{-1}\text{kpc}^{-2}$.
$h_{\text{s}}^{\text{o-disc}}$(g, r, i J, K, I1, I2, I3) | (2.0, 2.0, 2.0, 1.72,
---|---
| 1.77, 1.95, 1.98, 2.00)$\pm$13%
$h_{\text{s}}^{\text{o-tdisc}}$ | 1.45$\pm$0.2
$h_{\text{d}}^{\text{o-disc}}$ | 3.6${}^{+1.4}_{-1.0}$
Table 5: Main geometric parameters of the outer disc
### 6.3 A model with a spatial variation in dust properties
Inspection of the fitted profiles from Figs. 2-3 shows that the axi-symmetric
model provides a reasonable fit to the data, within $R_{l}$.
However, when deriving the spatially integrated SED (see Fig. 11, top left-
hand panel), we noticed that our predicted intrinsic (de-attenuated) stellar
SED shows a spike in the NUV, exactly at the position of the well-known “2200
Å bump” of the extinction curve. It can be seen that the inner disc
contributes most significantly to this spike, while the main disc has
negligible contribution, if any. The intrinsic stellar SED should be smooth,
with no feature at this wavelength. Because our model is in good agreement
with the data at all wavelengths, as seen in Figs. 2-3, the spike in the
intrinsic SED is indicative that the inner disc is being too heavily
attenuated in the NUV. This could be due to a spatial variation in the dust
properties, in the sense that the dust properties may be different in the
inner disc than assumed in our model. As mentioned before, our model for M51
uses the Milky Way-type dust from Weingartner & Draine (2001), described on
line 7 in Table 1 from Weingartner & Draine (2001). The extinction curve of
the Milky Way dust model along with the contributions from the different grain
types is shown in the left panel of Fig. 12, together with its prominent “2200
Å bump”. This feature is primarily due to the PAH molecules, which the model
considers to be the carbonaceous grains with sizes $a\leq 0.01\,\mu$m. The
over-attenuation of the NUV photons in the inner disc could suggest that the
extinction curve corresponding to the dust model that we use has a too strong
bump, meaning an overabundance of PAH in the inner disc, than is present in
M51. We note that it is extremely hard to constrain the shape of the UV
extinction curve of galaxies, especially in the inner region where the
attenuation is high and the geometry is complex, with just the GALEX FUV and
NUV bands. Several authors have shown that these bands alone are insufficient
to disentangle the shape of the continuum and the strength of the 2200A bump
(Hoversten et al. 2011; Hutton et al. 2015; Hagen et al. 2017; Decleir et al.
2019).
Nonetheless, within the framework of a model with varying dust properties, our
fits at $8\,\mu$m (see Fig. 11, bottom right-hand panel) provide additional
evidence that we may need a different type of dust in the inner disc. The
model profile is clearly over-predicted in the inner region. Since the
$8\,\mu$m is a strong tracer of PAH abundance, in addition to tracing the
radiation fields that are self-consistently calculated in our RT model, the
over-prediction could again be indicative of an overabundance of PAH in our
model of the inner disc.
As an alternative, we sought a dust model with a reduced PAH abundance and
chose one of the LMC dust models of Weingartner & Draine (2001), namely the
LMC average (line 1 from Table 3 of Weingartner & Draine 2001). This LMC-type
dust features the same optical constants but different grain size
distributions, particularly less PAH grains. Thus, the extinction curve
exhibits a less pronounced 2200 Å bump, as seen in the right panel of Fig. 12.
As the attenuation in the main and outer discs produced smooth intrinsic SEDs,
we chose to modify the dust properties only in the inner disc (using the LMC-
type dust). We thus fitted a new model of M51 with variable dust grain
properties. We refer to this alternative as the “hybrid model”. The new fits
produce the intended effect, reducing the NUV peak in the de-attenuated
stellar SED of the inner disc, as seen in Fig. 11. The new fits to the
$8\,\mu$m profiles (see Fig. 11, bottom right-hand panel) also provide the
right level of emission at this wavelength. We conclude that using the LMC-
type dust for the inner disc could provide an alternative to the standard
model.
The finding that PAHs could be less abundant in the inner disc than in the
rest of the galaxy raises the question of what physical processes could cause
this variation. Theoretical and observational studies suggest that strong UV
radiation fields could destroy PAH molecules (e.g., Boulanger et al. 1988;
Helou et al. 1991; Contursi et al. 2000; Siebenmorgen & Krügel 2010). Taking
into account that the inner region of M51 has the highest surface density of
SFR, and therefore intense UV radiation fields, it is plausible that PAHs are
more readily destroyed in this region, thus explaining the reduced abundance
inferred by the hybrid model. Several studies have suggested that the fraction
of PAHs becomes low in HII regions (e.g., Pety et al. 2005; Lebouteiller et
al. 2007; Thilker et al. 2007; Chastenet et al. 2023) and is a strong function
of environment, with higher fractions in less harsh environments due to star
formation (Chastenet et al. 2023). It is beyond the scope of our paper to
provide a quantitative model for PAH destruction in different ISM
environments, but we note the possibility of PAH abundance variation within
galaxies.
Figure 11: Top panels: the standard model (constant dust properties) for the
intrinsic global SED (top left) and the surface brightness profile at 8 $\mu$m
(top right). Bottoms panels: same but for the hybrid model (with spatially
varying dust properties). The left hand plots show the intrinsic global SED
(solid black line) plotted together with the contribution from the individual
morphological components: the inner (dashed black line) and main (dotted black
line) discs as well as the bulge (dashed purple line). The right hand plots
show the $8\,\mu$m profile of the observation (solid black line) compared to
the model (solid red line) along with the contributions from the dust discs
(dotted grey lines), and the HII components (dashed cyan lines). Residuals
between observed and model profiles are also shown on the bottom of each
panel.
The fits to the multiwavelength data using the hybrid model do not provide any
big differences from the standard model, with the exception of the 8 $\mu$m
profile, which is largely improved (see examples in the Appendix, in Figs.
13-14). The energy balance between direct and re-radiated stellar light seems
to be the same as in the standard model, with no major improvements in the
fit. The values of the ${\rm chi}^{2}$ are also similar. The individual values
at selected wavelengths are listed in Table 12. The values of the derived
intrinsic parameters are different, but the quality of the fit is the same.
This is perhaps not surprising, since the optical properties of the grains are
not changed. There are some other dust models available in the literature, in
particular "The Heterogeneous dust Evolution Model for Interstellar Solids"
(THEMIS) (Jones et al. 2013; Köhler et al. 2014; Ysard et al. 2015), which
consider different carbonaceous grains, with aliphatic rather than aromatic
molecules. The THEMIS model has enhanced submm efficiencies for grains, by a
factor of about 2.5 with respect to the Draine model. The THEMIS model would
produce a different energy balance and therefore different fits.
To conclude, the hybrid model with a reduced PAH abundance in the inner disc,
together with a modified grain size distribution in the central region,
provides a consistent solution to the panchromatic data of M51, and is based
on observational evidence coming from both dust extinction and dust emission.
Nonetheless, we cannot prove that these observational constraints are uniquely
met by our hybrid model.
## 7 Summary
Using the generic radiative transfer code and formalism of PT11 we derived an
axisymmetric model for M51, fitted to multiwavelength imaging data ranging
from FUV to submm. We find that, despite the interaction with the companion
galaxy M51b, M51 preserves a regular spiral structure within a radial distance
of $R_{\rm l}$=7 kpc, making it thus suitable for the assumption of axi-
symmetry within $R_{\rm l}$. The model fits to the azimuthally-averaged
surface brightness profiles were found to be in reasonable agreement with the
observations.
We find three distinct morphological components out to $R_{\rm l}$: a bulge,
an inner disc and a main disc. The bulge has an effective radius $R_{\rm
eff}=350$ pc and a Sérsic index $n=4$. The inner disc is prominent in the UV,
with a scale-length of the thin stellar disc, $h_{\rm s}^{\rm i-disc}$, of 600
pc. The distribution of dust in the inner disc is very flat, with a
scalelength of the dust disc, $h_{\rm d}^{\rm disc}$, of 5 kpc. The main disc
resides in the region between $\sim$ 1.8 kpc and $\sim$6.8 kpc. Both the
stellar and the dust distributions of this morphological component are
relatively flat, with a scale-length for the thin stellar disc, $h_{\rm
s}^{\rm t-disc}$ of 4.3 kpc and $h_{\rm d}^{\rm disc}=6$ kpc for the dust
disc.
The main intrinsic properties of M51, as derived from the model are as
follows:
* •
The global star-formation rate is
$\text{SFR}=4.11^{+0.40}_{-0.38}\,$$\text{M}_{\odot}\text{yr}^{-1}$. The inner
disc has
$\text{SFR}=1.38^{+0.15}_{-0.14}\,$$\text{M}_{\odot}\text{yr}^{-1}$and the
main disc has
$\text{SFR}=2.73^{+0.28}_{-0.26}\,$$\text{M}_{\odot}\text{yr}^{-1}$.
* •
The global surface density in SFR is
$\Sigma_{\text{SFR}}=2.83^{+0.28}_{-0.27}\times
10^{-2}\,$$\text{M}_{\odot}\text{yr}^{-1}\text{kpc}^{-2}$.
* •
The specific star formation rate, sSFR, is rather constant over much of the
extent of M51, except for the very centre.
* •
The young stars account for 69% of the dust heating. In the inner regions dust
heating has a significant contribution from old stellar populations, while the
rest of the dust in the disc is mainly heated by young stellar populations,
with their contribution rather constant or slowly increasing towards the outer
disc.
We discuss the properties of the faint and non axi-symmetric emission
extending beyond $R_{\rm l}$, also containing the bridge between M51 and its
companion M51b. We model this outer emission with another morphological
component that we call “outer disc” . The outer disc has $\text{SFR}=0.57\pm
0.06\,$$\text{M}_{\odot}\text{yr}^{-1}$and $\Sigma_{\text{SFR}}=5.11\pm
0.56\times 10^{-4}\,$$\text{M}_{\odot}\text{yr}^{-1}\text{kpc}^{-2}$.
We also show a model with varying dust properties, with the inner disc having
a reduced PAH abundance with respect to the main disc. This hybrid model was
chosen to solve the over-prediction of the standard model at 8 $\mu$m in the
inner disc, and to alleviate a residual bump in the derived intrinsic SED in
the NUV band, at the position of the 2200 Åbump.
## Acknowledgements
We would like to thank an anonymous referee for very useful and constructive
comments, that helped improve the manuscript. This work is based in part on
observations made with the NASA Galaxy Evolution Explorer. GALEX is operated
for NASA by the California Institute of Technology under NASA contract
NAS5-98034. This research has made use of the NASA/IPAC Infrared Science
Archive, which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration.
This work is also based on the Sloan Digital Sky Survey (SDSS) data. Funding for the SDSS IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss4.org. SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, Center for Astrophysics | Harvard & Smithsonian (CfA), the Chilean Participation Group, the French Participation Group, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatório Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.
This work has also made use of data products from the Two Micron All Sky
Survey, which is a joint project of the University of Massachusetts and the
Infrared Processing and Analysis Center/California Institute of Technology,
funded by the National Aeronautics and Space Administration and the National
Science Foundation. This work is based in part on observations made with the
Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory,
California Institute of Technology under a contract with NASA. We also utilise
observations performed with the ESA Herschel Space Observatory (Pilbratt et
al. 2010), in particular to do photometry using the PACS (Poglitsch et al.
2010) and SPIRE (Griffin et al. 2010) instruments.
## Data Availability
The data underlying this article are made available at the CDS database via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/MNRAS/
## References
* Aihara et al. (2011) Aihara H., et al., 2011, ApJS, 193, 29
* Aniano et al. (2011) Aniano G., Draine B. T., Gordon K. D., Sandstrom K., 2011, PASP, 123, 1218–1236
* Baes & Dejonghe (2001) Baes M., Dejonghe H., 2001, MNRAS, 326, 733
* Baes et al. (2010) Baes M., et al., 2010, A&A, 518, L39
* Baes et al. (2011) Baes M., Verstappen J., De Looze I., Fritz J., Saftly W., Vidal Pérez E., Stalevski M., Valcke S., 2011, ApJS, 196, 22
* Bendo et al. (2012) Bendo G. J., et al., 2012, MNRAS, 419, 1833
* Bersier et al. (1994) Bersier D., Blecha A., Golay M., Martinet L., 1994, A&A, 286, 37
* Bianchi (2008) Bianchi S., 2008, A&A, 490, 461
* Bianchi et al. (1996) Bianchi S., Ferrara A., Giovanardi C., 1996, ApJ, 465, 127
* Boulanger et al. (1988) Boulanger F., Beichman C., Desert F. X., Helou G., Perault M., Ryter C., 1988, ApJ, 332, 328
* Brown et al. (2014) Brown M. J. I., et al., 2014, ApJS, 212, 18
* Burbidge & Burbidge (1964) Burbidge E. M., Burbidge G. R., 1964, ApJ, 140, 1445
* Byun et al. (1994) Byun Y. I., Freeman K. C., Kylafis N. D., 1994, ApJ, 432, 114
* Calzetti et al. (2005) Calzetti D., et al., 2005, ApJ, 633, 871
* Camps & Baes (2020) Camps P., Baes M., 2020, Astronomy and Computing, 31, 100381
* Chastenet et al. (2023) Chastenet J., et al., 2023, ApJ, 944, L11
* Condon (1992) Condon J. J., 1992, ARA&A, 30, 575
* Considere & Athanassoula (1982) Considere S., Athanassoula E., 1982, A&A, 111, 28
* Contursi et al. (2000) Contursi A., et al., 2000, A&A, 362, 310
* Cunow (2001) Cunow B., 2001, MNRAS, 323, 130
* Danver (1942) Danver C.-G., 1942, Annals of the Observatory of Lund, 10, 7
* De Geyter et al. (2015) De Geyter G., et al., 2015, MNRAS, 451, 1728
* De Looze et al. (2014) De Looze I., et al., 2014, A&A, 571, A69
* Decleir et al. (2019) Decleir M., et al., 2019, MNRAS, 486, 743
* Disney et al. (1989) Disney M., Davies J., Phillipps S., 1989, MNRAS, 239, 939
* Dobbs et al. (2010) Dobbs C. L., Theis C., Pringle J. E., Bate M. R., 2010, MNRAS, 403, 625
* Draine & Lee (1984) Draine B. T., Lee H. M., 1984, ApJ, 285, 89
* Draine & Li (2007) Draine B. T., Li A., 2007, ApJ, 657, 810
* Durrell et al. (2003) Durrell P. R., Mihos J. C., Feldmeier J. J., Jacoby G. H., Ciardullo R., 2003, ApJ, 582, 170
* Dwek & Werner (1981) Dwek E., Werner M. W., 1981, ApJ, 248, 138
* Eskew et al. (2012) Eskew M., Zaritsky D., Meidt S., 2012, AJ, 143, 139
* Evans (1994) Evans R., 1994, MNRAS, 266, 511
* Fazio et al. (2004) Fazio G. G., et al., 2004, ApJS, 154, 10
* Ferrara et al. (1999) Ferrara A., Bianchi S., Cimatti A., Giovanardi C., 1999, ApJS, 123, 437
* Fischera & Dopita (2004) Fischera J., Dopita M. A., 2004, ApJ, 611, 919
* Fitzpatrick (1999) Fitzpatrick E. L., 1999, PASP, 111, 63
* Gadotti et al. (2010) Gadotti D. A., Baes M., Falony S., 2010, MNRAS, 403, 2053
* Giard et al. (2008) Giard M., Montier L., Pointecouteau E., Simmat E., 2008, AAP, 490, 547
* Gil de Paz et al. (2007) Gil de Paz A., et al., 2007, ApJS, 173, 185
* Greenberg (1963) Greenberg J. M., 1963, ARA&A, 1, 267
* Griffin et al. (2010) Griffin M. J., et al., 2010, A&A, 518, L3
* Gunn et al. (2006) Gunn J. E., et al., 2006, AJ, 131, 2332
* Hagen et al. (2017) Hagen L. M. Z., Siegel M. H., Hoversten E. A., Gronwall C., Immler S., Hagen A., 2017, MNRAS, 466, 4540
* Helou et al. (1985) Helou G., Soifer B. T., Rowan-Robinson M., 1985, ApJ, 298, L7
* Helou et al. (1991) Helou G., Ryter C., Soifer B. T., 1991, ApJ, 376, 505
* Hoversten et al. (2011) Hoversten E. A., et al., 2011, AJ, 141, 205
* Hu et al. (2013) Hu T., Shao Z., Peng Q., 2013, ApJL, 762, L27
* Hutton et al. (2015) Hutton S., Ferreras I., Yershov V., 2015, MNRAS, 452, 1412
* Jarrett et al. (2003) Jarrett T. H., Chester T., Cutri R., Schneider S. E., Huchra J. P., 2003, AJ, 125, 525
* Jones et al. (2013) Jones A. P., Fanciullo L., Köhler M., Verstraete L., Guillet V., Bocchio M., Ysard N., 2013, AAP, 558, A62
* Köhler et al. (2014) Köhler M., Jones A., Ysard N., 2014, AAP, 565, L9
* Kuchinski et al. (1998) Kuchinski L. E., Terndrup D. M., Gordon K. D., Witt A. N., 1998, AJ, 115, 1438
* Kylafis & Bahcall (1987) Kylafis N. D., Bahcall J. N., 1987, ApJ, 317, 637
* Kylafis & Xilouris (2005) Kylafis N., Xilouris E., 2005, in Popescu, CC and Tuffs, RJ ed., AIP CONFERENCE PROCEEDINGS Vol. 761, Spectral Energy Distributions of Gas-Rich Galaxies: Confronting Models with Data. pp 3–16
* Lebouteiller et al. (2007) Lebouteiller V., Brandl B., Bernard-Salas J., Devost D., Houck J. R., 2007, ApJ, 665, 390
* Lee et al. (2011) Lee J. H., Hwang N., Lee M. G., 2011, The Astrophysical Journal, 735, 75
* MacLachlan et al. (2011) MacLachlan J. M., Matthews L. D., Wood K., Gallagher J. S., 2011, ApJ, 741, 6
* Martin et al. (2005) Martin D. C., et al., 2005, ApJl, 619, L1
* McQuinn et al. (2016) McQuinn K. B. W., Skillman E. D., Dolphin A. E., Berg D., Kennicutt R., 2016, ApJ, 826, 21
* Mentuch Cooper et al. (2012) Mentuch Cooper E., et al., 2012, ApJ, 755, 165
* Misiriotis et al. (2001) Misiriotis A., Popescu C. C., Tuffs R., Kylafis N. D., 2001, A&A, 372, 775
* Möllenhoff et al. (2006) Möllenhoff C., Popescu C. C., Tuffs R. J., 2006, A&A, 456, 941
* Monnet et al. (1981) Monnet G., Paturel G., Simien F., 1981, A&A, 102, 119
* Montier & Giard (2004) Montier L. A., Giard M., 2004, AAP, 417, 401
* Morrissey et al. (2007) Morrissey P., et al., 2007, ApJS, 173, 682
* Mosenkov et al. (2016) Mosenkov A. V., et al., 2016, A&A, 592, A71
* Mosenkov et al. (2018) Mosenkov A. V., et al., 2018, A&A, 616, A120
* Natale et al. (2010) Natale G., et al., 2010, ApJ, 725, 955
* Natale et al. (2014) Natale G., Popescu C. C., Tuffs R. J., Semionov D., 2014, MNRAS, 438, 3137
* Natale et al. (2015) Natale G., Popescu C. C., Tuffs R. J., Debattista V. P., Fischera J., Grootes M. W., 2015, MNRAS, 449, 243
* Natale et al. (2017) Natale G., et al., 2017, A&A, 607, A125
* Natale et al. (2021) Natale G., Popescu C. C., Rushton M., Yang R., Thirlwall J. J., Pricopi D., 2021, MNRAS,
* Nersesian et al. (2020a) Nersesian A., et al., 2020a, A&A, 637, A25
* Nersesian et al. (2020b) Nersesian A., et al., 2020b, A&A, 643, A90
* Niederwanger et al. (2019) Niederwanger F., Reimer O., Kissmann R., Strong A. W., Popescu C. C., Tuffs R., 2019, Astroparticle Physics, 107, 1
* Pastrav et al. (2013a) Pastrav B. A., Popescu C. C., Tuffs R. J., Sansom A. E., 2013a, A&A, 553, A80
* Pastrav et al. (2013b) Pastrav B. A., Popescu C. C., Tuffs R. J., Sansom A. E., 2013b, A&A, 557, A137
* Pety et al. (2005) Pety J., Teyssier D., Fossé D., Gerin M., Roueff E., Abergel A., Habart E., Cernicharo J., 2005, AAP, 435, 885
* Pierini et al. (2003) Pierini D., Popescu C. C., Tuffs R. J., Völk H. J., 2003, A&A, 409, 907
* Pierini et al. (2004) Pierini D., Gordon K. D., Witt A. N., Madsen G. J., 2004, ApJ, 617, 1022
* Pilbratt et al. (2010) Pilbratt G. L., et al., 2010, A&A, 518, L1
* Poglitsch et al. (2010) Poglitsch A., et al., 2010, A&A, 518, L2
* Popescu (2021) Popescu C. C., 2021, Star Formation Rates of Galaxies. Cambridge University Press
* Popescu & Tuffs (2002) Popescu C. C., Tuffs R. J., 2002, MNRAS, 335, L41
* Popescu & Tuffs (2010) Popescu C. C., Tuffs R. J., 2010, in Debattista V. P., Popescu C. C., eds, American Institute of Physics Conference Series Vol. 1240, Hunting for the Dark: the Hidden Side of Galaxy Formation. pp 35–46 (arXiv:1006.4606), doi:10.1063/1.3458534
* Popescu et al. (2000) Popescu C. C., Misiriotis A., Kylafis N. D., Tuffs R. J., Fischera J., 2000, A&A, 362, 138
* Popescu et al. (2004) Popescu C. C., Tuffs R. J., Kylafis N. D., Madore B. F., 2004, A&A, 414, 45
* Popescu et al. (2011) Popescu C. C., Tuffs R. J., Dopita M. A., Fischera J., Kylafis N. D., Madore B. F., 2011, A&A, 527, A109
* Popescu et al. (2017) Popescu C. C., Yang R., Tuffs R. J., Natale G., Rushton M., Aharonian F., 2017, MNRAS, 470, 2539
* Rieke et al. (2004) Rieke G. H., et al., 2004, ApJS, 154, 25
* Robitaille et al. (2012) Robitaille T. P., Churchwell E., Benjamin R. A., Whitney B. A., Wood K., Babler B. L., Meade M. R., 2012, A&A, 545, A39
* Rushton et al. (2022) Rushton M. T., Popescu C. C., Inman C., Natale G., Pricopi D., 2022, MNRAS, 514, 113
* Salim et al. (2018) Salim S., Boquien M., Lee J. C., 2018, ApJ, 859, 11
* Salo & Laurikainen (2000) Salo H., Laurikainen E., 2000, MNRAS, 319, 377
* Savchenko et al. (2023) Savchenko S. S., et al., 2023, MNRAS, 524, 4729
* Schlafly & Finkbeiner (2011) Schlafly E. F., Finkbeiner D. P., 2011, ApJ, 737, 103
* Shetty et al. (2007) Shetty R., Vogel S. N., Ostriker E. C., Teuben P. J., 2007, ApJ, 665, 1138
* Siebenmorgen & Krügel (2010) Siebenmorgen R., Krügel E., 2010, A&A, 511, A6
* Skrutskie et al. (2006) Skrutskie M. F., et al., 2006, AJ, 131, 1163
* Steinacker et al. (2013) Steinacker J., Baes M., Gordon K. D., 2013, ARA&A, 51, 63
* Tamburro et al. (2008) Tamburro D., Rix H. W., Walter F., Brinks E., de Blok W. J. G., Kennicutt R. C., Mac Low M. M., 2008, AJ, 136, 2872
* Thilker et al. (2007) Thilker D. A., et al., 2007, ApJS, 173, 572
* Thirlwall et al. (2020) Thirlwall J. J., Popescu C. C., Tuffs R. J., Natale G., Norris M., Rushton M., Grootes M., Carroll B., 2020, MNRAS, 495, 835
* Toomre & Toomre (1972) Toomre A., Toomre J., 1972, ApJ, 178, 623
* Trumpler (1930) Trumpler R. J., 1930, PASP, 42, 214
* Tuffs et al. (2004) Tuffs R. J., Popescu C. C., Völk H. J., Kylafis N. D., Dopita M. A., 2004, A&A, 419, 821
* Tully (1974) Tully R. B., 1974, ApJS, 27, 437
* Turner & Ho (1994) Turner J. L., Ho P. T. P., 1994, ApJ, 421, 122
* Verstocken et al. (2020) Verstocken S., et al., 2020, A&A, 637, A24
* Viaene et al. (2017) Viaene S., et al., 2017, A&A, 599, A64
* Walter et al. (2008) Walter F., Brinks E., de Blok W. J. G., Bigiel F., Kennicutt Robert C. J., Thornley M. D., Leroy A., 2008, AJ, 136, 2563
* Weingartner & Draine (2001) Weingartner J. C., Draine B. T., 2001, ApJ, 548, 296
* Williams et al. (2019) Williams T. G., Baes M., De Looze I., Relaño M., Smith M. W. L., Verstocken S., Viaene S., 2019, MNRAS, 487, 2753
* Witt & Gordon (1996) Witt A. N., Gordon K. D., 1996, ApJ, 463, 681
* Witt & Gordon (2000) Witt A. N., Gordon K. D., 2000, ApJ, 528, 799
* Witt et al. (1992) Witt A. N., Thronson Harley A. J., Capuano John M. J., 1992, ApJ, 393, 611
* Wunderlich et al. (1987) Wunderlich E., Klein U., Wielebinski R., 1987, A&AS, 69, 487
* Xilouris et al. (1997) Xilouris E. M., Kylafis N. D., Papamastorakis J., Paleologou E. V., Haerendel G., 1997, A&A, 325, 135
* Xilouris et al. (1998) Xilouris E. M., Alton P. B., Davies J. I., Kylafis N. D., Papamastorakis J., Trewhella M., 1998, A&A, 331, 894
* Xilouris et al. (1999) Xilouris E. M., Byun Y. I., Kylafis N. D., Paleologou E. V., Papamastorakis J., 1999, A&A, 344, 868
* York et al. (2000) York D. G., et al., 2000, AJ, 120, 1579
* Ysard et al. (2015) Ysard N., Köhler M., Jones A., Miville-Deschênes M. A., Abergel A., Fanciullo L., 2015, AAP, 577, A110
* de Jong et al. (1985) de Jong T., Klein U., Wielebinski R., Wunderlich E., 1985, A&A, 147, L6
* van der Giessen et al. (2022) van der Giessen S. A., Leslie S. K., Groves B., Hodge J. A., Popescu C. C., Sargent M. T., Schinnerer E., Tuffs R. J., 2022, A&A, 662, A26
## Appendix A Amplitude and geometrical parameters
In our model we express the amplitude parameters of the stellar discs with
spectral luminosity densities $L_{\nu}$, and the dust discs with central face-
on optical depths in the B-band, $\tau_{\text{B}}$. Table 6 lists the spectral
luminosity densities for each stellar disc and the bulge. Table 7 lists the
B-band face-on optical depth for each of the morphological components. Table 8
lists other parameters, like the $R_{\text{in}}$, $R_{\text{tin}}$ and
$R_{\text{t}}$, which are fixed from data. The table also lists the $\chi$
values for the main disc. The latter parameter serves to more cohesively join
the inner to the main disc.
Table 6: The intrinsic spectral luminosity densities of the stellar and thin stellar disc, for each morphological component. $\lambda$ | $L_{\nu}^{\text{bulge}}$ | $L_{\nu}^{\text{disc, i}}$ | $L_{\nu}^{\text{disc ,m}}$ | $L_{\nu}^{\text{tdisc, i}}$ | $L_{\nu}^{\text{tdisc, m}}$
---|---|---|---|---|---
(Å) | (W Hz-1) | (W Hz-1) | (W Hz-1) | (W Hz-1) | (W Hz-1)
1542 | - | - | - | 1.07$\times 10^{21}$ | 2.11$\times 10^{21}$
2274 | - | - | - | 3.62$\times 10^{21}$ | 4.97$\times 10^{21}$
3562 | 8.31$\times 10^{19}$ | 7.42$\times 10^{20}$ | 1.48$\times 10^{21}$ | 1.75$\times 10^{21}$ | 3.42$\times 10^{21}$
4719 | 4.16$\times 10^{20}$ | 4.36$\times 10^{21}$ | 8.20$\times 10^{21}$ | 1.87$\times 10^{21}$ | 3.39$\times 10^{21}$
6185 | 6.05$\times 10^{20}$ | 6.26$\times 10^{21}$ | 1.21$\times 10^{22}$ | 1.96$\times 10^{21}$ | 2.95$\times 10^{21}$
7500 | 6.64$\times 10^{20}$ | 6.08$\times 10^{21}$ | 1.24$\times 10^{22}$ | 1.66$\times 10^{21}$ | 2.26$\times 10^{21}$
12000 | 1.13$\times 10^{21}$ | 9.26$\times 10^{21}$ | 2.11$\times 10^{22}$ | 8.17$\times 10^{20}$ | 1.48$\times 10^{21}$
22000 | 1.02$\times 10^{21}$ | 8.94$\times 10^{21}$ | 2.12$\times 10^{22}$ | 6.40$\times 10^{20}$ | 1.16$\times 10^{21}$
35070 | 5.25$\times 10^{20}$ | 5.25$\times 10^{21}$ | 1.23$\times 10^{22}$ | 1.25$\times 10^{20}$ | 5.63$\times 10^{20}$
44370 | 3.73$\times 10^{20}$ | 2.75$\times 10^{21}$ | 6.86$\times 10^{21}$ | 1.25$\times 10^{20}$ | 5.63$\times 10^{20}$
57390 | 4.47$\times 10^{20}$ | 5.74$\times 10^{21}$ | 2.05$\times 10^{22}$ | 1.50$\times 10^{20}$ | 5.63$\times 10^{20}$
Table 7: The face-on optical depth in the B-band at the inner radius of their respective morphological component. | $\tau_{\text{B}}^{\text{f}}\left(R_{\text{in}}\right)$
---|---
Inner | 5.33$\pm 0.33$
Main | 2.62$\pm 0.08$
Fitting parameters |
---|---
$\chi_{\text{s}}^{\text{m-tdisc}}$ | -0.7$\pm 0.5$
$\chi_{\text{s}}^{\text{m-disc}}$ | 1.5$\pm 0.25$
$\chi_{\text{d}}^{\text{m-disc}}$ | 1$\pm 0.25$
Parameters fixed by data |
$R_{\text{in,s}}^{\text{(i-tdisc, m-tdisc)}}$ | (0.7, 2.7)
$R_{\text{in,s}}^{\text{(i-disc, m-disc)}}$ | (0, 2.7)
$R_{\text{in,d}}^{\text{(i-disc, m-disc)}}$ | (0.8, 2.7)
$R_{\text{tin,s}}^{\text{(i-tdisc, m-tdisc)}}$ | (0, 1.8)
$R_{\text{tin,s}}^{\text{(i-disc, m-disc)}}$ | (0, 1.8)
$R_{\text{tin,d}}^{\text{(i-disc, m-disc)}}$ | (0, 1.8)
$R_{\text{t,s}}^{\text{(i-tdisc, m-tdisc)}}$ | (1.9, 6.8)
$R_{\text{t,s}}^{\text{(i-disc, m-disc)}}$ | (1.9, 6.8)
$R_{\text{t,d}}^{\text{(i-disc, m-disc)}}$ | (1.9, 6.8)
Parameters fixed from model |
$z_{\text{s}}^{\text{(i-tdisc, m-tdisc)}}$ | (0.09, 0.09)
$z_{\text{s}}^{\text{(i-disc, m-disc)}}$ | (0.19, 0.19)
$z_{\text{d}}^{\text{(i-disc, m-disc)}}$ | (0.16, 0.16)
Table 8: Other parameters of the model. All length parameters are given in
kpc.
## Appendix B Star formation and dust masses
In this appendix we present tables with the calculated values for SFR (Table.
9), $\Sigma_{\text{SFR}}$ (Table. 10), and dust mass (Table. 11). All these
values are given for both the global emission and for each morphological
component.
Table 9: Star formation rates for the global emission and for the morphological components of M51. Values are in (M⊙ yr-1). | SFR (M⊙yr-1)
---|---
Global | 4.11${}^{+0.41}_{-0.39}$
Inner | 1.38${}^{+0.15}_{-0.14}$
Main | 2.73${}^{+0.28}_{-0.26}$
| $\Sigma_{\text{SFR}}$ (M⊙yr-1kpc-2)
---|---
Global | 2.83${}^{+0.28}_{-0.27}\times 10^{-2}$
Inner | 1.21${}^{+0.12}_{-0.11}\times 10^{-1}$
Main | 2.00${}^{+0.18}_{-0.17}\times 10^{-2}$
Table 10: Surface density of star formation for the global emission and for the morphological components of M51. Values are in (M⊙ yr-1 kpc2). | dust mass M${}_{\text{d}}$ (M⊙)
---|---
Global | 3.74${}^{+0.34}_{-0.30}\times 10^{7}$
Inner | 6.39${}^{+0.37}_{-0.21}\times 10^{6}$
Main | 3.10${}^{+0.26}_{-0.18}\times 10^{7}$
Table 11: Dust masses for the global emission and for the morphological components of M51. Values are in (M⊙). Band | $\text{chi}^{2}_{r}$
---|---
NUV | 0.97
I2 | 2.60
PACS70 | 2.01
SPIRE500 | 0.640
Global | 1.89
Table 12: The $\text{chi}^{2}_{r}$ values at selected wavelengths, as well as
the total over all wavelengths for the hybrid model.
## Appendix C Extinction curves of the Milky Way and LMC dust models
In this paper we use the Milky Way and LMC dust models given in Weingartner &
Draine (2001) in Table 1, line 7 and Table 3, line 1 respectively. Fig. 12
shows the extinction curve for each grain composition as well as the total
extinction curve for the Milky Way and LMC dust models. These plots
demonstrate that the LMC type dust exhibits a reduced 2200 Å bump compared to
the Milky Way type dust. Figs. 13-14 show the fits at various wavelengths
using the hybrid model.
Figure 12: Extinction curves for the Milky Way (left) and the LMC (right) dust
models, with the grain size distribution and optical constants from
Weingartner & Draine (2001). We also plot the contributions from the different
grain compositions: Si (dotparted), Gra (dashed) and PAH (dot-dashed) as well
as the mean extinction curve for the Milky Way (double dot-dashed)
(Fitzpatrick 1999). It can be seen that the LMC dust model features a reduced
2000 bump wh It can be seen that the LMC dust model features a reduced 2200 Å
bump when compared to the Milky Way dust.
Figure 13: Comparison between the azimuthally averaged surface brightness
profiles of the observations (solid black line) at selected UV/optical/NIR
wavelengths and of the corresponding model for dust attenuated stellar light
(solid red line) using the combined dust models of Milky Way and LMC
(Weingartner & Draine 2001). The blue shaded region around the observed
profile represents the corresponding errors in the averaged surface
brightness, as described in Sect. 2. The contribution from each morphological
component is plotted with dashed lines and colour-coded as follows: bulge in
purple, stellar disc in green and thin stellar disc in blue. The lower panels
show the residuals between the observations and our model, with the dashed
blue lines showing the $\pm 20\%$ residuals to guide the eye. Figure 14: Same
as Fig. 13 but for selected wavelengths in the NIR/MIR/FIR/submm, where the
emission from the dust discs is plotted with the grey dotted lines and the
dust emission from the HII regions is shown by the dashed cyan lines.
|
# MeerKLASS L-band deep-field intensity maps: entering the Hi dominated regime
MeerKLASS Collaboration: Matilde Barberi-Squarotti1,2,3,4, José L. Bernal5,
Philip Bull6,7, Stefano Camera1,8,9,7, Isabella P. Carucci10,11, Zhaoting
Chen12, Steven Cunnington6, Brandon N. Engelbrecht7, José Fonseca7,13,14,
Keith Grainge6, Melis O. Irfan15,7, Yichao Li16,7, Aishrila Mazumder6, Sourabh
Paul6, Alkistis Pourtsidou12,17,7, Mario G. Santos7,18, Marta Spinelli19,7,
Jingying Wang20,7, Amadeus Witzemann7,6, Laura Wolz6
1Dipartimento di Fisica, Università degli Studi di Torino, via P. Giuria 1,
10125, Torino, Italy
2Dipartimento di Fisica, Università degli Studi di Milano, via G. Celoria 16,
20133, Milano, Italy
3INFN – Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via G.
Celoria 16, 20133, Milano, Italy
4INAF – Istituto Nazionale di Astrofisica, Osservatorio Astrofisico di Brera-
Merate, via Brera 28 20121 Milano, Italy
5 Instituto de Física de Cantabria (IFCA), CSIC-Univ. de Cantabria, Avda. de
los Castros s/n, E-39005 Santander, Spain
6Jodrell Bank Centre for Astrophysics, Department of Physics & Astronomy, The
University of Manchester, Manchester M13 9PL, UK
7Department of Physics and Astronomy, University of the Western Cape, Robert
Sobukwe Road, Cape Town 7535, South Africa
8INFN – Istituto Nazionale di Fisica Nucleare, Sezione di Torino, via P.
Giuria 1, 10125, Torino, Italy
9INAF – Istituto Nazionale di Astrofisica, Osservatorio Astrofisico di Torino,
10025 Pino Torinese, Italy
10INAF – Istituto Nazionale di Astrofisica, Osservatorio Astronomico di
Trieste, Via G.B. Tiepolo 11, 34131 Trieste, Italy
11IFPU – Institute for Fundamental Physics of the Universe, Via Beirut 2,
34151 Trieste, Italy
12Institute for Astronomy, The University of Edinburgh, Royal Observatory,
Edinburgh EH9 3HJ, UK
13Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto CAUP,
4150-762 Porto, Portugal
14Departamento de Física e Astronomia, Faculdade de Ciências, Universidade do
Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal
15Institute of Astrophysics, University of Cambridge, Madingley Road, CB3 0HA
16Department of Physics, College of Sciences, Northeastern University, Wenhua
Road, Shenyang, 11089, China
17Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The
University of Edinburgh, Edinburgh EH9 3FD, UK
18South African Radio Astronomy Observatory (SARAO), 2 Fir Street, Cape Town,
7925, South Africa
19Observatoire de la Côte d’Azur, Laboratoire Lagrange, Bd de l’Observatoire,
CS 34229, 06304 Nice cedex 4, France
20Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai, 200030, China Corresponding
authors:steven.cunnington<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
We present results from MeerKAT single-dish Hi intensity maps, the final
observations to be performed in L-band in the MeerKAT Large Area Synoptic
Survey (MeerKLASS) campaign. The observations represent the deepest single-
dish Hi intensity maps to date, produced from 41 repeated scans over
$236\,\deg^{2}$, providing 62 hours of observational data for each of the 64
dishes before flagging. By introducing an iterative self-calibration process,
the estimated thermal noise of the reconstructed maps is limited to
${\sim}\,1.21$ mK ($1.2\,\times$ the theoretical noise level). This thermal
noise will be sub-dominant relative to the Hi fluctuations on large scales
($k\,{\lesssim}\,0.15\,h\,\text{Mpc}^{-1}$), which demands upgrades to power
spectrum analysis techniques, particularly for covariance estimation. In this
work, we present the improved MeerKLASS analysis pipeline, validating it on
both a suite of mock simulations and a small sample of overlapping
spectroscopic galaxies from the Galaxy And Mass Assembly (GAMA) survey.
Despite only overlapping with ${\sim}\,25\%$ of the MeerKLASS deep field, and
a conservative approach to covariance estimation, we still obtain a
${>}\,4\,\sigma$ detection of the cross-power spectrum between the intensity
maps and the 2269 galaxies at the narrow redshift range
$0.39\,{<}\,z\,{<}\,0.46$. We briefly discuss the Hi auto-power spectrum from
this data, the detection of which will be the focus of follow-up work. For the
first time with MeerKAT single-dish intensity maps, we also present evidence
of Hi emission from stacking the maps onto the positions of the GAMA galaxies.
###### keywords:
cosmology: large scale structure of Universe – cosmology: observations – radio
lines: general – methods: data analysis – methods: statistical
††pubyear: 2024††pagerange: MeerKLASS L-band deep-field intensity maps:
entering the Hi dominated regime–D
## 1 Introduction
Probing density fluctuations in large-scale cosmic structure is a cornerstone
of precision cosmology. Constructing maps that span wide areas and reach deep
redshifts provides a rich resource for testing and constraining the standard
$\Lambda$CDM cosmological model. Surveys at optical and near-infrared
wavelengths have made impressive progress in this pursuit (eBOSS
Collaboration, 2021; Heymans et al., 2021; DES Collaboration, 2022; DESI
Collaboration et al., 2024). However, extending experiments to a broader range
of wavelengths boosts the quantity of available data. Furthermore, multi-
wavelength observations provide robust checks of systematic influence due to
the alternative instrumental approach required to survey other wavelengths.
There can also be vastly different complex astrophysics for tracers at
different wavelengths. Thus alternative tracers can validate the consistency
of cosmological conclusions.
Of particular interest for surveying large-scale structure is mapping
redshifted neutral hydrogen (Hi) gas at 21cm radio wavelengths. In the post-
reionisation Universe, at redshifts $z\,{\lesssim}\,6$, the remaining Hi is
overwhelmingly contained within dark matter halos (Villaescusa-Navarro et al.,
2018), meaning the 21cm radiation spontaneously emitted from Hi acts as a
tracer of the large-scale structure. However, at the redshifts of interest for
large-scale structure cosmology ($z\,{\gtrsim}\,0.1$), the 21cm emission
becomes faint and hard to detect with the high signal-to-noise required for a
comprehensive mapping survey. This problem can be circumvented by abandoning
the requirement to resolve individual galaxies and instead integrate the
combined unresolved emission with low-aperture radio telescopes. This process
is now commonly known as Hi intensity mapping (Bharadwaj et al., 2001; Battye
et al., 2004; Wyithe et al., 2008; Chang et al., 2008).
The novel approach of intensity mapping presents some challenges. Since it
records all radiation entering the telescope receiver within its frequency
bandpass, the contribution to the signal will not come from cosmological Hi
alone; in fact, only a small percentage is. There are multiple sources of
foregrounds in the frequency ranges corresponding to redshifted 21cm emission.
These mostly originate from our Galaxy and come from diffuse synchrotron and
free-free emission, but there can also be strong extragalactic point sources
(Mozdzen et al., 2019; Lian et al., 2020; Spinelli et al., 2021; Irfan et al.,
2021; Hurley-Walker et al., 2022; Matshawule et al., 2021). These sources
dominate the cosmological Hi brightness temperature by three to five orders of
magnitude. Furthermore, instrumental effects can distort the foreground
spectra, making their removal additionally challenging (Shaw et al., 2015;
Carucci et al., 2020; Matshawule et al., 2021; Spinelli et al., 2022). In
addition to foregrounds, the telescopes will detect terrestrial, human-made
radio frequency interference (RFI) (Harper & Dickinson, 2018; Engelbrecht et
al., 2024). This can dominate even more than the foregrounds, but vary with
time and sky position so can therefore be flagged and deleted from the final
maps. Residual RFI will also average down if repeated scans are performed over
the same patch of sky, as is the case with MeerKLASS observations.
Despite foreground contamination and RFI challenges, the potential benefits
from Hi intensity mapping are too large to ignore, thus justifying the ongoing
endeavour to develop the technique into a tool for precision cosmology.
Experiments have already delivered promising results, demonstrating how
detections are possible even without purpose-built instruments (Masui et al.,
2013; Wolz et al., 2017; Anderson et al., 2018; Wolz et al., 2022). New
experiments are starting to arrive (e.g. Nan et al., 2011; Battye et al.,
2012; Chen, 2012; Newburgh et al., 2016; PUMA Collaboration et al., 2019;
Vanderlinde et al., 2019; CHIME Collaboration et al., 2022; Pal et al., 2022),
many designed specifically with Hi intensity mapping in mind, creating a
bright future ahead.
The MeerKAT telescope,111www.sarao.ac.za/science/meerkat an array of 64 dishes
in the Upper Karoo region of South Africa, has already showcased its
capability to perform Hi intensity mapping both as an interferometer, claiming
the first detection of the Hi power spectrum (Paul et al., 2023), and also in
single-dish mode, where the field-of-view can reach cosmological scales. Using
the single-dish pilot survey (Wang et al., 2021), a detection of the cross-
correlation power spectrum was achieved with overlapping WiggleZ galaxies
(Cunnington, Li et al., 2022; Drinkwater et al., 2010).
The MeerKAT Large Area Synoptic Survey (MeerKLASS) (Santos et al., 2017) will
continue adding to these observations until it is merged with the SKA
Observatory (SKAO) (SKA Cosmology SWG et al., 2020) on the same
site.222www.skao.int/en MeerKAT can observe in two frequency bands relevant
for 21cm-cosmology: L-band
($900\,\text{MHz}\,{<}\,{\nu}\,{<}\,1420\,\text{MHz}$, reaching redshifts
$z\,{<}\,0.58$) and UHF-band
($580\,\text{MHz}\,{<}\,{\nu}\,{<}\,1000\,\text{MHz}$,
$1.45\,{>}\,z\,{>}\,0.4$).
In this work, we present results from MeerKAT single-dish observations in the
MeerKLASS L-band deep field, a set of observations performed over a single
$236\,\deg^{2}$ patch with 41 repeated scans for each of the 64 dishes.
Integrating over the sky in this way will average down time-varying
contributions from thermal noise and RFI, thus boosting the signal-to-noise of
time-constant sky signals, which includes the foregrounds and the cosmological
Hi. Previous analysis (e.g. Wolz et al., 2022; Cunnington, Li et al., 2022)
exploited the fact that the foreground cleaned intensity maps were thermal
noise dominated when estimating error bars on the power spectrum measurements.
For this MeerKLASS deep-field and future MeerKLASS observations, it can no
longer be assumed that the thermal noise will be dominant, therefore demanding
more emphasis on covariance estimation. We therefore explore using the scatter
in the foreground transfer function (Cunnington et al., 2023), primarily used
to correct for signal loss from foreground cleaning, as a measure of the
covariance in measurements. The transfer function is constructed using mock
signal injection into the real data, hence its scatter should encapsulate
cosmic variance, thermal noise fluctuations, residual RFI contributions,
foreground residuals and signal loss correction uncertainties.
We validate the precision and accuracy of the analysis pipeline to ensure
estimator accuracy is within the statistical error associated with these
MeerKLASS observations. We formalise the main steps in our pipeline, such as
beam reconvolution, foreground cleaning and sampling to a Cartesian grid,
presenting validation steps on a suite of simulations which emulate the
MeerKLASS L-band deep-field observations. We also use a small sample of
overlapping spectroscopic galaxies from the Galaxy And Mass Assembly (GAMA)
(Drinkwater et al., 2010) survey to study the cross-power spectrum and its
covariance under different analysis choices.
The paper is outlined as follows: Section 2 outlines the observations for the
MeerKLASS deep field, our new calibration strategy (with an iterative self-
calibration process) and overlapping GAMA galaxies. Section 3 presents the
methods for generating our suite of mock data which emulate the observations.
Section 4 provides details of our foreground cleaning technique. Section 5
focuses on the power spectra, formalising our estimation and modelling
process, and presents the results. Section 6 shows the results from stacking
the MeerKLASS deep-field maps onto the locations of GAMA galaxies and
discusses the process and complications that come with this. We then finally
conclude in Section 7.
## 2 Observational data
### 2.1 MeerKLASS L-band deep-field
Using the same scan strategy as the MeerKLASS L-band pilot survey in Wang et
al. (2021), a series of observations were conducted using the MeerKAT
telescope between September 1st and December 29th 2021. Observations with
single dishes typically require a scanning strategy where the dishes are
rapidly moved across the sky. The MeerKAT antennas were set to scan in azimuth
at constant elevation to minimise fluctuations of ground spill and airmass. To
make sure the projected speed on the sky is $5\,\mathrm{arcmin\,s^{-1}}$, the
telescope scan speed was set to $5/\cos(el)\,\mathrm{arcmin\,s^{-1}}$ (where
$el$ is the elevation angle) along the azimuth. This speed also ensures that
the telescope pointing does not move significantly compared to the width of
the primary beam (${\sim}\,1\,$deg) during a single time dump
($2\,\mathrm{s}$).
With a time resolution of $2\,\mathrm{s}$, this gives a scan speed of no more
than $10\,\mathrm{arcmin}$ per time sample, which is well within the beam
size. Noise diodes attached to each receiver were fired for
$0.585\,\mathrm{s}$ once every $19.5\,\mathrm{s}$ during the observation to
provide a relative time-ordered data (TOD) calibration reference. The dishes
are moved back and forth with a slew of about 10 deg sky projection in each
direction, corresponding to an observing time of about $120\,\mathrm{s}$ per
stripe. The duration of each set of scans with 48 strips is about 100 minutes,
and before and after each scan we spent ${\sim}\,8$ minutes tracking a nearby
celestial point source to use as a bandpass calibrator and absolute flux
calibrator.
At fixed elevation, two scans can be performed per night, corresponding to
when the field is rising and setting respectively. The two scans will cross
each other to achieve good sky coverage in the region of overlap. There were
41 observations in total (each one referred as a block), the details of each
are listed in the Appendix in Table 2. The observations repeated scans
covering a patch of sky approximately $236\,\deg^{2}$ in the southern sky,
spanning in R.A. $(330^{\circ},360^{\circ})$ and Dec.
$(-36^{\circ},-25^{\circ})$. The 62 hours obtained for each of the 64 dishes
before flagging, provide a cumulative 3968 hours of observations,333This
number is reduced for the final useable data since some observation blocks
were entirely flagged and some dishes failed in observation. We provide
details of this later in this section. likely to be the deepest single-dish
intensity maps, in terms of observational hours per square degree, in the
MeerKLASS campaign.
### 2.2 Standard MeerKLASS calibration
Figure 1: Flowchart showing each step in the calibration pipeline. This is an
extension beyond the Wang et al. (2021) pipeline since it includes the new
self-calibration strategy.
We employ the MeerKAT single-dish calibration pipeline KATcali to calibrate
the deep-field observations and create the combined 3D data cubes. KATcali
includes several cycles of RFI flagging, calibration, and map-making. The
details of the calibration are thoroughly described in Wang et al. (2021).
Here, we provide a brief overview of the main points. We take a two-step
approach to the calibration process. The first step is to use the flux model
of a strong point source during tracking observations to calibrate the
bandpass and absolute flux, as well as the injection power of the noise diode.
The second step then uses the noise diode injections to calibrate the TOD
during the constant-elevation scanning phase.
To calibrate the scanning data, we build the model as
$\displaystyle T_{\rm model}(t,\nu)=T_{\rm diffuse}(t,\nu)+T_{\rm
el}(t,\nu)+T_{\rm diode}(t,\nu)+T_{\rm rec}(t,\nu)\,,$ (1)
where $T_{\rm el}$ is the antenna temperature model of elevation-dependent
terrestrial emission (atmosphere and ground spill), $T_{\rm diode}$ is the
noise diode contribution, $T_{\rm rec}$ is the receiver temperature, while the
diffuse celestial components $T_{\rm diffuse}$ includes Galactic emission and
the cosmic microwave background (CMB),
$\displaystyle T_{\rm diffuse}(t,\nu)=T_{\rm Gal}(t,\nu)+T_{\rm CMB}\,,$ (2)
where the CMB emission is set to be a uniform $T_{\rm CMB}\,{=}\,2.725\,$K
(Fixsen, 2009), and the Galactic emission $T_{\rm Gal}(t,\nu)$ is derived from
PySM (Thorne et al. 2017) that is based on the reprocessed Haslam 408 MHz
total intensity map (Remazeilles et al. 2015).
To compare with the raw data from the correlator, we multiply the signal model
by the gain, $g(t,\nu)$, so that
$\displaystyle\hat{T}_{\rm model}(t,\nu)=g(t,\nu)\,T_{\rm model}(t,\nu)\,,$
(3)
where the hat indicates a temperature in the (arbitrary) correlator units.
The signal and gain models are then fitted (with a Bayesian framework; see
more details in Section 3.3 of Wang et al. 2021) to the TOD for each
polarisation, frequency channel, dish, and observation scan, all of which are
treated independently. Using the gain solution, we can then obtain the
calibrated temperature,
$\displaystyle T_{\rm cal}(t,\nu)\equiv\frac{\hat{T}_{\rm
raw}(t,\nu)}{g(t,\nu)}\,,$ (4)
where ${\hat{T}_{\rm raw}}$ is the raw data collected by MeerKAT. The residual
temperature difference between the signal model and calibrated data is then
$\displaystyle T_{\rm res}(t,\nu)\equiv T_{\rm cal}(t,\nu)-T_{\rm
model}(t,\nu)\,.$ (5)
In standard calibration, we assumed that $T_{\rm res}$ is noise-dominated
during the fitting, ignoring the faint point sources in the target field. This
was a fair assumption when the noise level was relatively high (Wang et al.,
2021). In this work, we seek a better calibration approach by implementing a
self-calibration method.
Figure 2: The standard deviations of $T_{\text{res}}$ for different self-
calibration cycles (Loops 1-5) in the 971–1023 MHz region. The standard
calibration value is also shown for reference (first point).
### 2.3 Self-calibration based on the MeerKLASS sky map
The higher signal-to-noise from the MeerKLASS deep-field data may allow us to
deal more carefully with the point sources, which were ignored in the standard
calibration, to achieve a better calibration result. Additionally, we aim to
use a more accurate sky model to replace the $T_{\rm Gal}$ model that is based
on the Haslam map. The 408 MHz radio continuum all-sky map, which has been
widely used to study diffuse synchrotron radiation from our Galaxy and works
as a Galactic foreground model for cosmological research, combines data from
four different surveys that happened more than 40 years ago (Haslam et al.
1981; Haslam et al. 1982). The instrument performance in the 1980s limited the
data quality, especially for the Galactic substructures and the zero level of
the maps (Remazeilles et al., 2015). The selection of the spectral index map,
used to generate Galactic maps at MeerKAT frequencies, is also a matter of
debate (Spinelli et al., 2022).
We therefore extend the calibration process described in Section 2.2 and test
the behaviour of the results by introducing an iterative self-calibration
process with several loops. This includes
(1) replacing $T_{\rm diffuse}(t,\nu)$ with $T_{\rm sky}(t,\nu)$, obtained
from the previous Level-6 $T_{\rm sky}$ result, when the corresponding Level-6
pixel has a hit-count$\,{>}\,40$ (which ensures that $T_{\rm sky}$ values are
obtained from high signal-to-noise data);
(2) setting the time-order weight $w({t_{i}})\,{=}\,1$ for all the scan time
samples; instead of setting a higher weight for the diode (main calibrator
during the standard calibration) injection samples (see more details in
Section 3.3 of Wang et al. 2021). Thus Equation 1 becomes
$T^{\rm upd}_{\rm model}(t,\nu)=T_{\rm sky}(t,\nu)+T_{\rm el}(t,\nu)+T_{\rm
diode}(t,\nu)+T_{\rm rec}(t,\nu)\,.$ (6)
In this way, we have all components (especially point sources and Galactic
substructures) included in the input “model”. The difference between the data
and our calibrated sky data is
$T^{\rm upd}_{\rm res}(t,\nu)\equiv T_{\rm cal}(t,\nu)-T^{\rm upd}_{\rm
model}(t,\nu)\,,$ (7)
which should be noise-dominated compared to that in Equation 5. The self-
calibration process is iterative for five loops (see Figure 1), in which the
$T_{\rm sky}(t,\nu)$ always uses the Level-6 result of the previous
calibration loop. Each iteration refines the calibration, reducing the
discrepancies between the observed data and the model. We stopped the self-
calibration iteration after five loops because the standard deviation of
$T_{\rm res}$ converged to a stable value, as shown in Figure 2.
Figure 3: Number count of good (flagged time $<50\%$) channels within the
971–1075 MHz band (L-band channels 550–1050) for each dish ($x$-axis) and
observation block ($y$-axis). We only show the 27 observation blocks used, out
of the possible 41. The other blocks were entirely removed mainly due to RFI
contamination, and one failed observation run. The details of all 41 blocks
are listed in Table 2. Figure 4: MeerKLASS L-band deep-field map at
$1023\,$MHz (middle panel) in comparison with the Galactic emission (top
panel) derived from PySM. Blue circles mark the positions of known point
sources with flux ${>}\,1$ Jy at 1.4 GHz. The bottom panel shows the number of
observational recordings i.e. timestamps in each pixel at 1023 MHz, from the
surviving 27 blocks. Each timestamp is 2 seconds and we use this for noise
estimation and weighting in the power spectrum analysis. Blue scatter points
are the positions of all overlapping GAMA galaxies. Figure 5: Amplitude of
each map pixel along the line-of-sight for the previous MeerKLASS pilot survey
(top panel) and the deep field used in this work (bottom panel. We only show
the ${\sim}\,250$ channels most clean of RFI used in later cosmological
analysis. This has the frequency range
$971.2\,{<}\,\nu\,{<}\,1023.6\,\text{MHz}$ corresponding to
$0.46\,{>}\,z\,{>}\,0.39$ for redshifted 21cm emission.
### 2.4 Characteristics of final MeerKLASS data cubes
#### 2.4.1 Final data cubes
The combined calibrated temperature cubes (with a pixel size of 0.3 deg and
frequency resolution of 0.2 MHz) are derived from all available scans and
dishes. However, predominantly due to RFI, an identified repeated source of
which we discuss in the following sub-section, only 27 of the 41 observation
blocks yielded well-calibrated results. We mark them in the last column of
Table 2. In Figure 3 we present the number count of good (flagged time
$<50\%$) channels within the 971–1075 MHz band (channels 550–1050) for each
block and each dish. In particular, we found that antenna m059 has a higher
chance of surviving calibration even when all other dishes fail.444For
consistency with other MeerKAT work, we indicate MeerKAT’s 64 antennae with
this notation spanning m000-m063. This may be because m059 is the easternmost
dish of MeerKAT, located far from the array core, providing it with a more
isolated environment.
We show the final sky temperature map from the combination of the remaining 27
scans after 5 loops of self-calibration at 1023 MHz in the middle panel of
Figure 4. For comparison, the top panel shows the PySM diffuse Galactic
emission model with the same sky coverage. We also show in the bottom panel
the total number of time samples (across all dishes and observations) that
were combined in each pixel.
To demonstrate the improvement in data quality from the deep-field intensity
maps, Figure 5 plots the amplitude of each pixel along the line-of-sight as a
function of frequency. We compare this to the previous MeerKLASS L-band pilot
survey (top-panel) relative to the deep field used for this work. The maps
should be dominated by the continuum synchrotron spectra, thus their
smoothness is a rough proxy for how dominant any systematics are. There are
numerous perturbations in the pilot survey spectra that have consequences for
the effectiveness of foreground cleaning, required to access the cosmological
Hi. However, for the deep field, the spectra appear smoother and well-
calibrated. In addition to the improvements shown in Figure 2, we also find a
marginal performance improvement on the line-of-sight rms when using the self-
calibration technique relative to the previous standard calibration in the
deep-field. The average variance along the line-of-sight of each pixel
decreases to 977.2 mK2 for the self-calibration, down from 984.3 mK2 for the
standard calibration.
#### 2.4.2 RFI contaminations
From the 41 observation blocks, we had to remove 14 blocks due to excessive,
low-level RFI contaminations. In affected blocks, these contaminations were
sometimes recognisable on the map level of individual dish data as horizontal
stripes. After careful investigations, we have now identified the source of
the RFI contamination as the downlink of the mobile communication tower
(${\sim}\,54\,\text{km}$ away) that enters through the sidelobes and causes a
non-linear gain effect in the entire band. Our aggressive flagging approach in
this study, with the entire removal of 14 blocks, was to ensure high data
quality for our analysis. We are currently developing a method to correct for
this effect, which should allow us to use the currently flagged data for
future projects.
#### 2.4.3 Noise level estimation
Here we estimate the noise levels from the final maps to check if they are
consistent with theoretical expectations. We use the final data cube, $T_{\rm
sky}$, to estimate the noise level using the difference between four
neighbouring channels (the so-called ABBA technique). By adopting the ABBA
pattern, this difference statistic removes both constant and linearly varying
signals to cancel signals detected by the receiver, leaving only the noise.
The noise level is estimated by
$\displaystyle\Delta T_{i}(\nu^{\star})=\frac{1}{2}\left(T^{i}_{\rm
sky}(\nu)+T^{i}_{\rm sky}(\nu+\delta\nu)\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{1}{2}\left(T^{i}_{\rm
sky}(\nu-\delta\nu)+T^{i}_{\rm sky}(\nu+2\delta\nu)\right),$ (8)
where the index $i$ goes over all pixels in the map and ⋆ means the combined
result from the four channels
$[\nu{-}\delta\nu,\,\nu,\,\nu{+}\delta\nu,\,\nu{+}2\delta\nu]$. From the
observation data, we take a weighted rms,
$\displaystyle\Delta T_{\rm
RMS}^{2}(\nu^{\star})=\frac{N_{\text{pix}}}{N_{\text{pix}}-1}\left(\frac{\sum_{i}w_{i}\Delta
T^{2}_{i}}{\sum_{i}w_{i}}-\left[\frac{\sum_{i}w_{i}\Delta
T_{i}}{\sum_{i}{w_{i}}}\right]^{2}\right),$ (9)
where the sum is over the number of pixels in the map, $N_{\text{pix}}$, and
we have suppressed the dependence of $w_{i}$ and $\Delta T_{i}$ on
$\nu^{\star}$. For the weight, $w_{i}$, we used the theoretical expected
variance itself, $w_{i}(\nu^{\star})=1/\sigma^{2}_{{\rm th},i}(\nu^{\star})$.
The corresponding variance in each pixel would be,
$\displaystyle\sigma^{2}_{{\rm
th},i}(\nu^{\star})=\frac{1}{4}\Big{(}\sigma^{2}_{{\rm
th},i}(\nu)+\sigma^{2}_{{\rm th},i}(\nu{+}\delta\nu)+\sigma^{2}_{{\rm
th},i}(\nu{-}\delta\nu)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sigma^{2}_{{\rm
th},i}(\nu{+}2\delta\nu)\Big{)}.$ (10)
Here the noise in the initial polarised TOD is given by the radiometer
equation
$\sigma^{2}_{\rm th,pol}(t,\nu)=\frac{T^{2}_{\rm
sys}(t,\nu)}{\delta\nu\,\delta t},$ (11)
where $\delta\nu\,{=}\,0.209\,\text{MHz}$ is the frequency width of the
MeerKAT L-band channels and $\delta t\,{=}\,2$ s is the length of each time
stamp recording. For the system temperature we use the calibrated data itself,
$T_{\rm sys}\,{=}\,T_{\rm cal,HH}(t,\nu)$ or $T_{\rm cal,VV}(t,\nu)$ for
horizontal and vertical polarisation data, respectively. Thus the noise in the
initial intensity TOD is
$\sigma^{2}_{\rm th}(t,\nu)=\frac{T^{2}_{\rm cal,HH}(t,\nu)+T^{2}_{\rm
cal,VV}(t,\nu)}{4\,\delta\nu\,\delta t}.$ (12)
In order to get the final data cube, $T^{i}_{\rm sky}(\nu)$, the data goes
through several stages of averaging. We propagate the variance taking into
account this averaging in order to get to the final $\sigma^{2}_{{\rm
th},i}(\nu)$.
The theoretical rms which can be compared to $\Delta T_{\rm RMS}(\nu^{\star})$
in Equation 9, is then simply
$\displaystyle\sigma^{2}_{\rm
th}(\nu^{\star})=N_{\text{pix}}\left(\sum_{i}\sigma^{-2}_{{\rm
th},i}(\nu^{\star})\right)^{-1}.$ (13)
More details can be found in Section 5.6 of Wang et al. (2021).
Figure 6: Comparison between noise levels derived from calibrated
observational data ($\Delta T_{\rm RMS}(\nu^{\star})$) and from theory
equation ($\sigma_{\rm th}(\nu^{\star})$) in 971–1075 MHz (upper panel), a
zoom-in for above in 971–1023 MHz (middle panel), and the histogram of the
ratio $R_{\rm RMS}=\Delta T_{\rm RMS}(\nu^{\star})/\sigma_{\rm
th}(\nu^{\star})$ in 971–1023 MHz (lower panel).
In the upper panel of Figure 6, we show the noise levels (from data and
theory) in the 971–1075 MHz frequency band (channels 550–1050). Note here both
the real and theory values are related to the real $T_{\rm sys}$ and hit
counts in corresponding map pixels. The count reduction due to RFI flagging,
and unflagged low-level RFIs that are hiding in the data would cause wiggles.
Four groups of spikes are present at higher frequencies, while below 1023 MHz,
there is much less scatter compared to the previous MeerKLASS pilot survey
result. We therefore chose to restrain our cosmological analysis to the
971–1023 MHz region, for which we show the noise level in the middle panel,
finding a median of $\Delta T_{\rm RMS}\,{=}\,1.21\,$mK. In the lower panel of
Figure 6, we also show the histogram of the ratio $R_{\rm RMS}\,{=}\,\Delta
T_{\rm RMS}(\nu^{\star})/\sigma_{\rm th}(\nu^{\star})$ in the 971–1023 MHz
band. We find a median value of $R_{\rm RMS}\,{=}\,1.2$, while some channels
are noisy and $R_{\rm RMS}$ can reach $1.5$.
### 2.5 GAMA spectroscopic galaxies
Overlapping with the MeerKLASS deep field, is a catalogue of spectroscopic
galaxies recorded in the Galaxy and Mass Assembly (GAMA) survey. GAMA began
observing in February 2008 (Driver et al., 2011; Liske et al., 2015). The
survey aimed to construct catalogues of galaxies to test the cold dark matter
paradigm and probe galaxy evolution (Driver et al., 2009). For our purposes,
we exploit the fact that these galaxies will trace dark matter and should
therefore correlate with the overlapping MeerKLASS intensity maps after
foreground cleaning.
We use the fourth and final data release from the Galaxy and Mass Assembly
survey, GAMA DR4 (Driver et al., 2022). This includes galaxies from the GAMA
II survey phase which enlarged the GAMA I equatorial survey, by including
extra survey regions. One of these extra regions was the 23 hr (G23) field,
which completely overlaps in area with the MeerKLASS L-band deep field, shown
by the blue scatter points in the bottom panel of Figure 4, which represent
the GAMA G23 galaxies relative to the observational time footprint map for the
L-band deep field. The G23 field covered
$339\,\deg\,{<}\,\text{R.A.}\,{<}\,351\,\deg$ and
${-}35\,\deg\,{<}\,\text{Dec.}\,{<}\,{-}30\,\deg$. The magnitude limit of the
field was reduced from the planned 19.8 in $r$-band to 19.2 in $i$-band due to
limited observational time, but this allowed the GAMA II survey to obtain a
high $94.2\%$ redshift completion in the G23 region (Liske et al., 2015).
The GAMA DR2 only provides random catalogues that emulate the survey selection
function of the real galaxies for the G09, G12, and G15 regions, (not the G23
used in this work). Ideally, these would be used to construct a survey
completion window function, which would be used in the cross-power spectrum
estimation. In the absence of this, we assume a uniform survey completion
across the G23 area and a constant distribution with redshift, given the very
flat distribution for the redshift range used in this analysis (green region
of Figure 7). Given our signal-to-noise levels, survey masks will be a sub-
dominant effect in the cross-power, thus this is an adequate approximation to
make.
Due to our aggressive flagging strategy on the MeerKLASS L-band data, the
overlap region in redshift is restricted to $0.39\,{<}\,z\,{<}\,0.46$. As
shown by Figure 7, this catches the GAMA survey galaxy redshift distribution,
$N_{\text{g}}(z)$, near the end of its optimal range, but still provides 2269
galaxies to cross-correlate with. At this redshift range, the comoving
(volumetric) number density of the GAMA galaxies is
$\bar{n}_{\text{g}}\,{=}\,4.8\,{\times}\,10^{-4}\,h^{3}\,\text{Mpc}^{-3}$.
Figure 7: Redshift distribution of GAMA G23 galaxies in the 23hr field. The
accessible redshifted 21cm emission for the full L-band range is shown by the
light-pink shading. The region used in this work between
$0.39\,{<}\,z\,{<}\,0.46$ ($1023.6\,{<}\,\nu\,{<}\,971.2\,\text{MHz}$) is
shown by the darker green shading, in which 2269 GAMA galaxies exist.
## 3 Simulation of mock data
To accompany the calibrated observations outlined in Section 2, we produce a
suite of 500 mock simulations, used for validating the analysis pipeline,
calculating signal loss from foreground cleaning and estimating covariance in
the power spectrum measurements. To create this large number of mocks we
generate lognormal density fields (Coles & Jones, 1991) from which a map of Hi
brightness temperatures can be cut, and galaxies Poisson sampled onto for the
spectroscopic galaxy counterpart.
An important stage of the analysis pipeline is regridding the intensity
mapping voxels with celestial sky coordinates (R.A., Dec., $\nu$) onto a
regular 3D Cartesian grid with lengths $l_{\text{x}}$, $l_{\text{y}}$,
$l_{\text{z}}$ in $h^{-1}\text{Mpc}$, on which a Fast Fourier Transform (FFT)
can be applied for the power spectrum estimation (Cunnington & Wolz, 2024). We
therefore require the mocks to begin in celestial coordinates so we can
validate the regridding process. To achieve this, we begin by fitting a
Cartesian cube around the full MeerKLASS deep-field footprint, creating a grid
with dimensions
$l_{\text{x}},l_{\text{y}},l_{\text{z}}\,{=}\,774.9,335.7,252.2\,h^{-1}\text{Mpc}$
over $n_{\text{cell}}\,{=}\,512^{3}$ cells. We use a large number of cells for
this initial grid to approximately emulate a continuous sky, aiming to avoid
the simulated maps resolving the finite input grid structure. Onto this grid,
we simulate a lognormal density field. This is done by interpolating
$\boldsymbol{k}$-modes from a model power spectrum onto the FFT grid. The Hi
model power spectrum is given by
$P_{\textrm{H{i}}}(\boldsymbol{k})\equiv
P_{\textrm{H{i}}}(k,\mu)=\frac{\overline{T}_{\textrm{H{i}}}^{2}\,b_{\textrm{H{i}}}^{2}\,\left(1+f\,\mu^{2}/b_{\textrm{H{i}}}\right)^{2}\,P_{\text{m}}(k)}{1+\left(k\,\mu\,\sigma_{\text{v}}/H_{0}\right)^{2}}\,.$
(14)
The anisotropy in this model makes it convenient to parameterise
$\boldsymbol{k}$ using $\mu$, defined as the cosine of the angle between the
line-of-sight and $\boldsymbol{k}$. The Hi parameters are the mean Hi gas
temperature ($\overline{T}_{\textrm{H{i}}}$) and the Hi bias
($b_{\textrm{H{i}}}$). $f$ is the growth rate of structure and is included to
model linear redshift space distortions (RSD) which enhance clustering across
all scales. The denominator is included to approximately model the non-linear
RSD; the so-called Fingers-of-God (FoG), which suppresses clustering at small
scales. The FoG are modulated by $\sigma_{\text{v}}$, which is the velocity
dispersion of the tracer, in this case, the Hi gas. All of these quantities
have a redshift dependence, which we omit for brevity; they will also have
little variation given our narrow redshift range. Lastly, $H_{0}$ is the
Hubble parameter at $z\,{=}\,0$, known as the Hubble constant. An alternative
process for generating brightness temperatures is pursued for our stacking
analysis discussed later in the paper. More realistic lognormal line intensity
mapping mocks are available. For example, in future work, we intend to adopt
the approach in Niemeyer et al. (2023) where luminosities from the spectral
lines are painted given a luminosity function. This has been shown to
reproduce better RSD for galaxy distributions and models the whole temperature
distribution expected in the maps.
Inverse Fourier transforming the model in Equation 14 provides the Hi
correlation function $\xi_{\textrm{H{i}}}$ which is then log-transformed via
$\xi^{\prime}_{\textrm{H{i}}}\,{=}\,\ln(1\,{+}\,\xi_{\textrm{H{i}}})$. Fourier
transforming $\xi^{\prime}_{\textrm{H{i}}}$ back gives the modified lognormal
power spectrum $P^{\prime}_{\textrm{H{i}}}$. Gaussian random numbers are then
sampled into each cell of the Fourier grid $\delta_{\text{G}}(\boldsymbol{k})$
with variance $\sigma^{2}(\boldsymbol{k})\propto
P^{\prime}_{\textrm{H{i}}}(\boldsymbol{k})$. This Gaussian random field is
then inverse Fourier transformed to real space giving
$\delta_{\text{G}}(\boldsymbol{x})$ from which the lognormal Hi brightness
temperature fluctuation field is given by
$\delta
T_{\textrm{H{i}}}(\boldsymbol{x})=\exp\left[\delta_{\text{G}}(\boldsymbol{x})-\frac{\sigma^{2}_{\text{G}}}{2}\right]-1\,,$
(15)
where $\sigma_{\text{G}}^{2}$ is the measured variance of the Gaussian random
number field $\delta_{\text{G}}$. Further details on this process can be found
in the Appendix A of Beutler et al. (2011). For the galaxy over-density field
$\delta_{\text{g}}(\boldsymbol{x})$, we follow the same procedure except the
input model is for the galaxy power spectrum, given by
$P_{\text{g}}(\boldsymbol{k})\equiv
P_{\text{g}}(k,\mu)=\frac{b_{\text{g}}^{2}\,\left(1+f\,\mu^{2}/b_{\text{g}}\right)^{2}\,P_{\text{m}}(k)}{1+\left(k\,\mu\,\sigma_{\text{v}}/H_{0}\right)^{2}}\,,$
(16)
where $b_{\text{g}}$ is the bias for the galaxies. We assume that the velocity
dispersion $\sigma_{\text{v}}$ for the Hi gas and galaxies is the same for
simplicity. This will be sufficient for this work where signal-to-noise is low
at the small scales where $\sigma_{\text{v}}$ has the main effect.
The galaxy over-density $\delta_{\text{g}}$ is unit-less whereas the intensity
maps $\delta T_{\textrm{H{i}}}$ are brightness temperature fluctuations
measured in mK. We use the same seed in the random number generation to ensure
that both the $\delta T_{\textrm{H{i}}}$ and $\delta_{\text{g}}$ correlate.
The lognormal code we adopt utilises some functions from the publically
available package powerbox (Murray, 2018).555github.com/steven-murray/powerbox
### 3.1 Transforming between Cartesian grids and sky maps
As already mentioned, the galaxies and the intensity mapping voxels from
observations are in celestial coordinates at a certain frequency (or redshift)
i.e.
$(\boldsymbol{\theta},\nu)\,{\equiv}\,(\theta_{\text{R.A.}},\theta_{\text{Dec.}},\nu)$.
However, for the final power spectrum estimation, we require the fields be in
a Cartesian space with dimensions ($l_{\text{x}},l_{\text{y}},l_{\text{z}}$)
in $h^{-1}\text{Mpc}$. For this we follow the recent work in Cunnington & Wolz
(2024) which we discuss further in Section 5.
Creating lognormal mocks in celestial coordinates that emulate observations
requires the inverse of this step i.e. $\delta
T_{\textrm{H{i}}}(\boldsymbol{x})\,{\rightarrow}\,\delta
T_{\textrm{H{i}}}(\boldsymbol{\theta},\nu)$, since the lognormal fields are
generated onto Cartesian grids measured in $h^{-1}\text{Mpc}$. Transforming
mock galaxy coordinates (the Poisson sampling of which is discussed in Section
3.3) from
$(l_{\text{x}},l_{\text{y}},l_{\text{z}})\,[h^{-1}\text{Mpc}]\,{\rightarrow}\,(\theta_{\text{R.A.}},\theta_{\text{Dec.}},\nu)$
is trivial. We can use astropy666docs.astropy.org (Astropy Collaboration et
al., 2022) transformation routines on each galaxy’s coordinate. However,
transforming the tessellating field of Cartesian cells, each with an
associated brightness temperature, into a gap-less map of voxels with
identical celestial coordinates and frequency binning as the real MeerKAT
maps, requires a resampling of the field. For this we also utilise the
techniques and public code777github.com/stevecunnington/gridimp outlined
Cunnington & Wolz (2024).
This process involves a Monte-Carlo particle sampling technique where each
Cartesian cell is filled with $N_{\text{p}}$ particles, randomly distributed
within the cell. The particles are assigned the brightness temperature of the
cell they fall within. The particles are then transformed to celestial
coordinates ($\theta_{\text{R.A.}},\theta_{\text{Dec.}},z$) and added into the
map voxels associated with their celestial coordinate and the frequency
channel corresponding to their redshift. Each voxel is normalised by the
number of particles falling within it. Providing enough sampling particles per
cell are used (in this work we use $N_{\text{p}}\,{=}\,10$) and the Cartesian
grid is of sufficient resolution (hence our choice of
$n_{\text{cell}}\,{=}\,512^{3}$), it will provide an intensity map in
celestial sky coordinates, that traces the fluctuations generated by the
lognormal field.
Throughout this work, we do not require the necessary extensions of higher-
order particle assignment or field interlacing to suppress the effects of
aliasing (as trialled in Cunnington & Wolz, 2024), since these effects are
below the current statistical noise from this data. We are however, still
required to implement a correction to the power spectrum estimation from the
Nearest-Grid-Point assignment which smooths the field, which we formalise in
Section 5.
### 3.2 Hi intensity map mocks
Using the regridding process described in the previous section we are provided
with Hi intensity mapping mocks $\delta
T_{\textrm{H{i}}}({\boldsymbol{\theta},\nu})$, with identical coordinates and
dimensions as the observed MeerKAT intensity maps. From here we can add in
observational effects and simulate various systematics. The most dominant
effect comes from foreground contamination. Systematics will have a large
influence on foreground contamination, and this can be extremely complex.
Therefore, this is not something we attempt to directly simulate in this work
since we risk under-estimating the problem. Instead, we use the observational
data itself which has the relevant systematics contained within, albeit only
one realisation of them. However, adding these directly onto different
realisations of Hi mocks, still recreates stochasticity in the Hi signal’s
response to systematics. We discuss this further in Section 4.
Another important addition to the mocks will be a smoothing to the intensity
maps corresponding to the angular pixelisation. This is necessary to emulate
the radio telescope’s beam, which for single-dish intensity mapping is quite
broad giving a low angular resolution to the maps and damping small-scale
fluctuations perpendicular to the line-of-sight. In this work, we assume the
MeerKAT beam profile to be Gaussian, which is a simplification but still valid
given the signal-to-noise at small scales where the beam suppresses signal
below the noise. We will investigate implementing a more realistic MeerKAT
beam profile in future work (see e.g. Asad et al., 2021; Matshawule et al.,
2021; Spinelli et al., 2022). The assumed Gaussian beam profile has a
frequency ($\nu$) dependent standard deviation in radians given by
$\sigma_{\text{beam}}(\nu)=\frac{1}{2\sqrt{2\ln 2}}\frac{c}{\nu
D_{\text{dish}}}\,,$ (17)
where we use $D_{\text{dish}}\,{=}\,13.5\,\text{m}$ for the diameter of the
MeerKAT dishes and $c$ is the speed of light. To emulate the beam effects we
smooth the mock maps by creating a 2D Gaussian array kernel centred on the
median angular pixel $\theta_{0}$ given by
$\pazocal{B}_{\text{beam}}(\boldsymbol{\theta},\nu_{i})=\exp\left[-\frac{1}{2}\left(\frac{\boldsymbol{\theta}-\theta_{0}}{\sigma_{\text{beam}}(\nu_{i})}\right)^{2}\right]\,.$
(18)
This is normalised such that
$\sum_{\boldsymbol{\theta}}\pazocal{B}_{\text{beam}}\,{=}\,1$, so that
convolution with this beam kernel does not change the total brightness
temperatures within the maps. The Hi intensity map mocks are convolved with
Equation 18 at each frequency $\nu_{i}$ to give the beam-smoothed Hi map;
$\delta T_{\circledast\textrm{H{i}}}(\boldsymbol{\theta},\nu_{i})=\delta
T_{\textrm{H{i}}}(\boldsymbol{\theta},\nu_{i})\circledast\pazocal{B}_{\text{beam}}(\boldsymbol{\theta},\nu_{i})\,,$
(19)
where the $\circledast$ symbol represents a convolution.
To realistically emulate additional components such as foregrounds, noise and
their response to systematics, we add the real observations to each mock
iteration. This is to avoid the requirement of simulating the complex and
poorly understood systematics.
In some cases, mostly for studying the Hi auto-correlation, it will be useful
to simulate just the thermal noise originating from the thermal signature of
the receivers. We do this by generating Gaussian random noise the power of
which can then be measured with the same pipeline as the rest of the analysis,
providing a robust estimate of the thermal noise floor. We begin with an
analytical description of the thermal noise fluctuations per pixel using the
radiometer equation (as in Section 2 but now at map-level)
$\sigma_{\text{N}}(\boldsymbol{\theta},\nu)=\frac{T_{\text{sys}}(\nu)}{\sqrt{2\,\delta\nu\,t_{\text{p}}(\boldsymbol{\theta},\nu)}}\,,$
(20)
where $t_{\text{p}}$ is the summed observation time contained in each voxel,
which can be taken directly from the time stamp counts as shown in Figure 4,
with each count representing $\delta t\,{=}\,2\,$sec. This emulates the real
data by creating a non-uniform noise map, generating higher noise around the
edges where scan coverage is lower. For the system temperature, we follow
Cunnington (2022) and use a model tuned to coincide with the estimated system
temperature of the MeerKAT telescope (Wang et al., 2021)888See
skaafrica.atlassian.net/wiki/x/AYCHE, assuming
$T_{\mathrm{sys}}(\nu)=T_{\mathrm{rx}}(\nu)+T_{\mathrm{spl}}+T_{\mathrm{CMB}}+T_{\mathrm{gal}}(\nu)\,,$
(21)
where we assume the contribution from our own galaxy is
$T_{\mathrm{gal}}\,{=}\,25\text{K}\,(408\mathrm{MHz}/\nu)^{2.75}$, the CMB
temperature is $T_{\text{CMB}}\,{=}\,2.725\,\text{K}$, as in Section 2. The
spill-over temperature is assumed as $T_{\text{spl}}\,{=}\,3\,\text{K}$ and
the receiver temperature we model as
$T_{\mathrm{rx}}(\nu)=7.5\mathrm{K}+10\mathrm{K}\left(\frac{\nu}{\mathrm{GHz}}-0.75\right)^{2}\,.$
(22)
Due to the narrow frequency band we use, this gives a near-constant system
temperature with the range
$15.97\,{<}\,T_{\text{sys}}\,{<}\,16.02\,[\text{K}]$. Gaussian random numbers
can then be generated across the MeerKLASS deep-field map with the voxel
dependent rms of $\sigma_{\text{N}}(\boldsymbol{\theta},\nu)$.
### 3.3 Galaxy mock catalogues
To accompany the mock intensity maps, we also require corresponding galaxy
mocks with the same correlated clustering. Rather than transforming the
Cartesian galaxy over-density field $\delta_{\text{g}}(\boldsymbol{x})$ into
sky coordinates, it is more realistic to instead Poisson sample the field to
generate a catalogue of galaxies with the same number density as the GAMA G23
survey. This also allows additional validity checks such as testing the
stacking pipeline which cannot be done without a catalogue of galaxy
coordinates.
We generate the galaxy catalogue coordinates initially in the Cartesian domain
to exploit the high $n_{\text{cell}}\,{=}\,512^{3}$ resolution. We first
calculate the average galaxy counts in cells $\langle{n}_{\text{g}}\rangle$
for the real GAMA G23 region in the redshift range where there is overlap with
the intensity map frequency range. We then take the galaxy survey selection
window function, which for the GAMA G23 region, we have to assume is constant
between $339\,{<}\,\text{R.A.}\,{<}\,351\,\deg$ and
${-}35\,{<}\,\text{Dec.}\,{<}\,{-}30\,\deg$ (see discussion in Section 2.5).
We then transform this window function into the high resolution Cartesian
space. A galaxy number counts field $n_{\text{g}}(\boldsymbol{x})$ is then
given by
$n_{\text{g}}(\boldsymbol{x})=\langle
n_{\text{g}}\rangle\left(\delta_{\text{g}}(\boldsymbol{x})+1\right)W_{\text{g}}(\boldsymbol{x})\,,$
(23)
where we use the transformed window function $W_{\text{g}}(\boldsymbol{x})$
and the real GAMA average counts $\langle n_{\text{g}}\rangle$ to ensure the
mock galaxy counts will share the same survey selection as the real galaxy
survey, with the same number density. In this process we normalise the window
function $W_{\text{g}}$ such that its maximum value is $1$, so for the GAMA
G23 survey, this becomes a binary mask of 0 and 1. We then Poisson sample the
$n_{\text{g}}$ field to introduce some stochasticity which we expect to be
present in the real galaxy catalogues. This provides an integer value in each
Cartesian cell for the galaxy counts. The last step is to assign random
Cartesian coordinates within the cell boundaries for the
$n_{\text{g}}(\boldsymbol{x})$ galaxies within each cell. At this stage we
rely on the Cartesian field having sufficiently high resolution such that the
random assignment of a coordinate within the cell does not shift the galaxies
away from the peaks of the density fluctuations. The catalogue of galaxy
coordinates (x, y, z [$h^{-1}$Mpc]) can then be transformed into celestial
coordinates $(\theta_{\text{R.A.}},\theta_{\text{Dec.}},z)$.
In this work, we generate 500 mock intensity maps with corresponding mock GAMA
galaxy catalogues. As some early validation of these mocks, we show in Figure
8 how the power spectra for both the Hi intensity map and galaxy auto-
correlations, along with their cross-correlation, agree with the fiducial
model (black dashed line). This demonstrates that our estimator is unbiased
and processes such as the regridding are performing to within an acceptable
tolerance. The power spectrum estimation process used here on the mocks is
identical to that used on the real data and is detailed later in Section 5,
along with the modelling.
Figure 8: Averaged power spectra (blue solid line) of the 500 regridded mocks
and agreement with the fiducial model (black dashed line). Mocks include the
beam but no other systematics. Green dotted lines in the bottom panels mark
the edges of the $10\%$ residual region, while the shaded regions mark the
diagonal variance inferred from the simulations.
The mocks used in the test of Figure 8 do not include any additional
systematics beyond the telescope beam in the intensity maps. We will
investigate the inclusion of foregrounds and systematics in Section 4. The
large scatter shown by the shaded blue region, which represents the
$1\,\sigma$ range across the 500 mocks, is therefore driven by cosmic variance
due to the small volume covered by the data. We see that the errors for the
galaxy power spectrum appear larger than the Hi intensity maps. This is
because we have not introduced any thermal noise (or other observational or
systematic effects) into the intensity maps. However, the galaxy catalogues
will have shot noise contributing to the uncertainties. This nicely showcases
the potential for Hi intensity mapping if noise and systematics can be
controlled, however, this is not the purpose of this exercise so is not a
point we will labour. There are also additional caveats when considering high-
density tracers such as Hi. As shown by Foreman et al. (2024), despite the low
shot noise, stochastic noise and non-linear bias in the Hi field can limit the
potential of constraints inferred from the power spectrum.
The green dashed lines in the residual panels mark the regions where there is
${>}\,90\%$ accuracy relative to the model which we can see is achieved at all
scales for all three cases. This is also well within the statistical
uncertainty (blue-shaded regions). The largest residual comes in the cross-
correlation where discrepancies seem to increase with $k$ nearly reaching the
$-10\%$ boundary. We speculate that this may be caused by the Poisson sampling
of mock galaxies followed by random assignment of coordinates within the cell.
This extra stochasticity is not repeated in the Hi intensity map mocks and
could therefore be generating a pseudo-cross-correlation coefficient, which
would depart from $r\,{=}\,1$ at smaller scales (higher-$k$). However, this
appears to be a minor effect. Furthermore, this could be seen as a feature,
since we expect $r\,{<}\,1$ in the real observations, which will also have a
scale dependence. For these reasons, we deem this to be a suitable result for
the purposes of this work, and defer a higher precision investigation and
validation to future work.
## 4 Foreground cleaning
Foreground cleaning Hi intensity maps is a well-studied problem (Santos et
al., 2005; Ansari et al., 2012; Wolz et al., 2014; Alonso et al., 2015; Shaw
et al., 2015; Olivari et al., 2016; Carucci et al., 2020; Cunnington et al.,
2021; Spinelli et al., 2022; Soares et al., 2022; Irfan & Bull, 2021), and
generally blind cleaning techniques are the currently preferred method for
single-dish experiments. As their name suggests, blind methods do not rely on
prior knowledge of the foreground structure. This is advantageous for current
experiments because the foregrounds at these frequencies lack observational
constraints. Furthermore, blind techniques avoid placing overwhelmingly
stringent calibration requirements on the data, since the dominant foregrounds
require beyond sub-percent precision modelling for a successful subtraction.
Blind techniques instead work by exploiting the foreground dominance and their
frequency-correlated structure to assume that they can be reduced to a
distinct set of correlated modes separated from the oscillating, sub-dominant
fluctuations which will contain most of the Hi information.
We outline the formalism for our blind foreground cleaning approach in Section
4.3, but first we discuss two pre-processing steps before foreground cleaning.
These are designed to improve the foreground clean by reducing the impact from
residual RFI or systematics, plus any known frequency-dependent observational
effects.
### 4.1 Reconvolution
All radio intensity mapping experiments will have a chromatic beam whose size
evolves with frequency. This can add to the number of degrees of freedom in
the radial foreground structure, inhibiting foreground removal since more
modes need to be removed to perform a suitable clean. Smoothing the maps
differently in each frequency channel can homogenise the angular resolution,
giving them a consistent effective beam size at all frequencies. This
reconvolution should also suppress residual contamination manifesting as
localised amplitude spikes in the temperature maps, smoothing them out so
there are fewer discontinuous fluctuations which will impact the PCA and
Fourier transforms for power spectrum estimation.
The reconvolution step involves convolving the intensity maps, one frequency
channel $\nu_{i}$ at a time with the kernel
$\pazocal{K}(\boldsymbol{\theta},\nu_{i})=\exp\left[-\frac{1}{2}\frac{\left(\boldsymbol{\theta}-\theta_{0}\right)^{2}}{\gamma\sigma_{\max}^{2}-\sigma_{\text{beam}}^{2}(\nu_{i})}\right]\,,$
(24)
where $\sigma_{\text{beam}}$ is the beam size as previously defined in
Equation 17, $\sigma_{\text{max}}$ is the maximum
$\sigma_{\text{beam}}(\nu_{\text{min}})$ value at the lowest frequency, and
$\gamma$ is a scaling factor which governs how much the final effective
resolution decreases. The reconvolution kernel is normalised such that the sum
over all pixels is unity i.e.
$\Sigma_{\boldsymbol{\theta}}\pazocal{K}(\boldsymbol{\theta},\nu_{i})\,{=}\,1$.
As in previous work (Cunnington, Li et al., 2022), we use a weighted
reconvolution to mitigate the impact of edge pixels with low scan coverage.
From hereon, the weights for the intensity maps derive directly from the time
stamp count in each pixel, as shown in Figure 4. The weighted reconvolved
intensity maps are given by
$\delta T_{\circledast}(\boldsymbol{\theta},\nu_{i})=\frac{\left[\delta
T(\boldsymbol{\theta},\nu_{i})w(\boldsymbol{\theta},\nu_{i})\right]\circledast\pazocal{K}(\boldsymbol{\theta},\nu_{i})}{w(\boldsymbol{\theta},\nu_{i})\circledast\pazocal{K}(\boldsymbol{\theta},\nu_{i})}$
(25)
The weights themselves must also be reconvolved to keep them consistent and
optimal for the new $\delta T_{\circledast}$ maps. The reconvolved weights are
given by
$w_{\circledast}(\boldsymbol{\theta},\nu_{i})=\frac{\left[w(\boldsymbol{\theta},\nu_{i})\circledast\pazocal{K}(\boldsymbol{\theta},\nu_{i})\right]^{2}}{w(\boldsymbol{\theta},\nu_{i})\circledast\pazocal{K}^{2}(\boldsymbol{\theta},\nu_{i})}\,.$
(26)
In this work we use a value of $\gamma\,{=}\,1.4$ for the reconvolution kernel
in Equation 24. We find that neglecting this resmoothing step entirely
increases the influence of systematics in the results which is likely due to
residual RFI or contaminants not being smoothed and depressing their
dominance. We discuss this further in Appendix B.1 but also plan a dedicated
study into this aspect of the analysis where a more robust model of the
MeerKAT beam will also be implemented.
The resmoothed, uncleaned map is shown in Figure 9 (top panel). Careful
comparison with Figure 4 reveals how some of the foreground structure has been
smoothed, albeit subtly, which is the aim. Too much reconvolution will
suppress power in high $k_{\perp}$ modes in the final power spectrum estimate.
The reconvolution is more noticeable in the time stamp count field, which we
use as the weights $w_{\circledast}$, shown by the bottom panel of Figure 9.
Figure 9: Reconvolved MeerKLASS deep field (top-panel) using
$\gamma\,{=}\,1.4$ (see Equation 25). The bottom panel shows the reconvolved
time-stamp field which we use for weighting, thus also requires reconvolution
(using Equation 26) for consistency with the intensity maps. Both panels have
been averaged along frequency.
### 4.2 Map trimming
Shown by the time stamp count field of Figure 4 (and the resmoothed version in
Figure 9), the MeerKLASS scanning strategy of combining numerous rising and
setting scans does not generate a uniform coverage. The edges of the patch
result in relatively low recorded time stamps due to a lack of overlapping
scans in these regions.
We found that implementing an edge trim to the field improved the quality of
the foreground clean and final power spectrum estimates. This is motivated and
explored in greater detail in Carucci et al. (in prep.). The improvement from
map trimming is because pixels are removed where residual time-varying
contamination has been averaged down less, due to fewer scans.
In Figure 9, the black contour shows the cut to the patch we make, removing
everything outside the trimmed lines. This amounts to all pixels at
$334^{\circ}\,{<}\,\text{R.A.}\,{<}\,357^{\circ}$ being set to zero in both
the weights and intensity map field.
### 4.3 Blind foreground cleaning with PCA
We represent the intensity maps with matrices X which are equivalent to the
$T(\boldsymbol{\theta},\nu)$ temperature fields, discretised into arrays on
which matrix algebra can be performed.
Blind foreground cleaning assumes the matrix of the observational data cube
$\textbf{{X}}_{\mathrm{obs}}$, which contains the intensity maps with
$n_{\theta}$ pixels at each of the $n_{\nu}$ frequency channels, can be
generalised by the linear equation
$\textbf{{X}}_{\mathrm{obs}}=\textbf{{A}}\textbf{{S}}+\textbf{{R}}\,.$ (27)
A is referred to as the mixing matrix and is the set of $N_{\text{fg}}$
spectral functions estimated to project out the foregrounds. S are the
$N_{\text{fg}}$ source maps obtained by projecting A along the data;
$\textbf{{S}}=\textbf{{A}}^{\text{T}}\textbf{{X}}_{\mathrm{obs}}\,.$ (28)
The dot product AS thus has shape $[n_{\nu},n_{\theta}]$, along with the other
terms in Equation 27, and represents a subset of the observations estimated to
contain the bulk of the foreground contamination. The residual term R will
contain the remaining Hi signal and thermal noise. In reality, the situation
is more complex and there is cross-term leakage. Both Hi signal will leak into
the subtracted AS term causing signal loss, and foreground will leak into the
remaining R causing foreground residuals. The control of both these is crucial
and we discuss them further throughout the remaining sections.
Different blind foreground techniques only vary in their approach to
estimating the mixing matrix A. In this work, we adopt the straightforward
principal component analysis (PCA) approach, since it is well-tested and
understood. Implementing more sophisticated blind techniques is planned in
future work. PCA foreground cleaning begins by computing the frequency
covariance matrix of the weighted observations
$\textbf{{C}}=\frac{\left(\textbf{{w}}\circ\textbf{{X}}_{\mathrm{obs}}\right)^{\mathrm{T}}\left(\textbf{{w}}\circ\textbf{{X}}_{\mathrm{obs}}\right)}{n_{\theta}-1}\,,$
(29)
where we have used the intensity mapping weights to optimise the estimation
and $\textbf{{w}}\circ\textbf{{X}}_{\mathrm{obs}}$ represents the element-wise
product between the weights and the observational data matrix. Since the
weights are the stacked average of the counts along the line-of-sight, they
are constant in frequency by design to avoid increasing the rank of
$\textbf{{X}}_{\mathrm{obs}}$ i.e. not inject additional structure into the
frequency spectra. Weighting in this way means C is not a formally normalised
covariance estimation, however, this is not a concern for us since we are only
extracting unitless eigenvectors from the covariance and using these to
project out modes in the real observed data so sensible amplitudes are
recovered in the component separated pieces. We discuss this further, with
some examples in Appendix B.2. C is then eigen-decomposed by
$\textbf{{C}}=\textbf{{U}}\boldsymbol{\Lambda}\textbf{{U}}^{\text{T}}\,,$ (30)
where U is the matrix of eigenvectors with the corresponding eigenvalues
contained in $\boldsymbol{\Lambda}$. The mixing matrix is then formed by
taking the first $N_{\text{fg}}$ column vectors $\boldsymbol{u}_{i}$ from U
i.e.
$\textbf{{A}}\,{=}\,[\boldsymbol{u}_{1},\boldsymbol{u}_{2},...,\boldsymbol{u}_{N_{\text{fg}}}]$.
The first 9 eigenmodes are shown in the Appendix plot Figure 22. It is at this
stage a choice is required for $N_{\text{fg}}$, which determines how many
modes are removed and deemed as foregrounds. If the $N_{\text{fg}}$ is too
high, the Hi signal loss increases; on the contrary, if $N_{\text{fg}}$ is too
low, residual contamination will dominate more. The estimate of the
foregrounds is provided by
$\textbf{{X}}_{\mathrm{fg}}\,{\equiv}\,\textbf{{A}}\textbf{{S}}$ which is
subtracted to provide the residuals, revealing the final cleaned data used in
the analysis i.e. $\textbf{{X}}_{\text{clean}}\,{\equiv}\,\textbf{{R}}$ where
$\textbf{{X}}_{\text{clean}}=\textbf{{X}}_{\mathrm{obs}}-\textbf{{X}}_{\mathrm{fg}}\,.$
(31)
Figure 10 shows the resulting cleaned maps by removing
$N_{\text{fg}}\,{=}\,10$ PCA modes. We show the maps split into six frequency
sub-bands (given in the bottom-left of each panel) to illustrate the changing
intensity structures. Each panel is the frequency average of the 42 channels
in each sub-band showing a reasonably consistent amplitude of temperature
fluctuations across the sub-bands. The amplitudes in the maps are still likely
driven by non-cosmological contributions, mostly thermal noise, but they are
not far from the $0.1\,\text{mK}$ fluctuations we expect from Hi. As we will
show statistically in Section 5, over large-scale fluctuations, the
predominant contribution comes from the cosmological Hi.
Figure 10: Foreground cleaned intensity maps for the MeerKLASS L-band deep
field in frequency sub-bands (stated in bottom left). Maps have been cleaned
by removing $N_{\text{fg}}\,{=}\,10$ PCA modes.
### 4.4 Foreground cleaning tests using mocks
In order to validate our foreground removal and estimate the signal loss it
induces, we run the same PCA cleaning technique on mock intensity maps. As
discussed in Section 3.2, simulating foreground contamination that emulates
the systematics and complex observational effects is beyond current
capabilities. We therefore use the real observations themselves and inject
mock Hi intensity maps into these. We then apply the foreground removal
pipeline on the mock-injected data and estimate the signal loss from the
injected mock whose original signal is known.
We follow the formalism outlined in (Cunnington, Li et al., 2022), beginning
with a Hi mock, whose matrices we notify by M to distinguish from the real
observation X. The mock is injected into the real observations i.e.
$\textbf{{X}}_{\text{obs}}+\textbf{{M}}_{\textrm{H{i}}}$. This is then cleaned
by projecting out a subset of modes defined by the mixing matrix, as in the
previous sections, such that
$\textbf{{M}}_{\text{clean}}=\left(\textbf{{X}}_{\text{obs}}+\textbf{{M}}_{\textrm{H{i}}}\right)-\textbf{{A}}\textbf{{A}}^{\text{T}}\left(\textbf{{X}}_{\text{obs}}+\textbf{{M}}_{\textrm{H{i}}}\right)\,,$
(32)
The data in $\textbf{{M}}_{\text{clean}}$ will now contain signal loss
emulating that of the real data clean, along with residual systematics and
noise from the real data.
We show the results of mocks contaminated by the real data in Figure 11. We
show measured power spectra averaged over all 500 mocks under different
scenarios (details of power spectrum estimation are provided in the following
Section 5). Firstly, the grey dotted line shows the Hi-only mocks,
$\textbf{{M}}_{\textrm{H{i}}}$, with no foreground cleaning effects,
representing the true power. The red solid line shows the result of adding the
real observations to the mock, then PCA cleaning as in Equation 32. The power
is higher than the grey line, despite the inevitable signal loss the clean
will have caused. This is evidence that the $\textbf{{M}}_{\text{clean}}$ maps
contain residual systematics and thermal noise, which are additively biasing
the power999There will also be a small contribution from the true Hi signal in
the observed data.. We can demonstrate this by considering the orange solid
line which is the cross-correlation of the cleaned mocks
$\textbf{{M}}_{\text{clean}}$ with the Hi-only mocks
$\textbf{{M}}_{\textrm{H{i}}}$, a test only possible with simulations. Since
the residual systematics and noise will be uncorrelated with the Hi-only
mocks, the distortion in power comes purely from signal loss, hence why it is
lower than the Hi-only power at all scales.
To isolate the contribution from unknown systematics, we need to account for
thermal noise and signal loss from foreground cleaning. The thermal noise
contribution can be approximated using Equation 20 to generate 500 noise maps
and add these to the Hi-only mocks. The black-dashed line shows this and
demonstrates the boost in power relative to the grey-dotted line, more so at
small scales where, due to the beam, the Hi signal is low relative to the
scale-independent noise101010The noise is not completely scale-independent
because it has gone through the same regridding and reconvolution processes as
the data which causes some subtle damping.. The last result presented in
Figure 11 is the reconstructed power (i.e. correcting for foreground cleaning
signal loss) shown by the blue solid line. This utilises the signal-loss
evaluated from the cross-power (orange line) to reconstruct the
$\textbf{{M}}_{\text{clean}}$ auto-power. This is discussed more in the
following sub-section. The discrepancy between the blue-solid and black-dashed
line must be caused by systematics, since signal-loss and thermal noise have
now been accounted for. We find a similar excess power with the real
observations, which we discuss in the following Section 5. The results in
Figure 11 are to validate the realism of the mock pipeline, and also to
motivate the signal loss reconstruction process, which we formalise in the
next sub-section.
Figure 11: Power spectra for the Hi intensity map mocks under different
scenarios, each averaged over 500 mock realisations. Including: Hi-only mock
(grey dotted line). Hi plus thermal noise realisations (black dashed). The Hi
mock with added foreground and systematics from the real MeerKLASS data, then
cleaned as in Equation 32 (red solid). The cross-correlation between Hi-only
and cleaned mock (orange solid). Lastly, the blue solid line is the auto-
correlation of the cleaned mock but with signal loss reconstructed.
### 4.5 Signal loss correction with transfer functions
We can utilise the method of mock signal injection from the previous Section
4.4 for estimating the signal loss in the power spectrum of the real data.
This is now a well-understood process (Switzer et al., 2015; Cunnington et
al., 2023) and used in previous single-dish intensity mapping analyses (Masui
et al., 2013; Anderson et al., 2018; Wolz et al., 2022; Cunnington, Li et al.,
2022).
We begin with the $\textbf{{M}}_{\text{clean}}$ term in Equation 32, which as
demonstrated, will contain signal loss that should emulate the real data. In
some cases, the cleaned real data $\textbf{{X}}_{\text{clean}}$ can be
subtracted from $\textbf{{M}}_{\text{clean}}$. This accelerates the
convergence of the transfer function calculation by removing the additional
noise caused by the real data but has no bearing on the signal loss
calculation. Leaving this piece in the calculation of
$\textbf{{M}}_{\text{clean}}$ is however crucial when using the transfer
function for estimating errors in the power spectrum. We also find that due to
the large number of mocks used in this work, an equivalent accuracy of the
transfer function is obtained whether subtracting
$\textbf{{X}}_{\text{clean}}$ or not. For simplicity, we therefore only use
the case as defined by Equation 32, without the $\textbf{{X}}_{\text{clean}}$
subtraction.
The transfer function is defined by the power spectrum, signified by the
operator $\pazocal{P()}$, of the cleaned mock $\textbf{{M}}_{\text{clean}}$
with the foreground-free Hi-only mock $\textbf{{M}}_{\textrm{H{i}}}$, and the
ratio of this with the power spectrum of the Hi-only mock
$\pazocal{T}_{\textrm{H{i}}}(\boldsymbol{k})=\left\langle\frac{\pazocal{P}\left(\textbf{{M}}_{\text{clean}},\textbf{{M}}_{\textrm{H{i}}}\right)}{\pazocal{P}\left(\textbf{{M}}_{\textrm{H{i}}},\textbf{{M}}_{\textrm{H{i}}}\right)}\right\rangle_{N_{\text{mock}}}\,.$
(33)
The angled brackets indicate an averaging over a large number of mocks such
that a converged result is obtained. In this work, we use all mocks in our
suite, thus $N_{\text{mock}}\,{=}\,500$. The operator $\pazocal{P()}$ can
represent any power spectrum estimator i.e. we use this same recipe for
estimating signal loss in the spherically averaged power and also in the
cylindrical ($k_{\perp}$, $k_{\parallel}$) power spectrum, which we later
utilise in some analysis.
In this work, we will mostly focus on the MeerKLASS deep field’s cross-
correlation with the overlapping GAMA galaxies. The signal loss from PCA
cleaning in the cross-power spectrum should be identical to that in the auto-
power. This was established empirically in Cunnington et al. (2023) which
showed how removing a subset of PCA modes entirely but leaving their
contribution in the other half of the cross-power product (e.g. in the galaxy
maps which have no cleaning, thus no signal loss) generates no more power than
the auto-Hi. We present further reasoning for why this is the case in Appendix
B.3. To summarise, we believe it is explained by the PCA modes being
orthogonal, thus maximally uncorrelated. Therefore, the additional remaining
modes in the uncleaned field are linearly uncorrelated with what survives in
the cleaned Hi, producing no extra power.
Despite the approximately equal signal loss between auto and cross-
correlations, we still define a separate transfer function for the cross-power
spectrum purely for covariance estimation purposes whereby the only difference
is we switch in the GAMA galaxy mocks $\textbf{{M}}_{\text{g}}$ instead which
will emulate the signal strength and shot-noise of the GAMA galaxies, thus
providing a more robust error estimate in the cross-power.
$\pazocal{T}_{\textrm{H{i}},\text{g}}(\boldsymbol{k})=\left\langle\frac{\pazocal{P}\left(\textbf{{M}}_{\text{clean}},\textbf{{M}}_{\text{g}}\right)}{\pazocal{P}\left(\textbf{{M}}_{\textrm{H{i}}},\textbf{{M}}_{\text{g}}\right)}\right\rangle_{N_{\text{mock}}}\,.$
(34)
Any measured power spectra should be divided by $\pazocal{T}(\boldsymbol{k})$,
thus the transfer function represents a reconstruction of signal at the power
spectrum level.
## 5 Power spectrum estimation
In this section we present power spectra estimations from the foreground
cleaned MeerKLASS L-band deep field. We first outline the pipeline for
performing these measurements then present the results. By default, we use
maps that have been foreground cleaned by removing $N_{\text{fg}}\,{=}\,10$
PCA modes, as outlined in the previous section. However, we also examine the
results from varying $N_{\text{fg}}$.
### 5.1 Transforming to a Cartesian grid
The first task to perform in order to measure a power spectrum is to transform
the foreground cleaned intensity maps onto a Cartesian grid with comoving
$h^{-1}\text{Mpc}$ coordinates. This allows for a Fast Fourier transform to be
calculated on the Cartesian array, from which the power at different Fourier
bandpasses $\boldsymbol{k}$, can be averaged and presented.
We discussed the transformation between sky maps and Cartesian grids already
in Section 3.1 when we performed the inverse routine of sampling Cartesian
cells from lognormal mocks into celestial sky map voxels that emulate the
MeerKLASS intensity maps. We again utilise the recent work and code in
Cunnington & Wolz (2024) and first define a grid of size
$l_{\text{x}},l_{\text{y}},l_{\text{z}}\,{=}\,632.2,341.5,231.5\,h^{-1}\text{Mpc}$
with $n_{\text{x}},n_{\text{y}},n_{\text{z}}\,{=}\,88,48,168$ cells along each
dimension. The choice of cells is a factor of $1.5$ smaller than the number of
map voxels in R.A., Dec., and $\nu$. Making the Cartesian grid slightly lower
resolution than the map and using 10 sampling particles per map voxel ensures
each Cartesian cell is well-sampled, avoiding empty cells.
The foreground cleaned maps $\delta
T_{\textrm{H{i}}}(\boldsymbol{\theta},\nu)$ are transformed to this Cartesian
grid, giving $\delta T_{\textrm{H{i}}}(\boldsymbol{x})$. The intensity mapping
weights also follow an identical transformation process
$w_{\textrm{H{i}}}(\boldsymbol{\theta},\nu)\,{\rightarrow}\,w_{\textrm{H{i}}}(\boldsymbol{x})$.
The catalogue of GAMA galaxies can be directly gridded by transforming their
celestial coordinates into Cartesian, then assigning them onto the Cartesian
grid $n_{\text{g}}(\boldsymbol{x})$. For all sampling particles and galaxies
we use a straightforward nearest grid point (NGP) assignment. Cunnington &
Wolz (2024) presented extensions to this method for avoiding aliased power and
obtaining sub-percent accuracy. However, given the current signal-to-noise of
our data at small scales, aliasing effects will be negligible, thus higher-
order interpolations and interlacing are overkill and serve only to increase
compute demand. However, we correct for the dominant effect which is the
field’s smoothing by the (NGP) particle assignment. This requires deconvolving
the field, achieved by dividing through by the NGP mass assignment window
function in Fourier space (Jing, 2005)
$\tilde{\delta
T}_{\textrm{H{i}}}(\boldsymbol{k})\rightarrow\frac{\tilde{\delta
T}_{\textrm{H{i}}}(\boldsymbol{k})}{W_{\text{ngp}}(\boldsymbol{k})}\,,$ (35)
where the window function $W_{\text{ngp}}$ accounts for the sampling particles
discretisation into cells of size $H$. These are separable functions along
each Cartesian dimension, thus modelled by a Fourier transform of a top-hat
function, which becomes a sinc function;
$\operatorname{sinc}(x)\,{=}\,\sin(x)/x$ and therefore
$\tilde{W}_{\mathrm{ngp}}(\boldsymbol{k})=\operatorname{sinc}\left(\frac{k_{\mathrm{x}}H_{\mathrm{x}}}{2}\right)\operatorname{sinc}\left(\frac{k_{\mathrm{y}}H_{\mathrm{y}}}{2}\right)\operatorname{sinc}\left(\frac{k_{\mathrm{z}}H_{\mathrm{z}}}{2}\right)\,.$
(36)
Previous work (e.g. Switzer et al., 2013; Cunnington, Li et al., 2022)
employed a tapering along the radial direction of the regridded field with the
primary purpose of reducing edge effects in the Fourier transform. Our
modelling approach should however account for these effects since the model is
also convolved with the window function (Blake, 2019). We still found results
were worse when omitting a radial tapering though, which we suspect is instead
caused by edge-effects from the foreground clean (Alonso et al., 2015) which
are suppressed by the tapering. We experimented with a less aggressive taper
by using a Tukey window, also known as a cosine window (this is also explored
in Carucci et al. in prep). This has a shape parameter $\alpha$ which is
simply a top-hat with no apodisation when $\alpha\,{=}\,0$ and equivalent to a
Hann window when $\alpha\,{=}\,1$. We show these window functions in Figure
12. By opting for less aggressive Tukey windows we still effectively suppress
the extreme edges where foreground cleaning effects may be present but avoid
unnecessary tapering on the central data. We will explore this further with
future data in MeerKAT’s UHF band where the larger useable frequency ranges
may render this less of an issue. In this work we found optimal results when
using $\alpha\,{=}\,0.8$ for the Tukey window which is applied to the
intensity map weights $w_{\textrm{H{i}}}(\boldsymbol{x})$. We do not apply
this to the galaxy data as we found this increased the correlations between
$k$-bins in the power spectrum.
Figure 12: Tukey window functions with differing shape parameters $\alpha$.
For comparison, we also show the Blackman window used in previous work.
### 5.2 Power spectrum formalism
With the fields assigned to a Cartesian space and appropriately tapered, the
power spectra can then be measured. The Fourier transformed field of the Hi
temperature fluctuation maps $\delta T_{\textrm{H{i}}}$ is given by
$\tilde{F}_{\textrm{H{i}}}(\boldsymbol{k})=\sum_{\boldsymbol{x}}\delta
T_{\textrm{H{i}}}(\boldsymbol{x})w_{\textrm{H{i}}}(\boldsymbol{x})\exp(i\boldsymbol{k}{\cdot}\boldsymbol{x})\,,$
(37)
where the weights, now denoted by $w_{\textrm{H{i}}}$ to distinguish them from
the galaxy weights, still derive from the field of time-stamp counts, which
are regridded following the same process as outlined in the previous sub-
section. The Fourier transformed field of the the galaxy count field
$n_{\text{g}}$ is given by
$\tilde{F}_{\text{g}}(\boldsymbol{k})=\sum_{\boldsymbol{x}}n_{\text{g}}(\boldsymbol{x})w_{\text{g}}(\boldsymbol{x})\exp(i\boldsymbol{k}{\cdot}\boldsymbol{x})-N_{\text{g}}\tilde{W}_{\text{g}}(\boldsymbol{k})\,,$
(38)
where $N_{\text{g}}\,{=}\,\sum n_{\text{g}}$ is the total number of galaxies
in the optical maps and $\tilde{W}_{\text{g}}$ is the weighted Fourier
transform of the selection function
$\tilde{W}_{\text{g}}(\boldsymbol{k})=\sum_{\boldsymbol{x}}W_{\text{g}}(\boldsymbol{x})w_{\text{g}}(\boldsymbol{x})\exp(i\boldsymbol{k}{\cdot}\boldsymbol{x})\,.$
(39)
The galaxy window function $W_{\text{g}}$ is normally constructed by stacking
mock galaxies that carefully follow the survey selection characteristics of
the experiment to account for its incompleteness. As discussed previously,
there are no GAMA G23 mocks available for this purpose so we therefore
construct a simple window function which is 1 within the survey boundaries and
0 everywhere else. Some inclusion of the redshift evolution of $N(z)$ could
have been included, but as Figure 7 showed, the variation over our narrow
redshift range is minimal. The window function is normalised so that
$\sum_{\boldsymbol{x}}W_{\text{g}}(\boldsymbol{x})\,{=}\,1$. Since we lack
this proper galaxy window function we do not implement the optimal weighting
as per Feldman et al. (1994), but instead use uniform weighting, so equivalent
of assuming
$w_{\text{g}}(\boldsymbol{x})\,{\equiv}\,W_{\text{g}}(\boldsymbol{x})$.
The cross-power spectrum estimator is then given by
$\hat{P}_{\textrm{H{i}},\text{g}}(\boldsymbol{k})=\frac{V_{\text{cell}}}{\sum_{\boldsymbol{x}}w_{\textrm{H{i}}}(\boldsymbol{x})w_{\text{g}}(\boldsymbol{x})W_{\text{g}}(\boldsymbol{x})}\operatorname{Re}\left\\{\tilde{F}_{\textrm{H{i}}}(\boldsymbol{k})\tilde{F}^{*}_{\text{g}}(\boldsymbol{k})\right\\}\frac{1}{N_{\text{g}}}\,,$
(40)
where
$V_{\text{cell}}\,{=}\,l_{\text{x}}l_{\text{y}}l_{\text{z}}/(n_{\text{x}}n_{\text{y}}n_{\text{z}})$.
The hat notation $\left(\hat{P}\right)$ indicates that this is the estimated
power from the data, to distinguish from the modelled power, which we discuss
shortly. The Hi auto power spectrum estimator is more straightforwardly given
by
$\hat{P}_{\textrm{H{i}}}(\boldsymbol{k})=\frac{V_{\text{cell}}}{\sum_{\boldsymbol{x}}w_{\textrm{H{i}}}^{2}(\boldsymbol{x})}\lvert\tilde{F}_{\textrm{H{i}}}(\boldsymbol{k})\rvert^{2}\,,$
(41)
These power spectra are either spherically averaged into bandpowers
$\lvert\boldsymbol{k}\rvert\,{\equiv}\,k$ to provide 1D power spectrum results
or alternatively cylindrically averaged into 2D bandpowers defined by modes
perpendicular ($k_{\perp}$) and parallel ($k_{\parallel}$) to the radial
direction z, which we assume is the line-of-sight in a plane-parallel
approximation, valid for the current MeerKLASS sky sizes. They are finally
divided by their relevant foreground transfer functions (Equation 33 and
Equation 34) to reconstruct signal loss for each $k$-bin.
#### 5.2.1 Modelling the power spectrum
Here we outline the model of the power spectrum estimations for which we
mostly follow the work in Blake (2019), with some extensions. We model the Hi
power spectrum using the equation already introduced in Equation 14 when
discussing mock simulations. We repeat the model here for convenience;
$P_{\textrm{H{i}}}(\boldsymbol{k})\equiv
P_{\textrm{H{i}}}(k,\mu)=\frac{\overline{T}_{\textrm{H{i}}}^{2}\left(b_{\textrm{H{i}}}+f\mu^{2}\right)^{2}P_{\text{m}}(k)}{1+\left(k\mu\sigma_{\text{v}}/H_{0}\right)^{2}}\,.$
(42)
Additional modelling of observational and mapping effects is also necessary
since these will distort the power. To model the MeerKAT beam and the
additional smoothing from the reconvolution process we assume the field has
undergone a constant smoothing by a 2D Gaussian kernel whose standard
deviation is given by
$R_{\text{beam}}\,{=}\,\sqrt{\gamma}\,R^{\max}_{\text{beam}}$ in comoving
units. $\gamma\,{=}\,1.4$ accounts for the reconvolution to a common effective
resolution and $R^{\max}_{\text{beam}}$ is the Gaussian approximation of the
MeerKAT beam size at the deepest redshift (and lowest frequency
$\nu_{\text{min}}$)
$R^{\max}_{\text{beam}}=\chi(z_{\max})\,\sigma_{\text{beam}}(\nu_{\min})\,,$
(43)
where $\sigma_{\text{beam}}$ is given as before from Equation 17 and $\chi(z)$
is the comoving distance to redshift $z$, which in this case, is the maximum
redshift for our frequency range. The perpendicular modes will then be damped
according to
$\tilde{\pazocal{B}}_{\text{beam}}(\boldsymbol{k})=\exp\left[\frac{-k^{2}\left(1-\mu^{2}\right)R_{\text{beam
}}^{2}}{2}\right]\,.$ (44)
For this work, we find $R_{\text{beam}}\,{=}\,14.1\,h^{-1}\text{Mpc}$.
Similar to the beam, there will also be distortions to the power caused by the
finite resolution of the intensity maps. We approximate the pixel size in
comoving units by $s_{\text{pix}}\,{=}\,\chi(z_{\text{eff}})\Delta\theta$
where $\Delta\theta\,{=}\,0.3\,\deg$ is the pixel size used for the MeerKLASS
L-band mapping. The frequency channel depth is approximated by taking the mean
across all spacings between redshifts corresponding to channel boundaries i.e.
$s_{\text{chan}}\,{=}\,\Sigma_{i}\left[\chi(z_{i})-\chi(z_{i{+}1})\right]/N_{\text{chan}}$.
These quantities are used in damping functions to model the suppressed
resolution from mapping into pixels and frequency channels, the former being
the far more dominant effect for the wide-pixel MeerKLASS intensity maps.
These functions are given as
$\tilde{\pazocal{B}}_{\text{pix}}(\boldsymbol{k})=\operatorname{sinc}\left(\frac{k\sqrt{1-\mu^{2}}\,s_{\text{pix}}}{2}\right)\,,$
(45)
and
$\tilde{\pazocal{B}}_{\text{chan
}}(\boldsymbol{k})=\operatorname{sinc}\left(\frac{k\,\mu\,s_{\text{chan
}}}{2}\right)\,,$ (46)
The model and damping functions are discretised onto the same 3D grid of
$k$-modes as the FFT for the real data, then convolved with the survey window
function. The model for the observed Hi auto-power spectrum therefore becomes
$P_{\textrm{H{i}}}^{\text{obs}}(\boldsymbol{k})=\frac{W^{2}_{\textrm{H{i}}}(\boldsymbol{k})\circledast\left[P_{\textrm{H{i}}}(\boldsymbol{k})\,\pazocal{B}_{\text{beam}}^{2}(\boldsymbol{k})\,\pazocal{B}_{\text{pix}}^{2}(\boldsymbol{k})\,\pazocal{B}_{\text{chan}}^{2}(\boldsymbol{k})\right]}{\sum_{\boldsymbol{x}}w^{2}_{\textrm{H{i}}}(\boldsymbol{x})W^{2}_{\textrm{H{i}}}(\boldsymbol{x})}\,,$
(47)
where we have normalised by the sum of the weights and window function,
$W_{\textrm{H{i}}}$, to obtain an amplitude that matches the weighted and
windowed estimator. For the intensity mapping window function, we assume a
simple binary function; 1 where we have scan coverage and 0 everywhere else.
There is scope for improving this approximation and further work is needed to
examine how intensity mapping surveys are impacted by any selection effects.
A similar approach is followed for modelling the cross-power with GAMA. The
cross-correlation power spectrum is given by
$P_{\textrm{H{i}},\mathrm{g}}(\boldsymbol{k})=\frac{\overline{T}_{\textrm{H{i}}}\left[rb_{\textrm{H{i}}}b_{\mathrm{g}}+b_{\textrm{H{i}}}f\mu^{2}+b_{\mathrm{g}}f\mu^{2}+f^{2}\mu^{4}\right]P_{\mathrm{m}}(k)}{1+\left(k\mu\sigma_{\text{v}}/H_{0}\right)^{2}}\,.$
(48)
where we introduce $r$, the correlation coefficient between the Hi
fluctuations and the GAMA galaxy over-densities. This accounts for the
different astrophysics in each tracer and non-linear clustering, which can
mean the two fields are not exact multiples of each other and there is instead
some stochasticity which will suppress their cross-power (see e.g.
Moradinezhad Dizgah et al., 2022; Sato-Polito et al., 2023; Obuljen et al.,
2023, for applications in the context of line-intensity mapping). We assume
$r$ to be constant, which is sufficient for the current signal-to-noise, but
in reality it will exhibit some scale dependence. The correlation coefficient
only enters on the single term in Equation 48 because it only influences the
biased tracer fields, not their unbiased velocity fields which are modulated
by $f$. The observational cross power is then similarly convolved with the
survey window functions but only requires single factors of the damping
functions $\pazocal{B}$;
$\displaystyle P_{\textrm{H{i}}\text{,g}}^{\text{obs}}$
$\displaystyle(\boldsymbol{k})=$ (49)
$\displaystyle\frac{W_{\textrm{H{i}}}(\boldsymbol{k})W^{*}_{\text{g}}(\boldsymbol{k})\circledast\left[P_{\textrm{H{i}}\text{,g}}(\boldsymbol{k})\,\pazocal{B}_{\text{beam}}(\boldsymbol{k})\,\pazocal{B}_{\text{pix}}(\boldsymbol{k})\,\pazocal{B}_{\text{chan}}(\boldsymbol{k})\right]}{\sum_{\boldsymbol{x}}w_{\textrm{H{i}}}(\boldsymbol{x})w_{\text{g}}(\boldsymbol{x})W_{\textrm{H{i}}}(\boldsymbol{x})W_{\text{g}}(\boldsymbol{x})}\,.$
We do not include modelling of uncertainties in the GAMA galaxy positioning,
such as redshift estimation. The GAMA radial kernel equivalent of Equation 46
will likely have a larger impact due to spectroscopic redshift estimation
errors. However, this will still be a sub-dominant contribution to the overall
modelling relative to the beam and intensity map pixelisation, thus it is
reasonable to neglect observational effects from the GAMA galaxies.
We perform a simple least-squares fit for an overall amplitude to our cross-
power, equivalent to assuming fixed values for all parameters except for a
fitted $\overline{T}_{\textrm{H{i}}}$. This is deliberately simplistic to keep
the model and fitting in a low-dimensional form. The small overlap sample in
the cross-correlation means the statistical noise from such measurements is
not yet suited to high-dimensional modelling. We are not therefore extending
much from previous work (e.g. Wolz et al., 2022; Cunnington, Li et al., 2022)
in this regard. Furthermore, the auto-Hi still has evidence of additive bias,
as we will discuss, and therefore can only be trusted for upper limits in
parameter inference.
For the power amplitude fitting, we assume the parameter values outlined by
Table 1. The Hi bias comes from interpolated results from hydrodynamical
simulations (Villaescusa-Navarro et al., 2018), which can be approximated by
the polynomial
$b_{\textrm{H{i}}}(z)=0.842+0.693z-0.046z^{2}\,.$ (50)
The galaxy bias is obtained by fitting the amplitude of the GAMA auto-power
spectrum. The growth rate is obtained from
$f\,{=}\,\Omega_{\text{m}}(z_{\text{eff}})^{0.545}$ for our Planck 18
cosmology. The correlation coefficient $r$ and velocity dispersion
$\sigma_{\text{v}}$ are poorly constrained nuisance parameters tuned by eye to
provide an adequate fit. We discuss model and parameter fitting further in
Section C where we present results from a simultaneous MCMC fit on several
parameters, demonstrating the degeneracies between them and discuss plans for
breaking such degeneracies, but this is not a central focus of this work.
Parameter | $b_{\textrm{H{i}}}$ | $b_{\text{g}}$ | $r$ | $f$ | $\sigma_{\text{v}}$
---|---|---|---|---|---
Fiducial value | 1.13 | 1.90 | 0.90 | 0.74 | 400 km s-1
Table 1: Fiducial parameters assumed in model fitting.
### 5.3 Error estimation with the transfer function
In previous analysis (Cunnington, Li et al., 2022), an analytical approach was
primarily used for estimating errors on the power spectrum measurements.
However, this approach makes various assumptions, mainly that thermal noise
dominates the error budget and that Gaussian statistics are valid. Since we
are now reaching regions in $k$-space where thermal noise no longer dominates
with MeerKLASS, we are required to relax these assumptions. Furthermore, the
assumption of Gaussianity is arguably not solid, given the additive
systematics we know to be still be present within the data.
A more robust approach of estimating the errors is to use the transfer
function scatter across the different mock realisations
$\pazocal{T}_{i}(\boldsymbol{k})$. This approach has been used in previous
analysis (e.g. Anderson et al., 2018) and has also been validated in
simulations (Cunnington et al., 2023). For it to perform effectively, the
mocks in the transfer function calculation must emulate the real data
effectively which we have strived to do so, as outlined in Section 3.
The transfer function for one single realisation is defined by
$\pazocal{T}_{i}^{\textrm{H{i}},\text{g}}(\boldsymbol{k})=\frac{\pazocal{P}\left(\textbf{{M}}_{\text{clean}},\textbf{{M}}_{\text{g}}\right)}{\pazocal{P}\left(\textbf{{M}}_{\textrm{H{i}}},\textbf{{M}}_{\text{g}}\right)}\,,$
(51)
i.e. equivalent to Equation 34 but without the averaging over
$N_{\text{mock}}$. To reconstruct signal in a measured power spectrum, we take
the ensemble average of Equation 34 i.e.
$\pazocal{T}(\boldsymbol{k})\,{=}\,\langle\pazocal{T}_{i}(\boldsymbol{k})\rangle_{N_{\text{mock}}}$,
and divide the estimated power by this;
$\hat{P}_{\text{rec}}(\boldsymbol{k})\,{=}\,\hat{P}(\boldsymbol{k})/\pazocal{T}(\boldsymbol{k})$.
Note we are dropping the subscript Hi,g for brevity and also because the
formalism we present here is identically applicable to the Hi-auto. The
assumption made when using the transfer function for error estimation is that
the scatter in the reconstructed power comes from the transfer function
distribution
$\pazocal{T}(\boldsymbol{k})\,{+}\,\delta\pazocal{T}_{i}(\boldsymbol{k})$
contributed by each $i$th mock. We can isolate the scatter component from the
array of calculated transfer functions by simply subtracting the ensemble mean
for each $i$th mock
$\delta\pazocal{T}_{i}(\boldsymbol{k})=\pazocal{T}_{i}(\boldsymbol{k})-\pazocal{T}(\boldsymbol{k})\,.$
(52)
Assuming any errors add in quadrature, error propagation suggests
$\frac{\delta\hat{P}^{\text{rec}}_{i}(\boldsymbol{k})}{\hat{P}_{\text{rec}}(\boldsymbol{k})}=\sqrt{\left(\frac{\delta\hat{P}_{i}(\boldsymbol{k})}{\hat{P}(\boldsymbol{k})}\right)^{2}+\left(\frac{\delta\pazocal{T}_{i}(\boldsymbol{k})}{\pazocal{T}(\boldsymbol{k})}\right)^{2}}\,,$
(53)
however, we assume all quantifiable error lies in the transfer function
distribution, hence $\delta\hat{P}_{i}\,{=}\,0$ and so
$\delta\hat{P}^{\text{rec}}_{i}(\boldsymbol{k})=\hat{P}_{\text{rec}}(\boldsymbol{k})\frac{\delta\pazocal{T}_{i}(\boldsymbol{k})}{\pazocal{T}(\boldsymbol{k})}\,.$
(54)
Our setup thus provides $N_{\text{mock}}$ realisations of
$\delta\hat{P}_{i}^{\text{rec}}$ and the covariance can be computed from this
for any $k$-bin. We reiterate that this formalism is valid whichever way the
$\boldsymbol{k}$-modes are binned e.g. for a 2D cylindrical averaged or 1D
spherically averaged power. In this work, our error bars in any 1D power
spectrum estimate represent the limits of the 68th percentile regions of the
distribution of $\delta\hat{P}^{\text{rec}}_{i}(k)$. If the uncertainties are
Gaussian symmetric, this is equivalent to $1\,\sigma$ error bars, however, as
we will later show, this is not perfectly the case for all $k$-modes.
### 5.4 Spherical averaging $k$-cuts
Before presenting spherically averaged power spectra from the MeerKLASS data,
we first present results from the cross-correlation with GAMA galaxies in
cylindrical-($k_{\perp},k_{\parallel})$ space. This is shown in Figure 13 and
motivates a discussion on where in $k$-space we have the most signal-to-noise
so we can make optimal spherically averaged $\hat{P}(k)$ measurements. The
dashed gray lines on all panels represent the $|\boldsymbol{k}|$-bin
boundaries we use for the spherically averaged power spectra, restricting this
to $k\,{<}\,0.3\,h\,\text{Mpc}^{-1}$ to avoid highly non-linear cosmology and
increasingly poor signal-to-noise. The orange solid lines represent boundaries
for potential $k$-cuts which we will discuss.
Figure 13: Cylindrical ($k_{\perp},k_{\parallel}$) analysis of power, signal-
loss and signal-to-noise. The first panel from the left shows the cross power
between the MeerKLASS deep field and GAMA galaxies. The second panel is the
transfer function used to reconstruct the power in
$\hat{P}_{\textrm{H{i}},\text{g}}$. The third panel shows the signal-to-noise
by dividing the cross power by the error estimation discussed in the text. The
final fourth panel counts the number of discrete $\boldsymbol{k}$ wavenumber
in the Fourier grid, demonstrating the concentration of available modes. The
gray contours mark the boundaries of the $k$-bins used in spherically averaged
power $P(k)$. The orange lines indicate regions used for $k$-cuts discussed in
the text.
The first panel of Figure 13 shows the cross-power between GAMA G23 galaxies
and the MeerKLASS deep field, cleaned by removing $N_{\text{fg}}\,{=}\,10$ PCA
modes. The $k_{\perp}k_{\parallel}$ scaling is for demonstrative purposes so
the high power at low-$k$ does not saturate the plot. At
$k\,{\gtrsim}\,0.15\,h\,\text{Mpc}^{-1}$ the large MeerKAT beam begins to have
an impact and will damp Hi fluctuations, which explains why the cross-power
signal tends to zero at high-$k_{\perp}$. Except for the extreme low-$k$
regions, the rest of the $k$-space reveals a reasonably consistent positive
cross-correlation signal, providing strong evidence of consistent clustering
between the MeerKLASS maps and GAMA galaxies. As discussed in the previous
Section 4.5, the signal loss in this power has been corrected with the
foreground transfer function. The second panel shows the corresponding
transfer function for the $N_{\text{fg}}\,{=}\,10$ clean. Low values of
$\pazocal{T}_{\textrm{H{i}},\text{g}}$ signify high signal loss. This is
evidence that nearly all signal is lost for the lowest-$k_{\parallel}$ modes
which is expected since these modes should be highly degenerate with the
removed long radial modes of the continuum foregrounds. The high signal loss
at low-$k_{\perp}$ is interesting and not seen in previous MeerKLASS work
(Cunnington, Li et al., 2022) with a similar method, albeit with less robust
mocks. This is likely caused by the small GAMA footprint (see Figure 4) which
means the large-ranging modes at these low-$k$ are not adequately resolved and
deliver little signal which is easily coupled to foreground modes and stripped
away in the clean. This can be explored further in follow-up work with wider
surveys where we expect this not to be a problem.
Due to the noisy power at low-$k$ and damped power at high-$k_{\perp}$ in
Figure 13, we can choose to cut these regions of $k$-space from our
spherically averaged power to obtain a more robust result. The orange solid
lines represent chosen boundaries, where modes beyond these boundaries are
excluded in the averaging of each $k$-bin (shown by the dashed gray lines).
Providing the modelling of any power spectra also incorporates the $k$-cuts,
then it will not bias any results. The $k$-cuts approach is further motivated
by considering the signal-to-noise of the cross-power which we show in the
third panel of Figure 13. Here the errors
$\sigma(\hat{P}_{\textrm{H{i}},\text{g}})$ are computed using the transfer
function by taking the standard deviation from the
$\delta\hat{P}_{i}^{\text{rec}}(k_{\perp},k_{\parallel})$ distribution (see
previous Section 5.3). This result provides a strikingly clear window where
signal-to-noise is optimal; the orange to red regions in the third panel. The
orange $k$-cut boundary lines have been chosen to target this region. One
limitation of this approach is that it excludes the regions of $k$-space with
the most modes, as shown by the fourth and final panel. We will present
comparisons between the spherically averaged $P(k)$ with and without these
$k$-cuts in the upcoming Section 5.6.
### 5.5 Hi auto-correlation power spectrum
We next examine results for the auto-correlation Hi power spectrum from the
MeerKLASS L-band deep field. Previous single-dish intensity mapping results
have been thermal noise-dominated across all scales. We are now crossing an
important threshold with MeerKLASS where the Hi fluctuations should be
dominating the noise at large scales. We demonstrate this in Figure 14, where
we show the estimated Hi power spectrum from the MeerKLASS deep field (blue
data points) and overlay an estimate of the thermal noise (red solid line). As
outlined in Section 3.2, this noise is modelled by generating Gaussian random
fields with the expected rms given by Equation 20 which utilises the
observational hit map. Despite being the Hi auto-power, we still restrict the
measurement to the GAMA-field footprint, for consistency with the cross-
correlation results. We tested using the full field but found no improvement
in the auto-power. We leave exploration of the full-field to follow-up work
which focuses on the Hi-auto power spectrum. The results in Figure 14 include
the scale $k$-cuts discussed in the previous section.
Figure 14 also shows the expected Hi power (grey dotted line) from Equation
47. For this auto-Hi model, we do not fit the amplitude to avoid additive bias
from residual systematics over inflating the model estimate. Instead we use
the scale-independent $\overline{T}_{\textrm{H{i}}}$ fit from the upcoming
cross-power measurement. This model shows how the cosmological Hi power
surpasses the thermal noise on large scales
($k\,{\lesssim}\,0.15\,h\,\text{Mpc}^{-1}$). The thermal noise will be present
even in the absence of clustering, thus the total expected power can be
reasonably approximated as a simple addition of the Hi power, plus the thermal
noise (shown by the black dashed line).
Figure 14: The Hi auto-power spectrum for the MeerKLASS L-band deep-field
foreground cleaned by removing $N_{\text{fg}}\,{=}\,10$ PCA modes. The red
solid line is the noise estimate from measuring the average power of 10
Gaussian random fields with rms given by $\sigma_{\text{N}}$ (Equation 20).
The grey dotted line is the observational Hi power spectrum model (Equation
47) and the black dashed line is the combination of this Hi model plus the
thermal noise estimate.
The small discrepancy between the deep-field power estimate and the model
represents the lowest upper limit from a Hi auto-correlation on these large
scales. We find the mean residual factor between model and data to be just
$2.36$ across all $k$. Whilst the model and data are close in agreement,
inferred parameter values for the bias and mean Hi temperature from fitting
the amplitude of the auto power yields inconsistent values with the GAMA
cross-correlation (discussed next). This is therefore evidence of low-level
residual systematics that are additively biasing the power. In this
evaluation, we inherently assume that the transfer function reconstructs any
signal loss in the Gaussian thermal noise. If signal loss in the Gaussian
noise is underestimated, the residual between the model and data may be higher
than we report here. We therefore intend to robustly test this assumption when
pursuing a more dedicated study of the Hi auto-correlation.
In follow-up work, we will focus on improving this upper limit from the Hi
auto power which can be done in two ways: Firstly, by implementing some
corrections to remove low-level systematic contributions such as the residual
RFI structures discussed in Section 2.4.2. Secondly, we can split the
observations into sub-sets i.e. combinations of different maps from different
scan times and/or dishes. This is useful for isolating dish- and time-
dependent systematics and removing their biasing contributions to the power.
Since each scan and dish is observing the same patch of sky, we would ideally
expect all maps to agree, thus any differences will represent thermal noise,
or systematics such as residual RFI, varying polarisation leakage etc. We do
not pursue these ideas in this paper since they warrant their own detailed
investigation.
### 5.6 Cross-correlation power with GAMA
We now focus on the cross-correlation with the GAMA galaxies for the remainder
of the paper. In Figure 15 we present the cross-correlation power spectrum
between the GAMA galaxies and the MeerKLASS deep field. All points are
positive so none are lost to the log scaling, indicating a strong, positive
correlation. Generally, a good agreement with the model is obtained. We
performed a null test on the result by substituting the GAMA mock galaxies
(from Section 3.3) in place of the real galaxies. The mocks should not
correlate with the intensity maps, and indeed, the resulting cross-power
converges to zero once averaged over a sufficient number of mocks. The fitted
amplitude of the model is equivalent to an inferred parameter of
$\overline{T}_{\textrm{H{i}}}\,{=}\,0.166\,\text{mK}$, with all other values
in Table 1 assumed. We discuss this further in Appendix C, but do not focus on
parameter inference in detail due to the large statistical noise owing to the
small overlap footprint between MeerKLASS and GAMA.
Figure 15: The cross-correlation power spectrum between GAMA galaxies and the
MeerKLASS L-band deep field, foreground cleaned by removing
$N_{\text{fg}}\,{=}\,10$ PCA modes. The black dashed line represents a model
given by Equation 49 with a fitted scale-independent amplitude.
The error bars for the cross-power in Figure 15 are obtained using the scatter
from the foreground transfer function as discussed in Section 5.3. The error
size appears to be over-predicted based on the scatter of the data points,
which have a reasonably tight agreement with the model relative to what the
error bars reflect. This is supported by examining the reduced $\chi^{2}$
which we find to be
$\chi^{2}_{\text{dof}}\,{\equiv}\,\chi^{2}{/}N_{\text{dof}}\,{=}\,0.42$. A
$\chi^{2}_{\text{dof}}\,{<}\,1$ is conventionally indicative of overinflated
errors. Using the reduced $\chi^{2}$ however is not a reliable test where
there are non-linear contributions, as is likely the case for these
observations that are still prone to residual systematics. In these cases, the
number of degrees of freedom cannot be robustly determined (Andrae et al.,
2010). For our case we simply assume $N_{\text{dof}}\,{=}\,13$, for our 14
$k$-bins minus a single fitted degree of freedom, representing the amplitude
fit. Furthermore, noise within the data causes high uncertainty in $\chi^{2}$
estimates, which despite the relatively low noise levels we are now reaching,
may still be a concern. Alternatives to $\chi^{2}$ have been proposed, and
specifically for 21cm applications (e.g. Tauscher et al., 2018), but these
have not been pursued in this work. We only use the $\chi^{2}$ as an
approximate guide, mainly for relative comparisons between different cases.
For the estimation of $\chi^{2}$, we include the full contribution from off-
diagonals in the $k$-bin covariance C, which can be trivially estimated for
our error estimation method, simply the covariance over
$\hat{P}_{i}^{\text{rec}}(k)\,{=}\,\hat{P}_{\text{rec}}(k)\,{+}\,\delta\hat{P}^{\text{rec}}_{i}(k)$
(see Section 5.3). The $\chi^{2}$ is then estimated from
$\chi^{2}=(\hat{\boldsymbol{p}}-\boldsymbol{p}_{\rm
mod})^{\text{T}}\,\textbf{{C}}^{-1}\,(\hat{\boldsymbol{p}}-\boldsymbol{p}_{\rm
mod})\,,$ (55)
where $\hat{\boldsymbol{p}}$ is the data vector and
$\boldsymbol{p}_{\text{mod}}$ the corresponding model. We present the
normalised covariance matrix (or correlation matrix) in Figure 16, showing
evidence that the $k$-bins are slightly correlated, especially at low-$k$.
This could be contributing to the large error bars, hence why we include the
full covariance in the $\chi^{2}$ estimation.
Figure 16: $k$-bin correlation matrix for the cross-power between the
MeerKLASS deep field and GAMA galaxies, shown in Figure 15.
To investigate the errors from the cross-power further, we plot the full
distributions for the reconstructed power spectra $\hat{P}_{i}^{\text{rec}}$
from each $i$th mock in the foreground transfer function computation. Figure
17 shows this for each $k$-bin, including an additional low-$k$ bin which we
otherwise exclude due to the large error on these scales. We centre the
distributions on the model and normalise by it, to aid comparison with the
case of no $k$-cuts (red results), which includes the areas of $k$-space with
low signal-to-noise (discussed in Section 5.4). The plot shows how the
distributions have some non-Gaussian features, which could also impact error
estimation, along with the correlated $k$-bins shown by the covariance. We
find the average kurtosis across all $k$-bins for the $k$-cut case to be
$-0.476$, indicating the distribution is flatter than a normal distribution
with more contribution in the tails. This could also widen the central
percentiles and thus increase the error bar size.
Figure 17: Estimated power distribution for each $k$-bin from the cross-
correlation between MeerKLASS intensity maps and GAMA. This is computed using
the power distribution in the 500 mocks used in the foreground transfer
function. $k$-bin value is given in the top-left corner of each panel. The
power distributions are centred on the fitted model at each $k$ so a zero
median in the distribution represents perfect accuracy. Shaded regions show
the 68th percentiles for the distributions which are used as the limits for
the $1\,\sigma$ error bars in the power spectra plots.
Figure 17 shows a mild improvement in performance for the case with $k$-cuts.
The distributions around the model are less broad at low-$k$, which is
expected since this was one of the regions where signal-to-noise was poorest
(see third panel of Figure 13). In terms of the Gaussianity of the
distributions, we find a better kurtosis score for the case with $k$-cuts,
suggesting the profiles are more Gaussian. In contrast, we find the results
without $k$-cuts to be less skewed, although this is marginal and in general,
the skewness of the distributions is not concerning. Any improvement coming
from the $k$-cuts is small and we also find little difference in detection
significance, as we will explore shortly. As the volume of data grows beyond
this small overlap sample, the $k$-cut results may prove more optimal since we
will no longer be cutting modes from an already sample variance-limited data
set. This is something that future data sets with MeerKLASS will reveal.
#### 5.6.1 Varying the foreground clean
So far we have only considered the chosen case of removing
$N_{\text{fg}}\,{=}\,10$ PCA modes for the foreground clean. Figure 18 shows
how the $\chi^{2}$ (which we continue to use as a rough goodness-of-fit
metric) varies with $N_{\text{fg}}$. As seen in previous work (Cunnington, Li
et al., 2022), too low an $N_{\text{fg}}$ does not sufficiently clean the
foregrounds, causing a poor fit. Going too high with $N_{\text{fg}}$ also
diminishes the fit. Ideally, we would expect high $N_{\text{fg}}$ to cause
more signal loss which should be reconstructed by the transfer function,
giving unbiased results and thus not diminishing the fit. A higher uncertainty
could be expected with greater signal loss, but this should be accounted for
by the increasing error bars i.e. the $\chi^{2}$ should stabilise and not
start to rise again at high $N_{\text{fg}}$. On further scrutiny, we found
that it is a single data point at $k\,{\sim}\,0.2\,h\,\text{Mpc}^{-1}$ with
very little power distorting the fits at higher $N_{\text{fg}}$. This is
evidence of low-level residual systematics that become more influential as
signal is stripped away in the more aggressive foreground clean. Therefore,
with future results where systematics are more controlled, this is expected to
be less of a problem. For the case with no $k$-cuts (thin dashed lines), there
is a plateau in $\chi^{2}$ at higher $N_{\text{fg}}$, as we argued should be
the expected situation. This is because the no $k$-cut power spectra were less
impacted by the $k\,{\sim}\,0.2\,h\,\text{Mpc}^{-1}$ anomaly at high
$N_{\text{fg}}$ which is potentially suggesting that the $k$-cuts are overly
aggressive for this scale. With larger overlapping fields, the more accessible
$k$-bins will naturally mitigate this discrepency. The no $k$-cut results in
Figure 18 are therefore encouraging that a model and data fit should reach a
plateau for high $N_{\text{fg}}$ as expected with the current analysis
pipeline.
Figure 18: Variation of cross-correlation results with changing
$N_{\text{fg}}$ i.e. the number of PCA modes removed in the foreground clean.
The left (black) axis shows the $\chi^{2}$, the lower values generally
represent a better fit between cross-power data and model. The right (red)
axis shows the cross-correlation power spectrum detection significance. Solid
and thin-dashed lines represent the case with and without $k$-cuts
respectively.
The right hand side (red) $y$-axis in Figure 18, also shows the detection
significance defined by
$\sqrt{\Delta\chi^{2}}\,{\equiv}\,\sqrt{\chi^{2}-\chi_{\text{null}}^{2}}$,
where $\chi^{2}_{\text{null}}$ is the result’s $\chi^{2}$ with a null model
with zero power. The cross-correlation with GAMA is returning a modest
${>}\,4\sigma$ detection for the most optimal choices of foreground clean,
lower than previous MeerKLASS detections (Cunnington, Li et al., 2022). This
is however still impressive, given that galaxies only overlap with
approximately a quarter of the full MeerKLASS deep field. The detection
significance is also restricted by the more robust approach to error
estimation, which as we have already discussed, provides conservative
estimates potentially inflated by residual systematics and correlated
$k$-bins. To demonstrate this point, we artificially construct a scenario
where a $\chi^{2}_{\text{dof}}\,{\sim}\,1$ is obtained for the
$N_{\text{fg}}\,{=}\,10$ case. This is achieved by assuming a perfectly
diagonal covariance and decreasing the diagonal errors by $40\%$. This boosts
the detection significance to $9.8\,\sigma$, verifying that it is the
conservative approach to errors lowering the detection significance, rather
than the data-model residual. This is of course only a toy demonstration but
it validates the argument that if large errors, which are suspected to be
driven by systematics and $k$-bin correlations, can be contained, then
detection robustness should improve. This is expected in future work where we
will pursue dedicated studies into systematic mitigation. Some mode coupling
is always likely, driven at least by non-linear clustering. However, we are
confident that improvements can be made in covariance. MeerKLASS observations
ongoing now are covering wider areas with more overlap with Stage-IV optical
galaxy surveys. These observations are also using MeerKAT’s UHF-band, which
will allow a deeper usable redshift range, extending the
$0.39\,{<}\,z\,{<}\,0.46$ used in this work to approximately
$0.4\,{<}\,z\,{<}\,1.4$. This will naturally improve the large statistical
noise due to the small volume overlap in this work.
## 6 Stacking
In this section, we explore the possibility of a stacking detection of the Hi
signal in the MeerKLASS L-band deep-field intensity map. The technique of
spectral line stacking (Zwaan et al., 2001) can be used to detect the Hi
content of galaxies below noise limit. In combination with spectroscopic
galaxy surveys, spectral line stacking probes the average emission line
profile of selected galaxy samples. Complimentary to the intensity mapping
technique, the stacking measurement of the emission line profile probes the Hi
mass function of galaxy samples (e.g. Bera et al. 2022), the relation of Hi
content with its host environment (e.g. Guo et al. 2020; Bera et al. 2023; Dev
et al. 2023; Rhee et al. 2023), and the baryonic Tully-Fisher relation (e.g.
Meyer et al. 2016). Interferometric observations of MeerKAT using the MeerKAT
International GigaHertz Tiered Extragalactic Exploration survey (MIGHTEE;
Jarvis et al. 2016; Maddox et al. 2021) have been used to produce Hi
measurements with regard to the scaling relation (Sinigaglia et al., 2022)
with galaxy properties and the dependency on the large-scale structure
environment (Sinigaglia et al., 2024). See also Ponomareva et al. (2021);
Tudorache et al. (2022); Pan et al. (2023); Ponomareva et al. (2023) for
direct detections of Hi galaxies in the MIGHTEE survey.
For intensity mapping experiments, stacking analysis has been performed on the
Hi intensity maps using the Parkes telescope (Anderson et al., 2018; Tramonte
et al., 2019; Tramonte & Ma, 2020) and the CHIME telescope (CHIME
Collaboration et al., 2023), as well as on CO intensity maps from the CO
Mapping Array Project (Dunne et al., 2024) and Ly$\alpha$ emission of [O
III]-selected galaxies from the Hobby-Eberly Telescope Dark Energy Experiment
(Lujan Niemeyer et al., 2022a, b).
MeerKLASS, on the other hand, faces unique challenges. The $\,{\sim}\,1\,$deg
instrument power beam introduces heavy contamination from nearby sources. For
the GAMA galaxy sample used, the number density of the sources is
$\,{\sim}\,35\,{\rm deg^{-2}}$ when integrated along the frequency sub-band.
Therefore, when stacking each source, the stacking procedure will inevitably
include emissions from more than one GAMA galaxy. The velocity width of the Hi
sources further scatter the 21 cm emission into nearby channels. The double-
counting of sources from the beam creates a plateau of excess signal as shown
in Figure 26, and can be resolved by PCA cleaning, which we discuss in
Appendix D. However, the residual after PCA cleaning also incorporates
systematic effects. The chromatic structure of the systematic effects produces
oscillations in the stacked profile, with their amplitudes comparable to the
signal. For this work, we are only interested in using stacking as part of the
validation test. As discussed later, we obtain a detection of a stacked signal
in both angular and spectral space, which can be validated by performing a
null test with random galaxy positions. The existence of a stacked signal
further demonstrates that we have reached the depth needed for the possible
detection of the Hi power spectrum using the MeerKLASS L-band intensity maps.
We leave a more detailed study into refining the stacking technique for
probing Hi science to follow-up work.
### 6.1 The stacking procedure of the intensity maps
We use the PCA-cleaned map described in Section 4 and the positions of the
GAMA galaxies to perform a 3D stacking in
$(\delta\alpha,\delta\phi,\delta\nu)$ space so that
$I(\delta\alpha,\delta\phi,\delta\nu)=\frac{\sum_{i}I(\alpha_{i}{+}\delta\alpha,\,\phi_{i}{+}\delta\phi,\,\nu_{i}{+}\delta\nu)w_{i}}{N_{\rm
g}\sum_{i}w_{i}},$ (56)
where $(\alpha_{i},\phi_{i})$ are the right ascension and declination of the
map pixel in which the $i^{\rm th}$ galaxy resides, $I$ is the flux density,
$N_{\rm g}$ is the number of galaxies, and
$w_{i}\,{=}\,w_{\textrm{H{i}}}(\alpha_{i}{+}\delta\alpha,\phi_{i}{+}\delta\phi,\nu_{i}{+}\delta\nu)$
is the reconvolved weight of each pixel defined in Section 4.1. We choose the
angular and the spectral bins to be consistent with the map resolution with
$\Delta\alpha\,{=}\,\Delta\phi\,{=}\,0.3\,$deg. The frequency resolution
corresponds to an effective velocity resolution of $\Delta v\,{=}\,62.83\,$km
s-1 at the centre frequency of the sub-band $997.37\,$MHz. Note that no
Doppler correction has been performed on the data. For our analysis, the
intensity maps are coherently averaged in frequencies for PCA cleaning and
therefore can not be averaged in velocity channels.
In order to calculate the statistical noise for the stacked cube, we perform
the same stacking routine on random galaxy positions, similar to the shuffling
null test in Section 5.6. The positions are randomly sampled uniformly within
the region of the GAMA galaxy sample. The redshifts are randomly sampled
following the redshift distribution of the survey. The random process is
repeated for 100 realisations to calculate the mean and the standard deviation
of the reference cube. The reference cube is used for performing the null
test, i.e. checking that the stacked emission signal in the reference cube is
consistent with zero.
After obtaining the stacked cubes, we create the angular and spectral stacking
by summing over the cubes along the radial and transverse directions. To
control the systematics generated by the foreground cleaning, we limit the
voxels that go into the summation. For angular stacking, only voxels that are
within 150 km s-1 of the central channel are considered, corresponding to the
typical width of the HI emission line profile. For spectral stacking, only the
voxels whose centre positions are within the $1\,$deg area of the centre voxel
in each channel are included, corresponding to the size of the primary beam.
The same summation is performed on the random realisation to calculate the
mean and the standard deviation of the reference image and spectrum. We note
that there is signal loss due to an insufficient number of voxels included.
However, as the random realisations follow the same procedure, it does not
affect the validation of the detection which is our focus in this work.
Figure 19: Left panel: The angular stacking of the GAMA galaxies on the Hi
intensity map. The dashed circle denotes the FWHM of the beam. Central panel:
The reference null detection image from 100 realisations of random galaxy
positions. Right panel: The standard deviation of the reference image
multiplied by 3 to visualise the 99.7% confidence interval.
### 6.2 The detection of stacked Hi signal
In Figure 19, we show the results for the angular stacking of the GAMA
galaxies on the Hi intensity map. The stacked map shows a clear detection at
the centre of the image, with the centre pixel achieving a detection with a
statistical significance of $8.35\,\sigma$. The stacked Hi emission follows
the structure of the beam. The extended structure of emission outside the beam
FWHM is a result of the relatively large thermal noise fluctuation and
contamination from nearby sources in the stacking procedure as discussed
above. We note that, as shown in Figure 4, the lower part of the GAMA galaxy
sample region sits near the edge of the observed intensity maps. As a result,
the lower half of the stacked image has a higher level of statistical
fluctuations as shown in the right panel of Figure 19. Furthermore, the
reference image is consistent with a null detection, demonstrating that the
stacked signal does not arise from systematic effects in the map.
Figure 20: The stacked spectrum of the GAMA galaxies on the Hi intensity map.
The blue solid line marks the measured stacked spectrum. The orange dashed
line denotes the reference spectrum from 100 random realisations, and the
shaded region denotes the standard deviation of the realisations. The black
dotted line denotes the $5\,\sigma$ limit of the reference spectrum rms and is
smoothed with a Gaussian kernel of 300 km/s for better illustration.
In Figure 20, we show the stacked spectrum along the line-of-sight. The peak
of the stacked spectrum shows a detection with a significance of
$6.30\,\sigma$. The detection is validated by the reference spectrum which is
consistent with a null detection. However, as seen in the reference spectrum,
there is a systematic noise component with fluctuations on large frequency
intervals. The systematics can be most visibly seen at the peaks and troughs
of the stacked signal at $v\,{\sim}\,2500,\,4500,\,$ and $7000\,$km/s. We have
identified the source of this fluctuation as the residual systematics in the
map after PCA cleaning, which is demonstrated in Appendix D. As stated
previously, we are only interested in validating the detection against the
null test and leave the suppressing of the systematic effects for future work.
The stacking results show a clear detection of Hi signal in the positions of
the GAMA galaxies. The detection of an emission line signal is statistically
significant against the null test. At the same time, the residual systematics
in the data, while being sub-dominant comparing to the Hi signal, induce
structures in the stacked spectrum. We intend to follow up on the findings in
this work and develop analysis techniques to extract the underlying emission
line signal in the future, as well as using the stacked signal to understand
the residual systematics.
## 7 Conclusions
In this work we have presented L-band observations from the MeerKLASS
intensity mapping campaign. The survey covered a single patch of approximately
$200\,\deg^{2}$, repeatedly scanned 41 times (27 scans remained after
aggressive RFI flagging), which we have referred to as the MeerKLASS L-band
deep field. The observations used in our analysis came from only a small
section of the full L-band ($971.2\,{<}\,\nu\,{<}\,1023.6$ at redshift
$0.39\,{<}\,z\,{<}\,0.46$) to avoid RFI which we find is more present in this
band relative to the lower frequency UHF-band ($580\,{<}\,\nu\,{<}\,1000$)
which all future MeerKLASS surveys will be conducted in. We detailed the
calibration strategy used on these observations which included a new approach
of self-calibration (Section 2.3), exploiting the data’s higher signal-to-
noise to devise a less model-dependent calibration. The improvement in data
quality relative to previous MeerKLASS pilot surveys is evident (see e.g.
Figure 5) and encouraging for future MeerKLASS surveys gathering ever-larger
data sets.
We demonstrated the enhanced signal-to-noise from the calibrated deep-field
intensity maps by presenting the auto-correlation power spectrum (Figure 14)
which revealed the lowest upper-limit from a Hi power spectrum at these
cosmological scales $k\,{<}\,0.3\,h\,\text{Mpc}^{-1}$. This remains a factor
of ${\sim}\,2$ above where we would expect the Hi power to lie. We will pursue
this limit further in follow-up work where we will explore a systematic
removal strategy to reduce the quantity of flagged data plus the cross-
correlation of divided data sets to mitigate the additive bias from time-
dependent residual systematics.
We also presented results from the cross-correlation power spectrum between
the MeerKLASS deep-field and 2269 overlapping GAMA galaxies with spectroscopic
redshifts. Despite the small field of overlap (${\sim}25\%$ of the deep field
patch), we still obtained a clear ${>}\,4\,\sigma$ detection of the cross-
power spectrum (Figure 15). Furthermore, we presented a detection of Hi
emission from stacking the MeerKLASS maps onto the positions of the GAMA
galaxies. This yielded a $8.35\,\sigma$ and $6.30\,\sigma$ detection from
angular and spectral stacking respectively, the first stacking detections with
MeerKAT in single-dish mode (see Section 6).
We implemented several novel strategies into the power spectrum analysis
pipeline, including a selective procedure to target the region of $k$-space
with the highest signal-to-noise. The third panel of Figure 13 demonstrated
the clear window of Fourier modes where power dominates over the error. By
implementing some minima and maxima cuts on $k_{\perp}$ and $k_{\parallel}$ we
were able to exclude regions of $k$-space dominated by error. This provides a
more robust result since error-dominated modes will not be included in final
spherical averaging. Comparison of the error profiles for each $k$-bin (Figure
17) demonstrated a mild improvement in terms of Gaussianity when using the
$k$-cuts, however, the sacrifice of many modes contained at high-$k$ could be
restricting this improvement. We plan to investigate this strategy in future
work where larger data sets will reveal the optimal approach.
We formalised our approach to covariance estimation for the power spectra
measurements which utlises the foreground transfer function distribution from
each mock realisation. This distribution will be determined by all foreseeable
sources of error; residual systematics, thermal noise fluctuations, galaxy
shot-noise, cosmic variance and any uncertainty in signal-loss correction.
This should therefore be a robust method for error estimation. However, we
found the errors in the spherically averaged cross-power spectrum appear to be
large relative to their scatter about the model. This was supported by a
$\chi^{2}$ analysis. Further inspection revealed that the $k$-bins are mildly
correlated and the error profiles for each $k$-bin are not perfectly Gaussian,
both of which could be inflating the errors. We hypothesise that this should
be naturally mitigated with future MeerKLASS analysis where systematics can be
more reliably controlled.
This work also presented validation tests of our improved analysis pipeline.
|
# Bredon homological stability for configuration spaces of $G$-manifolds
Eva Belmont Case Western Reserve University<EMAIL_ADDRESS>, J.D.
Quigley University of Virginia<EMAIL_ADDRESS>and Chase Vogeli
Cornell University<EMAIL_ADDRESS>
###### Abstract.
McDuff and Segal proved that unordered configuration spaces of open manifolds
satisfy homological stability: there is a stabilization map
$\sigma:C_{n}(M)\to C_{n+1}(M)$ which is an isomorphism on
$H_{d}(-;\mathbb{Z})$ for $n\gg d$. For a finite group $G$ and an open
$G$-manifold $M$, under some hypotheses we define a family of equivariant
stabilization maps $\sigma_{G/H}:C_{n}(M)\to C_{n+|G/H|}(M)$ for $H\leq G$. In
general, these do not induce stability for Bredon homology, the equivariant
analogue of singular homology. Instead, we show that each $\sigma_{G/H}$
induces isomorphisms on the ordinary homology of the fixed points of
$C_{n}(M)$, and if the group is Dedekind (e.g. abelian), we obtain the
following Bredon homological stability statement: $H^{G}_{d}(\bigsqcup_{n\geq
0}C_{n}(M))$ is finitely generated over $\mathbb{Z}[\sigma_{G/H}:H\leq G]$.
This reduces to the classical statement when $G=e$.
###### Contents
1. 1 Introduction
2. 2 Equivariant homotopy theory
3. 3 Configuration spaces and stabilization maps
4. 4 Equivariant homological stability
5. 5 Generalizations and variants
## 1\. Introduction
Let $X$ be a topological space and let $C_{k}(X)$ denote the configuration
space of $k$ unordered points in $X$. If $X$ is the interior of a manifold
with nonempty boundary, then there are stabilization maps
(1.0.1) $\sigma:C_{k}(X)\to C_{k+1}(X)$
defined by “adding a point near the boundary" (cf. Section 3.1). In the
1970’s, McDuff [McD75] and Segal [Seg79] observed that these stabilization
maps induce isomorphisms in a range of integral homology groups:
###### Theorem 1.0.2 (McDuff–Segal, Stability for Configuration Spaces
(Strong Form)).
Assume $M$ is the interior of a manifold with nonempty boundary. The
stabilization map (1.0.1) induces an isomorphism
$\sigma_{*}:H_{d}(C_{k}(M))\to H_{d}(C_{k+1}(M))$
for $d\leq k/2$.
### 1.1. Statement of main results
In this paper, we investigate an equivariant analogue of the McDuff–Segal
theorem where the manifold $M$ is replaced by a $G$-manifold (where $G$ is a
finite abelian group) and singular homology is replaced by Bredon homology.
The Bredon homology of a $G$-space $X$ with coefficients in the constant
Mackey functor $\underline{\mathbb{Z}}$, written
$H^{G}_{*}(X;\underline{\mathbb{Z}})$, is an equivariant analogue of singular
homology with integer coefficients. We recall this definition of Bredon
homology, as well as its representation by a $G$-spectrum, in Section 2.1.
If $X$ is a $G$-space, then the space of unordered configurations of $k$
points in $X$
$C_{k}(X):=\\{(x_{1},\ldots,x_{k})\in X^{k}:x_{i}\neq x_{j}\text{ for all
}i\neq j\\}/\Sigma_{k}$
is also a $G$-space with $G$ acting diagonally. If $X$ is a $G$-manifold which
is the interior of a $G$-manifold with nonempty boundary containing a fixed
point, then we can define $G$-equivariant stabilization maps
(1.1.1) $\sigma_{G/G}:C_{k}(X)\to C_{k+1}(X)$
by “adding a fixed point near the boundary” (see Section 3.3). In this
situation, we say $X$ is _$G$ -stabilizable_ (Definition 3.3.1). We can then
ask if these maps induce isomorphisms in Bredon homology for $k$ sufficiently
large. Surprisingly, this is not the case:
###### Theorem A (Failure of Strong Equivariant Form of McDuff–Segal,
Proposition 4.2.1).
Let $G=C_{p}$ and let $M$ be a $G$-manifold which is $G$-stabilizable. The map
(1.1.2) $(\sigma_{G/G})_{*}:H_{0}^{G}(C_{n}(M);\underline{\mathbb{Z}})\to
H_{0}^{G}(C_{n+1}(M);\underline{\mathbb{Z}})$
is not surjective for any $n\geq 1$.
Although the strong form of the McDuff–Segal Theorem fails equivariantly,
there is a weaker form which we can extend to the equivariant setting. Let
$C(X):=\bigsqcup_{k\geq 0}C_{k}(X)$
denote the space of all unordered configurations of finitely many points in
$X$ equipped with the disjoint union topology. The nonequivariant
stabilization maps (1.0.1) give rise to a self-map
(1.1.3) $\sigma:C(X)\to C(X)$
which allows us to state the following corollary of the McDuff–Segal Theorem:
###### Theorem 1.1.4 (McDuff–Segal, Nonequivariant Stability (Finite
Generation Form)).
Assume $M$ is an open, connected manifold with $\dim(M)\geq 2$. Moreover,
suppose that $H_{d}(M)$ is finitely generated as an abelian group for all
$d\geq 0$. Then the homology group $H_{d}(C(M))$ is finitely generated as a
$\mathbb{Z}[\sigma_{*}]$-module for all $d\geq 0$.
###### Remark 1.1.5.
The assumption that $H_{d}(M)$ is finitely generated ensures that
$H_{d}(C_{k}(M))$ is finitely generated for all $k\geq 0$ (Proposition 4.1.5),
so the finite generation form of nonequivariant stability is equivalent (cf.
Proposition 4.1.2) to saying that the map (1.0.1) is an isomorphism for all
$k\gg 0$.
We note that the philosophy of encoding stability as a finite generation
condition is not new. It has become increasingly prevalent in the literature
on homological stability (e.g., [ADCK20, Theorem 1.1]) and representation
stability (e.g. [CEF15, Theorem 6.2.1]).
The obvious finite generation form of equivariant stability fails since the
equivariant stabilization map (1.1.2) does not induce isomorphisms for all
$k\gg 0$. In other words, $H_{d}^{G}(C(M);\underline{\mathbb{Z}})$ is _not_
finitely generated as a $\mathbb{Z}[(\sigma_{G/G})_{*}]$-module.
However, there is a naturally occurring polynomial ring over which
$H_{d}^{G}(C(M);\underline{\mathbb{Z}})$ is a finitely generated module. If
$X$ is a $G$-manifold which is the interior of a $G$-manifold with boundary
containing an orbit of the form $G/H$, then we can define a new equivariant
stabilization map
(1.1.6) $\sigma_{G/H}:C_{k}(M)\to C_{k+|G/H|}(M)$
by “adding an _orbit_ of type $G/H$ near the boundary.” In this situation, we
say $X$ is _$H$ -stabilizable_ (Definition 3.3.1).
We adopt the philosophy that since orbits are equivariant analogues of points,
the equivariant analogue of stabilization maps should be the family of
stabilization maps $\sigma_{G/H}$. As such, we replace the ring
$\mathbb{Z}[\sigma_{*}]$ by the ring
$P_{G}=\mathbb{Z}[(\sigma_{G/H})_{*}:(H)\leq G]$
of all equivariant stabilization maps. By considering all possible equivariant
stabilization maps, we obtain our stability result.
###### Theorem B (Bredon Homological Stability, Theorem 4.5.5).
Let $G$ be an abelian group and $M$ a $G$-manifold that is the interior of a
compact $G$-manifold with boundary. Assume that for all $H\leq G$, $M$ is
$H$-stabilizable and $M^{H}$ is connected. Then
$H_{d}^{H}(C(M);\underline{\mathbb{Z}})$ is finitely generated over $P_{G}$
for all $H\leq G$.
For example, these hypotheses are satisfied in the case where $M$ is a sum of
copies of the regular representation (see Example 4.5.6).
###### Remark 1.1.7.
See Lemma 4.4.2 for the main reason for the abelian hypothesis, and Remark
4.4.3 for a mild relaxation of this hypothesis. Group-theoretic hypotheses are
also used in Lemma 4.5.1 and 4.5.3.
###### Remark 1.1.8.
We discuss two natural extensions of B in Section 5: in Theorem 5.1.3, we show
that stability holds for $RO(G)$-graded Bredon homology, and in Theorem 5.2.5,
we show that stability holds for Mackey functor-valued Bredon homology. The
$RO(G)$-graded Bredon homology of _ordered_ configuration spaces of
$G$-manifolds will also be studied in forthcoming work of Dugger and Hazel
[DH].
### 1.2. Motivation
Before discussing the proofs of A and B, we discuss some motivating ideas and
directions for future work.
#### 1.2.1. Equivariant loop spaces and Dyer–Lashof operations
Nonequivariantly, May’s recognition principle [May72] implies that $n$-fold
loop spaces are equivalent to algebras over the $E_{n}$-operad
$\mathcal{E}_{n}$ whose $k$-th space is $C_{k}(\mathbb{R}^{n})$. Using Cohen’s
computation of $H_{*}(C_{k}(\mathbb{R}^{n}))$ in [CLM76, Ch. III], one can
then define Dyer–Lashof operations on the mod $p$ homology of any $n$-fold
loop space.
In [GM17], Guillou and May proved an equivariant analogue of the recognition
principle: if $V$ is a real orthogonal $G$-representation, then $V$-fold loop
spaces are equivalent to algebras over the $E_{V}$-operad $\mathcal{E}_{V}$
whose $k$-th space is $C_{k}(V)$. They remark that in contrast with the
nonequivariant situation, very little is known about $H_{d}^{G}(C_{k}(V))$. B
gives the first step toward a more systematic analysis of these Bredon
homology groups which could shed light on equivariant Dyer–Lashof operations
(cf. [Wil19] for $G=C_{2}$). We verify that the hypotheses of B are satisfied
in the case of sums of regular representations in Example 4.5.6.
#### 1.2.2. Equivariant diffeomorphism groups of smooth $G$-manifolds
Configuration spaces of ordinary manifolds appear in the study of
diffeomorphism groups of smooth manifolds via the “decoupling theorems" of
Bödigheimer–Tillmann [BT01] and Bonatto [Bon22]. It would be interesting to
see if the configuration spaces of $G$-manifolds arise analogously in the
equivariant diffeomorphism groups of smooth $G$-manifolds.
### 1.3. Overview of key ideas
We now summarize the key ideas going into the proofs of A and B.
#### 1.3.1. Equivariant stabilization maps
A map between $G$-spaces induces a map on Bredon homology only if it is
$G$-equivariant, so we must be careful about how we add points. One natural
option is to add a _fixed_ point near infinity (provided our space has such a
fixed point), which yields the equivariant stabilization map (1.1.1) mentioned
above with underlying map (1.0.1). However, A shows that this map does not
produce isomorphisms in Bredon homology after iteration.
One of our key observations is that under suitable hypotheses on the manifold
$M$, there is a stabilization map (1.1.2) for each subgroup $H\leq G$. B says
that the Bredon homology stabilizes when all of these equivariant
stabilization maps are taken into account simultaneously.
###### Remark 1.3.1.
The stabilization map corresponding to $H\leq G$ inserts an orbit of the form
$G/H$, i.e., inserts $|G/H|$ points at once. Other stabilization maps which
add multiple points at once have appeared in the context of higher order
representation stability [MW19].
#### 1.3.2. Formulating stability in terms of finite generation
While none of the orbit stabilization maps (1.1.2) yield Bredon homological
stability for a fixed $H$ (A), the collection of maps $\\{\sigma_{G/H}:(H)\leq
G\\}$ yield a form of stability (B). To make this precise, we translate
homological stability from a statement about maps in homology to a statement
about finite generation over a polynomial ring (Proposition 4.1.2).
#### 1.3.3. Reduction from Bredon homology to singular homology of fixed
points
The analogue of cellular homology in the context of $G$-spaces is _Bredon
homology_ , $H^{G}_{d}(-)$ (cf. [May96, XIII.4.1], Section 2.1). The Bredon
homology of a $G$-space is not simply the homology of its fixed points, but in
Section 2 we use equivariant stable homotopy theory to relate these notions.
The main lemma enabling this comparison is Lemma 2.4.2, together with
statements in Section 2.5 relating the geometric localization of Bredon
homology to singular homology.
#### 1.3.4. Expressing fixed points in terms of nonequivariant configuration
spaces
We prove that the fixed points $C_{n}(M)^{G}$ satisfy homological stability by
reducing to 1.0.2. To do so, we express $C_{n}(M)^{G}$ in terms of
configuration spaces of other manifolds using the notion of _unordered
$S$-configurations_. We note that we are not the first to consider equivariant
configuration spaces (cf. [RS00]), but we use a new convention here.
Let $X$ be a $G$-space and let $S$ be a finite $G$-set. We define the _space
of unordered $S$-configurations in $X$_ (Definition 3.2.3) by
$C_{S}^{G}(X):=\operatorname{Emb}^{G}(S,X)/\operatorname{Aut}(S).$
When $M$ is $H$-stabilizable, the equivariant stabilization map (1.1.6)
restricts to a stabilization map
$\sigma_{G/H}:C_{S}^{G}(M)\to C_{S+G/H}^{G}(M),$
where $S+G/H$ denotes the disjoint union of $S$ and $G/H$ as $G$-sets.
###### Example 1.3.2.
If $S$ is a trivial $G$-set of cardinality $k$, then we have that
$C_{S}^{G}(X)=C_{k}(X^{G})$
is the space of unordered configurations of $k$ points in the $G$-fixed points
of $X$. If $X$ is a $G$-stabilizable $G$-manifold, the stabilization map
$\sigma_{G/G}:C_{S}^{G}(X)\to C_{S+[G/G]}^{G}(X)$
is the classical stabilization map
$\sigma:C_{k}(X^{G})\to C_{k+1}(X^{G}).$
Passing through the $S$-configuration spaces $C_{S}^{G}(X)$ allows us to
identify the fixed points $C_{n}(X)^{G}$ in terms of ordinary configuration
spaces:
###### Theorem C (Proposition 3.2.10 and Corollary 3.2.9).
1. (1)
For a $G$-space $X$ and $n\geq 0$, there is a disjoint union decomposition
$C_{n}(X)^{G}\cong\coprod_{|S|=n}C^{G}_{S}(X),$
where the disjoint union runs over isomorphism classes of finite $G$-sets $S$
with cardinality $n$.
2. (2)
Let $S$ be the finite $G$-set
$S=\bigsqcup_{(H)}k_{(H)}[G/H],$
where $k_{(H)}\geq 0$ for each conjugacy class of subgroups $(H)$. Then
$C_{S}^{G}(X)\cong\prod_{(H)}C_{k_{(H)}}(X_{(H)}/G),$
where $X_{(H)}$ is the subspace of $X$ consisting of points with stabilizer
conjugate to $(H)$ (Definition 3.2.4).
We apply part (1) of C to prove A; the essential point is that when $G$ is
nontrivial, the number of summands appearing in $C_{n}(X)^{G}$ increases as
$n$ increases, so the stabilization map cannot be surjective on homology. We
apply parts (1) and (2) to prove D.
#### 1.3.5. Stability for $S$-configurations and Bredon homology
Since they can be expressed in terms of ordinary configuration spaces, the
spaces of $S$-configurations often satisfy an analogue of the McDuff–Segal
theorem:
###### Theorem D (Theorem 4.3.1).
Let $M$ be a $G$-manifold and $S$ a finite $G$-set containing $k$ copies of
the orbit $[G/H]$ for a conjugacy class of subgroups $(H)$. If $M$ is
$H$-stabilizable and $M_{(H)}/G$ is connected, then the $(H)$-stabilization
map of (3.3.3) induces an isomorphism
$(\sigma_{G/H})_{*}:H_{d}(C^{G}_{S}(M))\to H_{d}(C^{G}_{S+[G/H]}(M))$
in integral homology in degrees $d\leq k/2$.
As explained above, B follows from stability for the fixed points
$C_{n}(M)^{G}$. C allows us to express these fixed points in terms of
$S$-configurations, and D states that the equivariant stabilization map
$\sigma_{G/H}$ from (1.1.6) induces an isomorphism in the homology of
$C_{S}^{G}(M)$ when $S$ contains sufficiently many copies of $[G/H]$. In
particular, once $S$ contains sufficiently many copies of $[G/K]$ for _any_
conjugacy class of subgroups $(K)$, there is some equivariant stabilization
map which induces an isomorphism on homology. Roughly speaking, B then follows
from the fact that there are finitely many finite $G$-sets $S$ for which none
of the equivariant stabilization maps induce isomorphisms in homology.
### 1.4. Outline
In Section 2, we recall the equivariant homology theory (Bredon homology) we
work with throughout the paper. We also recall the isotropy separation
sequence and prove some key results (Lemma 2.4.2, Proposition 2.5.1) needed to
reduce from the equivariant to nonequivariant setting.
In Section 3, we recall configuration spaces, stabilization maps, and their
equivariant analogues. We then introduce the more refined spaces of
$S$-configurations and use them to prove C which identifies the fixed points
of the configuration spaces of $G$-manifolds with (sums of products of)
configuration spaces in nonequivariant manifolds.
In Section 4, we prove A, B, and D, using the results of the previous section
to reduce to classical homological stability results.
In Section 5, we extend our results to the $RO(G)$-graded and Mackey functor-
valued settings.
### 1.5. Conventions
Throughout, $G$ denotes a finite group. If $M$ is an abelian group, then
$\underline{M}$ denotes the constant Mackey functor on $M$. In this work,
“manifold” means “smooth manifold.”
### 1.6. Acknowledgments
The authors thank Dan Dugger, Jeremy Hahn, Ben Knudsen, Irakli Patchkoria,
Oscar Randal-Williams, Bridget Schreiner, Inna Zakharevich, and Mingcong Zeng
for helpful discussions. The first author was partially supported by NSF grant
DMS-2204357. The second author was partially supported by NSF grants
DMS-2039316 amd DMS-2314082, as well as an AMS-Simons Travel Grant. The second
author also thanks the Max Planck Institute in Bonn for providing a wonderful
working environment and financial support during the beginning of this
project.
## 2\. Equivariant homotopy theory
In this section, we discuss the ideas from equivariant homotopy theory which
we will need in the sequel. Section 2.1 contains a rapid recollection of
Bredon homology with coefficients in a Mackey functor, which is the central
equivariant homology theory we use throughout this work. Section 2.2 and
Section 2.3 recall two standard constructions from equivariant stable homotopy
theory: geometric fixed points and proper homotopy orbits. These auxiliary
constructions appear in the isotropy separation sequence, which we recall in
Section 2.4. Our main technical result, Lemma 2.4.2, applies the isotropy
separation sequence to pass from Bredon homology to a simpler nonequivariant
homology theory. This simpler theory is identified as a sum of singular
homology theories in Section 2.5.
### 2.1. Bredon homology
Bredon homology associates to a $G$-space a sequence of Mackey functors (see
[May96, §IX.4]). As in nonequivariant homotopy theory, Bredon homology with
coefficients in a Mackey functor $\underline{M}$ is representable in the
equivariant stable homotopy category. By [May96, Theorem XIII.4.1], there is
an Eilenberg–MacLane $G$-spectrum $H\underline{M}$ with the property that its
homotopy Mackey functors satisfy
$\underline{\pi}_{0}(H\underline{M})\cong\underline{M}$ and
$\underline{\pi}_{i}(H\underline{M})=0$ for $i>0$. If $X$ is a $G$-space, then
we have
$\underline{H}_{*}^{(-)}(X;\underline{M})\cong\underline{\pi}_{*}(\Sigma^{\infty}_{G}X\wedge
H\underline{M}),$
where $\Sigma^{\infty}_{G}X$ is the _suspension $G$-spectrum of $X$_, and
$-\wedge-$ is the smash product of $G$-spectra.
The Bredon homology of a $G$-CW complex can equivalently be defined using an
equivariant analogue of the cellular chain complex [May96, §I.4]. A
_coefficient system_ is a contravariant functor
$\mathcal{O}_{G}\to\mathcal{A}b$, where $\mathcal{O}_{G}$ is the _orbit
category_ of $G$. If $X$ is a $G$-CW complex, we may define a coefficient
system by
$\underline{C}_{n}(X)(G/H):=\underline{H}_{n}(X^{n},X^{n-1};\mathbb{Z})(G/H):=H_{n}((X^{n})^{H},(X^{n-1})^{H};\mathbb{Z}).$
The connecting homomorphisms for the triples
$((X^{n})^{H},(X^{n-1})^{H},(X^{n-2})^{H})$ define a map of coefficient
systems
$d:\underline{C}_{n}(X)\to\underline{C}_{n-1}(X).$
If $\underline{M}$ is a _Mackey functor_ , we may define cellular chains with
coefficients in $\underline{M}$ by
$C_{n}^{G}(X;\underline{M}):=\underline{C}_{n}(X)\otimes\underline{M}:=\bigoplus_{G/H\in\mathcal{O}_{G}}\left(\underline{C}_{n}(X)(G/H)\otimes\underline{M}(G/H)\right)/\sim,$
where $f^{*}c\otimes m\sim c\otimes f_{*}m$ for a map $f:G/H\to
G/K\in\mathcal{O}_{G}$ and elements $c\in\underline{C}_{n}(X)(G/H)$ and
$m\in\underline{M}(G/H)$. This is a chain complex of abelian groups, with
differential given by $\partial=d\otimes 1$. By forgetting structure, we may
analogously define
$C_{n}^{H}(X;\underline{M}):=\underline{C}_{n}(i^{*}_{H}X)\otimes
i^{*}_{H}\underline{M}$
for each subgroup $H<G$; these assemble into a chain complex of coefficient
systems whose homology
$\underline{H}_{*}^{(-)}(X;\underline{M}):=H_{*}(C_{n}^{(-)}(X;\underline{M}))$
is the _Bredon homology of $X$ with coefficients in $\underline{M}$_.
For much of the paper, we will be concerned with $\mathbb{Z}$-graded Bredon
homology _groups_. By this, we mean the $G/G$-levels
$\underline{H}_{*}^{G}(X;\underline{M})$ of the Bredon homology Mackey
functors constructed here. We return to the level of generality of Bredon
homology Mackey functors in Section 5. It is also possible to extend Bredon
homology with coefficients in a Mackey functor from a $\mathbb{Z}$-graded
collection of groups to a $RO(G)$-graded collection of groups, where $RO(G)$
denotes the representation ring of $G$ [May96, §IX.5]. We review this
extension in Section 5 as well.
### 2.2. Geometric fixed points
As outlined in the previous section, the Bredon homology groups of a $G$-space
$X$ are the (categorical) fixed points of the $G$-spectrum
$\Sigma^{\infty}_{G}X\wedge H\underline{M}$. The _geometric fixed points_
construction is related to categorical fixed points (in a way that will be
made precise in Section 2.4), but has properties that make it more amenable to
computation. We refer the reader to [HHR16, Sec. 2.5] for proofs of the
results in this section.
###### Definition 2.2.1.
Let $G$ be a finite group and let $X$ be a $G$-spectrum. Let $\mathcal{P}$ be
the family of proper subgroups of $G$, and $E\mathcal{P}$ be the unique
$G$-homotopy type such that
$E\mathcal{P}^{H}\simeq\begin{cases}\emptyset&H=G,\\\ *&H<G.\end{cases}$
Consider the collapse map $E\mathcal{P}_{+}\to S^{0}$ and denote by
$\widetilde{E\mathcal{P}}$ its cofiber. The _geometric fixed points_ of $X$
are given by
$\Phi^{G}X=(\widetilde{E\mathcal{P}}\wedge X)^{G}.$
The key properties of the geometric fixed points functor are that it commutes
with taking suspension spectra and is strong symmetric monoidal. These
properties are the content of the following propositions.
###### Proposition 2.2.2.
For any $G$-space $X$, there is an equivalence of spectra
$\Phi^{G}\Sigma^{\infty}_{G}X\simeq\Sigma^{\infty}(X^{G}).$
###### Proposition 2.2.3.
For any two $G$-spectra $X$ and $Y$, there is an equivalence of spectra
$\Phi^{G}(X\wedge Y)\simeq\Phi^{G}X\wedge\Phi^{G}Y.$
### 2.3. Proper homotopy orbits
In order to relate geometric fixed points to categorical fixed points, we will
need the following auxiliary construction.
###### Definition 2.3.1.
The _proper homotopy orbits_ of a $G$-spectrum $X$ are given by
$X_{h\mathcal{P}}=(E\mathcal{P}_{+}\wedge X)^{G}.$
### 2.4. Isotropy separation sequence
In this section, we prove Lemma 2.4.2, a central tool in our analysis of
Bredon homology. It is based on the following cofiber sequence, which allows
us to understand categorical fixed points, provided we can understand
geometric fixed points and proper homotopy orbits.
###### Lemma 2.4.1 (Isotropy Separation Sequence [HHR16, Section 2.5.2]).
The collapse map of Definition 2.2.1 induces a cofiber sequence of spectra
$X_{h\mathcal{P}}\to X^{G}\to\Phi^{G}X.$
###### Lemma 2.4.2.
Let $G$ be a finite group, let $d\geq 0$, and let $X$ be a $G$-spectrum. Let
$\mathcal{C}$ be a Serre class. Assume
$\pi_{q}\Phi^{K}(H\underline{\mathbb{Z}}\wedge X)\in\mathcal{C}$ for all
$K\leq G$, $q\leq d+1$. Then
$H_{q}^{K}(X;\underline{\mathbb{Z}})\in\mathcal{C}$ for all $K\leq G$, $q\leq
d$.
The proof relies on the following homological fact, the proof of which we
learned from Piotr Pstrągowski.
###### Lemma 2.4.3 (Four Lemma mod Serre).
Suppose we have a commutative diagram
${A}$${B}$${C}$${D}$${A^{\prime}}$${B^{\prime}}$${C^{\prime}}$${D^{\prime}}$$\scriptstyle{f}$$\scriptstyle{\ell}$$\scriptstyle{g}$$\scriptstyle{m}$$\scriptstyle{h}$$\scriptstyle{n}$$\scriptstyle{p}$$\scriptstyle{r}$$\scriptstyle{s}$$\scriptstyle{t}$
in an abelian category $\mathcal{A}$. Let $\mathcal{C}\subseteq\mathcal{A}$ be
a Serre subcategory. Suppose the rows are exact, $\ell$ is an epimorphism mod
$\mathcal{C}$ and $p$ is a monomorphism mod $\mathcal{C}$. If $n$ is an
epimorphism mod $\mathcal{C}$ then so is $m$. If $m$ is a monomorphism mod
$\mathcal{C}$ then so is $n$.
###### Proof.
We just prove the first statement, as the second is analogous. As
$\mathcal{C}$ is a Serre subcategory, there exists an abelian category
$\mathcal{A}/\mathcal{C}$ and an exact functor
$F:\mathcal{A}\to\mathcal{A}/\mathcal{C}$ which is essentially surjective with
kernel $\mathcal{C}$ [Sta18, Lemma 02MS]. Applying $F$ yields a commutative
diagram
${FA}$${FB}$${FC}$${FD}$${FA^{\prime}}$${FB^{\prime}}$${FC^{\prime}}$${FD^{\prime}}$$\scriptstyle{Ff}$$\scriptstyle{F\ell}$$\scriptstyle{Fg}$$\scriptstyle{Fm}$$\scriptstyle{Fh}$$\scriptstyle{Fn}$$\scriptstyle{Fp}$$\scriptstyle{Fr}$$\scriptstyle{Fs}$$\scriptstyle{Ft}$
in the abelian category $\mathcal{C}/\mathcal{A}$ in which the rows are exact,
$F\ell$ and $Fn$ are epimorphisms, and $Fp$ is a monomorphism. The classical
Four Lemma implies that $Fm$ is an epimorphism, so
$F\operatorname{coker}(m)=\operatorname{coker}(Fm)=0$, i.e.,
$\operatorname{coker}(m)$ lies in $\mathcal{C}$, i.e., $m$ is an epimorphism
mod $\mathcal{C}$. ∎
###### Lemma 2.4.4.
Let $\mathcal{C}\subseteq\mathcal{A}$ be a Serre category. Let
$\sigma:E^{\prime}_{r}\to E^{\prime\prime}_{r}$ be a map of spectral sequences
that is an isomorphism modulo $\mathcal{C}$ on the $E_{r}$ page for some $r$.
Then $E^{\prime}_{r+1}\overset{\sigma}{\to}E^{\prime\prime}_{r+1}$ is an
isomorphism mod $\mathcal{C}$.
###### Proof.
This is a straightforward consequence of Lemma 2.4.3. First apply Lemma 2.4.3
to the diagram
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z^{\prime}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\prime}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{r}}$$\scriptstyle{\sigma}$$\textstyle{E^{\prime}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Z^{\prime\prime}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{E^{\prime\prime}_{r}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d_{r}}$$\textstyle{E^{\prime\prime}_{r}}$
(where $Z^{\prime}_{r}$ denotes the submodule of cycles) to show that
$Z^{\prime}_{r}\to Z^{\prime\prime}_{r}$ is an isomorphism mod $\mathcal{C}$.
Similar arguments involving the exact sequences $0\to Z^{\prime}_{r}\to
E^{\prime}_{r}\to B^{\prime}_{r}\to 0$ and $0\to B^{\prime}_{r}\to
Z^{\prime}_{r}\to Z^{\prime}_{r}/B^{\prime}_{r}=E^{\prime}_{r+1}\to 0$ show
that the induced maps $Z^{\prime}_{r}\to Z^{\prime\prime}_{r}$ and
$E^{\prime}_{r+1}\to E^{\prime\prime}_{r+1}$, respectively, are isomorphisms
mod $\mathcal{C}$. ∎
###### Proof of Lemma 2.4.2.
By induction up the subgroup lattice. For the base case,
$H^{e}_{q}(X;\underline{\mathbb{Z}})=\pi_{q}((H\underline{\mathbb{Z}}\wedge
X)^{e})=\pi_{q}(\Phi^{e}(H\underline{\mathbb{Z}}\wedge X))\in\mathcal{C}.$
Now let $\\{e\\}<K\leq G$ and let $\mathcal{P}$ denote the family of proper
subgroups of $K$. The isotropy separation sequence
$(H\underline{\mathbb{Z}}\wedge
X)_{h\mathcal{P}}\to(H\underline{\mathbb{Z}}\wedge
X)^{K}\to\Phi^{K}(H\underline{\mathbb{Z}}\wedge X)$
gives rise to a long exact sequence
$\displaystyle\cdots\to\pi_{q+1}(\Phi^{K}(H\underline{\mathbb{Z}}\wedge
X))\to\pi_{q}((H\underline{\mathbb{Z}}\wedge X)_{h\mathcal{P}})\to
H^{K}_{q}(X;\underline{\mathbb{Z}})$
$\displaystyle\to\pi_{q}(\Phi^{K}(H\underline{\mathbb{Z}}\wedge
X))\to\pi_{q-1}((H\underline{\mathbb{Z}}\wedge X)_{h\mathcal{P}})\to\cdots.$
To show that the middle term is in $\mathcal{C}$, we claim it suffices to show
that all the other terms are in $\mathcal{C}$: apply Lemma 2.4.3 to the map
from the above exact sequence to the zero exact sequence.
The assumption says that the first and fourth terms are in $\mathcal{C}$, so
we focus on showing the second term is in $\mathcal{C}$, which would also show
the fifth term is in $\mathcal{C}$ because $q\leq d$ is arbitrary. There is a
cell decomposition of $E\mathcal{P}_{+}$ by a finite collection of cells
$\\{{K/H_{i}}_{+}\wedge S^{n_{i}}\\}_{i\in\mathcal{I}}$
with every $H_{i}<K$ (cf. [HHR16, Section 2.5.2]). Filtering
$E\mathcal{P}_{+}\wedge X$ by the cells of $E\mathcal{P}_{+}$ gives rise to a
spectral sequence
$E_{1}^{i,p-n_{i}}\cong H_{p}^{K}({K/H_{i}}_{+}\wedge S^{n_{i}}\wedge
X)\Rightarrow H^{K}_{p-n_{i}}(E\mathcal{P}_{+}\wedge X).$
We have
$\displaystyle H^{K}_{p}({K/H}_{+}\wedge S^{n}\wedge X)$
$\displaystyle\cong[S,H\underline{\mathbb{Z}}\wedge K/H\wedge
X]^{K}_{p-n}\cong[{K/H}_{+},H\underline{\mathbb{Z}}\wedge X]^{K}_{p-n}$
$\displaystyle\cong[S,H\underline{\mathbb{Z}}\wedge X]^{H}_{p-n}\cong
H^{H}_{p-n}(X;\underline{\mathbb{Z}})$
using self-duality of $K/H_{+}$. In particular, since $H_{i}<K$, we have
$E_{1}^{i,q}\in\mathcal{C}$ by the induction hypothesis. By Lemma 2.4.4,
$H^{K}_{q}(E\mathcal{P}_{+}\wedge X)$ is also in $\mathcal{C}$. ∎
### 2.5. Geometric fixed points of Eilenberg–MacLane $G$-spectra
In light of the previous section, one piece of understanding the Bredon
homology $(H\underline{M}\wedge\Sigma^{\infty}_{G}X)^{G}$ of a $G$-space $X$
is understanding a geometric fixed points term of the form
$\Phi^{G}(H\underline{M}\wedge\Sigma^{\infty}_{G}X)\simeq\Phi^{G}H\underline{M}\wedge\Phi^{G}\Sigma^{\infty}_{G}X\simeq\Phi^{G}H\underline{M}\wedge\Sigma^{\infty}X^{G},$
which is the $\Phi^{G}H\underline{M}$-homology of the space $X^{G}$. In many
cases of interest (see Proposition 2.5.1), this splits into (nonequivariant)
Eilenberg-Maclane spaces, allowing us to reduce to studying ordinary homology
of fixed point spaces.
Recall that the category of Mackey functors is symmetric monoidal with unit
the Burnside Mackey functor $\underline{A}$ and monoidal product the _box
product_. A Green functor is a monoid in the category of Mackey functors
[Lew].
###### Proposition 2.5.1.
Let $\underline{M}$ be a Green functor. For all $H\leq G$, the spectrum
$\Phi^{H}H\underline{M}$ is a connective generalized Eilenberg–MacLane
spectrum, i.e., it splits as a wedge of nonnegative suspensions of
Eilenberg–MacLane spectra. Moreover, if $\underline{M}(G/K)$ is finitely
generated as an abelian group for all $K\leq H$, then
$\pi_{k}\Phi^{H}H\underline{M}$ is a finitely generated abelian group for each
$k\in\mathbb{Z}$.
###### Proof.
The lax monoidal transformation $(-)^{H}\to\Phi^{H}(-)$ induces a map of
commutative ring spectra (cf. [HHR16, §B.10.5])
$(H\underline{M})^{H}\to\Phi^{H}H\underline{M}.$
Since $H\underline{M}$ is Eilenberg–MacLane, $(H\underline{M})^{H}\simeq
H\underline{M}(G/H)$, and since $\underline{M}$ is a Green functor,
$\underline{M}(G/H)$ is a commutative ring (see, e.g., [Shu10, §2.2.5]).
Therefore we have a map of commutative ring spectra
$H\mathbb{Z}\to
H\underline{M}(G/H)\simeq(H\underline{M})^{H}\to\Phi^{H}H\underline{M}$
which determines an $H\mathbb{Z}$-module structure on
$\Phi^{H}H\underline{M}$. The claim that $\Phi^{H}H\underline{M}$ is
generalized Eilenberg–MacLane then follows from the classical fact (usually
attributed to Adams) that any $H\mathbb{Z}$-module is generalized
Eilenberg–MacLane.
That $\Phi^{H}H\underline{M}$ is connective follows from the isotropy
separation sequence: we have
$\Phi^{H}H\underline{M}\simeq\operatorname{cofib}((H\underline{M})_{h\mathcal{P}}\to(H\underline{M})^{H})$
and both $(H\underline{M})_{h\mathcal{P}}$ and $(H\underline{M})^{H}$ are
connective, so $\Phi^{H}H\underline{M}$ is connective.
Finite generation also follows from the isotropy separation sequence: the
finite generation hypothesis on $\underline{M}(G/K)$, along with the fact that
$E\mathcal{P}$ admits an $H$-CW structure with finitely cells in each
dimension, implies that each homotopy group of
$(H\underline{M})_{h\mathcal{P}}$ and $(H\underline{M})^{H}\simeq
H(\underline{M}(G/H))$ is finitely generated, so the same is true for
$\Phi^{H}H\underline{M}$. ∎
###### Example 2.5.2.
The Burnside Mackey functor $\underline{A}$ is the Mackey functor given by
$\underline{A}(G/K)=A(K),$
where $A(K)$ denotes the Grothendieck ring of finite $K$-sets. The
restrictions and transfers in $\underline{A}$ are induced by restriction and
induction of $K$-sets, respectively.
The constant Mackey functor $\underline{\mathbb{Z}}$ has
$\underline{\mathbb{Z}}(G/K)=\mathbb{Z}$ in every level. The restrictions in
$\underline{\mathbb{Z}}$ are the identity map $\mathbb{Z}\to\mathbb{Z}$ and
the transfer $\underline{\mathbb{Z}}(G/K)\to\underline{\mathbb{Z}}(G/L)$ is
given by multiplication by the index $[L:K]$.
Both $\underline{A}$ and $\underline{\mathbb{Z}}$ are Green functors, so
$\Phi^{H}H\underline{A}$ and $\Phi^{H}H\underline{\mathbb{Z}}$ are both
connective generalized Eilenberg–MacLane spectra. Moreover,
$\underline{A}(G/K)=A(K)$ and $\underline{\mathbb{Z}}(G/K)=\mathbb{Z}$ are
finitely generated abelian groups for all $K\leq H$, so
$\pi_{k}\Phi^{H}H\underline{A}$ and $\pi_{k}\Phi^{H}H\underline{\mathbb{Z}}$
are both finitely generated abelian groups for all $k\geq 0$.
###### Remark 2.5.3.
It is possible to describe $\Phi^{G}H\underline{M}$ explicitly in many cases
of interest using previous computations of $\pi_{*}\Phi^{G}H\underline{M}$
(e.g., [HHR16, Kri20]), but this will not be necessary for our applications.
We conclude by recording the following fact which can be used to reduce to
$p$-Sylow subgroups for coefficients in a constant Mackey functor. We will not
need it later, but it can be used to simplify some arguments in the sequel if
one works with constant Mackey functor coefficients.
###### Proposition 2.5.4.
Let $\underline{R}$ be a constant Green functor and let $\underline{M}$ be an
$\underline{R}$-algebra. If $G$ is not a $p$-group, then
$\Phi^{G}H\underline{M}\simeq*.$
###### Proof.
The $G$-spectrum $H\underline{M}$ is an $H\underline{R}$-algebra, and since
geometric fixed points are strong symmetric monoidal, the spectrum
$\Phi^{G}H\underline{M}$ is a $\Phi^{G}H\underline{R}$-algebra. But [Kri20,
Prop. 11] implies that when $G$ is not a $p$-group,
$\Phi^{G}H\underline{R}\simeq*$, from which we conclude
$\Phi^{G}H\underline{M}\simeq*$. ∎
###### Example 2.5.5.
The constant Green functor $\underline{\mathbb{Z}}$ is an algebra over itself,
so $\Phi^{G}H\underline{\mathbb{Z}}\simeq*$ if $G$ is not a $p$-group. On the
other hand, one can show that $\pi_{0}\Phi^{G}H\underline{A}\cong\mathbb{Z}$
for any finite group $G$, so $\Phi^{G}H\underline{A}\not\simeq*$ for any $G$.
###### Remark 2.5.6.
For simplicity, we will work with $\underline{\mathbb{Z}}$ coefficients in the
sequel. If $\underline{M}$ is a constant Mackey functor which is levelwise
finitely generated, then it is straightforward to deduce $\underline{M}$
coefficient analogues of all of the $\underline{\mathbb{Z}}$ coefficient
statements below using the presentation of $\underline{M}$ as a
$\underline{\mathbb{Z}}$-module (cf. deducing homology isomorphisms with $M$
coefficients from homology isomorphisms with $\mathbb{Z}$ coefficients).
For certain groups, e.g., any cyclic $p$-group $C_{p^{n}}$, it is also
straightforward to pass from $\underline{\mathbb{Z}}$ coefficients to
$\underline{A}$ coefficients using induction up the subgroup lattice, the
short exact sequence of Mackey functors
$0\to\underline{I}\to\underline{A}\to\underline{\mathbb{Z}}\to 0,$
and the fact that $H_{*}^{C_{p^{n}}}(X;\underline{I})\cong
H_{*}^{C_{p^{n-1}}}(X^{C_{p}};\underline{I}^{\prime})$, where
$\underline{I}^{\prime}$ is the $C_{p^{n-1}}$ Mackey functor with
$\underline{I}^{\prime}(C_{p^{n-1}}/C_{p^{m}})=\underline{I}(C_{p^{n}}/C_{p^{m+1}})$.
Since every Mackey functor is an $\underline{A}$-module, one can deduce
analogues of our results with coefficients in any Mackey functor which is
finitely generated for cyclic $p$-groups.
We leave the extension to other coefficients and more complicated groups to
the interested reader.
## 3\. Configuration spaces and stabilization maps
As in the previous section, $G$ is any finite group. In this section, we
discuss configuration spaces and stabilization maps. We recall the
nonequivariant theory in Section 3.1 and discuss the equivariant analogues in
Section 3.2 and Section 3.3. In Section 3.2, we define the more nuanced notion
of $S$-configuration spaces (Definition 3.2.2 and Definition 3.2.3) and their
stabilization maps. Our main results are Proposition 3.2.10, which expresses
the $G$-fixed points of configuration spaces in terms of $S$-configurations,
and Corollary 3.2.9 which expresses each $S$-configuration space in terms of
products of ordinary configuration spaces. Combining these results yields C.
### 3.1. Nonequivariant configuration spaces and stabilization maps
###### Definition 3.1.1.
Let $M$ be a space. The _configuration space of $n$ ordered points in $M$_ is
$\operatorname{Conf}_{n}(M):=\operatorname{Emb}(\\{1,\ldots,n\\},M)=\\{(x_{1},\dots,x_{n})\in
M^{\times n}\hskip 2.0pt:\hskip 2.0ptx_{i}\neq x_{j}\text{ for all }i\neq
j\\}$
topologized as a subspace of $M^{\times n}$.
The _configuration space of $n$ unordered points in $M$_ is the quotient space
$C_{n}(M):=\operatorname{Conf}_{n}(M)/\Sigma_{n},$
where the symmetric group $\Sigma_{n}$ acts by permuting points.
When $M$ is an open, connected manifold with $\dim M\geq 2$, then $M$ is the
interior of a manifold with boundary, and there there are well-defined
_stabilization maps_ $C_{n}(M)\to C_{n+1}(M)$ (cf. [Pal18, Secs. 2.1-2.2]).
Since we will describe the equivariant analogue of these maps in some detail
in Section 3.3, we review the non-equivariant construction.
###### Definition 3.1.2.
We say that an $n$-dimensional manifold $M$ is _stabilizable_ if $M$ is
homeomorphic to the interior of a $n$-dimensional manifold with nonempty
boundary.
###### Construction 3.1.3.
When $M$ is stabilizable, we can define a stabilization map on unordered
configurations as follows. Let $M$ be stabilizable with $M\cong W^{\circ}$,
where $W$ is an $n$-dimensional manifold with nonempty boundary. Let
$p\in\partial W$. By the definition of manifold with boundary, there exists an
open neighborhood $U\subset W$ containing $p$ together with a diffeomorphism
$\phi:U\xrightarrow{\cong}\mathbb{R}^{n}_{+}=\\{x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}:x_{1}\geq
0\\}$
sending $p$ to $0$. Let $b:\mathbb{R}^{n}_{+}\to\mathbb{R}^{n}_{+}$ be a
smooth bump function which sends $0$ to $(1,0,\ldots,0)$. Let $e:W\to W$ be
the self-embedding defined by
$e(x)=\begin{cases}x\quad&\text{ if }x\notin U,\\\ (\phi^{-1}\circ
b\circ\phi)(x)\quad&\text{ if }x\in U.\end{cases}$
Then by construction, $e(M)\subseteq M$ and $e(p)\notin e(M)$, so we may
define
(3.1.4) $\sigma:C_{n}(M)\to C_{n+1}(M)$
by
$\sigma(m_{1},\ldots,m_{n})=(e(m_{1}),\ldots,e(m_{n}),e(p)).$
### 3.2. Equivariant configuration spaces
In this section, we study equivariant configuration spaces. Our main result
(Corollary 3.2.9) will allow us to relate these spaces back to nonequivariant
configuration spaces.
###### Definition 3.2.1.
Let $X$ be a $G$-space and let $n\geq 1$. The configuration space of $n$
ordered points in $X$ is the $G$-space of nonequivariant embeddings
$\operatorname{Conf}_{n}(X)=\operatorname{Emb}(\\{1,\dots,n\\},X).$
The configuration space of $n$ unordered points in $X$ is the quotient
$G$-space
$C_{n}(X)=\operatorname{Conf}_{n}(X)/\Sigma_{n}.$
Let $C(X)=\bigsqcup_{n\geq 1}C_{n}(X)$ equipped with the disjoint union
topology. In $\operatorname{Conf}_{n}(X)$ and $C_{n}(X)$, the $G$-action is
induced by the action on $X$.
These are the same definitions as in the nonequivariant case, but now they are
$G$-spaces. In Proposition 3.2.10, we will show that their fixed points
$C_{n}(X)^{G}$ split into pieces $C_{S}^{G}(X)$ which are described below.
###### Definition 3.2.2.
Let $X$ be a $G$-space and let $S$ be a finite $G$-set of cardinality $n$. The
_space of ordered $S$-configurations in $X$_, denoted
$\operatorname{Conf}^{G}_{S}(X)$, is the space of $G$-equivariant embeddings
$S\to X$,
$\operatorname{Conf}_{S}^{G}(X):=\operatorname{Emb}^{G}(S,X).$
This is a $G$-equivariant analogue of the ordinary notion of an ordered
configuration space. Indeed, if $G=e$ is the trivial group, we have
$\operatorname{Conf}^{e}_{S}(X)\cong\operatorname{Conf}_{n}(X^{e}).$
Precomposing a $G$-equivariant embedding $S\to X$ by an automorphism of $S$
yields another such embedding. In this way, $\operatorname{Conf}^{G}_{S}(X)$
admits an action by the automorphism group $\operatorname{Aut}(S)$.
###### Definition 3.2.3.
The _space of unordered $S$-configurations in $X$_ is the quotient
$C^{G}_{S}(X):=\operatorname{Conf}^{G}_{S}(X)/\operatorname{Aut}(S)$
of $\operatorname{Conf}^{G}_{S}(X)$ by the $\operatorname{Aut}(S)$-action.
Similarly, this is a $G$-equivariant analogue of the ordinary notion of
unordered configuration space. If $G=e$ is the trivial group, then
$\operatorname{Aut}(S)\cong\Sigma_{n}$, and we have
$C^{e}_{S}(X)=\operatorname{Conf}^{e}_{S}(X)/\operatorname{Aut}(S)\cong\operatorname{Conf}_{n}(X^{e})/\Sigma_{n}=C_{n}(X^{e}).$
We can express $C^{G}_{S}(X)$ in terms of nonequivariant configuration spaces
of a related space.
###### Definition 3.2.4 ([tD87, Section I.5]).
For a $G$-space $X$ and subgroup $H\leq G$, we denote by $X_{(H)}$ the
subspace
$X_{(H)}=\\{x\in X:(G_{x})=(H)\\}\subseteq X$
consisting of points with stabilizer conjugate to $H$.
Since the stabilizers of points in a fixed orbit are conjugate, the $G$-action
on $X$ restricts to a $G$-action on $X_{(H)}$. The preimage of any point under
the projection $X_{(H)}\to X_{(H)}/G$ is a full orbit isomorphic to $G/H$.
###### Proposition 3.2.5.
Given a subgroup $H\subseteq G$ and $k\geq 0$, the space of unordered
$k[G/H]$-configurations in $X$ satisfies
$C^{G}_{k[G/H]}(X)\cong C_{k}\left(X_{(H)}/G\right).$
###### Proof.
A point in $C^{G}_{k[G/H]}(X)$ is a subset $S\subseteq X$ which is isomorphic
as a $G$-set to $k[G/H]$. Each point in $S$ has stabilizer conjugate to $H$,
so $S$ is furthermore a subset of $X_{(H)}$. The set $S$ consists of $k$
orbits of type $G/H$, so its image under the projection $X_{(H)}\to X_{(H)}/G$
is a size $k$ subset of $X_{(H)}/G$. This defines a map
$C^{G}_{k[G/H]}(X)\to C_{k}\left(X_{(H)}/G\right),$
which is a homeomorphism. Indeed, the preimage of any $k$ element subset of
$X_{(H)}/G$ is a subset $S\subseteq X_{(H)}\subseteq X$ of the above form,
which defines an inverse map. ∎
The space $X_{(H)}/G$ can also be expressed in terms of $H$-fixed points of
$X$ and their $W_{G}H$-action, as we show in the following lemma.
###### Lemma 3.2.6.
For any $G$-space $X$,
$X_{(H)}/G\cong\Big{(}X^{H}\setminus\bigcup_{K>H}X^{K}\Big{)}/W_{G}H.$
###### Proof.
There is a $G$-equivariant homeomorphism
(3.2.7) $\varphi:X_{(H)}\to
G\times_{N_{G}H}\Big{(}X^{H}\setminus\bigcup_{K>H}X^{K}\Big{)}$
and hence on orbits we have
$\displaystyle X_{(H)}/G$
$\displaystyle\cong\Big{(}X^{H}\setminus\bigcup_{K>H}X^{K}\big{)}/N_{G}H\cong\Big{(}X^{H}\setminus\bigcup_{K>H}X^{K}\big{)}/W_{G}H$
since $H\leq N_{G}H$ acts trivially on $X^{H}$. ∎
In Corollary 3.2.9 we generalize Proposition 3.2.5 to a formula for
$C^{G}_{S}(X)$ for an arbitrary $G$-set $S$. It follows from the next lemma,
whose proof is omitted.
###### Lemma 3.2.8.
Let $S$ and $T$ be finite $G$-sets such that no orbit of $S$ is isomorphic to
an orbit of $T$. Then,
$C^{G}_{S+T}(X)\cong C^{G}_{S}(X)\times C^{G}_{T}(X).$
Combining Proposition 3.2.5 and Lemma 3.2.8, we obtain our desired
decomposition formula.
###### Corollary 3.2.9.
Let $S$ be the finite $G$-set
$S=\coprod_{(H)}k_{(H)}[G/H],$
where $k_{(H)}\geq 0$ for each conjugacy class of subgroups $(H)$. Then,
$C^{G}_{S}(X)\cong\prod_{(H)}C_{k_{(H)}}(X_{(H)}/G).$
Recall that $C_{n}(X)$ denotes the $G$-space of nonequivariant embeddings of
$n$ points. The next proposition is the main result of this section, which
describes its fixed points in terms of the spaces $C_{S}^{G}(X)$ we have been
studying above, in the case that $X$ is a manifold.
###### Proposition 3.2.10.
For a $G$-manifold $M$ and $n\geq 0$, there is a disjoint union decomposition
$C_{n}(M)^{G}\cong\coprod_{|S|=n}C^{G}_{S}(M),$
where the disjoint union runs over isomorphism classes of finite $G$-sets $S$
with cardinality $n$.
###### Proof.
Let $S=\coprod_{i\in I}G/H_{i}$ be a $G$-set of cardinality $n$. Since
$C_{S}^{G}(M)$ and $C_{n}(M)^{G}$ are given compatible quotient topologies, it
suffices to show that $\operatorname{Conf}_{S}^{G}(M)$ is open in
$\operatorname{Conf}_{n}(M)^{G}$. Let $x\in\operatorname{Conf}_{S}^{G}(M)$ be
an ordered configuration consisting of orbits $\mathcal{O}_{i}=\\{g\cdot
x_{i}\\}_{g\in G/H_{i}}$ for $i\in I$. By the equivariant slice theorem (see
e.g. [Aud91, Theorem I.2.1.1]), there are open neighborhoods $U_{i,g}\ni
g\cdot x_{i}$ and diffeomorphisms $\phi_{i}:\bigsqcup_{g}U_{i,g}\cong
G\times_{H_{i}}T_{x_{i}}M$ with $\phi_{i}(g\cdot x_{i})=(g,0)$. Then
$U=\prod_{i,g}U_{i,g}\subseteq M^{\times n}$ is an open neighborhood of $x$.
An element of $\operatorname{Conf}_{n}^{G}(M)\cap U$ is an $n$-tuple
$(y_{i,g})$ where $y_{i,g}\in U_{i,g}$, and we may assume that the $U_{i,g}$’s
are disjoint in $M$. It suffices to show that $y_{i,g}$ has isotropy $H_{i}$.
Since each $U_{i,g}$ contains a single $y_{i,g}$ and $G$ acts on
$\\{y_{i,g}\\}$, an element $g^{\prime}\in G$ fixes $y_{i,g}$ if and only if
it preserves $U_{i,g}\cong\\{g\\}\times T_{x_{i}}M\subseteq
G\times_{H_{i}}T_{x_{i}}M$, and this occurs if and only if $g^{\prime}\in
H_{i}$. ∎
###### Remark 3.2.11.
Though Proposition 3.2.10 uses the theory of (finite-dimensional) manifolds,
we will investigate an analogous statement for $\rho^{\infty}$ in future work
and identify the pieces:
(3.2.12) $C^{G}_{G/H}(\rho^{\infty})\simeq B(W_{G}H).$
These spaces are interesting because Guillou and May [GM17, Lem. 1.2] identify
$C_{n}(\rho^{\infty})$ with the $n$-th space in the $G$-equivariant little
$\rho^{\infty}$-disks operad, which is well-known to be $G$-equivalent to
$B_{G}\Sigma_{n}$ (cf. [CW91, Pg. 489]). In particular, the splitting here
recovers the Lashof–May splitting [May96, Theorem VII.2.7] of
$(B_{G}\Sigma_{n})^{G}$.
### 3.3. Equivariant stabilization maps
Our goal in this subsection is to define an equivariant version of
Construction 3.1.3.
###### Definition 3.3.1.
Let $H\leq G$ be a subgroup. A $G$-manifold $M$ is _$H$ -stabilizable_ if it
is equivariantly homeomorphic to the interior of a $G$-manifold $W$ with
boundary containing a point of isotropy group $H$.
###### Construction 3.3.2.
Suppose $M$ is an $H$-stabilizable $G$-manifold of dimension $n$, and write
$M\cong W^{\circ}$ where $W$ is a $G$-manifold with boundary containing a
point $p\in\partial W$ with isotropy group $H$. By the equivariant slice
theorem for manifolds with boundary [Kan07, Theorem 3.6], there exists an open
subset $U\subseteq W$ containing the orbit $\\{g\cdot p\hskip 2.0pt:\hskip
2.0ptg\in G/H\\}$ together with a $G$-equivariant diffeomorphism
$\phi:U\to\mathbb{R}_{\geq 0}\times(G\times_{H}V)$
sending $g\cdot p$ to $(0,g,0)$, where $\mathbb{R}_{\geq 0}$ has the trivial
action and $V\cong T_{p}(\partial M)$ is the $G$-representation determined by
the action of $G$ on the tangent space of $\partial M$ at $p$. Let
$b:\mathbb{R}_{\geq 0}\times(G\times_{H}V)\to\mathbb{R}_{\geq
0}\times(G\times_{H}V)$ be a smooth, $G$-equivariant bump function which is
the identity in the $G\times_{H}V$ component and sends $(0,g,0)$ to $(1,g,0)$.
Let $e:W\to W$ be the $G$-equivariant self-embedding defined by
$e(x)=\begin{cases}x\quad&\text{ if }x\notin U,\\\ (\phi^{-1}\circ
b\circ\phi)(x)\quad&\text{ if }x\in U.\end{cases}$
Let $g_{1},\ldots,g_{r}$ range over the elements of $G/H$. We define
(3.3.3) $\sigma_{G/H}:C_{n}(M)\to C_{n+|G/H|}(M)$
by
$\sigma(m_{1},\ldots,m_{n})=(e(m_{1}),\ldots,e(m_{n}),e(p),g_{1}e(p),\ldots,g_{r}e(p)).$
###### Remark 3.3.4.
Observe that in 3.3.2, the bump function $\phi$, and hence $\sigma_{G/H}$,
preserves isotropy type. Moreover, since $\sigma_{G/H}$ adds a single $G/H$
orbit, it descends to maps
$\displaystyle C_{n}(M/G)$ $\displaystyle\to C_{n+1}(M/G)$ $\displaystyle
C_{n}(M)^{G}$ $\displaystyle\to C_{n+|G/H|}(M)^{G}$ $\displaystyle
C_{S}^{G}(M)$ $\displaystyle\to C_{S+[G/H]}^{G}(M).$
Putting together Corollary 3.2.9 and Proposition 3.2.10, we expect equivariant
homological stability of $M$ should be related to nonequivariant homological
stability of $M_{(H)}/G$. First (Lemma 3.3.5) we check that $M_{(H)}/G$
satisfies the hypotheses of nonequivariant homological stability except for
connectedness, which is an additional assumption and not needed for the
definition of stabilization maps. Then (Lemma 3.3.6) we check that the
stabilization maps agree. In the next section (Theorem 4.3.1) we will put
these together to prove a form of equivariant homological stability.
###### Lemma 3.3.5.
If $M$ is an $H$-stabilizable smooth $G$-manifold for $H\leq G$, then
$M_{(H)}/G$ is an open smooth manifold.
###### Proof.
Let $W=W_{H}G$. To see that $M_{(H)}/G$ is a manifold, note that
$M_{(H)}/G=M_{(H)}/W$ since $M_{(H)}$ consists entirely of $G/H$-orbit types.
Thus $W=\operatorname{Aut}(G/H)$ acts freely on the manifold $M_{(H)}$, and
its acts properly because $W$ is finite. Thus $M_{(H)}/W$ is a (smooth)
manifold.
For noncompactness, first we show noncompactness of $M_{(H)}$. The
stabilizability assumption says that there is a neighborhood of the manifold
with boundary $W$ that is diffeomorphic to $\mathbb{R}_{\geq
0}\times(G\times_{H}V)$, where $V$ is the tangent space in $\partial W$ of a
point $p\in\partial W$ of isotropy $H$ (see Construction 3.3.2). The subspace
of isotropy $H$ in this neighborhood is $r_{\geq 0}\times(G\times_{H}V^{H})$,
and taking away the boundary, we see that $M_{(H)}$ is missing a limit point,
namely $\\{0\\}\times(G\times_{H}\\{0\\})=p$. Now passing to orbits
$M_{(H)}/G$, we have a neighborhood $\mathbb{R}_{>0}\times V^{H}$ with limit
point $(0,0)$ that is also not in $M_{(H)}/G$. ∎
###### Lemma 3.3.6.
Suppose $M$ is $H$-stabilizable. Then the equivariant stabilization map
(3.3.3) induces a map
$\bar{\sigma}_{G/H}:C_{n}(M_{(H)}/G)\to C_{n+1}(M_{(H)}/G)$
that agrees with the nonequivariant stabilization map (3.1.4).
Note that Lemma 3.3.5 is used to guarantee the map (3.1.4) exists.
###### Proof.
Since $\sigma_{G/H}:C_{n}(M)\to C_{n+|G/H|}(M)$ preserves isotropy type and
adds a $G/H$ orbit, it induces a map
$\bar{\sigma}_{G/H}^{e}:C_{n}(M_{(H)}/G)\to C_{n+1}(M_{(H)}/G)$, and the
equivariant bump function $b$ induces a function $\bar{b}$ on $M_{(H)}/G$.
First note from Lemma 3.3.5 that $M_{(H)}/G$ is a manifold. To check
$\bar{\sigma}_{G/H}$ agrees with the classical stabilization map, we must
check that $\bar{b}$ is a bump function. It is clear that it has compact
support. To check it is smooth, first note that the restriction of the smooth
function $b$ to $M_{(H)}$ is smooth. To check it is smooth after taking
orbits, recall that every orbit in $M_{(H)}$ has a neighborhood diffeomorphic
to $G\times_{H}\mathbb{R}^{n}$, and taking orbits simply collapses the $|G/H|$
Euclidean components to one Euclidean component. ∎
## 4\. Equivariant homological stability
In this section, we prove our main theorems, A and B. Our proof is by
reduction to the classical stability result of McDuff–Segal (Theorem 4.1.1),
which we recall in Section 4.1. We also discuss a weaker notion of homological
stability (Corollary 4.1.3) in terms of finite generation; the relationship
between finite generation stability and sequential stability is made precise
in Proposition 4.1.2. Using nonequivariant homological stability, we prove D
(Theorem 4.3.1) that the equivariant stabilization maps eventually induce
isomorphisms in the homology of $S$-configuration spaces in Section 4.3. We
then use our decomposition of the $G$-fixed points of configuration spaces
into $S$-configuration spaces, along with our Bredon-to-geometric-fixed-point
reduction result (Lemma 2.4.2), to prove Bredon homological stability in
Section 4.4.
### 4.1. Nonequivariant homological stability
###### Theorem 4.1.1 (McDuff–Segal, as stated in [Pal18, Thm. 1.2] with
$X=*$).
Assume $M$ is an open, connected manifold with $\dim(M)\geq 2$. The
stabilization map $\sigma:C_{n}(M)\to C_{n+1}(M)$ induces an isomorphism on
integral homology in degrees $*\leq\frac{n}{2}.$
Our goal for the rest of this subsection is to relate homological stability
with respect to iterated applications of $\sigma$ with finite generation over
$\mathbb{Z}[\sigma_{*}]$. Since this does not depend on the fact that the
groups that stabilize are homology groups, we take an abstract approach,
presenting the translation for arbitrary sequences of modules that stabilize.
Fix a commutative noetherian ring $R$ and let
$A_{0}\xrightarrow{\sigma}A_{1}\xrightarrow{\sigma}A_{2}\xrightarrow{\sigma}\cdots$
be a sequence of $R$-modules $A_{n}$ for $n\geq 0$ connected by maps
$\sigma_{n}:A_{n}\to A_{n+1}$. Write
$A=\bigoplus_{n\geq 0}A_{n}.$
The collection of maps $\sigma_{n}$ assemble into an endomorphism $\sigma:A\to
A$, and this map endows $A$ with the structure of a $R[\sigma]$-module. The
next proposition relates finiteness conditions on $A$ and $\sigma$ to a
stability condition on the sequence.
###### Proposition 4.1.2.
With $R$, $A$ and $\sigma$ as above, the following are equivalent:
1. (1)
$A_{n}$ is a finitely generated $R$-module for each $n$, and the sequence
stabilizes: that is, there exists $N\geq 0$ such that for all $n\geq N$,
$\sigma:A_{n}\to A_{n+1}$ is an isomorphism; and
2. (2)
$A$ is finitely generated as an $R[\sigma]$-module.
###### Proof.
To see (1) $\Rightarrow$ (2), let $f_{n}:R^{r_{n}}\to A_{n}$ be a surjection.
Define a map
$f:R[\sigma]^{r_{0}}\oplus\cdots\oplus R[\sigma]^{r_{N}}\to A$
which maps the generators of $R[\sigma]^{r_{n}}$ to $A_{n}\subseteq A$
according to $f_{n}$ and extends $R[\sigma]$-linearly. To see that $f$ is
surjective, note that by construction it surjects onto
$\bigoplus_{n=0}^{N}A_{n}$, and it surjects onto
$\bigoplus_{n=N}^{\infty}A_{n}\cong A_{N}\otimes\mathbb{Z}[\sigma]$ because it
is a $\sigma$-linear map that surjects onto $A_{N}$.
To see (2) $\Rightarrow$ (1), let $f:R[\sigma]^{r}\to A$ be a surjection from
a free $R[\sigma]$-module of finite rank $r$. Since $f(\sigma
R[\sigma]^{r})\subseteq\sigma A$, $f$ descends to a surjective map on
quotients
$\bar{f}:R^{r}\cong R[\sigma]^{r}/\sigma R[\sigma]^{r}\to A/\sigma
A=\operatorname{coker}\sigma$
which exhibits $\operatorname{coker}\sigma$ as a finitely generated
$R$-module. Note that
$\operatorname{coker}\sigma\cong\bigoplus_{n\geq
0}(\operatorname{coker}\sigma_{n}).$
That $\operatorname{coker}\sigma$ is finitely generated thus implies that
$(\operatorname{coker}\sigma)_{n}=0$ for sufficiently large $n$. It follows
that the sequence is eventually surjective, that is, there exists a $M$ such
that $\sigma_{n}$ is surjective for $n\geq M$. If we let
$K_{i}=\operatorname{ker}(A_{M}\to A_{i}),$
for $i\geq M$, we obtain an increasing chain $K_{M}\subseteq
K_{M+1}\subseteq\cdots$ of submodules of $A_{M}$. As a finitely generated
module over the noetherian ring $R$, $A_{M}$ is noetherian. Therefore, there
must exist an $N\geq 0$ at which point this increasing chain stabilizes, that
is $K_{N}=K_{N+1}=\cdots$. It follows that for $n\geq N$, $\sigma_{n}$ is an
isomorphism.
Now we show that $A_{n}$ is finitely generated over $R$ by induction on $n$.
Since $\operatorname{coker}(\sigma)|_{A_{0}}=A_{0}$, we must have that $A_{0}$
is finitely generated. Now assume $n\geq 1$ and assume inductively that
$A_{n-1}$ is finitely generated. Consider the exact sequence
$A_{n-1}\overset{\sigma}{\to}A_{n}\overset{\sigma}{\to}\operatorname{coker}(\sigma)|_{A_{n}}\to
0.$
Since $A_{n-1}$ and $\operatorname{coker}(\sigma)|_{A_{n}}$ are finitely
generated by assumption, we have that $A_{n}$ is also finitely generated. ∎
###### Corollary 4.1.3.
Let $M$ be a stabilizable open, connected manifold with $\dim(M)\geq 2$. If
$H_{d}(C_{k}(M))$ is finitely generated as an abelian group for all $d\geq 0$,
then $H_{*}(\bigsqcup_{k\geq 0}C_{k}(M))$ is finitely generated as a
$\mathbb{Z}[\sigma_{*}]$-module.
We note that the finite generation of $H_{d}(C_{k}(M))$ in Corollary 4.1.3
follows from the finite generation of $H_{d}(M)$. First we need a lemma.
###### Lemma 4.1.4.
Suppose $X$ has a free $G$-action, and $H_{d}(X;\mathbb{Z})$ is a finitely
generated abelian group for all $d$. Then $H_{d}(X/G;\mathbb{Z})$ is finitely
generated.
###### Proof.
There is a fibration $X\to EG\times_{G}X\to BG$, and since the action of $G$
on $X$ is free, we have $EG\times_{G}X\simeq X/G$. There is a Serre spectral
sequence with local coefficients
$E_{2}^{*,*}=H_{*}(BG;H_{*}(X;\mathbb{Z}))\Rightarrow H_{*}(X/G).$
The $E_{2}$-page is the cohomology of the complex
$C_{*}(EG;\mathbb{Z})\otimes_{\mathbb{Z}[G]}H_{*}(X;\mathbb{Z})$ which is
finitely generated in each degree by assumption. Thus the $E_{2}$-term is
finitely generated in each degree, and so is the target. ∎
###### Proposition 4.1.5.
If $H_{d}(M)$ is finitely generated for all $d\geq 0$, then so is
$H_{d}(C_{n}(M))$ for all $d,n\geq 0$.
###### Proof.
By Lemma 4.1.4, it suffices to show that
$H_{d}(\operatorname{Conf}_{n}(M);\mathbb{Z})$ is finitely generated. Since
$\operatorname{Conf}_{n}(M)$ is a manifold, its homology is Poincaré dual to
cohomology with compact support
$H^{d}_{c}(\operatorname{Conf}_{n}(M);\mathcal{O})$ with coefficients in the
orientation sheaf $\mathcal{O}$ (see [Bre97, Example 2.9]). To study this, we
apply [Pet20, Theorem 5.18], which gives a spectral sequence that in our case
specializes to
$E_{1}^{p,q}=\bigoplus_{\begin{subarray}{c}T\in J_{U_{0}}\\\
|T|=n-p\end{subarray}}\bigoplus_{i+j=p+q}\widetilde{H}^{i}(J(T);H^{j}_{c}(\overline{M(T)};\mathcal{O}))\implies
H_{c}^{p+q}(\operatorname{Conf}_{n}(M);\mathcal{O})$
where $J_{U_{0}}$ is a subset of the set of partitions of $\\{1,\dots,n\\}$,
$J(T)$ is a finite CW complex, and $\overline{M(T)}$ is a closed submanifold
of
$\\{(x_{1},..,x_{n})\in M^{n}\hskip 2.0pt:\hskip
2.0ptx_{i}=x_{j}\iff\text{$x_{i}=x_{j}$ if $i$ and $j$ are in the same block
of $T$}\\},$
which is itself a closed submanifold of $M^{n}$. Closed manifolds have
cohomology that is finitely generated in each degree. Since the $E_{1}$ page
of the spectral sequence is finitely generated for each $(p,q)$ and $p,q\geq
0$ we have that the target is finitely generated in each degree. ∎
### 4.2. Instability for adding a trivial orbit
If $M$ is $G$-stabilizable, then we may add a fixed point to obtain a
$G$-equivariant stabilization map
$\sigma_{G/G}:C_{n}(M)\to C_{n+1}(M)$
whose underlying map
$\sigma_{G/G}^{e}=\sigma:C_{n}(M)\to C_{n+1}(M)$
is the classical stabilization map. In this section, we show that the Bredon
homology of $C(M)$ cannot be stable with respect to this stabilization map
alone. For simplicity of exposition, we consider the degree 0 case where
$G=C_{p}$.
###### Proposition 4.2.1.
For $G=C_{p}$ and $M$ an open $G$\- and $e$-stabilizable manifold, the
$\mathbb{Z}[(\sigma_{G/G})_{*}]$-module
$H_{0}^{G}(C(M);\underline{\mathbb{Z}})$ is not finitely generated.
###### Proof.
The cellular chain complex $C^{G}_{*}(C(M))$ computing Bredon homology (cf.
Section 2.1) is given by
$\displaystyle C^{G}_{*}(C(M))$
$\displaystyle=\Big{(}\sum_{G/H}C_{*}(C(M)^{H})\otimes\underline{\mathbb{Z}}(G/H)\Big{)}/\sim$
$\displaystyle=\Big{(}C_{*}(C(M)^{e})\otimes\underline{\mathbb{Z}}(C_{p}/e)\oplus
C_{*}(C(M)^{G})\otimes\underline{\mathbb{Z}}(C_{p}/C_{p})\Big{)}/\sim,$
In the case $G=C_{p}$, the orbit category is generated by the projection
$r:C_{p}/e\to C_{p}/C_{p}$ and automorphism $\gamma:C_{p}/e\to C_{p}/e$ which
corresponds to the action of the generator of $C_{p}$. Therefore, in
$C^{G}_{*}(C(M))$, we have the relations:
* •
$\gamma\cdot x\sim x$ for $x\in C_{*}(C(M)^{e})$ and $\gamma\in C_{p}$ the
generator of $C_{p}$, and
* •
$r^{*}x\sim px$ for $x\in C_{*}(C(M)^{G})$, where $r^{*}:C(M)^{G}\to C(M)^{e}$
is the inclusion of fixed points.
From Proposition 3.2.10, we have that the components of $C(M)^{G}$ are indexed
by finite $G$-sets, so we have
$H_{0}(C(M)^{C_{p}})\cong\mathbb{Z}[\sigma]\\{x_{0},x_{1},x_{2},\ldots\\},$
where $x_{i}$ represents the component corresponding to the $G$-set $i\cdot
C_{p}/e$. The components of $C(M)$ are indexed by natural numbers, so we have
$H_{0}(C(M)^{e})\cong\mathbb{Z}[\sigma]\\{y_{0}\\}.$
We conclude that $H^{G}_{0}(C(M))$ is given by
$H^{G}_{0}(C(M))\cong\frac{\mathbb{Z}[\sigma]\\{x_{0},x_{1},x_{2},\cdots,y_{0}\\}}{px_{i}=\sigma^{i}y_{0}},$
which is not finitely generated as a $\mathbb{Z}[\sigma]$-module. ∎
### 4.3. Homological stability for $S$-configurations
As above, let $M$ be a $G$-manifold. In this section, we fix a conjugacy class
of subgroups $(H)$ and study stabilization maps for $[G/H]$ orbits. We will
combine Corollary 3.2.9 with classical homological stability applied to the
nonequivariant space $M_{(H)}/G$, to obtain homological stability for
$C_{S}^{G}(M)$.
###### Theorem 4.3.1.
Let $M$ be a $G$-manifold and $S$ a finite $G$-set containing $k$ copies of
the orbit $[G/H]$ for a conjugacy class of subgroups $(H)$. If $M$ is
$H$-stabilizable and $M_{(H)}/G$ is connected, then the $(H)$-stabilization
map of (3.3.3) induces an isomorphism
$(\sigma_{G/H})_{*}:H_{d}(C^{G}_{S}(M))\to H_{d}(C^{G}_{S+[G/H]}(M))$
in integral homology in degrees $d\leq k/2$.
###### Proof.
Decompose the finite $G$-set $S$ as
$S\cong k[G/H]\sqcup S^{\prime},$
where $S^{\prime}$ is a finite $G$-set with no orbits isomorphic to $[G/H]$.
Corollary 3.2.9 allows us to decompose the space of unordered
$S$-configurations in $M$ as
$C^{G}_{S}(M)\cong C^{G}_{k[G/H]}(M)\times C^{G}_{S^{\prime}}\cong
C_{k}(M_{(H)}/G)\times C^{G}_{S^{\prime}}(M).$
By Corollary 3.2.9, we have
$\displaystyle H_{*}(C^{G}_{S}(M))$ $\displaystyle\cong
H_{*}(C_{k}(M_{(H)}/G)\times C^{G}_{S^{\prime}}(M)),$ $\displaystyle
H_{*}(C^{G}_{S+[G/H]}(M))$ $\displaystyle\cong H_{*}(C_{k+1}(M_{(H)}/G)\times
C^{G}_{S^{\prime}}(M)).$
By naturality, we obtain a map of Künneth short exact sequences
${\displaystyle\bigoplus_{i+j=d-1}\operatorname{Tor}_{1}\left(H_{i}(C_{k}(M_{(H)}/G)),H_{j}(C^{G}_{S^{\prime}}(M))\right)}$${\displaystyle\bigoplus_{i+j=d-1}\operatorname{Tor}_{1}\left(H_{i}(C_{k+1}(M_{(H)}/G)),H_{j}(C^{G}_{S^{\prime}}(M))\right)}$${H_{d}(C^{G}_{S}(M))}$${H_{d}(C^{G}_{S+[G/H]}(M))}$${\displaystyle\bigoplus_{i+j=d}H_{i}(C_{k}(M_{(H)}/G))\otimes
H_{j}(C^{G}_{S^{\prime}}(M))}$${\displaystyle\bigoplus_{i+j=d}H_{i}(C_{k+1}(M_{(H)}/G))\otimes
H_{j}(C^{G}_{S^{\prime}}(M))}$$\scriptstyle{(\sigma_{G/H})_{*}}$
By Lemma 3.3.6, the top and bottom horizontal arrows in the above diagram are
given by the classical stabilization maps $\sigma:C_{k}(M_{(H)}/G)\to
C_{k+1}(M_{(H)}/G)$ for the manifold $M_{(H)}/G$, and so Theorem 4.1.1 implies
that these maps are isomorphisms for $d\leq k/2$. (Note that $M_{(H)}/G$ is
not empty because we have assumed that there are $k$ points of isotropy type
$(H)$.) It follows from the five lemma that $(\sigma_{G/H})_{*}$ is an
isomorphism in the range $d\leq k/2$. ∎
### 4.4. Bredon homological stability
We now return to the situation of ordinary configuration spaces.
For $K\leq G$, let
$P_{K}=\mathbb{Z}[\sigma_{K/H}:H\subseteq K].$
Clearly, $P_{K}$ acts on $K$-Bredon cohomology $H^{K}_{*}(C(M))$, but we can
also consider the action of $P_{G}$ on $H^{K}_{*}(C(M))$ via the restriction
map $\operatorname{res}^{G}_{K}:P_{G}\to P_{K}$, which is defined as follows.
Given $H,K\leq G$, the double coset formula gives an isomorphism of $K$-sets
$G/H\ \ \cong\bigsqcup_{KgH\in K\backslash G/H}K/(K\cap gHg^{-1}),$
and so
(4.4.1) $\operatorname{res}^{G}_{K}(\sigma_{G/H})\ \ =\prod_{KgH\in
K\backslash G/H}\sigma_{K/(K\cap gHg^{-1})}.$
###### Lemma 4.4.2.
Suppose $G$ is a Dedekind group (i.e., every subgroup is normal). Then for all
$K\leq G$, $P_{K}$ is finitely generated as a $P_{G}$-module, where the action
is via the restriction map $\operatorname{res}^{G}_{K}:P_{G}\to P_{K}$.
###### Proof.
As $P_{K}$ is a finitely generated algebra, it suffices to show that $P_{K}$
is integral over $P_{G}$. We will show that there is a power of every
generator $\sigma_{K/H}$ (for $K\leq H$) in the image of
$\operatorname{res}^{G}_{K}$, which implies the same for every monomial.
Combining the Dedekind assumption with (4.4.1) shows that
$\operatorname{res}^{G}_{K}(\sigma_{G/H})=\sigma_{K/H}^{[G:K]}$. ∎
###### Remark 4.4.3.
The “Dedekind” hypothesis is needed to ensure that
$\operatorname{res}^{G}_{K}(\sigma_{G/H})=\sigma_{K/H}^{[G:K]}$ in the last
line of the previous proof. It is known [Ded97] that a finite group is
Dedekind if and only if it is abelian or isomorphic to $Q_{8}\times A$ where
$A$ is a finite abelian group whose 2-component has the form
$(\mathbb{Z}/2)^{r}$ for $r\geq 0$.
We remark that the conclusion of Lemma 4.4.2 holds for a more general class of
nonabelian groups than those mentioned in the previous paragraph, but an
explicit characterization is not known. For example, it can be checked that it
holds for dihedral groups of order $2n$ for $n$ squarefree.
###### Lemma 4.4.4.
Assume $M$ is $H$-stabilizable and $M_{(H)}/G$ is connected for every $H\leq
G$. For every $d$ there exists $N_{d}$ such that
$\pi_{d}(\Phi^{G}(H\underline{\mathbb{Z}}\wedge C(M)))\cong
P_{G}\cdot\pi_{d}(\Phi^{G}(H\underline{\mathbb{Z}}\wedge C_{\leq N_{d}}(M))).$
Moreover, if $H_{i}(M_{(H)}/G)$ is a finitely generated abelian group for all
$i\leq d$ and $(H)\leq G$, then
$\pi_{d}(\Phi^{G}(H\underline{\mathbb{Z}}\wedge C_{\leq N_{d}}(M)))$ is a
finitely generated abelian group.
###### Proof.
Using the splitting in Corollary 3.2.9 we have
$\pi_{d}(\Phi^{G}(H\underline{\mathbb{Z}}\wedge
C(M)))\cong\pi_{d}(\Phi^{G}H\underline{\mathbb{Z}}\wedge
C(M)^{G})\cong\bigoplus_{G\text{-sets
}S}\pi_{d}(\Phi^{G}(H\underline{\mathbb{Z}})\wedge C_{S}^{G}(M)).$
By Proposition 2.5.1, there is a finite collection of nonnegative integers
$n_{i}$ and finitely generated abelian groups $A_{i}$ such that
$\pi_{d}(\Phi^{G}(H\underline{\mathbb{Z}})\wedge
C_{S}^{G}(M))\cong\bigoplus_{i}H_{d-n_{i}}(C_{S}^{G}(M);A_{i}).$
For fixed $i$ and $H$, Theorem 4.3.1 says that there exists $N_{d,i,H}$ such
that
$H_{d-n_{i}}(C_{S}^{G}(M);A_{i})=\sigma_{G/H}\cdot
H_{d-n_{i}}(C_{S-[G/H]}^{G}(M);A_{i})$
whenever $S$ contains at least $N_{d,i,H}$ copies of $G/H$. For fixed $i$,
there are finitely many $G$-sets $S$ such that
$H_{d-n_{i}}(C_{S}^{G}(M);A_{i})$ is not in the image of $\sigma_{G/H}$ for
some $H$. Let $N_{d,i}$ be the maximum cardinality of the exceptional
$G$-sets, and let $N_{d}=\max_{i}\\{N_{d,i}\\}$. Then for every $S$ and
$n_{i}$, we have
$\displaystyle H_{d-n_{i}}(C_{S}^{G}(M);A_{i})$ $\displaystyle\subseteq
P_{G}\cdot\bigoplus_{|S^{\prime}|\leq
N_{d}}H_{d-n_{i}}(C_{S^{\prime}}^{G}(M);A_{i}).$
Summing over $i$, we obtain $\pi_{d}(\Phi^{G}(H\underline{\mathbb{Z}}\wedge
C^{G}_{S}(M)))\subseteq
P_{G}\cdot\pi_{d}(\Phi^{G}(H\underline{\mathbb{Z}}\wedge C_{\leq N_{d}}(M)))$.
This shows the first statement. For the second statement, combine Corollary
3.2.9 and Proposition 4.1.5, which holds for homology with
$A_{i}$-coefficients because $A_{i}$ is a finitely generated abelian group. ∎
###### Proposition 4.4.5.
Let $G$ be a Dedekind group. For $K\leq G$, let $i_{K}^{*}$ denote restriction
from $G$-spaces to $K$-spaces. Assume that for all $H\leq K\leq G$, $M$ is
$H$-stabilizable, $(i_{K}^{*}M)_{(H)}/K$ is connected, and
$H_{i}(i_{K}^{*}(M)_{(H)}/K)$ is a finitely generated abelian group for all
$i\leq d+1$. Then $H_{d}^{K}(C(M);\underline{\mathbb{Z}})$ is finitely
generated over $P_{G}$ for all $K\leq G$.
###### Proof.
We will apply Lemma 2.4.2 with the Serre class of finitely generated
$P_{G}$-modules. It suffices to check the hypothesis, namely that
$\pi_{i}(\Phi^{K}(H\underline{\mathbb{Z}}\wedge C(M)))$ is finitely generated
over $P_{G}$ for all $K\leq G$ and $i\leq d+1$. We will apply Lemma 4.4.4 to
$i^{*}_{K}(M)$. Checking the hypotheses, first note that if $M$ is
$H$-stabilizable as a $G$-manifold, then $i^{*}_{K}(M)$ is $H$-stabilizable.
Thus $\pi_{i}(\Phi^{K}(H\underline{\mathbb{Z}}\wedge
C(i^{*}_{K}(M))))=\pi_{i}(\Phi^{K}(H\underline{\mathbb{Z}}\wedge C(M)))$ is
finitely generated over $P_{K}$. Now use Lemma 4.4.2 to get finite generation
over $P_{G}$. ∎
### 4.5. Simplifying the hypotheses of Proposition 4.4.5
We will show (Lemma 4.5.3) that the finite generation hypothesis in
Proposition 4.4.5 follows from the assumption that $M$ the interior of a
compact $G$-manifold $W$. Such finite type hypotheses are common in the
literature on this subject; see for example [RW13] for the non-equivariant
version of our result. We also translate the connectedness hypothesis of
Proposition 4.4.5 into a condition which involves $M^{H}$ instead of $M_{(H)}$
(see Lemma 4.5.1). The results are combined in Theorem 4.5.5.
###### Lemma 4.5.1.
Suppose $G$ is Dedekind or a $p$-group, and let $W=W_{G}H$ denote the Weyl
group. If $M^{H}/W$ is connected then $M_{(H)}/G$ (if nonempty) is connected.
Moreover,
$M_{(H)}/G\cong\Big{(}M^{H}\backslash\bigcup_{e<L\leq W}(M^{H})^{L}\Big{)}/W.$
###### Proof.
Applying Lemma 3.2.6 to the $G$-manifold $M$ and the $W$-manifold $M^{H}$,
respectively, we have
$\displaystyle M_{(H)}/G$
$\displaystyle\cong\Big{(}M^{H}\backslash\bigcup_{H<K\leq
G}M^{K}\Big{)}\big{/}W$ $\displaystyle(M^{H})_{(e)}/W$
$\displaystyle\cong\Big{(}M^{H}\backslash\bigcup_{e<L\leq
W}(M^{H})^{L}\Big{)}\big{/}W\cong\Big{(}M^{H}\backslash\bigcup_{H<K\leq
N_{G}(H)}M^{K}\Big{)}\Big{/}W.$
Because of the group-theoretic conditions, $H<K\leq G$ implies $H<N_{K}(H)$,
and hence $M_{(H)}/G\cong(M^{H})_{(e)}/W$.
If $M_{(H)}$ is nonempty, then $(e)$ is the principal orbit type of $M^{H}$ as
a $W$-manifold. Then [tD87, Theorem I.5.14] applied to $M^{H}$ says that
$(M^{H})_{(e)}/W\cong M_{(H)}/G$ is connected. ∎
The next lemma is used to prove Lemma 4.5.3.
###### Lemma 4.5.2.
Suppose $M$ is $G$-manifold that is the interior of a compact $G$-manifold
with boundary. Let $H\leq G$. If $A$ is a $\pi_{1}(M^{H})$-module that is
finitely generated as an abelian group, then the homology with local
coefficients $H_{q}(M^{H};A)$ is a finitely generated abelian group for every
$q$.
###### Proof.
Let $Z$ be a compact $G$-manifold whose interior is $M$. First, we observe
that $M^{H}$ is the interior of $Z^{H}$; this can be checked on Euclidean
neighborhoods. Since $Z^{H}$ is a compact manifold with boundary, it has a
collar neighborhood, and this can be used to produce a homotopy equivalence
$M^{H}\simeq Z^{H}$. Since $\pi_{1}(M^{H})=\pi_{1}(Z^{H})$, we may view $A$ as
a sheaf of abelian groups over $Z^{H}$, and consider an open cover
$\mathscr{U}$ that trivializes $A$. There is a refinement
$\mathscr{U}^{\prime}$ that is a good cover (see [BT82, Corollary 5.2]) and
since $Z^{H}$ is compact, we may take $\mathscr{U}^{\prime}$ to be finite.
Since this is a good cover, homology with local coefficients
$H_{q}(Z^{H};A)=H_{q}(M^{H};A)$ agrees with Čech cohomology
$\check{H}_{q}(\mathscr{U}^{\prime};A)$. The Čech complex
$\check{C}_{q}(\mathscr{U}^{\prime};A)=\bigoplus_{(i_{0},\dots,i_{q})}A(U_{i_{0}}\cap\dots\cap
U_{i_{q}})=\bigoplus_{(i_{0},\dots,i_{q})}A$
is a finitely generated abelian group in each degree, and hence so is its
homology. ∎
###### Lemma 4.5.3.
Assume $G$ is abelian, and $M$ is the interior of a compact $G$-manifold. Then
$H_{d}(M_{(H)}/G;\mathbb{Z})$ is finitely generated for all
$d\in\mathbb{Z},H\leq G$.
###### Proof.
Let $W=W_{G}H$ be the Weyl group. Lemma 4.5.1 says that $M_{(H)}/G\cong X/W$
where
(4.5.4) $X=M^{H}\backslash\bigcup_{e<L\leq W}(M^{H})^{L}.$
By Lemma 4.1.4, it suffices to show that $H_{*}(X;\mathbb{Z})$ is finitely
generated in each degree.
If $q:G\to W=G/H$ is the quotient, then $(M^{H})^{L}=M^{q^{-1}(L)}$. Abusing
notation, we write $M^{L}=(M^{H})^{L}$; also let $U_{L}=M^{H}\backslash
M^{L}$. Since $M^{L}$ is a submanifold, there is a normal bundle $N(M^{L})$
which is open, and $N(M^{L})\cup U_{L}\cong M^{H}$. First we will show
$H_{d}(U_{L})$ is finite using a Mayer–Vietoris sequence.
The intersection term $N(M^{L})\cap U_{L}$ is homotopy equivalent to the
sphere bundle $S(N(M^{L}))$. There is a Serre spectral sequence associated to
the fibration $S^{n}\to S(N(M^{L}))\to M^{L}$,
$E_{2}^{*,*}=H_{*}(M^{L};H_{*}(S^{n};\mathbb{Z}))\Rightarrow
H_{*}(S(N(M^{L}));\mathbb{Z})$
where $\pi_{1}(M^{L})$ may act nontrivially on $H_{*}(S^{n};\mathbb{Z})$. By
Lemma 4.5.2, the $E_{2}$ term is finitely generated in each degree, and hence
so is the target.
Now consider the Mayer–Vietoris sequence
$\dots\to H_{d+1}(M^{H})\to H_{d}(S(N(M^{L})))\to H_{d}(N(M^{L}))\oplus
H_{d}(U_{L})\to\dots.$
The first term is finitely generated by Lemma 4.5.2 and above we showed the
second is as well. Hence the third term is finitely generated; in particular
so is $H_{d}(U_{L})$.
Since $G$ is abelian, write $W=\prod_{i}\mathbb{Z}/p_{i}^{r_{i}}$. In the
union (4.5.4), it suffices to restrict $L$ to products of elementary abelian
subgroups $\mathbb{Z}/p_{i}\cong\langle
p_{i}^{r_{i}-1}\rangle\leq\mathbb{Z}/p_{i}^{r_{i}}$, since any nontrivial
subgroup $H\leq\mathbb{Z}/p_{i}^{r_{i}}$ contains this $\mathbb{Z}/p_{i}$. If
$L\times L^{\prime}\leq W$, we have
$(M^{H})^{L}\cap(M^{H})^{L^{\prime}}=(M^{H})^{L\times L^{\prime}}$. More
generally, given $L=\prod_{i}L_{i}$ and $L^{\prime}=\prod_{i}L^{\prime}_{i}$,
note that $LL^{\prime}\leq W$ is a product of all $L_{i}$ and $L^{\prime}_{i}$
factors, with duplicates removed. Thus we have
$U_{L}\cup
U_{L^{\prime}}=M^{H}\backslash((M^{H})^{L}\cap(M^{H})^{L^{\prime}})=M^{H}\backslash(M^{H})^{LL^{\prime}}.$
We will show that
$X=\bigcap_{\begin{subarray}{c}L=L_{1}\times\dots\times L_{n}\\\ L_{i}\text{
elem.ab.}\end{subarray}}U_{L}$
has finitely generated homology, by induction on the number of sets
intersected. The base case was shown above. Suppose any intersection of $n$
such sets has finitely generated homology, and consider
$U_{L_{1}}\cap\dots\cap U_{L_{n}}\cap U_{L}$. There is a Mayer-Vietoris
sequence
$\dots\to H_{d+1}((\cap_{i=1}^{n}U_{L_{i}})\cup U_{L})\to
H_{d}((\cap_{i=1}^{n}U_{L_{i}})\cap U_{L})\to
H_{d}(\cap_{i=1}^{n}U_{L_{i}})\oplus H_{d}(U_{L})\to\dots.$
The inductive hypothesis implies that the third term is finitely generated, so
in order to show the second term is finitely generated, it suffices to show
the first term is finitely generated. We have
$\Big{(}\bigcap_{i=1}^{n}U_{L_{i}}\Big{)}\cup
U_{L}=\bigcap_{i=1}^{n}(U_{L_{i}}\cup U_{L})=\bigcap_{i=1}^{n}U_{L_{i}L}$
and the inductive hypothesis shows this has finitely generated homology. ∎
###### Theorem 4.5.5.
Let $G$ be an abelian group and let $M$ be a $G$-manifold that is the interior
of a compact $G$-manifold with boundary. Assume that for all $H\leq G$, $M$ is
$H$-stabilizable and $M^{H}$ is connected. Then
$H_{d}^{H}(C(M);\underline{\mathbb{Z}})$ is finitely generated over $P_{G}$
for all $H\leq G$.
###### Proof.
Combine Proposition 4.4.5, Lemma 4.5.1, and Lemma 4.5.3. ∎
###### Example 4.5.6.
If $G$ is abelian and $V=n\rho_{G}$ is a sum of regular representations, the
hypotheses of Theorem 4.5.5 are satisfied. One could also check the hypotheses
of Proposition 4.4.5 directly without using the results of this subsection: by
Lemma 3.2.6 the space $V_{(H)}$ arises from the complement of hyperplanes in
$n\rho_{G}$. This complement is Alexander dual to a union of spheres, one for
each hyperplane.
## 5\. Generalizations and variants
Throughout this section, $G$ denotes a finite group in which every subgroup is
normal. We have established stability for _$\mathbb{Z}$ -graded_ Bredon
homology _groups_. In this section, we discuss natural extensions of our
results to
1. (1)
$RO(G)$-graded Bredon homology groups, and
2. (2)
$\mathbb{Z}$-graded Bredon homology Mackey functors.
Here, $RO(G)$ is the (group completion) of the ring of isomorphism classes of
real orthogonal representations of $G$.
### 5.1. $RO(G)$-graded Bredon homological stability
Let $V\in RO(G)$. We write $-V$ for the additive inverse of $V$ in $RO(G)$ and
write $S^{V}$ for the representation sphere associated to $V$.
###### Definition 5.1.1.
Let $V\in RO(G)$. The $V$-th Bredon homology group of a $G$-space $X$ with
coefficients in the Mackey functor $\underline{M}$ is the group
$\pi_{0}^{G}\left(S^{-V}\wedge H\underline{M}\wedge X\right)\cong
H_{0}^{G}(S^{-V}\wedge X;\underline{M}).$
The following is well-known:
###### Lemma 5.1.2.
If $V\in RO(G)$, then we have
$(S^{V})^{e}=\Phi^{e}(S^{V})=S^{|V|},\quad\Phi^{G}(S^{V})=S^{|V^{G}|}.$
Recall that $P_{G}=\mathbb{Z}[(\sigma_{G/H})_{*}:(H)\leq G]$.
###### Theorem 5.1.3.
Let $V\in RO(G)$ and let $M$ be a $G$-manifold satisfying the hypotheses of
Theorem 4.5.5. Then the $P_{G}$-module
$H_{V}^{G}(C(M);\underline{\mathbb{Z}})$ is finitely generated.
###### Proof.
Applying Lemma 2.4.2 with $X=S^{-V}\wedge C(M)$ and $\mathcal{C}$ the Serre
class of finitely generated $P_{G}$-modules, it suffices to show that
$\pi_{q}\Phi^{K}(H\underline{\mathbb{Z}}\wedge S^{-V}\wedge C(M))$ is in
$\mathcal{C}$ for all $K\leq G$ and all $q\in\mathbb{Z}$. Lemma 5.1.2 implies
that
$\pi_{q}\Phi^{K}\left(H\underline{\mathbb{Z}}\wedge S^{-V}\wedge
C(M)\right)\cong\pi_{q+|V^{K}|}\Phi^{K}\left(H\underline{\mathbb{Z}}\wedge
C(M)\right),$
and this is finitely generated by combining the same ingredients as the proof
of Proposition 4.4.5 and Theorem 4.5.5. ∎
### 5.2. Mackey functor stability
If $L\leq G$ is a subgroup, we may view any $G$-manifold $M$ as an
$L$-manifold by restriction. If the $G$-manifold $M$ satisfies the hypotheses
of B, then its underlying $L$-manifold clearly also satisfies the conditions
of B (with $L$ in place of $G$) and we obtain the following:
###### Corollary 5.2.1.
Suppose $M$ satisfies the hypotheses of Theorem 4.5.5. Then
$H_{d}^{L}(C(M);\underline{\mathbb{Z}})$ is finitely generated over $P_{L}$
for all integers $d\geq 0$.
Recall that Bredon homology is actually a Mackey functor, i.e., an additive
functor from the Burnside category of $G$ to abelian groups. The previous
corollary implies that the value of the Mackey functor
$H_{d}^{(-)}(C(M);\underline{\mathbb{Z}})$
stabilizes when evaluated on any object in the Burnside category (i.e., any
finite $G$-set). We can encode this levelwise stability, and more, with the
following notion:
###### Definition 5.2.2.
A _module_ over a Green functor $\underline{R}$ is a Mackey functor
$\underline{M}$ equipped with an action map
$\underline{R}\boxtimes\underline{M}\to\underline{M}$ satisfying the usual
associativity and unitality conditions.
An $\underline{R}$-module $\underline{M}$ is _finitely generated_ if there is
a surjective $\underline{R}$-linear map
$\underline{R}\\{x_{T}\\}\to\underline{M}$, where $\underline{R}\\{x_{T}\\}$
is the free $\underline{R}$-module on a generator $x_{T}$ with $T$ a finite
$G$-set.
According to [Shu10, Lem. 2.17], a map
$\underline{R}\boxtimes\underline{M}\to\underline{M}$ determines, and is
determined by, its _Dress pairing_ : a collection of maps
$\underline{R}(S)\otimes\underline{M}(S)\to\underline{M}(S)$ for $S$ in the
Burnside category of $G$ satisfying certain compatibility axioms. Using Dress
pairings, we can relate levelwise finite generation to Mackey finite
generation:
###### Lemma 5.2.3.
An $\underline{R}$-module $\underline{M}$ is finitely generated if and only if
$\underline{M}(S)$ is finitely generated as a $\underline{R}(S)$-module for
each finite $G$-set $S$.
###### Proof.
Surjectivity for Mackey functors is checked levelwise. ∎
To express stability in terms of Mackey functors, we need a Green functor to
encode the action of the equivariant stabilization maps for each subgroup of
$G$. This is accomplished through the following:
###### Definition 5.2.4.
For each finite group $G$, let $\underline{P}_{G}$ be the constant Green
functor on $P_{G}=\mathbb{Z}[(\sigma_{G/H})_{*}:(H)\leq G]$. (_A priori_ ,
$\underline{P}_{G}$ is only a Mackey functor. It is a Green functor since it
is the fixed point functor of a commutative ring with trivial action.) We
define an action of $\underline{P}_{G}$ on
$H_{*}^{(-)}(C(M);\underline{\mathbb{Z}})$ via Dress pairing: the map
$\underline{P}_{G}(G/H)\otimes H_{*}^{H}(C(M);\underline{\mathbb{Z}})\cong
P_{G}\otimes H^{H}_{*}(C(M);\underline{\mathbb{Z}})\to
H_{*}^{H}(C(M);\underline{\mathbb{Z}})$
comes from the restriction map $\operatorname{res}^{G}_{H}:P_{G}\to P_{H}$ as
defined in Section 4.4.
In particular, if $H_{*}^{H}(C(M);\underline{\mathbb{Z}})$ is finitely
generated over $\underline{P}_{H}(H/H)$ for every subgroup $H\leq G$, and if
$\underline{P}_{H}(H/H)$ is finitely generated as a
$\underline{P}_{G}(G/H)$-module via the restriction map (e.g., if every
subgroup of $G$ is normal), then $H_{*}^{H}(C(M);\underline{\mathbb{Z}})$ is
finitely generated as a $\underline{P}_{G}(G/H)$-module. Combined with Lemma
5.2.3, this proves:
###### Theorem 5.2.5.
Let $M$ be as in Theorem 4.5.5. Then
$H_{d}^{(-)}(C(M);\underline{\mathbb{Z}})$ is finitely generated over
$\underline{P}_{G}$ for all integers $d\geq 0$.
## References
* [ADCK20] Byung Hee An, Gabriel Drummond-Cole, and Ben Knudsen. Edge stabilization in the homology of graph braid groups. Geom. Topol., 24(1):421–469, 2020.
* [Aud91] Michèle Audin. The topology of torus actions on symplectic manifolds, volume 93 of Progress in Mathematics. Birkhäuser Verlag, Basel, 1991. Translated from the French by the author.
* [Bon22] Luciana Basualdo Bonatto. Decoupling decorations on moduli spaces of manifolds. Mathematical Proceedings of the Cambridge Philosophical Society, pages 1–36, 2022.
* [Bre97] Glen E. Bredon. Sheaf theory, volume 170 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1997.
* [BT82] Raoul Bott and Loring W. Tu. Differential Forms in Algebraic Topology. Springer New York, 1982.
* [BT01] Carl-Friedrich Bödigheimer and Ulrike Tillmann. Stripping and splitting decorated mapping class groups. In Cohomological methods in homotopy theory (Bellaterra, 1998), volume 196 of Progr. Math., pages 47–57. Birkhäuser, Basel, 2001.
* [CEF15] Thomas Church, Jordan S. Ellenberg, and Benson Farb. FI-modules and stability for representations of symmetric groups. Duke Math. J., 164(9):1833–1910, 2015.
* [CLM76] Frederick R. Cohen, Thomas J. Lada, and J. Peter May. The homology of iterated loop spaces. Lecture Notes in Mathematics, Vol. 533. Springer-Verlag, Berlin-New York, 1976.
* [CW91] Steven R. Costenoble and Stefan Waner. Fixed set systems of equivariant infinite loop spaces. Trans. Am. Math. Soc., 326(2):485–505, 1991.
* [Ded97] R. Dedekind. Ueber Gruppen, deren sämtliche Teiler Normalteiler sind. Math. Ann., 48:548–561, 1897.
* [DH] Daniel Dugger and Christy Hazel. The ${R}{O}({G})$-graded Bredon cohomology of Euclidean configuration spaces. To appear.
* [GM17] Bertrand J. Guillou and J. Peter May. Equivariant iterated loop space theory and permutative $G$-categories. Algebr. Geom. Topol., 17(6):3259–3339, 2017.
* [HHR16] Michael A Hill, Michael J Hopkins, and Douglas C Ravenel. On the nonexistence of elements of Kervaire invariant one. Annals of Mathematics, 184(1):1–262, 2016.
* [Kan07] Marja Kankaanrinta. Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions. Algebr. Geom. Topol., 7:1–27, 2007.
* [Kri20] Sophie Kriz. Notes on equivariant homology with constant coefficients. Pacific J. Math., 309(2):381–399, 2020.
* [Lew] L. Gaunce Lewis, Jr. The theory of Green functors. Unpublished manuscript.
* [May72] J. P. May. The geometry of iterated loop spaces. Lecture Notes in Mathematics, Vol. 271. Springer-Verlag, Berlin-New York, 1972.
* [May96] J. P. May. Equivariant homotopy and cohomology theory, volume 91 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With contributions by M. Cole, G. Comezaña, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner.
* [McD75] Dusa McDuff. Configuration spaces of positive and negative particles. Topology, 14:91–107, 1975.
* [MW19] Jeremy Miller and Jennifer Wilson. Higher-order representation stability and ordered configuration spaces of manifolds. Geom. Topol., 23(5):2519–2591, 2019.
* [Pal18] Martin Palmer. Twisted homological stability for configuration spaces. Homology Homotopy Appl., 20(2):145–178, 2018.
* [Pet20] Dan Petersen. Cohomology of generalized configuration spaces. Compos. Math., 156(2):251–298, 2020.
* [RS00] Colin Rourke and Brian Sanderson. Equivariant configuration spaces. J. Lond. Math. Soc., II. Ser., 62(2):544–552, 2000.
* [RW13] Oscar Randal-Williams. Homological stability for unordered configuration spaces. Q. J. Math., 64(1):303–326, 2013.
* [Seg79] Graeme Segal. The topology of spaces of rational functions. Acta Math., 143(1-2):39–72, 1979.
* [Shu10] Megan Elizabeth Shulman. Equivariant local coefficients and the ${R}{O}({G})$-graded cohomology of classifying spaces. PhD thesis, The University of Chicago, 2010.
* [Sta18] The Stacks Project Authors. Stacks Project. https://stacks.math.columbia.edu, 2018.
* [tD87] Tammo tom Dieck. Transformation groups, volume 8 of De Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1987.
* [Wil19] Dylan Wilson. ${C}_{2}$-equivariant Homology Operations: Results and Formulas. arXiv preprint arXiv:1905.00058, 2019.
|
# Busemann process and semi-infinite geodesics in Brownian last-passage
percolation
Timo Seppäläinen Timo Seppäläinen, University of Wisconsin-Madison,
Mathematics Department, Van Vleck Hall, 480 Lincoln Dr., Madison WI
53706-1388, USA<EMAIL_ADDRESS>and Evan Sorensen Evan Sorensen,
University of Wisconsin-Madison, Mathematics Department, Van Vleck Hall, 480
Lincoln Dr., Madison WI 53706-1388, USA<EMAIL_ADDRESS>
###### Abstract.
We prove the existence of semi-infinite geodesics for Brownian last-passage
percolation (BLPP). Specifically, on a single event of probability one, there
exist semi-infinite geodesics started from every space-time point and
traveling in every asymptotic direction. Properties of these geodesics include
uniqueness for a fixed initial point and direction, non-uniqueness for fixed
direction but random initial points, and coalescence of all geodesics
traveling in a common, fixed direction. Along the way, we prove that for fixed
northeast and southwest directions, there almost surely exist no bi-infinite
geodesics in the given directions. The semi-infinite geodesics are constructed
from Busemann functions. Our starting point is a result of Alberts, Rassoul-
Agha and Simper that established Busemann functions for fixed points and
directions. Out of this, we construct the global process of Busemann functions
simultaneously for all initial points and directions, and then the family of
semi-infinite Busemann geodesics. The uncountable space of the semi-discrete
setting requires extra consideration and leads to new phenomena, compared to
discrete models.
###### Key words and phrases:
Brownian motion, Busemann function, semi-infinite geodesic, coalescence, last-
passage percolation, midpoint problem
###### 2020 Mathematics Subject Classification:
60K30,60K35,60K37
###### Contents
1. 1 Introduction
2. 2 Definitions and previous results
3. 3 Main results
4. 4 Construction and properties of the semi-infinite geodesics
5. 5 Connections to other models
6. 6 Construction and proofs for the Busemann process
7. 7 Proofs of the results for semi-infinite geodesics
8. A Deterministic facts about continuous functions
9. B Probabilistic results
10. C The Brownian queue and stationary last-passage process
11. D Time reversal
## 1\. Introduction
### 1.1. Brownian last-passage percolation
Brownian last-passage percolation (BLPP) dates back to the 1991 work of Glynn
and Whitt [GW91], where the model appeared as a large-scale limit of multiple
queues in series under heavy traffic conditions. Harrison and Williams [Har85,
HW90, HW92] also studied what is known as the Brownian queue and developed a
stability result for that model. More specifically, if the arrivals process is
given by increments of Brownian motion and the service process is given by
increments of an independent Brownian motion with drift, the departures
process is also given by increments of Brownian motion. The connection between
BLPP and the Brownian queue is expounded on by O’Connell and Yor [OY01], who
also introduced the positive temperature version of this model, known as the
Brownian polymer. We discuss the connection to queuing theory in Section 5.3
and Appendix C.
### 1.2. BLPP in the Kardar-Parisi-Zhang universality class
In the early 2000s, Gravner, Tracy, and Widom [GTW01] and Baryshnikov [Bar01]
discovered that the Brownian last-passage value $L_{(1,0),(n,1)}(\mathbf{B})$
(to be defined in (2.2) below) has the same distribution as the largest
eigenvalue of an $n\times n$ GUE random matrix. Soon after, O’Connell and Yor
[OY02] provided an alternate proof relying on the queuing interpretation of
the model. Since then, Brownian last-passage percolation and the Brownian
polymer have been widely studied as gateways to properties of the KPZ
universality class. As a semi-discrete model with one continuous and one
discrete parameter, BLPP serves as an intermediary between discrete models
such as LPP on the planar integer lattice and continuum models such as the
stochastic heat equation and the KPZ equation. In [CH14], Corwin and Hammond
constructed the Airy line ensemble and proved that certain statistics of
Brownian last-passage percolation converge in distribution to the Airy line
ensemble. Dauvergne, Nica, and Virág ([DNV19], Corollary 6.4) provided an
alternate method to prove this fact, which couples Brownian last-passage
percolation with geometric random walks.
Recently, Dauvergne, Ortmann, and Virág [DOV18] constructed the directed
landscape, a central object in the KPZ universality class, as the scaling
limit of Brownian last-passage percolation. Another central object is the KPZ
fixed point, constructed by Matetski, Quastel, and Remenik [MQR21], as the
limit of the totally asymmetric simple exclusion process. In [DOV18], a
variational duality is described between the directed landscape and KPZ fixed
point, which was rigorously proven in [NQR20]. Even more recently, convergence
to the KPZ fixed point was shown for a larger class of models, including the
height function of the KPZ equation. This was done independently by Quastel
and Sarkar [QS20] and Virág [Vir20].
### 1.3. Semi-infinite geodesics in discrete models
The study of infinite geodesics in planar random growth models has gone
through a number of stages over the last 30 years, beginning with the work of
Licea and Newman [LN96, New95] on first-passage percolation with i.i.d. edge
weights. Under a global curvature assumption on the limit shape, for
continuously distributed edge weights, they proved existence of a
deterministic full-Lebesgue measure set of directions in which there is a
unique semi-infinite geodesic out of every lattice point. They also showed
that, for each direction in this set, the semi-infinite geodesics in that
direction all coalesce.
A separate strand of work consists of long-term efforts to prove the
nonexistence of bi-infinite geodesics. In first-passage percolation, Licea and
Newman [LN96] showed that there are no bi-infinite geodesics in fixed
northeast and southwest directions. Howard and Newman [HN01] later proved
similar results for Euclidean last-passage percolation. Around this time, Wehr
and Woo [WW98] proved that, under a first moment assumption on the edge
weights, there are no bi-infinite geodesics for first-passage percolation that
lie entirely in the upper-half plane.
In 2016, Damron and Hanson [DH17] strengthened the result of Licea and Newman
by proving that, if the weights have continuous distribution and the boundary
of the limit shape is differentiable, for each fixed direction, there are no
bi-infinite geodesics with one end having that direction. The conjectured
nonexistence was finally resolved in exponential last-passage percolation
(LPP), known also as the exponential corner growth model (CGM). The proofs
came in two independent works: first by Basu, Hoffman, and Sly [BHS19] and
shortly thereafter by Balázs, Busani, and the first author [BBS20]. The latter
proof is in spirit aligned with the development in the present paper, as it
rested on understanding the joint distribution of the Busemann functions from
[FS20].
Another focus of research has been the coalescence structure of geodesics.
Ferrari and Pimentel [FP05] imported the Licea–Newman approach [LN96, New95]
to exponential LPP on the lattice. Later Pimentel [Pim16] developed a
probabilistic duality between the coalescence time of two semi-infinite
geodesics and their last exit times from the initial boundary, again in
exponential LPP. A key idea was the equality in distribution of the tree of
directed semi-infinite geodesics and the dual tree of southwest-directed
geodesics. From this, he obtained a lower bound on tail probabilities of the
coalescence time for two geodesics starting from $(-\lfloor
k^{2/3}\rfloor,\lfloor k^{2/3}\rfloor)$ and $(\lfloor k^{2/3}\rfloor,-\lfloor
k^{2/3}\rfloor)$. Coalescence bounds have seen significant recent improvement
in [BSS19] and [SS20]. A study of the coalescence structure of finite
geodesics in BLPP was undertaken by Hammond in four papers [Ham16, Ham19a,
Ham19b, Ham20]. The existence of semi-infinite geodesics in the Airy line
ensemble, for a countable dense set of directions and initial points, was
proven recently by Sarkar and Virág in [SV21].
Hoffman was the first to use ideas of Busemann functions to study geodesics in
first-passage percolation. In [Hof08] he showed the existence of disjoint
semi-infinite geodesics. Later, Damron and Hanson [DH14] constructed
generalized Busemann functions from weak subsequential limits of first-passage
times. This allowed them to develop results for semi-infinite geodesics under
weaker assumptions than the global curvature. Under the assumption that the
limit shape is strictly convex and differentiable, they proved that every
semi-infinite geodesic has an asymptotic direction and that, in every
direction, there exists a semi-infinite geodesic out of every lattice point.
On the side of discrete last-passage percolation with general i.i.d. weights,
Georgiou, Rassoul-Agha, and the first author [GRAS17b, GRAS17a] used the
stationary LPP process to prove existence of Busemann functions under mild
moment conditions. The Busemann functions were then used to construct semi-
infinite geodesics. They also showed that if the shape function is strictly
concave, every semi-infinite geodesic has an asymptotic direction. Further in
this direction, in [Sep20] the first author introduced a new proof of the
coalescence of semi-infinite geodesics that utilizes the stationary LPP
process.
Our work is related to this last approach. We use Busemann functions to
construct and study semi-infinite geodesics in BLPP. Existence of Busemann
functions in BLPP, for fixed initial points and directions, was recently
established by Alberts, Rassoul-Agha and Simper [ARAS20], along with Busemann
functions and infinite polymer measures for the semi-discrete Brownian
polymer. Their BLPP result is the starting point of our study.
### 1.4. New techniques and phenomena in the semi-discrete model
The present paper develops the global setting of Busemann functions and semi-
infinite geodesics in the semi-discrete BLPP, with a view to future study of
their finer properties. The novelty lies in going beyond the discrete set-up.
In discrete last-passage percolation, one can prove a probability-one
statement about semi-infinite geodesics out of a fixed initial point, and then
that statement extends to all initial points by a simple union bound. This is
not the case in BLPP. To overcome this difficulty, we need new methods of
proof. Additionally, the continuum of points gives rise to new results
regarding non-uniqueness of semi-infinite geodesics, as seen in Item (3)
below.
Specific items proved in this paper include the following, and are recorded as
Theorems 3.1 and 3.5.
1. (1)
The Busemann functions of BLPP from [ARAS20] are extended to a global process
on a single event of probability one, for all the uncountably many initial
points and directions.
2. (2)
Once the Busemann process is in place, we use it to construct semi-infinite
geodesics for BLPP. With probability one, there exists a family of semi-
infinite geodesics, starting from each initial point and in each asymptotic
direction.
3. (3)
With probability one, all semi-infinite geodesics, whether constructed by the
Busemann functions or not, have an asymptotic direction. For a fixed initial
point and direction, there is almost surely a unique semi-infinite geodesic
starting from the given point and traveling asymptotically in the given
direction. We also show that for a fixed direction, there is a countably
infinite set of initial points whose geodesic in that direction is not unique.
This non-uniqueness into a fixed direction is a new phenomenon that is not
present in exponential last-passage percolation on the lattice.
4. (4)
For each fixed direction, we prove that all semi-infinite geodesics, traveling
in that common direction, coalesce.
5. (5)
For fixed northeast and southwest directions, we prove the almost sure
nonexistence of bi-infinite geodesics in those directions.
To construct an infinite up-right path on the lattice $\mathbb{Z}^{2}$, one
chooses at each step whether to move upward or to the right. In the BLPP
setting, one chooses a real-valued increment to the right and then takes a
unit-size upward step. Section 4.1 explains informally this construction. The
locations of the upward steps of a Busemann geodesic are determined by a
variational problem for Brownian motion with drift. This formulation is
significant in at least two ways. First, this step is where non-uniqueness of
geodesics can arise. Understanding it involves properties of Brownian paths.
Furthermore, this variational construction can be potentially and profitably
adapted to other models. This includes both continuum models of the KPZ class,
such as the directed landscape, and lattice LPP. We address the latter in
Section 5.2.
The proof of the coalescence in Item (4) above requires technical novelties.
The underlying idea from [Sep20] is to construct a dual environment from the
original environment and a Busemann function. If two geodesics in a given
direction do not coalesce, there exists a bi-infinite geodesic in the dual
environment with given northwest and southeast directions. Then, it is proven
that there are almost surely no bi-infinite geodesics in fixed directions.
A key point is a non-intersection property between the original semi-infinite
geodesics and southwest-travelling semi-infinite geodesics in the dual
environment. This is in Theorem 7.12, which is the analogue of Lemma 4.4 in
[Sep20]. The dual environment in BLPP is constructed through dual queuing
mappings, first presented in [OY01] and further studied in [SV10, ARAS20]. See
Equation (3.9) for the precise definition. While the general approach is not
new, our contribution comes first from Theorem D.1 that shows that joint
queuing mappings can be inverted by reverse-time queuing mappings. Then, we
use this theorem and variational formulas to construct southwest-travelling
semi-infinite geodesics in the dual environment. Lastly, the non-intersection
of Theorem 7.12 is proved by comparing the maximizers of the variational
formulas.
### 1.5. Future work
The present paper lays the foundation for future work on Busemann functions
and semi-infinite geodesics, which most immediately will cover the following.
1. (1)
To study the full family of semi-infinite geodesics in BLPP, we derive the
joint distribution of Busemann functions across all directions. The analogous
result was achieved in [FS20] for the exponential CGM, which led to important
advances in the global structure of geodesic trees [JRS21], coalescence
[SS20], convergence [BBS21, BF18], and bi-infinite geodesics [BBS20]. We
likewise study the geometry of geodesics in BLPP and give a full
characterization of exceptional directions where geodesics do not coalesce, in
terms of the Busemann process.
2. (2)
We define the competition interface for each initial point of BLPP. This has a
straightforward definition in discrete models, but requires more care in the
semi-discrete setting. Specifically, we show that, on each level $m$ of the
semi-discrete space $\mathbb{Z}\times\mathbb{R}$, outside a random set of
Hausdorff dimension $\frac{1}{2}$, all points $(m,t)$ have a degenerate
competition interface. For each of the points in this set of Hausdorff
dimension $\frac{1}{2}$, there exists a random direction $\theta^{\star}$ such
that there are two semi-infinite geodesics starting at $(m,t)$ in direction
$\theta^{\star}$ that only share the common initial point.
As can be seen from Item (2), there are many rich questions to explore related
to non-uniqueness of semi-infinite geodesics in Brownian last-passage
percolation. Other questions related to Hausdorff dimension have recently been
explored in the KPZ fixed point and the directed landscape in [BGH21, BGH19,
GH21, CHHM21]. Our longer-term goal is to extend the approach of this paper to
the continuum models of the KPZ class.
### 1.6. Organization of the paper
Section 2 defines the Brownian last-passage percolation model and related
objects. We also state results from other papers, including the existence of
Busemann functions from [ARAS20]. Section 3 contains the main results on the
existence and properties of the global Busemann process and semi-infinite
geodesics. Section 4 states more technical theorems related to the specific
construction of semi-infinite geodesics from the Busemann functions. In
Section 5, we discuss connections to infinite polymer measures in the
O’Connell-Yor polymer, semi-infinite geodesics in discrete last-passage
percolation, and to queuing theory. The proofs of this paper are contained in
Sections 6 and 7. Section 6 constructs the global Busemann process and derives
its properties. All results for semi-infinite geodesics are proved in Section
7. The Appendices collect some standard material, background from the
literature, and some technical theorems.
## 2\. Definitions and previous results
### 2.1. Preliminaries
The following notation and conventions are used throughout the paper.
1. (i)
For a function $f:\mathbb{R}\rightarrow\mathbb{R}$ , we write
$f(s,t)=f(t)-f(s)$ and $\widetilde{f}(t)=-f(-t).$
2. (ii)
$\mathbb{Z}$, $\mathbb{Q}$ and $\mathbb{R}$ are restricted by subscripts, as
in for example $\mathbb{Z}_{>0}=\\{1,2,3,\dotsc\\}$.
3. (iii)
Whenever $m\leq n\in\mathbb{Z}$ and $s\leq t\in\mathbb{R}$, we say that
$(m,s)\leq(n,t)$.
4. (iv)
Let $X\sim\mathcal{N}(\mu,\sigma^{2})$ indicate that the random variable $X$
has normal distribution with mean $\mu$ and variance $\sigma^{2}$. For
$\alpha>0$, let $X\sim\operatorname{Exp}(\alpha)$ indicate that $X$ has
exponential distribution with rate $\alpha$, or equivalently, mean
$\alpha^{-1}$.
5. (v)
Equality in distribution between random variables and processes is denoted by
$\overset{d}{=}$.
6. (vi)
A two-sided Brownian motion is a continuous random process
$\\{B(t):t\in\mathbb{R}\\}$ such that $B(0)=0$ almost surely and such that
$\\{B(t):t\geq 0\\}$ and $\\{B(-t):t\geq 0\\}$ are two independent standard
Brownian motions on $[0,\infty)$.
7. (vii)
For $\lambda\in\mathbb{R}$, $\\{Y(t):t\in\mathbb{R}\\}$ is a two-sided
Brownian motion with drift $\lambda$ if the process $\\{Y(t)-\lambda
t:t\in\mathbb{R}\\}$ is a two-sided Brownian motion.
8. (viii)
The square ${\scaleobj{0.9}{\boxempty}}$ as a superscript represents a sign
$+$ or $-$.
### 2.2. Geodesics in Brownian last-passage percolation
The Brownian last-passage process is defined as follows. On a probability
space $(\Omega,\mathcal{F},\mathbb{P})$, let
$\mathbf{B}=\\{B_{r}\\}_{r\in\mathbb{Z}}$ be a field of independent, two-sided
Brownian motions. For $(m,s)\leq(n,t)$, define the set
$\Pi_{(m,s),(n,t)}:=\\{\mathbf{s}_{m,n}=(s_{m-1},s_{m},\ldots,s_{n})\in\mathbb{R}^{n-m+2}:s=s_{m-1}\leq
s_{m}\leq\cdots\leq s_{n}=t\\}.$
Denote the energy of a sequence $\mathbf{s}_{m,n}\in\Pi_{(m,s),(n,t)}$ by
(2.1) $\mathcal{E}(\mathbf{s}_{m,n})=\sum_{r=m}^{n}B_{r}(s_{r-1},s_{r}).$
Now, for $\mathbf{x}=(m,s)\leq(n,t)=\mathbf{y}$, define the Brownian last-
passage time as
(2.2)
$L_{\mathbf{x},\mathbf{y}}(\mathbf{B})=\sup\\{\mathcal{E}(\mathbf{s}_{m,n}):\mathbf{s}_{m,n}\in\Pi_{\mathbf{x},\mathbf{y}}\\}.$
Whenever the specific field of Brownian motions used is either clear from or
not important in the context, we write $L_{(m,s),(n,t)}$.
$s$$s_{0}$$s_{1}$$s_{2}$$s_{3}$$t$$0$$1$$2$$3$$4$ Figure 2.1. The Brownian
increments $B_{r}(s_{r-1},s_{r})$ for $r=0,\dotsc,4$ in (2.1) that make up the
energy of the path depicted in Figure 2.2.
The elements of $\Pi_{\mathbf{x},\mathbf{y}}$ are in bijection with paths in
$\mathbb{R}^{2}$ between $\mathbf{x}$ and $\mathbf{y}$, that move to the right
in real-valued increments, and move upwards in integer increments. For a given
$\mathbf{s}_{m,n}\in\Pi_{(m,s),(n,t)}$, the path consists of the following
points:
(2.3)
$\bigcup_{r=m}^{n}\bigl{\\{}(r,u):u\in[s_{r-1},s_{r}]\bigr{\\}}\cup\bigcup_{r=m}^{n-1}\bigl{\\{}(v,s_{r}):v\in[r,r+1]\bigr{\\}}.$
This set consists of horizontal and vertical line segments, such that the
vertical segments occur at the points $s_{r}$. Because of this bijection, we
sometimes say $\Gamma\in\Pi_{\mathbf{x},\mathbf{y}}$ for such an up-right
path. For $(m,t)\in\mathbb{Z}\times\mathbb{R}$, we graphically represent the
$t$-coordinate as the horizontal coordinate and the $m$-coordinate as the
vertical coordinate in the plane. Since $\Pi_{\mathbf{x},\mathbf{y}}$ is a
compact set and Brownian motion is continuous, on a single event of
probability one, for all
$(m,t)=\mathbf{x}\leq\mathbf{y}=(n,t)\in\mathbb{Z}\times\mathbb{R}$, there
exists a sequence $\mathbf{s}_{m,n}\in\Pi_{\mathbf{x},\mathbf{y}}$ such that
$\mathcal{E}(\mathbf{s}_{m,n})=L_{\mathbf{x},\mathbf{y}}$. The associated path
is called a geodesic between the points.
$(0,s)$$(4,t)$$s_{0}$$s_{1}$$s_{2}$$s_{3}$$0$$1$$2$$3$$4$ Figure 2.2. Example
of a planar path from $(0,s)$ to $(4,t)$, represented by the sequence
$(s=s_{-1},s_{0},s_{1},s_{2},s_{3},s_{4}=t)\in\Pi_{(0,s),(4,t)}$.
To an infinite sequence, $s=s_{m-1}\leq s_{m}\leq s_{m+1}\leq\cdots$ we
similarly associate a semi-infinite path. It is possible that $s_{r}=\infty$
for some $r\geq m$, in which case the last segment of the path is the ray
$[s_{r-1},\infty)\times\\{r\\}$, where $r$ is the first index with
$s_{r}=\infty$. The infinite path has direction $\theta\in[0,\infty]$ or is
$\theta$-directed if
$\lim_{n\rightarrow\infty}\frac{s_{n}}{n}\qquad\text{exists and equals
}\theta.$
We call an up-right semi-infinite path a semi-infinite geodesic if, for any
two points $\mathbf{x}\leq\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$ that lie
along the path, the portion of the path between the two points is a geodesic
between the two points. Similarly, a bi-infinite geodesic is a bi-infinite
path that forms a geodesic between any two of its points. Two up-right, semi-
infinite paths coalesce if there exists
$\mathbf{z}\in\mathbb{Z}\times\mathbb{R}$ such that the two paths agree above
and to the right of $\mathbf{z}$, as shown in Figure 2.3. Alternatively, if
the paths are defined by sequences of jump times $s^{1}=s_{m_{1}-1}^{1}\leq
s_{m_{1}}^{1}\cdots$ and $s^{2}=s_{m_{2}-1}^{2}\leq s_{m_{2}}^{2}\leq\cdots$,
then the two paths coalesce if and only if there exists $N\in\mathbb{Z}$ such
that $s_{r}^{1}=s_{r}^{2}$ for all $r\geq N$.
$\mathbf{x}$$\mathbf{y}$$\mathbf{z}$ Figure 2.3. Coalescence of semi-infinite
paths
The following lemma, due to Hammond [Ham19b], establishes uniqueness of
geodesics for a fixed initial and terminal point.
###### Lemma 2.1 ([Ham19b], Theorem B.1).
Fix endpoints $\mathbf{x}\leq\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$. Then,
there is almost surely a unique path whose energy achieves
$L_{\mathbf{x},\mathbf{y}}(\mathbf{B})$.
However, it is also true that for each fixed initial point
$\mathbf{x}\in\mathbb{Z}\times\mathbb{R}$, with probability one, there exist
points $\mathbf{y}\geq\mathbf{x}$, such that the geodesic between $\mathbf{x}$
and $\mathbf{y}$ is not unique. Hence, the following lemma is important for
our understanding. It is a deterministic statement which holds for last-
passage percolation across any field of continuous functions.
###### Lemma 2.2 ([DOV18], Lemma 3.5).
Between any two points $(m,s)\leq(n,t)\in\mathbb{Z}\times\mathbb{R}$, there is
a rightmost and a leftmost Brownian last-passage geodesic. That is, there
exist
$\mathbf{s}_{m,n}^{L},\mathbf{s}_{m,n}^{R}\in\mathbf{\Pi}_{(m,s),(n,t)}$, that
are maximal for $\mathcal{E}(\mathbf{s}_{m,n})$, such that, for any other
maximal sequence $\mathbf{s}_{m,n}$, $s_{r}^{L}\leq s_{r}\leq s_{r}^{R}$ holds
for $m\leq r\leq n$.
### 2.3. Busemann functions
In the present paper, semi-infinite geodesics are constructed from Busemann
functions. Busemann functions are defined to be the asymptotic difference of
last-passage times from two different starting points to a common terminal
point that is traveling to $\infty$ in a given direction. See Figure 2.4. The
direction is indexed by a parameter $\theta>0$.
$\mathbf{y}$$\mathbf{x}$$(n,n\theta)$$n$$n\theta$ Figure 2.4. Two geodesics
with a common terminal point $(n,n\theta)$.
Existence of these Busemann functions was proven in [ARAS20], both in the
positive temperature and zero temperature cases. We state the zero-temperature
result:
###### Theorem 2.3 ([ARAS20], Theorem 4.2).
Fix $\theta>0$ and $\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$. Then,
there exists a random variable $\mathcal{B}^{\theta}(\mathbf{x},\mathbf{y})$
and an event $\Omega_{\mathbf{x},\mathbf{y}}^{(\theta)}$ of probability one,
on which
(2.4)
$\mathcal{B}^{\theta}(\mathbf{x},\mathbf{y})=\lim_{n\rightarrow\infty}\bigl{[}L_{\mathbf{x},(n,t_{n})}-L_{\mathbf{y},(n,t_{n})}\bigr{]}$
holds for any sequence $\\{t_{n}\\}\subseteq\mathbb{R}$ satisfying
${t_{n}}/{n}\to\theta$. Further, if
(2.5) $\displaystyle
v_{m}^{\theta}(t):=\mathcal{B}^{\theta}((m-1,t),(m,t)),\text{ and}$ (2.6)
$\displaystyle h_{m}^{\theta}(t):=\mathcal{B}^{\theta}((m,0),(m,t)),$
then
$v_{m}^{\theta}(t)\sim\operatorname{Exp}\bigl{(}\frac{1}{\sqrt{\theta}}\bigr{)}$
and
$h_{m}^{\theta}(s,t)\sim\mathcal{N}\bigl{(}\frac{t-s}{\sqrt{\theta}},|t-s|\bigr{)}$
for all $s,t\in\mathbb{R}$ and $m\in\mathbb{Z}$.
## 3\. Main results
### 3.1. Semi-infinite geodesics
The following result summarizes the contributions of this paper related to
semi-infinite geodesics in BLPP.
###### Theorem 3.1.
The following hold
1. (i)
With probability one, for every initial point
$\mathbf{x}\in\mathbb{Z}\times\mathbb{R}$ and every direction $\theta>0$,
there exists a $\theta-$directed semi-infinite geodesic starting from
$\mathbf{x}$.
2. (ii)
(Uniqueness for fixed initial points and directions) For each fixed
$\mathbf{x}\in\mathbb{Z}\times\mathbb{R}$ and $\theta>0$, there exists an
event, $\Omega_{\mathbf{x}}^{(\theta)}$, of probability one, on which there is
exactly one $\theta$-directed semi-infinite geodesic starting from
$\mathbf{x}$.
3. (iii)
(Non-uniqueness for fixed direction and random initial points) For each
$\theta>0$, there exists an event $\widetilde{\Omega}^{(\theta)}$, of
probability one, on which the set
$\\{\mathbf{x}\in\mathbb{Z}\times\mathbb{R}:\text{the }\theta\text{-directed
semi-infinite geodesic starting from }\mathbf{x}\text{ is not unique}\\}$
is countably infinite. For every $(m,t)$ in this set, at most one of the
$\theta$-directed semi-infinite geodesics passes through the point
$(m,t+\varepsilon)$ for some $\varepsilon>0$. All the others pass through
$(m+1,t)$.
4. (iv)
With probability one, every semi-infinite geodesic is $\theta$-directed for
some $\theta\in[0,\infty]$. That is, for any infinite sequence $t=t_{m-1}\leq
t_{m}\leq t_{m+1}\leq\cdots$ defining a semi-infinite geodesic starting from
some point $(m,t)\in\mathbb{Z}\times\mathbb{R}$, the limit
(3.1) $\lim_{n\rightarrow\infty}\frac{t_{n}}{n}\ \text{ exists in
}[0,\infty].$
5. (v)
With probability one, if for any $(m,t)\in\mathbb{Z}\times\mathbb{R}$ and any
such sequence, the limit (3.1) equals $0$, then $t_{r}=t$ for all $r\geq m$.
Similarly, if this limit is $\infty$, then $t_{m}=t_{m+1}=\cdots=\infty$. That
is, the only semi-infinite geodesics that are asymptotically vertical or
horizontal are trivial (i.e. straight lines).
6. (vi)
(Non-existence of bi-infinite geodesics for fixed directions) Fix
$\theta,\eta>0$. Then, there exists an event, $\Omega^{(\theta,\eta)}$, of
probability one, on which there are no bi-infinite geodesics defined by jump
times $\cdots\leq\tau_{-1}\leq\tau_{0}\leq\tau_{1}\leq\cdots$ such that
(3.2)
$\lim_{n\rightarrow\infty}\frac{\tau_{n}}{n}=\theta\qquad\text{and}\qquad\lim_{n\rightarrow\infty}\frac{\tau_{-n}}{n}=-\eta.$
7. (vii)
(Coalescence of geodesics in a fixed direction) For each $\theta>0$, there
exists an event $\widehat{\Omega}^{(\theta)}$, of probability one, on which
all $\theta$-directed semi-infinite geodesics coalesce.
###### Remark 3.2.
As discussed in the introduction, the non-uniqueness stated in Part (iii) is a
new phenomenon that arises from the semi-discrete nature of the model. We
refer the reader to Section 4.3 for further discussion on non-uniqueness.
Theorem 3.1(vi) is proven by first solving the “midpoint problem.” This
problem first appeared in the context of first-passage percolation in a paper
of Benjamini, Kalai, and Schramm [BKS03]. In that context the problem asks
whether
$\lim_{n\rightarrow\infty}\mathbb{P}\Bigl{(}\Bigl{\lfloor}\frac{n}{2}\Bigr{\rfloor}e_{1}\text{
lies on some geodesic between }0\text{ and }ne_{1}\Bigr{)}=0.$
In 2016, Damron and Hanson [DH17] proved that this is true, under the
assumption that the boundary of the limit shape is differentiable. Later
Ahlberg and Hoffman [AH16] proved this result without the assumption of
differentiability. The following formulation of the midpoint problem more
closely matches that for exponential last-passage percolation in the arXiv
version of [Sep20], Theorem 4.12.
###### Lemma 3.3 (Midpoint problem).
Let $\theta,\eta>0$ and $(m,t)\in\mathbb{Z}\times\mathbb{R}$. Then, the
following subset of $\Omega$ is contained in an event of probability zero:
$\displaystyle\Big{\\{}\text{there exists a sequence
}\\{t_{n}\\}_{n\in\mathbb{Z}}\text{ satisfying
}\lim_{n\rightarrow\infty}\frac{t_{n}}{n}=\theta\text{ and
}\lim_{n\rightarrow\infty}\frac{t_{-n}}{-n}=\eta\text{ and }$
$\displaystyle\qquad\text{such that, for each }n\in\mathbb{Z}_{>0},\text{ some
geodesic between }(-n,t_{-n})\text{ and }(n,t_{n})\text{ passes through
}(m,t)\Big{\\}}$
### 3.2. Existence and properties of the Busemann process
To prove Theorem 3.1, we extend the individual Busemann functions of Theorem
2.3 to a global Busemann process. The following transformations are used to
understand the structure of this process. For functions
$Z,B:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $Z(0)=B(0)=0$ and
$\limsup_{s\rightarrow\infty}(B(s)-Z(s))=-\infty$, define
(3.3) $\displaystyle Q(Z,B)(t)$ $\displaystyle=\sup_{t\leq
s<\infty}\\{B(t,s)-Z(t,s)\\},$ (3.4) $\displaystyle D(Z,B)(t)$
$\displaystyle=Z(t)+Q(Z,B)(0)-Q(Z,B)(t),$ (3.5) $\displaystyle R(Z,B)(t)$
$\displaystyle=B(t)+Q(Z,B)(t)-Q(Z,B)(0).$
Reverse-time analogues of these transformations are defined for continuous
functions $Y,C:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $Y(0)=C(0)=0$ and
$\limsup_{s\rightarrow-\infty}(Y(s)-C(s))=-\infty$:
(3.6) $\displaystyle\overleftarrow{Q}(Y,C)(t)$
$\displaystyle=\sup_{-\infty<s\leq t}\\{C(s,t)-Y(s,t)\\},$ (3.7)
$\displaystyle\overleftarrow{D}(Y,C)(t)$
$\displaystyle=Y(t)+\overleftarrow{Q}(Y,C)(t)-\overleftarrow{Q}(Y,C)(0),$
(3.8) $\displaystyle\overleftarrow{R}(Y,C)(t)$
$\displaystyle=C(t)+\overleftarrow{Q}(Y,C)(0)-\overleftarrow{Q}(Y,C)(t).$
These transformations originate from the Brownian queue, first studied by
Glynn and Whitt [GW91], and further expounded on by Harrison and Williams
[Har85, HW90, HW92] and O’Connell and Yor [OY01]. See Section 5.3 and Appendix
C for more about the queuing interpretation. The following lemma is a
straightforward exercise. We state it for completeness, as we will refer to it
later in the paper.
###### Lemma 3.4.
Let $B,Z:\mathbb{R}\rightarrow\mathbb{R}$ be continuous functions satisfying
$\limsup_{s\rightarrow\infty}(B(s)-Z(s))=-\infty$. Then, $Q(Z,B)$, $D(Z,B)$,
and $R(Z,B)$ are continuous. Similarly, if
$Y,C:\mathbb{R}\rightarrow\mathbb{R}$ are continuous functions satisfying
$\limsup_{s\rightarrow-\infty}(Y(s)-C(s))=-\infty$, then
$\overleftarrow{Q}(Y,C),\overleftarrow{D}(Y,C)$, and $\overleftarrow{R}(Y,C)$
are continuous.
The next theorem summarizes the existence and properties of the Busemann
process. This process has discontinuities in the direction parameter $\theta$.
Instead of a single cadlag process, it is useful to retain both a left- and a
right-continuous version indicated by $\theta-$ and $\theta+$ because this
distinction captures spatial limits.
###### Theorem 3.5.
There exists a process,
$\\{\mathcal{B}^{\theta{\scaleobj{0.8}{\boxempty}}}(\mathbf{x},\mathbf{y}):\theta>0,\,{\scaleobj{0.9}{\boxempty}}\in\\{+,-\\},\,\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R}\\},$
and for $\theta>0$, there exist events
$\Omega^{(\theta)}\subseteq\Omega_{1}\subseteq\Omega$, each of probability
one, such that the following hold. Here,
$v_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}},h_{m}^{\theta{\scaleobj{0.8}{\boxempty}}}$
are defined as in (2.5) and (2.6), placing ${\scaleobj{0.9}{\boxempty}}$ in
the appropriate superscripts.
1. (i)
(Additivity) On $\Omega_{1}$, whenever
$\mathbf{x},\mathbf{y},\mathbf{z}\in(\mathbb{Z}\times\mathbb{R}),\theta>0$,
and ${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$,
$\mathcal{B}^{\theta{\scaleobj{0.8}{\boxempty}}}(\mathbf{x},\mathbf{y})+\mathcal{B}^{\theta{\scaleobj{0.8}{\boxempty}}}(\mathbf{y},\mathbf{z})=\mathcal{B}^{\theta{\scaleobj{0.8}{\boxempty}}}(\mathbf{x},\mathbf{z}).$
2. (ii)
(Monotonicity) On $\Omega_{1}$, whenever $0<\gamma<\theta<\infty$,
$m\in\mathbb{Z}$, and $s<t\in\mathbb{R}$,
$0\leq v_{m}^{\gamma-}(s)\leq v_{m}^{\gamma+}(s)\leq v_{m}^{\theta-}(s)\leq
v_{m}^{\theta+}(s),\text{ and }$ $B_{m}(s,t)\leq h_{m}^{\theta+}(s,t)\leq
h_{m}^{\theta-}(s,t)\leq h_{m}^{\gamma+}(s,t)\leq h_{m}^{\gamma-}(s,t).$
3. (iii)
(Convergence) On $\Omega_{1}$, for every $m\in\mathbb{Z}$, $\theta>0$ and
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$,
1. (a)
As $\gamma\nearrow\theta$, $h_{m}^{\gamma{\scaleobj{0.8}{\boxempty}}}$ and
$v_{m}^{\gamma{\scaleobj{0.8}{\boxempty}}}$ converge uniformly, on compact
subsets of $\mathbb{R}$, to $h_{m}^{\theta-}$ and $v_{m}^{\theta-}$,
respectively.
2. (b)
As $\delta\searrow\theta$, $h_{m}^{\delta{\scaleobj{0.8}{\boxempty}}}$ and
$v_{m}^{\delta{\scaleobj{0.8}{\boxempty}}}$ converge uniformly, on compact
subsets of $\mathbb{R}$, to $h_{m}^{\theta+}$ and $v_{m}^{\theta+}$,
respectively.
3. (c)
As $\gamma\rightarrow\infty$, $h_{m}^{\gamma{\scaleobj{0.8}{\boxempty}}}$
converges uniformly, on compact subsets of $\mathbb{R}$, to $B_{m}$.
4. (d)
As $\delta\searrow 0$, $v_{m}^{\delta{\scaleobj{0.8}{\boxempty}}}$ converges
uniformly, on compact subsets of $\mathbb{R}$, to $0$.
4. (iv)
(Continuity) On $\Omega_{1}$, for any $r,m\in\mathbb{Z}$, $\theta>0$, and
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$,
$(s,t)\mapsto\mathcal{B}^{\theta{\scaleobj{0.8}{\boxempty}}}((m,s),(r,t))$ is
a continuous function $\mathbb{R}^{2}\rightarrow\mathbb{R}$.
5. (v)
(Limits) On $\Omega_{1}$, for each $\theta>0$ and
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$,
$\lim_{s\rightarrow\pm\infty}\bigl{[}B_{m}(s)-h_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\bigr{]}=\mp\infty.$
6. (vi)
(Queuing relationships between Busemann functions) On $\Omega_{1}$, for all
$m\in\mathbb{Z},\theta>0$, and signs
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$,
$v_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}}=Q(h_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}},B_{m})\qquad\text{and}\qquad
h_{m}^{\theta{\scaleobj{0.8}{\boxempty}}}=D(h_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}},B_{m}).$
7. (vii)
(Independence) For any $m\in\mathbb{Z}$,
$\\{h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}:\theta>0,{\scaleobj{0.9}{\boxempty}}\in\\{+,-\\},r>m\\}\text{
is independent of }\\{B_{r}:r\leq m\\}$
8. (viii)
(Equality for fixed directions) Fix $\theta>0$. Then, on the event
$\Omega^{(\theta)}$, for all
$\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$ and all sequences
$\\{t_{n}\\}$ with $t_{n}/n\rightarrow\theta$,
$\mathcal{B}^{\theta-}(\mathbf{x},\mathbf{y})=\lim_{n\rightarrow\infty}\bigl{[}L_{\mathbf{x},(n,t_{n})}-L_{\mathbf{y},(n,t_{n})}\bigr{]}=\mathcal{B}^{\theta+}(\mathbf{x},\mathbf{y}).$
9. (ix)
(Shift invariance) For each $\mathbf{z}\in\mathbb{Z}\times\mathbb{R}$,
$\\{\mathcal{B}^{\theta{\scaleobj{0.8}{\boxempty}}}(\mathbf{x},\mathbf{y}):\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R},\theta>0,{\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}\\}\overset{d}{=}\\{\mathcal{B}^{\theta{\scaleobj{0.8}{\boxempty}}}(\mathbf{x}+\mathbf{z},\mathbf{y}+\mathbf{z}):\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R},\theta>0,{\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}\\}.$
###### Remark 3.6.
On account of Part (viii), when working on the event $\Omega^{(\theta)}$ we
write $\mathcal{B}^{\theta}=\mathcal{B}^{\theta-}=\mathcal{B}^{\theta+}$. The
fact that the limits exist for all initial points
$\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$ and fixed $\theta>0$ on a
single event of probability one gives a generalization of Theorem 2.3. As will
be seen in Section 6, the key is Lemma C.5, which generalizes a proof in
[ARAS20].
We introduce a dual field of Brownian motions used later in the proof of
coalescence of semi-infinite geodesics. Let $\theta>0$. Recall the definition
(3.5) of the mapping $R$, and on the event $\Omega^{(\theta)}$, set
(3.9) $X_{m}^{\theta}=R(h_{m}^{\theta},B_{m-1}).$
Denote the field of these random functions by
$\mathbf{X}^{\theta}:=\\{X_{m}^{\theta}\\}_{m\in\mathbb{Z}}$. The following
theorem describes the distribution of the Busemann functions for a fixed
direction $\theta$.
###### Theorem 3.7.
Fix $\theta>0$.
1. (i)
The process $t\mapsto h_{m}^{\theta}(t)$ is a two-sided Brownian motion with
drift $\frac{1}{\sqrt{\theta}}$.
2. (ii)
$\mathbf{X}^{\theta}$ is a field of independent two-sided Brownian motions.
For each $n\in\mathbb{Z}$, $\\{X_{m}^{\theta}\\}_{m>n}$ is independent of
$\\{h_{m}^{\theta}\\}_{m\leq n}$.
3. (iii)
The process $t\mapsto v_{m}^{\theta}(t)$ is a stationary and reversible strong
Markov process. For each $t\in\mathbb{R}$,
$v_{m}^{\theta}(t)\sim\operatorname{Exp}\bigl{(}\frac{1}{\sqrt{\theta}}\bigr{)}$.
4. (iv)
(Burke property) Fix $m<n\in\mathbb{Z}$ and $-\infty<t_{n}\leq
t_{n-1}\leq\cdots\leq t_{m+1}<\infty$. Then, the following random variables
and processes are mutually independent:
$\\{B_{m}(u,t_{m+1}):u\leq
t_{m+1}\\},\;\;v_{m+1}^{\theta}(t_{m+1}),\;\;\\{h_{m}^{\theta}(t_{m+1},u):u\geq
t_{m+1}\\},\;\\\\[10.00002pt] \\{B_{r}(u,t_{r+1}):u\leq
t_{r+1}\\},\;v_{r+1}^{\theta}(t_{r+1}),\;\\{h_{r}^{\theta}(t_{r+1},u):t_{r+1}\leq
u\leq t_{r}\\},\\\\[10.00002pt] \text{ and }\\{X_{r}^{\theta}(t_{r},u):u\geq
t_{r}\\},\;\text{ for }\;m+1\leq r\leq
n-1,\;\;\\{h_{n}^{\theta}(u,t_{n}):u\leq t_{n}\\},\;\text{ and
}\;\\{X_{n}^{\theta}(t_{n},u):u\geq t_{n}\\}.$
As a special case, $\\{v_{m}^{\theta}(t)\\}_{m\in\mathbb{Z}}$ is an i.i.d
sequence for each $t\in\mathbb{R}$. Refer to Figure 3.1 for clarity.
###### Remark 3.8.
Using the representation of Equation (3.12) below, many formulas for the
process $t\mapsto v_{m}^{\theta}(t)$ are well known. See [BS02], page 129 and
[SN01], specifically equations (4) and (5), for more on this process,
including the transition density. However, we caution that when comparing
formulas, in the setting of [BS02, SN01], the process is
$t\mapsto\sup_{-\infty<s\leq t}\\{B(s,t)-\mu(t-s)\\},$
where $B$ is a two-sided Brownian motion and $\mu>0$. In our setting, there is
a factor of $\sqrt{2}$ multiplied to $B$.
$m$$m+1$$m+2$$n-3$$n-2$$n-1$$n$$B_{m}$$v_{m+1}^{\theta}$$B_{m+1}$$v^{\theta}_{m+2}$$h_{m+1}^{\theta}$$X_{m+1}^{\theta}$$h_{m}^{\theta}$$B_{m+2}$$h_{m+2}^{\theta}$$v^{\theta}_{m+3}$$X^{\theta}_{m+2}$$X^{\theta}_{m+3}$$v_{n-2}^{\theta}$$B_{n-3}$$\ddots$$v^{\theta}_{n-1}$$h^{\theta}_{n-2}$$B_{n-2}$$X^{\theta}_{n-2}$$v_{n}^{\theta}$$B_{n-1}$$h^{\theta}_{n-1}$$X^{\theta}_{n-1}$$h_{n}^{\theta}$$X_{n}^{\theta}$$t_{n}$$t_{n-1}$$t_{n-2}$$\cdots$$t_{m+3}$$t_{m+2}$$t_{m+1}$
Figure 3.1. Independence structure for Busemann functions. Each process
$h^{\theta}_{r}$ is associated to the (purple/dotted) segment on level $r$,
processes $B_{r}$ and $X^{\theta}_{r}$ cover the remaining portions of
horizontal level $r$, and the process $v_{r}^{\theta}$ is associated to the
(red/vertical) edge from level $r-1$ to $r$ at time point $t_{r}$.
The construction of the Busemann process and the proof of Theorem 3.5 can be
found in Section 6. We prove Theorem 3.7 here, assuming Theorem 3.5 and with
the help of the results of the appendix.
###### Proof of Theorem 3.7.
Part (i): As will be seen from the construction in Section 6, $t\mapsto
h_{m}^{\theta}(t)$ has the proper finite-dimensional distributions and is
continuous by Theorem 3.5(iv).
Part (ii): Fix integers $p<n<m$. By Theorem 3.5(vii),
$(B_{p},\ldots,B_{n},\dots,B_{m-2},B_{m-1},h_{m}^{\theta})\quad\text{are
independent. }$
Theorem 3.5(vi) and definition (3.9) give
$h_{m-1}^{\theta}=D(h_{m}^{\theta},B_{m-1})$ and
$X_{m}^{\theta}=R(h_{m}^{\theta},B_{m-1})$, so by Theorem C.2,
$h_{m-1}^{\theta}$ and $X_{m}^{\theta}$ are independent and $X_{m}^{\theta}$
is a two-sided Brownian motion. In particular, now
$(B_{p},\ldots,B_{n},\dots,B_{m-2},h_{m-1}^{\theta},X_{m}^{\theta})\quad\text{are
independent. }$
Continue inductively by applying the transformation $(D,R)$ to successive
pairs $(B_{j-1},h_{j}^{\theta})$ for $j=m-1,m-2,\dotsc,n+1$ after which
$(B_{p},\ldots,B_{n-1},h_{n}^{\theta},X_{n+1}^{\theta},\dotsc,X_{m}^{\theta})\quad\text{are
independent. }$
To conclude, note that $(h_{p}^{\theta},\ldots,h_{n}^{\theta})$ is a function
of $(B_{p},\ldots,B_{n-1},h_{n}^{\theta})$ through iteration of
$h_{k}^{\theta}=D(h_{k+1}^{\theta},B_{k})$.
Part (iii): By Theorem 3.5(vi) and (3.9), on the event $\Omega^{(\theta)}$ we
have these relations $\forall m\in\mathbb{Z}$:
(3.10) $h_{m-1}^{\theta}=D(h_{m}^{\theta},B_{m-1}),\qquad
X_{m}^{\theta}=R(h_{m}^{\theta},B_{m-1}),\qquad\text{and}\qquad
v_{m}^{\theta}=Q(h_{m}^{\theta},B_{m-1}).$
The fact that $v_{m}^{\theta}(t)$ is exponential with rate
$\frac{1}{\sqrt{\theta}}$ then follows from Lemma B.1. Theorem D.1 allows us
to reverse these mappings, so $\forall m\in\mathbb{Z}$:
(3.11) $\displaystyle
h_{m}^{\theta}=\overleftarrow{D}(h_{m-1}^{\theta},X_{m}^{\theta}),\qquad
B_{m-1}=\overleftarrow{R}(h_{m-1}^{\theta},X_{m}^{\theta}),\qquad\text{and}\qquad
v_{m}^{\theta}=\overleftarrow{Q}(h_{m-1}^{\theta},X_{m}^{\theta}).$
Then, for $t\in\mathbb{R}$,
(3.12) $v_{m}^{\theta}(t)=\sup_{-\infty<u\leq
t}\\{X_{m}^{\theta}(u,t)-h_{m-1}^{\theta}(u,t)\\}.$
By Parts (i) and (ii), $t\mapsto X_{m}^{\theta}(t)-h_{m-1}^{\theta}(t)$ is
equal in distribution to a two-sided Brownian motion with negative drift,
multiplied by a factor of $\sqrt{2}$. Represented this way, $t\mapsto
v_{m}^{\theta}(t)$ is known as a stationary, reflected Brownian motion with
drift. Stationarity follows from the stationarity of increments. The fact that
$X$ is a reversible strong Markov process is proven in [Har85], pg. 81 (see
also pg. 49-50 in [Har85] and Equations (4) and (5) in [SN01] for a more
directly applicable statement).
Part (iv): By Part (ii) and (3.11), for any initial level $m$, the process
$\bigl{\\{}h_{r+m}^{\theta},v_{r+m+1}^{\theta},X_{r+m+1}^{\theta},B_{r+m}\bigr{\\}}_{r\geq
0}$ has the same distribution as
$\bigl{\\{}Y_{r}^{\frac{1}{\sqrt{\theta}}},q_{r+1}^{\frac{1}{\sqrt{\theta}}},B_{r+1},W_{r}^{\lambda}\bigr{\\}}_{r\geq
0}$ as defined in (C.1). Note that $B_{r+m}$ now plays the role of the
$W_{r}^{\lambda}$, as stated in the definition. Therefore, the independence
structure of Theorem C.3 holds. ∎
## 4\. Construction and properties of the semi-infinite geodesics
### 4.1. Heuristic for construction of semi-infinite geodesics
The next task is the construction of semi-infinite geodesics from each initial
point and in each asymptotic direction. For each given point
$(m,t)\in\mathbb{Z}\times\mathbb{R}$ and direction parameter $\theta>0$, we
want to find a semi-infinite geodesic, defined by jump times
$t=\tau_{m-1}\leq\tau_{m}\leq\cdots$ that satisfies
$\lim_{n\rightarrow\infty}\frac{\tau_{n}}{n}=\theta.$
We argue heuristically to motivate the useful construction. Start by finding a
maximal path for $L_{(m,t),(n,n\theta)}$ for a large value of $n$. Note that
$L_{(m,t),(n,n\theta)}=\max_{s\in[t,n\theta]}\bigl{(}B_{m}(t,s)+L_{(m+1,s),(n,n\theta)}\bigr{)},$
and the maximizer $s=\tau_{m}$ is the location where the geodesic jumps from
level $m$ to $m+1$. For all $t\leq s\leq n\theta$,
$B_{m}(t,s)+L_{(m+1,s),(n,n\theta)}\leq
B_{m}(t,\tau_{m})+L_{(m+1,\tau_{m}),(n,n\theta)}.$
Rearranging yields
$B_{m}(\tau_{m},s)\leq
L_{(m+1,\tau_{m}),(n,n\theta)}-L_{(m+1,s),(n,n\theta)}.$
As $n$ changes, so could $\tau_{m}$, but for the sake of heuristic we hold
$\tau_{m}$ constant. Take limits as $n\rightarrow\infty$ and rearrange again
to get, for some sign ${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$,
$B_{m}(s)-h_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\leq
B_{m}(\tau_{m})-h_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{m}).$
### 4.2. Busemann geodesics
The discussion of the previous section motivates this rigorous definition.
###### Definition 4.1.
On the event $\Omega_{1}$, for all $(m,t)\in\mathbb{Z}$, $\theta>0$ and
${\scaleobj{0.8}{\boxempty}}\in\\{+,-\\}$, let
$\mathbf{T}^{\theta{\scaleobj{0.8}{\boxempty}}}_{(m,t)}$ denote the set of
sequences
$t=\tau_{m-1}\leq\tau_{m}\leq\tau_{m+1}\leq\cdots$
that satisfy
(4.1)
$B_{r}(\tau_{r})-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{r})=\sup_{s\in[\tau_{r-1},\infty)}\\{B_{r}(s)-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\\}\qquad\text{for
each $r\geq m$.}$
Theorem 3.5(iv)–(v) imply that such sequences exist. At each level $r$, there
exist leftmost and rightmost maximizers. Let
$t=\tau_{(m,t),m-1}^{\theta{\scaleobj{0.8}{\boxempty}},L}\leq\tau_{(m,t),m}^{\theta{\scaleobj{0.8}{\boxempty}},L}\leq\tau_{(m,t),m+1}^{\theta{\scaleobj{0.8}{\boxempty}},L}\leq\cdots\qquad\text{and}\qquad
t=\tau_{(m,t),m-1}^{\theta{\scaleobj{0.8}{\boxempty}},R}\leq\tau_{(m,t),m}^{\theta{\scaleobj{0.8}{\boxempty}},R}\leq\tau_{(m,t),m+1}^{\theta{\scaleobj{0.8}{\boxempty}},R}\leq\cdots$
denote the leftmost and rightmost sequences in
$\mathbf{T}_{(m,t)}^{\theta{\scaleobj{0.8}{\boxempty}}}$. Furthermore, define
$\mathbf{T}_{(m,t)}^{\theta}:=\mathbf{T}_{(m,t)}^{\theta+}\cup\mathbf{T}_{(m,t)}^{\theta-}.$
###### Remark 4.2.
Since every non-decreasing sequence in $\mathbf{T}_{(m,t)}^{\theta}$ defines a
semi-infinite up-right path, $\mathbf{T}_{(m,t)}^{\theta}$ will be used to
denote the set of up-right paths constructed in this way. Theorem 4.3(i) below
shows that all sequences in $\mathbf{T}_{(m,t)}^{\theta}$ are semi-infinite
geodesics starting from $(m,t)$. By Theorem 3.5(viii), on the event
$\Omega^{(\theta)}\subseteq\Omega_{1}$,
$\mathbf{T}_{\mathbf{x}}^{\theta-}=\mathbf{T}_{\mathbf{x}}^{\theta+}=\mathbf{T}_{\mathbf{x}}^{\theta}$
for all $\mathbf{x}\in\mathbb{Z}\times\mathbb{R}$. This does not imply that
$\mathbf{T}_{\mathbf{x}}^{\theta}$ contains only one element, so the leftmost
and rightmost distinction is still necessary in general for a fixed direction
$\theta$. See Theorem 4.7 below. However, by Theorem 3.1(ii), for fixed
$\theta>0$ and fixed $(m,t)\in\mathbb{Z}\times\mathbb{R}$,
$\mathbf{T}_{(m,t)}^{\theta}$ almost surely contains a single element. In this
case, Theorem B.3 gives the distribution of the first jump time $\tau_{m}$.
$(m,t)$$\tau_{m}^{\theta}$$\tau_{m+1}^{\theta}$$\tau_{m+2}^{\theta}$$\cdots$$m$$m+1$$m+2$$\vdots$
Figure 4.1. Example of an element of $\mathbf{T}_{(m,t)}^{\theta}$
Two elements $\\{\tau_{r}\\}_{r\geq m-1}$ and $\\{\tau^{\prime}_{r}\\}_{r\geq
m-1}$ of $\mathbf{T}_{\mathbf{x}}^{\theta{\scaleobj{0.8}{\boxempty}}}$ are
distinct if $\tau_{r}\neq\tau^{\prime}_{r}$ for at least one index $r$.
Uniqueness of the $\theta{\scaleobj{0.8}{\boxempty}}$ Busemann geodesic from
$\mathbf{x}$ means that
$\mathbf{T}_{\mathbf{x}}^{\theta{\scaleobj{0.8}{\boxempty}}}$ contains exactly
one sequence.
The following theorems collect the properties of the Busemann semi-infinite
geodesics, to be proved in Section 7.
###### Theorem 4.3.
There exists an event, $\Omega_{2}$, of full probability, on which the
following hold.
1. (i)
(Existence) For all $\mathbf{x}\in\mathbb{Z}\times\mathbb{R},\theta>0$, and
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$, every element of
$\mathbf{T}_{\mathbf{x}}^{\theta{\scaleobj{0.8}{\boxempty}}}$ defines a semi-
infinite geodesic starting from $\mathbf{x}$. More specifically, for any two
points $\mathbf{y}\leq\mathbf{z}$ along a path in
$\mathbf{T}_{\mathbf{x}}^{\theta}$, the energy of this path between
$\mathbf{y}$ and $\mathbf{z}$ is
$\mathcal{B}^{\theta{\scaleobj{0.8}{\boxempty}}}(\mathbf{y},\mathbf{z})$, and
this energy is maximal over all paths between $\mathbf{y}$ and $\mathbf{z}$.
2. (ii)
(Leftmost and rightmost finite geodesics along paths) If, for some $\theta>0$,
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$, and
$\mathbf{x}\in\mathbb{Z}\times\mathbb{R}$, the points
$\mathbf{y}\leq\mathbf{z}\in\mathbb{Z}\times\mathbb{R}$ both lie on the
leftmost semi-infinite geodesic in
$\mathbf{T}_{\mathbf{x}}^{\theta{\scaleobj{0.8}{\boxempty}}}$, then the
portion of this geodesic between $\mathbf{y}$ and $\mathbf{z}$ coincides with
the leftmost finite geodesic between these two points. Similarly, the
rightmost semi-infinite geodesic is the rightmost geodesic between any two of
its points.
3. (iii)
(Monotonicity) The following inequalities hold.
1. (a)
For all $0<\gamma<\theta$, all $(m,t)\in\mathbb{Z}\times\mathbb{R}$, and
$r\geq m$,
$t\leq\tau_{(m,t),r}^{\gamma-,L}\leq\tau_{(m,t),r}^{\gamma+,L}\leq\tau_{(m,t),r}^{\theta-,L}\leq\tau_{(m,t),r}^{\theta+,L}\qquad\text{and}\qquad
t\leq\tau_{(m,t),r}^{\gamma-,R}\leq\tau_{(m,t),r}^{\gamma+,R}\leq\tau_{(m,t),r}^{\theta-,R}\leq\tau_{(m,t),r}^{\theta+,R}.$
2. (b)
For all $\theta>0$, $m\leq r\in\mathbb{Z}$, $s<t\in\mathbb{R}$, and
${\scaleobj{0.8}{\boxempty}}\in\\{+,-\\}$,
$\tau_{(m,s),r}^{\theta{\scaleobj{0.8}{\boxempty}},L}\leq\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},L}\qquad\text{and}\qquad\tau_{(m,s),r}^{\theta{\scaleobj{0.8}{\boxempty}},R}\leq\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},R}.$
3. (c)
For $\theta>0$, on the $\theta$-dependent full-probability event
$\widetilde{\Omega}^{(\theta)}$ of Theorem 3.1(iii), for all pairs of initial
points $(m,s)$ and $(m,t)$ in $\mathbb{Z}\times\mathbb{R}$ that satisfy $s<t$,
we have
$\tau_{(m,s),r}^{\theta,R}\leq\tau_{(m,t),r}^{\theta,L}\quad\text{ for all
$r\geq m$.}$
4. (iv)
(Convergence) The following limits hold.
1. (a)
For all $(m,t)\in\mathbb{Z}\times\mathbb{R}$, $r\geq m$, $\theta>0$, and
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$,
$\lim_{\gamma\nearrow\theta}\tau_{(m,t),r}^{\gamma{\scaleobj{0.8}{\boxempty}},L}=\tau_{(m,t),r}^{\theta-,L}\qquad\text{and}\qquad\lim_{\delta\searrow\theta}\tau_{(m,t),r}^{\delta{\scaleobj{0.8}{\boxempty}},R}=\tau_{(m,t),r}^{\theta+,R}.$
2. (b)
For all $(m,t)\in\mathbb{Z}\times\mathbb{R}$, $r\geq m$,
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$, and $S\in\\{L,R\\}$,
$\lim_{\theta\searrow
0}\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},S}=t\qquad\text{and}\qquad\lim_{\theta\rightarrow\infty}\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},S}=\infty.$
3. (c)
For all $(m,t)\in\mathbb{Z}\times\mathbb{R}$, $r\geq m$, $\theta>0$, and
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$,
$\lim_{s\nearrow
t}\tau_{(m,s),r}^{\theta{\scaleobj{0.8}{\boxempty}},L}=\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},L}\qquad\text{and}\qquad\lim_{u\searrow
t}\tau_{(m,u),r}^{\theta{\scaleobj{0.8}{\boxempty}},R}=\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},R},$
5. (v)
(Directness) For all $\mathbf{x}\in\mathbb{Z}\times\mathbb{R}$, $\theta>0$,
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$, and all $\\{\tau_{r}\\}_{r\geq
m}\in\mathbf{T}_{\mathbf{x}}^{\theta{\scaleobj{0.8}{\boxempty}}}$,
$\lim_{n\rightarrow\infty}\frac{\tau_{n}}{n}=\theta.$
###### Remark 4.4 (A look ahead).
In future work, we will use the joint distribution of Busemann functions to
build on these results and strengthen parts of Theorem 4.3. Specifically, Part
(iv)(iv)(a) can be made stronger in the following way: There exists an event
of full probability on which, for all $\theta>0,m\leq
r\in\mathbb{Z},s<t\in\mathbb{R},S\in\\{L,R\\}$ and
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$, there exists $\varepsilon>0$ such
that
$\tau_{(m,s),r}^{\gamma{\scaleobj{0.8}{\boxempty}},S}=\tau_{(m,s),r}^{\theta-,S}\text{
for all
}\theta-\varepsilon<\gamma<\theta,\qquad\text{and}\qquad\tau_{(m,s),r}^{\delta{\scaleobj{0.8}{\boxempty}},S}=\tau_{(m,s),r}^{\theta+,S}\text{
for all }\theta<\delta<\theta+\varepsilon.$
This is used to strengthen Part (iii)(iii)(c) to show that, on this event, for
each $m\in\mathbb{Z}$, $m\leq r\in\mathbb{Z},s<t\in\mathbb{R},\theta>0$, and
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$,
$\tau_{(m,s),r}^{\theta{\scaleobj{0.8}{\boxempty}},R}\leq\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},L}.$
Part (iii)(iii)(a) cannot be strengthened to compare
$\tau_{(m,t),r}^{\gamma{\scaleobj{0.8}{\boxempty}},R}$ and
$\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},L}$ for general
$\gamma<\theta$. Specifically, there exists $\gamma<\theta$ such that
$\tau_{(m,t),r}^{\gamma-,R}=\tau_{(m,t),r}^{\gamma+,R}>\tau_{(m,t),r}^{\theta-,L}=\tau_{(m,t),r}^{\theta+,L}.$
The following theorem shows that the Busemann geodesics give control over all
semi-infinite geodesics.
###### Theorem 4.5.
The following hold on the full probability event $\Omega_{2}$.
1. (i)
(Control of finite geodesics) Let
$\theta>0,(m,t)\in\mathbb{Z}\times\mathbb{R}$, and let $\\{t_{n}\\}$ be any
sequence that has direction $\theta$. For all $n$ sufficiently large so that
$n\geq m$ and $t_{n}\geq t$, let $t=t_{n,m-1}\leq t_{n,m}\leq\cdots\leq
t_{n,n}=t_{n}$ be any sequence that defines a (finite) geodesic between
$(m,t)$ and $(n,t_{n})$. Then, for each $r\geq m$,
$\tau_{(m,s),r}^{\theta-,L}\leq\liminf_{n\rightarrow\infty}t_{n,r}\leq\limsup_{n\rightarrow\infty}t_{n,r}\leq\tau_{(m,s),r}^{\theta+,R}.$
2. (ii)
(Control of semi-infinite geodesics) If, for some $\theta>0$ and
$(m,t)\in\mathbb{Z}\times\mathbb{R}$, any other geodesic (constructed from the
Busemann functions or not) is defined by the sequence $t=t_{m-1}\leq
t_{m}\leq\cdots$, starts at $(m,t)$, and has direction $\theta$, then for all
$r\geq m$,
$\tau_{(m,t),r}^{\theta-,L}\leq t_{r}\leq\tau_{(m,t),r}^{\theta+,R}.$
3. (iii)
(Convergence of finite geodesics) Assume that
$\mathbf{T}_{\mathbf{(}m,t)}^{\theta}$ contains a single element
$\\{\tau_{r}\\}_{r\geq m-1}$. If $\\{t_{n}\\}$ is a $\theta$-directed sequence
and, for each $n$, the sequence $t=t_{n,m-1}\leq t_{n,m}\leq\cdots\leq
t_{n,n}$ defines a finite geodesic between $(m,t)$ and $(n,t_{n})$, then
$\lim_{n\rightarrow\infty}t_{n,r}=\tau_{r}\qquad\text{for all }r\geq m.$
###### Remark 4.6.
In Part (iii), the assumption of uniqueness holds, for example, on the event
$\Omega_{(m,t)}^{(\theta)}$ of Theorem 3.1(ii). However, this assumption does
not extend to all $(m,t)\in\mathbb{Z}\times\mathbb{R}$ and $\theta>0$
simultaneously with probability one, as discussed in the following section.
### 4.3. Non-uniqueness of semi-infinite geodesics
There are two types of non-uniqueness of semi-infinite geodesics from an
initial point $\mathbf{x}$ into an asymptotic direction $\theta$.
(i) The first type, described in Theorem 3.1(iii) and in the next Theorem 4.7,
is caused by the continuum time variable and does not appear in the lattice
corner growth model. It is captured by the $L/R$ distinction. For each fixed
direction $\theta>0$ and level $m\in\mathbb{Z}$, this happens with probability
one at infinitely many locations. To illustrate, let $s^{\star}\geq 0$ be the
maximizer below:
$B_{m}(s^{\star})-h_{m+1}^{\theta}(s^{\star})=\sup_{0\leq
s<\infty}\\{B_{m}(s)-h_{m+1}^{\theta}(s)\\}.$
By Theorem 3.5(vii) and Lemma 7.1, with probability one, the maximizer
$s^{\star}$ is unique. By Theorem B.3, $s^{\star}>0$ with probability one. By
Theorem 3.5(v),
$t^{\star}:=\sup\\{t<0:B_{m}(t)-h_{m+1}^{\theta}(t)=B_{m}(s^{\star})-h_{m+1}^{\theta}(s^{\star})\\}\qquad\text{exists
in }\mathbb{R}_{<0}.$
Then, both $t^{\star}$ and $s^{\star}$ are maximizers of
$B_{m}(s)-h_{m+1}^{\theta}(s)$ on $[t^{\star},\infty)$. This gives at least
two distinct sequences in the set $\mathbf{T}_{(m,t^{\star})}^{\theta}$, with
$\tau_{(m,t^{\star}),m}^{\theta,L}=t^{\star}$ and
$\tau_{(m,t^{\star}),m}^{\theta,R}=s^{\star}$.
This presents a new type of non-uniqneness that is not present in discrete
last-passage percolation with exponential weights. However, when $\theta$ is
fixed, $\theta$-directed geodesics can disagree only for a finite amount of
time, because Theorem 3.1(vii) forces them to eventually come back together.
(ii) The second type of non-uniqueness of semi-infinite geodesics is captured
by the $\theta\pm$ distinction. Hence, it happens with probability zero at a
fixed $\theta$ and thereby requires investigation of the full Busemann process
and the full collection of all semi-infinite geodesics. In contrast to the
first type of non-uniqueness, this bears some similarity to the behavior
present in discrete last-passage percolation shown in [JRS21].
In future work, we show that there exists a random countable set of directions
$\theta$ such that, out of every initial point, there are two
$\theta$-directed geodesics. These geodesics may initially stay together for a
while, but eventually they separate for good and never come back together.
Furthermore, there is a distinguished subset of initial points at which
geodesics with the same direction split immediately. This set will be shown to
have almost surely Hausdorff dimension $\frac{1}{2}$.
The following theorem clarifies the non-uniqueness described by (i) above. Fix
$\theta>0$. On a full probability event where the $\theta\pm$ distinction is
not present, define the following sets:
$\displaystyle\operatorname{NU}_{0}^{\theta}$
$\displaystyle=\\{(m,t)\in\mathbb{Z}\times\mathbb{R}:\tau_{(m,t),r}^{\theta,L}<\tau_{(m,t),r}^{\theta,R}\text{
for some }r\geq m\\},\qquad\text{and}$
$\displaystyle\operatorname{NU}_{1}^{\theta}$
$\displaystyle=\\{(m,t)\in\operatorname{NU}_{0}^{\theta}:\tau_{(m,t),m}^{\theta,L}<\tau_{(m,t),m}^{\theta,R}\\}.$
Since $\theta>0$ is fixed, by Theorem 4.5(ii),
$\operatorname{NU}_{0}^{\theta}$ is almost surely the set of points
$\mathbf{x}\in\mathbb{Z}\times\mathbb{R}$ such that the $\theta$-directed
semi-infinite geodesic from $\mathbf{x}$ is not unique. Its subset
$\operatorname{NU}_{1}^{\theta}$ is the set of initial points from which two
$\theta$-directed geodesics separate on the first level.
###### Theorem 4.7.
There exists a full probability event $\widetilde{\Omega}^{(\theta)}$ on which
the following hold.
1. (i)
The sets $\operatorname{NU}_{0}^{\theta}$ and $\operatorname{NU}_{1}^{\theta}$
are countably infinite and can be written as
$\displaystyle\operatorname{NU}_{0}^{\theta}$
$\displaystyle=\\{(m,t)\in\mathbb{Z}\times\mathbb{R}:t=\tau_{(m,t),r}^{\theta,L}<\tau_{(m,t),r}^{\theta,R}\text{
for some }r\geq m\\},\qquad\text{and}$
$\displaystyle\operatorname{NU}_{1}^{\theta}$
$\displaystyle=\\{(m,t)\in\operatorname{NU}_{0}^{\theta}:t=\tau_{(m,t),m}^{\theta,L}<\tau_{(m,t),m}^{\theta,R}\\}.$
For each $(m,t)\in\mathbb{Z}\times\mathbb{R}$ and $\theta>0$, at most one
geodesic in $\mathbf{T}_{(m,t)}^{\theta}$ passes horizontally through
$(m,t+\varepsilon)$ for some $\varepsilon>0$.
2. (ii)
The set $\operatorname{NU}_{1}^{\theta}$ is neither discrete nor dense in
$\mathbb{Z}\times\mathbb{R}$. More specifically, for each point
$(m,t)\in\operatorname{NU}_{1}^{\theta}$ and every $\varepsilon>0$, there
exists $s\in(t-\varepsilon,t)$ such that
$(m,s)\in\operatorname{NU}_{1}^{\theta}$. For each
$(m,t)\in\operatorname{NU}_{1}^{\theta}$, there exists $\delta>0$ such that,
for all $s\in(t,t+\delta)$, $(m,s)\notin\operatorname{NU}_{0}^{\theta}$.
###### Remark 4.8.
Part (i) states that, on $\widetilde{\Omega}^{(\theta)}$, if there exist
multiple $\theta$-directed geodesics out of $(m,t)$, then these geodesics
separate one by one from the upward vertical ray at $(m,t)$. The set
$\operatorname{NU}_{1}^{\theta}$ is the subset of
$\operatorname{NU}_{0}^{\theta}$ such that two geodesics separate immediately
at the initial point. See Figures 4.2 and 4.3.
$(r,\tau_{(m,t),r}^{\theta,R})$$(r,t)=(r,\tau_{(m,t),r}^{\theta,L})$$(m,t)$
Figure 4.2. In this figure,
$(m,t)\in\operatorname{NU}_{0}^{\theta}\setminus\operatorname{NU}_{1}^{\theta}$.
The two geodesics split on the vertical line containing the initial point, but
they must come back together to coalesce.
$(m,t)=(m,\tau_{(m,t),m}^{\theta,L})$$(m,\tau_{(m,t),m}^{\theta,R})$ Figure
4.3. In this figure, $(m,t)\in\operatorname{NU}_{1}^{\theta}$. The two
geodesics split immediately from the initial point, but later coalesce.
### 4.4. Dual geodesics and coalescence
To prove the coalescence of Theorem 3.1(vii), we use the dual field
$\mathbf{X}^{\theta}$ of independent Brownian motions from (3.9) and their
southwest-directed semi-infinite geodesics. We use this to construct the BLPP
analogue of Pimentel’s dual tree [Pim16] and then adapt the argument of
[Sep20].
Since Brownian motion is symmetric, in distribution, about the origin, there
exist $\theta$-directed dual southwest semi-infinite geodesics for the
environment $\mathbf{X}^{\theta}$. These are constructed in a very similar
manner as the northeast geodesics in Definition 4.1. Specifically, for
$(m,t)\in\mathbb{Z}\times\mathbb{R}$, let $\mathbf{T}_{(m,t)}^{\theta,\star}$
be the set of sequences $t=\tau_{m}^{\star}\geq\tau_{m-1}^{\star}\geq\cdots$
satisfying
(4.2)
$h_{r-1}^{\theta}(\tau_{r-1}^{\star})-X_{r}^{\theta}(\tau_{r-1}^{\star})=\sup_{-\infty<s\leq\tau_{r}^{\star}}\\{h_{r-1}^{\theta}(s)-X_{r}^{\theta}(s)\\}\qquad\text{for
each }r\leq m.$
Define the leftmost and rightmost sequences similarly, by
$\tau_{(m,t),r}^{\theta,L\star}$ and $\tau_{(m,t),r}^{\theta,R\star}$. These
sequences define southwest semi-infinite paths, similar as for the northeast
paths. We graphically represent southwest paths on the plane, where the
continuous coordinate is not changed, but the discrete coordinate is shifted
down by $\frac{1}{2}$. That is, $\text{for }m\in\mathbb{Z},\text{ denote
}m^{\star}=m-\frac{1}{2}$, and for
$\mathbf{x}=(m,t)\in\mathbb{Z}\times\mathbb{R},\text{ denote
}\mathbf{x}^{\star}=\bigl{(}m-\frac{1}{2},t\bigr{)}$. Then, for
$\\{\tau_{r}^{\star}\\}_{r\leq m}\in\mathbf{T}_{(m,t)}^{\theta,\star}$, the
southwest path consists of horizontal and vertical line segments, where
$\tau_{r}^{\star}$ denotes the position of the vertical segment connecting
levels $(r+1)^{\star}$ and $r^{\star}$. Specifically, the path consists of the
points
$\bigcup_{r=-\infty}^{m}\\{(r^{\star},u):u\in[\tau_{r-1}^{\star},\tau_{r}^{\star}]\\}\cup\bigcup_{r=-\infty}^{m-1}\\{(v,\tau_{r}^{\star}):v\in[(r-1)^{\star},r^{\star}]\\}.$
Figure 4.4 shows the regular axes and the dual axes together, with a southwest
dual geodesic traveling on this dual plane. Each element of
$\mathbf{T}_{\mathbf{(}m,t)}^{\theta\star}$ is a southwest semi-infinite
geodesic for the dual environment $\mathbf{X}^{\theta}$. This fact is recorded
in Theorem 7.11.
$(m-1)^{\star}$$m-1$$m^{\star}$$m$$\vdots$$t$$\tau_{(m,t),m-1}^{\star}$$\tau_{(m,t),m-2}^{\star}$$\cdots$
Figure 4.4. The original discrete levels (solid) and the dual levels (dashed).
Dual levels are labeled with a $\star$.
Since $\mathbf{X}^{\theta}$ is an environment of i.i.d. Brownian motions, the
following theorem allows us to conclude Part (vii) of Theorem 3.1 from Part
(vi). Full details of this connection are found in the proofs. Refer to Figure
4.5 for clarity.
$\mathbf{x}$$\mathbf{y}$ Figure 4.5. The outcome of Theorem 4.9: two disjoint
semi-infinite northeast paths (red/thick) and the dual bi-infinite path
(blue/thin)
###### Theorem 4.9.
Fix $\theta>0$. With probability one, if for any
$\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$, the rightmost semi-
infinite geodesics in $\mathbf{T}_{\mathbf{x}}^{\theta}$ and
$\mathbf{T}_{\mathbf{y}}^{\theta}$ are disjoint (i.e. the paths share no
points), then there exists a bi-infinite, upright path defined by jump times
$\cdots\leq\tau_{-1}^{\star}\leq\tau_{0}^{\star}\leq\tau_{1}^{\star}\leq\cdots$
(where $\tau_{i}^{\star}$ denotes the jump time from level $i$ to level $i+1$)
that satisfies the following:
1. (i)
For any point $\mathbf{x}^{\star}$ along the path, the portion of that path to
the south and west of $\mathbf{x}^{\star}$ is the leftmost semi-infinite
geodesic in the set $\mathbf{T}_{\mathbf{x}}^{\theta,\star}$. Specifically,
when shifted back up by $\frac{1}{2}$, the path is a bi-infinite geodesic for
the environment $\mathbf{X}^{\theta}$.
2. (ii)
The sequence $\\{\tau_{n}^{\star}\\}_{n\in\mathbb{Z}}$ satisfies
$\lim_{n\rightarrow\infty}\frac{\tau_{n}^{\star}}{n}=\theta=\lim_{n\rightarrow\infty}\frac{\tau_{-n}^{\star}}{-n}.$
The analogous result holds if we assume that two leftmost $\theta$-directed
semi-infinite geodesics are disjoint. In this case, the portion of the path to
the south and west of each of its points is the rightmost dual semi-infinite
geodesic.
## 5\. Connections to other models
### 5.1. Connection to infinite-length polymer measures for the Brownian
polymer
Recalling the definitions at the beginning of Section 2.2, for
$(m,s)\leq(n,t)\in\mathbb{Z}\times\mathbb{R}$, the point-to-point partition
function of the Brownian polymer with unit temperature is
$Z_{(m,s),(n,t)}(\mathbf{B})=\int
e^{\mathcal{E}(\mathbf{s}_{m,n})}\mathbf{1}\\{\mathbf{s}_{m,n-1}\in\Pi_{(m,s),(n,t)}\\}d\mathbf{s}_{m,n-1}.$
The associated quenched polymer measure on $\Pi_{(m,s),(n,t)}$ is
$Q^{\mathbf{B}}(\tau_{m}\in ds_{m},\ldots,\tau_{n-1}\in
ds_{n-1})=\frac{1}{Z_{(m,s),(n,t)}(\mathbf{B})}\exp\Bigl{(}\sum_{k=m}^{n}B_{k}(s_{k-1},s_{k})\Bigr{)}1\\{\mathbf{s}_{m,n-1}\in\Pi_{(m,s),(n,t)}\\}d\mathbf{s}_{m,n-1}.$
Brownian last-passage percolation is the zero-temperature analogue of the
Brownian or O’Connell-Yor polymer. This is made precise by the limit
$\lim_{\beta\rightarrow\infty}\frac{1}{\beta}\log
Z_{(m,s),(n,t)}(\beta\mathbf{B})=L_{(m,s),(n,t)}(\mathbf{B}),$
which, by convergence of $L^{p}$ norms, holds in a deterministic sense as long
as $\mathbf{B}$ is a field of continuous functions
$\mathbb{R}\rightarrow\mathbb{R}$. The parameter $\beta$ is used to denote
inverse temperature.
Alberts, Rassoul-Agha, and Simper [ARAS20] showed the existence of Busemann
functions and infinite-length limits of the quenched measures when the right
endpoint $(n,t_{n})$ satisfies $t_{n}/n\rightarrow\theta$ for some fixed
$\theta>0$. The Busemann functions are defined as
$\widehat{\mathcal{B}}^{\theta}(\mathbf{x},\mathbf{y}):=\lim_{n\rightarrow\infty}\log\frac{Z_{\mathbf{x},(n,t_{n})}(\mathbf{B})}{Z_{\mathbf{y},(n,t_{n})}(\mathbf{B})}.$
Similarly as in (2.6), define
$\widehat{h}_{m}^{\theta}(t)=\widehat{\mathcal{B}}^{\theta}((m,0),(m,t))$.
Using the “Proof of Theorem 2.5, assuming Theorem 3.1” on page 1937 of
[ARAS20] and an analogous construction of the Busemann functions for all
initial points as in Section 6 of the present paper, it can be shown that, the
process $t\mapsto\widehat{h}_{m}^{\theta}(t)$ is a two-sided Brownian motion
with drift, independent of $\\{B_{r}\\}_{r\leq m}$. By Theorems 3.5(vii) and
3.7(i), the same is true for the zero temperature Busemann process $t\mapsto
h_{m}^{\theta}(t)$. An infinite, up-right path under the quenched infinite-
length polymer measure is a continuous-time Markov chain, starting from a
point $(m,t)$ and defined by jump times $t=\tau_{m-1}\leq\tau_{m}\leq\cdots$.
By Equation (2.3) in [ARAS20], the quenched conditional distribution of
$\tau_{r}$ given $\tau_{r-1}$ is
$\displaystyle Q^{\mathbf{B}}_{(m,s),(n,t_{n})}(\tau_{r}\in
ds_{r}|\tau_{r-1}=s_{r-1})$
$\displaystyle=e^{B_{r}(s_{r-1},s_{r})}\frac{Z_{(r+1,s_{r}),(n,t_{n})}(\mathbf{B})}{Z_{(r,s_{r-1}),(n,t_{n})}(\mathbf{B})}\mathbf{1}\\{s_{r-1}\leq
s_{r}\\}\,ds_{r}.$
The proof of existence of infinite length measures requires a rigorous
tightness argument, but to motivate the connection to BLPP, we formally take
limits as $n\rightarrow\infty$ to yield the conditional measure
$\displaystyle\;\exp\bigl{(}B_{r}(s_{r-1},s_{r})+\widehat{\mathcal{B}}^{\theta}((r+1,s_{r}),(r,s_{r-1}))\bigr{)}\mathbf{1}\\{s_{r-1}\leq
s_{r}\\}\,ds_{r}$
$\displaystyle=\exp\bigl{(}B_{r}(s_{r-1},s_{r})-\widehat{h}_{r+1}(s_{r})+\widehat{\mathcal{B}}^{\theta}((r+1,0),(r,s_{r-1})\bigr{)}\mathbf{1}\\{s_{r-1}\leq
s_{r}\\}\,ds_{r}.$
Now, note that the term $\widehat{\mathcal{B}}^{\theta}((r+1,0),(r,s_{r-1})$
does not depend on $s_{r}$. We can then think of the connection between BLPP
and the O’Connell-Yor polymer in the following sense: In the positive
temperature case, given the environment $\mathbf{B}$, the transition density
from $\tau_{r-1}$ to $\tau_{r}$ is given by
$C\exp\bigl{(}B_{r}(s)-\widehat{h}_{r+1}^{\theta}(s)\bigr{)}\mathbf{1}\\{\tau_{r-1}\leq
s\\}\,ds,$
where $C$ is a normalizing constant depending on $r$ and $\tau_{r-1}$. On the
other hand, by Definition 4.1, in the zero-temperature case, given the
environment $\mathbf{B}$ and the previous jump $\tau_{r-1}$, the jump
$\tau_{r}$ is chosen in a deterministic fashion by maximizing
$B_{r}(s)-h_{r+1}^{\theta}(s)$ over $s\in[\tau_{r-1},\infty)$.
### 5.2. Connection to semi-infinite geodesics in discrete last-passage
percolation
The corner growth model, or discrete last-passage percolation, is defined as
follows. Let $\\{Y_{\mathbf{x}}\\}_{\mathbf{x}\in\mathbb{Z}^{2}}$ be a
collection of nonnegative i.i.d random variables, each associated to a vertex
on the integer lattice. For
$\mathbf{x}\leq\mathbf{y}\in\mathbb{Z}\times\mathbb{Z}$, define the last-
passage time as
$G_{\mathbf{x},\mathbf{y}}=\sup_{\mathbf{x}_{\centerdot}\in\Pi_{\mathbf{x},\mathbf{y}}}\sum_{k=0}^{|\mathbf{y}-\mathbf{x}|_{1}}Y_{\mathbf{x}_{k}},$
where $\Pi_{x,y}$ is the set of up-right paths
$\\{\mathbf{x}_{k}\\}_{k=0}^{n}$ that satisfy
$\mathbf{x}_{0}=\mathbf{x},\mathbf{x}_{n}=\mathbf{y}$, and
$\mathbf{x}_{k}-\mathbf{x}_{k-1}\in\\{\mathbf{e}_{1},\mathbf{e}_{2}\\}$. Under
the assumption that $Y_{0}$ has finite second moment, Theorem 3.1 and
Corollary 3.1 of [GW91] introduced BLPP as a universal scaling limit of the
corner growth model, where one variable is scaled, and the other is held
constant. That is, if $Y_{0}$ is normalized to have unit mean and variance,
$\\{L_{(m,s),(n,t)}:(m,s)\leq(n,t)\in\mathbb{Z}\times\mathbb{R}\\}$
is the functional limit, as $k\rightarrow\infty$, of the properly interpolated
version of the process
$\bigl{\\{}\frac{1}{\sqrt{k}}G_{(m,\lfloor sk\rfloor),(n,\lfloor
tk\rfloor)}-(t-s)k:(m,s)\leq(n,t)\in\mathbb{Z}\times\mathbb{R}\bigr{\\}}.$
The most tractable case of discrete last-passage percolation is the case where
$Y_{0}$ has the exponential distribution with rate $1$. In this case, Busemann
functions exist and are indexed by a direction vector $\mathbf{u}$. They are
defined by
$U^{\mathbf{u}}(\mathbf{x},\mathbf{y}):=\lim_{n\rightarrow\infty}G_{(\mathbf{x},\mathbf{z}_{n})}-G_{(\mathbf{y},\mathbf{z}_{n})},$
where $\mathbf{z}_{n}$ satisfies $\mathbf{z}_{n}/n\rightarrow\mathbf{u}$ for a
fixed direction $\mathbf{u}$. For a given $\omega\in\Omega$,
$\mathbf{x}\in\mathbb{Z}^{2}$, and direction $\mathbf{u}$, a semi-infinite
geodesic $\mathbf{\gamma}$ is defined by the sequence
$\\{\mathbf{\gamma}_{k}^{\mathbf{u},\mathbf{x}}\\}_{k\in\mathbb{Z}_{\geq 0}}$.
At each step, a choice is made to move upward or to the right. First, set
$\mathbf{\gamma}^{\mathbf{u},\mathbf{x}}_{0}=\mathbf{x}$, and for $k\geq 0$,
(5.1)
$\mathbf{\gamma}_{k+1}^{\mathbf{u},\mathbf{x}}=\begin{cases}\mathbf{\gamma}_{k}^{\mathbf{u},\mathbf{x}}+\mathbf{e}_{1},&\text{if}\quad
U^{\mathbf{u}}\bigl{(}\mathbf{\gamma}_{k}^{\mathbf{u},\mathbf{x}},\mathbf{\gamma}_{k}^{\mathbf{u},\mathbf{x}}+\mathbf{e}_{1}\bigr{)}\leq
U^{\mathbf{u}}\bigl{(}\mathbf{\gamma}_{k}^{\mathbf{u},\mathbf{x}},\mathbf{\gamma}_{k}^{\mathbf{u},\mathbf{x}}+\mathbf{e}_{2}\bigr{)},\\\\[10.00002pt]
\mathbf{\gamma}_{k}^{\mathbf{u},\mathbf{x}}+\mathbf{e}_{2},&\text{if}\quad
U^{\mathbf{u}}\bigl{(}\mathbf{\gamma}_{k}^{\mathbf{u},\mathbf{x}},\mathbf{\gamma}_{k}^{\mathbf{u},\mathbf{x}}+\mathbf{e}_{2}\bigr{)}<U^{\mathbf{u}}\bigl{(}\mathbf{\gamma}_{k}^{\mathbf{u},\mathbf{x}},\mathbf{\gamma}_{k}^{\mathbf{u},\mathbf{x}}+\mathbf{e}_{1}\bigr{)}.\end{cases}$
In the case of exponential weights, this sequence is a semi-infinite geodesic
with direction $\mathbf{u}$. This construction is inherently discrete, so it
does not extend directly to the case of BLPP. However, this construction is
equivalent to taking a sequence of maximizers of the appropriate function,
analogous to Definition 4.1. The following is a discrete analogue of Theorem
3.5(vi) that holds in the case of exponential weights.
(5.2) $U^{\mathbf{u}}((m,r),(m,r+1))=\max_{\begin{subarray}{c}m\leq
k<\infty\\\
k\in\mathbb{Z}\end{subarray}}\Big{\\{}\sum_{i=m}^{k}Y_{(i,r)}+U^{\mathbf{u}}((k,r+1),(m,r+1))\Big{\\}}.$
Now, suppose that, starting at the point $(m,r)$, the semi-infinite geodesic
constructed in (5.1) makes an $\mathbf{e}_{2}$ step from $(k,r)$ to $(k,r+1)$
for some $k\geq m$. Then, we show that $k$ is maximal for (5.2). Indeed, by
(5.1), if the path $\mathbf{\gamma}$ makes an $\mathbf{e}_{2}$ step from
$(k,r)$ to $(k,r+1)$, then
$\displaystyle U^{\mathbf{u}}((i,r),(i+1,r))\leq
U^{\mathbf{u}}((i,r),(i,r+1))\qquad\text{ for }\qquad m\leq i\leq
k-1,\qquad\text{and }$ $\displaystyle
U^{\mathbf{u}}((k,r),(k+1,r))>U^{\mathbf{u}}((k,r),(k,r+1)).$
Using the identity $Y_{\mathbf{x}}=U^{\mathbf{u}}(\mathbf{x,x+e_{1}})\wedge
U^{\mathbf{u}}(\mathbf{x,x+e_{2}})$ and additivity of the Busemann functions,
$\displaystyle\quad\sum_{i=m}^{k}Y_{(i,r)}+U^{\mathbf{u}}((k,r+1),(m,r+1))$
$\displaystyle=\sum_{i=m}^{k-1}U^{\mathbf{u}}((i,r),(i+1,r))+U^{\mathbf{u}}((k,r),(k,r+1))+U^{\mathbf{u}}((k,r+1),(m,r+1))$
$\displaystyle=U^{\mathbf{u}}((m,r),(m,r+1)),$
so $k$ is indeed maximal in for the right hand-side of (5.2). The inductive
step follows in the same manner. Hence, we see that the construction of semi-
infinite geodesics in Definition 4.1 is a continuous analogue of the procedure
for the discrete case when viewed from the perspective of maximizers.
### 5.3. Connection to queuing theory
Fix $\theta>0$, and consider the almost surely unique $\theta$-directed semi-
infinite geodesic (Theorem 3.1(ii)) starting from $(0,0)$ and defined by the
sequence of jump times $\\{\tau_{r}\\}_{r\geq-1}$ with $\tau_{-1}=0$. For each
$r\geq 0$, associate a queuing station as follows. For $s<t$, let
$\frac{1}{\sqrt{\theta}}(t-s)-X_{r+1}^{\theta}(s,t)$ denote the service
available in the interval $(s,t]$ and let
$\frac{1}{\sqrt{\theta}}(t-s)-h_{r}^{\theta}(s,t)$ denote the arrivals to the
queue in the interval $(s,t]$. Here, the parameter $\frac{1}{\sqrt{\theta}}$
dictates the rate of service. See Appendix C for more details about the
queuing setup. By (3.11),
$h_{r+1}^{\theta}=\overleftarrow{D}(h_{r}^{\theta},X_{r+1})$, so as in the
proof of Theorem C.2, $\frac{1}{\sqrt{\theta}}(t-s)-h_{r+1}^{\theta}(s,t)$
gives the departures from the $r$th station in the interval $(s,t]$, and the
departures process from the $r$th station becomes the arrivals process for the
$r+1$st station. In other words, once a customer is served at the $r$th
station, they move into the queue at the $r+1$st station. By Equations (3.10)
and (3.11), for all $r\in\mathbb{Z}$ and $t\in\mathbb{R}$,
$v_{r+1}^{\theta}(t)=\sup_{t\leq
u<\infty}\\{B_{r}(t,u)-h_{r+1}^{\theta}(t,u)\\}=\sup_{-\infty<u\leq
t}\\{X_{r+1}^{\theta}(t,u)-h_{r}^{\theta}(t,u)\\}=\overleftarrow{Q}(h_{r}^{\theta},X_{r+1}^{\theta})(t).$
In queuing terms, $v_{r+1}^{\theta}(t)$ gives the length of the $r$th queue at
time $t$. As $\tau_{r}$ is the maximizer of $B_{r}(u)-h_{r+1}^{\theta}(u)$
over $u\in[\tau_{r-1},\infty)$, $\tau_{r}$ is the first time $u\geq\tau_{r-1}$
such that $v_{r+1}^{\theta}(u)=0$ (see Lemma 7.4). In queuing terms,
$\tau_{r}$ is the first time greater than or equal to $\tau_{r-1}$ at which
the queue is empty. Thus, the semi-infinite geodesic represents the movement
of a customer through the infinite series of queuing stations: the customer
starts at station $0$ at time $0$ and is served at the $r$th station at a time
no later than $\tau_{r}$.
## 6\. Construction and proofs for the Busemann process
The remainder of the paper is devoted to the proofs of the theorems. This
section constructs the global Busemann process, with the proof of Theorem 3.5
as the ultimate result. Section 7 proves all results about the semi-infinite
geodesics, culminating in the proof of Theorem 3.1.
### 6.1. Construction of Busemann functions for a fixed direction
###### Definition 6.1 (Definition of Busemann functions for fixed $\theta$
and a countable dense set of points).
For a fixed $\theta>0$, and
$\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{Q}$, let
$\Omega_{0}^{(\theta)}=\bigcap_{\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{Q}}\Omega_{\mathbf{x},\mathbf{y}}^{(\theta)},$
where the $\Omega_{\mathbf{x},\mathbf{y}}^{(\theta)}$ are the events of
Theorem 2.3. Then, $\mathbb{P}(\Omega_{0}^{(\theta)})=1$. On this event, for
$\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{Q}$, define
$\mathcal{B}^{\theta}(\mathbf{x},\mathbf{y})$ by (2.4). By definition in terms
of limits, it is clear that the Busemann functions are additive, that is, for
$\omega\in\Omega_{0}^{(\theta)}$ and
$\mathbf{x},\mathbf{y},\mathbf{z}\in\mathbb{Z}\times\mathbb{Q}$,
(6.1)
$\mathcal{B}^{\theta}(\mathbf{x},\mathbf{y})+\mathcal{B}^{\theta}(\mathbf{y},\mathbf{z})=\mathcal{B}^{\theta}(\mathbf{x},\mathbf{y}).$
Hence, the entire collection of Busemann functions is constructed from the
collection of horizontal and vertical Busemann functions
$\\{h_{m}^{\theta},v_{m}^{\theta}\\}_{m\in\mathbb{Z}}$ (Equations (2.6) and
(2.5)).
Set $Y(t)=-B_{0}(t)+\frac{1}{\sqrt{\theta}}t$. By Lemma C.5, as elements of
the space of functions $\mathbb{Q}\rightarrow\mathbb{R}$, equipped with the
standard product $\sigma-$algebra,
$\bigl{\\{}h_{0}^{\theta}(t):t\in\mathbb{Q}\bigr{\\}}\overset{d}{=}\bigl{\\{}Y(t):t\in\mathbb{Q}\bigr{\\}}\qquad\text{and}\qquad\bigl{\\{}v_{1}^{\theta}(t):t\in\mathbb{Q}\bigr{\\}}\overset{d}{=}\bigl{\\{}\overleftarrow{Q}(Y,B_{1})(t):t\in\mathbb{Q}\bigr{\\}}.$
For $\omega\in\Omega_{0}^{(\theta)}$, $m\in\mathbb{Z}$, and $t\in\mathbb{Q}$,
$h_{m}^{\theta}(t)=\lim_{n\rightarrow\infty}L_{(m,0),(n,n\theta)}(\mathbf{B})-L_{(m,t),(n,n\theta)}(\mathbf{B})=\lim_{n\rightarrow\infty}L_{(m,0),(n+m,(n+m)\theta)}(\mathbf{B})-L_{(m,t),(n+m,(n+m)\theta)}(\mathbf{B}).$
Since the environment $\mathbf{B}=\\{B_{m}\\}_{m\in\mathbb{Z}}$ is a field of
i.i.d. two-sided Brownian motions, the right-hand side implies that
$h_{m}^{\theta}(t)$ has the same distribution for all $m$. By this same
reasoning, for each $m\in\mathbb{Z}$,
$\\{h_{m}^{\theta}(t),v_{m+1}^{\theta}(t):t\in\mathbb{Q}\\}\overset{d}{=}\\{h_{0}^{\theta}(t),v_{1}^{\theta}(t):t\in\mathbb{Q}\\}.$
Since Brownian motion satisfies $\lim_{s\rightarrow\infty}\frac{B(s)}{s}=0$
almost surely, for all $m\in\mathbb{Z}$, the following limits also hold almost
surely:
$\lim_{\mathbb{Q}\ni
s\rightarrow\pm\infty}\bigl{[}B_{m}(s)-h_{m+1}^{\theta}(s)\bigr{]}=\mp\infty.$
Set
$\displaystyle\Omega^{(\theta)}=$
$\displaystyle\bigcap_{m,r\in\mathbb{Z}}\bigl{\\{}(s,t)\mapsto\mathcal{B}^{\theta}((m,s),(r,t))\text{
is uniformly continuous on all bounded subsets of
}\mathbb{Q}\times\mathbb{Q}\bigr{\\}}$ (6.2)
$\displaystyle\qquad\qquad\cap\bigcap_{m\in\mathbb{Z}}\\{\lim_{\mathbb{Q}\ni
s\rightarrow\pm\infty}\bigl{[}B_{m}(s)-h_{m+1}^{\theta}(s)\bigr{]}=\mp\infty\\}.$
By almost sure continuity of the functions $Y$ and
$\overleftarrow{Q}(Y,B_{1})$ (Lemma 3.4), $\mathbb{P}(\Omega^{(\theta)})=1$.
###### Definition 6.2 (Definition of Busemann functions for fixed $\theta$,
arbitrary points).
On the event $\Omega^{(\theta)}$, for arbitrary
$\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$, define
$\mathcal{B}^{\theta}(\mathbf{x},\mathbf{y})$ such that, for each
$m,r\in\mathbb{Z}$,
$(s,t)\mapsto\mathcal{B}^{\theta}((m,s),(r,t))$
is the unique continuous extension of this function from
$\mathbb{Q}\times\mathbb{Q}$ to $\mathbb{R}^{2}$.
The following lemma states properties for a fixed $\theta$, as a precursor to
the more general Theorems 3.5 and Theorem 3.7.
###### Lemma 6.3.
Let $\theta>0$. Then, the following hold.
1. (i)
On the event $\Omega^{(\theta)}$, for all
$\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$ and all sequences
$\\{t_{n}\\}$ satisfying $t_{n}/n\rightarrow\theta$,
$\mathcal{B}^{\theta}(\mathbf{x},\mathbf{y})=\lim_{n\rightarrow\infty}\bigl{[}L_{\mathbf{x},(n,t_{n})}-L_{\mathbf{y},(n,t_{n})}\bigr{]}.$
2. (ii)
On the event $\Omega^{(\theta)}$, whenever
$\mathbf{x},\mathbf{y},\mathbf{z}\in\mathbb{Z}\times\mathbb{R}$,
$\mathcal{B}^{\theta}(\mathbf{x},\mathbf{y})+\mathcal{B}^{\theta}(\mathbf{y},\mathbf{z})=\mathcal{B}^{\theta}(\mathbf{x},\mathbf{z}).$
3. (iii)
For $0<\gamma<\theta<\infty$, on the event
$\Omega^{(\theta)}\cap\Omega^{(\gamma)}$, for all $m\in\mathbb{Z}$ and
$s<t\in\mathbb{R}$,
$0\leq v_{m}^{\gamma}(s)\leq v_{m}^{\theta}(s),\text{ and }B_{m}(s,t)\leq
h_{m}^{\theta}(s,t)\leq h_{m}^{\gamma}(s,t).$
4. (iv)
On the event $\Omega^{(\theta)}$, for each $m\in\mathbb{Z}$,
$\lim_{s\rightarrow\pm\infty}B_{m}(s)-h_{m+1}^{\theta}(s)=\mp\infty.$
5. (v)
For every $m\in\mathbb{Z}$, the process $t\mapsto h_{m}^{\theta}(t)$ is a two-
sided Brownian motion with drift $\frac{1}{\sqrt{\theta}}$. For each
$m\in\mathbb{Z}$ and $t\in\mathbb{R}$,
$v_{m}^{\theta}(t)\sim\operatorname{Exp}\bigl{(}\frac{1}{\sqrt{\theta}}\bigr{)}$.
6. (vi)
For any $m\in\mathbb{Z}$, $\\{h_{r}^{\theta}\\}_{r>m}$ is independent of
$\\{B_{r}\\}_{r\leq m}$.
7. (vii)
There exists a full-probability event
$\Omega_{1}^{(\theta)}\subseteq\Omega^{(\theta)}$ on which the following hold
for all $m\in\mathbb{Z}$ and $t\in\mathbb{R}:$
$v_{m+1}^{\theta}(t)=Q(h_{m+1}^{\theta},B_{m})(t)\qquad\text{and}\qquad
h_{m}^{\theta}(t)=D(h_{m+1}^{\theta},B_{m})(t).$
###### Proof.
Part (i): On $\Omega^{(\theta)}$, let
$(m,s),(r,t)\in\mathbb{Z}\times\mathbb{R}$, and let $\\{t_{n}\\}$ be any
sequence with $t_{n}/n\rightarrow\theta$. Note that, whenever
$t_{1}<t_{2},m\in\mathbb{Z}$ and $\mathbf{y}\geq(m,t_{2})$,
$L_{(m,t_{1}),\mathbf{y}}\geq B_{m}(t_{1},t_{2})+L_{(m,t_{2}),\mathbf{y}}.$
Then, let $q_{1},q_{2}\in\mathbb{Q}$ be such that $q_{1}<s$ and $q_{2}>t$.
Then,
$L_{(m,s),(n,t_{n})}-L_{(r,t),(n,t_{n})}\leq
L_{(m,q_{1}),(n,t_{n})}-B_{m}(q_{1},s)-L_{(r,q_{2}),(n,t_{n})}-B_{r}(t,q_{2}),$
and therefore,
$\limsup_{n\rightarrow\infty}\bigl{[}L_{(m,s),(n,t_{n})}-L_{(r,t),(n,t_{n})}\bigr{]}\leq\mathcal{B}^{\theta}((m,q_{1}),(r,q_{2}))-B_{m}(q_{1},s)-B_{r}(t,q_{2}).$
Taking $q_{1}\nearrow s$ and $q_{2}\searrow t$ and using the continuity of
Brownian motion and the Busemann functions,
$\limsup_{n\rightarrow\infty}\bigl{[}L_{(m,s),(n,t_{n})}-L_{(r,t),(n,t_{n})}\bigr{]}\leq\mathcal{B}^{\theta}((m,s),(r,t)).$
A similar procedure gives the appropriate lower bound.
Part (ii): This follows from the additivity of the process on the countable
dense set (Equation (6.1)) and the construction of Definition 6.2 as the
unique continuous extension.
Part (iii): By Lemma A.4, for $s,t\in\mathbb{R}$ and all $n$ such that $n\geq
m$ and $n\gamma\geq s\vee t$,
$\displaystyle 0$ $\displaystyle\leq
L_{(m,s),(n,n\gamma)}-L_{(m+1,s),(n,n\gamma)}\leq
L_{(m,s),(n,n\theta)}-L_{(m+1,s),(n,n\theta)}$ $\displaystyle\text{and}\qquad
B_{m}(s,t)$ $\displaystyle\leq L_{(m,s),(n,n\theta)}-L_{(m,t),(n,n\theta)}\leq
L_{(m,s),(n,n\gamma)}-L_{(m,t),(n,n\gamma)}.$
The proof is complete by Part (i), taking limits as $n\rightarrow\infty$.
Part (iv): This follows from the definition of $\Omega^{(\theta)}$ and
continuity.
Part (v): Observe that $t\mapsto h_{m}^{\theta}(t)$ is a continuous process
with the correct finite-dimensional distributions by Lemma C.5, taking limits
when necessary. Part (i) and Theorem 2.3 guarantee that
$v_{m}^{\theta}(t)\sim\operatorname{Exp}\bigl{(}\frac{1}{\sqrt{\theta}}\bigr{)}$
for all $t\in\mathbb{R}$.
Part (vi): By Part (i),
$\\{h_{r}^{\theta}(t):r>m,t\in\mathbb{R}\\}=\bigl{\\{}\lim_{n\rightarrow\infty}\bigl{[}L_{(r,0),(n,n\theta)}-L_{(r,t),(n,n\theta)}\bigr{]}:r>m,t\in\mathbb{R}\bigr{\\}}.$
The right-hand side is a function of $\\{B_{r}\\}_{r>m}$, which is independent
of $\\{B_{r}\\}_{r\leq m}$.
Part (vii): We start with the first statement. We need to show that
(6.3) $v_{m+1}^{\theta}(t)=\sup_{t\leq
s<\infty}\\{B_{m}(t,s)-h_{m+1}^{\theta}(t,s)\\}.$
By Part (i),
(6.4) $\displaystyle\sup_{t\leq
s<\infty}\\{B_{m}(t,s)-h_{m+1}^{\theta}(t,s)\\}=\sup_{t\leq
s<\infty}\\{B_{m}(t,s)+\lim_{n\rightarrow\infty}\bigl{[}L_{(m+1,s),(n,n\theta)}-L_{(m+1,t),(n,n\theta)}\bigr{]}\\}.$
On the other hand,
(6.5)
$v_{m+1}^{\theta}(t)=\lim_{n\rightarrow\infty}\bigl{[}L_{(m,t),(n,n\theta)}-L_{(m+1,t),(n,n\theta)}\bigr{]}.$
For each $s\geq t$ and all $n$ sufficiently large so that $n\theta\geq s$ and
$n\geq m+1$,
$B_{m}(t,s)+L_{(m+1,s),(n,n\theta)}-L_{(m+1,t),(n,n\theta)}\leq
L_{(m,t),(n,n\theta)}-L_{(m+1,t),(n,n\theta)}.$
Taking limits as $n\rightarrow\infty$ and comparing (6.4) and (6.5)
establishes
$v_{m+1}^{\theta}(t)\geq\sup_{t\leq
s<\infty}\\{B_{m}(t,s)-h_{m+1}^{\theta}(t,s)\\}.$
Next, by Part (v),
$v_{m+1}^{\theta}(t)\sim\operatorname{Exp}(\frac{1}{\sqrt{\theta}})$. By Parts
(v) and (vi) and Lemma B.1,
$\displaystyle\sup_{t\leq
s<\infty}\\{B_{m}(t,s)-h_{m+1}^{\theta}(t,s)\\}\overset{d}{=}\sup_{0\leq
s<\infty}\bigl{\\{}\sqrt{2}B(s)-\frac{1}{\sqrt{\theta}}s\bigr{\\}}\overset{d}{=}v_{m+1}^{\theta}(t),$
where $B$ is a standard Brownian motion. Thus, since
$v_{m+1}^{\theta}(t)\geq\sup_{t\leq
s<\infty}\\{B_{m}(t,s)-h_{m+1}^{\theta}(t,s)\\}$, equality holds with
probability one for each fixed $t\in\mathbb{R}$. Let
$\Omega_{1}^{(\theta)}\subseteq\Omega^{(\theta)}$, be the event of full
probability on which (6.3) holds for all $t\in\mathbb{Q}$. Continuity of both
sides of (6.3) (Definition 6.2 and Lemma 3.4) extend the result to all
$t\in\mathbb{R}$ on $\Omega_{1}^{(\theta)}$. The equality for $h_{m}^{\theta}$
then follows by the definitions and additivity of the Busemann functions, as
shown below.
$\displaystyle D(h_{m+1}^{\theta},B_{m})(t)$
$\displaystyle=h_{m+1}^{\theta}(t)+Q(h_{m+1}^{\theta},B_{m})(0)-Q(h_{m+1}^{\theta},B_{m})(t)$
$\displaystyle=h_{m+1}^{\theta}(t)+v_{m+1}^{\theta}(0)-v_{m+1}^{\theta}(t)$
$\displaystyle=\;\mathcal{B}^{\theta}((m+1,0),(m+1,t))+\mathcal{B}^{\theta}((m,0),(m+1,0))-\mathcal{B}^{\theta}((m,t),(m+1,t))$
$\displaystyle=\mathcal{B}^{\theta}((m,0),(m,t))=h_{m}^{\theta}(t).\qed$
### 6.2. Global construction of the Busemann process
We now have the proper framework to define the global Busemann process of
Theorem 3.5. Fix a countable dense subset $D$ of $(0,\infty)$. Let
$\Omega_{0}=\bigcap_{\theta\in D}\Omega_{1}^{(\theta)},$
and on the event $\Omega_{0}$, define $\mathcal{B}^{\theta}$ for all
$\theta\in D$, as in Definition 6.2. Then, the conclusions of Lemma 6.3 hold
for all $\theta\in D$.
###### Lemma 6.4.
On $\Omega_{0}$, for each $\theta\in D,m\in\mathbb{Z}$, and $t\in\mathbb{R}$,
let $\tau_{(m,t)}^{\theta}$ denote the rightmost maximizer of
$B_{m}(s)-h_{m+1}^{\theta}(s)$ over $s\in[t,\infty)$. Such a maximizer exists
by continuity and Lemma 6.3(iv). Let $\Omega_{1}$ be the subset of
$\Omega_{0}$ on which, for each $\theta\in D,m,N\in\mathbb{Z}$, and
$t\in\mathbb{R}$, the following limits hold:
1. (i)
$\lim_{D\ni\gamma\rightarrow\theta}h_{m}^{\gamma}(N)=h_{m}^{\theta}(N)$
2. (ii)
$\lim_{D\ni\gamma\rightarrow\infty}h_{m}^{\gamma}(N)=B_{m}(N)$.
3. (iii)
$\lim_{D\ni\delta\searrow 0}\tau_{(m,t)}^{\delta}=t$.
4. (iv)
$\lim_{D\ni\gamma\rightarrow\infty}\tau_{(m,t)}^{\gamma}=\infty$.
Then, $\mathbb{P}(\Omega_{1})=1$.
###### Proof.
The almost sure uniqueness of maximizers follows from Lemma 6.3(vi) and the
$n=m$ case of Lemma 7.1. By Lemma 6.3(iii),
$h_{m}^{\gamma}(N)=h_{m}^{\gamma}(0,N)$ is monotone as
$D\ni\gamma\nearrow\theta$, and for $N>0$,
(6.6) $\lim_{D\ni\gamma\nearrow\theta}h_{m}^{\gamma}(N)\geq
h_{m}^{\theta}(N),$
For negative $N$, the inequality flips. By Theorem 2.3,
$h_{m}^{\gamma}(N)\sim\mathcal{N}\bigl{(}\frac{N}{\sqrt{\gamma}},|N|\bigr{)}$.
As $\gamma\nearrow\theta$, this converges in distribution to
$\mathcal{N}\bigl{(}\frac{N}{\sqrt{\theta}},|N|\bigr{)}$. Then, by (6.6),
$\lim_{D\ni\gamma\nearrow\theta}h_{m}^{\gamma}(N)=h_{m}^{\theta}(N)\text{ with
probability one}$. An analogous argument proves the almost sure convergence
for limits from the right and the convergence to $B_{m}(N)$.
Next, we show that, on $\Omega_{0}$, Parts (iii) and (iv) hold if and only if
they hold for all $t\in\mathbb{Q}$. Assume the statement holds for all
$t\in\mathbb{Q}$. By Lemmas 6.3(iii) and A.1, $\tau_{(m,t)}^{\delta}$ is
monotone as $\delta\searrow 0$, and so the limit exists. By definition of
$\tau_{(m,t)}^{\theta}$, we have the inequality
$\tau_{(m,s)}^{\theta}\leq\tau_{(m,t)}^{\theta}$ whenever $s<t$. Then, for any
$t\in\mathbb{R}$ and any $q_{1},q_{2}\in\mathbb{Q}$ with $q_{1}<t<q_{2}$,
$q_{1}=\lim_{D\ni\delta\searrow
0}\tau_{(m,q_{1})}^{\delta}\leq\lim_{D\ni\delta\searrow
0}\tau_{(m,t)}^{\delta}\leq\lim_{D\ni\delta\searrow
0}\tau_{(m,q_{2})}^{\delta}=q_{2}.$
Taking limits as $q_{1}\nearrow t$ and $q_{2}\searrow t$ shows the statement
for all $t\in\mathbb{R}$. The same argument can be applied to the limits as
$\gamma\rightarrow\infty$.
Lastly, we show that the limit in (iii) holds with probability one for fixed
$t\in\mathbb{R}$. By Lemma 6.3, Parts (v) and (vi),
$\bigl{\\{}B_{m}(s)-h_{m+1}^{\delta}(s):s\in\mathbb{R}\bigr{\\}}\overset{d}{=}\bigl{\\{}\sqrt{2}B(s)-\frac{s}{\sqrt{\delta}}:s\in\mathbb{R}\bigr{\\}},$
where $B$ is a standard, two-sided Brownian motion. Then, by Theorem B.3, for
$(m,t)\in\mathbb{Z}\times\mathbb{R}$ and $s\geq 0$,
$\mathbb{P}(\tau_{(m,t)}^{\delta}>s+t)=\left(2+\frac{t}{\delta}\right)\Phi\left(-\sqrt{\frac{t}{2\delta}}\right)-\sqrt{\frac{t}{\pi\delta}}\,e^{-\frac{t}{4\delta}}.$
Taking limits as $\delta\searrow 0$, it follows that $\tau_{(m,t)}^{\theta}$
converges weakly to the constant $t$. Since the limit exists almost surely by
monotonicity, the desired conclusion follows. A similar argument applies to
show $\lim_{\gamma\rightarrow\infty}\tau_{(m,t)}^{\gamma}=\infty$ with
probability one. ∎
On $\Omega_{1}$, we extend the definition of $\mathcal{B}^{\theta}$ to all
$\theta>0$. We proceed similarly as shown for the exponential corner growth
model in [Sep18]. For BLPP, there is an additional step that must be taken to
guarantee the convergence and continuity of Theorem 3.5, Parts (iii) and (iv).
###### Lemma 6.5.
On $\Omega_{1}$, for each $\theta\in D$, $m\in\mathbb{Z}$, and
$t\in\mathbb{R}$,
$\lim_{D\ni\gamma\rightarrow\theta}h_{m}^{\gamma}(t)=h_{m}^{\theta}(t)\qquad\text{and}\qquad\lim_{D\ni\gamma\rightarrow\theta}v_{m}^{\gamma}(t)=v_{m}^{\theta}(t).$
For each $m\in\mathbb{Z}$, the convergence is uniform in $t$ on compact
subsets of $\mathbb{R}$. Additionally, for each $m\in\mathbb{Z}$,
$\lim_{D\ni\gamma\rightarrow\infty}h_{m}^{\gamma}(t)=B_{m}(t),\qquad\text{and}\qquad\lim_{D\ni\delta\searrow
0}v_{m}^{\theta}(t)=0,$
uniformly in $t$ on compact subsets of $\mathbb{R}$.
###### Proof.
We first prove the statements for the $h_{m}$. We show that
$\lim_{D\ni\gamma\nearrow\theta}h_{m}^{\gamma}(t)$
exists and equals $h_{m}^{\theta}(t)$, uniformly in $t$ on compact subsets of
$\mathbb{R}$. The limits from the right (as well as the case of $B_{m}$ in
place of $h_{m}^{\theta}$) follow by analogous arguments.
By rearranging the inequality of Lemma 6.3(iii), for $0<\gamma<\theta<\infty$
and $a<b$ and any $t\in[a,b]$,
(6.7) $h_{m}^{\gamma}(a)-h_{m}^{\theta}(a)\leq
h_{m}^{\gamma}(t)-h_{m}^{\theta}(t)\leq h_{m}^{\gamma}(b)-h_{m}^{\theta}(b),$
and the inequality still holds if we replace $h_{m}^{\theta}$ with $B_{m}$.
Thus, on the event $\Omega_{1}$, $h_{m}^{\gamma}$ converges to
$h_{m}^{\theta}$, uniformly in $t$ on compact subsets of $\mathbb{R}$. Now, we
prove the convergence statements for $v_{m}$. By Lemma 6.3(vii), on
$\Omega_{0}$, for all $t\in\mathbb{R}$, and all $\theta\in D$.
(6.8) $v_{m}^{\theta}(t)=Q(h_{m}^{\theta},B_{m-1})(t)=\sup_{t\leq
s<\infty}\\{B_{m-1}(t,s)-h_{m}^{\theta}(t,s)\\}.$
Let $\omega\in\Omega_{1}$, and $t\in\mathbb{R}$. By the monotonicity of Lemma
6.3(iii) and Lemma A.1, since $\tau_{(m,t)}^{\theta+1}$ is a maximizer of
$B_{m-1}(s)-h_{m}^{\theta+1}(s)$ over $s\in[t,\infty)$, for all $\gamma\in D$
with $\gamma<\theta+1$,
$v_{m}^{\gamma}(t)=\sup_{t\leq
s<\infty}\\{B_{m-1}(t,s)-h_{m}^{\gamma}(t,s)\\}=\sup_{t\leq
s\leq\tau_{(m,t)}^{\theta+1}}\\{B_{m-1}(t,s)-h_{m}^{\gamma}(t,s)\\}.$
Since $h_{m}^{\gamma}$ converges uniformly on compact sets to
$h_{m}^{\theta}$, $v_{m}^{\gamma}(t)$ converges pointwise to
$v_{m}^{\theta}(t)$. By Lemma 6.3(iii), the convergence from both right and
left is monotone, so by the continuity of Definition 6.2 and Dini’s Theorem,
the convergence is uniform.
Lastly, for limits as $\delta\searrow 0$, we again apply Lemmas 6.3(iii) and
A.1 so that, for $\delta\in D$ with $\delta\leq 1$,
$\displaystyle v_{m}^{\delta}(t)$ $\displaystyle=\sup_{t\leq
s<\infty}\\{B_{m-1}(t,s)-h_{m}^{\delta}(t,s)\\}=\sup_{t\leq
s\leq\tau_{(m,t)}^{\delta}}\\{B_{m-1}(t,s)-h_{m}^{\delta}(t,s)\\}$
$\displaystyle\leq\sup_{t\leq
s\leq\tau_{(m,t)}^{\delta}}\\{B_{m-1}(t,s)-h_{m}^{1}(t,s)\\},$
and the right-hand side converges to $0$ as $\delta\searrow 0$ by the
continuity of $B_{m-1}-h_{m}^{1}$ and the convergence of
$\tau_{(m,t)}^{\delta}$ to $t$ given in the definition of $\Omega_{1}$. Dini’s
Theorem again strengthens the pointwise convergence to uniform convergence. ∎
By the additivity of Lemma 6.3(ii), for arbitrary
$(m,s),(r,t)\in\mathbb{Z}\times\mathbb{R}$ with $m\leq r$,
$\mathcal{B}^{\theta}((m,s),(r,t))=h_{m}^{\theta}(s,t)+\sum_{k=m}^{r-1}v_{k+1}(t).$
For $m>r$,
$\mathcal{B}^{\theta}((m,s),(r,t))=-\mathcal{B}^{\theta}((r,t),(m,s))$, so
$\mathcal{B}^{\theta}((m,s),(r,t))$ is still a sum of horizontal and vertical
increments. Then, by Lemma 6.5, on the event $\Omega_{1}$, for all $\theta\in
D$ and $\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$,
(6.9)
$\lim_{D\ni\gamma\rightarrow\theta}\mathcal{B}^{\gamma}(\mathbf{x},\mathbf{y})=\mathcal{B}^{\theta}(\mathbf{x},\mathbf{y}).$
###### Definition 6.6.
On the event $\Omega_{1}$, for an arbitrary $\theta>0$ and
$\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$, define the following.
$\displaystyle\mathcal{B}^{\theta-}(\mathbf{x},\mathbf{y})=\lim_{D\ni\gamma\nearrow\theta}\mathcal{B}^{\gamma}(\mathbf{x},\mathbf{y}),\qquad\text{and}\qquad\mathcal{B}^{\theta+}(\mathbf{x},\mathbf{y})=\lim_{D\ni\delta\searrow\theta}\mathcal{B}^{\delta}(\mathbf{x},\mathbf{y}).$
###### Remark 6.7.
By additivity of Lemma 6.3(ii) and the monotonicity of Lemma 6.3(iii), these
limits exist for all $\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$ and
$\theta>0$. By (6.9), on $\Omega_{1}$, for all $\theta\in D$ and
$\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$,
$\mathcal{B}^{\theta+}(\mathbf{x},\mathbf{y})=\mathcal{B}^{\theta-}(\mathbf{x},\mathbf{y})=\mathcal{B}^{\theta}(\mathbf{x},\mathbf{y})$.
###### Proof of Theorem 3.5.
Parts (i)–(ii): These follow by taking limits in Parts (ii) and (iii) of Lemma
6.3.
Parts (iii)–(iv): By Part (ii), it suffices to take limits along the countable
dense set $D$. Then, Lemma 6.5 establishes Parts (iii)(c) and (iii)(d). The
monotonicity of Part (ii) can be rearranged, just as in Equation (6.7), which
strengthens the pointwise convergence of the
$h_{m}^{\gamma{\scaleobj{0.8}{\boxempty}}}$ to uniform convergence on compact
sets. Thus, since $h_{m}^{\gamma}$ is continuous for $\gamma\in D$,
$h_{m}^{\theta{\scaleobj{0.8}{\boxempty}}}$ is also continuous for
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$. Now, for the convergence of the
$v_{m}^{\theta{\scaleobj{0.8}{\boxempty}}}$, recall that on the event
$\Omega_{1}$, for all $t\in\mathbb{R}$ and $\gamma\in D$,
$v_{m}^{\gamma}(t)=Q(h_{m}^{\gamma},B_{m-1})(t)=\sup_{t\leq
s<\infty}\bigl{\\{}B_{m-1}(t,s)-h_{m}^{\gamma}(t,s)\bigr{\\}}.$
By Part (ii) and Lemma A.1, this supremum may be restricted to a common
compact set for all $\gamma<\theta$. Then, since $h_{m}^{\gamma}$ converges
uniformly to $h_{m}^{\theta-}$ as $D\ni\gamma\nearrow\theta$,
$v_{m}^{\gamma}(t)$ converges pointwise to $Q(h_{m}^{\theta-},B_{m-1})(t)$ as
$\gamma\nearrow\theta$. But, by definition,
$v_{m}^{\theta-}(t)=\lim_{D\ni\gamma\nearrow\theta}v_{m}^{\gamma}(t)$, so
$v_{m}^{\theta-}(t)=Q(h_{m}^{\theta-},B_{m-1})(t)$, which is continuous by
continuity of $h_{m}^{\theta-}$ and $B_{m-1}$. Since the convergence is
monotone, Dini’s Theorem implies the convergence is uniform on compact sets.
The proof for $\delta\searrow\theta$ is analogous. Part (iv) follows by
continuity of the Busemann functions from Definition 6.2, the uniform
convergence of Part (iii), and the additivity in Part (i).
Part (v): This follows from Lemma 6.3(iv) and the monotonicity of Part (ii).
Part (vi): The equality for $v_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}}$ was
shown in the proof of Part (iii). The equality for $h_{m}^{\theta}$ follows by
the additivity of Part (i) and the same argument as in the proof of Lemma
6.3(vii).
Part (vii): This follows because
$\\{h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}:\theta>0,{\scaleobj{0.9}{\boxempty}}\in\\{+,-\\},r>m\\}$
is a function of $\\{B_{r}\\}_{r>m}$.
Part (viii): Fix $\theta>0$. We first show that, for each $\theta>0$ and
$\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{R}$, there exists an event
$\Omega_{\mathbf{x},\mathbf{y}}^{(\theta)}$ on which
(6.10)
$\mathcal{B}^{\theta-}(\mathbf{x},\mathbf{y})=\lim_{n\rightarrow\infty}\bigl{[}L_{\mathbf{x},(n,t_{n})}-L_{\mathbf{y},(n,t_{n})}\bigr{]}=\mathcal{B}^{\theta+}(\mathbf{x},\mathbf{y}),$
for all sequences $\\{t_{n}\\}$ with
$\lim_{n\rightarrow\infty}\frac{t_{n}}{n}=\theta$. By Theorem 2.3, for each
$\mathbf{x},\mathbf{y}$, and $\theta>0$, there exists an event of full
probability on which the limit in (6.10) exists and is independent of the
choice of sequence. Then, by additivity, it is sufficient to show that, for
each fixed $\theta>0,m\in\mathbb{Z}$, and $t\in\mathbb{R}$, with probability
one,
$\displaystyle
h_{m}^{\theta-}(t)=\lim_{n\rightarrow\infty}\bigl{[}L_{(m,0),(n,n\theta)}-L_{(m,t),(n,n\theta)}\bigr{]}=h_{m}^{\theta+}(t),\text{
and}$ $\displaystyle
v_{m}^{\theta-}(t)=\lim_{n\rightarrow\infty}\bigl{[}L_{(m-1,t),(n,n\theta)}-L_{(m,t),(n,n\theta)}\bigr{]}=v_{m}^{\theta+}(t).$
We show that
(6.11)
$h_{m}^{\theta-}(t)=\lim_{n\rightarrow\infty}\left[L_{(m,0),(n,n\theta)}-L_{(m,t),(n,n\theta)}\right]\qquad\text{a.s.},$
and the other statements follow by analogous arguments. By Theorem 2.3, the
right-hand side of (6.11) has distribution
$\mathcal{N}(\frac{t}{\sqrt{\theta}},|t|)$. By definition,
$h_{m}^{\theta-}(t)$ is the limit, as $\gamma\nearrow\theta$, of
$h_{m}^{\gamma}(t)$, and
$h_{m}^{\gamma}(t)\sim\mathcal{N}(\frac{t}{\sqrt{\gamma}},|t|)$, which
converges weakly to $\mathcal{N}(\frac{t}{\sqrt{\theta}},|t|)$. Using Theorem
6.3(i) and Lemma A.4, for each fixed $t>0$ and all $D\ni\gamma<\theta$,
$h_{m}^{\gamma}(t)=\lim_{n\rightarrow\infty}\left[L_{(m,0),(n,n\gamma)}-L_{(m,t),(n,n\gamma)}\right]\geq\lim_{n\rightarrow\infty}\left[L_{(m,0),(n,n\theta)}-L_{(m,t),(n,n\theta)}\right]\qquad\text{a.s.}$
The inequality flips for $t<0$. Hence, both sides in (6.11) have the same
distribution, while one dominates the other, so they are equal with
probability one.
Next, set
$\Omega^{(\theta)}=\bigcap_{\mathbf{x},\mathbf{y}\in\mathbb{Z}\times\mathbb{Q}}\Omega_{\mathbf{x},\mathbf{y}}^{(\theta)}.$
Then, using the continuity of Part (iv), the desired conclusion follows by the
same reasoning as in the proof of Lemma 6.3(i).
Part (ix) This follows from the construction of the Busemann functions as
limits of BLPP times and the shift invariance of BLPP. ∎
## 7\. Proofs of the results for semi-infinite geodesics
Throughout this section, $\Omega_{1}$ is the event defined in Lemma 6.4 and
referenced in Theorem 3.5.
### 7.1. Key lemmas
For two fields of Brownian motions, $\mathbf{B}$ and $\overline{\mathbf{B}}$,
$\lambda>0$, and an initial point $(m,s)\in\mathbb{Z}\times\mathbb{R}$, define
the point-to-line last passage time:
(7.1)
$\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}})=\sup\Bigl{\\{}\sum_{r=m}^{n}B_{r}(s_{k-1},s_{k})-\overline{B}_{n+1}(s_{n})-\lambda
s_{n}:\mathbf{s}_{m,n}\in\Pi_{(m,s),n}\Bigr{\\}},$
where $\Pi_{(m,s),n}$ denotes the set of sequences $s=s_{m-1}\leq
s_{m}\leq\cdots\leq s_{n}$. The following is due to Hammond [Ham19b]
###### Lemma 7.1 ([Ham19b], Lemma B.2).
Let $\mathbf{B}$ be a field of independent, two-sided Brownian motions and
$\overline{\mathbf{B}}$ be any other field of Brownian motions such that, for
each $n$, $\overline{B}_{n+1}$ is independent of $\\{B_{m}\\}_{m\leq n}$. Fix
an initial point $(m,s)$ and let $n\geq m$. Then, with probability one, the
quantity in (7.1) is finite, and there is a unique sequence
$\mathbf{s}_{m,n}\in\Pi_{(m,s),n}$ that is maximal for (7.1).
###### Remark 7.2.
Lemma 7.1 is stated slightly differently in [Ham19b]. In that paper, the
function $s\mapsto-\overline{B}_{n+1}(s)-\lambda s$ is replaced by an
arbitrary deterministic and measurable function
$h:\mathbb{R}\rightarrow\mathbb{R}\cup\\{-\infty\\}$ satisfying $h(t)>-\infty$
for some $t>s$ and $\limsup_{t\rightarrow\infty}h(t)/t<0$. The assumption that
$\\{\overline{B}_{n+1}\\}_{n+1}$ is independent of $\\{B_{m}\\}_{m\leq n}$
allows us to condition on $\overline{B}_{n+1}$ and obtain the desired result.
###### Lemma 7.3.
Let $\omega\in\Omega_{1}$, $(m,t)\in\mathbb{Z}\times\mathbb{R}$, $\theta>0$,
and ${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$. Then, the following hold.
1. (i)
Let $\\{\tau_{r}\\}_{r=m-1}^{\infty}$ be any sequence in
$\mathbf{T}_{(m,t)}^{\theta{\scaleobj{0.8}{\boxempty}}}$. Then, for each
$n\geq m$, the jump times $t=\tau_{m-1}\leq\tau_{m}\leq\cdots\leq\tau_{n}$ are
a maximizing sequence for
(7.2)
$\overline{L}_{(m,t),n}^{\lambda}(\mathbf{B},\overline{\mathbf{B}}):=\sup\Biggl{\\{}\sum_{r=m}^{n}B_{r}(s_{r-1},s_{r})-h_{n+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{n}):\mathbf{s}_{m,n}\in\Pi_{(m,t),n}\Biggr{\\}}.$
Here, $\overline{L}_{(m,t),n}^{\lambda}(\mathbf{B},\overline{\mathbf{B}})$ is
as in (7.1), where $\lambda=\frac{1}{\sqrt{\theta}}$,
$\mathbf{B}=\\{B_{r}\\}_{r\in\mathbb{Z}}$, and
$\overline{\mathbf{B}}=\\{h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}(t)-\frac{t}{\sqrt{\theta}}:t\in\mathbb{R}\\}_{r\in\mathbb{Z}}$.
2. (ii)
Conversely, for each $n\geq m$, whenever $t=t_{m-1}\leq t_{m}\leq\cdots\leq
t_{n}$ is a maximizing sequence for (7.2), there exists
$\\{\tau_{r}\\}_{r=m-1}^{\infty}\in\mathbf{T}_{(m,t)}^{\theta{\scaleobj{0.8}{\boxempty}}}$
such that $t_{r}=\tau_{r}$ for $m\leq r\leq n$.
3. (iii)
For each $n\geq m$, the sequences
$t=\tau_{(m,t),m-1}^{\theta{\scaleobj{0.8}{\boxempty}},L}\leq\cdots\leq\tau_{(m,t),n}^{\theta{\scaleobj{0.8}{\boxempty}},L}$
and
$t=\tau_{(m,t),m-1}^{\theta{\scaleobj{0.8}{\boxempty}},R}\leq\cdots\leq\tau_{(m,t),n}^{\theta{\scaleobj{0.8}{\boxempty}},R}$
are, respectively, the leftmost and rightmost maximizing sequences for (7.2).
###### Proof.
Part (i): Recall by (4.1), that $\tau_{m}\geq t$ is a maximizer of
$B_{m}(t,s)-h_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\qquad\text{ over
}\qquad s\in[t,\infty).$
Hence, the statement holds for $n=m$. Now, assume that the statement holds for
some $n\geq m$. Then, $t=\tau_{m-1}\leq\cdots\leq\tau_{n}$ satisfies
(7.3)
$\sum_{r=m}^{n}B_{r}(\tau_{r-1},\tau_{r})-h_{n+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{n})=\sup\Biggl{\\{}\sum_{r=m}^{n}B_{r}(s_{r-1},s_{r})-h_{n+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{n}):\mathbf{s}_{m,n}\in\Pi_{(m,t),n}\Biggr{\\}}.$
Using Theorem 3.5(vi) and rearranging the terms in the definition (3.4) of the
operator $D$,
(7.4)
$h_{n+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{n})=B_{n+1}(s_{n})+\sup_{0\leq
s<\infty}\\{B_{n+1}(s)-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\\}-\sup_{s_{n}\leq
s_{n+1}<\infty}\\{B_{n+1}(s_{n+1})-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{n+1})\\},$
for $s_{n}\in\mathbb{R}$. Specifically, since $\tau_{n+1}$ is a maximizer of
$B_{n+1}(s)-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)$ over
$s\in[\tau_{n},\infty)$,
(7.5)
$h_{n+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{n})=-B_{n+1}(\tau_{n},\tau_{n+1})+h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{n+1})+\sup_{0\leq
s<\infty}\\{B_{n+1}(s)-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\\}$
Substituting (7.4) and (7.5) into (7.3) and discarding the term $\sup_{0\leq
s<\infty}\\{B_{n+1}(s)-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\\}$ on
both sides
$\displaystyle\qquad\sum_{r=m}^{n+1}B_{r}(\tau_{r-1},\tau_{r})-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{n+1})$
$\displaystyle=$
$\displaystyle\sup\Biggl{\\{}\sum_{r=m}^{n}B_{r}(s_{r-1},s_{r})-B_{n+1}(s_{n})+\sup_{s_{n}\leq
s_{n+1}<\infty}\\{B_{n+1}(s_{n+1})-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{n+1})\\}:\mathbf{s}_{m,n}\in\Pi_{(m,t),n}\Biggr{\\}}$
$\displaystyle=$
$\displaystyle\sup\Biggl{\\{}\sum_{r=m}^{n+1}B_{r}(s_{r-1},s_{r})-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{n+1}):\mathbf{s}_{m,n+1}\in\Pi_{(m,t),n+1}\Biggr{\\}}.$
Part (ii): We prove this part by induction. First, note that in the case
$n=m$, maximizers of
$B_{n}(t,s_{m})-h_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{m})$ over
$s_{m}\in[t,\infty)$ are precisely those that are the first jump times of the
Busemann semi-infinite geodesics. Now, assume that the statement holds for
$n$. We show that if $t=t_{m-1}\leq\ldots\leq t_{n+1}$ is a maximizing
sequence for (7.3) (with $n$ replaced by $n+1$), then $t_{n+1}$ is a maximizer
of $B_{n+1}(u)-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(u)$ over
$u\in[t_{n},\infty)$, and $t=t_{m-1}\leq\cdots\leq t_{n}$ is a maximizing
sequence for (7.3).
With this procedure mapped out, observe that, as in the proof of Part (i),
$\displaystyle\sup\Biggl{\\{}\sum_{r=m}^{n+1}B_{r}(s_{r-1},s_{r})-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{n+1}):\mathbf{s}_{m,n+1}\in\Pi_{(m,t),n+1}\Biggr{\\}}$
$\displaystyle=$
$\displaystyle\sup\Biggl{\\{}\sum_{r=m}^{n}B_{r}(s_{r-1},s_{r})-B_{n+1}(s_{n})+\sup_{s_{n}\leq
s_{n+1}<\infty}\\{B_{n+1}(s_{n+1})-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{n+1})\\}:\mathbf{s}_{m,n}\in\Pi_{(m,t),n}\Biggr{\\}}$
Hence, any maximizing sequence $t=t_{m-1}\leq\cdots\leq t_{n+1}$ must satisfy
$B_{n+1}(t_{n+1})-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(t_{n+1})=\sup_{t_{n}\leq
u<\infty}\\{B_{n+1}(u)-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(u)\\}.$
Furthermore, $t=t_{m-1}\leq\cdots\leq t_{n}$ is a maximizing sequence for
$\sum_{r=m}^{n}B_{r}(s_{r-1},s_{r})-B_{n+1}(s_{n})+\sup_{s_{n}\leq
u<\infty}\\{B_{n+1}(u)-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(u)\\}$
over all sequences $\mathbf{s}_{m,n}\in\Pi_{(m,t),n}$ Subtracting off a
constant, $t=t_{m-1}\leq\cdots\leq t_{n}$ is also a maximizing sequence for
$\displaystyle\qquad\sum_{r=m}^{n}B_{r}(s_{r-1},s_{r})-\Big{(}B_{n+1}(s_{n})+\sup_{0\leq
u<\infty}\\{B_{n+1}(u)-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(u)\\}-\sup_{s_{n}\leq
u<\infty}\\{B_{n+1}(u)-h_{n+2}^{\theta{\scaleobj{0.8}{\boxempty}}}(u)\\}\Big{)}$
$\displaystyle=\sum_{r=m}^{n}B_{r}(s_{r-1},s_{r})-h_{n+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{n}),$
where the last line comes by (7.4). This completes the inductive step.
Part (iii): This follows by Parts (i) and (ii) since
$t=\tau_{(m,t),m-1}^{\theta{\scaleobj{0.8}{\boxempty}},L}\leq\tau_{(m,t),m}^{\theta{\scaleobj{0.8}{\boxempty}},L}\leq\cdots$
and
$t=\tau_{(m,t),m-1}^{\theta{\scaleobj{0.8}{\boxempty}},R}\leq\tau_{(m,t),m}^{\theta{\scaleobj{0.8}{\boxempty}},R}\leq\cdots$
are respectively the leftmost and rightmost sequences in
$\mathbf{T}_{(m,t)}^{\theta{\scaleobj{0.8}{\boxempty}}}$. ∎
###### Lemma 7.4.
Let $\omega\in\Omega_{1}$, $(m,t)\in\mathbb{Z}\times\mathbb{R},\theta>0$, and
${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$, and $\\{\tau_{r}\\}_{r\geq
m-1}\in\mathbf{T}_{(m,t)}^{\theta{\scaleobj{0.8}{\boxempty}}}$. Then, for all
$r\geq m$,
$v_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{r})=0,\qquad\text{and}\qquad
h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}(u,v)=B_{r}(u,v)\text{ for all
}u,v\in[\tau_{r-1},\tau_{r}].$
Furthermore, the following identities hold for $r\geq m$.
(7.6)
$\displaystyle\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},L}=\inf\bigl{\\{}u\geq\tau^{\theta{\scaleobj{0.8}{\boxempty}},L}_{(m,t),r-1}:v_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(u)=0\bigr{\\}}\text{
and }$ (7.7)
$\displaystyle\tau^{\theta{\scaleobj{0.8}{\boxempty}},R}_{(m,t),r}=\sup\bigl{\\{}u\geq\tau^{\theta{\scaleobj{0.8}{\boxempty}},R}_{(m,t),r-1}:h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau^{\theta{\scaleobj{0.8}{\boxempty}},R}_{(m,t),r-1},u)=B_{r}(\tau^{\theta{\scaleobj{0.8}{\boxempty}},R}_{(m,t),r-1},u)\bigr{\\}}$
More specifically, if
$u\geq\tau^{\theta{\scaleobj{0.8}{\boxempty}},R}_{(m,t),r-1}$, then
$h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau^{\theta{\scaleobj{0.8}{\boxempty}},R}_{(m,t),r-1},u)=B_{r}(\tau^{\theta{\scaleobj{0.8}{\boxempty}},R}_{(m,t),r-1},u)$
if and only if $u\leq\tau^{\theta{\scaleobj{0.8}{\boxempty}},R}_{(m,t),r}$.
###### Remark 7.5.
The first part of Lemma 7.4 says that, along any semi-infinite path in
$\mathbf{T}_{(m,t)}^{\theta{\scaleobj{0.8}{\boxempty}}}$, the Busemann process
agrees with the energy of the semi-infinite geodesic. That is, at every
vertical jump from $(r,\tau_{r})$ to $(r+1,\tau_{r})$, the vertical Busemann
function $v_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}$ equals zero. This is as
it should because, according to (2.1), a vertical step of a path does not
collect any energy from the environment. Along each horizontal step from
$(r,\tau_{r-1})$ to $(r,\tau_{r})$, the increment of the horizontal Busemann
function $h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}$ agrees with the increment
of the Brownian motion $B_{r}$. Equations (7.6) and (7.7) are more subtle. Let
$s_{r-1}$ be the time when a $\theta{\scaleobj{0.8}{\boxempty}}$ geodesic
jumps from level $r-1$ to level $r$. Equation (7.6) says that the leftmost
$\theta{\scaleobj{0.8}{\boxempty}}$ geodesic jumps from level $r$ to $r+1$ at
the first time $u\geq s_{r-1}$ such that
$v_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(u)=0$. Equation (7.7) says that
the rightmost geodesic jumps from level $r$ to $r+1$ at the last time $u\geq
s_{r-1}$ such that
$B_{r}(s_{r-1},u)=h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{r-1},u)$. At
all subsequent times $v>u$ the equality is lost, and we have
$B_{r}(s_{r-1},v)<h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{r-1},v)$.
###### Proof of Lemma 7.4.
By Theorem 3.5(vi) and the definition of $\tau_{r}$ as a maximizer,
$\displaystyle 0$ $\displaystyle\leq
v_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{r})=\sup_{\tau_{r}\leq
s<\infty}\\{B_{r}(\tau_{r},s)-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{r},s)\\}$
$\displaystyle=\sup_{\tau_{r}\leq
s<\infty}\\{B_{r}(s)-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\\}-[B_{r}(\tau_{r})-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{r})]$
$\displaystyle\leq\sup_{\tau_{r-1}\leq
s<\infty}\\{B_{r}(s)-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\\}-[B_{r}(\tau_{r})-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{r})]=0.$
To establish (7.6), assume, by way of contradiction, that for some
$\tau_{(m,t),r-1}^{\theta{\scaleobj{0.8}{\boxempty}},L}\leq
u<\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},L}$,
$0=v_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(u)=\sup_{u\leq
s<\infty}\\{B_{r}(u,s)-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(u,s)\\}.$
Then,
$B_{m}(u)-h_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(u)\geq
B_{m}(s)-h_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)$
for all $s\geq u$, and specifically for
$s=\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},L}$. This contradicts the
definition of $\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},L}$ as the
leftmost maximizer of
$B_{r}(s)-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)$ over
$s\in[\tau_{(m,t),r-1}^{\theta{\scaleobj{0.8}{\boxempty}},L},\infty)$.
By Theorem 3.5(vi),
(7.8) $h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}(u,v)=B_{r}(u,v)+\sup_{u\leq
s<\infty}\\{B_{r}(s)-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\\}-\sup_{v\leq
s<\infty}\\{B_{r}(s)-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\\},$
so, since $\tau_{r}$ maximizes
$B_{r}(s)-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)$ on
$[\tau_{r-1},\infty)$, the two supremum terms above are both equal to
$B_{r}(\tau_{r})-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{r})$
whenever $u,v\in[\tau_{r-1},\tau_{r}]$. Therefore,
$h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}(u,v)=B_{r}(u,v)$. Now we establish
(7.7). If, for some $r\geq m$, (7.7) fails, then by (7.8), for some
$u>\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},R}$,
$\sup_{\tau^{\theta{\scaleobj{0.8}{\boxempty}},R}_{(m,t),r-1}\leq
s<\infty}\\{B_{r}(s)-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\\}=\sup_{u\leq
s<\infty}\\{B_{r}(s)-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)\\}.$
Thus, all maximizers of
$B_{r}(s)-h_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s)$ over
$s\in[u,\infty)$ are also maximizers over the larger set
$[\tau_{(m,t),r-1}^{\theta{\scaleobj{0.8}{\boxempty}},R},\infty)$. But this is
a contradiction because $\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},R}$
is the rightmost maximizer over
$s\in[\tau_{(m,t),r-1}^{\theta{\scaleobj{0.8}{\boxempty}},R},\infty)$. ∎
Recall, by Remark 4.2, that on the event
$\Omega^{(\theta)}\subseteq\Omega_{1}$ of Theorem 3.5,
$\mathbf{T}_{\mathbf{x}}^{\theta+}=\mathbf{T}_{\mathbf{x}}^{\theta-}=\mathbf{T}_{\mathbf{x}}^{\theta}$
for all $\mathbf{x}\in\mathbb{Z}\times\mathbb{R}$.
###### Lemma 7.6.
Fix $\theta>0$ and $\mathbf{x}=(m,t)\in\mathbb{Z}\times\mathbb{R}$. Then there
exists an event
$\widetilde{\Omega}_{\mathbf{x}}^{(\theta)}\subseteq\Omega^{(\theta)}$, of
probability one, on which the set $\mathbf{T}_{\mathbf{x}}^{\theta}$ contains
exactly one sequence $\\{\tau_{r}\\}_{r\geq m-1}$. This sequence satisfies
$\lim_{n\rightarrow\infty}\frac{\tau_{n}}{n}=\theta.$
To prove this lemma, we need some machinery from [ARAS20]. For two fields of
Brownian motions $\mathbf{B}$ and $\overline{\mathbf{B}}$ and a subset
$A\subset\mathbb{R}$, define
(7.9)
$\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}};s_{n}\in
A)=\sup\Bigl{\\{}\sum_{r=m}^{n}B_{r}(s_{r-1},s_{r})-\overline{B}_{n+1}(s_{n})-\lambda
s_{n}:\mathbf{s}_{m,n}\in\Pi_{(m,s),n},\,s_{n}\in A\Bigr{\\}}.$
The only difference between this definition and (7.1) is the restriction on
the $s_{n}$.
###### Lemma 7.7 ([ARAS20], page 1949).
Let $\mathbf{B}$ be a field of independent, two-sided Brownian motions and
$\overline{\mathbf{B}}$ an arbitrary field of Brownian motions. Fix
$s\in\mathbb{R}$, $0\leq S\leq T\leq\infty$, and $m\in\mathbb{Z}$. Then, with
probability one,
$\lim_{n\rightarrow\infty}n^{-1}\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}};s+nS\leq
s_{n}\leq s+nT)=\sup_{S\leq t\leq T}\bigl{\\{}2\sqrt{t}-\lambda t\bigr{\\}}.$
###### Remark 7.8.
The appearance of the term $2\sqrt{t}$ in Lemma 7.7 comes from the shape
theorem for BLPP. Namely, the following limit holds with probability one.
$\lim_{n\rightarrow\infty}n^{-1}L_{(0,0),(n,nt)}=2\sqrt{t}.$
This almost sure convergence was first proved in [HMO02] (see also [MO07] for
an alternative proof). If $\overline{\mathbf{B}}_{n+1}$ is independent of
$\mathbf{B}_{m},\ldots,\mathbf{B}_{n}$, then the quantity
$\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}})$ is
distributed as the sum of $n-m+1$ independent exponential random variables
with rate $\lambda$ and has the interpretation as the sum of vertical
increments in the increment-stationary BLPP model. (see Appendix C of the
present paper and Section 4 of [OY01]). Then, in the case $S=0$ and
$T=\infty$, Lemma 7.7 degenerates to an application of the law of large
numbers, namely
$\lim_{n\rightarrow\infty}n^{-1}\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}})=\lambda^{-1}.$
###### Lemma 7.9 ([ARAS20], Lemma 4.7).
Let $\mathbf{B}$ and $\overline{\mathbf{B}}$ satisfy the same conditions of
Lemma 7.7. Fix $\lambda>0$, $s\in\mathbb{R}$, and $m\in\mathbb{Z}$. If
$0<\theta<\lambda^{-2}$, then there exist a nonrandom
$\varepsilon=\varepsilon(\lambda,\theta)>0$ such that, with probability one,
for all sufficiently large $n$,
$\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}};s_{n}\leq
s+n\theta)+n\varepsilon<\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}}).$
Similarly, if $\theta>\lambda^{-2}$, there exists a nonrandom
$\varepsilon=\varepsilon(\lambda,\theta)>0$ such that, with probability one,
for all sufficiently large $n$,
$\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}};s_{n}\geq
s+n\theta)+n\varepsilon<\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}})$
Lemma 7.9 is slightly stronger than the result stated in [ARAS20], so we
include a proof.
###### Proof.
We prove the first statement, and the second follows analogously. The unique
maximum of $2\sqrt{t}-\lambda t$ for $t\in[0,\infty)$ is achieved at
$t=\lambda^{-2}$. Then, by Lemma 7.7 and the assumption $\theta<\lambda^{-2}$,
$\lim_{n\rightarrow\infty}n^{-1}\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}};s_{n}\leq
s+n\theta)=\sup_{0\leq t\leq\theta}\bigl{\\{}2\sqrt{t}-\lambda
t\bigr{\\}}<\sup_{0\leq t<\infty}\bigl{\\{}2\sqrt{t}-\lambda
t\bigr{\\}}=\lim_{n\rightarrow\infty}n^{-1}\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}}).$
Hence, for sufficiently large $n$,
$\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}};s_{n}\leq
s+n\theta)+n\varepsilon<\overline{L}^{\lambda}_{(m,s),n}(\mathbf{B},\overline{\mathbf{B}}),$
where
$\varepsilon=\frac{1}{2}\Bigl{(}\;{\sup_{0\leq
t<\infty}\bigl{\\{}2\sqrt{t}-\lambda t\bigr{\\}}-\sup_{0\leq
t\leq\theta}\bigl{\\{}2\sqrt{t}-\lambda t\bigr{\\}}}\Bigr{)}.\qed$
###### Proof of Lemma 7.6.
By Lemma 7.3(i), for a sequence $\\{\tau_{r}\\}_{r\geq
m-1}\in\mathbf{T}_{(m,t)}^{\theta}$, the jump times
$t=\tau_{m-1}\leq\cdots\leq\tau_{n}$ are a maximizing sequence for the point-
to-line last passage time (7.2). By Theorem 3.5(vii), $h_{n+1}^{\theta}$ and
$\\{B_{r}\\}_{r\leq n}$ are independent for each $n$. This allows us to apply
the almost sure uniqueness of point-to-line last-passage maximizers in Lemma
7.1. Let $\widetilde{\Omega}_{\mathbf{x}}^{(\theta)}$ be the event on which
these maximizers are unique and on which $\tau_{n}/n\to\theta$ for the (now)
almost surely unique sequence $\\{\tau_{r}\\}_{r\geq
m-1}\in\mathbf{T}_{(m,t)}^{\theta}$. It remains to show that
$\mathbb{P}\bigl{(}\lim_{n\rightarrow\infty}\frac{\tau_{n}}{n}=\theta\bigr{)}=1.$
For $\gamma<\theta=\lambda^{-2}$, Lemma 7.9 guarantees that, with probability
one, for all sufficiently large $n$,
$\overline{L}^{\lambda}_{(m,t),n}(\mathbf{B},\overline{\mathbf{B}};s_{n}\leq
s+n\gamma)<\overline{L}^{\lambda}_{(m,t),n}(\mathbf{B},\overline{\mathbf{B}}).$
Therefore, by Lemma 7.3(i), $\tau_{n}>s+n\gamma$ for all sufficiently large
$n$. Thus, for $\gamma<\theta$,
$\mathbb{P}\bigl{(}\liminf_{n\rightarrow\infty}\frac{\tau_{n}}{n}\geq\gamma\bigr{)}=1,\qquad\text{so
by taking
}\gamma\nearrow\theta,\qquad\mathbb{P}\bigl{(}\liminf_{n\rightarrow\infty}\frac{\tau_{n}}{n}\geq\theta\bigr{)}=1.$
A symmetric argument using the second statement of Lemma 7.9 shows that
$\mathbb{P}\bigl{(}\limsup_{n\rightarrow\infty}\frac{\tau_{n}}{n}\leq\theta\bigr{)}=1.\qed$
### 7.2. Proofs of Theorems 4.3–4.7:
We now begin to prove the theorems of Section 4.2. First, we define the event
$\Omega_{2}$ used in the theorems. Let
$\widetilde{\Omega}_{\mathbf{x}}^{(\gamma)}$ be the events of Lemma 7.6. For
the countable dense set $D\subseteq(0,\infty)$ of Section 6, set
(7.10) $\Omega_{2}:=\bigcap_{\gamma\in
D,\mathbf{x}\in\mathbb{Z}\times\mathbb{Q}}\widetilde{\Omega}_{\mathbf{x}}^{(\gamma)}$
Then, $\mathbb{P}(\Omega_{2})=1$ and
$\Omega_{2}\subseteq\bigcap_{\theta\in\mathbb{Q}_{>0}}\Omega^{(\theta)}\subseteq\Omega_{1}$.
###### Proof of Theorem 4.3.
Part (i): On the event $\Omega_{2}$, let $(m,t)\in\mathbb{Z}\times\mathbb{R}$,
$\theta>0$ and ${\scaleobj{0.9}{\boxempty}}\in\\{+,-\\}$, and let
$t=\tau_{m-1}\leq\tau_{m}\leq\cdots$ be a sequence in
$\mathbf{T}_{(m,t)}^{\theta{\scaleobj{0.8}{\boxempty}}}$. Let $\Gamma$ be the
associated path. By Lemma 7.4,
(7.11)
$v_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{r})=0,\qquad\text{and}\qquad
h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}(s,t)=B_{r}(s,t)\text{ for
}\tau_{r-1}\leq s\leq t\leq\tau_{r}.$
We take $\mathbf{x}=\mathbf{y}=(m,t)$ and $\mathbf{z}=(n,\tau_{n})$. The case
for general $\mathbf{y}\leq\mathbf{z}$ along the path $\Gamma$ follows by the
same argument. By (7.11) and additivity of Busemann functions (Theorem
3.5(i)), the energy of path $\Gamma$ between $\mathbf{x}$ and $\mathbf{y}$ is
given by
$\displaystyle\sum_{r=m}^{n}\bigl{(}B_{r}(\tau_{r-1},\tau_{r})\bigr{)}=\sum_{r=m}^{n-1}\bigl{(}h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{r-1},\tau_{r})+v_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{r})\bigr{)}+h_{n}^{\theta{\scaleobj{0.8}{\boxempty}}}(\tau_{n-1},\tau_{n})=\mathcal{B}^{\theta{\scaleobj{0.8}{\boxempty}}}((m,t),(n,\tau_{n})).$
Let jump times $t=s_{m-1}\leq s_{m}\leq s_{m+1}\leq\cdots\leq s_{n}=\tau_{n}$
define any other path between $(m,t)$ and $(n,\tau_{n})$. Then, by Theorem
3.5, Parts (i) and (ii), the energy of this path is
(7.12)
$\displaystyle\sum_{r=m}^{n}B_{r}(s_{r-1},s_{r})\leq\sum_{r=m}^{n-1}\bigl{(}h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{r-1},s_{r})+v_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{r})\bigr{)}+h_{n}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{n-1},\tau_{n})=\mathcal{B}^{\theta{\scaleobj{0.8}{\boxempty}}}((m,t),(n,\tau_{n})).$
Part (ii): By Theorem 3.5(ii), equality holds in Equation (7.12) only if
$B_{r}(s_{r-1},s_{r})=h_{r}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{r-1},s_{r})$
for $m\leq r\leq n$ and $v_{r+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(s_{r})=0$
for $m\leq r\leq n-1$. Then, the statement follows by the Equations (7.6) and
(7.7) of Lemma 7.4.
Part (iii)(iii)(a): The key is Theorem 3.5(ii). We show that, for $r\geq m$,
$\tau_{r}^{-}:=\tau_{(m,t),r}^{\theta-,L}\leq\tau_{(m,t),r}^{\theta+,L}=:\tau_{r}^{+},$
and all other inequalities of the statement follow by the same procedure. By
definition, $\tau_{m-1}^{-}=\tau_{m-1}^{+}=t$. Inductively, assume that
$\tau_{r}^{-}\leq\tau_{r}^{+}$ for some $r\geq m$. We use the notation
$L\arg\sup$ to denote leftmost maximizer. Then,
$\displaystyle\tau_{r+1}^{-}$ $\displaystyle=L\arg\sup_{\tau_{r^{-}}\leq
s<\infty}\\{B_{r}(s)-h_{r+1}^{\theta-,L}(s)\\}\leq L\arg\sup_{\tau_{r}^{+}\leq
s<\infty}\\{B_{r}(s)-h_{r+1}^{\theta-,L}(s)\\}$ $\displaystyle\leq
L\arg\sup_{\tau_{r}^{+}\leq
s<\infty}\\{B_{r}(s)-h_{r+1}^{\theta+,L}(s)\\}=\tau_{r+1}^{+}.$
The first inequality above holds because $\tau_{r}^{-}\leq\tau_{r}^{+}$. The
second inequality is an application of Lemma A.1, using the fact that
$h_{m}^{\theta-}(s,t)\leq h_{m}^{\theta+}(s,t)$ for all $s\leq t$ (Theorem
3.5(ii)).
Part (iii)(iii)(b): Let $s<t$. We show that
$\tau_{s,r}:=\tau_{(m,s),r}^{\theta{\scaleobj{0.8}{\boxempty}},L}\leq\tau_{(m,t),r}^{\theta{\scaleobj{0.8}{\boxempty}},L}=:\tau_{t,r},$
and the statement with ‘$L$’ replaced by ‘$R$’ has an analogous proof. Again,
the base case of $r=m-1$ follows by definition. Assume the inequality holds
for some $r\geq m-1$. Then,
$\tau_{s,r+1}=L\arg\sup_{\tau_{s,r}\leq
u<\infty}\\{B_{m}(u)-h_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(u)\\}\leq
L\arg\sup_{\tau_{t,r}\leq
u<\infty}\\{B_{m}(u)-h_{m+1}^{\theta{\scaleobj{0.8}{\boxempty}}}(u)\\}=\tau_{t,r+1}.$
Part (iii)(iii)(c): The proof of this item is postponed until the very end of
Section 7. This item is not used in any subsequent proofs.
Part (iv)(iv)(a): The monotonicity of Part (iii)(iii)(a) ensures that the
limits exist and that
(7.13)
$\tau_{r}:=\lim_{\gamma\nearrow\theta}\tau_{(m,t),r}^{\gamma{\scaleobj{0.8}{\boxempty}},L}\leq\tau_{(m,t),r}^{\theta-,L}.$
We prove equality in the above expression, and the other statements follow
analogously. By Lemma 7.3(i), for any $n\geq m$,
$t=\tau_{(m,t),m-1}^{\gamma{\scaleobj{0.8}{\boxempty}},L}\leq\ldots\leq\tau_{(m,t),n}^{\gamma{\scaleobj{0.8}{\boxempty}},L}$
is a maximizing sequence for
$\overline{L}_{(m,t),n}^{\frac{1}{\sqrt{\gamma}}}(\mathbf{B},\overline{\mathbf{B}}):=\sup\Bigl{\\{}\sum_{r=m}^{n}B_{r}(s_{r-1},s_{r})-h_{n+1}^{\gamma{\scaleobj{0.8}{\boxempty}}}(s_{n}):\mathbf{s}_{m,n}\in\Pi_{(m,t),n}\Bigr{\\}}.$
Using the monotonicity of Part (iii)(iv)(a) again, for all $\gamma\leq\theta$,
the supremum may be restricted to the compact subset of $\Pi_{(m,t),n}$ such
that $s_{n}\leq\tau_{(m,t),n}^{(\theta+1)-,L}$. By Theorem 3.5(iii)(iii)(a),
as $\gamma\nearrow\theta$, $h_{n+1}^{\gamma{\scaleobj{0.8}{\boxempty}}}(s)$
converges to $h_{n+1}^{\theta-}(s)$, uniformly over
$s\in[t,\tau_{(m,t),n}^{(\theta+1)-,L}]$. Then, by Lemma A.2,
$t=\tau_{m-1}\leq\ldots\leq\tau_{n}$ is a maximizing sequence for
$\overline{L}_{(m,t),n}^{\frac{1}{\sqrt{\theta}}}(\mathbf{B},\overline{\mathbf{B}}):=\sup\Bigl{\\{}\sum_{r=m}^{n}B_{r}(s_{r-1},s_{r})-h_{n+1}^{\theta-}(s_{n}):\mathbf{s}_{m,n}\in\Pi_{(m,t),n}\Bigr{\\}}.$
Part (iii) of Lemma 7.3 then implies the inequality (7.13) must be an
equality. The statement for limits as $\delta\searrow\theta$ follows by the
same reasoning.
Part (iv)(iv)(b): The proof of this item is postponed until the very end of
Section 7. This item is not used in any subsequent proofs.
Part (iv)(iv)(c): This follows the same proof as that of Part (iv)(iv)(a),
replacing the use of Part (iii)(iii)(a) with Part (iii)(iii)(b) and replacing
the use of Lemma A.2 with Lemma A.3.
Part (v): Let $\omega\in\Omega_{2}$, $\theta>0$,
$(m,t)\in\mathbb{Z}\times\mathbb{Q}$, $(\tau_{r})_{r\geq
m-1}\in\mathbf{T}_{(m,t)}^{\theta{\scaleobj{0.8}{\boxempty}}}$,
$\varepsilon>0$, and let $\gamma,\delta\in D$ be such that
$\theta-\varepsilon<\gamma<\theta<\delta<\theta+\varepsilon$. Part
(iii)(iii)(a) and Lemma 7.6 imply that
$\limsup_{n\rightarrow\infty}\frac{\tau_{n}}{n}\leq\limsup_{n\rightarrow\infty}\frac{\tau_{(m,t),n}^{\theta+,R}}{n}\leq\limsup_{n\rightarrow\infty}\frac{\tau_{(m,t),n}^{\delta+,R}}{n}=\delta<\theta+\varepsilon.$
By a similar argument, on $\Omega_{2}$,
$\liminf_{n\rightarrow\infty}\frac{\tau_{n}}{n}\geq\theta-\varepsilon.$
An analogous application of Part (iii)(iii)(b) extends Theorem 4.3(v) to all
$\mathbf{x}\in\mathbb{Z}\times\mathbb{R}$, on the event $\Omega_{2}$. ∎
###### Proof of Theorem 4.5.
Part (i): Let $\omega\in\Omega_{2}$,
$\theta>0,(m,t)\in\mathbb{Z}\times\mathbb{R}$, and $\\{t_{n,r}\\}_{n\geq
m,m-1\leq r\leq n}$ satisfy the given assumptions. Let
$0<\gamma<\theta<\delta$ be arbitrary. Then, by assumption that
$t_{n}/n\rightarrow\theta$ and Theorem 4.3(v), there exists $N\in\mathbb{Z}$
such that for all $n\geq N$,
(7.14) $\tau_{(m,t),n}^{\gamma-,L}<t_{n}<\tau_{(m,t),n}^{\delta+,R}.$
By Theorem 4.3(ii), the sequence
$t=\tau_{(m,t),m-1}^{\gamma-,L}\leq\tau_{(m,t),m}^{\gamma-,L}\leq\cdots\leq\tau_{(m,t),n}^{\gamma-,L}$
defines the leftmost geodesic between $(m,t)$ and
$(n,\tau_{(m,t),n}^{\gamma-,L})$, and
$t=\tau_{(m,t),m-1}^{\delta+,R}\leq\tau_{(m,t),m}^{\delta+,R}\leq\cdots\tau_{(m,t),n}^{\delta+,R}$
defines the rightmost geodesic between $(m,t)$ and
$(n,\tau_{(m,t),n}^{\delta+,R})$. Let $t=t_{m-1}^{L}\leq t_{m}^{L}\leq\cdots
|
# Federated Multi-Agent DRL for Radio Resource Management in Industrial 6G
in-X subnetworks ††thanks: This work is partly supported by the HORIZON-JU-
SNS-2022-STREAM-B-01-02 CENTRIC project (grant agreement No. 101096379) and
the HORIZON-JU-SNS-2022-STREAM-B-01-03 6G-SHINE project (grant agreement No.
101095738).
Bjarke Madsen and Ramoni Adeogun
Department of Electronic Systems, Aalborg University, Denmark
<EMAIL_ADDRESS>
###### Abstract
Recently, 6G in-X subnetworks have been proposed as low-power short-range
radio cells to support localized extreme wireless connectivity inside entities
such as industrial robots, vehicles, and the human body. Deployment of in-X
subnetworks within these entities may result in rapid changes in interference
levels and thus, varying link quality. This paper investigates distributed
dynamic channel allocation to mitigate inter-subnetwork interference in dense
in-factory deployments of 6G in-X subnetworks. This paper introduces two new
techniques, Federated Multi-Agent Double Deep Q-Network (F-MADDQN) and
Federated Multi-Agent Deep Proximal Policy Optimization (F-MADPPO), for
channel allocation in 6G in-X subnetworks. These techniques are based on a
client-to-server horizontal federated reinforcement learning framework. The
methods require sharing only local model weights with a centralized gNB for
federated aggregation thereby preserving local data privacy and security.
Simulations were conducted using a practical indoor factory environment
proposed by 5G-ACIA and 3GPP models for in-factory environments. The results
showed that the proposed methods achieved slightly better performance than
baseline schemes with significantly reduced signaling overhead compared to the
baseline solutions. The schemes also showed better robustness and
generalization ability to changes in deployment densities and propagation
parameters.
## I Introduction
Wireless communication has evolved significantly, with advancements in 5G
technology enabling faster and more reliable communication. However, the need
for more efficient and reliable networks has led to the development of 6G
technology, which promises ultra-reliable communication, higher data rates,
and lower latency. Wireless communication has limitless applications,
including virtual and augmented reality in entertainment, education,
healthcare, and manufacturing, personalized healthcare solutions, and real-
time environmental monitoring. To achieve these benefits, new communication
technologies must be developed, such as 6G short-range low-power in-X
subnetworks [1], which can support demanding requirements inside entities like
robots, production modules, vehicles, and human-body scenarios. Ensuring data
privacy and security is crucial for advanced wireless communication networks,
especially for sensitive personal or environmental data. Techniques for
resource allocation must intelligently guarantee privacy and security while
adapting to changing wireless environments.
Related work: Interference management solutions for wireless systems have been
proposed in the literature, see e.g., [2]. There has also been a significant
amount of work targetting 6G in-X subnetworks within the last few years. These
solutions can be categorized into heuristics, machine learning (ML) methods,
and reinforcement learning (RL) techniques. Heuristic methods generate near-
optimal solutions based on simple rules or algorithms, providing good results
with low computational complexity. In [3], three heuristic-based algorithms
are presented: $\epsilon$-greedy channel selection, nearest neighbor conflict
avoidance, and minimum Signal to Interference plus Noise Ratio (SINR)
guarantee. Centralized Graph Coloring (CGC) is presented as a benchmark
algorithm, but it is impractical due to its high signaling and computation
overhead. A distributed interference-aware dynamic channel selection is
presented in [4]. In [5], channel selection is approached based on centralized
selective graph constructions. ML-based solutions may offer a higher quality
of service compared to heuristics by leveraging historical data and adapting
to evolving network conditions.
In [6], a Deep Neural Network (DNN) was successfully trained in offline
simulations using CGC with mobile subnetworks. Subsequently, this DNN was
deployed for real-time distributed channel selection. A novel solution for
centralized power control is presented in [7], where decisions are based on
positioning information using Graph Neural Network (GNN).
In [8], joint allocation of channel and transmit power based on a distributed
multi-objective optimization problem is addressed, and an approach of
Q-learning for multiple agents based on limited sensing information is
proposed. The GA-Net framework presented in [9] is a distributed framework
Multi-Agent Reinforcement Learning (MARL) resource management, based on GNN.
Contributions: This paper investigates the potential of federated MARL
techniques for efficient, privacy-preserving channel allocation in 6G in-X
subnetworks. The authors present their first study on federated MARL for radio
resource management (RRM) in short-range low-power 6G in-X subnetworks,
presenting the first contribution on FRL for RRM in 6G in-X subnetworks. They
propose two novel federated MARL solutions - F-MADDQN and F-MADPPO - for
dynamic channel allocation in 6G in-X subnetworks. These techniques enable
privacy-preserving collaborative training of a Double Deep Q-Network (DDQN)
policy and proximal Policy Optimization (PPO) agent for dynamic channel
selection by multiple subnetworks in the presence of an umbrella network. The
training phase requires only offline connectivity to the umbrella network. The
paper also presents performance evaluation in industrial factory environment
simulations based on a realistic factory floor plan from 5G-ACIA and 3GPP
channel models for in-factory environments.
## II System Model and Problem Formulation
System Model: We consider downlink transmissions in a wireless network
comprising $N$ independent and mobile subnetworks, each responsible for
serving one or more devices, such as sensors and actuators. Each subnetwork
contains an Access Point (AP) which is responsible for managing transmissions
with its associated devices. We denote the collection of subnetworks as
$\mathcal{N}={1,\cdots,N}$, and the set of devices in the $n^{\text{th}}$
subnetwork as $\mathcal{M}_{n}={1,\cdots,M_{n}}$. It is assumed that each
subnetwork’s AP is equipped with a local Resource Manager (RM), which utilizes
data acquired from wireless environment sensing or received from its devices.
Within each mobile subnetwork, wireless transmission occurs over one of $K$
shared orthogonal channels. Given the limited availability of resources, the
number of bands is typically much less than the number of co-existing
subnetworks. We assume that transmissions within each subnetwork are
orthogonal, hence there is no intra-subnetwork interference. In practical
systems, it may be impossible to make transmissions completely orthogonal due
to limitations on the bandwidth of the channel allocated to the subnetwork
leading to intra-subnetwork interference. Consideration of the effects of such
interference is however left for future work.
The received SINR, denoted $\gamma_{n,m}$, on a link between the
$n^{\text{th}}$ AP and $m^{\text{th}}$ device can be expressed
$\gamma_{n,m}(t)=\dfrac{g_{n,m}(t)}{\sum_{i\in\mathcal{I}_{k}}g_{n,i}(t)+\sigma^{2}},$
(1)
where $g_{n,m}(t)$ is the received power on the link between the $n$th AP and
device $m$, $\mathcal{I}_{k}$ denotes the APs or devices transmitting on
channel $k$ and $\sigma^{2}$ denotes the noise power and is defined as
$\sigma^{2}=10^{(-174+\text{NF}+10\log_{10}(B_{k}))/10}$, where NF denotes the
noise figure and $B_{k}$ is the bandwidth of the channel. We consider
practical 6G in-X subnetworks featuring short packets (in the order of tens of
bytes) transmission, making the infinite block length assumption and the
Shannon approximation used in most of the existing studies unrealistic. To
compute the capacity, we instead use the finite block-length approximation as
[10]
$r_{n,m}=B_{k}\left[\log_{2}\left(1+\gamma_{n,m}(t)\right)-\sqrt{\frac{V_{n,m}(t)}{l}}Q^{-1}(\epsilon)\log
e\right],$
where $Q$ is the complementary Gaussian cumulative distribution function based
on the code-word decoding error probability $\epsilon$, $\log e\approx 0.434$
is a constant constraint of the loss, and $V$ is the channel dispersion
defined as [10].
$V_{n,m}(t)=1-\dfrac{1}{\left(1+\gamma_{n,m}(t)\right)^{2}}.$ (2)
When the length of the code-word block $l$ tends toward infinity, the achieved
rate approaches the classical Shannon’s approximation.
Problem Formulation: We focus on a resource allocation problem that involves
the distributed selection of channels to maximize the achieved sum rate while
ensuring that devices within each subnetwork achieve a specified minimum rate.
This optimization problem can be mathematically expressed as
${\text{P}}:\left\\{{{{\max}_{\left\\{{{{\mathbf{c}}^{t}}}\right\\}}}\sum\limits_{m=1}^{M}{{r_{n,m}}}\left({{{\mathbf{c}}^{t}}}\right)}\right\\}_{n=1}^{N}{\text{
st: }}\>{r_{n,m}}\geq{r_{{\text{target }}}},\>\forall n,m$ (3)
Here, $\mathcal{K}=\left[1,\cdots,K\right];k\in\\{1,2,\cdots,K\\}$ represents
the vector of channel indices selected by all subnetworks at time t.
$r_{\text{target}}$ denotes the target minimum rate, assumed to be the same
for all subnetworks. The problem described in (3) involves the joint
optimization of $N$ conflicting non-convex objective functions, making it a
challenging problem to solve. In this paper, federated MARL methods are
proposed as a solution to this problem.
Figure 1: Illustration of the training procedures for (a) multi-agent DDQN and
(b) multi-agent PPO and the federated learning concept (c).
## III MARL FOR RRM IN SUBNETWORKS
### III-A MARL Algorithms
Resource selection in scenarios with multiple in-X subnetworks is cast as a
decentralized partially observable Markov decision process (DEC-POMDP). A DEC-
POMDP is formally represented as
$\left(N,\mathcal{S},\\{\mathcal{A}^{n}\\}_{n=1}^{N},\mathcal{P},\\{\mathcal{R}^{n}\\}_{n=1}^{N}\right)$.
The set of all possible states for all agents is defined as the state space
$\mathcal{S}=\mathcal{S}_{1}\times\cdots\times\mathcal{S}_{N}$, and the joint
action space containing all possible actions for the $n^{\text{th}}$ agent
$\mathcal{A}^{n}$ is denoted
$\boldsymbol{\mathcal{A}}=\mathcal{A}^{1}\times\cdots\times\mathcal{A}^{N}$.
The reward signal for the $n^{\text{th}}$ agent is denoted
$\mathcal{R}^{n}:\mathcal{S}\times\boldsymbol{\mathcal{A}}\times\mathcal{S}$
and the transition probability from state $s\in\mathcal{S}$ to state
$s^{\prime}\in\mathcal{S}$ by joint action
$\boldsymbol{a}\in\boldsymbol{\mathcal{A}}$ is denoted
$\mathcal{P}=\mathcal{S}\times\boldsymbol{\mathcal{A}}$.
#### III-A1 MADDQN
Multi-Agent Double Deep Q Network (MADDQN) [11, 12] extends the Double Deep Q
Network (DDQN) algorithm to enable decentralized learning among multiple
agents. MADDQN allows each agent to have its own local Q-network and learn
from its individual experiences. To train the Q-network, each agent interacts
with the environment and collects experiences by executing actions according
to its current policy. The experiences, including the state, action, reward,
and next state, are stored in either a shared replay buffer, $\mathcal{B}_{c}$
in case of centralized learning or individual reply buffers,
$\left\\{\mathcal{B}_{n}\right\\}_{n=1}^{N}$ in distributed learning.
To stabilize the learning process, MADDQN utilizes target networks. These
networks are copies of the local Q-networks and their weights are periodically
updated by polyak averaging, which involves updating the target network
weights with a fraction of the local network weights.
The Q-networks are optimized using mini-batches of experiences sampled from
the replay buffer with the goal of learning decentralized policies that
maximize the expected cumulative rewards for each agent. For each agent, the
Q-network is trained to minimize the mean squared Bellman error, i.e., the
squared difference between the estimated Q-value and the target Q-value. The
loss function is defined as [12]
$\mathcal{L}(\theta)=\frac{1}{N}\sum_{i=1}^{N}(y_{i}-Q(s_{i},a_{i};\theta_{i}))^{2}$
(4)
where $N$ is the mini-batch size, $s_{i}$ is the state, $a_{i}$ is the action,
and $Q(s_{i},a_{i};\theta_{i})$ is the estimated Q-value from the local
Q-network. The variable, $y_{i}$ is the target Q-value which is calculated by
using the target network to select the best action for the next state and then
evaluating it with the local Q-network. The target Q-value is typically
defined as
$y_{i}=r_{i}+\gamma
Q^{\prime}\left(s^{\prime},\max_{a}Q(s^{\prime},a^{\prime};\theta_{i});\theta_{i}^{\prime}\right)$
(5)
where $r_{i}$ denotes the reward received by agent $i$, $s^{\prime}$ is the
next state, $a^{\prime}$ is the action selected by the target Q-network, and
$Q^{\prime}$ is the target Q-network.
During training or execution, each agent selects actions using its local
Q-network based on the observed state using the well-known $\epsilon$-greedy
strategy. The actions can be either random with probability, $\epsilon$, to
encourage exploration, or greedy with probability, $1-\epsilon$, to exploit
the learned policies.
#### III-A2 MAPPO
Multi-agent Proximal Policy Optimization (MAPPO) is a state-of-the-art
algorithm designed to address challenges in multi-agent environments. It is an
extension of the Proximal Policy Optimization algorithm, originally proposed
for single-agent reinforcement learning. PPO uses an actor-critic
architecture, where each agent has its own policy and value function, allowing
them to interact and learn decentralizedly. The core idea is to update the
policy while ensuring deviation from the previous policy remains within a
certain range. This is achieved by maximizing a surrogate objective function,
which approximates the expected improvement of the policy using the current
state-action distribution and the ratio between new and old policies. This
balances exploration and exploitation, enabling stable learning in multi-agent
environments.
In MAPPO, each agent interacts with the environment by executing actions
according to its policy. Trajectories of state-action pairs are collected and
stored in a replay buffer for efficient sampling. The collected trajectories
are used to estimate the value function for each agent, typically using a
separate critic network to predict cumulative rewards from each state. The
surrogate objective is computed based on the collected trajectories and the
current policy, quantifying the expected improvement and determining the
policy update direction. Denoting the ratio of the new policy,
$\pi_{\theta^{\prime}}^{\prime}$ and the old policy, $\pi(\theta)$ as
$r(\theta)$, the surrogate objective is defined as [13]
$\mathcal{L}(\theta)=\mathbb{E}\left[\min\left(r(\theta^{\prime})A,\operatorname{clip}\left(r(\theta^{\prime},1-\eta,1+\eta\right)A\right)\right],$
(6)
where $A$ denotes the advantage function which provides a measure of the
advantage of taking a specific action under the current policy compared to the
estimated value function. The parameter $\eta$ is a hyperparameter that
controls the range of the policy deviation.
The policy is updated by optimizing the surrogate objective function using
stochastic gradient descent (SGD) or a similar optimization technique. The
objective is to maximize the surrogate objective with respect to the policy
parameters, $\theta$. This is typically done by taking multiple gradient steps
on the objective function, considering the trajectories collected from the
replay buffer.
The update of the parameters, $\theta$ of the policy can be represented by
$\displaystyle\theta^{\prime}$
$\displaystyle=\arg\max_{\theta^{\prime}}\left\\{\frac{1}{N}\sum_{m=1}^{N}\min\left(r(\theta^{\prime})A_{m},C(\theta^{\prime})A_{m}\right)\right\\},$
(7) $\displaystyle C(\theta^{\prime})$
$\displaystyle=\operatorname{clip}\left(r(\theta^{\prime},1-\eta,1+\eta\right),$
(8)
where $N$ denotes the number of trajectories and $A_{m}$ is the advantage of
the $m^{\text{th}}$ state-action pair.
The value function is updated to improve its estimation accuracy. This is
typically done by minimizing the Mean-Squared Error (MSE) between the
predicted value function and the actual cumulative rewards. The value function
update is performed using SGD or another optimization algorithm, and the
target is to minimize the loss function:
$\mathcal{L}(\theta_{v})=\frac{1}{N}\sum_{i=1}^{N}(V(s_{i};\theta_{v})-G_{i})^{2},$
(9)
where $\theta_{v}$ represents the value function parameters, $s_{i}$ is the
$i^{\text{th}}$ state, $V_{i}$ is the predicted value function for state
$s_{i}$, and $G_{i}$ is the actual cumulative reward obtained from the
$i^{\text{th}}$ trajectory.
The steps above are repeated iteratively to improve the policy and value
function over multiple iterations. The process continues until a desired level
of performance or convergence is achieved.
### III-B Proposed Methods
We propose two algorithms combining the MARL techniques, i.e., MADDQN and
MAPPO described above with federated learning [14, 15] to solve the resource
optimization problem in (3). The methods are referred to as Federated MADDQN
(F-MADDQN) and Federated MAPPO (F-MAPPO). The algorithms require the
definition of the environment, action space, state space, and reward function.
The considered environment is the wireless network with $N$ in-X subnetworks.
The other components of the proposed solutions are described below.
#### III-B1 Action Space
As stated earlier, we assume that the available frequency band is divided into
a set of $K$ equally sized channels. This leads to a $K$-dimensional action
space for each subnetwork. Denoting the $k^{\text{th}}$ channel as $b_{k}$,
the action space for the $n^{\text{th}}$ subnetwork, $\mathcal{A}^{n}$ is
defined as
$\mathcal{A}^{n}=\left\\{b_{1},\ldots,b_{K}\right\\}\quad\forall
n\in\mathcal{N}$ (10)
#### III-B2 Observation Space
The local observation of subnetwork $n$ at time $t$ is defined as
$\mathbf{S}_{n}(t)=\left[f(\mathbf{s}_{n}^{1}(t)),\cdots,f(\mathbf{s}_{n}^{K}(t))\right]^{T},$
(11)
where
$\mathbf{s}_{n}^{k}(t)=[\mathrm{SIR}_{n1}^{k}(t),\cdots,\mathrm{SIR}_{nM}^{k}(t)]\in\mathbb{R}^{M\times
1}$ denotes the vector of SIRs measured at all devices in the $n^{\text{th}}$
subnetwork on channel $k$ and $f$ is the Observation Reduction Function (ORF)
which is applied to the measurements to obtain the state to be used as input
to an RL agent. The choice of ORF affects the input to the RL model and hence
plays a significant role in the learning process. Considering the minimum rate
constraint in (3), it appears reasonable to optimize for the worst case, i.e.,
allocate channels based on the minimum SIR over each channel. However, the
combinatorial nature of the optimization problem makes it difficult to
conclude whether this is the optimal choice. We therefore propose to use two
categories of ORF defined as
$f(\mathbf{x}):\begin{cases}\mathbf{x}\mapsto\mathbf{x}\in\mathbb{R}^{M\times
1}&\text{Case I:Full state}\\\ \mathbf{x}\mapsto
f_{A}(\mathbf{x})\in\mathbb{R}^{1}&\text{Case II:Reduced state}\end{cases},$
(12)
where $f_{A}$ is the mean, max, or median aggregation functions.
#### III-B3 Reward Signal
To guide the learning of the agents toward achieving this objective in (3), we
define the reward function for the $n$ subnetwork as
$R_{n}=\lambda_{1}\sum_{m=1}^{M}r_{n,m}-\lambda_{2}\sum_{m=1}^{M}\mathds{1}(r_{n,m})(r_{\mathrm{min}}-r_{n,m})$
(13)
where $\lambda_{i};i=1,2$ are scaling factors that are selected to create a
balance between rate maximization and satisfaction of the minimum rate
constraint. The function, $\mathds{1}(r)$, is a binary indicator function with
a value equal to unity if and only if $r\geq r_{\mathrm{min}}$.
#### III-B4 Policy
The main component of any reinforcement learning solution is the policy that
defines how state measurements from the environment are mapped into actions.
Representation of this state-action mapping and the associated optimization
framework determines the policy. In this paper, we propose two methods viz:
F-MADDQN and F-MAPPO. As stated earlier F-MADDQN and F-MAPPO combines
federated learning with MADDQN and MAPPO, respectively. In both algorithms,
the policy is modeled using a DNN.
#### III-B5 Training
In both F-MADDQN and F-MAPPO, the agents are trained using federated learning
(FL) [16] as shown in Figure 1c. Consider $N$ agents
$\left\\{\mathcal{F}_{i}\right\\}_{i=1}^{N}$, each storing their respective
data-sets in experience replay buffer, $\mathcal{B}_{i}$ from which a set of
model parameters $\mathbf{\theta}_{i}$ is learned. Denoting the loss function
for the $i^{\text{th}}$ agent as $\mathcal{L}_{i}(\mathbf{\theta}_{i})$. the
common global model loss can be defined as $\mathcal{L}_{g}(\mathbf{\theta})$
[16].
$\mathcal{L}_{g}(\mathbf{\theta})=\sum_{i=1}^{N}\eta_{i}\mathcal{L}_{i}(\mathbf{\theta}),$
(14)
where $|\cdot|$ denotes the size of the set and $\eta_{i}>0$ is the relative
impact of each agent. Typically, the term $\eta$ is constrained to
$\sum_{i=1}^{N}\eta_{i}=1$. To allocate equal priority to all agents, we use
$\eta_{i}=1/N;\forall i\in\mathcal{N}$. The goal is to find the optimal
parameters $w^{*}$, which minimizes the global loss function [16].
$\mathbf{\theta}^{*}=\arg\min_{\mathbf{\theta}}\mathcal{L}_{g}(\mathbf{\theta})$
(15)
One solution for (15) would be a gradient-descent approach, known as the
federated averaging algorithm. Each agent uses its local data to perform a
number of steps in gradient descent on current model parameters
$\bar{\mathbf{\theta}}(t)$. This gradient descent step is defined as
$\mathbf{\theta}_{i}(t+1)=\bar{\mathbf{\theta}}(t)-\gamma\nabla\mathcal{L}_{i}(\bar{\mathbf{\theta}}_{i}(t))\quad\forall
i\in\mathcal{N},$ (16)
where $\gamma>0$ is the learning rate, and $\nabla f(\mathbf{\theta})$ for any
scalar expression $f(\mathbf{\theta})$ denotes the vector of partial
derivatives with respect to the components of the parameters
$\mathbf{\theta}$. During federated training, the global model is updated as
$\mathbf{\theta}_{g}(t+1)=\sum_{i=1}^{N}\dfrac{1}{N}\mathbf{\theta}_{i}(t+1)$
(17)
Once updated, the global weight is sent back to the agents at specified
intervals referred to as _aggregation interval_ , $\tau_{\mathrm{agg}}$.
TABLE I: Simulation parameters.
Parameter | | Value
---|---|---
Total factory area | $\mathcal{R}$ | $180~{}\mathrm{m}\times 80~{}\mathrm{m}$
Clutter type table | | Sparse
Number of subnetworks | $N$ | 20
Timestep | $t$ | 0.005 s
Number of episode | | 2000
Number of steps per episode | $T$ | 200
Subnetwork separation distance | $d_{\text{min}}$ | 1 m
Subnetwork radius | $d_{r}$ | 1 m
Subnetwork velocity | | 3 m/s
Transmit power | $p_{n}(t)$ | -10 dBm
Number of frequency channels | $K$ | 4
Carrier frequency | $f_{c}$ | 6 GHz
Bandwidth per subnetwork | $BW$ | 10 MHz
Noise figure | NF | 10 dB
Shadowing decorrelation distance | $d_{\delta}$ | 10 m
Max action switch delay | $\tau_{\text{max}}$ | 10
Figure 2: Illustration of the (a) Deep Q-Network (DQN) architecture for MADDQN
and (b) Deep PPO architecture for MAPPO which are used for the simulations.
## IV PERFORMANCE EVALUATION
### IV-A Simulation Settings
We consider an indoor industrial factory scenario inspired by existing
production facilities of manufacturing companies as identified by the industry
initiative, 5G alliance for connected industries, and automation [17]. Such a
layout is implemented as a $180~{}\mathrm{m}\times 80~{}\mathrm{m}$ hall
containing several separate areas for production, assembly, storage, and human
work zones. Multiple in-robot subnetworks can be deployed with the task of
transporting materials or tools around the facility. The alleys separating
laboring areas are 5 m wide taking up $\sim 1600$ m2 of the factory area and
are outlined as two-lane roads in a right-handed traffic setting. In the
deployment, the in-robot subnetworks are separated with a minimum distance of
$d_{\text{min}}=1$ m and move with a speed of 3 m/s. The in-robot subnetworks
are modeled by a circular coverage area with a radius of $d_{r}=1$ m, making
it possible for robots to pass each other in the alleys. Collisions are
avoided by prioritizing robots with the shortest distances to a common
intersection, slowing down any other robot that is within minimum separation
distance.
Except where otherwise stated, we consider a total bandwidth of
$100~{}\mathrm{MHz}$ which is partitioned into $K=4$ channels. Transmission
within each subnetwork is then performed over a single channel at each time
instant. The channel gain is calculated as
$g_{n,m}=h_{n,m}\times\sqrt{10^{(PL_{n,m}+X^{\mathrm{SD}}_{n,m})/10}}$, where
$h_{n,m}$ denotes the small scale channel gain which is modeled as a
temporally correlated Rayleigh random variable, $PL_{n,m}$ denotes the path
loss and $X^{\mathrm{SD}}_{n,m}$ is the shadow fading. We simulate the
pathloss using the 3GPP Indoor Factory (InF) channel model for the Dense-
clutter Low-antenna (InF-DL) and the Sparse-clutter Low-antenna (InF-SL)
scenarios. The shadow fading $X^{\mathrm{SD}}_{n,m}$ is generated using the
spatially correlated shadowing model used in [18]. Other simulation parameters
are given in Table I. We consider different values of the aggregation
intervals, i.e., $T^{\text{Agg}}=[128,256,512,1024]$ for F-MAPPO and F-MADDQN.
The minimum required data rate is set as, $r_{\rm{min}}=11~{}\mathrm{bps/Hz}$.
### IV-B Benchmarks
We benchmark the performance of the proposed F-MADDQN and F-MAPPO algorithms
with the following methods.
1\. Centralized Graph Coloring (CGC): The CGC algorithm utilizes improper
coloring to assign colors equivalent to channels for all subnetworks [19]. At
each timestep $t$, the pair-wise interference power relationships among
subnetworks, $\boldsymbol{I}(t)\in\mathbb{R}^{N\times N}$, are collected, and
mapped to a mutual coupling graph $G_{t}$. Each vertex corresponds to a
subnetwork, and edges are created by connecting each vertex to its $K-1$
nearest neighbors, where the weights of edges are equivalent to the
interference power between the connected subnetworks.
2\. Greedy channel selection: At each switching instant, each subnetwork
selects the channel with the highest measured SINR from the previous time
step.
3\. Random channel selection: At the beginning of every episode, a random
channel is allocated to each subnetwork.
In addition, the federated learning solutions are also compared with previous
solutions based on centralized as well as distributed training. We denote the
centralized (distributed) MADDQN and MAPPO as C-MADDQN (D-MADDQN) and C-MAPPO
(D-MAPPO), respectively.
(a) MADDQN.
(b) MAPPO.
Figure 3: Averaged reward versus number of episodes. Figure 4: CDF of achieved
rate. Figure 5: Sensitivity evaluation. Figure 6: Rate versus number of
subnetworks.
### IV-C Training and Convergence
We consider the network architecture in Figure 2a and Figure 2b for F-MADDQN
and F-MAPPO, respectively. The networks are trained using the federated
learning procedure illustrated in Figure 1c. Figure 3a shows the averaged
reward signal achieved at each training episode by the MADDQN-based methods.
At convergence approximately after 1500 episodes, the MARL methods approach
the distributed greedy selection baseline. At convergence, the F-MADDQN
methods with greater aggregation intervals achieve a marginally better
performance than the centralized and distributed baselines, indicating a
potential advantage of the federated learning framework. The figure also shows
that too frequent aggregation of DQN weights due to a low value of aggregation
interval, $\tau_{\mathrm{agg}}$ can result in a comparatively slower
convergence rate. For example, while F-MADDQN with $\tau_{\mathrm{agg}}\geq
256$ converged at $\approx 1500$ episodes, the algorithm failed to converge
after $2000$ episodes with $\tau_{\mathrm{agg}}=128$.
We evaluate the convergence performance of F-MAPPO with the in Figure 3b.At
convergence, achieved approximately after $\approx 750$ episodes, the F-MAPPO
achieves similar performance to the benchmark for all values of the
aggregation interval, $\tau_{\mathrm{agg}}$. Compared to the methods based on
MADDQN, the MAPPO-based algorithms give a reduction of about $50\%$ in the
number of episodes required for convergence. A plausible explanation for this
is the effect of the decay rate of the $\epsilon$-greedy parameter of MADDQN
and the iterative training over multiple epochs at each training step of
MAPPO.
### IV-D Performance Comparison
The trained models are deployed in each in-robot subnetwork for distributed
channel allocation, and the performance is compared with random, greedy, CGC,
and centralized and distributed training frameworks. To evaluate the trained
models, the aggregation interval is set to $T^{\text{Agg}}=512$. Figure 4
shows the CDF of the achieved rate per device. The proposed F-MADDQN performs
marginally better than D-MADDQN and C-MADDQN below the $30^{\text{th}}$
percentile. Furthermore, the performance of F-MADDQN is also better than that
of the distributed greedy scheme. F-MAPPO similarly achieves marginally better
performance than the distributed baseline and approaches the centralized
baseline below the $30^{\text{th}}$ percentile. As expected, the centralized
graph coloring baseline shows significant performance superiority to all other
schemes which are based on distributed execution. This difference is expected
since CGC exploits the global information about the scenario, unlike the
MADDQN, MAPPO, and greedy schemes which rely solely on local measurements of
only the aggregate interference power. It should, however, be noted that the
CGC method has significantly higher sensing, signalling, and computational
complexity. Such high complexity makes CGC impractical for dense deployments
of subnetworks.
### IV-E Sensitivity and Robustness Evaluation
We evaluate the ability of the proposed methods to generalize to changes in
the deployment density and the wireless environment parameters.
#### IV-E1 Robustness to changes in deployment density
The models trained in the factory with $N=20$ subnetworks are deployed in
scenarios with different numbers of subnetworks. We vary the number of
subnetworks in the test scenarios between $N=10$ and $N=50$ and evaluate the
average per-device rate. As shown in Figure 6, when the number of subnetworks
increases, an overall decrease in performance with similar proportions across
all methods is observed. While the methods based on MADDQN appear to be robust
to the changes in deployment density for all values of $N$, there seems to be
a degradation in the performance of the methods based on MAPPO. This shows
that the proposed F-MADDQN is more robust to changes in deployment density
than the F-MAPPO.
#### IV-E2 Sensitivity to changes in environment parameters
To study the ability of the proposed methods to generalize to environments
with different parameters than those used for the training, we train the
models in a factory environment with sparse clutter with an average clutter
element size of $d_{\text{clutter}}=10$ m and a density of
$r_{\text{clutter}}=20\%$. The models are then introduced to environments with
different types of clutter. We consider sparse and dense clutter cases with
varying clutter density as defined in the 3GPP model for in-factory
environment [20]. Figure 5 shows the minimum, maximum, and average values of
the achieved rate per device for the different cases using the proposed
schemes and the baseline algorithms. The figure shows that the RL-based
methods can maintain their performance in comparison to the centralized and
distributed baselines. Furthermore, similar to the observation from the
sensitivity test with varying numbers of subnetworks, the MADDQN-based
solutions appear to be slightly more robust to changes in the environment and
maintain performance close to the greedy selection baseline.
## V CONCLUSION
We proposed two federated reinforcement learning-based schemes named Federated
Multi-Agent Double DQN (F-MADDQN) and Federated Multi-Agent Deep Proximal
Policy Optimization (F-MADPPO) for dynamic channel allocation in 6G in-X
subnetworks which overcome the inherent issues of convergence and high
signalling overhead with privacy challenges associated with distributed and
centralized multi-agent reinforcement learning techniques. Our performance
evaluation results using realistic in-factory models defined by 5G-ACIA and
3GPP have shown that the proposed schemes can achieve similar performance to
the best-performing baselines and are robust to changes in the deployment
density as well as wireless environment conditions.
## References
* [1] R. Adeogun, G. Berardinelli, P. E. Mogensen, I. Rodriguez, and M. Razzaghpour, “Towards 6g in-x subnetworks with sub-millisecond communication cycles and extreme reliability,” _IEEE Access_ , vol. 8, pp. 110 172–110 188, 2020.
* [2] H. Sun, X. Chen, Q. Shi, M. Hong, X. Fu, and N. D. Sidiropoulos, “Learning to optimize: Training deep neural networks for interference management,” _IEEE Trans on Signal Proc._ , vol. 66, no. 20, pp. 5438–5453, 2018.
* [3] R. Adeogun, G. Berardinelli, I. Rodriguez, and P. Mogensen, “Distributed dynamic channel allocation in 6g in-x subnetworks for industrial automation,” in _2020 IEEE Globecom Workshops (GC Wkshps_ , 2020, pp. 1–6.
* [4] R. Adeogun, G. Berardinelli, and P. E. Mogensen, “Enhanced interference management for 6g in-x subnetworks,” _IEEE Access_ , vol. 10, pp. 45 784–45 798, 2022.
* [5] G. Berardinelli and R. Adeogun, “Spectrum assignment for industrial radio cells based on selective subgraph constructions,” in _2021 IEEE 94th Vehicular Technology Conference (VTC2021-Fall)_ , 2021, pp. 01–05.
* [6] R. O. Adeogun, G. Berardinelli, and P. E. Mogensen, “Learning to dynamically allocate radio resources in mobile 6g in-x subnetworks,” 2021.
* [7] D. Abode, R. Adeogun, and G. Berardinelli, “Power control for 6g industrial wireless subnetworks: A graph neural network approach,” in _2023 IEEE WCNC_. IEEE, 2023, pp. 1–6.
* [8] R. Adeogun and G. Berardinelli, “Multi-agent dynamic resource allocation in 6g in-x subnetworks with limited sensing information,” _Sensors_ , vol. 22, no. 13, p. 5062, 2022.
* [9] X. Du, T. Wang, Q. Feng, C. Ye, T. Tao, L. Wang, Y. Shi, and M. Chen, “Multi-agent reinforcement learning for dynamic resource management in 6g in-x subnetworks,” _IEEE transactions on wireless communications_ , vol. 22, no. 3, pp. 1–1, 2023.
* [10] B. Lu, H. Zhang, T. Xue, S. Guo, and H. Gai, “Deep reinforcement learning-based power allocation for ultra reliable low latency communications in vehicular networks,” in _2021 IEEE/CIC International Conference on Communications in China (ICCC)_ , 2021, pp. 1149–1154.
* [11] W. Du and S. Ding, “A survey on multi-agent deep reinforcement learning: from the perspective of challenges and applications,” _Artificial Intelligence Review_ , vol. 54, pp. 3215–3238, 2021.
* [12] A. Feriani and E. Hossain, “Single and multi-agent deep reinforcement learning for ai-enabled wireless networks: A tutorial,” _IEEE Comm. Surveys & Tutorials_, vol. 23, no. 2, pp. 1226–1252, 2021.
* [13] J. G. Kuba, R. Chen, M. Wen, Y. Wen, F. Sun, J. Wang, and Y. Yang, “Trust region policy optimisation in multi-agent reinforcement learning,” _arXiv preprint arXiv:2109.11251_ , 2021.
* [14] L. Collins, H. Hassani, A. Mokhtari, and S. Shakkottai, “Fedavg with fine tuning: Local updates lead to representation learning,” 2022.
* [15] A. B. Mansour, G. Carenini, A. Duplessis, and D. Naccache, “Federated learning aggregation: New robust algorithms with guarantees,” 2022.
* [16] J. Qi, Q. Zhou, L. Lei, and K. Zheng, “Federated reinforcement learning: Techniques, applications, and open challenges,” _arXiv preprint arXiv:2108.11887_ , 2021.
* [17] “5G-ACIA - 5G Alliance for Connected Industries and Automation — 5g-acia.org,” http://www.5g-acia.org/, [Accessed 16-May-2023].
* [18] R. Adeogun and G. Berardinelli, “Multi-agent dynamic resource allocation in 6g in-x subnetworks with limited sensing information,” _Sensors_ , vol. 22, no. 13, p. 5062, 2022.
* [19] R. O. Adeogun, G. Berardinelli, and P. E. Mogensen, “Learning to dynamically allocate radio resources in mobile 6g in-x subnetworks,” 2021.
* [20] 3GPP, “Study on channel model for frequencies from 0.5 to 100 GHz,” 3rd Generation Partnership Project (3GPP), Technical Report (TR) 38.901, 01 2017, version 16.1.0 Release 16.
|
# Spontaneous symmetry breaking or non minimal coupling?
M. Novello and A.E.S. Hartmann Centro de Estudos Avançados de Cosmologia/CBPF
Rua Dr. Xavier Sigaud 150, Urca 22290-180 Rio de Janeiro, RJ-Brazil
###### Abstract
We show that the combined minimal and non minimal interaction with the
gravitational field may produce the generation of a cosmological constant
without self-interaction of the scalar field. In the same vein we analyze the
existence of states of a scalar field that by a combined interaction of
minimal and non minimal coupling with the gravitational field can exhibit an
unexpected property, to wit, they are acted on by the gravitational field but
do not generate gravitational field. In other words, states that seems to
violate the action-reaction principle. We present explicit examples of this
situation in the framework of a spatially isotropic and homogeneous universe.
## I Introduction
The distribution of the energy of a scalar field dependent only on time in a
space time endowed with a spatially isotropic and homogeneous metric may be
represented as a perfect fluid even in the case of a non minimal coupling to
gravity. This allows a very unexpected result to appear, that is the
possibility of the scalar field to be acted upon by the gravitational field
that does not drive the gravitational field, as a sort of violation of the
action-reaction principle,in the realm of General Relativity theory. We will
show here how this is possible.
From the quantum theory of fields the mechanism of spontaneous symmetry
breaking set in evidence a sort of dynamical origin of a cosmological
constant. In a sense, this opened the way to allow the proposed scenario of
inflation guth , starobinski , novellosalim . The idea is very simple. Start
with a scalar field the dynamics of which, besides its kinematical term is
provided by a potential $V(\varphi)$ that is a polynomial. A typical case,
transformed in a paradigm, is $V=c_{1}\,\varphi^{2}+c_{2}\,\varphi^{4}$ for a
convenient choice of constants $c_{1}$ and $c_{2}.$ The density of energy is
associated to the constant $\varphi_{0}$ that extremizes $V.$ A further
restriction on the constants makes this extremum a stable one for the case
$\varphi_{0}.$ Then this constant field shows a non trivial energy
distribution having the appearance of a cosmological constant.
In the present paper we show an alternative of associating the energy
distribution of the scalar field (without self-interaction) to a cosmological
constant related to a combined interaction of gravity to the scalar field.
## II Dynamics: Minimal coupling
We start by making a short resume of properties that will be used in our text.
In the framework of standard cosmology the geometry of the universe is given
by a spatially isotropic and homogeneous metric with Euclidean section
$ds^{2}=dt^{2}-a^{2}(t)\,(dx^{2}+dy^{2}+dz^{2})$ (1)
In this case
$R_{00}=\dot{\theta}+\frac{1}{3}\,\theta^{2},$
$R_{ij}=\frac{1}{3}\,(\dot{\theta}+\,\theta^{2})\,g_{ij}$
$R=2\,\dot{\theta}+\frac{4}{3}\,\,\theta^{2}$
where we have defined the expansion factor $\theta=3\,\dot{a}/a$ and a dot
means time derivative.
Let us first examine the simplest case of the Lagrangian of a massless scalar
field
$L=\frac{1}{2\kappa}\,R+\frac{1}{2}\,\partial_{\mu}\varphi\,\partial^{\mu}\varphi-\frac{\Lambda}{\kappa}$
(2)
The equation of motion of the scalar field and the metric are given by:
$\Box\,\varphi=0$ (3)
$R_{\mu\nu}-\frac{1}{2}\,R\,g_{\mu\nu}=-\,\kappa\,T_{\mu\nu}^{0}-\Lambda\,g_{\mu\nu}$
(4)
where (note that we use indiscriminately
$\varphi_{\mu}\equiv\varphi_{,\,\mu}$)
$T_{\mu\nu}^{0}=\varphi_{,\,\mu}\,\varphi_{,\,\nu}-\frac{1}{2}\,\varphi_{\alpha}\,\varphi^{\alpha}\,g_{\mu\nu}.$
(5)
The unique non trivial equations reduces to this case to the conservation of
energy and the evolution of the expansion factor, that is
$\dot{\rho}+(\rho+p)\,\theta=0.$ (6)
$\rho+\Lambda=\frac{1}{3}\,\theta^{2}$ (7)
where, in our case
$\rho=p=\frac{1}{2}\,\dot{\varphi}^{2}.$
It then follows that the density of energy
$\rho=\rho_{0}\,a^{-6}.$
Thus, for very small values of $a(t)$ the cosmological constant can be
completely neglected. In fact, comparison of the energy of the scalar field to
others forms of matter, like the energy of an electromagnetic fluid for
instance, where the density of the photons is provided by
$\rho_{\gamma}=constant\,a^{-4},$ shows that such scalar field dominates the
cosmical scenario in the primordial era where the scalar function $a(t)$ is
very small.
In this case the solution of these equations are immediate
$\varphi=\left(\frac{2}{3}\right)^{1/2}\,\ln t$ $a(t)=a_{0}\,t^{1/3}.$
Let us now introduce a non minimal coupling and examine its consequences. We
shall see that in certain cases, the combined coupling minimal plus non
minimal can supress totally the effect of the scalar field on the geometry.
## III Non minimal coupling: suppressing the energy
Let us set the Lagrangian
$L=\frac{1}{2\kappa}\,R+\frac{1}{2}\,\partial_{\mu}\varphi\,\partial^{\mu}\varphi+B(\varphi)\,R-\frac{1}{\kappa}\,\Lambda$
(8)
where $B(\varphi)$ is for the time being an arbitrary function of the scalar
field. Then, it yields the equation for the metric as
$R_{\mu\nu}-\frac{1}{2}\,R\,g_{\mu\nu}=-\,T_{\mu\nu}^{0}-\,Z_{\mu\nu}-\Lambda\,g_{\mu\nu}$
(9)
where $T_{\mu\nu}^{0}$ is given by (5) and
$Z_{\mu\nu}=2B\,(R_{\mu\nu}-\frac{1}{2}\,R\,g_{\mu\nu})+2\\!\mathrel{\raisebox{1.49994pt}{\reflectbox{\rotatebox[origin={c}]{180.0}{$\triangle$}}}}\\!\\!_{\mu}\\!\mathrel{\raisebox{1.49994pt}{\reflectbox{\rotatebox[origin={c}]{180.0}{$\triangle$}}}}\\!\\!_{\nu}B-2\Box
B\,g_{\mu\nu}$ (10)
In the case the scalar field depends only on the cosmological time we obtain
for the density of energy the forms
$p_{0}=\rho_{0}=\frac{1}{2}\,\dot{\varphi}^{2}$
and for the non minimal part
$\rho_{Z}=-\,\frac{2}{3}\,B\,\theta^{2}-2\,\dot{B}\,\theta.$
$p_{Z}=2\,\ddot{B}+\frac{4}{3}\,\dot{B}\,\theta+\,\frac{2}{3}\,B\,\theta^{2}+\frac{4}{3}\,B\,\dot{\theta}$
We are looking for solution of the scalar field such that its total energy
$\rho=\rho_{0}+\rho_{Z}$ and pressure $p=p_{0}+p_{Z}$ vanishes, although its
field satisfies the equation of motion of the curved space-time. The
responsible for the curvature is the cosmological constant that generates a
constant expansion factor $\theta_{0}.$ Note that this situation can be
generalized for other curved space-times.
We start by solving the equation of motion for $\varphi.$ The equation is
$\Box\,\varphi-R\,B^{{}^{\prime}}=0$ (11)
where $B^{{}^{\prime}}$ represents derivative of $B$ with respect to
$\varphi.$ Let us choose, just to present an specific case the non minimal
term as given by
$B=b\,\varphi^{2}$
and the equation of motion (in the case the field depends only on time)
reduces to
$\ddot{\varphi}+\theta_{0}\,\dot{\varphi}-b\,\frac{8}{3}\,\theta^{2}_{0}\,\varphi=0.$
(12)
We set a solution of the form
$\varphi=\varphi_{0}\,e^{m\,t}$
The equation of motion is satisfied if $m$ and $b$ are related by
$m^{2}+\theta_{0}\,m-\frac{8}{3}\,b\,\theta_{0}^{2}=0.$ (13)
Using this proposal into the two independent conditions
$\rho_{0}+\rho_{Z}=0$ $p_{0}+p_{Z}=0$
yields, respectively, the equations:
$m^{2}-\frac{4}{3}b\,\theta_{0}^{2}-8\,b\,m\,\theta_{0}=0.$ (14)
$m^{2}+16\,b\,m^{2}+\frac{4}{3}b\,\theta_{0}^{2}+\frac{16}{3}\,b\,m\,\theta_{0}=0.$
(15)
The solution of these equations gives the value
$m=-\,\sqrt{3\,\Lambda}/3=-\,1/3\,\theta_{0},$ and the value of the constant
$b=-\,1/12.$ This means that the non minimal term is nothing but a conformal
coupling.
We can then state the Lemma: _The combined coupling of a minimal and conformal
coupling of a massless scalar field to gravity can annihilate the effect of
the scalar field on gravity in an universe of constant expansion._
### III.1 The case of a variable expansion factor
The suppressed process of the energy by non minimal gravitational interaction
can be generalized for the case the expansion factor is not a constant. We set
the Lagrangian
$L=\frac{1}{2\kappa}\,R+\frac{1}{2}\,\partial_{\mu}\varphi\,\partial^{\mu}\varphi+b\,\varphi^{2}\,R+V(\varphi)+L_{m}$
(16)
and consider the matter term as generating an expanding universe with
expansion factor $\theta=\theta_{0}/t.$
The equation of motion of the scalar field is
$\Box\varphi-2\,b\,\varphi\,R-\frac{\delta V}{\delta\varphi}=0$
Let us consider the case in which $V=\lambda\,\varphi^{4}.$ Then, a solution
is given by
$\varphi=\frac{\varphi_{0}}{t}.$
In the case of a conformal interaction $b=-1/12$ it follows that
$\lambda\,\varphi_{0}^{2}=\frac{1}{2}\,(1-\frac{\theta_{0}}{3})^{2}.$
It is then immediate to show that for the density and pressure of the scalar
field we obtain
$\rho(\varphi)=p(\varphi)=0.$
That is, the non minimal coupling term annihilates the free part of the
energy-momentum tensor of the scalar field.
### III.2 Second example
Let us now consider the case that there is an electromagnetic field and set
the Lagrangian as given by
$L=\frac{1}{2\kappa}\,R+\frac{1}{2}\,\partial_{\mu}\varphi\,\partial^{\mu}\varphi+B(\varphi)\,R-\frac{1}{4}\,F_{\mu\nu}\,F^{\mu\nu}$
(17)
We are interested in analyze the effects of the non minimal coupling in the
standard spatially isotropic and homogeneous metric. To be consistent with the
symmetries of this choice of the metric, an averaging procedure must be
performed if electromagnetic fields are to be taken as the source of the
gravitational field, according to the standard procedure tolmanbook . As a
consequence, the components of the electric $E_{i}$ and magnetic $H_{i}$
fields must satisfy the following relations:
$\displaystyle<{E}_{i}>=0,\qquad<{H}_{i}>$ $\displaystyle=$ $\displaystyle
0,\qquad<{E}_{i}\,{H}_{j}>=0,$ (18) $\displaystyle<{E}_{i}\,{E}_{j}>$
$\displaystyle=$ $\displaystyle-\,\frac{1}{3}{E}^{2}\,g_{ij},$ (19)
$\displaystyle<{H}_{i}\,{H}_{j}>$ $\displaystyle=$
$\displaystyle-\,\frac{1}{3}{H}^{2}\,g_{ij}.$ (20)
where $E$ and $H$ depends only on time.
Using the above average values it follows that the Maxwell energy-momentum
tensor $T_{\mu\nu}^{m}$ reduces to a perfect fluid configuration with energy
density $\rho_{\gamma}$ and pressure $p_{\gamma}$ given by
$<T_{\mu\nu}^{\gamma}>=(\rho_{\gamma}+p_{\gamma})\,v_{\mu}\,v_{\nu}-p_{\gamma}\,g_{\mu\nu},$
(21)
where
$\rho_{\gamma}=3p_{\gamma}=\frac{1}{2}\,(E^{2}+{H}^{2}).$ (22)
From the conservation of the energy-momentum tensor (note that there is no
direct interaction between the scalar and the electromagnetic fields) it
follows that the density of energy goes like $a^{-4}$. Consequently we obtain
the standard value for the scale factor once, as we shall see, there is no
contribution coming from the scalar field to change the curvature of space-
time
$a(t)=a_{0}\,\sqrt{t}$
The equation of motion for the scalar field reduces to
$\Box\,\varphi=0$
that is
$\varphi=\frac{\varphi_{0}}{\sqrt{t}}$
Let us set
$B=b\,\varphi^{2}$
Then we obtain for the annihilation of the density of energy of the scalar
field the two conditions
$\rho_{0}+\rho_{Z}=\frac{1}{2}\,\dot{\varphi}^{2}-\,\frac{2}{3}\,B\,\theta^{2}-2\,\dot{B}\,\theta=0.$
$p_{0}+p_{Z}=\frac{1}{2}\,\dot{\varphi}^{2}+2\,\ddot{B}+\frac{4}{3}\,\dot{B}\,\theta+\,\frac{2}{3}\,B\,\theta^{2}+\frac{4}{3}\,B\,\dot{\theta}=0.$
These equations are satisfied only in the case $b=-1/12$ that is in the case
the scalar field is coupled conformal to gravity.
We can then state the Lemma: _The combined coupling of a minimal and conformal
coupling of a massless scalar field to gravity can annihilate the effect of
the scalar field on gravity in the presence of a cosmological scenario driven
by a gas of photons._
## IV Curvature induced cosmological constant
Let us now follow another path and show how it is possible that a combination
of the minimal and a specific non minimal interaction of a scalar field to
gravity can generate a cosmological constant. We start by a very short review
of the original framework of inflationary universe linde .
In order to explore the consequences of a primordial phase in which the
universe has a constant expansion factor, $\theta=\theta_{0}$ in our notation,
it was suggested to make appeal to the spontaneous symmetry breaking mechanism
that allows a simple configuration of a constant scalar field controlled by
the minimum of a given potential $V(\varphi)$ drive by the Lagrangian
$L=\frac{1}{2\kappa}\,R+\frac{1}{2}\,\partial_{\mu}\varphi\,\partial^{\mu}\varphi+V(\varphi)$
(23)
where we set $\hbar=c=1.$ In a spatially homogeneous and isotropic geometry
the scalar field must depend only on the cosmical time. Then its energy
density and pressure are given by
$\rho=\frac{1}{2}\,\dot{\varphi}^{2}-V$
$p=\frac{1}{2}\,\dot{\varphi}^{2}+V.$
The equation of motion takes the form
$\Box\varphi-\frac{\delta V}{\delta\varphi}=0$
For constant solution, in the extremum of the potential, one obtains trivially
the condition that turns the distribution of the energy of the field into a
cosmological constant. Thus the true origin of the cosmological constant in
this framework rests on the self-interaction contained in the potential
$V(\varphi).$
Let us now show that there is another possibility that does not need such
self-interaction but instead use the non minimal coupling of the scalar field
to gravity to generate a cosmological constant.
The Lagrangian for a massive scalar field coupled non minimally to gravity is
given by
$L=\frac{1}{2\kappa}\,R+\frac{1}{2}\,\partial_{\mu}\varphi\,\partial^{\mu}\varphi+b\,\varphi^{2}\,R-\frac{1}{2}\,\mu^{2}\,\varphi^{2}$
(24)
In the framework of a spatially isotropic and homogeneous space time with
Euclidean section driven by the cosmological constant (where the expansion
factor is constant $\theta_{0}$ ) the minimal coupling part of the energy-
momentum of the field takes the form
$\rho_{0}=\frac{1}{2}\,\dot{\varphi}^{2}+\frac{1}{2}\,\mu^{2}\,\varphi^{2}$
$p_{0}=\frac{1}{2}\,\dot{\varphi}^{2}-\frac{1}{2}\,\mu^{2}\,\varphi^{2}$
and for the non minimal coupling it yields
$\rho_{Z}=-\,\frac{2}{3}\,B\,\theta^{2}-2\,\dot{B}\,\theta.$
$p_{Z}=2\,\ddot{B}+\frac{4}{3}\,\dot{B}\,\theta+\,\frac{2}{3}\,B\,\theta^{2}+\frac{4}{3}\,B\,\dot{\theta}$
in which $B=b\,\varphi^{2}.$
The case of a constant field $\varphi=\varphi_{0}$ produces an expanding
universe (Euclidean section) with a constant expansion $\theta=\theta_{0}.$
The equation of motion of the scalar field combined with the equation of the
metric $\kappa\,\rho=\theta^{2}/3c^{2}$ implies that
$\rho_{0}+\rho_{Z}+p_{0}+p_{Z}=0$
$\theta_{0}^{2}=\frac{3}{8}\,\frac{\mu^{2}}{b}$
$\varphi_{0}^{2}=\frac{4}{3}\,\left(\frac{\theta_{0}}{\mu}\right)^{2}.$
In other words, a cosmological constant appears thanks to the non minimal
coupling given by
$\Lambda=\frac{1}{4}\,\mu^{2}\,\varphi^{2}_{0}=\frac{1}{3}\,\theta_{0}^{2}.$
This solution is stable. This can be easily shown by making the perturbation
$\varphi_{0}\rightarrow\varphi_{0}+\delta\varphi$ and
$\theta_{0}\rightarrow\theta_{0}+\delta\theta.$ The time dependance is given
by
$\delta\varphi=\delta\varphi_{i}\,\exp{(-\theta_{0}\,t)}.$
Note that in the present case the non minimal coupling cannot be a conformal
one com .
### IV.1 Spontaneous symmetry breaking or non minimal coupling?
Let us analyze now a generalization of both cases examined in the previous
sections. We set for the Lagrangian
$L=\frac{1}{2\kappa}\,R+\frac{1}{2}\,\partial_{\mu}\varphi\,\partial^{\mu}\varphi+V(\varphi)\left(1-\frac{2\,b}{\mu^{2}}\,R\right)$
(25)
where $b$ is a dimensionless constant and for comparision to the standard
framework we set for the potential the form
$V(\varphi)=-\,\frac{\mu^{2}}{2}\,\varphi^{2}+\lambda\,\varphi^{4}.$
We limit the analysis to the cosmological scenario of spatially homogeneous
and isotropic metric with Euclidean section for constant expansion where the
scalar of curvature reduces to
$R=\frac{4}{3}\,\theta^{2}.$
The equation of motion is given by
$\Box\varphi-\left(1-\frac{2\,b}{\mu^{2}}\,R\right)\,\frac{\delta
V}{\delta\varphi}=0$
In the case of a constant solution $\varphi=\varphi_{0}$ the energy-momentum
of the field reduces to the density and pressure given by
$\rho=-V+\frac{4}{3}\,\frac{b\,\theta^{2}}{\mu^{2}}\,V$
$p=+V-\frac{4}{3}\,\frac{b\,\theta^{2}}{\mu^{2}}\,V$
that is, the distribution of energy is nothing but an effective cosmological
constant, that is, $\rho+p=0$
In this case, the dynamics of $\varphi$ admits two class of solutions:
$\frac{\delta V}{\delta\varphi}=0$ (26)
or
$1-\frac{2\,b}{\mu^{2}}\,R=0$ (27)
The first case led to the break of symmetry scenario of standard inflation.
The second one represents the combined effect of minimal and non minimal
coupling of $\varphi$ field to gravity where the expansion factor is given by
$\theta_{0}^{2}=\frac{3}{8}\,\frac{\mu^{2}}{b}.$
In both cases the energy distribution of the scalar field is identical to a
cosmological constant.
In the gravitational case the equation
$\rho=\frac{\theta^{2}}{3}$
yields c1
$\varphi_{0}^{2}=\frac{\mu^{2}}{4\,\lambda}\,\left(1+\sqrt{1-\frac{32}{3}\,\frac{\lambda\,\theta^{2}_{0}}{\mu^{4}}}\right).$
(28)
## V Final comments
We have presented some cases in which the effect of non minimal coupling of a
scalar field to the curvature of space-time allows the existence of states
such that the net effect of the free part of the energy added to the
gravitational interaction annihilate each other. This means that although the
scalar field is embedded in a curved space its contribution to the metric
vanishes. A similar interaction allows the generation of a cosmological
constant without self interaction of the scalar field. In a forth coming work
we examine these properties for other fields, in particular for the
electromagnetic field novellohartmann , hartmannenovello .
### V.1 Acknowledgements
We would like to acknowledge the financial support from brazilian agencies
Finep, Faperj and CNPq.
## References
* (1) A. Guth, Inflationary Universe in Phys Rev D23 (1981) 347.
* (2) A. Starobinsky, A new type of isotropic cosmological models without singularity in Phys Lett 91 B (1980) 99
* (3) M Novello and J M Salim in Phys Rev D (1979)
* (4) M Novello and S E P Bergliaffa: Bouncing Cosmologies in Physics Reports vol 463, no 4, july 2008.
* (5) R. Tolman _Relativity, Thermodynamics, and Cosmology_ , Oxford, Clarendon (1934).
* (6) A. D. Linde: The inflationary universe in Rep. Progr. Phys. vol 47, pp 925-986, 1984.
* (7) M. Novello and A.Hartmann in Is the elctromagnetic field responsible for the cosmic acceleration in late times? (to be submited)
* (8) A.Hartmann and M.Novello in A new bifurcation in the universe, to be submitted for publication.
* (9) We note that the expression of the constant value of the field reads $\varphi_{0}^{2}=4\theta_{0}^{2}/3\mu^{2}L_{Pl}^{2}.$ In the paper we are using units such that the Planck lenght is equal to $1.$
* (10) Compare with the case of SSB in which the condition to extremize the potential yields the same result for the particular choice of constant $\lambda$ given by $\lambda=3\,b\,\mu^{2}/4.$
|
# Global Optimization for Trajectory Design via Invariant Manifolds in the
Earth-Moon Circular Restricted Three-Body Problem
Flavio Tagliaferri Advanced Concepts Team, European Space Agency, European
Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk,
The Netherlands Sapienza University of Rome, Via Eudossiana 18, 00184, Rome,
Italy Emmanuel Blazquez Advanced Concepts Team, European Space Agency,
European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201 AZ
Noordwijk, The Netherlands Giacomo Acciarini Advanced Concepts Team,
European Space Agency, European Space Research and Technology Centre (ESTEC),
Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands Surrey Space Centre,
University of Surrey, GU2 7XH, Guildford, United Kingdom Dario Izzo Advanced
Concepts Team, European Space Agency, European Space Research and Technology
Centre (ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands
_Abstract_ \- This study addresses optimal impulsive trajectory design within
the Circular Restricted Three-Body Problem (CR3BP), presenting a global
optimization-based approach to identify minimum $\Delta V$ transfers between
periodic orbits, including heteroclinic connections. By combining a Monotonic
Basin Hopping (MBH) algorithm with a sequential quadratic solver in a parallel
optimization framework, a wide range of minimum $\Delta V$ transfers are
efficiently found. To validate this approach, known connections from the
literature are reproduced. Consequently, three-dimensional periodic orbits are
explored and a systematic search for minimum propellant trajectories is
conducted within a selected interval of Jacobi constants and a maximum time of
flight. Analysis of the results reveals the presence of very low $\Delta V$
solutions and showcases the algorithm’s effectiveness across various mission
scenarios.
## I. Introduction
Nowadays, as multiple space agencies and private companies are becoming more
interested in developing missions far beyond the Earth orbit, trajectory
design in multi-body environments becomes a fundamental asset. In particular,
the Moon is considered the cornerstone of the near future space exploration
[1], with several missions aiming to bring scientific discovery, technology
advancement, and acquire the essential skills needed to thrive in a different
planetary environment, specifically in preparation for upcoming human missions
to other planets, e.g. Mars. In this context, the Earth-Moon Circular
Restricted Three-Body Problem (CR3BP) represents a key mathematical framework
for preliminary mission analysis studies, offering families of periodic orbits
about the lunar libration points with peculiar operational characteristics,
and providing a potential low-energy transport network to move between these
orbits, by means of the invariant manifolds [2]. In the CR3BP, the process of
optimal trajectory design can be challenging and traditionally involves the
use of Poincaré maps (PM), a tool from dynamical systems theory, which allows
the mission designer to visualize the solution space and select promising
candidate trajectories. In [3] Koon et al. exploited this tool to compute
propellant-free heteroclinic transfers, between $L_{1}$ and $L_{2}$ periodic
orbits in the planar system. However, when the full 6-dimensional CR3BP is
considered, the use of PMs becomes more demanding due to the higher
dimensionality of the system which does not enable a qualitative analysis of
the maps [4]. Several optimization strategies were developed to tackle this
problem. For example, in [5] Haapala and Howell implemented a combination of
interactive and automated search strategies to locate maneuver-free and low
$\Delta V$ transfers between vertical Lyapunov, halo and axial orbits with
different Jacobi constant in the Earth-Moon systems, and consequently
established a catalog of these transfers. In [6] Zhaoyu Li, together with the
other authors, proposed a global search strategy based on comparing the
orbital state of the unstable and stable manifolds, incorporated with low-
thrust techniques, to seek a suitable matching point for maneuver application.
The obtained velocity increments were then refined to achieve more fuel-
optimal transfers using Sequential Quadratic Programming techniques, also
deriving the gradient of the constraint to enhance the algorithm convergence
accuracy and rapidity. Recently, in [7] D. B. Henry and D.J. Scheeres proposed
a robust multiparameter continuation scheme to retrieve the families of
transfers between both periodic and quasi-periodic orbits, using the PMs to
identify the starting solution.
In this work, we propose a black-box approach based on combining the use of a
global optimization algorithm, the Monotonic Basin Hopping (MBH), with the
sequential quadratic solver SNOPT [8] in a parallel optimization framework to
perform trajectory design in the CR3BP. More specifically, we focus on
impulsive minimum propellant transfers via invariant manifolds, between halo
and vertical Lyapunov orbits at the same value of Jacobi constant ($C$), about
the $L_{1}$ and $L_{2}$ libration points of the Earth-Moon system. To validate
the approach, we reproduce trajectories from the literature, both in the
planar and spatial scenarios. We consequently perform a systematic search of
minimum propellant trajectories in a region of Jacobi constants and accounting
for a maximum time of flight. We finally report on the best trajectories found
with low propellant consumption, for each transfer type. The proposed approach
can be used to retrieve initial low $\Delta V$ trajectories without employing
preliminary orbit cartography.
## II. Theoretical background
In this section, we begin by describing the dynamical system considered.
Subsequently, we delve into the method for retrieving the selected periodic
orbits, followed by a discussion on computing stable and unstable manifolds.
### A. The Circular restricted three-body problem
The circular restricted three-body problem describes the motion of an object
(satellite) with a negligible mass under the gravitational influence of two
other masses $m_{1}$ and $m_{2}$. The larger of these two, $m_{1}$ (Earth) is
called the primary, while $m_{2}$ (Moon) is called the secondary. The primary
and secondary are assumed to orbit their center of mass in a circular orbit.
This dynamical system is usually implemented in a non-dimensional coordinate
system with distance unit (DU) equal to the distance between $m_{1}$ and
$m_{2}$ (i.e., $DU=384,400$ km for the Earth-Moon system). The time unit (TU)
is such that the mean angular motion of the primary and secondary bodies about
the system’s center of mass is unitary ($TU=375,699.79375$ s for the Earth-
Moon system). The origin lies at the center of mass of $m_{1}$ and $m_{2}$,
and the $\hat{x}$ axis points from the primary to the secondary. Aligning
parallel to the angular momentum vector of the system is the $\hat{z}$ axis,
while the $\hat{y}$ axis completes the right-handed system. This reference
frame, known as Synodic reference frame, undergoes a constant rotation around
the $\hat{z}$ axis, sharing the same angular velocity as the orbital motion of
the primary and secondary, as depicted in Fig.1. Thus, the coordinates of both
the primary and secondary remain fixed. The dynamical system can be fully
described by (1):
$\left\\{\begin{array}[]{@{}l@{}@{}}\ddot{x}-2\dot{y}=\pdv{U}{x}\\\
\ddot{y}+2\dot{x}=\pdv{U}{y}\\\ \ddot{z}=\pdv{U}{z}\\\ \end{array}\right.\ $
(1)
Where:
$U(x,y,z)=\frac{(x^{2}+y^{2})}{2}+\frac{1-\mu}{r_{1}}+\frac{\mu}{r_{2}}$ (2)
represents the potential function, $\mu=\frac{m_{2}}{m_{1}+m_{2}}$ is the mass
ratio parameter, which relates the masses of the primary and secondary
($\mu$=0.01215058 for the Earth-Moon system), $r_{1}$ the distance of the
object to the primary and $r_{2}$ its distance to the secondary. The CR3BP has
five equilibrium points, known as Lagrange points. Three of these points are
known as the collinear points and lie on the $\hat{x}$ axis. The two collinear
points closest to the moon, $L_{1}$ and $L_{2}$ are used for this study. This
system admits a first integral of motion, called the Jacobi constant, defined
through (3):
$C=2U-({\dot{x}}^{2}+{\dot{y}}^{2}+{\dot{z}}^{2})\ $ (3)
This parameter is used in designing heteroclinic connections between different
periodic orbits since it is required that the departure and arrival orbits
must have the same Jacobi constant. Finally, by constraining the object to
move in the xy plane, a simplified framework is retrieved, called Planar
Circular Restricted Three-Body Problem (PCR3BP).
Fig. 1: Synodic reference frame and location of the libration points, $L_{1}$
and $L_{2}$
### B. Periodic orbits
The computation of periodic orbits is performed using a predictor-corrector
method. Initially, the state is numerically integrated, which results in a
deviation from periodicity. This initial state is then corrected based on
information obtained from the monodromy matrix, defined as the state
transition matrix evaluated over one complete period T of the orbit. To
achieve this, routines from SEMpy, a Python library specialized in non-
Keplerian astrodynamics [9], were utilized and adapted to retrieve the desired
values of Jacobi constants for halo, planar, and vertical Lyapunov orbits.
### C. Invariant manifolds
Invariant Manifolds are higher-dimensional dynamical structures that govern
the asymptotic flow of motion towards and away from unstable orbits. Those can
be computed by exploiting the properties of the monodromy matrix. For each
point $\boldsymbol{X_{i}}=[x,\ y,\ z,\ \dot{x},\dot{y},\dot{z}]$ on the
periodic orbit, the initial state of the stable and unstable manifolds, can be
computed using (4) and (5):
$\boldsymbol{X_{i}^{S}}=\boldsymbol{X_{i}}\pm\epsilon\frac{\Phi_{i}{,_{0}}\nu_{0}^{S}}{\left\lVert\Phi_{i}{,_{0}}\nu_{0}^{S}\right\rVert.}$
(4)
$\boldsymbol{X_{i}^{U}}=\boldsymbol{X_{i}}\pm\epsilon\frac{\Phi_{i}{,_{0}}\nu_{0}^{U}}{\left\lVert\Phi_{i}{,_{0}}\nu_{0}^{U}\right\rVert.}$
(5)
Where $\Phi_{i}{,_{0}}$ is the state transition matrix computed from a
reference state $\boldsymbol{X_{0}}$ up to $\boldsymbol{X_{i}}$, while
$\nu_{0}^{S}$ and $\nu_{0}^{U}$ represent the stable ad unstable eigenvectors
of the monodromy matrix evaluated at $\boldsymbol{X_{0}}$, corresponding to
the smallest and largest eigenvalue, respectively. The value of the step
parameter $\epsilon$ is applied to the whole state in the positive and
negative directions, i.e. interior and exterior w.r.t. CR3BP’s center of mass
[10]. In this study, its value is taken constant and equal to 50km for both
manifolds, which is a common choice for the Earth-Moon system [11]. Finally,
the manifolds can be retrieved by numerically integrating the state backward
and forward in time for the stable and unstable manifolds, respectively.
Intersections of the unstable and stable manifolds result in low-energy
transfers.
## III. Trajectory design
In this section, the proposed approach is explained in detail. First, the
chosen problem formulation is addressed, then the specific optimization
architecture used is illustrated.
### A. Problem formulation
In the proposed approach, the search for low $\Delta V$ trajectories between
periodic orbits is formulated as a single-objective constrained optimization
problem through (6), (7) and (8):
minimize $\displaystyle J=\lVert\Delta V(\textbf{x})\rVert$ (6) subject to
$\displaystyle\Delta x=0,\Delta y=0,\Delta z=0$ (7) $\displaystyle
l_{B}\leq\textbf{x}\leq u_{B}$ (8)
where the Euclidean norm of the difference in velocity and the components of
the difference in position, i.e. $\Delta x$, $\Delta y$ and $\Delta z$, both
computed at the patching point of the unstable and stable manifolds branches,
represent the objective function $J$ to be minimized and the three equality
constraints to be fulfilled. The decision vector x is composed of four
optimization variables: $[\theta_{U},\tau_{U},\theta_{S,}\tau_{S}]$. In
particular $(\theta_{U},\theta_{S})$, called the departure and arrival angles,
are defined as fractions of the orbit’s periods $T_{d_{\text{ep}}}$ and
$T_{a_{\text{rr}}}$ through (9) and (10), respectively:
$\theta_{U}=\frac{t_{d{\text{ep}}}}{T_{d{\text{ep}}}}$ (9)
$\theta_{S}=\frac{t_{a{\text{rr}}}}{T_{a{\text{rr}}}}$ (10)
where $t_{d{\text{ep}}}$ and $t_{a{\text{rr}}}$ are the coasting times taken
from the reference starting points $\boldsymbol{X_{0}{{}^{U}}}$ and
$\boldsymbol{X_{0}{{}^{S}}}$ along the departure and arrival orbits. They are
used to reach the states $\boldsymbol{X_{i}{{}^{U}}}$ and
$\boldsymbol{X_{i}{{}^{S}}}$ from which the unstable and stable manifolds
branches are computed. The coasting times on the unstable and stable manifold
branches are represented by $(\tau_{U},\tau_{S})$. We will refer the sum of
$\tau_{U}$ and $\tau_{S}$ as the time of flight ($TOF$) of a given trajectory.
The optimization variables are bounded between $u_{L}=[0,0,0,0]$ and
$u_{B}=[1,8,1,8]$. An example of a transfer scenario is depicted in Fig.2.
Fig. 2: Example of a simplified mission scenario, with all the optimization
variables represented
### B. Optimization architecture
To solve the presented optimization problem a nested optimization architecture
is employed, where a Monotonic Basin Hopping (MBH) algorithm uses sequential
quadratic programming (and in particular SNOPT) as its local optimizer. MBH is
a stochastic global optimization algorithm [12] that works by continuously
applying a perturbation vector to the problem’s optimization variables in a
method called hopping. Following each hop, a local optimization is conducted
on the new configuration. If this step yields an improved solution, that
solution updates the set of optimization variables for the next perturbation.
This iterative process is repeated until a maximum number of hops is reached.
Here we adopted a modified version of the standard MBH. In particular, the
initial decision vector is obtained as the best individual among a population
of 200, generated from a random seed. A tolerance of $10^{-}6$ is set for the
three equality constraints. The architecture is then integrated in a parallel
optimization framework, where a number of 100 different computational unit,
called islands, are used to solve one optimization problem. The best solution
is then extracted among the best individual of each island. Note that in
principle, multiple trajectories can be retrieved with the solution of a
single optimization problem. This could allow the mission designer to make
further trade offs based on geometry and time of flight. The resulted
optimization architecture as well as the parallel framework was created using
the open-source Python library pyGMO [13]. Moreover, to increase the speed and
efficiency of the optimization process, the open-source Python library Heyoka
[14] was employed for fast integration of ODE systems.
## IV. Validation
In this section we validate the proposed approach by reproducing established
trajectories from the literature. In particular, the algorithm is first tested
in the PCR3BP, where heteroclinic connections between planar Lyapunov orbits
are sought. Then the CR3BP framework is considered, where heteroclinic
connections between periodic orbits become challenging to find. For this
reason, a minimum fuel trajectory retrieved through a state-of-the-art method
in [7] is set as a reference solution for the validation of the algorithm.
Furthermore, each trajectory is characterized by the Euclidean norm of the
velocity and position differences at the patching point, i.e $\Delta V$ and
$\Delta R$, along with the $TOF$. The specific values of the key parameters
used in the two validation cases can be found in Tab. 1.
Tab. 1: Key parameters used in the validation cases
$Case$ | $\mu$ | $\epsilon$ | $TOF[TU]$
---|---|---|---
PCR3BP | 0.012150585 | $1\times 10^{-6}$ | -
CR3BP | 0.01215 | $1\times 10^{-5}$ | $[0,6]$
### A. Planar trajectories
The first task we tackle is the computation of heteroclinic connections
between planar Lyapunov orbits, belonging to three different values of the
Jacobi constant coming from [15]. The detailed numerical results are reported
in Tab. 2., while the trajectories are represented in Fig.3., where the
unstable and stable branches are depicted in red and green, respectively. As
one can see, in all the three cases, the results successfully led to the
reconstruction of trajectories with the same qualitative geometry as the ones
in the reference paper, up to a precision of about $10^{-4}$ m/s in velocity
and of hundreds of meters in position at the patching point.
Fig. 3: Representation of the obtained numerical heteroclinic connections
between planar Lyapunov orbits for cases (a), (b) and (c) in Tab. 2. Tab. 2:
Numerical results of the validation in the PCR3BP
$Case$ | $C$ | $\Delta V$ $[m/s]$ | $\Delta R$ $[m]$ | $TOF[dd]$
---|---|---|---|---
(a) | 3.130459 | $4.0\times 10^{-5}$ | 25.53 | 48.075
(b) | 3.097474 | $1.3\times 10^{-4}$ | 491.80 | 52.787
(c) | 3.025554 | $4.3\times 10^{-4}$ | 360.08 | 75.079
### B. Spatial trajectories
As a successive test, spatial trajectories are considered. Since finding
heteroclinic connections between three-dimensional periodic orbits is very
challenging due to the low dimension of their manifolds relative to the phase
space, we perform the validation in the CR3BP case by considering the example
presented in [7]. This trajectory connects with minimum propellant a $L_{1}$
vertical Lyapunov and a $L_{2}$ southern halo orbits at a value of $C$ equal
to 3.1328. The results led to the computation of a comparable trajectory in
terms of geometry, propellant consumption and time of flight, as showed in
Tab. 3. Then, the best minimum fuel solution found by our approach is also
retrieved. As one can see from Tab. 3., the best minimum fuel solution found
by the presented algorithm requires about 45% less fuel and about 13% longer
$TOF$. The corresponding trajectories are depicted in Fig.4. and in Fig.5.
Tab. 3: Numerical results of the validation in the CR3BP
$Case$ | $C$ | $\Delta V$ [m/s] | $\Delta R$ [m] | $TOF[dd]$
---|---|---|---|---
Reference [7] | 3.1328 | 280.47 | - | 39.570
Reproduced | 3.1328 | 280.70 | 14.38 | 39.564
Best min-fuel | 3.1328 | 149.10 | 64.04 | 44.789
Fig. 4: Representation of the trajectory from the reproduced case in Tab. 3.
from [7]
(a) Three-dimensional trajectory
(b) Projections in the xy and xz planes
Fig. 5: Representation of the trajectory from the best min-fuel case in Tab.
3., retrieved with the proposed algorithm
## V. Applications
In this section, we apply the algorithm to three different transfer types:
halo to halo, vertical Lyapunov to vertical Lyapunov and halo to vertical
Lyapunov. Our specific goal is to identify the global optimal trajectories
that the method can produce, for each transfer type. To achieve this, we
systematically explore minimum fuel trajectories between halo and vertical
Lyapunov orbits. This investigation covers a Jacobi constant range of
$[3.057,3.152]$ and a maximum time of flight of approximately $70$ days.
Notably, both the interior and exterior manifolds are considered in this
study. The upper bound of the Jacobi constant corresponds to the value of the
bifurcation of the $L_{2}$ halo family from the Lyapunov family [5]. A step of
$6\times 10^{-4}$ is used to determine the orbit’s Jacobi constant and it
comes from a trade-off between the number of trajectories obtained and the
dimension of successive orbits, i.e. the z-amplitudes. Each mission scenario
is identified by:
1. 1.
The departure and arrival orbits characterized by a value of $C$ and their
respective libration points;
2. 2.
The Family of each orbit;
3. 3.
The types of manifolds branches used.
Based on these characteristics, a compact notation is introduced in Tab. 4. to
identify a particular mission scenario.
Tab. 4: Description of the notation adopted to identify a mission scenario $Notation$ | $Description$
---|---
NH1 | $L_{1}$ northern halo
SH1 | $L_{1}$ southern halo
V1 | $L_{1}$ vertical Lyapunov
NH2 | $L_{2}$ northern halo
SH2 | $L_{2}$ southern halo
V2 | $L_{2}$ vertical Lyapunov
+ | interior manifold
- | exterior manifold
outward | $L_{1}$ to $L_{2}$
return | $L_{2}$ to $L_{1}$
For example, with $[NH1-,V2+]$ we describe a trajectory obtained departing
from a $L_{1}$ northern halo orbit moving on the exterior unstable manifold
branch and arriving to a $L_{2}$ vertical Lyapunov orbit using the interior
stable manifold branch. Trajectories which either exhibit a $\Delta R\geq 1$
km or impact on the Moon surface are discarded and excluded from the analysis.
In the following subsections, we report the global optimal trajectories
discovered with this algorithm, for each transfer type.
### A. Halo to halo trajectories
The best solutions in terms of velocity increment for outward and return
trajectories are reported in Tab. 5. Observing the results, we found several
trajectories with remarkably low fuel requirements. Among them, the
trajectories with the lowest $\Delta V$ are then represented in Fig.6. and
Fig.7. Furthermore, the algorithm exhibits a convergence toward outcomes that
align with those achievable through the application of CR3BP’s symmetries
[16]. For example, the successive application of the image of the xy-plane and
backward image of the xz-plane to the $[NH1+,SH2-]$ would produce the
$[NH2-,SH1+]$ trajectory, while the use of only the backward image of the xz-
plane would result in the $[SH2-,NH1+]$, with the same $C$, $\Delta V$ and
$TOF$ values. Consistent findings emerged from solving the optimization
problems corresponding to these three mission scenarios. This observed
behavior validates the obtained results and could offer an opportunity to
streamline the investigation of mission scenarios when employing this
numerical method.
Tab. 5: Numerical results for the optimal halo to halo trajectories, in
outward and return mission scenarios
Mission scenario | $C$ | $\boldsymbol{\Delta V}$ [m/s] | $\boldsymbol{\Delta R}$ [m] | $TOF$ [dd]
---|---|---|---|---
$[NH1-,NH2-]$ | 3.132397 | 3.000 | 110.76 | 50.976
$[NH1+,SH2-]$ | 3.080797 | 2.980 | 9.02 | 64.410
$[SH1-,SH2-]$ | 3.131797 | 2.684 | 144.10 | 51.077
$[SH1+,NH2-]$ | 3.071197 | 4.991 | 0.06 | 61.276
$[NH2+,NH1-]$ | 3.129397 | 3.903 | 446.72 | 49.83
$[SH2-,NH1+]$ | 3.080797 | 2.980 | 44.49 | 64.410
$[SH2+,SH1+]$ | 3.132397 | 2.999 | 414.86 | 50.976
$[NH2-,SH1+]$ | 3.080797 | 5.029 | 31.70 | 64.405
Fig. 6: Representation of the global optimum solution for outward
trajectories, corresponding to the mission scenario $[SH1-,SH2-]$ in Tab. 5.
Fig. 7: Representation of the global optimum solution for return trajectories,
corresponding to the mission scenario $[SH2-,NH1+]$ in Tab. 5.
### B. Vertical Lyapunov to vertical Lyapunov trajectories
The second application we addressed involves computing trajectories connecting
vertical Lyapunov orbits. The best solutions in terms of velocity increment
for the outward and return scenarios are reported in Tab. 6. Among them, the
trajectories with the lowest $\Delta V$ are then represented in Fig.8. and
Fig.9. In particular, the mission scenario $[V2+,V1+]$ shows a $\Delta V$ of
4.128 m/s which is, to the best of our knowledge and based of the literature
we read, a value remarkably low for this type of transfer. As for halo to halo
trajectories, the algorithm demonstrates adherence to the symmetry properties
of the CR3BP.
Tab. 6: Numerical results for the optimal vertical to vertical Lyapunov
trajectories, in outward mission scenarios
Mission scenario | $C$ | $\boldsymbol{\Delta V}$ [m/s] | $\boldsymbol{\Delta R}$ [m] | $TOF$ [dd]
---|---|---|---|---
$[V1+,V2+]$ | 3.109600 | 39.629 | 95.48 | 41.583
$[V1-,V2+]$ | 3.086799 | 13.068 | 35.23 | 23.253
$[V1+,V2-]$ | 3.082600 | 9.150 | 1.16 | 38.803
$[V1-,V2-]$ | 3.089200 | 6.961 | 430.75 | 49.406
$[V2+,V1+]$ | 3.084400 | 4.128 | 0.50 | 63.482
$[V2-,V1+]$ | 3.104800 | 61.037 | 3.38 | 26.467
$[V2+,V1-]$ | 3.089200 | 6.962 | 2.27 | 49.406
$[V2-,V1-]$ | 3.086799 | 13.0776 | 25.45 | 23.251
Fig. 8: Representation of the global optimum solution for outward
trajectories, corresponding to the mission scenario $[V1-,V2-]$ in Tab. 6.
Fig. 9: Representation of the global optimum solution for return trajectories,
corresponding to the mission scenario $[V2+,V1+]$ in Tab. 6.
### C. Halo to vertical Lyapunov trajectories
The last application concerns transfers associated with halo orbits and
vertical Lyapunov orbits. The best solutions in terms of velocity increment
for the outward and return scenarios are reported in Tab. 7. Among them, the
trajectories with the lowest $\Delta V$ are then represented in Fig.10. and
Fig.11. Higher values of the $\Delta V$ are noticed compared to previous
transfer types. Also in this scenario, the algorithm follows symmetry
properties, as one can see in Tab. 7.
Tab. 7: Numerical results for the optimal halo to vertical Lyapunov
trajectories, in outward mission scenarios
Mission scenario | $C$ | $\boldsymbol{\Delta V}$ [m/s] | $\boldsymbol{\Delta R}$ [m] | TOF [dd]
---|---|---|---|---
$[NH1+,V2-]$ | 3.084997 | 29.338 | 6.37 | 58.522
$[NH1-,V2-]$ | 3.138997 | 37.159 | 269.83 | 63.321
$[SH1+,V2-]$ | 3.084397 | 29.283 | 0.11 | 58.630
$[SH1-,V2-]$ | 3.108397 | 32.678 | 187.17 | 63.670
$[V2+,NH1+]$ | 3.083197 | 31.950 | 13.89 | 58.866
$[V2+,NH1-]$ | 3.086197 | 30.731 | 38.40 | 58.317
$[V2+,SH1+]$ | 3.106597 | 20.939 | 49.32 | 64.209
$[V2+,SH1-]$ | 3.086197 | 30.731 | 0.14 | 58.317
Fig. 10: Representation of the global optimum solution for outward
trajectories, corresponding to the mission scenario $[SH1+,V2-]$ in Tab. 7.
Fig. 11: Representation of the global optimum solution for return
trajectories, corresponding to the mission scenario $[V2+,SH1+]$ in Tab. 7.
## VI. Conclusions
In this study, we introduce a general black-box methodology for trajectory
design within the framework of the CR3BP. Our validation demonstrates that the
proposed approach yields low-cost trajectories that are comparable to those
found in the literature. Leveraging a direct problem formulation and an
optimized computational architecture, we efficiently compute a large number of
trajectories across various scenarios in an automated manner. The numerical
results reveal very low $\Delta V$ solutions, but also alternative
trajectories that could serve as initial approximations for existing
methodologies. Remarkably, despite its reliance on a numerical approach, the
algorithm demonstrates convergence toward the symmetries of the CR3BP,
achieving a certain precision. Furthermore, our proposed approach suggests an
exploration of a broader search space compared to traditional methods that
rely on Poincaré maps. Indeed, the algorithm we employed operates without the
need for defining a surface of section or imposing problem-dependent
constraints to reduce system dimensionality. These unique characteristics
could facilitate the uncovering of new fuel-optimal solutions, as evidenced by
the numerical results for trajectories from vertical Lyapunov to southern halo
orbits. Our future research will delve into a more detailed study and analysis
of this property, along with an investigation into the convergence limits of
the algorithm.
## References
* [1] European Space Agency. “ESA Strategy for Science at the Moon.” https://exploration.esa.int/web/moon/-/61371-esa-strategy-for-science-at-the-moon, 2019.
* [2] Szebehely, V. Theory of Orbits: The Restricted Problem of Three Bodies. Academic Press, New York, 1967.
* [3] Koon, W. S., Lo, M. W., Marsden, J. E., and Ross, S. D. “Heteroclinic Connections between Periodic Orbits and Resonance Transitions in Celestial Mechanics.” Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 10, No. 2, pp. 427–469, 2000.
* [4] Haapala, A. F. and Howell, K. C. “Representations of Higher-Dimensional Poincaré Maps with Applications to Spacecraft Trajectory Design.” Acta Astronautica, Vol. 96, pp. 23–41, March 2014.
* [5] Haapala, A. and Howell, K. “A framework for construction of transfers linking periodic libration point orbits in the Earth-Moon spatial circular restricted three-body problem.” Int. J. Bifurcat. Chaos, Vol. 26, No. 5, 2016.
* [6] Li, Z., Zeng, H., Xu, R., Peng, K., and Huang, Z. “Hierarchical global search for fuel-optimal impulsive transfers between lunar libration orbits.” Adv. Space Res., Vol. 67, No. 6, pp. 1773–1786, 2021.
* [7] Henry, D. B. and Scheeres, D. J. “Quasi-periodic orbit transfer design via whisker intersection sets.” J Guid Control Dyn, pp. 1–16, 2023.
* [8] Gill, P. E., Murray, W., and Saunders, M. A. “SNOPT: An SQP Algorithm for Large-Scale Constrained Optimization.” SIAM Review, Vol. 47, No. 1, pp. 99–131, 2005.
* [9] Blazquez, E. “SEMPy: a python open-source toolbox for non-Keplerian astrodynamics.” “International Conference on Astrodynamics Tools and Techniques (ICATT),” Virtual, 2021.
* [10] Blazquez, E. Rendezvous Optimization and GNC Design for Proximity Operations on cis-lunar Near Rectilinear Halo Orbits. Ph.D. thesis, 2021.
* [11] Gómez, G., Jorba, A., Masdemont, J., and Simó, C. “Study refinement of semi-analytical halo orbit theory.” Final Report, ESOC Contract No: 8625/89/D/MD (SC), 1991.
* [12] Eason, G., Noble, B., and Sneddon, I. N. “On certain Wales, D. J. & Doye, J. P. K. Global optimization by basin-hopping and the lowest energy structures of Lennard–Jones clusters containing up to 110 atoms.” J. Phys. Chem. A, Vol. 101, pp. 5111–5116, 1997.
* [13] Biscani, F. and Izzo, D. “A parallel global multiobjective framework for optimization: pagmo.” Journal of Open Source Software, Vol. 5, No. 53, p. 2338, 2020.
* [14] Biscani, F. and Izzo, D. “Revisiting high-order Taylor methods for astrodynamics and celestial mechanics.” Monthly Notices of the Royal Astronomical Society, Vol. 504, No. 2, pp. 2614–2628, 2021.
* [15] Barrabés, E., Mondelo, J. M., and Ollé, M. “Numerical continuation of families of heteroclinic connections between periodic orbits in a Hamiltonian system.” Nonlinearity, Vol. 26, No. 10, p. 2747, 2013.
* [16] Miele, A. “Revisit of the Theorem of Image Trajectories in the Earth-Moon Space.” Optimization Theory and Applications, Vol. 147, No. 3, pp. 483–490, 2010.
|
$\displaystyle=F\Big{[}-h\,dt^{2}+\frac{dr^{2}}{h}+r^{2}d\theta^{2}\Big{]}+\frac{r^{2}\sin^{2}\theta}{F}\bigl{(}d\phi-\Omega\,dt\bigr{)}^{2}\,,$
(C.6a) $\displaystyle A$
$\displaystyle=A_{0}\,dt+\frac{A_{3}}{F}\bigl{(}d\phi-\Omega\,dt\bigr{)}\,,$
(C.6b)
where
$\displaystyle h$
$\displaystyle=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}+\frac{H^{2}}{r^{2}}\,,$
(C.7a) $\displaystyle F$
$\displaystyle=1+\jmath^{2}\Bigl{[}r^{2}\sin^{2}\theta+\bigl{(}Q^{2}+H^{2}\bigr{)}\cos^{2}\theta\Bigr{]}^{2}\,,$
(C.7b) $\displaystyle\Omega$ $\displaystyle=-4\,\jmath\,h\,r\cos\theta\,,$
(C.7c) $\displaystyle A_{0}$ $\displaystyle=-\frac{Q}{r}-\jmath
Hr\Bigl{[}1+h\cos^{2}\theta\Bigr{]}\,,$ (C.7d) $\displaystyle A_{3}$
$\displaystyle=\cos\theta\Bigl{[}H+Q\sqrt{F-1}\Bigr{]}\,.$ (C.7e)
From the components of the electromagnetic potential, we can see that, in a
certain sense, the swirling parameter $\jmath$ generates a magnetic charge
from an electric one, and vice-versa.
## Appendix D Embedding of the Event Horizon in $\mathbb{E}^{3}$
An event horizon can be embedded in the Euclidian three-dimensional space
$\mathbb{E}^{3}$ as a surface of revolution, by comparing the induced metric
on the horizon with that of the flat three-dimensional space in cylindrical
coordinates.
Indeed, the induced metric of an event horizon of radius $r_{H}$ is obtained
by setting $r=r_{H}$ and $t=constant$ in the complete spacetime metric, which
yields
$ds^{2}_{H}=f_{1}d\theta^{2}+f_{2}d\phi^{2}\,,$ (D.1)
where
$\displaystyle f_{1}$ $\displaystyle=g_{\theta\theta}(\theta,r=r_{H})\,,$
(D.2a) $\displaystyle f_{2}$ $\displaystyle=g_{\phi\phi}(\theta,r=r_{H})\,.$
(D.2b)
On the other hand, the flat three-dimensional metric in cylindrical
coordinates is given by
$ds_{3}^{2}=dz^{2}+dR^{2}+R^{2}d\phi^{2}\,.$ (D.3)
Therefore, considering $z$ and $R$ as functions of $\theta$, and comparing the
flat three-dimensional metric in cylindrical coordinates (D.3) with that
induced on the event horizon (D.1), it follows that
$R=f_{2}\,,$ (D.4)
while $z$ is determined by the following differential equation:
$\biggl{[}\frac{dR}{d\theta}\biggr{]}^{2}+\biggl{[}\frac{dz}{d\theta}\biggr{]}^{2}=f_{1}\,.$
(D.5)
In conclusion, the embedding of an event horizon in $\mathbb{E}^{3}$ is
determined once the functions $R=R\bigl{(}\theta\bigr{)}$ and
$z=z\bigl{(}\theta\bigr{)}$ are obtained.
## Appendix E Majorana Representation of the Gamma Matrices
The (real) Majorana representation of the gamma matrices
$\gamma_{a}=\bigl{(}\gamma_{0},\gamma_{1},\gamma_{2},\gamma_{3}\bigr{)}$ can
be obtained by applying the unitary transformation
$U=\frac{1}{\sqrt{2}}\begin{pmatrix}\mathds{1}_{2}&\sigma_{2}\\\
\sigma_{2}&-\mathds{1}_{2}\end{pmatrix}\,,$ (E.1)
to the gamma matrices $\gamma^{\prime}_{a}$ in the standard Dirac
representation:
$\gamma_{a}=U^{-1}\gamma^{\prime}_{a}U\,,$ (E.2)
which yields
$\displaystyle\gamma_{0}$ $\displaystyle=\begin{pmatrix}0&-i\sigma_{2}\\\
-i\sigma_{2}&0\end{pmatrix}\,,$ (E.3) $\displaystyle\gamma_{1}$
$\displaystyle=\begin{pmatrix}-\sigma_{3}&0\\\ 0&-\sigma_{3}\end{pmatrix}\,,$
(E.4) $\displaystyle\gamma_{2}$
$\displaystyle=\begin{pmatrix}0&-i\sigma_{2}\\\ i\sigma_{2}&0\end{pmatrix}\,,$
(E.5) $\displaystyle\gamma_{3}$ $\displaystyle=\begin{pmatrix}\sigma_{1}&0\\\
0&\sigma_{1}\end{pmatrix}\,,$ (E.6) $\displaystyle\gamma_{5}$
$\displaystyle=\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}=\begin{pmatrix}i\sigma_{2}&0\\\
0&-i\sigma_{2}\ \end{pmatrix}\,.$ (E.7)
Where the $\sigma_{j}$ denoted the standard representation of the two-
dimensional Pauli matrices
$\sigma_{j}=\begin{pmatrix}\delta_{j3}&\delta_{j1}-i\delta_{j2}\\\
\delta_{j1}+i\delta_{j2}&-\delta_{j3}\end{pmatrix}\,.$ (E.8)
## Bibliography
* [1] A. Einstein. Zur Allgemeinen Relativitätstheorie. [On the General Theory of Relativity]. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 1915:778–786, (1915). [Addendum: Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 799–801 (1915)].
* [2] A. Einstein. Die Grundlage der Allgemeinen Relativitätstheorie. [The Basis of the General Theory of Relativity.]. Annalen der Physik, 1916:769–822, (1916). doi:10.1002/andp.19163540702.
* [3] G.F.B. Riemann. Ueber die Hypothesen, welche der Geometrie zu Grunde liegen. [On the Hypotheses on which Geometry is Based]. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13:133–159, (1868).
* [4] A. Einstein. Zur Elektrodynamik bewegter Körper. [On the Electrodynamics of Moving Bodies]. Annalen der Physik, 322:891–921, (1905). doi:10.1002/andp.19053221004.
* [5] J.A. Wheeler and K. Ford. Geons, Black Holes, and Quantum Foam: A Life in Physics. W. W. Norton & Company, (1998).
* [6] C.W. Misner, K.S. Thorne, and J.A. Wheeler. Gravitation. W. H. Freeman and Company, (1973).
* [7] G.M. Clemence. The Relativity Effect in Planetary Motions. Rev. Mod. Phys., 19:361–364, (1947). doi:10.1103/RevModPhys.19.361.
* [8] F.W. Dyson, A.S. Eddington, and C. Davidson. A Determination of the Deflection of Light by the Sun’s Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919. Phil. Trans. Roy. Soc. Lond. A, 220:291–333, (1920). doi:10.1098/rsta.1920.0009.
* [9] R.V. Pound and G.A. Rebka. Gravitational Red-Shift in Nuclear Resonance. Phys. Rev. Lett., 3:439–441, (1959). doi:10.1103/PhysRevLett.3.439.
* [10] B.P. Abbott et al. Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett., 116:061102, (2016). arXiv:1602.03837, doi:10.1103/PhysRevLett.116.061102.
* [11] K. Schwarzschild. "Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie" [On the Gravitational Field of a Mass Point According to Einstein’s Theory]. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.), 7:189–196, (1916). arXiv:physics/9905030.
* [12] E.T. Newman and A.I. Janis. Note on the Kerr Spinning-Particle Metric. J. Math. Phys., 6:915–917, (1965). doi:10.1063/1.1704350.
* [13] E.T. Newman et al. Metric of a Rotating, Charged Mass. J. Math. Phys., 6:918–919, (1965). doi:10.1063/1.1704351.
* [14] M. Astorino, R. Martelli, and A. Viganò. Black Holes in a Swirling Universe. Phys. Rev. D, 106:064014, (2022). arXiv:2205.13548, doi:10.1103/PhysRevD.106.064014.
* [15] M.A. Melvin. Pure Magnetic and Electric Geons. Phys. Lett., 8:65–70, (1964). doi:10.1016/0031-9163(64)90801-7.
* [16] S.M. Carroll. Spacetime and Geometry: an Introduction to General Relativity. Addison Wesley, (2004). doi:10.1017/9781108770385.
* [17] R.M. Wald. General Relativity. Chicago Press, (1984). doi:10.7208/chicago/9780226870373.001.0001.
* [18] F.J. Ernst. New Formulation of the Axially Symmetric Gravitational Field Problem. Phys. Rev., 167:1175–1179, (1968). doi:10.1103/PhysRev.167.1175.
* [19] F.J. Ernst. New Formulation of the Axially Symmetric Gravitational Field Problem. II. Phys. Rev., 168:1415–1417, (1968). [Erratum: F.J. Ernst, Phys. Rev., 172(5):1850, (1968)]. doi:10.1103/PhysRev.168.1415.
* [20] A. Viganó. Black Holes and Solution Generating Techniques. Ph.D. Thesis, Milan U., (2022). arXiv:2211.00436.
* [21] R. Martelli. The Action of the Axisymmetric and Stationary Symmetry Group of General Relativity on a Static Black Hole. Bachelor Thesis, Milan U., (2022). URL: https://pcteserver.mi.infn.it/~molinari/TESI/Tesi_Riccardo_Martelli.pdf.
* [22] G. Dall’Agata and M. Zagermann. Supergravity: From First Principles to Modern Applications. Springer Berlin, (2021). doi:10.1007/978-3-662-63980-1.
* [23] L.J. Romans. Supersymmetric, Cold and Lukewarm Black Holes in Cosmological Einstein-Maxwell Theory. Nucl. Phys. B, 383:395–415, (1992). arXiv:hep-th/9203018, doi:10.1016/0550-3213(92)90684-4.
* [24] D. Klemm and M. Nozawa. Supersymmetry of the C-metric and the General Plebanski-Demianski Solution. JHEP, 05:123, (2013). arXiv:1303.3119, doi:10.1007/JHEP05(2013)123.
* [25] M.M. Caldarelli and D. Klemm. Supersymmetry of Anti-de Sitter Black Holes. Nucl. Phys. B, 545:434–460, (1999). arXiv:hep-th/9808097, doi:10.1016/S0550-3213(98)00846-3.
* [26] N. Alonso-Alberca, P. Meessen, and T. Ortin. Supersymmetry of Topological Kerr-Newman-Taub-NUT-adS Spacetimes. Class. Quant. Grav., 17:2783–2798, (2000). arXiv:hep-th/0003071, doi:10.1088/0264-9381/17/14/312.
* [27] D. Hilbert. Die Grundlagen der Physik. 1. [Foundations of Physics. 1]. Gott. Nachr., 27:395–407, (1915).
* [28] B. Carter. Killing Horizons and Orthogonally Transitive Groups in Space-time. J. Math. Phys., 10:70–81, (1969). doi:10.1063/1.1664763.
* [29] H. Weyl. Zur Gravitationstheorie [On the Theory of Gravity]. Annalen der Physik, 359:117–145, (1917). doi:10.1002/andp.19173591804.
* [30] T. Lewis. Some Special Solutions of the Equations of Axially Symmetric Gravitational Fields. Proc. Roy. Soc. Lond. A, 136:179–192, (1932). doi:10.1098/rspa.1932.0073.
* [31] A. Papapetrou. Eine Rotationssymmetrische Lösung in der Allgemeinen Relativitätstheorie [A Rotationally Symmetric Solution in General relativity]. Annalen der Physik, 447:309–315, (1953). doi:10.1002/andp.19534470412.
* [32] A. Papapetrou. Champs Gravitationnels Stationnaires à Symétrie axiale [Stationary Gravitational Fields with Axial Symmetry]. Ann. Inst. Henri Poincaré, 4:83–105, (1966).
* [33] M. Heusler. Black Hole Uniqueness Theorems. Cambridge Lecture Notes in Physics. Cambridge University Press, (1996). doi:10.1017/CBO9780511661396.
* [34] S. Chandrasekhar. The Mathematical Theory of Black Holes. Oxford University Press, (1983).
* [35] O. Aharony, M. Fabinger, G.T. Horowitz, and E. Silverstein. Clean Time-Dependent String Backgrounds from Bubble Baths. JHEP, 07:007, (2002). arXiv:hep-th/0204158, doi:10.1088/1126-6708/2002/07/007.
* [36] S.W. Hawking. Black Hole Explosions ? Nature, 248:30–31, (1974). doi:10.1038/248030a0.
* [37] J.D. Bekenstein. Black Holes and the Second Law. Lett. Nuovo Cim., 4:737–740, (1972). doi:10.1007/BF02757029.
* [38] S.W. Hawking. Particle Creation by Black Holes. Commun. Math. Phys., 43:199–220, (1975). [Erratum: Hawking, S.W. , Commun. Math. Phys. 46:206 (1976)]. doi:10.1007/BF02345020.
* [39] J.M. Bardeen, B. Carter, and S.W. Hawking. The Four Laws of Black Hole Mechanics. Commun. Math. Phys., 31:161–170, (1973). doi:10.1007/BF01645742.
* [40] J. Ehlers. Konstruktionen und Charakterisierung von Losungen der Einsteinschen Gravitationsfeldgleichungen [Constructions and Characterization of Solutions to Einstein’s gravitational Field Equations]. Ph.D. Thesis, Hamburg University, (1957).
* [41] B.K. Harrison. New Solutions of the Einstein-Maxwell Equations from Old. Journal of Mathematical Physics, 9:1744–1752, (1968). doi:10.1063/1.1664508.
* [42] H. Stephani, D. Kramer, et al. Exact Solutions of Einstein’s Field Equations. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, (2003). doi:10.1017/CBO9780511535185.
* [43] G. Neugebauer and D. Kramer. Eine Methode zur Konstruktion stationärer Einstein-Maxwell-Felder [A Method for Constructing Stationary Einstein-Maxwell fields]. Annalen der Physik, 479:62–71, (1969). doi:10.1002/andp.19694790108.
* [44] W. Kinnersley. Generation of Stationary Einstein-Maxwell Fields. J. Math. Phys., 14:651–653, (1973). doi:10.1063/1.1666373.
* [45] M. Astorino. Charging Axisymmetric Space-times with Cosmological Constant. JHEP, 06:086, (2012). arXiv:1205.6998, doi:10.1007/JHEP06(2012)086.
* [46] M.S. Illy. Accelerated Reissner–Nordström Black Hole in a Swirling, Magnetic Universe. Bachelor Thesis, Milan U., (2023). arXiv:2312.14995.
* [47] J. Barrientos, A. Cisterna, et al. Mixing "Magnetic” and "Electric” Ehlers–Harrison Transformations: The Electromagnetic Swirling Spacetime and Novel Type I Backgrounds. (2024). arXiv:2312.17347.
* [48] F.J. Ernst. Black Holes in a Magnetic Universe. J. Math. Phys., 17:54–56, (1976). doi:10.1063/1.522781.
* [49] A.H. Taub. Empty Space-times Admitting a Three Parameter Group of Motions. Annals Math., 53:472–490, (1951). doi:10.2307/1969567.
* [50] E. Newman, L. Tamburino, and T. Unti. Empty Space Generalization of the Schwarzschild Metric. J. Math. Phys., 4:915, (1963). doi:10.1063/1.1704018.
* [51] H. Reissner. Über die Eigengravitation des Elektrischen Feldes nach der Einsteinschen Theorie [About the Self-Gravity of the Electric Field According to Einstein’s theory]. Annalen der Physik, 355:106–120, (1916). doi:10.1002/andp.19163550905.
* [52] G. Nordström. On the Energy of the Gravitation Field in Einstein’s Theory. Koninklijke Nederlandse Akademie van Wetenschappen Proceedings Series B Physical Sciences, 20:1238–1245, (1918).
* [53] C. Reina and A. Treves. NUT-like Generalization of Axisymmetric Gravitational Fields. Journal of Mathematical Physics, 16:834–835, (1975). doi:10.1063/1.522614.
* [54] G.W. Gibbons and D.L. Wiltshire. Space-Time as a Membrane in Higher Dimensions. Nucl. Phys. B, 287:717–742, (1987). arXiv:hep-th/0109093, doi:10.1016/0550-3213(87)90125-8.
* [55] G.W. Gibbons and C.A.R. Herdeiro. The Melvin Universe in Born-Infeld Theory and Other Theories of Nonlinear Electrodynamics. Class. Quant. Grav., 18:1677–1690, (2001). arXiv:hep-th/0101229, doi:10.1088/0264-9381/18/9/305.
* [56] R. Emparan and M. Gutperle. From p-branes to Fluxbranes and Back. JHEP, 12:023, (2001). arXiv:hep-th/0111177, doi:10.1088/1126-6708/2001/12/023.
* [57] M. Astorino. Removal of Conical singularities From Rotating C-metrics and Dual CFT Entropy. JHEP, 10:074, (2022). arXiv:2207.14305, doi:10.1007/JHEP10(2022)074.
* [58] G.W. Gibbons, A.H. Mujtaba, and C.N. Pope. Ergoregions in Magnetised Black Hole Spacetimes. Class. Quant. Grav., 30:125008, (2013). arXiv:1301.3927, doi:10.1088/0264-9381/30/12/125008.
* [59] P.A.M. Dirac. Quantised Singularities in the Electromagnetic Fiels. Proc. Roy. Soc. Lond. A, 133:60–72, (1931). doi:10.1098/rspa.1931.0130.
* [60] P.A.M. Dirac. The Theory of Magnetic Poles. Phys. Rev., 74:817–830, (1948). doi:10.1103/PhysRev.74.817.
* [61] K. Gödel. An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation. Rev. Mod. Phys., 21:447–450, (1949). doi:10.1103/RevModPhys.21.447.
* [62] C.W. Misner. The Flatter Regions of Newman, Unti and Tamburino’s Generalized Schwarzschild Space. J. Math. Phys., 4:924–938, (1963). doi:10.1063/1.1704019.
* [63] D. Astefanesei, R.B. Mann, and E. Radu. Nut Charged Space-times and Closed Timelike Curves on the Boundary. JHEP, 01:049, (2005). arXiv:hep-th/0407110, doi:10.1088/1126-6708/2005/01/049.
* [64] M. Alcubierre. The Warp Drive: Hyperfast Travel Within General Relativity. Class. Quant. Grav., 11:L73–L77, (1994). arXiv:gr-qc/0009013, doi:10.1088/0264-9381/11/5/001.
* [65] M. Astorino. Magnetised Kerr/CFT Correspondence. Phys. Lett. B, 751:96–106, (2015). arXiv:1508.01583, doi:10.1016/j.physletb.2015.10.017.
* [66] H.M. Siahaan. Magnetized Kerr-Taub-NUT Spacetime and Kerr/CFT Correspondence. Phys. Lett. B, 820:136568, (2021). arXiv:2102.04345, doi:10.1016/j.physletb.2021.136568.
* [67] G. Compère. The Kerr/CFT Correspondence and its Extensions. Living Rev. Rel., 15:11, (2012). arXiv:1203.3561, doi:10.1007/s41114-017-0003-2.
* [68] D.Z. Freedman and A.K. Das. Gauge Internal Symmetry in Extended Supergravity. Nucl. Phys. B, 120:221–230, (1977). doi:10.1016/0550-3213(77)90041-4.
* [69] E.S. Fradkin and M.A. Vasiliev. Unpublished. Lebedev Institute, Preprint 197, (1976).
* [70] P. van Nieuwenhuizen and N.P. Warner. Integrability conditions for Killing spinors. Commun. Math. Phys., 93:277–284, (1984). doi:10.1007/BF01223747.
|
[a]Atreya Acharyya
# Identifying multiwavelength counterparts to astrophysical neutrino events
Marcos Santander
###### Abstract
High-energy neutrinos originating in astrophysical sources should be
accompanied by gamma-rays at production. Depending on the properties of the
emission environment and the distance of the source to the Earth, these gamma-
rays may be observed directly, or through the detection of lower energy
photons that result from interactions with the intervening radiation fields.
In this work, we present an automated tool that aims at using data from the
_Fermi_ -Large Area Telescope to identify multiwavelength counterparts to
astrophysical neutrino events. The main goal of this tool is to enable prompt
follow-up observations with ground-based and space-based observatories in
order to help pinpoint the neutrino source.
## 1 Introduction
The extragalactic gamma-ray sky is dominated by blazars [1], a subclass of
radio-loud active galactic nuclei (AGN) powered by a central supermassive
black hole (SMBH), with relativistic jets pointed close to our line-of-sight.
The spectral energy distribution (SED) of a typical blazar comprises two
distinct peaks. While the first peak, occurring in the radio to the X-ray
regime, has been attributed to synchrotron emission from electrons and
positrons within the jet, the physical mechanisms responsible for the second
peak, produced in the X-ray to gamma-ray regime, is still a matter of debate.
Towards understanding this, the _Fermi_ -Large Area Telescope (LAT, [2]) is a
pair conversion telescope capable of detecting gamma-ray photons in the energy
range between 20 MeV to above 500 GeV. Primarily operating in survey mode, it
scans the entire sky every three hours.
Leptonic models [3, 4] attribute the high-energy peak of the SED to the
inverse Compton (IC) scattering of electrons off a source of seed photons,
either the same photons emitted through synchrotron emission (synchrotron
self-Compton [SSC] model) or photon populations external to the jet (external
Inverse Compton [EIC] model). On the other hand, lepto-hadronic models [5, 6]
suggest that the second peak may be a result of either proton synchrotron or
the decay of high-energy mesons produced in cosmic-ray interactions. A
hadronic component of the gamma-ray emission [7] would potentially make AGN
prime candidate sources of astrophysical neutrinos [8, 9].
Since 2016, IceCube has been broadcasting automatic real-time alerts for
potential astrophysical neutrino candidate events in order to allow for prompt
follow-up observations with ground-based and space-based observatories. The
alerts are issued by the Astrophysical Multimessenger Observatory Network
(AMON111https://www.amon.psu.edu (accessed on 06/06/2023), [10]) and
circulated using the Gamma-ray Coordinates Network
(GCN)222https://gcn.gsfc.nasa.gov (accessed on 06/06/2023), an open-source
platform created by NASA to receive and transmit alerts about astronomical
transient phenomena. One of the main goals of these alerts is to allow for
follow-up observations with other observatories, in the hope of observing
multimessenger events.
In this work, we present an automated tool that aims at using _Fermi_ -LAT
data to identify multiwavelength counterparts to astrophysical neutrino events
with ground-based and space-based observatories in order to help pinpoint the
neutrino source. More specifically, we want to know what is in the region of
interest (RoI) around the neutrino alert, whether the RoI is observable from a
ground-based observatory and whether anything interesting, for example a
gamma-ray flare, is occurring at that particular location. In Section 2, we
introduce the main components of the analysis and processing pipeline. In
Section 3, we consider a typical example of a neutrino alert, IC230506A-gold-
rev1, and discuss the primary outputs from the analysis tool. We summarise our
conclusions and discuss some plans for future work in Section 4.
## 2 Methodology
Figure 1: A flowchart of the automated pipeline illustrating how each IceCube
GCN alert is processed and analyzed before the results are mirrored on to the
web server (https://multimessenger.ua.edu/fermi/, accessed on 06/06/2023).
The automated pipeline, illustrated in Fig. 1, listens for incoming
AMON_ICECUBE_GOLD and AMON_ICECUBE_BRONZE event alerts coming in through the
GCN. These are individual single-neutrino events in the IceCube detector with
a neutrino energy in the sub-PeV to 1 PeV energy regime and are shown in Fig.
2 for the time interval between June 19, 2019 and June 6, 2023.
More specifically, the analysis pipeline extracts information from the GCN
alert related to the type of event (GOLD or BRONZE), the sky coordinates of
the alert along with the associated confidence interval, the date and time of
the alert in UTC, and also the corresponding _Run_ID_ and _Event_ID_. The main
components of the tool include performing an automatic Fermi-LAT analysis of
the neutrino RoI, calculating the visibilities for common follow-up
instruments and collecting multiwavelength archival data for known sources in
the RoI. The entire process is then repeated for multiple revisions of a
particular alert.
The tool then runs an automatic analysis of the Fermi-LAT photons detected
from within the RoI of the neutrino alert over a 30 day interval prior to each
individual event. Each analysis uses the Fermi Science Tools version
$11-05-03$ 333http://fermi.gsfc.nasa.gov/ssc/data/analysis/software (accessed
on 06/06/2023), FERMIPY version 1.0.1 444http://fermipy.readthedocs.io
(accessed on 06/06/2023) [11] in conjunction with the PASS 8 instrument
response functions [12]. The contributions from the isotropic and Galactic
diffuse backgrounds are modeled using the most recent templates for isotropic
and Galactic diffuse emission, iso_P8R3_SOURCE_V2_v1.txt and gll_iem_v07.fits
respectively. Sources in the 4FGL-DR3 catalog [13] within a radius of
$20^{\circ}$ from the best-fit location of each alert are included in the
model with their spectral parameters fixed to their catalog values. The
gtfindsrc routine is also applied to search for any additional point sources
not accounted for in the model. Any source found to have a test statistic (TS,
[14]) $\geq$ 9 (roughly corresponding to a significance of $\sim$ 3$\sigma$)
is permanently added to the model at the position of its highest TS value.
Moreover, the normalization factor for both the isotropic and Galactic diffuse
emission templates are left free to vary, along with the spectral
normalization of all modeled sources within the RoI. Finally, the spectral
shape parameters of all modeled sources within 3∘ of the alert are left free
to vary while those of the remaining sources are fixed to the values reported
in the 4FGL-DR3 catalog. A binned likelihood analysis is then performed in
order to obtain the spectral parameters best describing the model during the
period of observation, using a spatial binning of 0.1∘ pixel-1 and two energy
bins per decade. The results, including HTML, FITS files, and images of plots,
are mirrored on to the web server (https://multimessenger.ua.edu/fermi/,
accessed on 06/06/2023).
Figure 2: A skymap, in celestial coordinates, showing AMON_ICECUBE_GOLD events
(in yellow) and AMON_ICECUBE_ BRONZE events (in red) between June 19, 2019 and
June 6, 2023, from the web server (https://multimessenger.ua.edu/fermi/,
accessed on 06/06/2023).
## 3 Discussion
Figure 3: Left: Top panel: A skymap of the neutrino alert, also containing all
sources in the confidence region from the 4FGL-DR3 and 3FHL catalog. Bottom
panel: These sources, listed separately, with links to the source entry in
_Simbad_ , as well as multiwavelength archival data repositories. Right: The
diagnostic plots produced in this _Fermi_ -LAT data analysis. These include a
skymap of the residual significance and the excess photon counts obtained in
the analysis over the RoI centred on the location of the alert. The colour
scales correspond to the excess significance of each pixel in the RoI in
Gaussian $\sigma$ and the number of excess photons at each pixel in the RoI
respectively.
Figure 4: Left: Top panel: The results obtained from the _Fermi_ -LAT
analysis, including the TS, flux and spectrum for both known 4FGL sources, as
well as potential new sources. Middle panel: A skymap and lightcurve of the
GeV photons emitted from within the RoI. Bottom panel: An example of a _Fermi_
-LAT spectrum for the source 4FGL J0325.7+2225 in red. The data are binned
into two energy bins per decade, with individual bins having a TS $<$ 4
(roughly corresponding to a significance of $\sim$ 2$\sigma$) considered as
upper limits. Also shown for comparison is the 4FGL-DR3 best-fit spectrum.
Right: Top panel: An interactive skymap containing markers for nearby sources
in the 4FGL, 3FHL, and 2WHSP catalog. Bottom panel: A visibility plot of the
RoI with VERITAS.
In this Section, we consider a typical example of a neutrino alert,
IC230506A-gold-
rev1555https://gcn.gsfc.nasa.gov/notices_amon_g_b/137910_29871391.amon
(accessed on 06/06/2023), received on 2023-05-06 at 15:53:45 UTC. This was a
revised AMON_ICECUBE_GOLD type alert, with _Run_ID_ = 137910, _Event_ID_ =
29871391, and the associated sky coordinates are RA = $50.19^{\circ}$, Dec =
$21.06^{\circ}$, with a corresponding error radius of $3.12^{\circ}$. The
plots produced by the automated tool are shown in Fig. 3 and Fig. 4.
The output results include a skymap of the neutrino alert, also containing all
sources in the confidence region from the 12-year _Fermi_ -LAT catalog (4FGL-
DR3, [13]), as well as the third _Fermi_ -LAT catalog of sources significantly
detected in the 10 GeV – 2 TeV energy range (3FHL, [15]). These sources are
also listed separately in a Table containing their 4FGL and 3FHL names, sky
coordinates, distance from the neutrino alert and links to the source entry in
_Simbad_ [16], as well as multiwavelength archival data repositories including
the _Fermi_ -LAT Light Curve Repository [17] and the _Fermi_ All-sky
Variability Analysis (FAVA, [18]) Light Curve Repository.
Also shown in Fig. 3, are the set of diagnostic plots produced in this _Fermi_
-LAT data analysis. These are obtained for each individual alert and
subsequent revision and include a skymap showing the modelled distribution of
the gamma-ray photons obtained after removing all sources found to have a TS <
9 followed by applying the gtfindsrc routine over the RoI, as well as a skymap
of the residual significance and the excess photon counts obtained in the
analysis over the RoI centred on the location of the alert.
Fig. 4 shows more results obtained from the _Fermi_ -LAT analysis, including
both known 4FGL sources having TS $\geq 9$, as well as new sources obtained
from the gtfindsrc routine. This includes the TS obtained for each source, the
measured flux values in the energy range 100 MeV – 300 GeV along with the
corresponding 4FGL flux values for known sources. There is also a link to
corresponding plots, such as the one shown in Fig. 4, of the _Fermi_ -LAT
spectrum for each source, with the 4FGL-DR3 best-fit spectrum [13] also shown
for comparison for known sources. Furthermore, we also obtain a skymap and
lightcurve of the GeV photons, emitted from within the RoI and having a $\geq$
99 $\%$ probability of originating from each individual source, over the one
month observation period investigated.
Moreover, the tool also produces an interactive skymap containing markers for
nearby sources in three catalogs, namely the 4FGL, 3FHL, and 2WHSP [19]. Other
catalogs, for example the 2RXS [20] and BZCat [21] plus a dynamic HiPS search
in _Simbad_ , can also be enabled. An additional _Simbad_ search can be made
to show a selection of objects detected in radio, X-rays, gamma-rays and
galaxies near the best fit position of the alert. The background sky can also
be changed to optical (Mellinger survey, [22]) using the radio buttons below
the image. Finally, we also produce visibility plots of the RoI for the four
ground-based instruments, namely MAGIC, H.E.S.S., VERITAS and HAWC, over the
time interval just after the alert, in order to help enable prompt follow-up
observations.
## 4 Conclusions and Future Work
In this work, we introduce an automated tool that aims at using _Fermi_ -LAT
data to identify multiwavelength counterparts to astrophysical neutrino events
and enable prompt follow-up observations with ground-based and space-based
observatories in order to help pinpoint the neutrino source. After discussing
the main components of the the analysis and processing pipeline, we walk-
through the primary outputs from the analysis tool for a typical example of a
neutrino alert, IC230506A-gold-rev1.
It should be noted that this is an early version of the automated tool with
plenty of scope for even further improvement. This includes, for example,
adding support for 1-year and full-mission _Fermi_ -LAT analysis alongside the
month long time period currently analyzed, gathering more multiwavelength data
and resources to enable SED construction, and finally improving compatibility
with the GCN Kafka Client setup 666https://gcn.nasa.gov/docs/client (accessed
on 06/06/2023).
## Acknowledgments
A.A. and M.S. acknowledge support through NASA grants 80NSSC22K1515,
80NSSC22K0950, 80NSSC20K1587, 80NSSC20K1494 and NSF grant PHY-1914579.
## References
* [1] C.D. Dermer and B. Giebels, _Active galactic nuclei at gamma-ray energies_ , _C R Phys._ 17 (2016) 594 [1602.06592].
* [2] W.B. Atwood, A.A. Abdo, M. Ackermann, W. Althouse, B. Anderson, M. Axelsson et al., _The Large Area Telescope on the Fermi Gamma-Ray Space Telescope Mission_ , _The Astrophysical Journal_ 697 (2009) 1071 [0902.1089].
* [3] R. Blandford and A. Levinson, _Pair Cascades in Extragalactic Jets. I. Gamma Rays_ , _The Astrophysical Journal_ 441 (1995) 79.
* [4] M. Georganopoulos, F.A. Aharonian and J.G. Kirk, _External Compton emission from relativistic jets in Galactic black hole candidates and ultraluminous X-ray sources_ , _The Astronomy and Astrophysics Journal_ 388 (2002) L25 [astro-ph/0110379].
* [5] G.E. Romero, D.F. Torres, M.M. Kaufman Bernadó and I.F. Mirabel, _Hadronic gamma-ray emission from windy microquasars_ , _The Astronomy and Astrophysics Journal_ 410 (2003) L1 [astro-ph/0309123].
* [6] M. Böttcher, A. Reimer, K. Sweeney and A. Prakash, _Leptonic and Hadronic Modeling of Fermi-detected Blazars_ , _The Astrophysical Journal_ 768 (2013) 54 [1304.0605].
* [7] K. Mannheim, _The proton blazar._ , _The Astronomy and Astrophysics Journal_ 269 (1993) 67 [astro-ph/9302006].
* [8] K. Mannheim, _High-energy neutrinos from extragalactic jets_ , _Astroparticle Physics_ 3 (1995) 295.
* [9] F. Halzen and E. Zas, _Neutrino Fluxes from Active Galaxies: A Model-Independent Estimate_ , _The Astrophysical Journal_ 488 (1997) 669 [astro-ph/9702193].
* [10] M.W.E. Smith, D.B. Fox, D.F. Cowen, P. Mészáros, G. Tešić, J. Fixelle et al., _The Astrophysical Multimessenger Observatory Network (AMON)_ , _Astroparticle Physics_ 45 (2013) 56 [1211.5602].
* [11] M. Wood, R. Caputo, E. Charles, M. Di Mauro, J. Magill, J.S. Perkins et al., _Fermipy: An open-source Python package for analysis of Fermi-LAT Data_ , in _35th International Cosmic Ray Conference (ICRC2017)_ , vol. 301 of _International Cosmic Ray Conference_ , p. 824, Jan., 2017 [1707.09551].
* [12] W. Atwood, A. Albert, L. Baldini, M. Tinivella, J. Bregeon, M. Pesce-Rollins et al., _Pass 8: Toward the Full Realization of the Fermi-LAT Scientific Potential_ , _arXiv e-prints_ (2013) arXiv:1303.3514 [1303.3514].
* [13] Fermi-LAT collaboration, :, S. Abdollahi, F. Acero, L. Baldini, J. Ballet et al., _Incremental Fermi Large Area Telescope Fourth Source Catalog_ , _arXiv e-prints_ (2022) arXiv:2201.11184 [2201.11184].
* [14] J.R. Mattox, D.L. Bertsch, J. Chiang, B.L. Dingus, S.W. Digel, J.A. Esposito et al., _The Likelihood Analysis of EGRET Data_ , _The Astrophysical Journal_ 461 (1996) 396.
* [15] M. Ajello, W.B. Atwood, L. Baldini, J. Ballet, G. Barbiellini, D. Bastieri et al., _3FHL: The Third Catalog of Hard Fermi-LAT Sources_ , _Astrophysical Journal Supplement Series_ 232 (2017) 18 [1702.00664].
* [16] M. Wenger, F. Ochsenbein, D. Egret, P. Dubois, F. Bonnarel, S. Borde et al., _The SIMBAD astronomical database. The CDS reference database for astronomical objects_ , _American Association of Pharmaceutical Scientists_ 143 (2000) 9 [astro-ph/0002110].
* [17] S. Abdollahi, M. Ajello, L. Baldini, J. Ballet, D. Bastieri, J. Becerra Gonzalez et al., _The Fermi-LAT Lightcurve Repository_ , _Astrophysical Journal Supplement Series_ 265 (2023) 31 [2301.01607].
* [18] S. Abdollahi, M. Ackermann, M. Ajello, A. Albert, L. Baldini, J. Ballet et al., _The Second Catalog of Flaring Gamma-Ray Sources from the Fermi All-sky Variability Analysis_ , _The Astrophysical Journal_ 846 (2017) 34 [1612.03165].
* [19] Y.L. Chang, B. Arsioli, P. Giommi and P. Padovani, _2WHSP: A multi-frequency selected catalogue of high energy and very high energy $\gamma$-ray blazars and blazar candidates_, _The Astronomy and Astrophysics Journal_ 598 (2017) A17 [1609.05808].
* [20] T. Boller, M.J. Freyberg, J. Trümper, F. Haberl, W. Voges and K. Nandra, _Second ROSAT all-sky survey (2RXS) source catalogue_ , _The Astronomy and Astrophysics Journal_ 588 (2016) A103 [1609.09244].
* [21] E. Massaro, P. Giommi, C. Leto, P. Marchegiani, A. Maselli, M. Perri et al., _Roma-BZCAT: a multifrequency catalogue of blazars_ , _The Astronomy and Astrophysics Journal_ 495 (2009) 691 [0810.2206].
* [22] A. Mellinger, _A Color All-Sky Panorama Image of the Milky Way_ , _Publications of the Astronomical Society of the Pacific_ 121 (2009) 1180 [0908.4360].
|
# Interplay of nonlocality and incompatibility breaking qubit channels
Swati Kumari<EMAIL_ADDRESS>Department of Physics and Center for
Quantum Frontiers of Research & Technology (QFort), National Cheng Kung
University, Tainan 701, Taiwan Javid Naikoo<EMAIL_ADDRESS>Centre
for Quantum Optical Technologies, Centre of New Technologies, University of
Warsaw, Banacha 2c, 02-097 Warsaw, Poland Sibasish Ghosh
<EMAIL_ADDRESS>Optics and Quantum Information Group, The Institute of
Mathematical Sciences, HNBI, CIT Campus, Taramani, Chennai 600113, India A.
K. Pan<EMAIL_ADDRESS>Department of physics, Indian Institute of
Technology Hyderabad Kandi,Telengana, India
###### Abstract
Incompatibility and nonlocality are not only of foundational interest but also
act as important resources for quantum information theory. In the Clauser-
Horne-Shimony-Holt (CHSH) scenario, the incompatibility of a pair of
observables is known to be equivalent to Bell nonlocality. Here, we
investigate these notions in the context of qubit channels. The Bell-CHSH
inequality has a greater perspective—compared to any genuine tripartite
nonlocality scenario—while determining the interplay between nonlocality
breaking qubit channels and incompatibility breaking qubit channels. In the
Bell-CHSH scenario, we prove that if the conjugate of a channel is
incompatibility breaking, then the channel is itself nonlocality breaking and
vice versa. However, this equivalence is not straightforwardly generalized to
multipartite systems, due to the absence of an equivalence relation between
incompatibility and nonlocality in the multipartite scenario. We investigate
this relation in the tripartite scenario by considering some well-known states
like Greenberger-Horne-Zeilinger and $W$ states and using the notion of Mermin
and Svetlichny nonlocality. By subjecting the parties in question to unital
qubit channels, we identify the range of state and channel parameters for
which incompatibility coexists with nonlocality. Further, we identify the set
of unital qubit channels that is Mermin or Svetlichny nonlocality breaking
irrespective of the input state.
## I Introduction
Nonlocality is one of the most profound notions in quantum mechanics Bell
(1964) and is often discussed in conjunction with incompatibility of
observables. Recent developments in quantum information theory have found
nonlocality a useful phenomenon underpinning many advantages afforded by
various quantum information processing tasks Brunner _et al._ (2014).
Nonlocality can also be considered as a potential quantum resource for
information processing, such as in developing quantum protocols to reduce the
amount of communication needed in certain computational tasks Brunner _et
al._ (2014) and providing secure quantum communications Scarani and Gisin
(2001); Ekert (1991). Incompatibility, like nonlocality, is not merely of
theoretical interest but of practical utility, for example, in order to
explore the advantage of entanglement shared by two parties in a cryptography
task, each party needs to carry out measurements that are incompatible, in the
sense that these cannot be carried out simultaneously by a single measurement
device. Incompatibility should not be confused with noncommutativity or the
related concept of uncertainty principle. The notion of incompatibility is
best understood in terms of joint measurability Heinosaari _et al._ (2016). A
collection of quantum measurements is jointly measurable, if it can be
simulated by a single common quantum measurement device. If such a single
common device cannot be constructed by a given set of quantum measurements, it
then enables the set to be used as a quantum resource. This was first noted in
Wolf _et al._ (2009) in the context of Clauser-Horne-Shimony-Holt (CHSH)
inequalities and later in the Einstein-Podolsky-Rosen steering, which is more
explicit, when incompatibility appears as a quantum resource. Incompatibility
is necessary and sufficient for the violation of the steering inequalities
Quintino _et al._ (2014); Uola _et al._ (2015). The relation between
incompatibility and contextuality has also been studied in references Kochen
and Specker (1975); Budroni _et al._ (2022). Further, a set of observables
that is pairwise incompatible, but not triplewise can violate the Liang-
Spekkens-Wiseman noncontextuality inequality Liang _et al._ (2011). Recently,
the connection between steerability and measurement incompatibility was
studied in Ku _et al._ (2022a) in the context of the so-called steerability
equivalent observables. Thus, both nonlocality and incompatibility can be
considered as quantum resources whose understanding is of utmost importance in
view of emerging quantum technologies.
The interplay of nonlocality and incompatibility has been a subject matter of
various studies. It is well known that any incompatible local measurements,
performed by the constituent parties of a system, lead to the violation of
Bell inequality provided they share a pure entangled state Bell (1964);
Brunner _et al._ (2014). Absence of either of them (i.e., entanglement or
incompatibility) will not allow the system to exhibit nonlocality. It is
important to mention here that the notion of quantum nonlocality without
entanglement has been proposed in Bennett _et al._ (1999) which is different
from Bell nonlocality Bell (1964) and amounts to the inability of
discriminating a set of product states by local operations and classical
communication, while mutual orthogonality of the states assures their perfect
global discrimination.
Further, for any pair of dichotomic incompatible observables, there always
exists an entangled state which enables the violation of a Bell inequality
Wolf _et al._ (2009). The relationship of incompatibility and nonlocality is
sensitive to the dimension of the system; for example, increasing the
dimension beyond $2$, the incompatible observables do not necessarily lead to
the violation of Bell-type inequalities, implying that the measurement
incompatibility can not guarantee nonlocality in general Bene and Vértesi
(2018); Hirsch _et al._ (2018). Here, we probe the interplay between
incompatibility and nonlocality in the tripartite case by using the well-known
Mermin and Svetlichny inequalities Svetlichny (1987). The Svetlichny
inequality, unlike the Mermin inequality, is a genuine measure of nonlocality
that assumes nonlocal correlations between two parties which are locally
related to a third party and is known to provide a suitable measure to detect
tripartite nonlocality for $W$ and Greenberger-Horne-Zeilinger (GHZ) classes
of states Ajoy and Rungta (2010). We refer the interested reader to Bancal
_et al._ (2013); Brunner _et al._ (2014) for various facets of the
multipartite nonlocality.
The extent to which a system can exhibit nonlocal correlations is also
sensitive to its interaction with the ambient environment. Such interaction is
usually accompanied with a depletion of various quantum features like
coherence, entanglement and nonlocality. The reduced dynamics of the system in
such cases is given by completely positive and trace preserving maps, also
known as quantum channels (QCs). On the other hand, the action of conjugate
channels on projective measurements turns them into unsharp positive operator-
valued measures (POVMs) which may be biased, in general. In light of the above
discussion, a study of the open system effects on the interplay of nonlocality
and incompatibility naturally leads to the notions of nonlocality breaking and
incompatibility breaking quantum channels Pal and Ghosh (2015); Heinosaari
_et al._ (2016). A nonlocality breaking channel (NBC) can be defined as a
channel which when applied to a system (or part of it) leads to a state which
is local Pal and Ghosh (2015), while the incompatibility breaking channel
(IBC) is the one that turns incompatible observables into compatible ones
Heinosaari _et al._ (2015); Heinosaari and Miyadera (2017). An IBC that
renders any set of $n$ incompatible observables compatible would be denoted by
$n$-IBC. The notion of the NBC has been introduced in a similar spirit of
well-studied entanglement breaking channels Horodecki _et al._ (2003). Every
entanglement breaking channel is nonlocality breaking but the converse is not
true. As an example, the qubit depolarizing channel
$\mathcal{E}\big{(}\rho\big{)}:=p\big{(}\rho\big{)}+(1-p)\mathbf{I}/2$ is CHSH
nonlocality breaking for all $\frac{1}{3}\leq p\leq\frac{1}{2}$, but not
entanglement breaking Pal and Ghosh (2015). Hence, based on this
classification, nonlocality and entanglement emerge as different resources.
The equivalence of the steerability breaking channels and the incompatibility
breaking channels was reported in Kiukas _et al._ (2017) and CHSH nonlocality
breaking channels were shown to be a strict subset of the steerability
breaking channels Ku _et al._ (2022b). The connection between Bell
nonlocality and incompatibility of two observable is well understood, however,
the question of the equivalence between NBC Pal and Ghosh (2015) and IBC
Heinosaari and Miyadera (2017) is rarely discussed. This motivates us to
explore the relation between CHSH nonlocality breaking channels (CHSH-NBC) and
$2$-IBC, such that the action of one may be replaced by the other. The
tripartite nonlocality has much richer and complex structure and less is known
about its synergy with incompatibility as compared to its bipartite
counterpart. Mermin inequality assumes local-realistic correlations among all
the three qubits; hence a violation would be a signature of the tripartite
nonlocality shared among the qubits. However, biseparable states were shown to
also violate the Mermin inequality Collins _et al._ (2002); Scarani and Gisin
(2001). This motivated Svetlichny to introduce the notion of genuine
tripartite nonlocality Svetlichny (1987) and provide a set of inequalities
sufficient to witness it. We make use of these notions of _absolute_ and
_genuine_ nonlocality to figure out the ranges of state and channels
parameters in which NBC and $2$-IBC coexist.
The paper is organized as follows. In Sec. II, we revisit some basic notions
and definitions used in this paper. Section III is devoted to results and
their discussion where we prove an equivalence between NBCs and $2$-IBCs in
the CHSH scenario. This is followed by an analysis of these notions in the
tripartite scenario, where we identify the state and channel parameters in
which NBCs and 2-IBCs co-exist. A conclusion is given in Sec. IV.
## II Preliminaries
In this section, we discuss the notion of incompatibility in the context of
observables and quantum channels and look at specific cases of bipartite and
tripartite scenarios.
### II.1 Incompatibility
#### II.1.1 Incompatibility of observables
A finite collection of observables $A_{1},\dots,A_{n}$ associated with the
respective outcome spaces $\Omega_{A_{1}},\dots,\Omega_{A_{n}}$, is said to be
compatible (or jointly measurable) if there exists a joint observable $G$,
defined over the product outcome space
$\Omega_{A_{1}}\times\cdots\times\Omega_{A_{n}}$, such that for all
$X_{1}\subset\Omega_{A_{1}},\dots,X_{n}\subset\Omega_{A_{n}}$, the following
marginal relations hold Busch (1986):
$\displaystyle\sum_{a_{i},i=1,\dots,n;i\neq
k}G(a_{1},\dots,a_{n})=A_{k}(a_{k})$ (1)
where $a_{k}$ is the outcome associated to observable $A_{k}$ and the
summation is carried out over all outcomes $a_{i}$ except for $i=k$. The
notion of the incompatibility of observables can be illustrated by a simple
example of Pauli matrices $\sigma_{x}$ and $\sigma_{z}$ which are noncommuting
and can not be measured jointly. However, consider the unsharp observables
$S_{x}(\pm)=\frac{1}{2}(\mathbf{I}\pm\frac{1}{\sqrt{2}}\sigma_{x})$ and
$S_{z}(\pm)=\frac{1}{2}(\mathbf{I}\pm\frac{1}{\sqrt{2}}\sigma_{z})$, with
$\mathbf{I}$ being the $2\times 2$ identity matrix. The joint observable
$G(i,j)=\frac{1}{4}(\mathbf{I}+\frac{i}{\sqrt{2}}\sigma_{x}+\frac{j}{\sqrt{2}}\sigma_{z})$
with $i,j=\pm 1$ jointly determines the probabilities of generalized
measurements $S_{x}$ and $S_{z}$, since the later can be obtained as marginals
$S_{x}(\pm)=\sum_{j}G(\pm,j)$ and $S_{z}(\pm)=\sum_{i}G(i,\pm)$.
#### II.1.2 Incompatibility breaking quantum channel
A QC, in the Schrödinger picture, is a completely positive trace preserving
map
$\mathcal{E}:\mathcal{L}({\mathcal{H^{A}}})\rightarrow{\mathcal{L}}({\mathcal{H^{B}}})$,
where ${\mathcal{L}}({\mathcal{H}^{i}})$ is the set of bounded linear
operators on Hilbert space $\mathcal{H}^{i}$ ($i=A,B$). One may write the
operator sum representation Nielsen and Chuang (2010)
$\displaystyle\rho^{\prime}=\mathcal{E}\big{(}\rho\big{)}=\sum_{j=1}^{n}K_{j}\rho
K_{j}^{\dagger}$ (2)
where $K_{i}$ are known as Kraus operators satisfying the completeness
relation $\sum_{j}K^{\dagger}_{j}K_{j}=\mathbf{I}$. The QCs which map the
identity operator to itself, i.e., $\mathcal{E}(\mathbf{I})=\mathbf{I}$, are
known as unital QCs. A quantum channel $\mathcal{E}$ in the Schrödinger
picture acting on quantum state $\rho$ can be thought of as conjugate channel
$\mathcal{E}^{*}$ acting on observable $A$ through the following duality
relation
$\Tr\left[\mathcal{E}(\rho)A\right]=\Tr\left[\rho~{}\mathcal{E}^{*}(A)\right].$
(3)
###### Definition 1.
A quantum channel $\mathcal{E}$ is said to be incompatibility breaking if the
outputs $\mathcal{E}^{*}(A_{1}),\dots,\mathcal{E}^{*}(A_{n})$ are compatible
for any choice of input observables $A_{1},\dots,A_{n}$ $(n\geq 2)$.
If a channel $\mathcal{E}$ breaks the incompatibility of every class of $n$
observables, it is said to be $n-$ IBC. For example, a channel would be
$2-{\rm IBC}$ if it breaks the incompatibility of a pair of observables. As an
example, the white noise mixing channel $\mathcal{W}_{\eta}$ is described as
$\rho^{\prime}=\mathcal{W}_{\eta}[\rho]=\eta\rho+(1-\eta)\frac{\mathbf{I}}{d}.$
(4)
Here, $\eta$ is the channel parameter, $d$ is the dimension of the underlying
Hilbert space. This channel is $n$-incompatibility breaking for all
$0\leq\eta\leq\frac{n+d}{n(d+1)}$. With $d=2$, $\mathcal{W}_{\eta}$ is $2-{\rm
IBC}$ and $3-{\rm IBC}$ for $\eta\leq 0.66$ and $0.55$, respectively
Heinosaari and Miyadera (2017).
#### II.1.3 Incompatibility of generalized spin-observables
Consider a spin observable $A=\hat{a}\cdot\vec{\sigma}$ with projectors
$P_{\pm}(\vec{a})=\frac{1}{2}(\mathbf{I}\pm A)$, with $|\vec{a}|=1$. In
presence of a noise channel $\mathcal{E}$, these projectors are mapped to the
noise induced POVM $P_{\pm}(\vec{\alpha})$, by the transformation
$A\xrightarrow[]{\mathcal{E}^{*}}A^{(x,\eta)}=x\mathbf{I}+\eta\hat{a}\cdot\vec{\sigma}.$
(5)
Here, $x$ and $\eta$ characterize the bias and sharpness such that un-biased
projective measurements correspond to $x=0$ and $\eta=1$. Two biased and
unsharp observables $A^{(x,\eta)}$ and $B^{(y,\xi)}$ are jointly measurable if
Yu _et al._ (2010); Heinosaari (2020)
$\left(1-S(x,\eta)^{2}-S(y,\xi)^{2}\right)\left(1-\frac{\eta^{2}}{S(x,\eta)^{2}}-\frac{\xi^{2}}{S(y,\xi)^{2}}\right)\leq\left(\vec{\eta}\cdot\vec{\xi}-xy\right),$
(6)
where $\vec{\eta}=\eta\hat{a}$ and $\vec{\xi}=\xi\hat{b}$ are the unsharpness
parameters, with
$S(p,q)=\frac{1}{2}\left(\sqrt{\left(1+p\right)^{2}-q^{2}}+\sqrt{\left(1-p\right)^{2}-q^{2}}\right).$
(7)
For unbiased observables $A^{(0,\eta)}$ and $B^{(0,\xi)}$, the necessary and
sufficient condition for compatibility simplifies to the following
$\displaystyle|\vec{\eta}+\vec{\xi}|+|\vec{\eta}-\vec{\xi}|\leq 2.$ (8)
When both the observables are subjected to identical noise channel i.e.,
$\xi=\eta$ and $\vec{\eta}$ and $\vec{\xi}$ are perpendicular, then the
pairwise joint measurability condition Eq. (8) becomes
$\eta\leq\frac{1}{\sqrt{2}}.$ (9)
This provides the condition for the incompatibility breaking of two
observables and the corresponding (unital) channel is called an
incompatibility breaking channel ($2-{\rm IBC}$).
### II.2 Nonlocality
#### II.2.1 Bipartite nonlocality: CHSH inequality
Consider the scenario of two spatially separated qubits with observables
$\hat{A}_{i}=\sum_{k=1}^{3}{\hat{a_{ik}}}\cdot{\hat{\sigma_{k}}}$ and
$\hat{B}_{j}=\sum_{l=1}^{3}{\hat{b_{jl}}}\cdot{\hat{\sigma_{l}}}$ acting on
each qubit respectively, where $\hat{a_{ik}}$ and $\hat{b_{jl}}$ are unit
vectors in $\mathbb{R}^{3}$; $i,j=1,2$; and the $\hat{\sigma}_{i}$’s are spin
projection operators. The Bell operator associated with the CHSH inequality
has the form
$\hat{\mathcal{B}}=\hat{A}_{1}\otimes(\hat{B}_{1}+\hat{B}_{2})+\hat{A}_{2}\otimes(\hat{B}_{1}-\hat{B}_{2}),$
(10)
Then the Bell-CHSH inequality, for any state $\rho$ is
$\Tr\left[\rho~{}\hat{\mathcal{B}}(\hat{A}_{1},~{}\hat{A}_{2},~{}\hat{B}_{1},~{}\hat{B}_{2})\right]\leq
2.$ (11)
where the observables on Alice’s and Bob’s side are pairwise compatible. The
violation of the above inequality (11) is sufficient to justify the
nonlocality of the quantum state. Since incompatible observables acting on
entangled particles enable nonlocality, thus incompatibility is necessary for
violation of (11).
The essence of Eq. (3) is that the effect of noise as decoherence of the
nonlocal resource $\rho$, can be interpreted as distortion of Alice’s local
measurement resource (incompatibility). This fact was exploited by Pal and
Ghosh Pal and Ghosh (2015) to introduce the notion of CHSH-NBC defined below.
###### Definition 2.
Any qubit channel
$\mathcal{E}:\mathcal{E}({\mathcal{H}}^{i})\rightarrow{\mathcal{E}}({\mathcal{H}}^{i})$
is said to be ${\rm IBC}$ if applying on one side of (arbitrary) bipartite
state $\rho_{AB}$, it produces a state
$\rho^{\prime}_{AB}=(\mathbf{I}\otimes\mathcal{E})(\rho_{AB})$ which satisfies
the Bell-CHSH inequality (11).
This means that for any choice of POVMs $\\{\pi_{a|x}^{A}\\}$ and
$\\{\pi_{b|y}^{B}\\}$ on subsystems $A$ and $B$, respectively, there exist
conditional distributions $P(a|x,\lambda)$ and $P(b|y,\lambda)$ and shared
variable $\lambda$, such that
$\displaystyle\Tr\left[(\pi_{a|x}^{A}\otimes\pi_{b|y}^{B})\sigma^{AB}\right]=\int
d\lambda~{}p(\lambda)~{}P(a|x,\lambda)~{}P(b|y,\lambda).$ (12)
The violation of the Bell inequalities in the measurement statistics
$P(ab|xy,\lambda)=\Tr[(\pi_{a|x}^{A}\otimes\pi_{b|y}^{B})\sigma^{AB}]$
indicates that $\sigma^{AB}$ is not a local state. A unital channel is
particularly important as it breaks the nonlocality for any state when it does
so for maximally entangled states Pal and Ghosh (2015).
#### II.2.2 Tripartite nonlocality: Svelitchny inequality
In a tripartite Bell scenario, with Alice, Bob, and Charlie performing
measurements $A$, $B$ and $C$ having outcome $a$, $b$ and $c$ respectively, if
the joint correlations can be written as
$\displaystyle P(abc|ABC)$
$\displaystyle=\sum_{l}c_{l}P_{l}(a|A)P_{l}(b|B)P_{l}(c|C)$ (13)
with $0\leq c_{l}\leq 1$ and $\sum_{l}c_{l}=1$, then they are local Mermin
(1990); Brunner _et al._ (2014). Svetlichny Svetlichny (1987) proposed the
hybrid local nonlocal form of probability correlations
$\displaystyle P(abc|ABC)$
$\displaystyle=\sum_{l}c_{l}P_{l}(ab|AB)P_{l}(c|C)+\sum_{m}c_{l}P_{m}(ac|AC)P_{m}(b|B)$
$\displaystyle+\sum_{n}c_{l}P_{n}(bc|C)P_{n}(a|A),$ (14)
with $0\leq c_{l},c_{m},c_{n}\leq 1$ and
$\sum_{l}c_{l}+\sum_{m}c_{m}+\sum_{n}c_{n}=1$. The quantum version involves
triplets of particles subjected to independent dichotomic measurements with
operators $\hat{A}_{1}$ and $\hat{A}_{2}$ for Alice, $\hat{B}_{1}$ and
$\hat{B}_{2}$ for Bob, and $\hat{C}_{1}$ and $\hat{C}_{2}$ for Charlie, with
each measurement resulting in outcome $\pm 1$. One defines the Mermin operator
Mermin (1990) as
$\hat{M}=\hat{A}_{1}\otimes\hat{B}_{1}\otimes\hat{C}_{2}+\hat{A}_{1}\otimes\hat{B}_{2}\otimes\hat{C}_{1}+\hat{A}_{2}\otimes\hat{B}_{1}\otimes\hat{C}_{1}-\hat{A}_{2}\otimes\hat{B}_{2}\otimes\hat{C}_{2}.$
(15)
Similarly, the Svetlichny operator is defined as Svetlichny (1987); Cereceda
(2002)
$\displaystyle\hat{S}$ $\displaystyle=$
$\displaystyle\hat{A}_{1}\otimes\big{[}(\hat{B}_{1}+\hat{B}_{2})\otimes\hat{C}_{1}+(\hat{B}_{1}-\hat{B}_{2})\otimes\hat{C}_{2}\big{]}$
(16) $\displaystyle+$
$\displaystyle\hat{A}_{2}\otimes\big{[}(\hat{B}_{1}-\hat{B}_{2})\otimes\hat{C}_{1}-(\hat{B}_{1}+\hat{B}_{2})\otimes\hat{C}_{2}\big{]}.$
The respective average values of these operators are classically upper bounded
in the form of the Mermin and Svetlichny inequalities written as
$|\langle\hat{M}\rangle|\leq 2,\qquad|\langle\hat{S}\rangle|\leq 4.$ (17)
The maximum quantum bounds are known to be $2\sqrt{2}$ and $4\sqrt{2}$ for
$|\langle\hat{M}\rangle|$ and $|\langle\hat{S}\rangle|$, respectively, and can
be attained by, for example, the GHZ state Mitchell _et al._ (2004). Several
approaches have been adopted to find the maximum quantum value of the
Svetlichny operator Ghose _et al._ (2009); Su _et al._ (2018). Recently,
Siddiqui and Sazim (2019); Li _et al._ (2017) have analytically found the
tight upper bound of the Mermin and Svetlichny operator as given below.
###### Definition 3.
For any three-qubit quantum state $\rho$, the maximum quantum value of the
Mermin and the Svetlichny operator is bounded as Li _et al._ (2017); Siddiqui
and Sazim (2019)
$\displaystyle\max|\langle\hat{M}\rangle_{\rho}|\leq
2\sqrt{2}\lambda_{1},\qquad\max|\langle\hat{S}\rangle_{\rho}|\leq
4\lambda_{1},$ (18)
where $\langle\hat{M}\rangle_{\rho}=\Tr[\hat{M}\rho]$,
$\langle\hat{S}\rangle_{\rho}=\Tr[\hat{S}\rho]$ and $\lambda_{1}$ is the
maximum singular value of the matrix $M=(M_{j,ik})$, with
$M=(M_{ijk}=\Tr[\rho(\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k})]$,
$i,j,k=1,2,3$.
We refer the reader to Siddiqui and Sazim (2019); Li _et al._ (2017) for the
class of states that saturate the above inequalities. We make use of above
mentioned bounds to study the nonlocality breaking property of (unital)
channels acting on one (or more) party in the tripartite scenario.
## III Results and discussion
In what follows, we will make use of the fact that every qubit channel
$\mathcal{E}$ can be represented in the Pauli basis
$\\{\sigma_{0},\sigma_{1},\sigma_{2},\sigma_{3}\\}$, where
$\sigma_{0}=\mathbf{I}$, by a unique $4\times 4$ matrix
$\mathbb{M}_{\mathcal{E}}=[1,\bm{0};\textbf{t},\textbf{T}]$ Beth Ruskai _et
al._ (2002); Naikoo _et al._ (2021). Here $\textbf{T}={\rm
diag.}[\eta_{1},\eta_{2},\eta_{3}]$ is a real diagonal matrix and
$\bm{0}=(0~{}0~{}0)$ and $\textbf{t}=(t_{1}~{}t_{2}~{}t_{2})^{T}$ are row and
column vectors, respectively. For $\mathcal{E}$ to be unital i.e.,
$\mathcal{E}(\mathbf{I})=\mathbf{I}$, we must have
$\textbf{t}=(0~{}0~{}0)^{T}$. The conjugate map $\mathcal{E}^{\dagger}$ is
characterized by
$\mathbb{M}^{\dagger}_{\mathcal{E}}=[1,\textbf{t}^{T};\bm{0}^{T},\textbf{T}]$,
such that the action on a state
$\rho=\frac{1}{2}(\mathbf{I}+\textbf{w}\cdot\bm{\sigma})$ is given by
$\displaystyle\mathbb{M}_{\mathcal{E}}:\mathbf{I}\rightarrow\mathbf{I}+\textbf{t}\cdot\bm{\sigma},\quad\sigma_{j}\rightarrow\eta_{j}\sigma_{j},$
(19)
$\displaystyle\mathbb{M}^{\dagger}_{\mathcal{E}}:\mathbf{I}\rightarrow\mathbf{I},\quad\sigma_{j}\rightarrow
t_{j}\mathbf{I}+\eta_{j}\sigma_{j}.$ (20)
### III.1 Equivalance of CHSH nonlocality breaking and incompatibility
breaking channels
Our first result establishes an equivalence of the CHSH nonlocality breaking
channel acting on one party, with its dual being an incompatibility breaking
channel – in the context of $2-{\rm IBC}$s. The result can be summarized by
the following two theorems:
###### Theorem 1.
If the conjugate of a qubit channel ${\rm\mathcal{E}}$ is $2-{\rm IBC}$ , then
the channel itself is CHSH-NBC.
###### Proof.
Consider the Bell-CHSH inequality given in (11), such that $[A_{1},A_{2}]\neq
0$ and $[B_{1},B_{2}]\neq 0$, i.e., the operators $A_{1}$ and $A_{2}$ and
$B_{1}$ and $B_{2}$ are incompatible in conjunction. Let
$\mathcal{E}^{\dagger}$ be the conjugate channel that is $2-{\rm IBC}$. Then
the action of this channel on Alice’s side makes $A_{1}$ and $A_{2}$
compatible, i.e., $[A_{1},A_{2}]=0$. Therefore, the Bell-CHSH inequality is
not violated Wolf _et al._ (2009), and we have
$\Tr\left[\rho\mathcal{B}\left(\mathcal{E}^{\dagger}[A_{1}],\mathcal{E}^{\dagger}[A_{2}],B_{1},B_{2}\right)\right]\leq
2.$ (21)
Alternatively,
$\displaystyle\Tr\left[\rho\mathcal{E}^{\dagger}[A_{1}]\otimes
B_{1}\right]+\Tr\left[\rho\mathcal{E}^{\dagger}[A_{2}]\otimes B_{1}\right]$
$\displaystyle+\Tr\left[\rho\mathcal{E}^{\dagger}[A_{1}]\otimes
B_{2}\right]-\Tr\left[\rho\mathcal{E}^{\dagger}[A_{2}]\otimes B_{2}\right]\leq
2.$ (22)
This can be viewed in the Schrödinger picture as
$\displaystyle\Tr\left[(\mathcal{E}\otimes\mathbf{I})[\rho]A_{1}\otimes
B_{1}\right]+\Tr\left[(\mathcal{E}\otimes\mathbf{I})[\rho]A_{2}\otimes
B_{1}\right]$
$\displaystyle+\Tr\left[(\mathcal{E}\otimes\mathbf{I})[\rho]A_{1}\otimes
B_{2}\right]-\Tr\left[(\mathcal{E}\otimes\mathbf{I})[\rho]A_{2}\otimes
B_{2}\right]\leq 2,$ (23)
which tells us that the CHSH inequality is satisfied even when operators
$A_{1}$ and $A_{2}$ and $B_{1}$ and $B_{2}$ are incompatible in conjunction.
Therefore, the action of $\mathcal{E}$ (to be precise of
$\mathcal{E}\otimes\mathbf{I}$) on state $\rho$ is solely responsible for
nonviolation of the CHSH inequality. We conclude that $\mathcal{E}$ is CHSH-
NBC. ∎
###### Theorem 2.
If a qubit channel $\mathcal{E}$ is CHSH-NBC, then its conjugate is $2-{\rm
IBC}$.
###### Proof.
Here we start with incompatible operators associated with the respective
subsystems, $[A_{1},A_{2}]\neq 0$ and $[B_{1},B_{2}]\neq 0$ and assume that
the channel $\mathcal{E}$ acting on Alice’s side does not allow for the
violation of the CHSH inequality, that is
$\Tr\left[\left(\mathcal{E}\otimes\mathbf{I}\right)[\rho]\mathcal{B}\left(A_{1},A_{2},B_{1},B_{2}\right)\right]\leq
2.$ (24)
In other word, looking from the measurement point of view, in the Heisenberg
picture, we have
$\displaystyle\Tr\left[\rho\mathcal{E}^{\dagger}[A_{1}]\otimes
B_{1}\right]+\Tr\left[\rho\mathcal{E}^{\dagger}[A_{1}]\otimes B_{2}\right]$
$\displaystyle+\Tr\left[\rho\mathcal{E}^{\dagger}[A_{2}]\otimes
B_{1}\right]-\Tr[\rho\mathcal{E}^{\dagger}[A_{2}]\otimes B_{2}]\leq 2$ (25)
The above inequality holds for arbitrary state $\rho$ which can even be an
entangled state. Thus the nonviolation of the CHSH inequality is coming from
the action of $\mathcal{E}^{\dagger}$ on the operators $A_{1}$ and $A_{2}$,
making them compatible,
$\left[\mathcal{E}^{\dagger}[A_{1}],\mathcal{E}^{\dagger}[A_{2}]\right]=0$. We
conclude that $(\mathcal{E}^{\dagger}\otimes\mathbf{I})$ is incompatibility
breaking. ∎
In order to verify the above results, let us consider the CHSH inequality
(11), such that the local observables are subject to some biased noise
characterized by $(x_{a},\eta_{a})$ and $(x_{b},\eta_{b})$. As a result, the
observables are modified as
$A_{k}=\mathbf{I}x_{a}+\eta_{a}\hat{a}_{k}\cdot\vec{\sigma}_{k}$,
$B_{k}=\mathbf{I}x_{b}+\eta_{b}\hat{b}_{k}\cdot\vec{\sigma}_{k}$, and $k=1,2$.
Averaging with respect to the singlet-state, the CHSH inequality is satisfied
if
$\displaystyle
2x_{a}x_{b}+\eta_{a}\eta_{b}\left|\cos\theta_{11}+\cos\theta_{12}+\cos\theta_{21}-\cos\theta_{22}\right|\leq
2.$ (26)
The modulus term has a maximum value of $2\sqrt{2}$ for
$\theta_{ab}=\theta_{a^{\prime}b}=\theta_{ab^{\prime}}=\theta$ and
$\theta_{a^{\prime}b^{\prime}}=3\theta$ with $\theta=\pi/4$. The above
inequality becomes Kar and Roy (1995)
$\eta_{a}\eta_{b}\leq\frac{1-x_{a}x_{b}}{\sqrt{2}}.$ (27)
For the unbiased noise on either side, i.e., $x_{a}=0$ or $x_{b}=0$, the
incompatibility breaking condition reduces to
$\eta_{a}\eta_{b}\leq\frac{1}{\sqrt{2}}.$ (28)
Further, if this unbiased noise is acting only on one side (for example, on
$A$) of the bipartite system, then $\eta_{b}=1$, and we have
$\eta_{a}\leq\frac{1}{\sqrt{2}}.$ (29)
Having seen the impact of noise on the violation of the CHSH inequality, we
now look at the incompatibility breaking condition for observables $A_{1}$ and
$A_{2}$ subjected to the same noise. As a result we have
$A_{1}^{(x_{a},\eta_{a})}$ and $A_{2}^{(x_{a},\eta_{a})}$ as biased and
unsharp observables. Using inequality (6), the incompatibility condition for
these observables reads
$\eta_{a}\leq\frac{1-x_{a}^{2}}{\sqrt{2}}.$ (30)
This coincides with the NB condition (29) only if $x_{a}=0$. Since the action
of the nonunital channel on a projector results in a biased observable Naikoo
_et al._ (2021), one finds that the conditions for nonlocality breaking and
incompatibility breaking agree as long as the dynamics is unital in the
Schrödinger picture.
### III.2 Nonlocality and incompatibility breaking channels in tripartite
scenario
In the tripartite scenario, the aforesaid relation between nonlocality and
incompatibility does not hold in general. In this section, we therefore first
obtain the NB condition for some well known tripartite states and then
identify the range of channel parameter where NB agrees with the $2$-IBC
condition. Let us first introduce the definition of the absolute and genuine
tripartite NBC as
###### Definition 4.
For any three-qubit state $\rho_{ABC}$, a given qubit channel $\mathcal{E}$ is
said to be absolute or genuine nonlocality breaking if acting on any qubit, it
gives a state (for example) $\rho^{\prime}_{ABC}=(\mathcal{E}\otimes I\otimes
I)(\rho_{ABC})$, which satisfies Mermin inequality
$\langle\hat{M}\rangle_{\rho^{\prime}_{ABC}}\leq 2$ or Svetlichny inequality
$\langle\hat{S}\rangle_{\rho^{\prime}_{ABC}}\leq 4$.
In terms of the largest singular value in Definition 3, the Mermin nonlocality
breaking condition (M-NBC) and Svetlichny nonlocality breaking condition
(S-NBC) are, respectively, given as
$\eta_{M}\leq\frac{1}{\sqrt{2}\lambda_{max}},\qquad\eta_{S}\leq\frac{1}{\lambda_{max}}.$
(31)
We now consider some well-known tripartite quantum states in which one party
is subjected to a noisy evolution, and use Def. 3 to obtain the conditions on
the noise parameter for breaking the Mermin and Svetlichny nonlocality.
Example 1. Our first example is the generalized GHZ state Dür _et al._
(2000); Greenberger _et al._ (2007)
$|\psi_{GHZ}\rangle=\alpha|000\rangle+\beta|111\rangle.$ (32)
The entanglement in this state is completely destroyed when any of the three
qubits is traced out, i.e.,
$\Tr_{k}\left[|\psi_{GHZ}\rangle\langle\psi_{GHZ}|\right]=\mathbf{I}/2$, with
$k=A,B,C$. Let us assume a unital noise channel $\mathcal{E}$ acting on one
qubit (for example, the first one) according to Eq. (19). This can be achieved
by expressing $|\psi_{GHZ}\rangle$ in the Pauli basis and invoking
$\sigma_{i}\rightarrow\eta\sigma_{i}$ ($i=x,y,z$, $0\leq\eta\leq 1)$ at the
first qubit
$\displaystyle\left(\mathcal{E}\otimes\mathbf{I}\otimes\mathbf{I}\right)\left[|\psi_{GHZ}\rangle\langle\psi_{GHZ}|\right]$
$\displaystyle=\frac{1}{8}\Big{[}\mathbf{I}\otimes\mathbf{I}\otimes\mathbf{I}+\mathbf{I}\otimes\sigma_{3}\otimes\sigma_{3}+\eta(\sigma_{3}\otimes\mathbf{I}\otimes\sigma_{3})$
$\displaystyle+(\alpha^{2}-\beta^{2})(\eta\sigma_{3}\otimes\mathbf{I}\otimes\mathbf{I}+\mathbf{I}\otimes\sigma_{3}\otimes\mathbf{I}$
$\displaystyle+\mathbf{I}\otimes\mathbf{I}\otimes\sigma_{3}+\eta\sigma_{3}\otimes\sigma_{3}\otimes\sigma_{3})+2\alpha\beta\eta(\sigma_{1}\otimes\sigma_{1}\otimes\sigma_{1}$
$\displaystyle-\sigma_{1}\otimes\sigma_{2}\otimes\sigma_{2}-\sigma_{2}\otimes\sigma_{1}\otimes\sigma_{2}-\sigma_{2}\otimes\sigma_{2}\otimes\sigma_{1})\Big{]}$
(33)
The matrix $M_{j,ik}$ in Def. 3 turns out to be
$\displaystyle
M_{j,ik}=\small\begin{pmatrix}2\eta\alpha\beta&0&0&0&-2\eta\alpha\beta&0&0&0&0\\\
0&-2\eta\alpha\beta&0&-2\eta\alpha\beta&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&\eta(\alpha^{2}-\beta^{2})\end{pmatrix}.$
This has singular values $2\sqrt{2}\eta\alpha\beta$,
$2\sqrt{2}\eta\alpha\beta$, and $\eta\left|\alpha^{2}-\beta^{2}\right|$. Thus
the condition for the channel to be Mermin and Svetlichny NB is given by,
$\displaystyle\eta_{M}\leq\frac{1}{4\alpha\beta},\qquad\eta_{S}\leq\frac{1}{2\sqrt{2}\alpha\beta}.$
(35)
Example 2. Next, we consider the well-known generalized W-state given by
$|\psi\rangle=\alpha|100\rangle+\beta|010\rangle+\gamma|001\rangle,$ (36)
with $\alpha,\beta$, and $\gamma$ real and
$\alpha^{2}+\beta^{2}+\gamma^{2}=1$. This state is special in the sense that
if one qubit is lost, the state of the remaining two qubits is still
entangled, unlike the GHZ state. The matrix $M_{j,ik}$ in Def.3 corresponding
to unital noise acting on the first qubit of the state is given by
$\displaystyle M_{j,ik}=\begin{pmatrix}0&0&\eta\omega&0&0&0&\eta\omega&0&0\\\
0&0&0&0&0&\eta\omega&0&\eta\omega&0\\\
\eta\omega^{\prime}&0&0&0&\eta\omega^{\prime}&0&0&0&\eta(2\alpha\gamma-\beta^{2})\end{pmatrix},$
with $\omega=\alpha\beta+\beta\gamma$,
$\omega^{\prime}=(\alpha^{2}+\gamma^{2})$. The largest singular value turns
out to be $\lambda=\eta\sqrt{1+8\beta^{2}\gamma^{2}}$. Now
$\beta^{2}+\gamma^{2}=1-\alpha^{2}=k$ (for example), so that
$\beta^{2}\gamma^{2}=\beta^{2}(k-\beta^{2})$. This quantity attains the
maximum $k^{2}/4$ at $\beta^{2}=k/2$, leading to
$\lambda_{max}=\eta\sqrt{1+2k^{2}}$. The Mermin and Svetlichny nonlocality
breaking conditions then read
$\eta_{M}\leq\frac{1}{\sqrt{2}\sqrt{1+2k^{2}}},\qquad\eta_{S}\leq\frac{1}{\sqrt{1+2k^{2}}}.$
(37)
The singular values attain a maximum at $k=2/3$, i.e.,
$\alpha=\beta=\gamma=1/\sqrt{3}$. Under this condition and for $\eta_{S}$
approximately in the range $[0.707-0.727]$, the channel is S-NBC but not
$2-{\rm IBC}$. Note that in the bipartite case, a unital channel is CHSH-NBC
for all $\eta\leq 1/\sqrt{2}$ Pal and Ghosh (2015). Thus in the range
$\eta\approx 0.707-0.727$, it is S-NBC but not CHSH-NBC.
Example 3. We next consider the three-qubit partially entangled set of maximal
slice (MS) states Carteret and Sudbery (2000); Liu _et al._ (2016)
$|\psi_{MS}\rangle=\frac{1}{\sqrt{2}}\bigg{[}|000\rangle+|11\bigg{(}\alpha|0\rangle+\beta|1\rangle\bigg{)}\bigg{]},$
(38)
where $\alpha$ and $\beta$ are real with $\alpha^{2}+\beta^{2}=1$. The matrix
$M_{j,ik}$ corresponding to Eq. (38) after the action of unital noise on the
first qubit is given by
$\displaystyle M_{j,ik}$
$\displaystyle=\begin{pmatrix}\eta\beta&0&\eta\alpha&0&-\eta\beta&0&0&0&0\\\
0&-\eta\beta&0&-\eta\beta&0&-\eta\alpha&0&0&0\\\
0&0&0&0&0&0&\eta\alpha\beta&0&\eta(\frac{1+\alpha^{2}-\beta^{2}}{2})\end{pmatrix}$
with three singular values
$\lambda_{1}=\eta\sqrt{\frac{1}{4}\left(\alpha^{2}-\beta^{2}+1\right)^{2}+(\alpha\beta)^{2}}$
and two equal singular values
$\lambda_{2}=\lambda_{3}=\eta\sqrt{(\alpha^{2}+2\beta^{2})}$ respectively,
leading to the condition for Mermin and Svetlichny nonlocality breaking as
$\displaystyle\eta_{M}\leq\frac{1}{\sqrt{2(\alpha^{2}+2\beta^{2})}},\qquad\eta_{S}\leq\frac{1}{\sqrt{(\alpha^{2}+2\beta^{2})}}.$
(39)
Figure 1: (Color online): The region below the dashed (red) and solid (black)
curve in (a), (b), (c), and (d) corresponds to M-NBC and S-NBC given by Ineqs.
(35), (37), (39), and (41), respectively, plotted against the (dimensionless)
state coefficients. The pairwise incompatibility breaking condition (9)
pertains to all points below the horizontal dashed line.
Example 4. Consider the quantum state Augusiak _et al._ (2015)
$\sigma_{GHZ}=p|GHZ\rangle\langle GHZ|+(1-p)I_{2}\otimes\tilde{I}.$ (40)
Here, $|GHZ\rangle\langle=(|000\rangle+|111\rangle)/\sqrt{2}$,
$\tilde{I}=diag.(1,0,0,1)$, is diagonal matrix, and $0\leq p\leq 1$. In fact,
$\tilde{I}=|\Phi^{+}\rangle\langle\Phi^{+}|+|\Phi^{-}\rangle\langle\Phi^{-}|$,
where $\Phi^{\pm}$ are Bell states, is a separable state, and it tells us that
$\sigma_{GHZ}$ has with probability $p$ the GHZ state and with probability
$(1-p)$ the first qubit is left in the mixed state and the second and third
qubit are in separable state $\tilde{I}$. The corresponding $M_{j,ik}$ matrix
with a unital noise acting on one qubit is given by
$\displaystyle M_{j,ik}=\begin{pmatrix}p\eta&0&0&0&-p\eta&0&0&0&0\\\
0&-p\eta&0&-p\eta&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0\end{pmatrix},$
with singular values $\lambda_{1}=\lambda_{2}=\sqrt{2}\eta p$. Thus the
condition for the channel to be Mermin and Svetlichny NBC becomes
$\displaystyle\eta_{M}\leq\frac{1}{2p},\qquad\eta_{S}\leq\frac{1}{\sqrt{2}p}.$
(41)
The conditions for Mermin and Svetlichny nonlocality breaking channel given by
Ineqs. (35), (36), (39), and (41) obtained by the application of a unital
quantum channel to one party of a tripartite system, are depicted in Fig. 1.
All the points below the solid (black) and dashed (red) curve correspond to
the nonlocality breaking channel, while as the points below the horizontal
dashed line, $\eta=1/\sqrt{2}$, pertain to pairwise incompatibility breaking.
In all the four examples, Fig. 1 (a)-(d), the minimum value of $\eta_{S}$ for
which SI is violated is $1/\sqrt{2}$, suggesting that _genuine_ nonlocal
correlations can not be established if at least one pair of observable is
compatible. The converse is not true, since there exist regions (above the
horizontal dashed line and below the solid (black) curve) of Svetlichny
nonlocality breaking even when the channel is not $2-{\rm IBC}$. Thus, these
examples illustrate that corresponding to $2$-IBC the conjugate channels are
definitely S-NLB; however, the conjugate of S-NB channels may not necessarily
be a $2$-IBC. However, in the context of Mermin nonlocality, even the first
statement does not hold, that is, existence of a $2$-IBC does not necessarily
guarantee a conjugate channel that is M-NB. Also, the minimum $\eta_{M}$ (that
is maximum noise) for which a channel is M-NB is always less by a factor of
$1/\sqrt{2}$ than the minimum noise below which that channel is S-NB. It is
worth pointing here to Fig. 1 (c), which illustrates that if a (unital)
channel breaks the Svetlichny nonlocality for the GHZ state (which in fact
violates the SI maximally) then it also does so for the mixture (40) for
$1/\sqrt{2}<p<1$. This is unlike the bipartite scenario where a unital channel
that breaks the CHSH nonlocality for the maximally entangled states is
guaranteed to do so for all other states Pal and Ghosh (2015). Summarizing, it
is clear from the above examples that the existence of M-NBC or S-NBC does not
guarantee the existence of a conjugate $2$-IBC, unlike the CHSH scenario. In
particular, with the (unital) noise acting on one party of the $W$-state,
$|\psi\rangle=(|100\rangle+|010\rangle+|001\rangle$)/$\sqrt{3}$, there exists
a range of the channel parameter $\eta\in(1/\sqrt{2},3/\sqrt{17})$, where the
channel is S-NBC but not CHSH NBC as depicted in the Fig. 2. Thus the Bell-
CHSH inequality seems to be more suitable for a study of the incompatibility
of observables than the multi-partite Bell-type inequalities.
Note that instead of one party, if two or all the three parties are subjected
to noise, the conditions (35), (37), (39), and (41), become
$\eta_{M}\leq\left(\frac{1}{\sqrt{2}\lambda_{max}}\right)^{1/n},\qquad\eta_{S}\leq\left(\frac{1}{\lambda_{max}}\right)^{1/n},$
(42)
where $n$ corresponds to the number of qubits subjected to noise. Since
$1/\lambda_{max}<(1/\lambda_{max})^{1/2}<(1/\lambda_{max})^{1/3}$ (with
$\lambda_{max}>1$), the solid (black) and dashed (red) curves in Fig. 1
(a)-(d) are shifted up, thereby decreasing the region of nonlocality with
increase in $n$.
General three qubit state subjected to general unital noise: The (unital)
noise acting on a single party considered in the above analysis involving the
tripartite system assumes identical effects on $\sigma_{x}$, $\sigma_{y}$, and
$\sigma_{z}$ corresponding to that party, in the sense that
$\sigma_{k}\rightarrow\eta\sigma_{k}$ for all $k=x,y,z$. A more general
transformation would take the particular party’s
$\sigma_{k}\rightarrow\eta_{k}\sigma_{k}$, with
$\sqrt{\eta_{x}^{2}+\eta_{y}^{2}+\eta_{z}^{2}}=\eta$, and $0\leq\eta\leq 1$,
such that
$\Phi\left(\frac{\mathbf{I}+\vec{r}\cdot\vec{\sigma}}{2}\right)=\frac{\mathbf{I}+(\textbf{T}\vec{r})\cdot\vec{\sigma}}{2},$
(43)
where $\textbf{T}={\rm diag.}[\eta_{x},\eta_{y},\eta_{z}]$, is a real diagonal
matrix. The map $\Phi$ is completely positive for King and Ruskai (2001)
$\left|\eta_{x}\pm\eta_{y}\right|\leq\left|1\pm\eta_{z}\right|,$ (44)
which is a set of four inequalities and defines a tetrahedron in
$\eta_{x}-\eta_{y}-\eta_{z}$ space. Under such a transformation, the singular
values for the GHZ in (32) and $W$ state in (36) are, respectively, given by
$\displaystyle\left(2\sqrt{2}\alpha\beta\eta_{x},2\sqrt{2}\alpha\beta\eta_{y},\left|\alpha^{2}-\beta^{2}\right|\eta_{z}\right),$
(45) $\displaystyle{\rm and}~{}~{}$
$\displaystyle\left(2\alpha\sqrt{\beta^{2}+\gamma^{2}}\eta_{x},2\alpha\sqrt{\beta^{2}+\gamma^{2}}\eta_{y},\sqrt{1+8\beta^{2}\gamma^{2}}\eta_{z}\right).$
(46)
with the three singular values depending linearly on the respective noise
parameters. Depending on which singular value is the largest, one can draw
similar conclusions about nonlocality and incompatibility breaking properties
of the noise channel as in the case with uniform noise action. However, for a
general three-qubit input state, the dependence of the singular values on
noise parameters tuns out to be complicated leading to different conclusions
regarding the nonlocality breaking property of such a channel. Let us consider
the general situation where one would like to make a statement about the
limiting noise beyond which no Mermin or Svetlichny nonlocal correlations can
be established _irrespective_ of the state chosen. In this direction, we make
use of the canonical five term decomposition of the three-qubit state Acín
_et al._ (2001):
$\ket{\psi}=\lambda_{0}\ket{000}+\lambda_{1}e^{i\phi}\ket{100}+\lambda_{2}\ket{101}+\lambda_{3}\ket{110}+\lambda_{4}\ket{111},$
(47)
with $\lambda_{i}$ and $\phi$ real parameters and
$\sum_{i}\left|\lambda_{i}\right|^{2}=1$ and $0<\phi<\pi$. In the Pauli basis,
one may denote the density matrix corresponding to $\ket{\psi}$ as
$|\psi\rangle\langle\psi|\left[\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k}\right]$
with Pauli matrices $\sigma_{i}$ $i=x,y,z$. If a unital noise acts on one
party (for example, the first) of such a state, we would have
$\displaystyle|\psi\rangle\langle\psi|\left[\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k}\right]\rightarrow|\psi\rangle\langle\psi|\left[\eta_{i}\sigma_{i}\otimes\sigma_{j}\otimes\sigma_{k}\right].$
(48)
Figure 2: (Color online): Depicting the range of channel parameter
$\eta\in(\frac{1}{\sqrt{2}},\frac{3}{\sqrt{17}})$ in which it is S-NBC (for
the $W$ state) but not CHSH-NBC.
Figure 3: (Color online) Singular values corresponding to a sample of $5\times
10^{6}$ randomly generated states of the form $(\ref{eq:SVAcin})$ plotted with
respect to the dimensionless parameter
$\eta=\sqrt{\eta_{x}^{2}+\eta_{y}^{2}+\eta_{z}^{2}}$. In terms of these
singular values, the conditions for establishing the Mermin and Svetlichny
nonlocal correlations are given in Def. 3 which demands that the largest
singular value be greater than $1/\sqrt{2}$ and $1$, respectively. The
corresponding bound on the unsharpness parameters $\eta_{M}$ and $\eta_{S}$
trivially follow and are summarized in the text.
We can now check the nonlocality breaking properties of such a general noise
channel using Def. 3 based on singular values of matrix $M$. For the above
state, the singular values calculated according to Def. 3, are plotted in Fig.
3 with respect to $\eta=\sqrt{\eta_{x}^{2}+\eta_{y}^{2}+\eta_{z}^{2}}$. The
simulation makes use of a sample of $5\times 10^{6}$ randomly generated states
and the corresponding noise parameters $\eta_{i}$ subjected to
$0\leq\sqrt{\eta_{x}^{2}+\eta_{y}^{2}+\eta_{z}^{2}}\leq 1$ and also satisfying
the completely positive condition (44). According to the condition (18), the
Mermin and Svetlichny nonlocal correlations are established only if the
largest singular value is greater than $\frac{1}{\sqrt{2}}$ for the former and
$1$ for the later. In Fig. 3 the three singular values
$\lambda_{1},\lambda_{2}$, and $\lambda_{3}$ are depicted with respect to
parameter $\eta$. One finds that below a minimum $\eta_{M}$ and $\eta_{S}$,
the singular values do not exceed $\frac{1}{\sqrt{2}}$ and $1$, respectively.
These values turn out to be $(\eta_{M},\eta_{S})=(0.090,0.128)$ for
$\lambda_{1}$, $(\eta_{M},\eta_{S})=(0.182,0.259)$ for $\lambda_{2}$, and
$(\eta_{M},\eta_{S})=(0.300,0.409)$ for $\lambda_{3}$, and one may conclude
that no Mermin (Svetlichny) nonlocal correlations are supported by the unital
noise channel if the noise parameter is below $0.090~{}(0.128)$, irrespective
of the input state.
## IV Conclusion
This paper is devoted to a study of the interplay between nonlocality breaking
and incompatibility breaking power of noisy quantum qubit channels. The action
of quantum channels on projective measurements transforms them into noisy
POVMs, characterized in particular by unsharpness parameters. As a
consequence, noise tends to increase the compatibility of observers that are
otherwise incompatible. In fact, pairwise incompatibility breaking is assured
if the channel parameter is less than or equal to $1/\sqrt{2}$. To be
specific, we consider bipartite and tripartite scenarios, with CHSH
nonlocality in the former and Mermin and Svetlichny nonlocality in the later
case. The degree of incompatibility breaking directly depends on the
unsharpness parameters. Here, we showed that in the Bell-CHSH scenario, if the
conjugate of a channel is incompatibility breaking then the channel is itself
nonlocality breaking and the converse is also true. In the tripartite
scenario, however, such an equivalence between nonlocality breaking and
incompatibility breaking does not exist. We then consider various examples of
three-qubit states and identify the state parameters for which the equivalence
of nonlocality breaking corroborates with the pairwise incompatibility. In
particular, it is illustrated that the conjugate of incompatibility breaking
channels is nonlocality breaking, however, the nonlocality breaking channels
do not guarantee the existence of conjugate channels that are incompatibility
breaking. This may be viewed as a useful feature of the Bell-CHSH inequality
when it comes to the study incompatibility of observables. Further, from
randomly generated three-qubit states subjected to general unital channels, we
conclude that no Mermin (Svetlichny) nonlocal correlations are supported for
$\eta_{M}<0.090$ ($\eta_{S}<0.128$), $\eta_{M/S}$ being the channel
unsharpness parameter.
The channel activation of nonlocality in the CHSH scenario has been studied in
Zhang _et al._ (2020). A future extension of this paper could be the study of
activation of Mermin and Svetlichny nonlocality under more general noise
models and with general three-qubit input state. This also invites for a
detailed analysis on the hierarchy of nonlocality-breaking, steerability-
breaking, and entanglement-breaking quantum channels in the tripartite
scenario.
## Acknowledgment
Authors thank Guruprasad Kar for the initial motivation of the work. SK
acknowledges Yeong-Cherng Liang for the fruitful comments and the support from
the National Science and Technology Council, Taiwan (Grants
no.109-2112-M-006-010-MY3,110-2811-M-006-511,111-2811-M-006-013). JN’s work
was supported by the Foundation for Polish Science within the “Quantum Optical
Technologies” project carried out within the International Research Agendas
programme cofinanced by the European Union under the European Regional
Development Fund. AKP acknowledges the support from the research grant
DST/ICPS/QuEST/2019/4.
## References
* Bell (1964) J. S. Bell, Physics Physique Fizika 1, 195 (1964).
* Brunner _et al._ (2014) N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, Rev. Mod. Phys. 86, 419 (2014).
* Scarani and Gisin (2001) V. Scarani and N. Gisin, Phys. Rev. Lett. 87, 117901 (2001).
* Ekert (1991) A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991).
* Heinosaari _et al._ (2016) T. Heinosaari, T. Miyadera, and M. Ziman, Journal of Physics A: Mathematical and Theoretical 49, 123001 (2016).
* Wolf _et al._ (2009) M. M. Wolf, D. Perez-Garcia, and C. Fernandez, Phys. Rev. Lett. 103, 230402 (2009).
* Quintino _et al._ (2014) M. T. Quintino, T. Vértesi, and N. Brunner, Phys. Rev. Lett. 113, 160402 (2014).
* Uola _et al._ (2015) R. Uola, C. Budroni, O. Gühne, and J.-P. Pellonpää, Phys. Rev. Lett. 115, 230402 (2015).
* Kochen and Specker (1975) S. Kochen and E. P. Specker, “The problem of hidden variables in quantum mechanics,” in _The Logico-Algebraic Approach to Quantum Mechanics: Volume I: Historical Evolution_, edited by C. A. Hooker (Springer Netherlands, Dordrecht, 1975) pp. 293–328.
* Budroni _et al._ (2022) C. Budroni, A. Cabello, O. Gühne, M. Kleinmann, and J.-A. Larsson, Rev. Mod. Phys. 94, 045007 (2022).
* Liang _et al._ (2011) Y.-C. Liang, R. W. Spekkens, and H. M. Wiseman, Physics Reports 506, 1 (2011).
* Ku _et al._ (2022a) H.-Y. Ku, C.-Y. Hsieh, S.-L. Chen, Y.-N. Chen, and C. Budroni, Nature Communications 13, 4973 (2022a).
* Bennett _et al._ (1999) C. H. Bennett, D. P. DiVincenzo, C. A. Fuchs, T. Mor, E. Rains, P. W. Shor, J. A. Smolin, and W. K. Wootters, Phys. Rev. A 59, 1070 (1999).
* Bene and Vértesi (2018) E. Bene and T. Vértesi, New Journal of Physics 20, 013021 (2018).
* Hirsch _et al._ (2018) F. Hirsch, M. T. Quintino, and N. Brunner, Phys. Rev. A 97, 012129 (2018).
* Svetlichny (1987) G. Svetlichny, Phys. Rev. D 35, 3066 (1987).
* Ajoy and Rungta (2010) A. Ajoy and P. Rungta, Phys. Rev. A 81, 052334 (2010).
* Bancal _et al._ (2013) J.-D. Bancal, J. Barrett, N. Gisin, and S. Pironio, Phys. Rev. A 88, 014102 (2013).
* Pal and Ghosh (2015) R. Pal and S. Ghosh, Journal of Physics A: Mathematical and Theoretical 48, 155302 (2015).
* Heinosaari _et al._ (2015) T. Heinosaari, J. Kiukas, D. Reitzner, and J. Schultz, 48, 435301 (2015).
* Heinosaari and Miyadera (2017) T. Heinosaari and T. Miyadera, Journal of Physics A: Mathematical and Theoretical 50, 135302 (2017).
* Horodecki _et al._ (2003) M. Horodecki, P. W. Shor, and M. B. Ruskai, Reviews in Mathematical Physics 15, 629–641 (2003).
* Kiukas _et al._ (2017) J. Kiukas, C. Budroni, R. Uola, and J.-P. Pellonpää, Phys. Rev. A 96, 042331 (2017).
* Ku _et al._ (2022b) H.-Y. Ku, J. Kadlec, A. Cernoch, M. T. Quintino, W. Zhou, K. Lemr, N. Lambert, A. Miranowicz, S.-L. Chen, F. Nori, and Y.-N. Chen, PRX Quantum 3, 020338 (2022b).
* Collins _et al._ (2002) D. Collins, N. Gisin, S. Popescu, D. Roberts, and V. Scarani, Phys. Rev. Lett. 88, 170405 (2002).
* Busch (1986) P. Busch, Phys. Rev. D 33, 2253 (1986).
* Nielsen and Chuang (2010) M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information: 10th Anniversary Edition_ (Cambridge University Press, 2010).
* Yu _et al._ (2010) S. Yu, N.-l. Liu, L. Li, and C. H. Oh, Phys. Rev. A 81, 062116 (2010).
* Heinosaari (2020) T. Heinosaari, Journal of Physics: Conference Series 1638, 012002 (2020).
* Mermin (1990) N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990).
* Cereceda (2002) J. L. Cereceda, Phys. Rev. A 66, 024102 (2002).
* Mitchell _et al._ (2004) P. Mitchell, S. Popescu, and D. Roberts, Phys. Rev. A 70, 060101 (2004).
* Ghose _et al._ (2009) S. Ghose, N. Sinclair, S. Debnath, P. Rungta, and R. Stock, Phys. Rev. Lett. 102, 250404 (2009).
* Su _et al._ (2018) Z. Su, L. Li, and J. Ling, Quantum Information Processing 17, 1 (2018).
* Siddiqui and Sazim (2019) M. A. Siddiqui and S. Sazim, Quantum Information Processing 18 (2019), 10.1007/s11128-019-2246-1.
* Li _et al._ (2017) M. Li, S. Shen, N. Jing, S.-M. Fei, and X. Li-Jost, Phys. Rev. A 96, 042323 (2017).
* Beth Ruskai _et al._ (2002) M. Beth Ruskai, S. Szarek, and E. Werner, Linear Algebra and its Applications 347, 159 (2002).
* Naikoo _et al._ (2021) J. Naikoo, S. Banerjee, A. K. Pan, and S. Ghosh, Phys. Rev. A 104, 042608 (2021).
* Kar and Roy (1995) G. Kar and S. Roy, Physics Letters A 199, 12 (1995).
* Dür _et al._ (2000) W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000).
* Greenberger _et al._ (2007) D. M. Greenberger, M. A. Horne, and A. Zeilinger, “Going beyond bell’s theorem,” (2007), arXiv:0712.0921 [quant-ph] .
* Carteret and Sudbery (2000) H. A. Carteret and A. Sudbery, Journal of Physics A: Mathematical and General 33, 4981 (2000).
* Liu _et al._ (2016) J. Liu, Z.-w. Mo, and S.-q. Sun, International Journal of Theoretical Physics 55, 2182 (2016).
* Augusiak _et al._ (2015) R. Augusiak, M. Demianowicz, J. Tura, and A. Acín, Phys. Rev. Lett. 115, 030404 (2015).
* King and Ruskai (2001) C. King and M. Ruskai, IEEE Transactions on Information Theory 47, 192 (2001).
* Acín _et al._ (2001) A. Acín, A. Andrianov, E. Jané, and R. Tarrach, Journal of Physics A: Mathematical and General 34, 6725 (2001).
* Zhang _et al._ (2020) Y. Zhang, R. A. Bravo, V. O. Lorenz, and E. Chitambar, New Journal of Physics 22, 043003 (2020).
|
# On the relevance of lift force modelling in turbulent wall flows with small
inertial particles
Wei Gao1,2<EMAIL_ADDRESS>Pengyu Shi3,4 Matteo Parsani1,2 Pedro
Costa5 1Mechanical Engineering, Physical Science and Engineering Division,
King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi
Arabia 2Applied Mathematics and Computational Science, Computer Electrical and
Mathematical Science and Engineering Division, Extreme Computing Research
Center, King Abdullah University of Science and Technology, Thuwal 23955-6900,
Saudi Arabia 3Institut de Mécanique des Fluides de Toulouse (IMFT), Université
de Toulouse, CNRS, Toulouse, France 4Helmholtz-Zentrum Dresden – Rossendorf,
Institute of Fluid Dynamics, 01328 Dresden, Germany 5Process & Energy
Department, TU Delft, Leeghwaterstraat 39, 2628 CB, The Netherlands
###### Abstract
In particle-laden turbulent wall flows, lift forces can influence the near-
wall turbulence. This has been recently observed in particle-resolved
simulations, which, however, are too expensive to be used in upscaled models.
Instead, point-particle simulations have been the method of choice to simulate
the dynamics of these flows during the last decades. While this approach is
simpler, cheaper, and physically sound for small inertial particles in
turbulence, some issues remain. In the present work, we address challenges
associated with lift force modelling in turbulent wall flows and the impact of
lift forces in the near-wall flow. We performed direct numerical simulations
(DNS) of small inertial point particles in turbulent channel flow for fixed
Stokes number and mass loading while varying the particle size. Our results
show that the particle dynamics in the buffer region, causing the apparent
particle-to-fluid slip velocity to vanish, raise major challenges for
accurately modelling lift forces. While our results confirm that lift forces
have little influence on particle dynamics for sufficiently small particle
sizes, for inner-scaled diameters of order one and beyond, lift forces become
quite important near the wall. The different particle dynamics under lift
forces result in the modulation of streamwise momentum transport in the near-
wall region. We analyze this lift-induced turbulence modulation for different
lift force models, and the results indicate that realistic models are critical
for particle-modeled simulations to correctly predict turbulence modulation by
particles in the near-wall region.
###### keywords:
particle-laden wall turbulence, lift force, turbulence modulation
MSC Codes (Optional) Please enter your MSC Codes here
## 1 Introduction
Wall-bounded turbulent flows laden with inertial particles abound in
environmental and industrial contexts, such as the transport of particulate
matter in the atmosphere, sediment transport in rivers, the separation of fine
particles within industrial cyclones, and fluidized bed reactors. The chaotic
and multiscale nature of the fluid turbulence coupled with the particle
dynamics results in fascinating phenomena, which, however, is still
challenging to understand and model (Balachandar & Eaton, 2010; Brandt &
Coletti, 2022).
In the past decades, many studies have been devoted to the dynamics of
turbulent wall transport of small inertial particles, driven by their
prevalent nature and rich physics. These systems have been traditionally
classified by how the dispersed phase influences the overall flow behaviour
(Elghobashi, 1994). Precisely, one-way coupling (1WC) corresponds to very
small particle loadings, so small that their influence on the dynamics of the
suspending fluid turbulence is negligible; two-way coupling (2WC) denotes flow
regimes where mass loading is high enough such that the particles modify flow
observably, but particle-particle interactions are negligible; finally regimes
where both particle–particle and particle–fluid interactions influence the
flow dynamics are grouped into the four-way coupling category.
In a first-principles, fully-resolved direct numerical simulation (DNS) of a
particle-laden flow, the flow around each particle needs to be explicitly
resolved (Balachandar & Eaton, 2010; Maxey, 2017) (so-called particle-resolved
DNS, PR-DNS). While this approach is free from modelling assumptions for the
dispersed phase dynamics, it is computationally expensive due to the explicit
imposition of no-slip and no-penetration boundary conditions at the surface of
many particles moving in a turbulent medium. This is incredibly challenging
when particles are tiny and there is a scale separation between the particle
size and the smallest (Kolmogorov) turbulence scale due to the need for
extreme resolution requirements. Fortunately, in this case, one may be able to
resort to the so-called point-particle approximation (PP-DNS), where
interphase coupling is considered to be localized to a point.
Indeed, the point-particle approximation has been the method of choice for
simulating the particle dynamics in turbulent wall flows. In these cases, it
is assumed that the local properties of an undisturbed flow at the particle
position drive the dispersed-phase dynamics (Maxey & Riley, 1983; Gatignol,
1983). In the case of highly inertial particles (i.e., large particle-to-fluid
density ratios), the so-called Maxey-Riley-Gatignol equations simplify to a
drag force term (Arcen et al., 2006), which nevertheless yields highly non-
trivial particle dynamics, even in isotropic turbulence (Toschi & Bodenschatz,
2009). In wall-bounded turbulent flows, the inhomogeneous turbulence results
in even richer dynamics, with the particle distribution driven by the
interplay between small-scale clustering, turbophoresis, and the interaction
between the particles and near-wall turbulence structures (Reeks, 1983;
Soldati & Marchioli, 2009; Sardina et al., 2012), resulting in very
inhomogeneous particle concentrations peaking at the wall, with strong
preferential sampling of low-speed regions, as reproduced in a plethora of
numerical studies, such as Fessler et al. (1994); Uijttewaal & Oliemans
(1996); Marchioli et al. (2003); Kuerten (2006); Marchioli et al. (2008);
Soldati & Marchioli (2009); Bernardini et al. (2013); Jie et al. (2022).
When two-way coupling effects are important, a point-particle DNS must
describe the back-reaction of the dispersed phase in the flow. This is a major
challenge, as the point-particle dynamics are driven by a local _undisturbed_
fluid velocity, while the local flow field is being disturbed by the particles
(see, e.g. Gualtieri et al., 2015). The classical approach, known as the
particle-in-cell method, was developed by Crowe et al. (1977) and is widely
used – even in the present study – but it requires sufficiently high number of
particles per grid cell and cannot reproduce simple benchmarks of a
sedimenting sphere in a quiescent medium. Indeed, approaches for a consistent
treatment are being actively investigated; see Gualtieri et al. (2015);
Horwitz & Mani (2016); Ireland & Desjardins (2017); Horwitz & Mani (2020);
Horwitz et al. (2022); Horwitz (2023). Investigations of particle-laden
turbulent flows in the two-way-coupling regime are found in e.g. Vreman et al.
(2009); Zhao et al. (2010); Kuerten (2016); Richter & Sullivan (2014);
Capecelatro et al. (2018); Wang & Richter (2019); Battista et al. (2023).
While employing the point-particle approximation to small inertial particles
is physically sound, validating the fidelity of the approximation in one- and
two-way coupling conditions remains a challenge. Experimental data are
available (Eaton & Fessler, 1994; Kaftori et al., 1995). Still there are few
parameter-matched numerical studies due to limitations in terms of Reynolds
number and well-controlled experiments, and only in recent years, efforts in
this direction have started to appear (Wang et al., 2019). Fortunately, PR-DNS
of small inertial particles in turbulence has become possible, thanks to the
continuous growth in available computer power and development of efficient
numerical methods, with the first direct comparisons between point-particle
models and particle-resolved simulations starting to appear for forced
homogeneous isotropic turbulence and decaying HIT with moving particles
(Schneiders et al., 2017b, a; Mehrabadi et al., 2018; Fröhlich et al., 2018),
and turbulent channel flow (Horne & Mahesh, 2019; Costa et al., 2020a).
These results from PR-DNS of particles in wall-bounded turbulence confirm that
a sole drag force may not suffice to accurately reproduce the particle
dynamics even for relatively small particles with a large density ratio under
one-way coupling conditions. Where the shear rate is high near the wall,
_lift_ forces are also important. This has already been suggested in early
works using PP-DNS, since the work by McLaughlin (1989) (see also Botto et
al., 2005). In this regard, while Arcen et al. (2006) reported that the lift
force has a negligible impact on the dispersed phase statistics, Marchioli et
al. (2007) and Shin et al. (2022) found that the inclusion of the lift force
in PP-DNS can lead to weakened near-wall particle accumulation in upward and
horizontal channels, respectively; McLaughlin (1989) showed that the inclusion
of the lift force resulted in a higher deposition rate; Wang et al. (1997)
found that neglecting the lift force resulted in a slight reduction in the
deposition rate.
While these findings are not necessarily contradictory, as there are some
variations in the governing parameters in the different studies and lift force
models, many questions remain elusive: _(1) which form of lift force model is
appropriate for reproducing with high fidelity the dynamics of small inertial
particles in a turbulent wall flow; (2) under which conditions do lift forces
matter in the particle dynamics; and (3) what are the consequences of choosing
an inaccurate lift force model in dispersed phase dynamics and near-wall
turbulence modulation._
In the recent direct comparison between PR-DNS and PP-DNS for small particles
in turbulent channel flow by Costa et al. (2020a) (see also Costa et al.,
2020b), it was shown that lift models are vital for reproducing the near-wall
particle dynamics, at least for inertial particles with a size of the order of
one viscous wall unit. Surprisingly, a modified Saffman lift model predicted
the particle dynamics perfectly, with the Saffman force scaled by a normalized
shear rate, while conventional lift models yield poorer predictions (Costa et
al., 2020b). In addition to reducing the near-wall concentration peak, it was
also seen that lift force causes a large increase in correlated
streamwise–wall-normal particle velocity fluctuations. This quantity has
dramatic drag-increasing effects for sufficiently high mass loading, as it
modulates the streamwise momentum budget; see Costa et al. (2021). This direct
link between lift force and drag increase makes their accurate modelling
crucial.
In the present work, we address the three questions presented above by
performing DNS of turbulent channel flow laden with small inertial particles,
using the point-particle approximation, with three different lift models from
the classical Saffman lift model to the one that perfectly predicts previous
PR-DNS data. We consider different inner-scaled particle sizes $D^{+}\sim
1-0.1$ in one and two-way coupling conditions for a Stokes number that is
known to feature strong wall accumulation and preferential concentration in
low-speed regions. Our analysis shows that currently available lift models are
bound to fail near the wall for small inertial particles due to a vanishing
particle-to-fluid slip velocity. We then use two-way coupling simulations to
illustrate how different lift force models can result in _qualitatively_
different turbulence modulation (i.e., turbulence attenuation vs. turbulence
enhancement). Near-wall accumulation is still significantly reduced for the
smallest particle size ($D^{+}=0.1$). Still, the lift-induced increase of
correlated velocity fluctuations near the wall is negligible, and thus, lift
force has little consequences in turbulence modulation.
This paper is organized as follows. Next, in §2, the numerical method, lift
force models, and computational setups are described. Then, in §3, the effects
of lift force models on the particle dynamics and near-wall accumulation are
investigated using one-way coupling DNS, based on which we try to explore the
reason for the failure of conventional lift force models. Following this, we
qualitatively demonstrate the effect of lift force models on turbulence
modulation and momentum transfer with two-way coupling DNS. Finally,
conclusions are drawn in §4.
## 2 Methodology
### 2.1 Governing equations and numerical method
The fluid phase is governed by the incompressible Navier–Stokes equations,
$\begin{gathered}\nabla\cdot\mathbf{u}=0,\\\
\rho\left(\partial_{t}\mathbf{u}+(\mathbf{u}\cdot\nabla)\mathbf{u}\right)=-\nabla
p+\mu\nabla^{2}\mathbf{u}+\mathbf{f},\end{gathered}$ (1)
where $\mathbf{u}$ denotes the fluid velocity vector, $p$ is the fluid
pressure, $\mu$ is the dynamic viscosity, $\rho$ is the fluid density, and
$\mathbf{f}$ is the particle feedback force to the fluid phase in the case of
two-way coupling point-particle simulations, here computed by using a standard
particle-in-cell (PIC) approach, which spreads the particle force to the
nearest Eulerian grid points with a linear kernel (Boivin et al., 1998; Lee &
Lee, 2019; Zhang et al., 2023).
We consider $x$, $y$, and $z$ as the streamwise, wall-normal, and spanwise
directions, respectively. No-slip and no-penetration boundary conditions are
specified at the domain walls, while periodicity is applied along the
streamwise and spanwise directions. These equations are discretized in space
using a pseudospectral approach along $x$ and $z$ and second-order finite
differences along $y$. Wray’s low-storage third-order Runge–Kutta scheme is
employed for time marching (see Wray, 1990).
Spherical particles with density $\rho_{p}\gg\rho$ and particle diameter $D$
in the absence of gravity are tracked with a standard Lagrangian point-
particle method, with their dynamics governed by
$m_{p}\dot{\mathbf{U}}_{p}=\mathbf{F}_{d}+\mathbf{F}_{l}\mathrm{,}\;\;\dot{\mathbf{X}}_{p}=\mathbf{U}_{p}\mathrm{,}$
(2)
with the Schiller-Naumann drag force
$\mathbf{F}_{d}=-3\pi\mu
D\mathbf{U}_{s}\left(1+0.15\Rey_{p}^{0.687}\right)\mathrm{,}$ (3)
where $\mathbf{U}_{p}$, $\mathbf{X}_{p}$, $\mathbf{F}_{d}$ and
$\mathbf{F}_{l}$ are the particle velocity, position, drag and lift forces,
$m_{p}$ is the particle mass,
$\mathbf{U}_{s}=\mathbf{U}_{p}-\mathbf{u}|_{\mathbf{x}=\mathbf{X}_{p}}$ is the
local slip velocity evaluated at the particle location, and $\Rey_{p}$
$=\left|\mathbf{U}_{s}\right|D/{\nu}$ is the particle Reynolds number. Since
the particle-to-fluid density ratio is high, other dynamic effects such as
added mass, fluid acceleration, and the Basset history force are negligible.
While in most practical scenarios of particle-laden wall transport, gravity is
important at high density ratios, we neglect it in the present work. We do
this to isolate the interplay between more intricate wall accumulation
mechanisms (e.g., turbophoresis) and lift forces, in a flow which could
otherwise feature significant settling effects. Finally, for the sake of
simplicity, a perfectly elastic hard-sphere rebound is employed for particle-
wall interactions.
#### Lift force models
We consider standard shear-induced lift force models where lift force
$\mathbf{F}_{l}$ acts perpendicular to the local shear sampled by the particle
and points along $\boldsymbol{\omega}\times\mathbf{U}_{s}$, where
$\boldsymbol{\omega}=\left(\nabla\times\mathbf{u}\right)|_{\mathbf{x}=\mathbf{X}_{p}}$
is the undisturbed flow vorticity evaluated at the particle position. For
convenience, let us express the lift force in terms of the dimensionless lift
coefficient, $C_{L}$, defined as
$C_{L}=\frac{\mathbf{F}_{l}}{\frac{1}{8}\pi\rho
D^{2}|\mathbf{U}_{s}|^{2}}\cdot\frac{\boldsymbol{\omega}\times\mathbf{U}_{s}}{|\boldsymbol{\omega}\times\mathbf{U}_{s}|}.$
(4)
Assuming the Oseen length $\ell_{u}=\nu/{\left|\mathbf{U}_{s}\right|}$ is much
larger than the Saffman length
$\ell_{\omega}=\sqrt{\nu/{\left|\boldsymbol{\omega}\right|}}$, i.e.
$\varepsilon\gg 1$ with
$\varepsilon=\sqrt{\left|\boldsymbol{\omega}\right|\nu}/\left|\mathbf{U}_{s}\right|$,
an explicit lift solution can be derived:
$C_{L}=\frac{18}{\pi^{2}}\varepsilon J(\varepsilon),$ (5)
with
$J(\varepsilon\gg 1)=J^{\infty}=2.255.$ (6)
Hereafter, the lift expression (5) together with the $J$ function by Eq. (6)
will be referred to as the Saffman model, which is expected to be valid in the
double limits $\Rey_{p}\to 0$ and $\Rey_{\omega}\to 0$ provided that
$\varepsilon\gg 1$, where
$\Rey_{\omega}=\left|\boldsymbol{\omega}\right|D^{2}/\nu$ is the shear
Reynolds number.
Still in the double limits $\Rey_{p}\to 0$ and $\Rey_{\omega}\to 0$, the $J$
function in Eq. (5) at finite $\varepsilon$ turns out to be a volume integral
in Fourier space. Its value cannot be put in closed form but was estimated
numerically in Asmolov (1989), McLaughlin (1991), and more recently, Shi &
Rzehak (2019). Based on these numerical data, various empirical correlations
of $J(\varepsilon)$ were proposed (see Shi & Rzehak (2019) for a comprehensive
review), with the most commonly used being the one proposed by Mei (1992),
i.e.,
$J(\varepsilon)=0.3J^{\infty}\left(1+\tanh\left[\frac{5}{2}\left(\log_{10}\varepsilon+0.191\right)\right]\right)\left(\frac{2}{3}+\tanh(6\varepsilon-1.92)\right).$
(7)
Hereafter, the lift expression (5) together with the $J$ function by Eq. (7)
will be referred to as the Mei model, which is expected to be valid
irrespective of $\varepsilon$, provided that $\Rey_{p}$ and $\Rey_{\omega}$
are small.
The flow is assumed to be unbounded for the two lift models introduced above.
In particle-laden channel flows, near-wall accumulation of particles is often
observed, as discussed above. Typically, the peak in the particle
concentration appears within the viscous sublayer, i.e., for $y^{+}\leq 5$,
wherein the wall effect on the lift is crucial if $D^{+}\geq 1$ (Balachandar &
Eaton, 2010; Shi et al., 2021). In this context, the explicit lift solution
can be derived if the wall lies in the “inner region” of the flow disturbance
in the low-$\Rey_{p}$ limit, specifically, if the separation between the
particle and the wall, $\ell$, is much smaller than the inertial lengths
($\ell\ll\min{(\ell_{u},\ell_{\omega})}$). The corresponding lift solutions
(see Shi & Rzehak (2020) for a comprehensive review) take the general form
$C_{L}=A+BSr+CSr^{2},$ (8)
where $Sr=\left|\boldsymbol{\omega}\right|D/\left|\mathbf{U}_{s}\right|$ is
the normalized shear rate, and $A$, $B$, and $C$ are pre-factors that are
independent of $Sr$. Hereafter, the terms proportional to $Sr$ and $Sr^{2}$
will be referred to as the linear and quadratic inner contributions
$C_{L,\text{in}}^{\text{linear}}$ and $C_{L,\text{in}}^{\text{quad}}$,
respectively.
Situations where the wall is in the “outer region” of the flow disturbance
($\ell\gg\max{(\ell_{u},\ell_{\omega})}$), while still in the limit
$\Rey_{p}\to 0$, were considered in Asmolov (1990), McLaughlin (1993) and
Takemura et al. (2009). Assuming that $Sr=\Rey_{\omega}^{1/2}\varepsilon\leq
O(1)$ (i.e. the same condition for the shear rate as considered in Saffman
(1965)), the outer-region lift approaches the Saffman solution (5), namely
$C_{L,\text{out}}^{\text{linear}}\propto\varepsilon$. Together with the
scaling in the inner-region that $C_{L,\text{in}}^{\text{linear}}\propto Sr$,
it appears that the inner-region lift contribution transitions into the outer-
region by a pre-factor $\varepsilon/Sr$. If one assumes that the quadratic
contribution follows the same transition, it may be speculated that
$C_{L,\text{in}}^{\text{quad}}$ scales as $\varepsilon Sr$ in the outer
region, i.e. $C_{L,\text{out}}^{\text{quad}}\propto\varepsilon Sr$. For the
particle-laden channel flow,
$Sr\approx\frac{(u_{\tau}^{2}/\nu)D}{u_{s}}=\left(\frac{u_{s}}{u_{\tau}}\right)^{-1}D^{+},$
(9)
where $u_{s}$ is the streamwise mean slip velocity, namely
$u_{s}=\langle\mathbf{U}_{s}\cdot\mathbf{e}_{x}\rangle$ with $\mathbf{e}_{x}$
the unit vector along the streamwise direction; $u_{\tau}$ is the conventional
wall friction velocity. As will be demonstrated in figure 5 in section 3.1,
$u_{s}$ might change its sign in the buffer layer, leading to extremely large
values of $Sr$ in the near-wall region.
The analysis above implies that there is also a quadratic lift contribution
$C_{L}^{\text{quad}}\propto\varepsilon Sr$, which might dominate the lift
generation in the inner wall region. This is consistent with the form of the
lift force proposed in Costa et al. (2020a, b), where:
$C_{L}=\frac{18}{\pi^{2}}\varepsilon SrJ^{\infty},$ (10)
with $J^{\infty}=2.255$ according to Saffman (1965). Despite its simplified
form, this correlation aligns well with their PR-DNS results. Hereafter, the
lift model by Eq. (10) will be referred to as the CBP model.
Finally, it should be noted that in practice, to avoid singularities in the
numerical calculation, we implemented a slight variant of the lift coefficient
as described in (4), where the quotient on the right-hand side to define the
unit vector is modified to
$\boldsymbol{\omega}\times\mathbf{U}_{s}/|\boldsymbol{\omega}||\mathbf{U}_{s}|$.
In practice, the results are not very sensitive to this choice, as
$\boldsymbol{\omega}\times\mathbf{U}_{s}$ is nearly aligned with the wall-
normal direction.
### 2.2 Validation
Figure 1: $(a)$ Normalized particle volume fraction $\phi/\Phi$ and outer-
scaled second-order moments of particle velocity: $(b)$ streamwise velocity
r.m.s., $(c)$ wall-normal velocity r.m.s., $(d)$ Reynolds shear stress
profile. The points (colour online) denote the corresponding lift-included 1WC
PP-DNS from Costa et al. (2020b). Here $U_{b}$ is the flow bulk velocity.
The present code has demonstrated successful applications in wall-bounded
particle-laden flows, including open channel flow (Wang & Richter, 2019; Gao
et al., 2023) and planar Couette flow (Richter & Sullivan, 2014; Richter,
2015) loaded with inertial particles, but the lift force is neglected. Hence,
for completeness, we present the validation of the one-way-coupling PP-DNS
code, including the lift force models. Specifically, we reproduced in figure 1
the PP-DNS of particle-laden turbulent channel flow in the one-way coupling
regime using these three lift models, as reported in Costa et al. (2020a, b)
in the one-way coupling regime. This corresponds to case CL1 in Table 1. The
agreement is excellent, which validates our implementation.
### 2.3 Computational parameters
We perform point-particle DNS of channel flow at $\Rey_{\tau}=180$ in a
computational domain $(L_{x},L_{y},L_{z})=(6h,2h,3h)$, where $h$ is the half
channel height. The domain is discretized on
$(N_{x},N_{y},N_{z})=(160,320,160)$ grid points; the grid is slightly
stretched to refine the near-wall resolution, corresponding to a grid spacing
of $(\Delta_{x}^{+},\Delta_{y}^{+},\Delta_{z}^{+})=(6.75,0.9,3.375)$, with the
conventional ‘$+$’ superscript denoting viscous wall scaling. Note that the
grid spacing near the wall is comparable to that of the largest particle
sizes. All simulations are carried out at a constant time-marching step fixed
at $\Delta t^{+}=0.1$ (normalized by viscous unit $\nu/u^{2}_{\tau}$), which
corresponds to $\text{CFL}\approx 0.4$. The DNS results confirm this time step
to be sufficiently small that particles and fluid elements could not pass
through a grid cell per time step (Zheng et al., 2021). The total simulation
time is $T\approx 450h/u_{\tau}$ (about $80\,000$ viscous time scales
$\nu/u_{\tau}^{2}$), which is long enough to guarantee converged statistics
(Sardina et al., 2012).
Case | $D^{+}$ | $\rho_{p}/\rho$ | $N_{p}$
---|---|---|---
CPF | particle-free case
CL1 | 3 | 100 | $5\times 10^{4}$
CM1 | 1 | 900 | $1.5\times 10^{5}$
CS1 | 0.1 | 90000 | $1.5\times 10^{6}$
Table 1: Physical parameters used in the present point-particle DNS campaign.
$N_{p}$ denotes the total number of particles, and $D^{+}$ is inner-scaled
particle diameter. The particle density ratio was adjusted to keep the same
value of the Stokes number $St^{+}=\tau_{p}/(\nu/u_{\tau}^{2})=50$ and bulk
solid mass fraction $\Psi_{m}=0.337$ for all cases.
We consider three different setups for varying particle diameter, while
$St^{+}=\tau_{p}/(\nu/u_{\tau}^{2})=50$ is fixed by varying the particle
density. These parameters are described in table 1. This target particle
Stokes number, was chosen since it is known to feature highly inhomogeneous
particle distributions in wall turbulence; see Sardina et al. (2012).
Moreover, it should be noted that a value of $D^{+}=3$ corresponds to about
one Kolmogorov length scale in the channel bulk, an order of magnitude which
is often investigated in the literature using PP-DNS (see, e.g., Bernardini,
2014; Motoori et al., 2022; Zhang et al., 2022). We performed one- and two-way
coupling simulations at a fixed mass fraction for different particle sizes and
lift force models. Naturally, in the one-way coupled simulations, the mass
fraction is not a governing parameter, and the high number of particles
ensures statistical convergence of the results. In the two-way coupling PP-
DNS, the bulk mass fraction is fixed to about $\Psi_{m}\approx 30\%$, to
ensure significant turbulence modulation and highlight the effects of lift
force model choice in the flow statistics.
## 3 Results and discussion
### 3.1 Particle dynamics
We start by describing the dynamics of the dispersed phase under the effect of
lift force models using a simple, one-way coupling approach. This allows us to
measure observables sampled by the particle that dictate the validity region
of the lift models (e.g., $Sr$, $\varepsilon$, and $\Rey_{p}$). As we will
see, current lift models are bound to fail in the buffer layer.
Figure 2 shows the profiles of the normalized mean particle concentration for
all particle sizes. When the lift force is not taken into account, the
preferential accumulation of particles in the wall region is significant,
which has been widely observed (Picano et al., 2009; Sardina et al., 2012; Lee
& Lee, 2015; Gao et al., 2023; Gualtieri et al., 2023) and termed as
turbophoresis (Reeks, 1983; Johnson et al., 2020). However, the particles show
less wall accumulation when subjected to lift forces, and this effect is more
pronounced the larger the particle size. The Saffman model yields a much
smaller peak, while the other two models feature smaller deviations, with the
highest peak corresponding to the Mei model. When the particles are very small
($D^{+}=0.1$), the differences in the lift force models become almost
indistinguishable, with all the models robustly showing the same reduction of
wall accumulation. Yet, the difference in accumulation in the absence of lift
is clear.
Figure 2: Local particle mass fraction $\phi$, normalized by the bulk value
$\Phi$ as a function of the wall-normal distance (1WC). $(a)$ CL1, $(b)$ CM1,
$(c)$ CS1. The inset shows the normalized particle volume fraction, versus the
inner-scaled wall distance.
Figure 3: Inner-scaled instantaneous streamwise velocity fluctuation and the
corresponding instantaneous snapshots of particle locations in $x$-$z$
($y^{+}=12$, left column) and $y$-$z$ ($x^{+}=540$, right column) planes (Case
CL1, 1WC). The wall-parallel plane shows all particles with inner-scaled wall
normal position $Y_{p}^{+}\leq 12$. $(a)$ Lift-free, $(b)$ Saffman model,
$(c)$ Mei model, $(d)$ CBP model.
To qualitatively describe the lift force effect on the spatial localization of
wall accumulation, figure 3 shows instantaneous snapshots of streamwise
velocity fluctuations in a wall-parallel plane within the buffer layer
($y^{+}=12$) and in a plane of constant streamwise location ($x^{+}=540$),
along with the corresponding particle positions for the case with largest
particles CL1. Expectedly, the particles show strong spatial localization and
inhomogeneous distribution, with larger local density occurring in elongated
clusters (Sardina et al., 2012). Effects of the lift force model on the gross
characteristics of the particle distribution are readily discernible. One
notable manifestation is the difference in particle numbers in the wall-
parallel ($x$-$z$) plane. Consistent with figure 2, the lift-free model yields
the highest number of particles, while the Saffman model yields the lowest.
Conversely, Mei and CBP models, fall somewhere between these two. This
indicates the weakening accumulation of particles near the wall by the lift
force, consistently with observations by Marchioli et al. (2007) and Shin et
al. (2022). The other manifestation concerns small-scale particle clustering.
At first glance, lift-free, Mei and CBP simulations show a more significant
tendency to over-sample low-speed regions than the Saffman one. This tendency
is checked in figure 4, showing the probability density functions (p.d.f.s) of
streamwise fluid velocity fluctuations sampled by the particles within the
$y^{+}\leq 12$ region. For larger particle sizes, particles under the Saffman
lift force sample a less focused range of streamwise velocity fluctuations
than other models for larger particle sizes. In particular, for $D^{+}=3$ and
$1$, the CBP and Mei cases show about the same tendency to sample low-speed
regions, differently than the Saffman model and comparable to the lift-free
case for $D^{+}=3$. Preferential accumulation gets more focused with
decreasing particle size, with the lift-free case producing the strongest
preferential accumulation for $D^{+}=1$ and $0.1$.
Figure 4: Probability density functions (p.d.f.s) of inner-scaled streamwise
fluid velocity fluctuations conditioned at the particle location within the
$y^{+}\leq 12$ region (1WC). $(a)$ CL1, $(b)$ CM1, $(c)$ CS1.
Figure 5: Inner-scaled streamwise mean particle velocity and slip velocity
profiles (1WC). $(a,d)$ CL1, $(b,e)$ CM1, $(c,f)$ CS1. The horizontal dashed
lines denote $\langle u_{s}\rangle=0$.
Figure 6: Second-order moments of mean particle (solid lines) and fluid
(dashed lines) velocity (1WC). $(a,b,c)$ Inner-scaled streamwise and $(d,e,f)$
wall-normal velocity variances, and $(g,h,i)$ Reynolds shear stress profiles.
The left $(a,d,g)$, middle $(b,e,h)$ and right $(c,f,i)$ columns denote Case
CL1, CM1 and CS1, respectively.
Figure 5 shows the inner-scaled profiles of streamwise mean particle velocity
$\langle u_{p}\rangle$ and local slip velocity $\langle u_{s}\rangle$.
Compared with the lift-free particles with $D^{+}\geq 1$, lift forces increase
$\langle u_{p}\rangle$ in the viscous sublayer and buffer layer, while the
differences between cases become negligible beyond $y^{+}\sim 30$. Once more,
particles under the Saffman lift force show the highest variation with respect
to the lift-free case, yielding the highest $\langle u_{p}\rangle$; particles
under the Mei lift force yield the lowest, and the ones under the CBP lift
force fall in between. Consistently, for the smallest particle sizes, the
overall modification of $\langle u_{p}\rangle$ by lift forces is quite small.
This tendency of lift forces becoming less important with decreasing particle
size is expected – as we will discuss later in more detail (§3.3), the
relative importance of lift to drag forces close to the wall roughly scales
with the inner-scaled particle diameter $D^{+}$.
At the wall, while the fluid velocity must vanish, particles can feature a
mean apparent slip where they flow faster than the fluid due to their inertia
(Zhao et al., 2012). This is clearly shown in the particle mean slip (figure
5$d,e,f$). At larger wall distances, the tendency of particles to over-sample
slower-than-average fluid velocity (Kiger & Pan, 2002; Baker & Coletti, 2021)
is also clearly reflected in the mean particle slip velocity. A similar trend
can be observed in the corresponding streamwise mean slip velocity profiles,
however, the difference in $\langle u_{s}\rangle$ is larger than $\langle
u_{p}\rangle$, and this is attributed to the negative apparent slip velocity
($\langle u|_{\mathbf{x}=\mathbf{X}_{p}}\rangle-\langle u\rangle<0$), which
reflects preferential sampling of slower-than-average fluid. Interestingly,
lift forces increase the apparent slip velocity near the wall. This can be
understood from the sign of the lift force in this region, which points
towards the wall due to the positive slip velocity. Particles sampling higher-
momentum regions will be driven towards the wall by the lift force, and their
inertia results in a higher mean slip.
The sign of $\langle u_{s}\rangle$ changes in roughly the same location of the
buffer layer, irrespective of the particle size and lift force model. This
mechanism seems relatively robust and has been observed in numerous other
studies with inertial particles at different Stokes number, and different flow
Reynolds number (see, e.g., Mortensen et al., 2008; Zhao et al., 2012; Wang &
Richter, 2020; Gao et al., 2023). It will be shown that this change in sign
makes lift force modelling in wall turbulence extremely challenging (see
discussions in §3.3).
Figure 6 shows the $x-y$ components inner-scaled velocity covariances for the
particle and fluid phases, respectively. The spanwise velocity variances are
hardly affected by lift forces, and thus not shown. Expectedly, away from the
wall where the mean shear is low, the particle statistics closely follow those
of the lift-free cases. Near the wall, instead, the lift force enhances the
streamwise and wall-normal particle velocity variances ($\langle u^{\prime
2}_{p}\rangle$ and $\langle v^{\prime 2}_{p}\rangle$) and Reynolds stress
($\langle u^{\prime}_{p}v^{\prime}_{p}\rangle$), even for the smallest
particle size ($D^{+}=0.1$). Enhancement of wall-normal velocity fluctuations
near the wall has indeed been reported in previous numerical and experimental
studies (Fong et al., 2019; Costa et al., 2020a). Indeed, as also shown in
Costa et al. (2020a), this enhancement of velocity fluctuations near the wall
cannot be reproduced when lift forces are not considered in the particle
dynamics. It seems that lift force is very effective at correlating streamwise
and wall-normal velocity fluctuations near the wall – under high shear rate,
small variations in streamwise velocity naturally translate to changes in
vertical acceleration through the lift forces, which in turn induce wall-
normal velocity fluctuations. Streamwise velocity fluctuations will also be
naturally amplified, since streamwise and wall-normal velocity fluctuations
are correlated through the mean shear. As we will see in §3.4, this
amplification of near-wall velocity fluctuations by lift forces has major
consequences on turbulent modulation and drag changes in the flow at high
particle mass loading.
### 3.2 Mechanism for near-wall accumulation
To understand the role of the lift force in altering the near-wall particle
distributions, it is essential to examine three primary mechanisms responsible
for wall-normal inertial particle transport. The first is the lift-induced
migration, which is the direct consequence of the inertial lift force. The
second is turbophoresis (Reeks, 1983), causing particle migration from regions
of higher to lower turbulence intensity. In wall-bounded flows, turbophoresis
results in strong particle accumulation in the viscous sublayer, where
turbulent fluctuations vanish. Note however that in most cases the peak in the
particle concentration does not appear at the wall but at a distance of $O(D)$
from the wall (Marchioli & Soldati, 2002; Costa et al., 2020a), owing to the
wall-particle collisions as well as hydrodynamic wall-particle interactions
(Goldman et al., 1967; Vasseur & Cox, 1977; Zeng et al., 2005). Finally, there
is a biased sampling effect, due to the tendency of particles to sample
ejection regions in the buffer layer featuring negative streamwise velocity
and positive (i.e. repelling) wall-normal velocity fluctuations (Marchioli &
Soldati, 2002; Sardina et al., 2012). As a result, biased sampling usually
provides a net slow particle drift away from the wall. The interaction among
the three mechanisms is sketched in figure 7 and will be elaborated in the
following.
Figure 7: Illustration of the interplay between different mechanisms
responsible for the wall-normal particle transport.
Lift-induced migration is directed along
$\boldsymbol{\omega}\times\mathbf{U}_{s}$, which in the case of a turbulent
wall flow is predominantly in the wall-normal direction. According to figure
5($d,e,f$), particles lead the liquid flow for $y^{+}\lesssim 10$ and lag
behind the flow further away from the wall. Hence, the lift-induced migration
is _towards_ the wall in the viscous sublayer and acts cooperatively with the
turbophoresis effect. The way these two mechanisms interact in the viscous
sublayer may seem surprising at first glance, as figure 2 clearly indicates a
suppression of the near-wall particle accumulation by lift force. The
reasoning behind this suppression may be understood by noting the following
two issues. First, the lift force is towards the channel center within the
buffer layer and beyond $y^{+}\approx 10$. The resulting outward migration,
together with the strong turbulence intensity in the buffer layer, tends to
entrain particles in this region outwards, leading to a net decrease in the
concentration in the region $y^{+}\lesssim 10$ as revealed from figure 2.
Second, the lift force influences the wall-normal particle transport by
turbophoresis. This phenomenon is linked to the particle wall-normal velocity
variance $\langle v^{\prime 2}_{p}\rangle$ (Reeks, 1983; Johnson et al.,
2020). Indeed, in the limit of vanishing particle Reynolds number, the
particle wall-normal momentum balance at steady state yields a turbophoresis
pseudo-force proportional to $\mathrm{d}\langle v^{\prime
2}_{p}\rangle/\mathrm{d}y$, which creates the migration of particles down
gradients in particle wall-normal velocity variance (Sikovsky, 2014; Johnson
et al., 2020). Recall figure 6($d,e,f$), where the variation of $\langle
v^{\prime 2}_{p}\rangle$ with the wall distance is depicted. Clearly, the lift
force tends to increase the “apparent inertia” of the particles, enhancing the
deviation between the particle velocity variance and that of the fluid within
the viscous sublayer. This leads to an attenuation in the corresponding
turbophoresis pseudo-force. In particular, for the two cases where $D^{+}\geq
1$ (figure 6$d,e$), the prediction with the Saffman lift model yields
$\mathrm{d}\langle v^{\prime 2}_{p}\rangle/\mathrm{d}y<0$ in the viscous
sublayer, indicating that the turbophoresis pseudo-force is outwards, driving
the particles away from the wall. Consequently, the lift force competes with
the turbophoresis pseudo-force in these two cases, leading to a highly
flattened near-wall peak in the particle fraction as seen in figure 2($a,b$).
In contrast, they cooperate in predictions using the Mei and the CBP models,
as figure 2($a$) shows. There, the lift force compensates the attenuation in
turbophoresis, making the negative slopes in $\phi(y^{+})$ from results using
Mei and the CBP models closely follow that in the lift-free case (highlighted
in the figure insets). Note, however, that the magnitude of the Mei lift (CBP
lift) is proportional to $Re_{\omega}^{1/2}$ ($Re_{\omega}$), which approaches
$D^{+}$ (${D^{+}}^{2}$) in the viscous sublayer. Consequently, for the two
cases where $D^{+}\leq 1$ the lift-induced migration is weak and incapable of
compensating the still-pronounced attenuation in turbophoresis revealed from
figure 6$(e,f)$. This is why in these two cases the slopes in $\phi(y^{+})$
from results employing the Mei and CBP models are smaller in magnitude than
that in the lift-free case. Still, the difference in accumulation with respect
to the lift-free case for smaller sizes is clear, due to near-wall particles
residing for long times near the wall while being subjected to small but
persistent lift forces.
Figure 8: The inner-scaled lift-induced acceleration (i.e., normalized by
$u^{3}_{\utau}/\nu$) profiles along the wall-normal direction (1WC). $(a)$
CL1, $(b)$ CM1, $(c)$ CS1. The horizontal dashed lines denote $\langle
F_{l,y}\rangle=0$.
Given the discussion above, it is worth examining in more detail the link
between the chosen lift model and the corresponding modulation in the
turbophoresis pseudo-force, particularly whether to which extent this
modulation relates to the lift force magnitude. Combing the turbophoresis
pseudo-force (figure 6$d,e,f$) and lift force as shown in figure 8, a
monotonic increase in the suppression of the turbophoresis pseudo-force with
increasing lift is observed at $D^{+}\leq 1$, while no such trend is evident
at $D^{+}=3$. In the latter case, the suppression of the turbophoresis pseudo-
force is most pronounced in the prediction using the Saffman model, even
though the magnitude of the corresponding lift force is only half that of the
prediction using the CBP model for $y^{+}\lesssim 5$. Hence, the differences
in peak concentration are not directly linked to the lift force magnitudes,
but rather to the modulation in the turbophoresis pseudo-force induced by
these models. In the present work, the turbophoresis pseudo-force is always
suppressed by the presence of lift, leading to a decrease in the peak
concentration. Hence, the precise relationship between the extent of this
suppression and the lift force magnitude remains unclear, making it
challenging to directly connect the chosen lift models with the changes in the
predicted concentration profiles. Finally, it would be worth investigating
these dynamics at lower Stokes numbers, as both turbophoresis and preferential
sweeping will be less pronounced and the nature of the interaction between
lift and turbulence may drastically change. Given the increased issues with
lift force modelling at lower Stokes number, discussed in the next section,
addressing this question should be the object of a future, dedicated study.
To gain insights into biased sampling of sweeps/ejection events under lift
forces, we carried out a quadrant analysis for the fluid flow experienced by
the particles. In the ($u^{\prime},v^{\prime}$) plane, with positive
$v^{\prime}$ directed away from the wall, the flow experienced by the particle
is categorized into four types of events: first quadrant events (Q1),
characterized by outward motion of high-speed fluid, with $u^{\prime}>0$ and
$v^{\prime}>0$; second quadrant events (Q2), characterized by outward motion
of low-speed fluid, with $u^{\prime}<0$ and $v^{\prime}>0$, which are usually
called ejections; third quadrant events (Q3), characterized by inward motion
of low-speed fluid, with $u^{\prime}<0$ and $v^{\prime}<0$; and finally,
fourth quadrant events (Q4), which represent motions of high-speed fluid
toward the wall, with $u^{\prime}>0$ and $v^{\prime}<0$, and are usually
called sweeps.
Figure 9: Joint p.d.f.s of the streamwise and wall-normal fluid fluctuating
velocities seen by ascending ($v_{p}>0$) and descending ($v_{p}<0$) particles
within the $y^{+}\leq 12$ region (1WC). $(a)$ CL1, $(b)$ CM1, $(c)$ CS1. The
numbers in panels denote the proportion of ascending particles.
Figure 9 shows the joint p.d.f.s in the ($u^{\prime},v^{\prime}$) plane of the
fluid fluctuation seen by ascending ($v_{p}>0$) and descending ($v_{p}<0$)
particles within the $y^{+}\leq 12$ region. The first observation is that a
larger proportion of particles tends to move outwards (ascending),
irrespective of the presence of lift force. The proportion of ascending
particles without lift force decreases as the particle size decreases, whereas
particles subject to lift force exhibit an inverse trend. Second, the lift-
free ascending particles are highly likely to sample ejections (Q2), while the
lift-free descending ones are prone to sample sweeps (Q4) for particles of
$D^{+}\geq 1$. This agrees with experimental observations in channel flow
laden with finite-size particles (Kiger & Pan, 2002; Yu et al., 2016; Baker &
Coletti, 2021). However, the particles under the effect of lift forces tend
not to over-sample sweeps, irrespective of the wall-normal particle velocity
sign, which indicates that the wall-normal velocity of the particles is less
correlated to the fluid one. This also confirms that inertial particles under
the effect of lift forces are not prone to slowly drive toward the wall in
low-speed regions, because the competing wall-repelling effect of lift forces
seems to dominate their dynamics. Close inspection of the high-intensity
region of the joint p.d.f.s on the $u^{\prime}$-axis, we note that particles
under the Saffman lift force tend to sample a wider range of negative
$u^{\prime}$ regions, compared with particles under the Mei and CBP lift
forces. This aligns with the p.d.f.s of $u^{\prime}$ as shown in figure 4.
Third, irrespective of the presence of the lift force or wall-normal particle
velocity sign, the smallest particles ($D^{+}=0.1$) over-sample ejection (Q2)
and inward motions of low-speed fluid regions (Q3). Sweep events, bringing
high-momentum fluid towards the wall are only weakly experienced by particles.
Overall, the percentage of particles sampling Q2 events only slightly
surpasses that of Q3, suggesting that biased sampling may play a minor role in
driving high inertial particles ($St^{+}=50$) to depart from the inner wall
region. This observation aligns with numerical results from Marchioli &
Soldati (2002), where biased sampling is found to be more efficient for
transferring particles with smaller inertia.
### 3.3 Emerging challenges in lift force modelling
The aforementioned results allow us to address several issues concerning the
effects of lift force in turbulent wall flows, which may have been previously
overlooked in PP-DNS simulations. Ideally, such discussions would be based on
results from PR-DNS simulations. However, in this instance, we will
extrapolate from PP-DNS results, in conjunction with the CBP model, as an
analogue to PR-DNS outcomes. This approach is justified by findings from Costa
et al. (2020a, b), which demonstrate that PP-DNS simulations are capable of
satisfactorily reproducing the corresponding PR-DNS results for the most
challenging case considered in this study, with $D^{+}=3$.
Let us first address the circumstances under which the effects of lift force
could be considered negligible for large inertial particles, and thereby
disregarded in PP-DNS simulations. Given that biased sampling does not play a
key role in the wall-normal transport of inertial particles, a criterion based
on the ratio of lift to the turbophoresis pseudo-force seems sufficient to
describe the relative significance of the lift force. In this context, the
dimensionless particle size $D^{+}$ seems a promising candidate, owing to the
following two reasons. Firstly, ${D^{+}}^{2}$ approaches the shear Reynolds
number $Re_{\omega}$ in the viscous sublayer; the latter directly measures the
magnitude of the shear lift irrespective of the choice of lift models
(Saffman, 1965; McLaughlin, 1991; Mei, 1992; Shi & Rzehak, 2019; Costa et al.,
2020b). Secondly, the turbophoresis effects seem to decay with increasing
$D^{+}$. This is clearly revealed by comparing the slopes of $\langle
v^{\prime 2}_{p}\rangle(y^{+})$ revealed from figure 6$(d,e,f)$ which
correspond to results with increasing $D^{+}$. Glueing these two aspects
together, it seems that the force ratio of lift to turbophoresis roughly
scales as ${D^{+}}^{k}$ with $k>0$. Although the exact value of $k$ is
infeasible to be determined, the results summarized in §3.1 made it clear that
by no means should the lift force be neglected for $D^{+}\geq 1$. On the other
hand, it is reasonable for the lift force to be neglected for $D^{+}\leq 0.1$,
except for the underprediction of near-wall accumulation discussed above
(recall figure 2).
Figure 10: The p.d.f.s of $u_{s}/(\omega_{z}D)$ along the wall-normal
direction (1WC DNS with the CBP model). $(a)$ CL1, $(b)$ CM1, $(c)$ CS1. The
vertical dashed lines denote $u_{s}/(\omega_{z}D)=\pm 1$.
We should now address a major issue in commonly applied lift force models for
PP-DNS simulations of turbulent wall flows. As outlined in section 2.1, any of
the shear-lift models originating from the pioneering work of Saffman (1965)
are rigorously applicable only in the double limits $\Rey_{p}\to 0$ and
$\Rey_{\omega}\to 0$. In the viscous sublayer, $\Rey_{p}\approx D^{+}(\langle
u_{s}\rangle/u_{\tau})\approx D^{+}$ for $D^{+}\geq 1$ according to figure 5,
whereas $\Rey_{\omega}\approx{D^{+}}^{2}$. Consequently, none of the two
limits are satisfied for $D^{+}\geq 1$. In addition to the double limits
above, there is a constraint concerning the dimensionless shear rate $Sr$.
This is made clear by noting that $Sr=\Rey_{\omega}^{1/2}\varepsilon$ and
hence $Sr\leq O(1)$ even in the strong shear limit, $\varepsilon\geq 1$,
considered in Saffman (1965). This is why in the wall-bounded situation the
“outer-region” lift solutions (Asmolov, 1990; McLaughlin, 1993; Takemura et
al., 2009) following the methodology of Saffman (1965) match with the
corresponding “inner-region” solutions (Cherukat & McLaughlin, 1994; Magnaudet
et al., 2003; Shi & Rzehak, 2020) only for $Sr=O(1)$. For particle-laden
channel flows with small inertial particles (Mortensen et al., 2008; Zhao et
al., 2012; Wang & Richter, 2020; Gao et al., 2023), this constraint is more
seriously violated than that for the double limits above, as the slip velocity
reverses at $y^{+}\approx O(10)$ (see, e.g., figure 5$(d,e,f)$ in the present
work), making $Sr\to\infty$ for particles within the buffer layer. This issue
is highlighted in figure 10, where we show the p.d.f.s of the inverse of $Sr$
(horizontal axis) as a function of the wall-normal distance $y^{+}$ (vertical
axis). Apparently, a large portion of particles experiences a strong shear
where $Sr\geq 1$, in particular for the two cases where $D^{+}\geq 1$.
Finally, it is worth noting that the probability of a particle experiencing
very high values of $Sr$ near the wall will increase with decreasing values of
$St^{+}$ as, for the same local shear rate, particles will experience an ever
smaller slip velocity. This exacerbates the violation of the $Sr=O(1)$
constraint, meaning that current lift force models are even less suitable for
turbulent channel flow laden with particles at lower Stokes numbers.
Additionally, the discussion above focuses on the lift force arising from the
shear in the ambient flow. In wall-bounded flows, as considered in this work,
the lift force may deviate significantly from its unbounded counterpart if the
particle is close to the wall. In brief, the presence of the wall leads to a
repulsive transverse force in the absence of the ambient shear (Vasseur & Cox,
1977; Zeng et al., 2005) and, in the presence of an ambient shear, a
suppression of the shear-induced lift (McLaughlin, 1993; Magnaudet et al.,
2003; Takemura et al., 2009). Although analytical solutions for the combined
lift force are achievable in some asymptotic limits (see, e.g., Wang et al.
(1997) and, more recently, Shi & Rzehak (2020) for an overview of these
solutions), there is so far no satisfactory way to glue these solutions
together to achieve a general treatment of the wall effect (Ekanayake et al.,
2020, 2021). Here, instead of trying to make any further progress in achieving
such a general solution, it is more feasible to discuss under which condition
may this intricate wall effect be neglected. Within the $O(D)$-depth layer
close to the wall, wall-particle collisions govern the wall-normal position of
the peak particle concentration. Hence, the hydrodynamic wall-particle
interaction may play a significant role only beyond $y^{+}\approx D^{+}$. On
the other hand, it is known that (McLaughlin, 1993) the amount of reduction of
the shear-induced lift due to the wall decreases as
$(\ell/\ell_{\omega})^{-5/3}$ and reduces to only $20\%$ of its magnitude in
the limit $\ell/\ell_{\omega}\to 0$ at $\ell/\ell_{\omega}\approx 3$. Hence,
the wall effect may be considered negligible if the position of the peak
particle concentration, which is of $O(D)$ from the wall, is larger than about
$3\ell_{\omega}$, which yields roughly $D^{+}\geq 3$. This finding, together
with the previous one where (shear-induced) lift force is non-negligible for
$D^{+}\geq 1$, indicates that such a wall effect can be disregarded except in
the overlap regime where $1\leq D^{+}\leq 3$.
In addition to the ambient shear, particle rotation is known to produce a lift
force as well (Rubinow & Keller, 1961). This lift contribution is often
neglected in prior PP-DNS simulations and not considered in the present work.
Nevertheless, it might be worthy to discuss in the following its relevant
importance. Assuming that the particle inertia is small enough for the
particle to always stay in the torque-free state, particle rotation leads to a
lift contribution of $C_{L}^{\Omega}=0.5Sr$ in the double limits considered by
Saffman (1965). The ratio between this spin-induced lift contribution and that
induced by ambient shear is proportional to $Re_{\omega}^{1/2}$, indicating
that $C_{L}^{\Omega}$ is merely a component of the second-order lift
contribution from the ambient shear. This is confirmed from the recent work by
Candelier et al. (2023), where second-order lift contributions (with respect
to $Re_{\omega}^{1/2}$) were obtained using matched asymptotic expansions. In
addition to the spin-induced contribution $C_{L}^{\Omega}=0.5Sr$, they also
obtained a second-order contribution from the ambient shear
$C_{L}^{{}^{\prime}(2)}=-0.505Sr$, which counterbalances the spin-induced
contribution. In other words, the effect of particle rotation on the lift is
negligible in the torque-free state. This conclusion is valid irrespective of
the presence of the wall (Cherukat & McLaughlin, 1994; Magnaudet et al., 2003;
Shi & Rzehak, 2020). However, large-inertia particles with a large density
ratio as considered in this work might not always stay torque-free.
Specifically, for particles with density ratio $\rho_{p}/\rho\geq 100$, the
relaxation time in response to torque is comparable with that to slip (see,
e.g., Bagchi & Balachandar (2002)). This finding, together with the pronounced
slip in the viscous sublayer revealed from figure 5$(a,b)$, indicates that
particles of $D^{+}\geq 1$ may experience a strong angular acceleration and
possibly a non-negligible lift contribution from forced rotation.
Investigating the effects of rotation by accounting for the particles’
conservation of angular momentum would therefore be interesting but is left
outside the scope of the present work due to the higher complexity associated
with drag and lift closures for angular momentum.
Finally, it is worth noting that while the CBP lift force model used in Costa
et al. (2020a, b) appears to provide reasonable predictions in some cases, it
is not universally valid. Specifically, it does not align with any of the
inner-region solutions (Shi & Rzehak, 2020), which are applicable for
particles extremely close to the wall (e.g., within the viscous sublayer), nor
does it approach the unbounded solution (Saffman, 1965; McLaughlin, 1991) for
particles located within the outer layer of the channel flow. The agreement
between the PP-DNS simulations utilizing this lift force model and PR-DNS
suggests that the model offers a satisfactory blending of results in the two
limits for the specific cases explored in Costa et al. (2020a) (see table 1).
A preliminary attempt to establish a more general lift force model can be
found in the work of Ekanayake et al. (2020, 2021), where a lift model
satisfying the above conditions was proposed in the limit of small shear
Reynolds numbers, i.e. $\Rey_{\omega}\ll 1$. However, the applicability of
this model is limited to $D^{+}\ll 1$ as, in the viscous layer,
$\Rey_{\omega}\approx(u_{\tau}^{2}/\nu)D^{2}/\nu=(D^{+})^{2}$. While this
provides some insights, its practicality may be limited since the lift force
no longer plays a pivotal role for $D^{+}<1$, as demonstrated here. Hence,
further work extending the work of Ekanayake et al. (2020, 2021) to finite
$\Rey_{\omega}$ (hence $D^{+}$) would be significant.
### 3.4 Lift-induced turbulence modulation
Finally, in this section, we use two-way coupling point-particle DNS to
illustrate that lift force models can have tremendous consequences in basic
integral quantities such as the overall drag. To achieve this, we use the same
parameters shown in table 1, which fixes the total mass fraction and particle
Stokes number while varying the particle diameter. We should reiterate that
high-fidelity two-way coupling point-particle DNS is highly challenging and is
currently being actively investigated. Still, the simple particle-in-cell 2WC
DNS employed here suffices to illustrate the impact of lift force model choice
on important turbulence metrics. In what follows, we illustrate the effects of
lift force on the overall drag and, analyze the streamwise momentum fluxes
that contribute to it.
Figure 11 shows the friction coefficient for the 2WC DNS, defined as
$C_{f}=2u^{2}_{\tau}/U^{2}_{b}$. While all small inertial particles have a
drag-increasing effect, the difference between lift force models is highly
amplified as the particle size increases. At $D^{+}=3$, in particular,
particles under the Saffman lift force show a drag reduction compared to the
other particle sizes, almost reaching an overall drag decrease. Instead, the
other lift models show a monotonic drag increase.
Figure 11: The wall friction coefficient $C_{f}$ versus the particle diameter
$D^{+}$ (2WC). The horizontal dashed line denotes the mean $C_{f}$ of the
particle-free case.
Figure 12: Budget of streamwise momentum $(a,c,e)$ and contribution of each
stress contribution $\tau_{i}$ to the mean wall friction coefficient
$C_{f}=2\tau_{w}/(\rho U_{b}^{2})$ $(b,d,f)$,
$1/(h^{2})\int_{0}^{h}6(h-y)\tau_{i}\mathrm{d}y/(\rho U_{b}^{2})$ normalized
by that of the particle-free case. $(a,b)$ CL1, $(c,d)$ CM1, $(e,f)$ CS1.
Different line colours in $(a,c,e)$ denote different cases, which match the
legend in $(c)$.
To better understand and quantify this difference, we follow the analysis in
Costa et al. (2021) and investigate the stress budget in the two-way coupling
limit of vanishing volume fraction, while keeping the mass fraction finite:
$\tau=\rho
u_{\tau}^{2}\left(1-\frac{y}{h}\right)\approx\underbrace{\left(\mu\frac{\mathrm{d}\left\langle
u\right\rangle}{\mathrm{d}y}-\rho\left\langle
u^{\prime}v^{\prime}\right\rangle\right)}_{\tau_{\nu}+\tau_{T,f}}-\underbrace{\left\langle\psi\right\rangle\left\langle
u_{p}^{\prime}v_{p}^{\prime}\right\rangle}_{\tau_{T,p}},$ (11)
where $\left<\psi\right>=\rho_{p}\left<\phi\right>$ is the mean local mass
fraction. The first two terms in the right-hand-side of this equation are the
viscous stress, fluid Reynolds stress, and particle Reynolds stress (hereafter
$\tau_{\nu}$, $\tau_{T,f}$, and $\tau_{T,p}$). This equation boils down to the
stress budget in the single-phase limit ($\tau_{\nu}$ and $\tau_{T,f}$),
corrected by an additional turbulent particle momentum flux, $\tau_{T,p}$ that
involves the local mass fraction and the particle counterpart of the Reynolds
shear stress. This term relates to the transfer of streamwise momentum by
correlated particle velocity fluctuations, and shows the importance of high
particle mass fraction near the wall when combined with correlated
streamwise–wall-normal particle velocity fluctuations. We have seen in §3.1
that, while the former decreases in the presence of lift force, the latter may
be increased by it.
The profiles of stress terms and the corresponding contributions of each
momentum transfer mechanism to the overall friction coefficient, normalized by
the overall friction coefficient of the particle-free case, are shown in
figure 12. On the left panels, the normalized shear stress profiles are shown,
while on the right we show the so-called FIK identity employed to a particle
suspension (Fukagata et al., 2002; Yu et al., 2021). In the case of a
turbulent plane channel flow, this identity decomposes the mean friction
coefficient ($C_{f}$) into a laminar and turbulent contribution. Here, the
latter can be further decomposed into a contribution due to the fluid phase
and one due to the particles. Each contribution is given by the following
weighted integral
$C_{f,i}=1/h^{2}\int_{0}^{h}6(h-y)\tau_{i}\,\mathrm{d}y/(\rho U_{b}^{2})$,
with $\tau_{i}$ being one the stress mechanisms described above. Note that the
residual of this stress budget is virtually zero, which validates the
approximation in equation (11).
As anticipated from figure 11, all 2WC DNS show increased drag compared to the
unladen flow. Yet, remarkably, different lift models show qualitatively
different turbulence modulation for the largest particle size ($D^{+}=3$).
Particles under the Saffman lift force have a major net turbulence attenuation
effect, which results in an almost drag-reducing flow with respect to the
unladen case. Conversely, particles under the Mei and CBP lift forces have a
net drag increase, mostly caused by an increase in the two-way coupling stress
term $\psi\left<u^{\prime}_{p}v_{p}^{\prime}\right>$. This results from the
amplification of particle Reynolds shear stresses
$\left<u^{\prime}_{p}v^{\prime}_{p}\right>$ discussed in section 3.1, which
compensate for the decrease in local mass fraction $\psi$. Naturally, as the
particle size decreases, the differences between lift models become less
pronounced but still noticeable at all particle sizes, with particles under
the Saffman lift force consistently showing the highest fluid turbulence
attenuation. The smallest particle size $D^{+}=0.1$, the two turbulent
stresses (fluid and particle) terms still show noticeable but in practice
negligible differences (recall figure 11).
Hence, it appears that the most important dynamical effects of lift forces in
two-way coupling conditions can be seen as finite-size effects that directly
modify single-point moments of particle velocity for $D^{+}\gtrsim 1$ (recall
figure 6). However, the lift force still affects near-wall turbulence
modulation when particles are relatively small, as they strongly affect the
mean particle concentration (recall figure 2), directly linked to turbulence
modulation via the two-way coupling stress term.
## 4 Conclusions
We have investigated the importance of lift forces in the dynamics of
particle-laden turbulent channel flow. Three relevant shear-induced lift force
models are analyzed using PP-DNS for varying particle size and fixed Stokes
number. Our results confirm that shear-induced lift forces strongly influence
particle dynamics. We have analyzed the main mechanisms for near-wall particle
accumulation under lift forces: lift-induced migration, turbophoresis, and
biased sampling. Very close to the wall, where the particle-to-fluid slip
velocity is positive and particle fluctuations are vanishing, lift forces and
turbophoresis cooperate in driving particles toward the wall. Particle-wall
interactions and biased sampling balance this effect. Further away from the
wall, lift forces and turbophoresis compete, with the former having a wall-
repelling effect. Concerning biased sampling, we observe that, for larger
particle sizes ($D^{+}\geq 1$), the lift-free ascending particles prefer to
sample ejections, while the descending ones tend to sample sweeps. Conversely,
particles under lift forces tend to sample ejection regions while experiencing
a marginal occurrence of sweep events, regardless of the employed lift model
or wall-normal particle velocity sign. This is again a consequence of the
competing wall-repelling effect of lift force. For the smallest particles
($D^{+}=0.1$), all cases exhibit over-sampling in ejection and inward motions
of low-speed fluid regions (i.e., Q3 events). Overall, the sampling percentage
of the ejections events only slightly surpasses that of Q3, suggesting that
biased sampling plays a minor direct role in near-wall particle transport.
However, the biased sampling of low-speed regions exposes particles for long
times to persistent lift forces, which results in significantly reduced
particle accumulation, even for small particle sizes.
These different dynamics have major consequences for turbulence modulation due
to a modification of the near-wall values of (1) the local particle
concentration, and (2) the correlated streamwise-wall-normal particle velocity
fluctuations. While the latter is only modified for sufficiently large
particle size and therefore can be seen as a finite-size effect, the former is
still non-negligible for relatively small particle sizes, and can still result
in small but visible changes in overall drag. Indeed, correlated particle
velocity fluctuations show little sensitivity to lift forces for sufficiently
small values of inner-scaled particle diameter ($D^{+}\lesssim 0.1$).
Conversely, for small particle sizes, the modification of near-wall particle
accumulation due to lift showed little sensitivity to the flavour of shear-
induced lift model. Hence, to accurately account for lift-induced turbulence
modulation with $D^{+}=O(1)$ particle sizes, reliable shear-induced lift
models are needed that correctly reproduce the particle turbulent momentum
flux. This could be seen as a necessary correction for finite-sized effects,
in the same spirit as the Faxén correction in the Maxey-Riley-Gatignol
equation.
While such correction is important for reliable particle-modeled simulations
of turbulent wall flows, our analysis has shown that current lift force models
are bound to lose their validity in the near-wall region for small inertial
particles. The reason is that the well-known tendency of particles to over-
sample low-speed regions in the buffer layer, along with their tendency to
flow faster than the fluid in the viscous sublayer, results in a region near
the wall of vanishing particle-to-fluid slip velocity. Naturally, this issue
will be even more pronounced at lower Stokes numbers, where the range of very
low slip velocities in regions of high shear is wider. Unfortunately, current
shear-induced lift force models are only available for non-dimensional shear
rates $Sr$ of order unity, which robustly tends to infinity within the buffer
layer. Hence, the condition of $Sr\leq 1$ required by these models is
seriously violated in wall turbulence with small inertial particles.
[Acknowledgements]The Cray XC40 Shaheen II at KAUST was used for all
simulations reported. PC thanks F. Picano from Padova University for
insightful discussions.
[Funding]This research was partially supported by the KAUST Office of
Sponsored Research (OSR) under award no. OSR-2019-CCF-3666 and under baseline
research funds of MP.
[Declaration of interests]The authors report no conflict of interest.
[Author ORCIDs] Wei Gao, https://orcid.org/0000-0001-7313-0058; Pengyu Shi,
https://orcid.org/0000-0001-6402-4720; Matteo Parsani,
https://orcid.org/0000-0001-7300-1280; Pedro Costa,
https://orcid.org/0000-0001-7010-1040.
## References
* Arcen et al. (2006) Arcen, B., Tanière, A. & Oesterlé, B. 2006 On the influence of near-wall forces in particle-laden channel flows. International Journal of Multiphase Flow 32, 1326–1339.
* Asmolov (1989) Asmolov, E. S. 1989 Lift force exerted on a spherical particle in a laminar boundary layer. Fluid Dynamics 24 (5), 710–714.
* Asmolov (1990) Asmolov, E. S. 1990 Dynamics of a spherical particle in a laminar boundary layer. Fluid Dynamics 25 (6), 886–890.
* Bagchi & Balachandar (2002) Bagchi, P. & Balachandar, S. 2002 Effect of free rotation on the motion of a solid sphere in linear shear flow at moderate Re. Physics of Fluids 14 (8), 2719–2737.
* Baker & Coletti (2021) Baker, L. J. & Coletti, F. 2021 Particle–fluid–wall interaction of inertial spherical particles in a turbulent boundary layer. Journal of Fluid Mechanics 908, A39.
* Balachandar & Eaton (2010) Balachandar, S. & Eaton, J. K. 2010 Turbulent dispersed multiphase flow. Annual Review of Fluid Mechanics 42, 111–133.
* Battista et al. (2023) Battista, F., Gualtieri, P., Mollicone, J.-P., Salvadore, F. & Casciola, C. M. 2023 Drag increase and turbulence augmentation in two-way coupled particle-laden wall-bounded flows. Physics of Fluids 35, 045133.
* Bernardini (2014) Bernardini, M. 2014 Reynolds number scaling of inertial particle statistics in turbulent channel flows. Journal of Fluid Mechanics 758, R1.
* Bernardini et al. (2013) Bernardini, M., Pirozzoli, S. & Orlandi, P. 2013 The effect of large-scale turbulent structures on particle dispersion in wall-bounded flows. International Journal of Multiphase Flow 51, 55–64.
* Boivin et al. (1998) Boivin, M., Simonin, O. & Squires, K. D. 1998 Direct numerical simulation of turbulence modulation by particles in isotropic turbulence. Journal of Fluid Mechanics 375, 235–263.
* Botto et al. (2005) Botto, L., Narayanan, C., Fulgosi, M. & Lakehal, D. 2005 Effect of near-wall turbulence enhancement on the mechanisms of particle deposition. International Journal of Multiphase Flow 31, 940–956.
* Brandt & Coletti (2022) Brandt, L. & Coletti, F. 2022 Particle-laden turbulence: Progress and Perspectives. Annual Review of Fluid Mechanics 54, 159–189.
* Candelier et al. (2023) Candelier, F., Mehaddi, R., Mehlig, B. & Magnaudet, J. 2023 Second-order inertial forces and torques on a sphere in a viscous steady linear flow. Journal of Fluid Mechanics 954, A25.
* Capecelatro et al. (2018) Capecelatro, J., Desjardins, O. & Fox, R. O. 2018 On the transition between turbulence regimes in particle-laden channel flows. Journal of Fluid Mechanics 845, 499–519.
* Cherukat & McLaughlin (1994) Cherukat, P. & McLaughlin, J. B. 1994 The inertial lift on a rigid sphere in a linear shear flow field near a flat wall. Journal of Fluid Mechanics 263, 1–18.
* Costa et al. (2020a) Costa, P., Brandt, L. & Picano, F. 2020a Interface-resolved simulations of small inertial particles in turbulent channel flow. Journal of Fluid Mechanics 883, A54.
* Costa et al. (2020b) Costa, P., Brandt, L. & Picano, F. 2020b Interface-resolved simulations of small inertial particles in turbulent channel flow – CORRIGENDUM. Journal of Fluid Mechanics 891, E2.
* Costa et al. (2021) Costa, P., Brandt, L. & Picano, F. 2021 Near-wall turbulence modulation by small inertial particles. Journal of Fluid Mechanics 922, A9.
* Crowe et al. (1977) Crowe, C. T., Sharma, M. P. & Stock, D. E. 1977 The particle-source-in cell (PSI-CELL) model for gas-droplet flows. Journal of Fluids Engineering 99 (2), 325–332.
* Eaton & Fessler (1994) Eaton, J. K. & Fessler, J. R. 1994 Preferential concentration of particles by turbulence. International Journal of Multiphase Flow 20, 169–209.
* Ekanayake et al. (2021) Ekanayake, N. I. K., Berry, J. D. & Harvie, D. J. E. 2021 Lift and drag forces acting on a particle moving in the presence of slip and shear near a wall. Journal of Fluid Mechanics 915, A103.
* Ekanayake et al. (2020) Ekanayake, N. I. K., Berry, J. D., Stickland, A. D., Dunstan, D. E., Muir, I. L., Dower, S. K. & Harvie, D. J. E. 2020 Lift and drag forces acting on a particle moving with zero slip in a linear shear flow near a wall. Journal of Fluid Mechanics 904, A6.
* Elghobashi (1994) Elghobashi, S. 1994 On predicting particle-laden turbulent flows. Applied Scientific Research 52, 309–329.
* Fessler et al. (1994) Fessler, J. R., Kulick, J. D. & Eaton, J. K. 1994 Preferential concentration of heavy particles in a turbulent channel flow. Physics of Fluids 6 (11), 3742–3749.
* Fong et al. (2019) Fong, K. O., Amili, O. & Coletti, F. 2019 Velocity and spatial distribution of inertial particles in a turbulent channel flow. Journal of Fluid Mechanics 872, 367–406.
* Fröhlich et al. (2018) Fröhlich, K., Schneiders, L., Meinke, M. & Schröder, W. 2018 Validation of Lagrangian two-way coupled point-particle models in large-eddy simulations. Flow, Turbulence and Combustion 101 (2), 317–341.
* Fukagata et al. (2002) Fukagata, K., Iwamoto, K. & Kasagi, N. 2002 Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows. Physics of Fluids 14 (11), L73–L76.
* Gao et al. (2023) Gao, W., Samtaney, R. & Richter, D. H. 2023 Direct numerical simulation of particle-laden flow in an open channel at ${Re}_{\tau}=5186$. Journal of Fluid Mechanics 957, A3.
* Gatignol (1983) Gatignol, R. 1983 The Faxén formulas for a rigid particle in an unsteady non-uniform Stokes-flow. J. Méc. Théor. Appl. 2 (2), 143–160.
* Goldman et al. (1967) Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall—I Motion through a quiescent fluid. Chemical Engineering Science 22 (4), 637–651.
* Gualtieri et al. (2023) Gualtieri, P., Battista, F., Salvadore, F. & Casciola, C. M. 2023 Effect of Stokes number and particle-to-fluid density ratio on turbulence modification in channel flows. Journal of Fluid Mechanics 974, A26.
* Gualtieri et al. (2015) Gualtieri, P., Picano, F., Sardina, G. & Casciola, C. M. 2015 Exact regularized point particle method for multiphase flows in the two-way coupling regime. Journal of Fluid Mechanics 773, 520–561.
* Horne & Mahesh (2019) Horne, W. J. & Mahesh, K. 2019 A massively-parallel, unstructured overset method to simulate moving bodies in turbulent flows. Journal of Computational Physics 397, 108790.
* Horwitz (2023) Horwitz, J. A. K. 2023 Improved force models for Euler–Lagrange computations. In Modeling Approaches and Computational Methods for Particle-Laden Turbulent Flows, pp. 265–298. Elsevier.
* Horwitz et al. (2022) Horwitz, J. A. K., Iaccarino, G., Eaton, J. K. & Mani, A. 2022 The discrete Green’s function paradigm for two-way coupled Euler–Lagrange simulation. Journal of Fluid Mechanics 931, A3.
* Horwitz & Mani (2016) Horwitz, J. A. K. & Mani, A. 2016 Accurate calculation of Stokes drag for point–particle tracking in two-way coupled flows. Journal of Computational Physics 318, 85–109.
* Horwitz & Mani (2020) Horwitz, J. A. K. & Mani, A. 2020 Two-way coupled particle-turbulence interaction: Effect of numerics and resolution on fluid and particle statistics. Physical Review Fluids 5, 104302\.
* Ireland & Desjardins (2017) Ireland, P. J. & Desjardins, O. 2017 Improving particle drag predictions in Euler–Lagrange simulations with two-way coupling. Journal of Computational Physics 338, 405–430.
* Jie et al. (2022) Jie, Y., Cui, Z., Xu, C. & Zhao, L. 2022 On the existence and formation of multi-scale particle streaks in turbulent channel flows. Journal of Fluid Mechanics 935, A18.
* Johnson et al. (2020) Johnson, P. L., Bassenne, M. & Moin, P. 2020 Turbophoresis of small inertial particles: theoretical considerations and application to wall-modelled large-eddy simulations. Journal of Fluid Mechanics 883, A27.
* Kaftori et al. (1995) Kaftori, D., Hetsroni, G. & Banerjee, S. 1995 Particle behavior in the turbulent boundary layer. I. Motion, deposition, and entrainment. Physics of Fluids 7 (5), 1095–1106.
* Kiger & Pan (2002) Kiger, K. T. & Pan, C. 2002 Suspension and turbulence modification effects of solid particulates on a horizontal turbulent channel flow. Journal of Turbulence 3, N19.
* Kuerten (2006) Kuerten, J. G. M. 2006 Subgrid modeling in particle-laden channel flow. Physics of Fluids 18 (2), 025108.
* Kuerten (2016) Kuerten, J. G. M. 2016 Point-particle DNS and LES of particle-laden turbulent flow - a state-of-the-art review. Flow, Turbulence and Combustion 97, 689–713.
* Lee & Lee (2015) Lee, J. & Lee, C. 2015 Modification of particle-laden near-wall turbulence: Effect of Stokes number. Physics of Fluids 27, 023303.
* Lee & Lee (2019) Lee, J. & Lee, C. 2019 The effect of wall-normal gravity on particle-laden near-wall turbulence. Journal of Fluid Mechanics 873, 475–507.
* Magnaudet et al. (2003) Magnaudet, J., Takagi, S. & Legendre, D. 2003 Drag, deformation and lateral migration of a buoyant drop moving near a wall. Journal of Fluid Mechanics 476, 115–157.
* Marchioli et al. (2003) Marchioli, C., Giusti, A., Salvetti, M. V. & Soldati, A. 2003 Direct numerical simulation of particle wall transfer and deposition in upward turbulent pipe flow. International Journal of Multiphase Flow 29 (6), 1017–1038.
* Marchioli et al. (2007) Marchioli, C., Picciotto, M. & Soldati, A. 2007 Influence of gravity and lift on particle velocity statistics and transfer rates in turbulent vertical channel flow. International Journal of Multiphase Flow 33, 227–251.
* Marchioli & Soldati (2002) Marchioli, C. & Soldati, A. 2002 Mechanisms for particle transfer and segregation in a turbulent boundary layer. Journal of Fluid Mechanics 468, 283–315.
* Marchioli et al. (2008) Marchioli, C., Soldati, A., Kuerten, J. G. M., Arcen, B., Taniere, A., Goldensoph, G., Squires, K. D., Cargnelutti, M. F. & Portela, L. M. 2008 Statistics of particle dispersion in direct numerical simulations of wall-bounded turbulence: Results of an international collaborative benchmark test. International Journal of Multiphase Flow 34 (9), 879–893.
* Maxey (2017) Maxey, M. 2017 Simulation methods for particulate flows and concentrated suspensions. Annual Review of Fluid Mechanics 49, 171–193.
* Maxey & Riley (1983) Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Physics of Fluids 26 (4), 883–889.
* McLaughlin (1989) McLaughlin, J. B. 1989 Aerosol particle deposition in numerically simulated channel flow. Physics of Fluids A: Fluid Dynamics 1 (7), 1211–1224.
* McLaughlin (1991) McLaughlin, J. B. 1991 Inertial migration of a small sphere in linear shear flows. Journal of Fluid Mechanics 224, 261–274.
* McLaughlin (1993) McLaughlin, J. B. 1993 The lift on a small sphere in wall-bounded linear shear flows. Journal of Fluid Mechanics 246, 249–265.
* Mehrabadi et al. (2018) Mehrabadi, M., Horwitz, J. A. K., Subramaniam, S. & Mani, A. 2018 A direct comparison of particle-resolved and point-particle methods in decaying turbulence. Journal of Fluid Mechanics 850, 336–369.
* Mei (1992) Mei, R. 1992 An approximate expression for the shear lift force on a spherical particle at finite Reynolds number. International Journal of Multiphase Flow 18 (1), 145–147.
* Mortensen et al. (2008) Mortensen, P. H., Andersson, H. I., Gillissen, J. J. J. & Boersma, B. J. 2008 Dynamics of prolate ellipsoidal particles in a turbulent channel flow. Physics of Fluids 20, 093302\.
* Motoori et al. (2022) Motoori, Y., Wong, C. & Goto, S. 2022 Role of the hierarchy of coherent structures in the transport of heavy small particles in turbulent channel flow. Journal of Fluid Mechanics 942, A3.
* Picano et al. (2009) Picano, F., Sardina, G. & Casciola, C. M. 2009 Spatial development of particle-laden turbulent pipe flow. Physics of Fluids 21, 093305.
* Reeks (1983) Reeks, M. W. 1983 The transport of discrete particles in inhomogeneous turbulence. Journal of Aerosol Science 14 (6), 729–739.
* Richter (2015) Richter, D. H. 2015 Turbulence modification by inertial particles and its influence on the spectral energy budget in planar Couette flow. Physics of Fluids 27, 063304.
* Richter & Sullivan (2014) Richter, D. H. & Sullivan, P. P. 2014 Modification of near-wall coherent structures by inertial particles. Physics of Fluids 26, 103304.
* Rubinow & Keller (1961) Rubinow, S. I. & Keller, J. B. 1961 The transverse force on a spinning sphere moving in a viscous fluid. Journal of Fluid Mechanics 11 (3), 447–459.
* Saffman (1965) Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. Journal of Fluid Mechanics 22 (2), 385–400.
* Sardina et al. (2012) Sardina, G., Schlatter, P., Brandt, L., Picano, F. & Casciola, C. M. 2012 Wall accumulation and spatial localization in particle-laden wall flows. Journal of Fluid Mechanics 699, 50–78.
* Schneiders et al. (2017a) Schneiders, L., Meinke, M. & Schröder, W. 2017a Direct particle–fluid simulation of Kolmogorov-length-scale size particles in decaying isotropic turbulence. Journal of Fluid Mechanics 819, 188–227.
* Schneiders et al. (2017b) Schneiders, L., Meinke, M. & Schröder, W. 2017b On the accuracy of Lagrangian point-mass models for heavy non-spherical particles in isotropic turbulence. Fuel 201, 2–14.
* Shi & Rzehak (2019) Shi, P. & Rzehak, R. 2019 Lift forces on solid spherical particles in unbounded flows. Chemical Engineering Science 208, 115145.
* Shi & Rzehak (2020) Shi, P. & Rzehak, R. 2020 Lift forces on solid spherical particles in wall-bounded flows. Chemical Engineering Science 211, 115264.
* Shi et al. (2021) Shi, P., Rzehak, R., Lucas, D. & Magnaudet, J. 2021 Drag and lift forces on a rigid sphere immersed in a wall-bounded linear shear flow. Physical Review Fluids 6, 104309\.
* Shin et al. (2022) Shin, H. H., Portela, L. M., Schaerer, C. E. & Mangiavacchi, N. 2022 Dynamics of suspended sediment transport: A Direct Numerical Simulation study. International Journal of Multiphase Flow 155, 104165.
* Sikovsky (2014) Sikovsky, D. P. 2014 Singularity of inertial particle concentration in the viscous sublayer of wall-bounded turbulent flows. Flow, Turbulence and Combustion 92, 41–64.
* Soldati & Marchioli (2009) Soldati, A. & Marchioli, C. 2009 Physics and modelling of turbulent particle deposition and entrainment: Review of a systematic study. International Journal of Multiphase Flow 35, 827–839.
* Takemura et al. (2009) Takemura, F., Magnaudet, J. & Dimitrakopoulos, P. 2009 Migration and deformation of bubbles rising in a wall-bounded shear flow at finite Reynolds number. Journal of Fluid Mechanics 634, 463–486.
* Toschi & Bodenschatz (2009) Toschi, F. & Bodenschatz, E. 2009 Lagrangian properties of particles in turbulence. Annual Review of Fluid Mechanics 41, 375–404.
* Uijttewaal & Oliemans (1996) Uijttewaal, W. S. J. & Oliemans, R. V. A. 1996 Particle dispersion and deposition in direct numerical and large eddy simulations of vertical pipe flows. Physics of Fluids 8 (10), 2590–2604.
* Vasseur & Cox (1977) Vasseur, P. & Cox, R. G. 1977 The lateral migration of spherical particles sedimenting in a stagnant bounded fluid. Journal of Fluid Mechanics 80 (3), 561–591.
* Vreman et al. (2009) Vreman, B., Geurts, B. J., Deen, N. G., Kuipers, J. A. M. & Kuerten, J. G. M. 2009 Two-and four-way coupled Euler–Lagrangian large-eddy simulation of turbulent particle-laden channel flow. Flow, Turbulence and Combustion 82 (1), 47–71.
* Wang et al. (2019) Wang, G., Fong, K. O., Coletti, F., Capecelatro, J. & Richter, D. H. 2019 Inertial particle velocity and distribution in vertical turbulent channel flow: A numerical and experimental comparison. International Journal of Multiphase Flow 120, 103105.
* Wang & Richter (2019) Wang, G. & Richter, D. H. 2019 Two mechanisms of modulation of very-large-scale motions by inertial particles in open channel flow. Journal of Fluid Mechanics 868, 538–559.
* Wang & Richter (2020) Wang, G. & Richter, D. H. 2020 Multiscale interaction of inertial particles with turbulent motions in open channel flow. Physical Review Fluids 5, 044307.
* Wang et al. (1997) Wang, Q., Squires, K. D., Chen, M. & McLaughlin, J. B. 1997 On the role of the lift force in turbulence simulations of particle deposition. International Journal of Multiphase Flow 23 (4), 749–763.
* Wray (1990) Wray, A. A. 1990 Minimal-storage time advancement schemes for spectral methods. In Tech. Rep.. NASA Ames Research Center.
* Yu et al. (2016) Yu, W., Vinkovic, I. & Buffat, M. 2016 Finite-size particles in turbulent channel flow: quadrant analysis and acceleration statistics. Journal of Turbulence 17 (11), 1048–1071.
* Yu et al. (2021) Yu, Z., Xia, Y., Guo, Y. & Lin, J. 2021 Modulation of turbulence intensity by heavy finite-size particles in upward channel flow. Journal of Fluid Mechanics 913, A3.
* Zeng et al. (2005) Zeng, L., Balachandar, S. & Fischer, P. 2005 Wall-induced forces on a rigid sphere at finite Reynolds number. Journal of Fluid Mechanics 536, 1–25.
* Zhang et al. (2023) Zhang, H., Cui, Y. & Zheng, X. 2023 How electrostatic forces affect particle behaviour in turbulent channel flows. Journal of Fluid Mechanics 967, A8.
* Zhang et al. (2022) Zhang, X., Nathan, G. J., Tian, Z. F. & Chin, R. C. 2022 The dominant mechanisms for each regime of secondary flows in horizontal particle-laden pipe flows. Journal of Fluid Mechanics 949, A10.
* Zhao et al. (2010) Zhao, L., Andersson, H. I. & Gillissen, J. J. J. 2010 Turbulence modulation and drag reduction by spherical particles. Physics of Fluids 22, 081702.
* Zhao et al. (2012) Zhao, L., Marchioli, C. & Andersson, H. I. 2012 Stokes number effects on particle slip velocity in wall-bounded turbulence and implications for dispersion models. Physics of Fluids 24, 021705.
* Zheng et al. (2021) Zheng, X., Feng, S. & Wang, P. 2021 Modulation of turbulence by saltating particles on erodible bed surface. Journal of Fluid Mechanics 918, A16.
|
AP and the percentage overlap between the training data of two languages. For
example, monolingual models trained on Zul and Nbl with $14.14\%$ and
$13.24\%$ overlap gives an AP of $62.9\%$ and $61.5\%$. respectively. On the
other hand, a model trained on Ssw and evaluated on Xho with only $6.84\%$
overlap gives an AP of $45.2\%$. Although all three languages belong to the
same language family as Xho, the two with a higher percentage overlap
outperform the one with less overlap. Moreover, we suspect that the model
trained on Sot outperforms the model trained on Afr (from the same language
family as Eng) when evaluating on Eng because of the high number of Eng words
present in the training set of Sot, showing a $9.69\%$ overlap.
This begs the question of whether the increased performance of training on a
language related to the target language (Figure 5.5) is related to the
phonetic properties shared among them, or because a model is exposed to
training examples from the target language. We investigate this uncertainty by
reproducing the experiment yielding the results in Figure 5.5, this time
removing the overlapping word segments from the training and validation sets.
The results without overlapping data are displayed in Figure 5.6. A
monolingual model is trained on data on each language where all words that
overlap with any other language is removed, then applied to every other
language where words from the training language are removed from the
evaluation set.
First, we observe a performance decrease when training and evaluating on the
same language (shown along the diagonal), especially for the Bantu languages.
Secondly, for most languages, the AP considerably decreased when training a
model on one language and applying it to another from the same language
family. Importantly, although the performance decreased, the best results are
still achieved by training on related languages, except for the one model
where training on Ven outperforms a model trained on Nbl by $0.6\%$ when
evaluated on Xho. Finally, for most languages, the AP decreased on Eng after
the Eng words were removed from the training data of all the other languages.
For example, the AP of Sot dropped from $57.2\%$ to $26.7\%$, with Afr (same
language family as Eng) now achieving the best AP ($11.1\%$ higher than second
best). This experiment indicates the potential of using related languages in
multilingual AWE modelling which we explore in the following section.
#### 5.5.2 Multilingual evaluation
The cross-lingual experiment above was in large part our inspiration for the
subsequent analysis of multilingual models. Concretely, we hypothesise that
even better performance can be achieved by training on multiple languages from
the same family as the target zero-resource language and that this would be
superior to multilingual models trained on unrelated languages. Focusing on
two evaluation languages, Zul and Sot from distinct language families, we
investigate this hypothesis by training multilingual models with different
language combinations, as shown in Table 5.2. Here we use the original data
without disregarding any particular words since it better simulates a real-
world scenario. Firstly, we see the best result on a language when all the
training languages come from the same family as the target. Secondly, we see
how the performance gradually decrease as the number of training languages
related to the target drop. Furthermore, notice the performance boost from
including even just one training language related to the target compared to
not including any. For instance, on Sot we see an increase of more than 12%
absolute when adding just one related language (from 52.5% and 51.9% to
64.8%).
To further demonstrate the benefit of using training languages from the same
family, we train a multilingual model for each evaluation language on all its
related languages using a 10% subset of the original data. For both Zul and
Sot, the subset models outperform the models where no related languages are
used. E.g. on Zul, the Xho+Ssw+Nbl subset model outperforms the full
Tsn+Nso+Eng model (no related languages) by more than 20% in AP. Moreover,
this subset model (58.6% in AP) even outperforms the Xho+Nso+Eng model (55.7%)
where all the training data from one related language are included and almost
matches the AP when using two related languages (Xho+Ssw+Eng, 60.9%).
These comparisons do more than just show the benefit of training on related
languages: they also show that it is beneficial to train on a diverse set of
related languages.
Table 5.2: AP (%) on test data for multilingual models trained on different combinations of well-resourced languages. Models are applied to two zero-resource languages from different language families, Nguni and Sotho-Tswana. For each training language, 100k word pairs were extracted. Multilingual model | Zul∗ | Sot†
---|---|---
Nguni: | |
Xho∗ \+ Ssw∗ \+ Nbl∗ | 68.6 | —
Xho∗ \+ Ssw∗ \+ Eng‡ | 60.9 | —
Xho∗ \+ Nso† \+ Eng‡ | 55.7 | —
Tsn∗ \+ Nso† \+ Eng‡ | 37.5 | —
Xho∗ \+ Ssw∗ \+ Nbl∗ ($10$% subset) | 58.6 | —
Sotho-Tswana: | |
Nso† \+ Tsn† | — | 76.7
Nso† \+ Eng‡ | — | 64.8
Xho∗ \+ Ssw∗ | — | 51.9
Xho∗ \+ Eng‡ | — | 52.5
Nso† \+ Tsn† ($10$% subset) | — | 58.4
#### 5.5.3 Adding more languages
In the above experiments, we controlled for the amount of data per language
and saw that training on languages from the same family improves multilingual
transfer. But this raises a question: will adding additional unrelated
languages harm performance? This question is of practical relevance in cases
where related data is either unavailable or the system developer is unsure for
which language AWEs will be extracted. To answer this, we systematically train
two sequences of multilingual models on all six well-resourced languages,
evaluating each target language as a new training language is added.
Same-different results for all five evaluation languages are shown in Figure
5.7 and QbE results for Zul, Sot and Afr are shown in Figure 5.8. Notice how
the trends in the same-different and QbE results track each other closely for
corresponding evaluation languages. In the first sequence of multilingual
models (green), we start by adding the three Nguni languages (Xho, Ssw, Nbl),
followed by the two Sotho-Tswana languages (Nso, Tsn), and lastly the Germanic
training language (Eng). The second sequence (orange) does not follow a
systematic procedure.
Figure 5.7: Same-different results from two sequences of multilingual models,
trained by adding one language at a time. For each training language, 100k
positive word pairs are used, which is indicated on the $x$-axis.
On Zul, the green sequence, which starts with a related language (Xho),
initially achieves a higher score compared to the orange sequence in both
Figures 5.7 and 5.8. Then, the score gradually increase by adding more related
languages (Ssw, Nbl). Thereafter, adding additional unrelated languages (Nso,
Tsn, Eng) show no performance increase. In fact, AP decreases slightly after
adding the two Sotho-Tswana languages (Nso, Tsn), but not significantly. The
orange sequence starts low on Zul until the first related language (Ssw) is
added, causing a sudden increase. Adding the Germanic language (Eng) has
little effect. Adding the last two related languages (Nbl, Xho) again causes
the score to increase. A similar trend follows for Sot and Afr, where adding
related languages causes a noticeable performance increase, especially when
adding the first related language; after this, performance seems to plateau
when adding more unrelated languages. (Afr is the one exception, with a drop
when adding the last language in the orange sequence). On Tso, which does not
have any languages from the same family in the training set, AP gradually
increases in both sequences without any sudden jumps. Although Ven is not in
the same family as Nso and Tsn at the lowest level of the tree in Figure 5.2,
it belongs to the same family (Sotho-Tswana) at a higher level. This explains
why it closely tracks the Sot results.
Figure 5.8: QbE results on Zul, Sot and Afr, using the same sequences of
multilingual models as in Figure 5.7. DTW result are shown for baseline
performance.
Summarising these results, we see that adding unrelated languages generally
does not decrease scores, but also does not provide a big benefit (except if
it is one of the earlier languages in the training sequence, where data is
still limited). In contrast, it seems that training on languages from the same
family is again beneficial; this is especially the case for the first related
language, irrespective of where it is added in the sequence.
### 5.6 Chapter summary
We investigated the effect of training language choice when applying a
multilingual AWE model to a zero-resource language. Using word discrimination
and QbE search tasks on languages spoken in South Africa, we showed that
training a multilingual model on languages related to the target is
beneficial. We observed gains in absolute scores, but also in data efficiency:
you can achieve similar performance with much less data when training on
multiple languages from the same family as the target. We showed that even
including just one related language already gives a large gain. From a
practical perspective, these results indicate that one should prioritise
collecting data from related languages (even in modest quantities) rather than
collecting more extensive datasets from diverse unrelated families, when
building multilingual acoustic word embedding models for a zero-resource
language. In the following chapter, we present a framework for producing
semantically organised AWEs using multilingual models developed in this
chapter.
## Chapter 6 Hate Speech Detection Using Multilingual Acoustic Word Embeddings
Leading up to this chapter, we mainly aimed at improving the quality of AWEs
for low-resource languages: we developed a new AWE model (Chapter 3), explored
multilingual adaptation techniques (Chapter 4), and considered language choice
in multilingual AWE modelling (Chapter 5). In Chapter 5 we briefly measured
AWE performance in a downstream query-by-example (QbE) speech search task.
However, this was performed with limited development data in a controlled
environment. In this chapter, we evaluate the practicality of using
multilingual AWEs for real-world applications, beyond the experimental
environments considered up to now.
The content of this chapter originates from a collaborative project with
VoxCroft, a company specialising in developing speech applications in low-
resource settings. We address a specific case study involving the development
of keyword spotting (KWS) systems to monitor radio broadcast audio for low-
resource languages. Here, we confront the challenge of lacking transcribed
data in the target language and dealing with radio recordings that differs
significantly from experimental audio data. In this challenging environment,
we develop an ASR-free approach, utilising a pre-trained multilingual AWE
model in a KWS system. Our main objective is to compare the performance of
this AWE-based KWS system to the conventional ASR-based KWS systems, which
mostly rely on transcribed data from the target language. We are particularly
concerned about the trade-off between accuracy and feasibility for rapid
deployment. To measure performance in this in-the-wild environment, we perform
our final experiments on real Swahili radio broadcast audio, scraped from
local radio stations in Kenya. Moreover, our KWS systems are specifically
developed to detect instances of hate speech in radio broadcasts. Therefore,
to search the audio, we use a set of keywords carefully selected by domain
experts that are potential markers of hate speech.
The content of this chapter was presented at Interspeech:
C. Jacobs, N. C. Rakotonirina, E. A. Chimoto, B. A. Bassett, and H. Kamper,
“Towards hate speech detection in low-resource languages: Comparing ASR to
acoustic word embeddings on Wolof and Swahili,” in Proceedings of Interspeech,
2023.
Contributions per author:
C. Jacobs developed the AWE-based KWS systems (Section 6.4.2).
N. C. Rakotonirina performed the ASR experiments (Section 6.4.1).
E. A. Chimoto evaluated radio recordings flagged by our KWS systems (Section
6.5.3.2).
B. A. Bassett and H. Kamper supervised the project and helped with editing the
paper.
### 6.1 Background
The rise of the internet and communication platforms are increasing the spread
of content, some of which include the use of derogatory or discriminatory
language targeting individuals or groups based on their identity or
characteristics. This form of expression is commonly referred to as hate
speech. Given the overwhelming volume of content being generated on online
platforms, several automatic monitoring systems have been explored to curb the
spread of hate speech (usually in the form of written text), especially for
resource-rich languages.
However, in many developing countries where there is a lack of technological
advancement, radio remains the primary medium for communicating news and
information to the public [111]. These radio stations play a vital role as
platforms for public discourse, facilitating freedom of expression [112].
Therefore, it is essential to establish appropriate mechanisms for monitoring
these radio stations to ensure responsible broadcasting practices.
A striking example of the consequences of hate speech over radio is the 1994
genocide in Rwanda where propaganda was disseminated through local radio
stations targeting individuals from an ethnic minority. The dehumanising
language used on air actively fuelled violence, resulting in devastating
consequences. Similarly, in the 2007-2008 post-election violence in Kenya,
vernacular radio stations played a significant role in exacerbating the
conflict [113].
In response to this ongoing issue, the United Nations has taken proactive
measures to address hate speech on radio [59]. By actively monitoring hate
speech claims, authorities can promptly respond and mitigate instances,
preventing tension escalation and the dissemination of harmful ideologies. The
efforts of this chapter are in line with the objective of combating hate
speech on radio broadcasts, specifically for low-resource languages.
Hate speech detection can be supported by using a keyword spotting (KWS)
system that can effectively search an audio corpus for a predetermined set of
keywords or phrases that are indicative of hate speech. Such a KWS system has
the potential ability to quickly scan through large amounts of audio content
and flag parts of the audio that have been identified as potential markers of
hate speech. It is important to note that a KWS system solely relying on
keyword or phrase identification may not be sufficient for comprehensive hate
speech detection. For instance, given the keyword “blood”, a KWS system may
retrieve utterances “we must spill their blood and eradicate them from our
society” and “she donated blood to help save lives during the blood drive.”
The former contains hate speech whereas the latter highlights a positive
action. Therefore, the context in which these potential hate speech markers
occur should be carefully considered and analysed by domain experts.
### 6.2 Related work
The conventional KWS approach involves transcribing the target audio with an
automatic speech recognition (ASR) system and then searching for keywords in
the output text [60, 61]. For low-resource languages, the absence of large
quantities of transcribed speech in the target language can make it difficult
to train high-quality ASR models. Several studies have looked into this
problem, proposing methods to train ASR models on limited amounts of labelled
data [114, 115, 55, 57, 116].
Menon et al. [56] were the first to evaluate and report results of a KWS
system monitoring radio broadcasts in low-resource languages spoken in Uganda.
Their system was tailored to identify specific keywords, contributing to the
progress of development and relief initiatives in rural areas. Their system
utilises an ASR model trained on around nine hours of transcribed data from
the target language. However, relying on this amount of data poses a challenge
as it will require a significant amount of time to collect and transcribe the
audio.
Saeb et al. [55] extended the work of Menon et al. [56] and implemented an
ASR-based KWS system that combines multilingual acoustic modelling and semi-
supervised training relying on much less transcribed data from the target
language. Here, they only use twelve minutes of annotated data from the target
language to train an ASR model. However, they use several hours of
untranscribed audio from the target language, coupled with a self-supervised
learning strategy to improve their system.
Rapidly deploying ASR KWS systems for low-resource languages remains
challenging due to the time-consuming process of collecting even a few minutes
of transcribed data, which can take several weeks. Moreover, multiple hours of
untranscribed speech are not necessarily available for all target languages
for self-supervised training.
When we want to rapidly develop and deploy a KWS system for hate speech
detection in a new setting, it might be better to opt for an ASR-free
approach. In a truly zero-resource setting, ASR-free KWS systems are
favourable, where KWS systems are developed without the need for transcribed
data. The most popular methodology extends QbE (Section 2.2.3), which uses a
spoken instance of a keyword to search through an audio corpus, without the
need for transcribing the audio into text. To use QbE for KWS, we need a small
number of spoken templates to serve as queries for the keywords of interest;
this is the approach followed in [57, 62] and the methodology we also use for
our AWE-based KWS system in this chapter.
DTW (Section 2.2.2) is typically used to match the speech features of a spoken
query to a search utterance [9, 48, 49, 50]. However, DTW has drawbacks in
terms of its high time complexity and limitations, such as poor speaker
independence, often requiring a large template set.
AWEs have been proposed as an alternative approach to finding matches in QbE
where search segments and query segments are jointly mapped to the same fixed
vector space [87, 6, 52, 53]. This allows for faster comparisons between query
and search segments in linear time complexity. Various neural networks have
been considered to obtain these AWEs as described in Section 2.3.
As demonstrated in Chapter 3 and Chapter 5, one approach to quickly obtain
robust AWE for a new language is to train a multilingual AWE model on labelled
data from multiple well-resourced languages, and then apply it to the target
low-resource language. Although these multilingual AWE models have proven
successful in controlled experiments, there has been limited work
investigating the effectiveness of these systems beyond the experimental
environment.
In this chapter, for the first time, we develop and evaluate a KWS system
utilising multilingual AWEs in a real-world scenario. For this experiment, we
apply the system to real radio broadcast audio, specifically targeting the
detection of instances of hate speech. The radio data was collected from
stations in Kenya, a Sub-Saharan country in Africa, where the predominant
language is Swahili. Additionally, we conducted developmental experiments
where we included Wolof, another low-resource language spoken mainly in
Senegal, Sub-Saharan Africa.
We compare the AWE-based KWS system to ASR-based KWS systems. We train our ASR
models, leveraging advancements in self-supervised speech representation
learning (SSRL) (Section 2.2.1). These models enable fine-tuning of ASR models
using limited transcribed data from the target language. Our main research
question is to compare the AWE-based KWS system with ASR-based KWS systems,
focusing on the trade-off between accuracy and rapid deployment.
Before we describe the different KWS systems, we first briefly give the
overall methodology that we follow towards our end goal of hate speech
detection on Swahili radio broadcasts.
### 6.3 Overview of methodology
We start by developing Wolof and Swahili KWS systems in a controlled
environment where we use parallel audio-text data from experimental datasets
to train and test our models. These datasets [117, 118] contain high-quality,
noise-free read speech. We then test our models on held-out test data from
these corpora. We refer to these experiments as in-domain because the test
data comes from the same domain as the training data. In this controlled
setup, we perform KWS using a set of keywords drawn from development data.
These are not necessarily hate speech keywords. The benefit of this controlled
setting is that we have transcriptions for the test data, and can therefore
quantify performance exactly.
We then apply the models to the out-of-domain Swahili radio broadcasts. These
recordings were obtained from three radio stations in Kenya. They consist of a
mix of read and spontaneous speech, and also include non-speech audio like
music and advertisements. Additionally, there may be instances of different
accents and languages present within the recordings. This is the type of real-
world audio that we would like to monitor for hate speech using KWS. As our
keywords in this context, we utilise actual hate speech keywords. These were
labelled as inflammatory words by native Swahili experts familiar with the
media environment. We should emphasise that, when we detect occurrences of
these words, they might not necessarily correspond to hate speech. For
example, the English sentence “do not kill your neighbour” is not hate speech
despite containing the word “kill”. In a deployed system the utterances
flagged by our KWS approach would be passed on to human moderators for further
review and hate speech assessment. In this work we also use a human moderator,
but only to mark whether a detected keyword actually occurred in an utterance.
We do this for our in-the-wild test since we do not have transcriptions for
the radio broadcasts.
In the following section, we provide a detailed overview of the KWS systems
that we developed. This will be followed by the implementation details of the
KWS systems (Section 6.5).
### 6.4 Low-resource keyword spotting
The ASR-based KWS systems of this chapter were developed by a colleague.
Therefore, in Section 6.4.1, we provide a high-level overview of this
approach. The AWE-based KWS system we developed as an ASR-free alternative, is
discussed in detail in the follow-up section.
#### 6.4.1 Keyword spotting with ASR
Figure 6.1: An illustration of ASR-based keyword spotting.
In an ASR-based KWS system, the target corpus is first transcribed to text.
Occurrences of a keyword in the corpus can be efficiently identified by simply
matching the keyword tokens with the output transcriptions. An example of this
ASR-based KWS is illustrated in Figure 6.1. In this example, when using the
keyword “hair” to search the output transcriptions, the KWS system will
correctly retrieve the first utterance (a). In the third utterance (c), “air”
was incorrectly transcribed as “hair” and will therefore be an incorrect
retrieval (although the keyword matches the transcription). In the second
utterance (b), the word “hair” in the audio was incorrectly transcribed as
“air” and will not be retrieved by the KWS system. Also, note that in this
example the ASR model correctly transcribed all the other words and would
therefore have a low word error rate (WER), but WER does not always correspond
to KWS performance. In Section 6.5.3 we give a more detailed explanation of
how we evaluate KWS performance.
Typically, to train an ASR model we need audio with transcriptions from the
target language. In a low-resource setting, a lack of large quantities of
transcribed speech data from the target languages prohibits the rapid
development of ASR-based KWS systems.
For our ASR systems, we build on recent advancements in ASR models leveraging
SSRL (Section 2.2.1). These SSRL models are trained in a self-supervised
fashion using large amounts of unlabelled audio recordings to produce high-
quality latent frame-level speech representations. It has been shown that
these pre-trained SSRL models can be fine-tuned in an end-to-end fashion for
ASR by mapping the latent frame representations to target text tokens. Fine-
tuning SSRL models for ASR requires using much less training data than
training an ASR model from scratch.
In our experiments, we use the pre-trained Wav2Vec2.0 XLSR model [76, 119].
This SSRL model is a large-scale cross-lingual speech model pre-trained on
half a million hours of unlabelled speech data in 128 languages (including
Swahili but not Wolof). The speech representation learned during cross-lingual
pre-training improves performance for low-resource languages by leveraging
data from the high-resource languages on which it is pre-trained. Using the
pre-trained model as a starting point, the model is fine-tuned on labelled
data from the language of interest.
Previous work has shown that this fine-tuning approach leads to competitive
ASR performance even with limited amounts of labelled data [119]. In our
experiments (Table 6.2 specifically) we show that the sample efficiency of the
XLSR model also extends to KWS when training on just five minutes of labelled
data.
#### 6.4.2 Keyword spotting with multilingual acoustic word embeddings
Figure 6.2: Search utterances ($u_{1},u_{2},\ldots,u_{N}$) are segmented and
mapped to a fixed-dimensional space, along with a query segment $q$, to
produce embeddings $\bigl{\\{}\mathbf{z}_{n}^{(i)}\bigr{\\}}$ and
$\mathbf{z}_{q}$, respectively. The subscript $n$ represents a unique
utterance within the search collection, and superscript $i$ represents a
segment extracted from utterance $\mathbf{z}_{n}$. Subsequently, search
utterances are ranked based on the shortest distance between a search segment
$\mathbf{z}_{n}^{(i)}$ and the query vector $\mathbf{z}_{q}$.
For a severely under-resourced language, there might not even be limited
amounts of labelled data available for ASR fine-tuning. In this scenario,
collecting only a few spoken instances of a keyword of interest would allow
for KWS through QbE search (Section 2.2.3). QbE is the task of retrieving
utterances from a search corpus using a spoken instance of a keyword, instead
of a written keyword.
In the previous chapter, our results (on development data) showed improved
performance in QbE search by using multilingual AWEs instead of DTW. We use a
similar setup to the QbE experiments described in Section 5.3 for experiments
in this chapter. The workings of a QbE system utilising AWEs are displayed in
Figure 6.2. First, the search corpus is segmented into word-like segments.
Search utterances are segmented by splitting into overlapping segments of some
minimum to some maximum duration (see Figure 5.3). Concretely, utterances from
the search corpus ($u_{1},u_{2},\ldots,u_{N}$) are segmented to produce a set
of variable-length speech segments
$S=\bigl{\\{}s_{n}^{(i)}\bigr{\\}}_{n=1}^{N}$, where $i$ is the $i$th search
segment extracted from utterance $n$.
A multilingual AWE model (trained on labelled data from multiple well-
resourced languages but not the target language) is then applied to the
segmented search corpus $S$ and query segment $q$, mapping them to the same
fixed-dimensional vector space. In this experiment, we use the CAE-RNN network
(Section 2.1.3) and train a multilingual CAE-RNN AWE model. This model is
trained on word pairs using labelled data from multiple languages (but does
not include the target language). This multilingual AWE model can be
transferred and applied to any unseen language to embed word segments of
arbitrary length. Utilising such a universal, language-independent model would
enable fast deployment of practical QbE-based KWS systems.
Applying the AWE model to search segments $S$ and query segment $q$ produces
search embeddings $\big{\\{}\mathbf{z}_{n}^{(i)}\bigr{\\}}$ and query
embedding $\mathbf{z}_{q}$, respectively. The utterances in the search corpus
are ranked based on comparing the query-segment distances
$d(\mathbf{z}_{q},\mathbf{z}_{n}^{(i)})$. For each query, the lowest distance
for each utterance is used as the final utterance score. We can then predict
that all query-utterance pairs with a distance below some threshold are
positive matches (this requires a threshold to be tuned on development data).
With these predictions, we can calculate common KWS metrics and compare them
to those achieved by the ASR-based KWS systems.
### 6.5 Experimental setup
In this section, we present the implementation details of both the ASR-based
(Section 6.4.1) and AWE-based (Section 6.4.2) KWS systems. It is important to
note that the overview provided in Section 6.3 serves as a reminder for the
reader. By the end of this section, the difference between the developmental
experiments conducted in a controlled environment and our in-the-wild KWS
testing, particularly concerning data and evaluation methods, should be clear.
#### 6.5.1 Data
Here we emphasise the difference between the training and evaluation data we
use during development and the data we use for our in-the-wild testing.
##### 6.5.1.1 Development
For our controlled experiments, we train and evaluate our models on Swahili
and Wolof data, collected from the Common Voice [118] and ALFFA [117]
datasets, respectively. Common Voice is an open-source data collection
initiative by Mozilla Foundation where volunteers can contribute by recording
their voices while reading prompts collected from a variety of sources. These
recordings are then verified for quality and accuracy. For our experiments we
use the Swahili data from the Common Voice Corpus 8.0 dataset which gives
train, test, and developing sets totalling 150 hours. In order to match the
sampling rate used during the training of the XLSR model, the original
recordings, which are sampled at 48 kHz, are downsampled to 16 kHz.
The current data repository of Common Voice does not include Wolof. We
therefore use a separate dataset from the ALFFA project, which contains Wolof
recordings and their corresponding transcriptions. Wolof is primarily spoken
in Senegal, another country in Sub-Saharan Africa. However, the official
language in Senegal is French, which serves as the lingua franca in all
governmental institutions. Due to the prevalence of French as the official
language, there are limited written documents available for Wolof to construct
parallel training data. For the Wolof language, we have designated training,
testing, and development sets, consisting only of 16 hours, 2 hours, and 2
hours, respectively.
For both datasets, we use the default train, development, and test splits. In
our controlled experiments, we evaluate in-domain KWS performance on the
Swahili and Wolof test splits using available transcriptions. The test split
of both languages serves as the search collection and is not used during model
development. The Swahili search collection contains 8 941 utterances and has a
duration of approximately 14 hours, while the Wolof set contains 2 000
utterances and has a duration of approximately 2 hours.
##### 6.5.1.2 In-the-wild
We perform our in-the-wild test on recorded radio broadcasts (collected by
VoxCroft) from three different radio stations in Kenya: Ghetto Radio, Radio
Salaam, and Radio Taifa. We do not have any transcriptions, speaker
information or content information on these audio clips. The nature of these
radio recordings has distinct characteristics that are different from the
prompted read speech we use during development. These broadcasts can contain a
variety of elements, including music in the background, fast-paced speech by
radio hosts, advertisements, background music, and dialogues involving
multiple speakers.
Our systems are only applied to these out-of-domain recordings and are nowhere
used during the development of our KWS systems. Before applying our systems to
this data we first apply a preprocessing step where we segment the recordings
using an unsupervised diarisation system that also attempts to remove music.
Apart from this preprocessing step, we simply apply our systems to this out-
of-domain data without further changes or calibration. After this
preprocessing step, 19 716 utterances are obtained, which is our out-of-domain
search collection. The duration of all utterances is between three and 30
seconds. Given that we do not have transcriptions of these recordings, we
cannot evaluate system performance in the same way we do in the controlled
experiments (where we have audio with corresponding transcriptions). As we
explain below (Section 6.5.3.2), we use a domain expert to evaluate our
system’s performance on this set.
#### 6.5.2 Models
We now provide the implementation details of the ASR model and the
multilingual AWE model for ASR-based (Section 6.4.1) and AWE-based KWS
(Section 6.4.2), respectively.
##### 6.5.2.1 ASR model
To train our ASR models, we fine-tune the pre-trained XLSR model by adding a
projection head to the final transformer layer. Subsequently, the model is
trained using transcribed data from the target language by mapping the frame-
level representations to character tokens. The model is optimised using the
connectionist temporal classification (CTC) loss [120]. The CTC loss is
effective for aligning input sequences and producing output sequences that may
not be temporally aligned, as is the case with speech and text. After the
fine-tuning step, the model produces a sequence of characters given an
arbitrary audio recording, that is then decoded to give the predicted
transcription.
For the ASR-based KWS systems, we control the amount of training data and
train three ASR models for Wolof and Swahili. Specifically, we fine-tune the
XLSR model [119] for each language on 30 hours, one hour, and five minutes of
labelled training data. These models are used for our development experiments
before applying them to the radio broadcast recordings.
##### 6.5.2.2 Multilingual acoustic word embedding model
For the AWE-based KWS system (Section 6.4.2), we train a multilingual AWE
model on five different Common Voice languages: Abkhazian (ab), Czech (cs),
Basque (eu), Swedish (sv), Tamil (ta). Word boundaries are obtained using the
Montreal forced-aligner [121]. We pool the data from all languages and extract
300k positive word pairs which we use to train a multilingual CAE-RNN AWE
model similar to the CAR-RNN model of Chapter 3. We use another well-resourced
language, Bulgarian, for validation during training using the same-different
evaluation (Section 2.5). The encoder and decoder each consist of three
unidirectional RNNs with 400-dimensional hidden vectors and an embedding size
of 100 dimensions. As inputs to the encoder, we use the XLSR model to extract
speech features (Section 2.2.1) instead of MFCCs used for AWE modelling in
previous chapters. We use the output of the $12$th transformer layer,
producing input features with 1024 dimensions. In the work of [122], they show
that the lower layers of the XLSR transformer model capture more
discriminative phonetic properties than the final layers when simply using the
model to extract frame-level speech representations. Preliminary experiments
showed using these features outperforms MFCCs in AWE modelling. We provide a
short analysis in Section 6.7.1 comparing multilingual AWEs trained on MFCC
and XLSR. All models are optimised using Adam [103] with a learning rate of
$0.001$.
#### 6.5.3 Evaluation
We outline our KWS evaluation methods for the ASR-based and AWE-based KWS
systems for the two scenarios: controlled in-domain experiments with available
transcriptions and out-of-domain in-the-wild tests where transcriptions are
unavailable for evaluation.
##### 6.5.3.1 Controlled experiments
To measure in-domain KWS performance, we apply the ASR-based and AWE-based KWS
systems to the same search corpora. Each utterance in the search corpus is
assigned a label of 1 if the keyword is detected by the respective KWS system,
or 0 if the keyword is not present. We evaluate our KWS systems by using the
predicted labels and compare them against the true labels obtained from
available transcriptions. Following this approach, we can calculate precision
and recall scores.
Precision measures the accuracy of a KWS system in predicting whether an
utterance contains a keyword. It is calculated as the number of utterances
correctly predicted as keyword-containing utterances (# hits) divided by the
total number of utterances predicted to contain a keyword (# keywords
detected):
$P=\frac{\text{\\# hits}}{\text{\\# keywords detected}}$ (6.1)
Recall quantifies the KWS system’s ability to retrieve keyword-containing
utterances from the search corpus. It is calculated as the number of correctly
predicted keyword-containing utterances (# hits) divided by the total number
of keyword-containing utterances in the search corpus (# keywords):
$R=\frac{\text{\\# hits}}{\text{\\# keywords}}$ (6.2)
In addition to precision and recall, we report the $F_{1}$-score to quantify
the trade-off between these two metrics. The $F_{1}$-score represents the
balanced combination of precision and recall, calculated as the harmonic mean
of the two metrics:
$F_{1}=2\times\frac{P\times R}{P+R}$ (6.3)
To evaluate our ASR KWS systems, we simply check whether a keyword occurs in
the predicted transcript after applying it to the search corpus and label each
utterance accordingly (1 or 0) to calculate the metrics above.
The AWE-based KWS system does not output transcribed text but a query-
utterance score for each query segment (bottom of Section 6.4.2). First, sub-
segments for utterances in the search collection are obtained by extracting
windows ranging from 20 to 35 frames with a 5-frame overlap (we experimented
with different window and overlap sizes on development data where this
configuration worked best). Then, an AWE model is applied to all the sub-
segments and query segment. Finally, the similarity between a query and search
segment is calculated using cosine distance.
We use a set of ten templates of 36 unique keywords with a minimum character
length of five for Swahili, and 15 unique keywords with a minimum character
length of four, for Wolof. These keywords were randomly sampled from a larger
set of words that appear at least ten times in both the development and test
sets. These keywords do not represent hate speech keywords. Query templates
are drawn from development data, embedded, and averaged to obtain a single AWE
embedding representing a keyword. Preliminary experiments showed that
averaging all templates from the same keyword improves KWS performance
compared to using a single template for searching. The threshold value in the
AWE-based KWS is tuned for the highest $F_{1}$-score across all keywords on
the controlled test data (more in Section 6.7.2).
Table 6.1: Swahili hate speech keywords used for in-the-wild KWS on Swahili radio broadcasts, with their English translations. Swahili | English | Swahili | English
---|---|---|---
vita | war | wezi | thieves
damu | blood | majimbo | states
hama | move | wakora | conmen
kabila | tribe | panga | machete
utapeli | fraud | takataka | garbage
mende | cockroaches | mjinga | stupid
kitendawili | riddle | fala | stupid
##### 6.5.3.2 In-the-wild
For the in-the-wild KWS, we apply the systems directly to the out-of-domain
Swahili radio broadcasts. We use a set of keywords labelled as inflammatory by
expert analysts from VoxCroft who are familiar with the media environment for
the purpose of hate speech detection. These keywords with their English
translations are given in Table 6.1. For the AWE-based KWS system, ten query
templates per keyword are extracted from the in-domain Swahili data. We
evaluate KWS performance by asking each approach to give 100 utterances out of
the search collection that are most likely to contain any of the hate speech
keywords. In the absence of per-word confidence scores for the ASR models,
here we simply take 100 random utterances that were predicted to contain a
keyword (for most models, the total number of utterances was around 150).
For the AWE-based KWS, we use the 100 highest-ranked utterances. Because we do
not have transcriptions for this data, we provide a native Swahili speaker
with untranscribed recordings of the utterances that were predicted to contain
a keyword. They then mark whether the keyword was indeed present. Note that
due to the absence of transcriptions for the radio recordings, we are unable
to calculate recall or the $F_{1}$-score. Therefore, we can only measure
performance by calculating precision, based on 100 retrieved utterances per
system (Table 6.4).
### 6.6 Experiments
We first look at the results from our controlled test on in-domain Swahili and
Wolof data. We then look at the results on the out-of-domain data, which is of
particular interest to us as it would indicate the effectiveness of both these
systems for real-world application.
#### 6.6.1 KWS results in a controlled environment
Table 6.2: ASR and AWE KWS results (%) on in-domain test data. For the ASR systems, the XLSR model is fine-tuned on each target language, controlling the amount of training data. For the AWE KWS system, a supervised multilingual AWE is trained on multiple well-resourced languages and applied to the two low-resource target languages. Model | Swahili | Wolof
---|---|---
| Prec. | Rec. | $F_{1}$ | Prec. | Rec. | $F_{1}$
ASR: | | | | | |
XLS-R (30-h) | 97.6 | 98.6 | 98.0 | 93.1 | 87.1 | 90.0
XLS-R (1-h) | 96.4 | 76.1 | 85.1 | 93.1 | 74.3 | 82.7
XLS-R (5-min) | 95.4 | 42.6 | 58.9 | 88.0 | 45.6 | 60.0
Multilingual AWE: | | | | | |
CAE-RNN (ab+cs+eu+sv+ta) | 57.1 | 56.1 | 56.6 | 44.0 | 58.4 | 50.2
The first three lines in Table 6.2 report the ASR KWS results on in-domain
Swahili and Wolof data. The results show that fine-tuning XLSR on only five
minutes of labelled data achieves decent KWS performance, with $F_{1}$-scores
of 59% and 60% for Swahili and Wolof, respectively. The multilingual AWE KWS
system using roughly one minute of template data, performs worse on most
metrics. The only metric on which the AWE system is better is recall, where it
achieves higher scores than the 5-minute ASR system on both languages. In
practice, the AWE system could therefore be a better option if recall is
important.
We briefly consider how ASR performance is affected by the amount of training
data. For reference, WERs for the 30-hour, 1-hour, and 5-minute models are
respectively 9%, 36%, and 62% on Swahili. For Wolof, the WERs are 27%, 44%,
and 68%. As expected, more training data gives better WER and KWS scores. But
it is noteworthy that with just five minutes of training data, we can already
spot keywords with high precision (95% and 88% for Swahili and Wolof,
respectively). This is especially useful for our use case, where we want to
rapidly develop KWS applications in severely low-resourced settings. The other
noteworthy finding from the ASR results is that scores for Swahili are not
notability higher than Wolof, although the former is one of XLSR’s pretraining
languages.
Table 6.3: Supervised AWE KWS results (%) on in-domain Swahili test data. The supervised monolingual CAE-RNN AWE model is trained using labelled Swahili data. It is applied with two segmentation configurations: using ground-truth word boundaries (true segm.), and using a variable-length window which is swept across the search collection (random segm.). Here we also report the standard QbE metrics $P@10$ and $P@N$ (Section 5.4.3.2). Model | Prec. | Rec. | $F_{1}$ | $P@10$ | $P@N$
---|---|---|---|---|---
Supervised monolingual: | | | | |
CAE-RNN (true segm.) | 95.5 | 90.8 | 93.1 | 98.6 | 94.3
CAE-RNN (random segm.) | 79.2 | 76.1 | 77.6 | 92.2 | 90.9
Supervised multilingual: | | | | |
CAE-RNN ab+cs+eu+sv+ta (random segm.) | 57.1 | 56.1 | 56.6 | 87.8 | 64.7
We now turn to the AWE-based approach. Specifically, we ask what the upper
bound on performance would be if we had more training data or a more idealised
search setting. Therefore, for a moment, we assume we have labelled data
available and perform top-line experiments on the in-domain Swahili data,
shown in Table 6.3. A supervised Swahili AWE model is trained on 30 hours of
labelled data and applied to the search collection that is segmented using
true word boundaries. Results are shown in the first row of Table 6.3, serving
as the top-line performance for AWE KWS, with a high $F_{1}$-score of 93%.
Compared to the 30-hour ASR system (Table 6.2), the idealised AWE system comes
closer to the $F_{1}$ of 98%. The second row in Table 6.3 shows KWS
performance using a supervised Swahili AWE model but applied to search
segments extracted by sliding a variable-length window across the search
collection—the way we apply the AWE approach in practice and in Table 6.2.
Segmentation without true word boundaries incurs a significant penalty,
resulting in an $F_{1}$-score drop to 77%. It is clear that the sliding window
approach has a large effect on downstream performance, so future work should
consider more sophisticated unsupervised word segmentation techniques.
#### 6.6.2 KWS results in the wild
We now turn to our main research question: comparing ASR to multilingual AWE-
based KWS on real-life, out-of-domain audio in a low-resource setting. As
mentioned in Section 6.5.3.2, systems are applied to out-of-domain Swahili
radio broadcasts, after which, for each system, the top 100 utterances
predicted to contain a hate speech keyword (Table 6.1) are manually reviewed.
The results are given in Table 6.4.
The table reports precision: the proportion of retrieved top-100 utterances
that correctly contain a hate speech keyword. Surprisingly, we see that the
multilingual AWE KWS system achieves a precision of 45%—better than the
5-minute and even the 1-hour ASR system in this in-the-wild test. This is in
contrast to the in-domain KWS results, where the 1-hour ASR model outperformed
the AWE KWS system (Table 6.2). Further investigation is required to
understand exactly why the relative performance of the ASR and AWE systems are
affected differently when applied to out-of-domain data. However, it is worth
noting that several studies have shown that ASR system performance can drop
dramatically when it is applied to data outside of its training domain [123,
124, 125]. One example where this can be seen in this case is how many of the
ASR retrievals contain music (which we asked the human annotator to mark).
The in-the-wild search collection has been diarised automatically, which
included a step to remove music segments. But this preprocessing step is not
perfect: the search collection still ends up with some music (which neither
the ASR nor AWE systems have seen in training). Table 6.4 shows that, out of
the top-100 utterances for the 30-hour ASR system, 30% contained music. This
decreases for the 1-hour (29%) and 5-minute (17%) ASR systems. But the AWE KWS
system only retrieves two utterances containing music. And of these two, one
utterance actually did contain a hate speech keyword in the music lyrics
(there are also examples of such correct matches in the ASR music retrievals).
Nevertheless, this shows that the AWE approach seems to be more robust to
domain mismatch compared to training an ASR system on one hour or five minutes
of labelled data. We therefore conclude that implementing an AWE-based KWS
system as an ASR-free alternative for rapid deployment in an unseen language
proves to be fruitful.
Table 6.4: In-the-wild KWS results (%) on Swahili radio broadcasts. For each system, we report the percentage of utterances retrieved containing a keyword (precision) and the percentage of utterances retrieved containing music. Model | Precision | Music
---|---|---
ASR: | |
XLS-R (30-h) | 52 | 30
XLS-R (1-h) | 42 | 29
XLS-R (5-min) | 36 | 17
Multilingual AWE: | |
CAE-RNN ab+cs+eu+sv+ta | 45 | 2
### 6.7 Further analysis
We briefly examine the transition from MFCCs to XLSR features as input to our
multilingual AWE model. We also investigate the potential of selecting a more
optimal threshold value for the AWE-based KWS system (Section 6.4.2).
(a) MFCC
(b) XLSR
Figure 6.3: UMAP [126] visualisation of the ten most frequent two-character
word segments in Bulgarian development data. These plots show the difference
between AWEs produced by a multilingual AWE model trained on (a) MFCC features
and (b) XLSR features.
#### 6.7.1 XLSR vs MFCCs
For our experiments in this chapter we substituted MFCCs for XLSR speech
features as input to our AWE models. Here we briefly look at the difference in
embedding quality produced between a multilingual AWE model trained on XLSR
features compared to MFCCs. We use the same training setup in Section 6.5.2.2
and train a multilingual AWE using MFCCs. For evaluation, we use the same
developmental language, Bulgarian, and perform the same-different word task.
On the same set of evaluation pairs, we obtain an AP of 29% and 58% for the
MFCC and XLSR multilingual AWE models, respectively. With further
investigation, we found that a model trained on XLSR features is much better
at learning higher-quality embeddings for shorter word segments. Figure 6.3
shows the difference between the quality of embeddings produced for two
character words when using (a) MFCCs and (b) XLSR as input features. Overall,
we see word segments from the same class to be much better clustered together
and separated from word segments from different classes. Using XLSR features
as input clearly improves the quality of AWEs, leading to improved performance
in the AWE-based KWS approach.
#### 6.7.2 Threshold
As mentioned in Section 6.4.2, the threshold value selected for the AWE-based
KWS system was tuned on the $F_{1}$-score by varying the threshold value over
all keywords. For interest, we tune the threshold value per keyword and plot a
precision-recall curve for each of the 36 keywords, shown in Figure 6.4. We
also plot the precision-recall curve (in black) using the same threshold value
for all keywords. From the figure we see that most per-keyword scores are
higher than the average (blue lines above black line). When we average the
$F_{1}$-scores using the optimal threshold value per keyword, we obtain an
$F_{1}$ score of 69% compared to 57% (bottom of Table 6.2) when using the same
threshold for all keywords. This raises the question of the selected threshold
value and whether simply treating keywords as equal might be a suboptimal
approach. Practically, however, a per-keyword threshold can not be calculated
given that for a new arbitrary keyword in a new unseen language we do not have
development data to tune this.
But there is a possibility that a threshold can be selected based on other
properties of a new keyword. These factors include the length of the speech
segment, the number of phones present, and the relative distances between
other unknown word segments in the acoustic space. For example, we may find
that, in general, the threshold value for shorter word segments is lower than
for longer speech segments. Future work should investigate the correlation
between the threshold value of keywords and the potential factors mentioned
above.
Figure 6.4: Precision-recall curves from in-domain AWE-based KWS on in-domain
Swahili test data using the same 36 keywords as in Section 6.6.1.
### 6.8 Chapter summary
This chapter addressed the problem of keyword spotting (KWS) for the purpose
of hate speech detection in low-resource languages. We compared two KWS
systems for data from Swahili and Wolof: a fine-tuned automatic speech
recognition (ASR) model using different amounts of training data, and an ASR-
free KWS system that utilises multilingual AWEs. The results show that fine-
tuning a pre-trained multilingual ASR model using even a small amount of
labelled data can outperform an AWE-based KWS system in controlled
environments where training and test data come from the same domain. However,
the AWE-based KWS system is more robust on out-of-domain radio broadcast data
and achieves comparable results to an ASR model fine-tuned on 30 hours of
labelled data. In the end, the merits of ASR vs AWE KWS will come down to the
practical setting: it will depend on whether labelled training data can be
collected from the target domain or not, and whether precision or recall is
more important.
## Chapter 7 Leveraging Multilingual Transfer for Unsupervised Semantic
Acoustic Word Embeddings
In the previous chapters, we focussed on producing vector representations for
spoken word segments where different acoustic realisations from the same word
type should have similar embeddings. This can clearly be seen in the way we
evaluated the embeddings: we use a word discrimination task (Section 2.5), a
QbE speech search task (Section 5.3), and a keyword spotting task (Section
6.4.2), all of which require exact matching of spoken word segments. The
performance of both these tasks relies on the discriminative ability between
embeddings from different word types. Specifically, we aimed at producing
these embeddings in low-resource settings, where annotated data in the target
language is not available; here a multilingual transfer approach demonstrated
superior performance.
In this chapter, we shift our focus to a different type of representation
learning: semantic acoustic word embeddings. In a semantic-acoustic space,
embeddings should not only be similar for different realisations of the same
word type. Additionally, embeddings from distinct word types should also be
close to each other if the respective words share meaning. These embeddings,
mapping to spoken words, should reflect the same semantic properties as word
embeddings trained on text data (Figure 1.1(a)). In other words, proximity in
this new acoustic-semantic space should reflect word-type similarity as well
as semantic relatedness between embeddings from different word types.
Again, we consider a low-resource scenario where word labels in the target
language are unavailable. In this setting, there have been limited efforts to
address the challenging problem of producing semantic AWEs. As a contribution,
we propose a novel semantic AWE modelling approach: leveraging a pre-trained
multilingual AWE model to assist semantic AWE modelling. This marks the first
instance of using a multilingual transfer strategy in this context.
Furthermore, existing research on semantic AWEs lacks comprehensive
evaluation. We therefore perform our experiments in a constrained setup and
evaluate our embeddings in an intrinsic word similarity task (measuring
semantic relatedness between words), and an extrinsic semantic QbE search. In
the latter, a system should retrieve utterances containing different
realisations of the query as well as utterances containing word segments that
are semantically related to the query.
Parts of this chapter are presented in the journal paper:
C. Jacobs and H. Kamper, “Leveraging multilingual transfer for unsupervised
semantic acoustic word embeddings,” IEEE Signal Processing Letters, 2023.
### 7.1 Background
We remind the reader of the properties and differences between AWEs and
textual word embeddings described in Section 1.2.1. In the zero-resource
setting, unsupervised representation learning of whole word speech segments
primarily focuses on producing embeddings that allow discrimination between
word segments of different word types. In this context, we train our AWE
models using isolated word pairs. For example, the CAE-RNN (Section 3.2.1) is
trained to reconstruct a different instance of the same word type as the
input, and the ContrastiveRNN (Section 3.3) is explicitly trained to minimise
the distance between instances of the same word type and maximise the distance
between all instances of different word types. Consequently, different
instances of the same word type should have similar embeddings and will be
different from instances from different word types. These representations are
learned without considering the context in which they appear within a spoken
corpus. From this point onwards, we refer to these representations, which only
encapsulate the acoustic properties linked to word type, as phonetic AWEs. An
example vector space containing these phonetic AWEs is shown in Figure 7.1(a).
Here, different spoken instances of the same word type, for example “bank”,
end up close to each other. Different spoken instances of the word “tank” will
also end up close to each other. Since “bank” and “tank” are phonetically
similar, instances of both word types might also end up close to each other in
vector space. These are the type of embeddings generated in all our AWE
experiments from previous chapters.
In contrast, in the text domain, word embedding models map written words to
fixed vectors, capturing the contextual meaning of words within a text corpus.
These models learn from the co-occurrence information in large unlabelled text
corpora. For example, the Skip-gram model of Mikolov et al. [127] employs a
neural network model that, given a target word, aims to predict the
neighbouring words in that sentence (more in Section 7.2). As a result, words
that are related in meaning end up having similar embeddings. Moreover, these
textual word embedding (TWE) models assign a unique embedding to each word
type within a fixed vocabulary. Figure 7.1(b) illustrates an example semantic
space, where the written words “finance”, “bank”, and “currency” are mapped
nearby, and “tank” and “infantry”, reflecting their shared contextual usage.
Instead of written words, our goal in this chapter is now to map spoken word
segments to vector representations that preserve this contextual relationship
among words described by Mikolov et al. [127] (Figure 7.1(b)). A unique aspect
of our problem lies in the continuous nature of speech, where each spoken word
segment will be different from another. Consequently, each spoken segment will
have a unique vector representation, which is different to TWEs where a single
vector is assigned to each word type.
For instance, we would now like to see spoken instances of “finance”,
“currency”, and “bank” mapped close to each other and, “tank” and “infantry”,
to be in close proximity in the resulting acoustic-semantic space, as
illustrated in Figure 7.2.
(a) Acoustic word embeddings
(b) Textual word embeddings
Figure 7.1: A visualisation of the fundamental difference between (a) AWEs
and (b) TWEs. (a) Preserves word-type similarity among different acoustic
realisations of five different word types. (b) Preserves semantic relatedness
among the same word types mapping to written words. Figure 7.2: A
visualisation of a semantic acoustic word embedding space.
As part of our approach to create semantic AWEs, we will extend the textual
word embedding approach called skip-gram. This approach has also been used as
the basis for other existing semantic AWE approaches. We therefore present the
skip-gram approach in detail below, before going into further details on
previous work on semantic AWEs.
### 7.2 Skip-gram
The Skip-gram model of Mikolov et al. [127] is trained on large amounts of
text corpora. The objective of this model is to learn word representations
that are useful for predicting neighbouring words in a sentence. Formally,
given a sequence of training words, $w_{1},w_{2},w_{3},\ldots,w_{T}$, the
model aims to maximise the average log probability [28],
$\frac{1}{T}\sum_{t=1}^{T}\sum_{-c\leq j\leq c,j\neq
0}\text{log}\,p(w_{t+j}|w_{t}),$ (7.1)
where $c$ is the number of words before and after the target centre word
$w_{t}$. Consider an example of multiple pairs of target and context words as
training samples for a single sentence in a text corpus, shown in Table 7.1.
Table 7.1: Training samples extracted from a sentence using a context size of two. Example sentence: “two boys are kicking a ball to each other in the park”. Context size = 2 | Target word | Context words
---|---|---
[two boys are] | two | boys, are
[two boys are kicking] | boys | two, are, kicking
[two boys are kicking a] | are | two, boys, kicking, a
[boys are kicking a ball] | kicking | boys, are, a, ball
… | … | …
[each other in the park] | in | each, other, the, park
[other in the park] | the | other, in, park
[in the park] | park | in, the
For the training sample with input (the target word) “kicking” the model will
learn to predict the words “boys”, “are”, “a”, “ball”.
Figure 7.3 shows the Skip-gram model architecture. The model consists of two
fully connected feedforward layers with weight parameters
$\mathbf{W}_{1}\in\mathbb{R}^{(V\times D)}$ between input and hidden layer,
and $\mathbf{W}_{2}\in\mathbb{R}^{(D\times V)}$ between hidden layer and
output, where $V$ is the vocabulary size (number of unique words in text
corpus) and $D$ is the dimension of the hidden layer.
Figure 7.3: The Skip-gram model architecture.
For each training sample, the target and context words are encoded into one-
hot vectors where the non-zero index corresponds to a word’s index in the
vocabulary set. Given a one-hot vector as input, the model produces a latent
embedding of size $D$. The embedding is then given as input to the second
layer to produce an output vector of size $V$. A softmax function is applied
to the output vector to normalise the output to a probability distribution
over $V$ classes, where the value at each index shows the prediction of the
model given the input word. The model then updates weight parameters
$\mathbf{W}_{1}$ and $\mathbf{W}_{2}$ to maximise the objective function in
Equation 7.1. Note that the one-hot input vector selects the $i$-th row from
weight parameter $\mathbf{W}_{1}$ to produce the latent embedding. The one-hot
vector essentially acts as a lookup table for each word in the vocabulary.
Therefore, after training, $\mathbf{W}_{1}$ represents the learned word
embeddings where the $i$th row of size $D$ represents the $i$th word in the
vocabulary set.
Keep in mind that the meaning of the words itself is not important, but rather
the context in which they appear with other words, such that the words can be
replaced with any unique character sequence (i.e. label) as long as the
sequence in which they appear are retained. This allows a model to be trained
on a text corpus with unknown word labels. In one of our speech models, we
make use of this property by inferring unknown word labels for spoken word
segments.
The training strategy of the Skip-gram model, which involves predicting
surrounding words from a target centre word, served as an inspiration to the
speech community as a means to obtain semantic embeddings from unlabelled
spoken word segments. In the following section, we explore the existing body
of work addressing this problem.
### 7.3 Related work
Learning semantic AWEs from speech is extremely difficult due to channel
variability, noise, and speaker-specific information that are not present in
written text. Moreover, the presence of inherent phonetic interference poses a
significant obstacle to semantic AWE modelling. For example, distinguishing
between spoken instances of “steal” and “steel” is challenging, even with the
context provided, since from a model’s perspective they would appear
identical. Some studies, therefore, consider the use of another modality as a
grounding signal, for example, images [128, 129, 91] or text labels [130] as a
weak form of supervision.
Only a handful of studies have looked at learning semantic AWEs from
unlabelled speech alone. Chung and Glass [63] were the first to present and
gain widespread recognition for producing semantic AWEs without word labels.
They introduced a model named Speech2Vec, borrowing the training methodology
of the Skip-gram model, by training on pairs of context words. Speech2Vec uses
an RNN decoder-encoder structure to accommodate variable-length speech
segments as input and target output (similar to the CAE-RNN), while the Skip-
gram model uses a two-layer fully connected feedforward network with one-hot
encoded vectors as input and output.
Speech2Vec is evaluated as follows: After training, all word segments from the
training data are embedded using the encoder RNN. Each word type is then
represented by a single embedding by averaging all instances of that type. The
averaged embeddings are then used in multiple intrinsic word similarity tasks
measuring semantic relatedness. With this setup, Speech2Vec achieved
surprisingly good results, even outperforming textual word embeddings (trained
on the corresponding transcripts) in some tasks. This certainly raised our
suspicion, as well as others [131], concerning the validity of the embeddings
produced by this model. In Section 7.4.1 we describe the model in more detail
and address some of the concerns with Speech2Vec. Nevertheless, Speech2Vec has
not been evaluated for single sample performance nor word segments that were
not included in the model’s training data. The quality of embeddings produced
for unseen segments is essential to determine the usefulness of embeddings for
downstream applications, such as semantic QbE.
Chen et al. [64] proposed a two-step framework towards semantic AWE modelling.
Speech segments are first given to a speaker disentanglement network (trained
using the unlabelled word segments), which outputs two embeddings, separating
speaker and phonetic information. The idea is that the phonetic embedding
should only contain information related to word type (the same property of our
multilingual AWEs). To produce semantic embeddings, they construct context
training pairs using the phonetic embeddings and train a network optimising a
contrastive loss similar to the ContrastiveRNN (Equation 3.6). Their results
are not compared to Speech2Vec and also use averaged embeddings from the
training data for evaluation.
Speech2Vec [63] and the work of Chen et al. [64] only consider the unlabelled
speech from the target language during model development. In this chapter, we
propose strategies for semantic AWE modelling, utilising a pre-trained
multilingual AWE model. We evaluate our models on an intrinsic word similarity
task, measuring the quality of embeddings for single word segments not used
during training. To demonstrate the practical applicability of our embeddings,
we also apply them to a downstream semantic QbE task.
In both cases [63, 64], the problem is simplified by assuming that we know
where words start and end in the target language (but the word classes are
still unknown). However, in a true zero-resource setting, word boundaries will
be unknown. In the experiments presented in this chapter, we also assume word
boundaries are known. This problem needs to be addressed in the future to
consider complete unsupervised semantic AWE modelling (Section 8.2).
### 7.4 Semantic AWEs trained from scratch
Until now, our AWE models mapped speech segments from the same word type
nearby in vector space (Figure 7.1(a)). In our notation, a speech segment
$X=\left(\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{T}\right)$ is
projected to a fixed vector $\mathbf{z}$. From this point onwards we add the
subscript p to the AWEs from previous chapters. In other words,
$\mathbf{z}_{\text{p}}$ indicates that the embedding preserves phonetic
information related to word type only.
The goal of this chapter is to produce semantic embeddings
$\mathbf{z}_{\text{s}}$. These embeddings should not only reflect phonetic
similarity but also capture word meaning (Figure 7.2). Speech2Vec [63] and the
work of Chen et al. [64] achieve this by only training on context pairs
extracted from unlabelled speech data (from the target language). We use the
training strategies presented in these two methods as a baseline for our
multilingual enhanced semantic AWE models (Section 7.5).
#### 7.4.1 Speech2Vec
Speech2Vec is a variant of the CAE-RNN AWE model (Section 3.2.1). The CAE-RNN
is an encoder-decoder model that takes a speech segment as input and aims to
reconstruct a different instance of the same word type. Speech2Vec uses the
same encoder-decoder framework, but instead of training on pairs of instances
from the same word type (as for phonetic AWE modelling), it uses context word
pairs $(X_{\text{trg}},X_{\text{ctx}})$. $X_{\text{trg}}$ is a target centre
word segment while $X_{\text{ctx}}$ is a context word appearing somewhere in a
window around the centre. These context pairs are constructed similarly to
those used for training the Skip-gram model on text data (see Table 7.1).
However, in this case, $X_{\text{trg}}$ and $X_{\text{ctx}}$ represent spoken
words. Importantly, these context pairs are constructed without word labels by
only considering the relative position of words within an utterance (requiring
word boundaries). Figure 7.4 shows the Speech2Vec model for a single training
pair. Speech2Vec optimises the reconstruction loss of Equation 3.1, where
$X^{\prime}$ in that equation is replaced with $X_{\text{ctx}}$. Ideally the
resulting embeddings $\mathbf{z}_{\text{s}}$ should therefore be similar for
words that co-occur in the speech data. Although this approach seems
theoretically sound, concerns have been raised regarding the performance of
this model. We address this now.
Figure 7.4: The Speech2Vec model architecture. This is similar to the CAE-RNN
model architecture (Figure 3.1) but adapted for context word pairs
$(X_{\text{trg}},X_{\text{ctx}})$ instead of positive pairs $(X,X^{\prime})$.
Dashed lines in the audio indicate true word boundaries.
Criticism. The authors of Speech2Vec [63] did not make their code available,
despite multiple requests from the public. They did however release a set of
embeddings produced by the
model.111https://github.com/iamyuanchung/speech2vec-pretrained-vectors With
numerous attempts to replicate the model’s embeddings exactly as described in
the paper, and reaching out to the authors without receiving a response, we
have not been able to achieve results close to theirs. We are also not aware
of any credible publications that successfully replicated Speech2Vec to obtain
similar quality semantic embeddings on the same evaluation tasks. Other failed
attempts to replicate the results of Speech2Vec include the works of
Jang222https://github.com/yjang43/Speech2Vec,
Wang333https://github.com/ZhanpengWang96/pytorch-speech2vec and
EarthSpecies444https://github.com/earthspecies/audio-embeddings/issues/6.
Chen and Cao [131] recently performed an in-depth analysis questioning the
authenticity of the original Speech2Vec model. On a homophone inspection
analysis, they show that Speech2Vec is immune to phonetic interference. For
example, the model will be able to perfectly discriminate between spoken
instances of “steal” and “steel” despite being phonetically identical, which
should inevitably affect model performance. By inspecting the vocabulary of
the released embeddings, they found it exactly matches the vocabulary of a
word embedding model trained on the corresponding transcripts. They show this
should not be true given that some information (words) are lost when applying
a forced-aligner to segment the audio. Based on these findings, there is
compelling evidence that the embeddings released by Speech2Vec have been
influenced by word labels in some way.
Moreover, the CAE-RNN AWE was originally designed to learn embeddings that are
invariant to speaker identity, channel characteristics, etc. (Section 3.2.1).
In the Speech2Vec setup, training pairs are from the same speaker and
utterance (the way context pairs are constructed), so that the model is not
explicitly able to learn embeddings invariant to those properties. Despite the
scepticism of Speech2Vec we include the model as a baseline in our
experiments. We use our own version of the model and perform experiments in a
different setup (but still refer to it as Speech2Vec).
#### 7.4.2 Semantic ContrastiveRNN
Similar to how the CAE-RNN AWE model is adapted for semantic AWE modelling
(Speech2Vec), we modify the ContrastiveRNN AWE model (Section 3.3) to learn
semantic AWEs. We substitute $X_{a}$, $X_{p}$ and $X_{n_{1}},\ldots,X_{n_{K}}$
in Equation 3.6, with $X_{\text{trg}}$ (target centre), $X_{\text{ctx}}$
(context), and out-of-context words. Now, by minimising the contrastive loss
of Equation 3.6, the model should optimise relative distances between
embeddings such that it reflect semantic similarity. This is similar to the
contrastive optimisation for semantic AWEs by Chen et al. [64], without
speaker disentanglement (requiring speaker labels).
For both Speech2Vec and semantic ContrastiveRNN, a semantic AWE model is
trained from scratch, therefore requiring the models to learn phonetic and
semantic similarity simultaneously. We now look at new semantic AWE modelling
strategies that we propose, utilising a pre-trained multilingual AWE model.
### 7.5 Our approach: Semantic AWEs using multilingual transfer
This section introduces a novel semantic AWE modelling approach. We propose
incorporating phonetic word knowledge through a pre-trained multilingual AWE
model during semantic modelling. Incorporating phonetic details may assist the
model to prioritise learning word semantics by filtering out unrelated
characteristics, for example, speaker information. Specifically, we propose
three strategies utilising a pre-trained multilingual AWE model in various
ways.
#### 7.5.1 ContrastiveRNN with multilingual initialisation
Instead of training semantic models from scratch (Section 7.4) we warm-start
them using the learned weights of a pre-trained multilingual AWE model. In our
experiments, we use the learned weights of a multilingual AWE model’s encoder
to initialise the encoder RNN of the ContrastiveRNN and subsequently train a
semantic ContrastiveRNN (Section 7.4.2). This learned parameter initialisation
method is similar to the multilingual adaptation strategy explored in Chapter
4: a pre-trained multilingual AWE model is adapted to a target low-resource
language using discovered word pairs. Here, we adapt the multilingual weights
to produce semantic AWEs by training on context pairs from the target
language.
#### 7.5.2 Projecting multilingual AWEs
Figure 7.5: An illustration of the Project AWE semantic strategy, as detailed
in Section 7.5.2.
An alternative approach would be to first apply the multilingual AWE model to
the segmented speech segments $\\{X^{(n)}\\}^{N}_{n=1}$ where $N$ is the
number of word tokens in the training corpus. Then, train a projection
network, mapping the resulting phonetic AWEs to a new semantic AWE space.
Figure 7.5 illustrates this mapping process. Applying the multilingual model
to the unlabelled speech segments $\\{X^{(n)}\\}$ produces a set of phonetic
AWEs $\\{\mathbf{z}^{(n)}_{\text{p}}\\}$ (upper part in figure). In the
phonetic space, we note that word segments are mapped to unknown word classes
(five in this simplified example). This implies $\mathbf{z}_{\text{p}}^{(35)}$
(blue) and $\mathbf{z}_{\text{p}}^{(71)}$ most likely belong to the same
unknown word class, as well as, $\mathbf{z}_{\text{p}}^{(18)}$ (green) and
$\mathbf{z}_{\text{p}}^{(87)}$. Groups containing
$\mathbf{z}_{\text{p}}^{(35)}$ and $\mathbf{z}_{\text{p}}^{(18)}$ are in close
proximity, indicating they might also share phonetic similarities.
In the semantic space, we would like to see embeddings of segment $X^{(35)}$
and $X^{(36)}$ to be positioned close to each other (assuming the unknown
classes associated with these segments frequently co-occur). We do this by
training a projection network, mapping the phonetic AWEs to a new set of
semantic embeddings $\\{\mathbf{z}^{(n)}_{\text{s}}\\}$. We train this network
using context pairs and the contrastive loss of Equation 3.6, optimising the
distances between the output embeddings $\mathbf{z}_{\text{s}}$. The lower
part of Figure 7.5 illustrates the transformation of the phonetic space into a
new semantic space, through optimising the contrastive loss. We now observe
neighbouring word segments $X^{(35)}$ and $X^{(36)}$ end up having similar
semantic representations $\mathbf{z}^{(35)}_{\text{s}}$ and
$\mathbf{z}^{(36)}_{\text{s}}$.
This approach is similar to the semantic AWE method presented by Chen et al.
[64]. Here they used data from the target language to train a network
separating speaker and phonetic information (which requires speaker labels).
Instead, we use a pre-trained multilingual AWE model designed for generating
phonetic representations for unlabelled word segments.
#### 7.5.3 Cluster+Skip-gram
Figure 7.6: Our Cluster+Skip-gram semantic AWE approach. Speech segments
$X^{(n)}$ are represented by soft pseudo-word label vectors $\mathbf{v}^{(n)}$
which are then used to train a Skip-gram model.
This new approach is based on the Skip-gram model (Section 7.2). The Skip-gram
model uses a fixed dictionary of discrete word class labels to create input
and output vectors for training on text data. In the low-resource setting we
do not have word labels for speech segments.
Instead, we use the phonetic similarities in the original AWE space to derive
a soft pseudo-word label for each speech segment. This process is illustrated
in Figure 7.6. First, a pre-trained multilingual AWE model is applied to the
segmented speech corpus $\\{X^{(n)}\\}$, producing a set of phonetic AWEs
$\\{\mathbf{z}_{\text{p}}^{(n)}\\}$. The colourless embedding space represents
the unsupervised setting where embeddings have no associated word types or
labels. We then apply $K$-means clustering to the phonetic embedding space,
producing a set of centroids $\\{\mathbf{c}_{k}\\}^{K}_{k=1}$, with $K$
representing the predetermined number of clusters for partitioning the vector
space.
Figure 7.7: Given an acoustic embedding $\mathbf{z}_{\text{p}}^{(n)}$
(assigned to cluster 2), instead of using a one-hot encoded input
$\mathbf{x}=\\{0,1,0,\cdots,0\\}$, the value at each index is set using
Equation 7.2, for example, $\mathbf{v}=\\{0.03,0.45,0.01,\cdots,0.43\\}$. Note
that the values at index $2$ and $K$ are the highest given that
$\mathbf{z}_{\text{p}}^{(n)}$ is the closest to cluster means $\mathbf{c}_{2}$
and $\mathbf{c}_{K}$. Black triangles indicate cluster means.
Ideally, these clusters should resemble distinct word classes. For example,
all spoken instances of “apple” should be assigned to cluster one (purple) and
all instances of “water” should be assigned to cluster two (green). Note, with
a perfect clustering, where all embeddings from an unknown word label are
exclusively assigned to a unique cluster, training a Skip-gram model using the
cluster labels would be equivalent to training on the corresponding text
labels (see bottom of Section 7.2). However, due to clustering inaccuracies,
not all embeddings belonging to the same unknown word class will be assigned
to the same cluster. The main challenge arises from embeddings of the same
word class being assigned to multiple clusters; a consequence of the
uncertainty regarding vocabulary size and word class frequency range. For
example, embeddings assigned to cluster $2$ (green) and cluster $K$ (blue) in
Figure 7.7 might represent the same word class given that there is no clear
separation between them. We experimented with cluster labels as training
labels for Skip-gram modelling, but this gave very poor performance on
development data.
Rather than assigning a hard label to each segment based on its associated
cluster number, we calculate a soft vector label of an AWE belonging to each
cluster:
$v_{k}^{(n)}=\frac{\exp\left(-\text{sim}(\textbf{z}_{\text{p}}^{(n)},\textbf{c}_{k})/\sigma^{2}\right)}{\sum_{j=1}^{K}\exp\left(-\text{sim}(\textbf{z}_{\text{p}}^{(n)},\textbf{c}_{j})/\sigma^{2}\right)}$
(7.2)
where $\text{sim}(\cdot)$ denotes cosine similarity and $\sigma$ is a
hyperparameter controlling the influence of distant centroids. This non-linear
scaling approach assigns more weight to embeddings closer to the target
embedding $\mathbf{z}^{(n)}_{\text{p}}$. Figure 7.7 extends the soft labelling
calculation illustration in the bottom right in Figure 7.6. Now, each word
segment $X^{(n)}$ is associated with a new unique vector
$\mathbf{v}^{(n)}\in\mathbb{R}^{K}$, where the value at index $k$ indicates
the likelihood of the segment being contained in cluster $k$. Even if segments
from the same unknown word class are assigned to multiple clusters, their
relative position in the embedding space is captured in $\mathbf{v}$.
Lastly, we use these soft vectors as inputs and targets to train a Skip-gram
model, illustrated in Figure 7.8. Compared to the Skip-gram model trained on
text data (Figure 7.3), there is a key difference: using these soft
representations as input, compared to a one-hot vector. This means that
$\textbf{W}_{1}$ do not act as a look-up table, instead, each row contributes
to the hidden embedding with corresponding weight (visually illustrated with
shading in Figure 7.8). In other words, the hidden layer contains the weighted
sum of all rows in $\textbf{W}_{1}$, which we take as the semantic embedding
$\textbf{z}_{\text{s}}$. The same applies to $\textbf{W}_{2}$ when using soft
targets.
Figure 7.8: The skip-gram model using soft inputs and targets.
During inference, a unique embedding $\textbf{z}_{\text{s}}$ is produced for a
word segment by first obtaining a phonetic AWE using the multilingual model,
then calculating representation $\mathbf{v}$ using the trained cluster means,
and finally applying it to the learned weight matrix $\mathbf{W}_{1}$.
### 7.6 Semantic query-by-example
In Chapter 5 and Chapter 6, we performed QbE experiments using AWEs to
calculate the phonetic similarity between speech segments. This involved
searching an unlabelled audio corpus with a query word segment, aiming to
retrieve utterances containing instances of the query. We refer to this as
exact QbE, where the query only retrieves utterances if it contains a word
sharing the same class label. In this chapter, apart from intrinsic AWE
evaluations, we consider the task of semantic QbE as a downstream application
for semantic AWEs. In semantic QbE the goal is to retrieve utterances from an
audio collection that not only match segments containing the exact query
segment but also utterances which contain word segments that are semantically
similar to the query. For example, using a query segment, “football”, the
system should retrieve utterances containing exact query matches for instance,
“the football player is injured”, as well as utterances containing word
segments semantically related to the query for instance, “two kids play soccer
in the field”, and “the player was offside”.
We perform our semantic QbE experiments using a similar setup as in [132].
They performed experiments to produce semantic embeddings on the same data
corpus we use for our semantic AWE experiments. However, during training, they
use images as a weak form of supervision to produce semantic embeddings. To
the best of our knowledge, we are the first to perform semantic QbE using
semantic AWEs trained on unlabelled speech segments.
### 7.7 Setup
In this section we outline the data corpus we use for our semantic AWE
modelling. We provide implementation details for the two semantic models,
Speech2Vec and ContrastiveRNN, trained from scratch (Section 7.4).
Subsequently, we present the specifics of the three semantic AWE models
leveraging a pre-trained multilingual AWE model we introduced in Section 7.5.
Finally, we discuss the evaluation process for our semantic AWEs.
#### 7.7.1 Data
Figure 7.9: Examples of images with corresponding spoken captions from the
FACC.
For this experiment, we use the Flickr8k Audio Captions Corpus (FACC) [128].
The corpus contains 40k spoken captions in English describing the content of a
Flickr image [133]. These images depict scenes for which humans have given
verbal descriptions. Example images with corresponding spoken captions are
shown in Figure 7.9. Note, the text in Figure 7.9 is only for the purpose of
illustration and not used during training (the corresponding speech are used).
The corpus is divided into train, development, and test splits containing
$30\,000$, $5\,000$, $5\,000$ utterances respectively.
Although the amount of training data in the Flickr8k corpus is relatively low,
the content of the data suits our semantic experiments well: images come from
a fairly narrow domain, with five different spoken captions describing the
content of each image. Therefore, semantic concepts reoccur in different
utterances. This dataset has also been used for acoustic semantic modelling by
others [128, 91, 132].
We train and evaluate our models using two sets of speech features. Firstly,
we employ 13-dimensional MFCCs. Secondly, as in Chapter 6, we conduct
experiments with self-supervised speech features: we use the 12th transformer
layer of the multilingual XLSR model [119] to get 1024-dimensional features.
Utterances are normalised per speaker and segmented using true word boundaries
from forced alignments [121]. We use these word segments from the training
data to construct context word pairs. For all the semantic models, we use a
context window of three words before and after a centre word: this yields
roughly two million context pairs in our training data.
#### 7.7.2 Semantic models trained from scratch
The encoder and decoder of our Speech2Vec (Section 7.4.1) implementation each
consists of three unidirectional RNNs with 400-dimensional hidden vectors and
an embedding size of 100. The semantic ContrastiveRNN (Section 7.4.2) uses the
same encoder structure. We train both these models using the context pairs
extracted from the training set. For the ContrastiveRNN we randomly sample
word segments outside the context window as negatives; for each positive, we
sample 20 negatives.
#### 7.7.3 Semantic models using multilingual transfer
All the multilingual transfer strategies presented in Section 7.5 use a pre-
trained multilingual AWE model. The multilingual model is trained in the same
way multilingual models were trained in previous chapters, with embedding
dimension set to 100. We train a multilingual CAE-RNN AWE model on labelled
data from four different well-resourced languages included in Common Voice
[118]: Italian, Dutch, Russian, and Czech. Note, the entire speech corpus is
in English and therefore not included as a training language. We pool the data
from all four languages and extract 300k training pairs.
Multilingual initialisation. (Section 7.5.1) For the multilingual weight
initialisation method, we use the trained parameters of the multilingual CAE-
RNN encoder to initialise the ContrastiveRNN’s encoder. Furthermore, we freeze
the weights of the first two encoder layers to preserve the phonetic structure
of input words during further training. This initialisation approach is
similar to how we initialised AWE models for adaptation in Chapter 4: a
multilingual AWE model is adapted using discovered word pairs from the target
language. Here, we adapt for semantic AWEs by training on context pairs from
the target corpus.
Project AWE. (Section 7.5.2) For our projection network, we use a feed-forward
network of two linear layers with an inner dimension of 1024 and input and
output dimensions of 100. The input dimension is set to match the embedding
dimension produced by the multilingual AWE model. The inner and output
dimensions are chosen based on experimenting on development data. Again, we
train on context pairs and sample 20 negatives for each positive. The network
optimises by minimising the contrastive loss of Equation 3.6.
Cluster+Skip-gram. (Section 7.5.3) For $K$-means clustering, we use $K=5\,000$
clusters and set $\sigma=0.01$ in Equation 7.2. The number of unique word
types in the target corpus is $6\,307$ where $2\,150$ appears once. We
experimented with different values for $K$ ranging from $1\,000$ to $10\,000$
which produced similar results. We initialise our cluster means with a random
selection of acoustic embeddings, perform 50 permutations, and keep the best
means. We reimplement the Skip-gram model [127] described in Section 7.2,
setting the latent embedding dimension to $D=100$. The model is trained using
the soft-vectors as input and optimises a negative log-likelihood with class
probabilities set to the context target word’s soft-vector. The model is
optimised using Adam [103], with a learning rate of $0.001$.
#### 7.7.4 Evaluation
We evaluate the quality of resulting semantic embeddings in an intrinsic word
similarity task. To measure the applicability for real-life applications, we
also apply the semantic AWEs in a downstream semantic QbE task for extrinsic
evaluation.
##### 7.7.4.1 Intrinsic semantic AWE evaluation: Word similarity
Table 7.2: Word-pair similarity ratings extracted from the WordSim353 [134] evaluation set. Similarity scores indicate word relatedness based on human judgements. money | bank | 8.50
---|---|---
physics | proton | 8.12
tennis | racket | 7.56
journey | car | 5.85
smart | student | 4.62
problem | airport | 2.38
professor | cucumber | 0.31
In the text domain, word embeddings are evaluated for semantics by measuring
the similarity between word pairs through vector distance calculations. These
vector distances are then compared to standard reference sets containing
similarity ratings assigned by humans. As an illustration, Table 7.2 shows
some reference word-pair similarities from the WordSim353 [134] evaluation
set. For instance, in a resulting semantic space, the distance between the
embedding of “tennis” and “racket” should be smaller than the distance between
“journey” and “car”, but greater than the distance between “physics” and
“proton.” Furthermore, “money” and “bank” should have the smallest distance
than all the shown word-pair distances. Spearman’s $\rho$ is used to quantify
the similarity between the set of word-pair distances
$(\text{WP}_{\text{embed}})$ and the set of word-pair human judgement ratings
$(\text{WP}_{\text{human}})$, calculated as follows:
$\rho=\frac{\text{cov}(R(\text{WP}_{\text{embed}}),R(\text{WP}_{\text{human}}))}{\sigma_{R(\text{WP}_{\text{embed}})}\sigma_{R(\text{WP}_{\text{human}})}}$
(7.3)
with $R$ converting the scores to ranks, and $\text{cov}(\cdot\,,\cdot)$ and
$\sigma_{\cdot}$ denotes, covariance and standard deviation, respectively.
[.8] For the sentence below, select all words that could be used to search for
the described scene.
Sentence: a skateboarder in a light green shirt.
$\square$ dogs $\square$ beach $\square$ children $\square$ white
$\square$ swimming $\square$ wearing $\square$ skateboard $\square$ None of
the above
Figure 7.10: An example annotation job presented to a human for manual
semantic labelling, with keyword options. In this case, “skateboard” was
selected by all five annotators and “wearing” by four. Reproduced from [132].
In our controlled experiment, our vocabulary is limited, causing many words
from the standard evaluation sets typically used to assess word embeddings
trained on large amounts of text to be absent from our training and evaluation
data. To address this, we compile our own list of word pairs that frequently
appear in our dataset. Instead of using human judgement, we assign similarity
scores to those pairs based on the distance between textual word embeddings
generated by an off-the-shelf Skip-gram model trained on the corresponding
transcribed utterances. Given that our goal is to obtain semantic embeddings
similar to Word2Vec, we believe this approach is reasonable.
To obtain a single semantic embedding for each word class, we calculate the
average of all embeddings from the same class and report $\rho_{\text{avg}}$.
Given that we are particularly interested in obtaining semantic embeddings for
individual word segments, single-sample performance is also measured by
randomly selecting one instance of each spoken word. This is repeated ten
times and averaged to get a single score $\rho_{\text{single}}$.
##### 7.7.4.2 Semantic QbE
Table 7.3: The number of annotators (out of five) that selected the keyword for that utterance is shown. The majority vote indicates whether the keyword is deemed semantically relevant to an utterance, according to the labelling. Adapted from [132]. Keyword | Example utterance | Count 0 | Majority vote
---|---|---|---
jumps | biker jumps off of ramp | 5 / 5 0 | Y
ocean | man falling off a blue surfboard in the ocean | 5 / 5 0 | Y
race | a red and white race car racing on a dirt racetrack | 5 / 5 0 | Y
snowy | a skier catches air over the snow | 5 / 5 0 | Y
bike | a dirt biker rides through some trees | 4 / 5 0 | Y
children | a group of young boys playing soccer | 4 / 5 0 | Y
young | a little girl in a swing laughs | 4 / 5 0 | Y
field | two white dogs running in the grass together | 3 / 5 0 | Y
carrying | small dog running in the grass with a toy in its mouth | 2 / 5 | Y
face | a man in a gray shirt climbs a large rock wall | 2 / 5 | N
large | a group of people on a zig path through the mountains | 1 / 5 | N
sitting | a baby eats and has food on his face | 1 / 5 | N
hair | two women and a man smile for the camera | 0 / 5 | N
We use the same setup as [91] to evaluate downstream semantic QbE performance.
Semantic labels for 1 000 test utterances from FACC were collected from human
annotators, using a set 67 keyword classes [132]. Specifically, each of the 1
000 utterances was labelled by five annotators, indicating whether a
particular keyword is semantically relevant to that utterance (regardless of
whether the word instance appears verbatim in the utterance). An example
annotation job is shown in Figure 7.10.
We use the majority decision to assign a hard label for whether a keyword is
relevant to an utterance as in [132]. Table 7.3 shows annotations for some of
the utterances. For instance, a correct retrieval using the keyword “jump”
would be “bike jumps of ramp”, while an incorrect retrieval using the keyword
“face” would be “a man in a gray shirt climbs a large rock wall”. Using these
hard labels, we calculate semantic $P@10$, $P@N$, EER (metric details in
Section 5.3).
Given the ambiguity among annotators when determining if a keyword is
semantically related to an utterance, we also use the counts as soft labels
and calculate Spearman’s $\rho$ (Equation 7.3). Here, $\rho$ measures the
correlation between a system’s ranking and the actual number of annotators who
deemed a query keyword relevant to an utterance.
To simplify the QbE task, we still assume that ground truth word boundaries
are known: a query AWE is therefore compared to AWEs for the word segments in
an unlabelled search utterance.
### 7.8 Experimental results
We first compare embedding quality through measuring word similarity between
the semantic AWE models trained from scratch (Section 7.4) and the semantic
AWE models we proposed leveraging multilingual AWEs (Section 7.5). We then
apply one of our proposed semantic AWE models in a semantic QbE search for
extrinsic evaluation. This is followed by a qualitative analysis of our
semantic AWEs.
#### 7.8.1 Intrinsic evaluation: Word similarity
Table 7.4: Spearman’s $\rho$ (%) for semantic word similarity of AWEs on development data. Embeddings are either averaged over word segments of the same class, or a single segment is sampled per class. AWE models take either MFCC or XLSR features as input. | MFCC | XLSR
---|---|---
Model | $\rho_{\text{single}}$ | $\rho_{\text{avg}}$ | $\rho_{\text{single}}$ | $\rho_{\text{avg}}$
Trained from scratch: (Section 7.4) | | |
Speech2Vec | 01.0 | 05.3 | 07.2 | 21.3
ContrastiveRNN | 00.7 | 05.1 | 04.6 | 25.9
Using multilingual transfer: (Section 7.5) | | |
ContrastiveRNN multilingual init. | 00.6 | 05.5 | 06.1 | 24.5
Project AWE | 03.6 | 18.2 | 18.3 | 33.6
Cluster+Skip-gram | 18.0 | 35.6 | 35.9 | 41.7
Table 7.4 presents the intrinsic scores of embeddings from the semantic AWE
models, trained either from scratch (top section) or using multilingual
transfer (bottom). The benefit of multilingual transfer is evident in the
scores of the projection and Cluster+Skip-gram approaches, with the latter
outperforming all other models regardless of the input features used or
whether single or averaged embeddings are evaluated. The single-sample
performance $\rho_{\text{single}}$ is particularly significant as it shows
that individual representations can be compared accurately—a useful property
for downstream applications such as semantic QbE (Section 7.6).
The ContrastiveRNN is the one exception that does not show a clear gain from
initialising with multilingual weights compared to training from scratch. As a
sanity check, we evaluate the phonetic multilingual AWEs before semantic
training, in other words, embeddings produced by the foundation multilingual
AWE model (top of Section 7.7.3) used for transfer in the bottom section. We
obtain results for $\rho_{\text{single}}=\text{0.59\%}$ and
$\rho_{\text{avg}}=-\text{0.13\%}$. As expected, this indicates that phonetic
multilingual AWEs do not capture semantic information.
Again, we see the benefit of using self-supervised speech representations as
input to AWEs instead of conventional features, as also in previous work [135,
122]. We therefore use XSLR features for all further experiments. We apply the
semantic AWE model achieving the best results on the word similarity task,
Cluster+Skip-gram, in our semantic QbE search.
#### 7.8.2 Extrinsic evaluation: Semantic QbE
Table 7.5 compares the Cluster+Skip-gram (semantic) and multilingual AWE
(phonetic) models when used in a downstream QbE system. We evaluate both exact
and semantic QbE, where the latter gets awarded for retrieving exact query
matches as well as utterances labelled as semantically related to the search
query. To situate results, we use a random baseline model that assigns a
random relevance score to each utterance. (The relatively high scores of the
random approach are due to the narrow domain of the evaluation data, Section
7.7.1.)
Table 7.5: Exact and semantic QbE results (%) on test data. | Exact QbE | Semantic QbE
---|---|---
Model | $P@10$ | $P@N$ | EER | $P@10$ | $P@N$ | EER | $\rho$
Baselines: | | | | | | |
Random | 05.0 | 05.0 | 50.0 | 09.1 | 09.1 | 50.0 | 05.7
Multilingual AWE | 90.7 | 82.8 | 04.7 | 92.8 | 59.4 | 24.1 | 17.0
Semantic AWE model: | | | | | | |
Cluster+Skip-gram | 85.8 | 71.3 | 09.3 | 88.2 | 52.1 | 21.6 | 28.2
Table 7.6: Semantic QbE results (%) on test data, where any instance of a query is masked from the search utterances. Model | $P@10$ | $P@N$ | EER | Spearman’s $\rho$
---|---|---|---|---
Baselines: | | | |
Random | 09.2 | 09.0 | 50.0 | 05.4
Multilingual AWE | 21.5 | 15.6 | 46.1 | 06.4
Semantic AWE model: | | | |
Cluster+Skip-gram | 29.9 | 23.1 | 32.3 | 22.9
Looking at the EER and Spearman’s $\rho$ for semantic QbE, we see that the
Cluster+Skip-gram model achieves the highest score, outperforming the purely
phonetic AWEs from the multilingual AWE model. The reason why the phonetic
multilingual AWE model outperforms the semantic model in $P@10$ and $P@N$ is
due to its proficiency in detecting exact matches (which are also correct
semantic matches).
In this chapter, we are mainly interested in observing information captured by
our embeddings related to word meaning. However, in the semantic QbE task, a
system also gets rewarded for retrieving utterances containing exact instances
of the query. To get a better sense of the ability of a model to retrieve non-
verbatim semantic matches, we mask out all exact occurrences of the query
before we segment the search collection. The system will now only be able to
retrieve utterances based on word meaning. The results are shown in Table 7.6.
Now we see a clear benefit in using the Cluster+Skip-gram model, with the
phonetic multilingual AWE model becoming close to a random search given that
there are no exact matches.
Our core goal is semantic QbE, but it is worth briefly touching on exact QbE
performance of the Cluster+Skip-gram model in Table 7.5. In exact QbE, the
system should only retrieve utterances containing instances of the query word
type. The Cluster+Skip-gram model learns to predict frequently co-occurring
words, without explicit training for preserving word-type information.
Although only trained for semantics, the Cluster+Skip-gram still achieves
reasonable exact retrieval performance, with only a drop of between 5% and 10%
in scores compared to the multilingual AWE model. It is therefore clear that
this semantic model is able to retain phonetic properties for word-type while
also capturing semantic information related to context.
#### 7.8.3 Qualitative analysis
Figure 7.11: PCA projection of semantic AWEs (averaged), produced by our Cluster+Skip-gram model on development data. We highlight the five nearest neighbours for “several” (pink), “bike” (green), and “orange” (blue). Table 7.7: The top ten closest embeddings for keywords, derived from text (left) and speech segments (right), respectively. Speech embeddings were produced using the Cluster+Skip-gram model and averaged per word-type before nearest neighbour calculation. american | | swimming | | boys | | football
---|---|---|---|---|---|---
flags | african | | pool | swimmers | | children | girls | | opponent | baseball
spiderman | waving | | swim | swims | | kids | children | | soccer | hockey
tin | older | | water | swim | | men | kids | | hockey | soccer
camouflage | costume | | wet | into | | girls | women | | player | uniform
muzzles | elderly | | diving | zooms | | boy | teens | | players | miami
participating | flags | | headfirst | swings | | players | men | | volleyball | player
elderly | asian | | wading | pool | | people | are | | hitting | approaches
tiny | leather | | paddling | scratching | | dolphins | traveller | | uniform | apportioned
instrument | establishment | | swims | scratches | | gateway | cans | | field | saxophone
asian | instruments | | patiently | splashes | | girl | adults | | swim | players
Figure 7.11 visualises the semantic embedding space produced by the
Cluster+Skip-gram model. It is clear that the acoustic realisations of
semantically related words end up in similar areas in the space. For example,
the model learned that spoken instances of “orange”, “red”, “blue”, “yellow”,
and “green” should be close to each other. For additional visualisation, we
direct the reader back to Figure 1.1 and Figure 1.2. Figure 1.1(a) displays a
selected number of TWEs produced by a Skip-gram model trained on the
transcribed speech we use for acoustic semantic modelling in this chapter
(Section 7.7.1). Figure 1.1(b) displays the AWE produced by the multilingual
AWE model we use in our semantic AWE approaches (Section 7.5). Finally, Figure
1.2 displays the embeddings (without averaging) produced by our Cluster+Skip-
gram model. From these visualisations, it is evident that the semantic AWEs
satisfactory capture of word-type and semantic information.
The intrinsic results in Table 7.4 and qualitative analysis in Figure 1.2 show
that our semantic embeddings, to some extent, resemble the semantic
information captured in the corresponding text embeddings. To get a better
understanding of the difference between the embeddings produced from speech
and the corresponding transcripts, we look at the variation in the nearest
embeddings of some words. Table 7.7 displays chosen keywords, comparing
semantic AWEs (on the right) with textual word embeddings (on the left).
Firstly, we notice that across all keywords some words are shared (highlighted
in bold). Secondly, we observe a strong phonetic influence on the semantic
embeddings trained on speech. They are more likely to have words higher ranked
that are phonetically similar to the keyword. For example, “american” and
“football”, have “african” and “baseball”, as their closest neighbour,
respectively. In this instance, we could say they are semantically similar
from our own judgement, but they do not appear among the nearest words for the
corresponding text embeddings. Looking at the keyword “swimming” we can see
where this phonetic disturbance has a negative impact. For“swimming”, the word
“swims” appear in the top neighbours for both speech and text. Because “swims”
are phonetically similar to “zooms” and “swings”, the model fails to separate
them in vector space. As a result, “swimming” also ends up close to “zooms”
and “swings”. In this instance, we would not regard them as semantically
relevant to each other.
### 7.9 Chapter summary
In this chapter, we addressed a new form of acoustic representation learning,
where word embeddings should capture the meaning of words, rather than
phonetic information. To achieve this, we presented several semantic AWE
modelling strategies. We specifically promoted transferring knowledge from a
pre-trained multilingual AWE model trained for word-class discrimination.
Through intrinsic and extrinsic evaluations, we demonstrated the effectiveness
of our strategies in learning semantic representations from unlabelled speech
data. Our best semantic AWE approach involves a soft clustering on the
original multilingual AWEs, serving as input to a Skip-gram model.
The main shortcoming of our work (as also in others [63, 64]), is that the
word segmentation is assumed to be known. This was reasonable given our goal
of comparing different semantic AWE approaches on a sensible benchmark, but
future work should look into incorporating unsupervised word segmentation
methods [136, 137, 138] in order to do fully unsupervised semantic AWE
modelling.
## Chapter 8 Summary and Conclusions
Developing robust speech systems for zero-resource languages—where no
transcribed speech resources are available for model training—remains a
challenge. Most of these systems, at their core, require speech segments of
different duration to be compared. Previous studies showed the benefit of AWEs
over alignment methods for this task. In a zero-resource setting, the two
existing training strategies are unsupervised monolingual learning (Section
2.4.1) and supervised multilingual transfer (Section 2.4.2). This thesis
mainly contributes to the development of AWEs following the multilingual
transfer strategy. We presented multiple approaches toward improving the
intrinsic quality of AWEs as well as the implementation of AWEs in a
downstream speech retrieval task. Furthermore, we extended the multilingual
transfer strategy to semantic representation learning, demonstrating the
versatility of multilingual AWEs in upstream tasks. We first summarise our
main findings. We then provide recommendations on how the work can be extended
and applied in future research.
### 8.1 Main findings
In Chapter 3, we presented the ContrastiveRNN AWE model. This model optimises
a distance metric function not considered for AWE modelling in previous work.
We compared the ContrastiveRNN to two existing AWE embedding models, CAE-RNN
[21] and SiameseRNN [32]. We re-implemented both existing models and compared
our model in the same experimental setup as Kamper et al. [23]. On a word-
discrimination task measuring the intrinsic quality of the AWEs, the
ContrastiveRNN significantly outperformed the CAE-RNN and SiameseRNN following
the unsupervised monolingual training strategy, reaching up to 17.8% absolute
increase in average precision on one of the evaluation languages. In the
supervised multilingual setting, we observed marginal performance increases
from the ContrastiveRNN compared to the CAE-RNN and SiameseRNN.
In Chapter 4, we proposed a new training strategy that extends the
multilingual transfer strategy. After training a multilingual model, instead
of only applying it to an unseen target language, we first fine-tune the
multilingual model using discovered word-like pairs obtained from applying an
unsupervised term discovery system (UTD) to unlabelled speech data in the
target language. For each of the three multilingual models, we adapt a model
to every zero-resource language before applying it. Here we observe that the
ContrastiveRNN generally adapts better to a target language compared to the
CAE-RNN and SiameseRNN.
In Chapter 5, we investigated the impact a particular choice of training
languages have when training an AWE model for a specific target language. We
specifically want to see the impact of including training languages that
belong to the same language family as the target language. We performed
multiple experiments on data from languages in South Africa where we train
multilingual AWEs with different language combinations while controlling for
the amount of data. We show that including related languages over unrelated
languages are beneficial, evaluated on an intrinsic word discrimination task.
We also evaluated the multilingual AWEs in a downstream zero-resource speech
task, query-by-example, supporting the results of the intrinsic evaluation
task. To summarise the findings in this chapter, including training data from
related languages, even in small quantities, are beneficial over using
unrelated languages. For instance, a multilingual model trained on
approximately 6 hours of related languages, outperform a model trained on
approximately 60 hours of unrelated languages. The findings in this chapter
would be of great interest to practitioners who want to develop speech
applications in a zero-resource setting.
In Chapter 6, we developed keyword spotting systems to monitor hate speech in
radio broadcast audio in a collaborative effort with a company, VoxCroft. Our
goal was to compare existing ASR-based KWS systems (requiring labelled
training data from the target language) to a new multilingual AWE-based KWS
system. We first developed our systems in a controlled environment where we
used parallel audio-text data from experimental datasets to train and test our
models. In this in-domain environment, an ASR model trained on only five
minutes of transcribed audio outperforms our AWE-based KWS system. We then
applied the models, without further calibrations, to the real radio broadcast
audio scraped from radio stations in Kenya. In this in-the-wild test, our AWE-
based KWS system proves to be more robust against domain mismatch and performs
similar to an ASR-KWS system using 30 hours of labelled target audio. Our
findings in this chapter suggest multilingual AWEs can be used as an
alternative to existing ASR systems for rapidly deploying KWS systems for
languages with limited transcribed audio.
In Chapter 7, we shifted our attention to a new form of representation
learning of spoken word segment: AWEs that reflect word-type information as
well as word meaning. We specifically proposed using a pre-trained
multilingual AWE model to assist in this task compared to existing approaches
that only use unlabelled data from the target language [63, 64]. One of our
models, Cluster+Skip-gram, show great improvement over previous models in an
intrinsic word similarity task, measuring semantic relatedness. For the first
time, we applied these semantic AWEs in a downstream semantic speech retrieval
task. Our results show that semantic information is indeed being captured in
the embeddings produced by our new models, indicating promise for future
downstream applications.
### 8.2 Future work
Throughout this thesis, we identified multiple opportunities to expand on in
future research.
* •
All our AWE models up to Chapter 5 took hand-engineered MFCCs as input
features. In Chapter 6 and Chapter 7 we incorporated self-supervised speech
features as an alternative to MFCCs for input to our AWE models. These learned
speech features showed improved performance in a variety of speech tasks [76,
77, 78, 79]. We also observed the benefits in AWE modelling using these
features: improved word discrimination (Section 6.7.1) and improved semantic
representations (Table 7.4). However, we did not systematically investigate
this but merely replaced features without considering alterations to our
models or training conditions. We hypothesise further improvement by
optimising the training setup to accommodate these speech features.
* •
We notice a large performance gap between the unsupervised monolingual models
trained on discovered word pairs (top of Table 3.1) and the supervised
monolingual models trained on true word segments (top of Table 4.2). A more
accurate unsupervised segmentation of unlabelled speech data, delivering
higher quality discovered pairs, could reduce the gap between the unsupervised
and supervised training strategies. Kamper et al. [93] presented a full-
coverage segmentation and clustering system—unlabelled speech data are
segmented and clustered into unknown word types. Their system employs
primitive acoustic word embeddings. By replacing the AWEs with higher quality
AWEs, for instance, embeddings produced by a multilingual AWE model, better
segmentation can be expected. Applying this segmentation system to unlabelled
speech data in a zero-resource language could potentially yield higher quality
discovered word pairs compared to the UTD system of Jansen and Van Durme [10]
used in this thesis.
* •
Up until now, the choice of training languages in a multilingual transfer
setting have not been considered, specifically for AWEs. We showed that a more
careful selection of training languages can improve performance when
developing a multilingual AWE model for a specific target language by
including training languages that belong to the same language family as the
target. We showed that the performance increase is not simply due to the
overlap of words in related languages. However, a more in-depth analysis of
the phonetic and syntactic properties common to languages from the same
language family could give more insight assisting the development of
multilingual transfer learning in general.
* •
In Chapter 6, we identified the need for a more optimal threshold value to
improve KWS performance. We hypothesise that a per-keyword threshold could be
determined based on factors including the length of the speech segment, the
number of phones present, and the relative distances between other unknown
word segments in the acoustic space. In the work of Matusevych et al. [104],
they analyse an AWE space and discover that the embeddings encode absolute
duration and speaker information. Additionally, they observe that the distance
between word embeddings increases with phonetic dissimilarity. With such
knowledge we could explore potential correlations between these factors stored
in AWEs and the distance between instances in the acoustic space.
* •
In Chapter 7, we introduced a novel approach for learning speech
representations that reflect semantic meaning instead of acoustic similarity.
Although our approach exhibits improved performance compared to existing
methods, certain limitations remain: Similar to others [63, 64], we used true
word boundaries for segmenting unlabelled audio. Future work should consider
unsupervised segmentation methods [136, 137, 138], or explore an end-to-end
semantic AWE approach that eliminates the need for explicit word boundaries.
Additionally, we performed our experiments within a constrained setup
involving a limited vocabulary and a smaller number of words when compared to
the millions or even billions used for training textual word embeddings.
Future work should therefore investigate the scalability of the semantic
approaches we have introduced.
## References
## References
* Besacier et al. [2014] L. Besacier, E. Barnard, A. Karpov, and T. Schultz, “Automatic speech recognition for under-resourced languages: A survey,” _Speech Commun._ , vol. 56, pp. 85–100, 2014.
* Eberhard et al. [2021] Eberhard, M. David, G. F. Simons, and C. D. Fenning, “Ethnologue: Languages of the world,” 2021. [Online]. Available: https://www.ethnologue.com
* Versteegh et al. [2016] M. Versteegh, X. Anguera, A. Jansen, and E. Dupoux, “The Zero Resource speech challenge 2015: Proposed approaches and results,” in _Proc. SLTU_ , 2016.
* Jansen et al. [2013] A. Jansen, E. Dupoux, S. Goldwater, M. Johnson, S. Khudanpur, K. Church, N. Feldman, H. Hermansky, F. Metze, R. Rose, M. Seltzer, P. Clark, I. McGraw, B. Varadarajan, E. Bennett, B. Borschinger, J. Chiu, E. Dunbar, A. Fourtassi, D. Harwath, C.-y. Lee, K. Levin, A. Norouzian, V. Peddinti, R. Richardson, T. Schatz, and S. Thomas, “A summary of the 2012 JHU CLSP workshop on zero resource speech technologies and models of early language acquisition,” in _Proc. ICASSP_ , 2013.
* Räsänen [2012] O. Räsänen, “Computational modeling of phonetic and lexical learning in early language acquisition: Existing models and future directions,” _Speech Commun._ , vol. 54, no. 9, pp. 975–997, 2012.
* Levin et al. [2015] K. Levin, A. Jansen, and B. Van Durme, “Segmental acoustic indexing for zero resource keyword search,” in _Proc. ICASSP_ , 2015.
* Huang et al. [2018] S.-F. Huang, Y.-C. Chen, H.-y. Lee, and L.-s. Lee, “Improved audio embeddings by adjacency-based clustering with applications in spoken term detection,” _arXiv preprint arXiv:1811.02775_ , 2018.
* Yuan et al. [2018] Y. Yuan, C.-C. Leung, L. Xie, H. Chen, B. Ma, and H. Li, “Learning acoustic word embeddings with temporal context for query-by-example speech search,” in _Proc. Interspeech_ , 2018.
* Park and Glass [2008] A. S. Park and J. R. Glass, “Unsupervised pattern discovery in speech,” _IEEE Trans. Audio, Speech, Language Process._ , vol. 16, no. 1, pp. 186–197, 2008.
* Jansen and Van Durme [2011] A. Jansen and B. Van Durme, “Efficient spoken term discovery using randomized algorithms,” in _Proc. ASRU_ , 2011.
* Ondel et al. [2019] L. Ondel, H. K. Vydana, L. Burget, and J. Černocký, “Bayesian subspace hidden markov model for acoustic unit discovery,” in _Proc. Interspeech_ , 2019.
* Räsänen and Blandón [2020] O. Räsänen and M. A. C. Blandón, “Unsupervised discovery of recurring speech patterns using probabilistic adaptive metrics,” _arXiv preprint arXiv:2008.00731_ , 2020.
* Kamper et al. [2017a] H. Kamper, K. Livescu, and S. Goldwater, “An embedded segmental K-means model for unsupervised segmentation and clustering of speech,” in _Proc. ASRU_ , 2017.
* Seshadri and Räsänen [2019] S. Seshadri and O. Räsänen, “SylNet: An adaptable end-to-end syllable count estimator for speech,” _IEEE Signal Process. Letters_ , vol. 26, no. 9, pp. 1359–1363, 2019.
* Kreuk et al. [2020] F. Kreuk, J. Keshet, and Y. Adi, “Self-supervised contrastive learning for unsupervised phoneme segmentation,” in _Proc. Interspeech_ , 2020.
* Rabiner et al. [1978] L. Rabiner, A. Rosenberg, and S. Levinson, “Considerations in dynamic time warping algorithms for discrete word recognition,” _IEEE Trans. Acoust., Speech, Signal Process._ , vol. 26, no. 6, pp. 575–582, 1978.
* Levin et al. [2013] K. Levin, K. Henry, A. Jansen, and K. Livescu, “Fixed-dimensional acoustic embeddings of variable-length segments in low-resource settings,” in _Proc. ASRU_ , 2013.
* Holzenberger et al. [2018] N. Holzenberger, M. Du, J. Karadayi, R. Riad, and E. Dupoux, “Learning word embeddings: Unsupervised methods for fixed-size representations of variable-length speech segments,” in _Proc. Interspeech_ , 2018.
* Chung et al. [2016] Y.-A. Chung, C.-C. Wu, C.-H. Shen, H.-Y. Lee, and L.-S. Lee, “Unsupervised learning of audio segment representations using sequence-to-sequence autoencoder,” in _Proc. Interspeech_ , 2016.
* Kamper et al. [2016a] H. Kamper, W. Wang, and K. Livescu, “Deep convolutional acoustic word embeddings using word-pair side information,” in _Proc. ICASSP_ , 2016\.
* Kamper [2019] H. Kamper, “Truly unsupervised acoustic word embeddings using weak top-down constraints in encoder-decoder models,” in _Proc. ICASSP_ , 2019.
* Kamper et al. [2020] H. Kamper, Y. Matusevych, and S. Goldwater, “Multilingual acoustic word embedding models for processing zero-resource languages,” in _Proc. ICASSP_ , 2020.
* Kamper et al. [2021] ——, “Improved acoustic word embeddings for zero-resource languages using multilingual transfer,” _IEEE Trans. Audio, Speech, Language Process._ , vol. 29, pp. 1107–1118, 2021.
* Hu et al. [2021] Y. Hu, S. Settle, and K. Livescu, “Acoustic span embeddings for multilingual query-by-example search,” in _Proc. SLT_ , 2021.
* Hu et al. [2020] ——, “Multilingual jointly trained acoustic and written word embeddings,” in _Proc. Interspeech_ , 2020.
* Ruder [2019] S. Ruder, “Neural transfer learning for natural language processing,” PhD diss., NUI Galway, 2019.
* Rumelhart et al. [1986] D. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” _Nature_ , vol. 323, pp. 533–536, 1986.
* Mikolov et al. [2013a] T. Mikolov, I. Sutskever, K. Chen, G. Corrado, and J. Dean, “Distributed representations of words and phrases and their compositionality,” in _Proc. NeurIPS_ , 2013.
* Pennington et al. [2014] J. Pennington, R. Socher, and C. Manning, “Glove: Global vectors for word representation,” in _Proc. EMNLP_ , 2014.
* Jacobs et al. [2021] C. Jacobs, Y. Matusevych, and H. Kamper, “Acoustic word embeddings for zero-resource languages using self-supervised contrastive learning and multilingual adaptation,” in _Proc. SLT_ , 2021.
* Chen et al. [2015] G. Chen, C. Parada, and T. N. Sainath, “Query-by-example keyword spotting using long short-term memory networks,” in _Proc. ICASSP_ , 2015.
* Settle and Livescu [2016] S. Settle and K. Livescu, “Discriminative acoustic word embeddings: Recurrent neural network-based approaches,” in _Proc. SLT_ , 2016.
* Doersch and Zisserman [2017] C. Doersch and A. Zisserman, “Multi-task self-supervised visual learning,” in _Proc. ICCV_ , 2017.
* Asano et al. [2020] Y. M. Asano, C. Rupprecht, and A. Vedaldi, “A critical analysis of self-supervision, or what we can learn from a single image,” in _Proc. ICLR_ , 2020.
* Doersch et al. [2015] C. Doersch, A. Gupta, and A. A. Efros, “Unsupervised visual representation learning by context prediction,” in _Proc. ICCV_ , 2015.
* Noroozi and Favaro [2016] M. Noroozi and P. Favaro, “Unsupervised learning of visual representations by solving jigsaw puzzles,” in _Proc. ECCV_ , 2016.
* Gidaris et al. [2018] S. Gidaris, P. Singh, and N. Komodakis, “Unsupervised representation learning by predicting image rotations,” in _Proc. ICLR_ , 2018.
* Pascual et al. [2019] S. Pascual, M. Ravanelli, J. Serrà, A. Bonafonte, and Y. Bengio, “Learning problem-agnostic speech representations from multiple self-supervised tasks,” 2019.
* Synnaeve et al. [2020] G. Synnaeve, Q. Xu, J. Kahn, T. Likhomanenko, E. Grave, V. Pratap, A. Sriram, V. Liptchinsky, and R. Collobert, “End-to-end ASR: From supervised to semi-supervised learning with modern architectures,” in _Proc. ICML_ , 2020\.
* Baevski et al. [2020a] A. Baevski, S. Schneider, and M. Auli, “vq-wav2vec: Self-supervised learning of discrete speech representations,” in _Proc. ICLR_ , 2020.
* Baevski et al. [2020b] A. Baevski, M. Auli, and A. Mohamed, “Effectiveness of self-supervised pre-training for speech recognition,” in _Proc. ICASSP_ , 2020.
* Wang et al. [2020] W. Wang, Q. Tang, and K. Livescu, “Unsupervised pre-training of bidirectional speech encoders via masked reconstruction,” in _Proc. ICASSP_ , 2020.
* Chen et al. [2020] T. Chen, S. Kornblith, M. Norouzi, and G. Hinton, “A simple framework for contrastive learning of visual representations,” in _Proc. ICML_ , 2020\.
* Sohn [2016] K. Sohn, “Improved deep metric learning with multi-class N-pair loss objective,” in _Proc. NeurIPS_ , 2016.
* van der Merwe [2020] R. van der Merwe, “Triplet entropy loss: Improving the generalisation of short speech language identification systems,” _arXiv preprint arXiv:2012.03775_ , 2020.
* Yi et al. [2019] J. Yi, J. Tao, Z. Wen, and Y. Bai, “Language-adversarial transfer learning for low-resource speech recognition,” _IEEE/ACM Trans. Audio, Speech, Language Process._ , vol. 27, no. 3, pp. 621–630, 2019.
* van der Westhuizen et al. [2021] E. van der Westhuizen, T. Padhi, and T. Niesler, “Multilingual training set selection for ASR in under-resourced Malian languages,” in _Proc. SPECOM_ , 2021.
* Hazen et al. [2009] T. J. Hazen, W. Shen, and C. White, “Query-by-example spoken term detection using phonetic posteriorgram templates,” in _Proc. ASRU_ , 2009.
* Zhang and Glass [2009] Y. Zhang and J. R. Glass, “Unsupervised spoken keyword spotting via segmental DTW on Gaussian posteriorgrams,” in _Proc. ASRU_ , 2009.
* Jansen and Durme [2012] A. Jansen and B. V. Durme, “Indexing raw acoustic features for scalable zero resource search,” in _Proc. Interspeech_ , 2012.
* Anastasopoulos et al. [2016] A. Anastasopoulos, D. Chiang, and L. Duong, “An unsupervised probability model for speech-to-translation alignment of low-resource languages,” in _Proc. EMNLP_ , 2016.
* Settle et al. [2017] S. Settle, K. Levin, H. Kamper, and K. Livescu, “Query-by-example search with discriminative neural acoustic word embeddings,” in _Proc. Interspeech_ , 2017.
* Yuan et al. [2019] Y. Yuan, C.-C. Leung, L. Xie, H. Chen, and B. Ma, “Query-by-example speech search using recurrent neural acoustic word embeddings with temporal context,” _IEEE Access_ , vol. 7, pp. 67 656–67 665, 2019.
* Ram et al. [2019] D. Ram, L. Miculicich, and H. Bourlard, “Neural network based end-to-end query by example spoken term detection,” _IEEE Trans. Audio, Speech, Lang. Process._ , vol. 28, no. 1, pp. 1416–1427, 2019.
* Saeb et al. [2017] A. Saeb, R. Menon, H. Cameron, W. Kibira, J. Quinn, and T. Niesler, “Very low resource radio browsing for agile developmental and humanitarian monitoring,” in _Proc. Interspeech_ , 2017.
* Menon et al. [2017] R. Menon, A. Saeb, H. Cameron, W. Kibira, J. Quinn, and T. Niesler, “Radio-browsing for developmental monitoring in Uganda,” in _Proc. ICASSP_ , 2017.
* Menon et al. [2018] R. Menon, H. Kamper, J. Quinn, and T. Niesler, “Fast ASR-free and almost zero-resource keyword spotting using DTW and CNNs for humanitarian monitoring,” in _Proc. Interspeech_ , 2018.
* Menon et al. [2019] R. Menon, H. Kamper, E. v. d. Westhuizen, J. Quinn, and T. Niesler, “Feature exploration for almost zero-resource ASR-free keyword spotting using a multilingual bottleneck extractor and correspondence autoencoders,” in _Proc. Interspeech_ , 2019.
* [59] “United Nations strategy and plan of action on hate speech.” [Online]. Available: https://www.un.org/en/genocideprevention/documents/advising-and-mobilizing/Action_plan_on_hate_speech_EN.pdf
* Larson and Jones [2012] M. Larson and G. Jones, “Spoken content retrieval: A survey of techniques and technologies,” _Found. Trends Inform. Retrieval_ , pp. 235–422, 2012\.
|
# Lithium in red giants: the roles of the He-core flash and the luminosity
bump
Deepak1,2 and David L. Lambert3
1Indian Institute of Astrophysics, Bangalore - 560034, India
2Pondicherry University, R. V. Nagara, Kalapet, Puducherry - 605014, India
3W.J. McDonald Observatory and Department of Astronomy, The University of
Texas at Austin, Austin, TX 78712, USA E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted 9th July 2021. Received 9th July 2021; in original form 31st March
2021)
###### Abstract
Lithium abundances for red giants in the GALAH DR3 survey are studied. The
rare examples of Li-enriched stars with abundances A(Li) $\geq 1.5$ are
confirmed to be He-core burning stars belonging to or evolved from the red
clump with similar masses and metallicity: $M\simeq 1.1\pm 0.2{\rm M_{\odot}}$
and [Fe/H] $\simeq-0.3\pm 0.3$. Li enrichment over the Li abundance present in
a star’s predecessor at the tip of the red giant branch likely occurs in all
these red clump stars. Examination of the elemental abundances (C to Eu) in
the GALAH catalogue shows no anomalous abundances in red clump giants and, in
particular, no dependence on the Li abundance, which ranges over at least five
dex. Lithium synthesis is attributed to the He-core flash occurring in stars
at the tip of the red giant branch. Models from the Modules for Experiments in
Stellar Astrophysics (MESA) match the observed evolution of these stars along
the red giant branch and to the red clump but only at the low effective
temperature end of the observed spread of red clump giants. Run of Li
abundance on the red giant branch is fairly well reproduced by MESA models. A
speculation is presented that the series of He-core flashes not only leads to
7Li synthesis from a star’s internal reservoir of 3He but also may lead to
internal restructuring leading to the observed effective temperature spread of
red clump stars at about a constant luminosity. Giants exhibiting marked Li
enrichments are not found at other evolutionary phases and, in particular, not
directly associated with the luminosity bump on the red giant branch for which
the Li abundance increase does not exceed 0.3 dex.
###### keywords:
Surveys – Hertzsprung-Russell and colour-magnitude diagrams – Stars: evolution
– Stars: abundances – Nuclear reactions, nucleosynthesis, abundances – Stars:
individual: Li-rich giants
††pubyear: 2021††pagerange: Lithium in red giants: the roles of the He-core
flash and the luminosity bump–A
## 1 Introduction
Lithium’s presence in low mass red giants has occasioned considerable interest
across the evolutionary phases from subgiant, the red giant branch, the red
clump He-core burning giant to the asymptotic giant branch and then to the
post-asymptotic giant branch followed by ejection of the envelope with the
stellar residual present as a white dwarf. Lithium present at low abundance in
a giant’s main sequence progenitor is depleted in the giant because the deep
convective envelope drastically dilutes the Li abundance. This major episode
of dilution occurs as the giant evolves to the base of the first red giant
branch (RGB). This stage is known as the first dredge-up (FDU). Continuing
depletion of Li occurs in the lowest mass RGB giants beyond the completion of
the FDU. Given that the effect of the FDU in low mass giants is to lower the
envelope’s Li abundance by approximately a factor of 100 (more precise
estimates are provided below), a red giant post-FDU of about solar metallicity
is expected to have a Li abundance of A(Li) $\sim 1.5$.111Elemental abundances
are given on the traditional scale: A(X) = $\log$[N(X)/N(H)]+ 12 where N(X) is
the number density of element X. A Li abundance of about this value represents
the anticipated upper limit for a giant’s Li abundance because Li may have
experienced depletion, dilution or diffusion in a main sequence star’s outer
envelope well prior to the FDU and the giant’s convective envelope may post-
FDU destroy additional Li. Thus, a Li-rich giant has been widely known as one
with an abundance A(Li)$\geq 1.5$.
Discovery of Li-rich K giants was initiated by Wallerstein & Sneden (1982)’s
finding of a strong 6707 Å Li i line in the post-FDU K III giant HD 112127
with the abundance A(Li) $\simeq 3.2$. Discoveries of additional Li-rich K
giants across a range of Li abundances trickled out after 1982, but, recently,
discoveries have been provided at an accelerated pace thanks in large part to
surveys such as the Gaia-ESO, GALAH and LAMOST surveys (Casey et al., 2016;
Deepak & Reddy, 2019; Gao et al., 2019; Martell et al., 2021). Li-rich giants
are rare, representing a per cent or so of K giants in the Galactic disk
(Deepak & Reddy, 2019). At present, the most Li-rich giant known may be TYC
429-2097-1 with A(Li) = 4.5 (Yan et al., 2018).
The discovery of HD 112127 and others led to speculations about Li production
in these red giants. Almost all proposed explanations involve Li production
tied to a giant’s interior layer of 3He. This He isotope at low abundance is
present in a star from its birth with additional amounts produced in low mass
stars by initial steps of the pp-chain in main sequence H-burning. Li
production from 3He is straightforward for a nuclear astrophysicist: the
reaction 3He($\alpha,\gamma$)7Be is followed by electron capture
7Be(e-,$\nu_{e}$)7Li. For the stellar astrophysicist, this link requires
coupling the hot reaction site where 7Li is created but also destroyed by
protons to the cool stellar atmosphere where the Li may survive and is
observed. This necessary coupling acknowledges too that the intermediary 7Be
has a half-life of just 53 days. The giant’s stellar convective envelope is
required to bring the fresh (7Be and) 7Li to the atmosphere with, of course,
also convection returning 7Li to the interior for destruction and
replenishment. This coupling of nuclear and stellar astrophysics first
recognized by Cameron (1955) is now widely known as the Cameron-Fowler
mechanism.
Cameron & Fowler (1971) suggested this mechanism involving He-burning shell
flashes to account for the rare carbon stars like WZ Cas (McKellar, 1940) and
WX Cyg (Sanford, 1950) with very strong 6707 Å lines. In modern parlance,
these carbon stars are examples of luminous AGB stars experiencing helium
shell burning episodes and the third dredge-up. Lithium, carbon and
$s$-process enrichment are predicted by contemporary modelling and observed in
luminous AGB giants (Smith & Lambert, 1989). However, for K giants like HD
112127, which are not luminous AGB stars, two evolutionary phases have
received the most attention with respect to Li enrichment: giants at the so-
called luminosity bump (LB) on the RGB and more luminous low mass giants at
the RGB’s tip, which experience a He-core flash before evolving to the red
clump (horizontal branch) as a He-core burning giant.
Suspicions about the association of the He-core flash with Li synthesis were
aired in the first survey for Li-rich giants by Brown et al. (1989) and raised
subsequently by others. For example, Deepak & Reddy (2019) found that most of
the Li-rich giants in the GALAH DR2 survey belong to the red clump phase.
These suspicions were wonderfully raised to a near-certainty by Kumar et al.
(2020) whose analysis of Li abundances of giants in the GALAH DR2 survey led
to the bold suggestion that only He-core burning giants were Li-rich and,
thus, that Li synthesis occurs at or immediately following the He-core flash.
They asserted that Li enrichment of He-core burning giants is ‘ubiquitous’ and
confined to these giants, that is – all low mass giants experiencing a He-core
flash synthesize Li at a level from very mild to severe to contaminate the RC
giant which follows the flash. Subsequent studies of asteroseismology of He-
core burning giants with A(Li) determinations suggest that Li abundances
following the He-core flash may decline with age (Deepak & Lambert, 2021;
Singh et al., 2021). The common assumption is that the Li results from the
Cameron-Fowler mechanism. Kumar et al. (2020) did not detail the mechanism’s
application to the He-core flash. A recent theoretical exploration (Schwab,
2020) is discussed below.
Adaptation of the 3He reservoir to Li production has also been connected to
RGB giants experiencing the LB. This ‘bump’ on the RGB occurs at the point
when the H-burning shell moves outward across an H-discontinuity. The
evolution of the star reverses course, and it moves slowly and slightly down
the RGB before reversing again and continuing to evolve up the RGB. This
manoeuvre introduces a clumping of stars on the RGB. At the time of this LB,
operation of the Cameron-Fowler mechanism occurs through a thermohaline
instability expected on theoretical grounds (Eggleton et al., 2008; Lagarde et
al., 2011) and suggested as a possible way to account for Li enriched giants
(e.g., Martell et al., 2021). But definitive observational identification of
Li enriched LB giants remains open.222Thermohaline instability at the LB has
been suggested also as a way to change a giant’s surface 12C/13C ratio.
In this paper, we concentrate on the exploitation of the recent GALAH survey
of stellar abundances, with the focus here on the Li abundances of low-mass
red giants. Our primary goal is to set Li production limits by the He-core
flash as a function of stellar mass and initial metallicity. Our secondary
goal is to search for Li enrichment by LB giants. Our observational inquiry
based on the latest data release DR3 from the GALAH survey is supported by
predictions of Li abundances from calculations performed using the Modules for
Experiments in Stellar Astrophysics (MESA; Paxton et al., 2011, 2013, 2015,
2018, 2019) suite of computer programmes.
## 2 Data sample
Recent observational advances in understanding Li-rich giants have largely
come from combining spectroscopic abundance surveys with astrometric data from
the Gaia satellite (Gaia Collaboration et al., 2016). Li (and other) elemental
abundances from the GALAH DR2 survey have been used by Deepak & Reddy (2019);
Deepak et al. (2020); Kumar et al. (2020) with Martell et al. (2021) using the
DR3 results. The LAMOST survey was considered by Deepak et al. (2020) for C, N
and O (but not for Li) for stars in common with the GALAH DR2. LAMOST’s Li
abundances were explored by Singh et al. (2019) for stars in the Kepler field
of view and in detail by Yan et al. (2021).
Here, we consider the third data release (DR3) of the GALAH survey. Our
analysis is devoted to low mass giants with a metallicity [Fe/H]
representative of the Galactic disk, say $-1\lesssim$ [Fe/H] $\lesssim+0.5.$
GALAH DR3 (and DR2) provides too few metal-poor halo giants to probe the
matter of Li-richness among the halo population. As a secondary interest, we
make a foray into possible abundance anomalies introduced into RC giants by
the He-core flash, which converts a giant at the RGB tip to a He-core burning
giant on the RC.
The Third Data Release of the Galactic Archaeology with the HERMES survey
provides stellar parameters and elemental abundances derived from 678,423
spectra for 588,571 unique stars observed with the HERMES spectrograph at the
Anglo-Australian Telescope between November 2013 and February 2019 (see Buder
et al., 2021, for more detail on the data release). It is also accompanied by
several value-added-catalogues (VAC) providing various other intrinsic,
kinematic and dynamical properties of the observed stars (hereafter, GALAH$+$
DR3).333The GALAH$+$ DR3 data is available at
https://cloud.datacentral.org.au/teamdata/GALAH/public/GALAH_DR3/
GALAH_DR3_main_allspec_v1.fits is one of the two main catalogues of GALAH$+$
DR3 and includes stellar parameters and overall abundances of about 30
chemical species derived for each of the observed spectra. This catalogue also
includes abundances derived for each individual line for all the elements. We
joined this catalogue with the VAC GALAH_DR3_VAC_ages_v1.fits which provides
other stellar parameters like age, mass, distances, luminosity, etc.,
estimated using Bayesian Stellar Parameter Estimation code (BSTEP, Sharma et
al., 2018). The two catalogues are joined using the recommended method of
using the internal ID for each GALAH observation provided in the sobject_id
column. To eliminate multiple entries of a star in the catalogue and to select
the entry with the highest signal-to-noise, we restrict our sample to stars
with flag_repeat = 0. We further restrict our sample to stars with reliable
stellar parameters (flag_sp = 0) and metallicity (flag_fe_h = 0) which lead to
abandoning of stars with $T_{\rm eff}<4100$ K. We also discarded all the stars
with uncertainty in temperature ($\sigma_{T\rm_{eff}}$) and luminosity
($\sigma_{\log(L/{\rm L_{\odot}})}$) of more than 200 K and 0.2, respectively.
This resulted in a sample of about 425 thousand stars. Sample stars are shown
in Figure 1 in the form of the HR diagram. Note that the sample is not mass
limited, and the large scatter in the figure can be attributed in part to the
presence of relatively massive stars (with masses 2$-$5 M⊙) in the sample. For
each sample star, uncertainties in luminosity and temperature are also shown
in the background, however, due to crowding, they are visible only for the
star on the boundaries. From this sample, we select giant stars with
temperature from 4100 to 5600 K and luminosity $\log(L/{\rm L_{\odot}})$ from
0.3 to 3.4 dex. This selection of 159,841 stars is shown within the dotted
green box in Figure 1 (hereafter, the RBS, which stands for the red-giant
branch sample).
Except for Martell et al. (2021), previous discussions of Li abundances in
giants based on the GALAH survey drew on the DR2 catalogue. Of interest to the
present study is the fact that our sample of RBS giants from the DR3 catalogue
is approximately doubled over our previous sample from DR2 (Deepak et al.,
2020), abundance quality flagging is also improved, including the provision of
upper limits and abundances are now provided for 31 rather than 21 elements
previously. In the case of the 335 Li-rich giants found by Deepak & Reddy
(2019) in the DR2 catalogue, Li abundances of common stars in DR2 and DR3 show
only a slight systematic change: the least Li-rich are slightly less Li-rich,
and the super Li-rich giants are slightly more Li-rich. Our assessment is that
the differences between the data releases have not affected principal
conclusions about Li in giants in the RBS collection studied in previous
works. The increase in the RBS sample offers the chance to search for new
properties of Li-rich giants and to explore already covered properties in more
detail.
Figure 1: Sample stars in a HR diagram also showing the selection of red
giants (the RBS) within the green box. For all the stars, uncertainty in
luminosity and temperature are also plotted in the background as red error
bars.
### 2.1 Elemental abundances
The GALAH$+$ DR3 survey provides the elemental abundance ratio [X/Fe]444For
two elements, $X$ and $Y$, with number densities $N_{X}$ and $N_{Y}$, the
elemental abundance ratio [$X/Y$] $\equiv$ $\log$($N_{X}/N_{Y}$)star $-$
$\log$($N_{X}/N_{Y}$)Sun. for up to 30 chemical elements per star with X = Li,
C, O, Na, Mg, Al, Si, K, Ca, Sc, Ti, V, Cr, Mn, Co, Ni, Cu, Zn, Rb, Sr, Y, Zr,
Mo, Ru, Ba, La, Ce, Nd, Sm, and Eu. Abundances for most of the elements were
considered in detail by Deepak et al. (2020) using the DR2 catalogue. The
global analysis of [X/Fe] will not be reexamined here. Previously, the only
detectable difference between [X/Fe] for Li-rich, Li super-rich and normal
giants was that Li super-rich giants were slightly underabundant in [C/Fe] but
see Martell et al. (2021).
For Li, the DR3 catalogue provides the non-LTE correction for the Li abundance
provided by the 6707 Å resonance doublet. We convert the [Li/Fe] using the
non-LTE Li and LTE Fe abundances into A(Li) using A(Li) = [Li/Fe] + [Fe/H] +
A(Li)⊙, where A(Li)⊙ is the solar Li abundance for which we adopt A(Li)⊙ =
1.05 $\pm$ 0.10 dex, which is the value used for the GALAH data (see Buder et
al., 2021, for more details). The DR3 catalogue also provides uncertainties
($\rm\sigma_{[X/Fe]}$ available in corresponding e_X_fe column) and a quality
flag (flag_X_fe) for each of abundance entry. flag_X_fe can have any value
from 0, 1 and 32, where flag 0 means provided abundance is reliable, 1 means
provided abundance is an upper limit, and 32 means no reliable measurement
reported. In this study, we will use flag_X_fe = 0 for elements except Li. In
the case of Li, flag_Li_fe = 0 will be used to identify a Li-enriched star,
while for the Li-normal stars, we will use all three flags to avoid discarding
stars with a relatively lower Li-abundance from the sample. Throughout the
paper, we will specify if we are considering a sample of stars for which DR3
notes a detection of Li or considering the larger sample combining stars with
a detection or an upper limit of the Li 6707 Å line.
## 3 Lithium-rich giants: overview
Figure 2: HR diagram and A(Li) as a function of luminosity for the RBS giants
with masses $M/{\rm M_{\odot}}$ = 1.10 $\pm$ 0.02 and metallicity [Fe/H] =
0.00 $\pm$ 0.10 dex. The left and middle panels show giants with both Li
detections and upper limits with light yellow colour indicating a higher
density of stars. In the right-most panel, giants with Li detections are shown
in red, while giants with Li upper limits are in blue. Figure 3: The HR
diagram for the RBS giants. Colour in the top and bottom panels represents the
linear density and maximum A(Li) abundances, respectively. Note that the
density visualization scale in the top panel is linear, but in Figure 1 scale
is logarithmic. Figure 4: HR diagrams for the RBS giants for Li abundances in
different abundance intervals. All giants shown here have a Li detection
(i.e., flag_Li_fe=0). Figure 5: Li abundance maps along with metallicity and
mass information in colour for the RBS giants in the HR diagram. All the
giants shown here have Li detections (i.e., flag_Li_fe=0). Figure 6: Top row:
Metallicity and mass histograms for giants plotted in Figure 5. Histograms are
shown for different Li abundance intervals, as shown in the keys in the two
panels. Bottom row: Metallicity and mass histograms for entire RBS sample with
$\log(L/{\rm L_{\odot}})>$ 1.5.
Our principal endeavour is to use the GALAH DR3 RBS sample (Figure 1) to (i)
explore closely the Li enrichment attributable to the He-core flash in low
mass red clump giants and (ii) probe more thoroughly than before the Li
enrichment predicted from thermohaline mixing at the luminosity bump on the
RGB. Li-enrichment by the He-core flash was empirically proven by Kumar et al.
(2020) from analysis of the GALAH DR2 Li abundances for giants but theoretical
predictions of Li-enrichment are lacking apart from the recent exploration by
Schwab (2020). In contrast to this observational–theoretical imbalance
regarding Li enrichment in red clump giants, theoretical modelling of Li-
enrichment at the luminosity bump (Eggleton et al., 2008; Lagarde et al.,
2011) may be more advanced than observational efforts.
To provide an intriguing glimpse into the relative roles of the two
evolutionary phases of possible Li enrichment, Figure 2 shows in the left
panel the HR diagram for the narrow range of masses $M=1.10\pm 0.02\ {\rm
M}_{\odot}$ and metallicities [Fe/H]$=0.00\pm 0.10$ shown below to be rich in
Li enriched giants. The middle and right-most panels show the relation between
A(Li) and luminosity. The left and middle panels show giants with both Li
detections and upper limits, while in the right-most panel, giants with Li
detections are shown in red while giants with Li upper limits are in blue.
Gross features of the three panels are readily recognizable. The HR diagram
begins with subgiants at $\log(L/{\rm L_{\odot}})\sim 0.5$ and leads into the
RGB with its tip at $\log(L/{\rm L_{\odot}})\simeq 3.5$. There is a steady
decrease of the Li abundance A(Li) up the RGB. The LB occurs on the RGB at a
luminosity just below that of the RC. At the LB, there is a noticeable
increase in the counts. Superimposed on the RGB in the HR diagram is the RC at
$\log(L/{\rm L_{\odot}})\simeq 1.7$ and this marks a large spread in lithium
abundance up to A(Li) $\sim 4$. Apart from a few outliers at luminosities
above and below the RC’s luminosity, this selection does not indicate Li
enrichment associated with a density increase on the RGB arising from the LB.
If this selection by mass and metallicity from the RBS is representative, the
outstanding message is that Li enrichment among RC giants, as Kumar et al.
(2020) noted, is a striking feature. But Figure 2 also suggests that a search
for Li enrichment associated with the LB may require subtle procedures because
at the mass and metallicity ranges explored in Figure 2 (and at neighbouring
mass and metallicities) Li enrichment from the LB appears not to be a major
process.
In our effort to advance the story of Li-enrichment in giants at the red clump
post-He-core flash stage and in other giants at the luminosity bump on the
RGB, we combine Li abundances from the GALAH DR3 survey with stellar evolution
calculations for low mass stars from the MESA suite of computer programmes.
Our focus on possibilities of Li-enrichment in RC and LB low mass Population I
giants does not mean that Li enrichment cannot occur in giants other than at
the RC or LB phases, as in mass transfer to a star from a Li-rich AGB
companion (again, Li synthesis occurs via 3He) or in Population II giants or
by other mechanisms such as engulfment of planetary bodies (Alexander, 1967).
### 3.1 He-core burning giants - the red clump
In Figure 1, the prominent feature superimposed on the red giant branch (RGB)
at $\log(L/{\rm L_{\odot}})\simeq 1.7$ is the red clump (RC) comprised of
giants burning He in their core. As noted in the Introduction, Kumar et al.
(2020)’s vital contribution was to show that observations demand that all RC
giants in the GALAH survey were enriched in Li to differing degrees relative
to stars at the tip of the RGB, the immediate predecessors of RC giants, that
is Li enrichment in RC giants is ‘ubiquitous’, i.e., even RC giants with the
lowest Li abundance, A(Li) $\simeq-0.5$, are Li enriched. Lithium production
was attributed to the He-core flash occurring at the RGB tip as the star
ignites its He core. Giants with masses less than about 2.25 M⊙ at solar
metallicity [Fe/H] = 0 experience a He-core flash at the tip of the RGB. The
upper mass limit for a He-core flash decreases slightly for metal-poor stars
(Karakas, 2010). The existence of the flash has stimulated theoretical
speculation on the possibility of internal mixing and associated
nucleosynthesis including production of Li from 3He and enrichment of the
surface of the RC giant with Li. Consideration of lithium abundances among RC
stars is the principal concern of our study. (Extension of the RC to hotter
temperatures, as in globular clusters, marks the horizontal branch (HB), but
here we use RC to refer to the entire temperature range.)
Lithium abundances among the RC giants span several dex (Kumar et al., 2020).
A simple map of Li abundance over the HR diagram is provided in Figure 3. The
top panel shows the number density of giants across the RBS. Presence of the
RC is obvious from about 4600 K to, perhaps, 5400 K with a peak ratio of
counts in the mid-RC strip to the counts on either side of about 7. The Li map
in the lower panel is colour-coded according to the Li abundance and makes
clear that the most Li-rich giants are on or maybe evolved from the RC.
More detailed maps of Li abundances are provided in the eight panels of Figure
4 showing HR diagrams for giants with A(Li) abundances in the intervals
$-0.5$-0.0, 0.0-0.5, 0.5-1.0, 1.0-1.5, 1.5-2.0, 2.0-2.5, 2.5-3.0 and $>3.0$.
The full RBS includes stars of mass from about 0.8 M⊙ to 2.5 M⊙. The low Li
abundance panels in Figure 4 clearly show the results of the first dredge-up
(FDU) affecting subgiants and early residents on the RGB, most notably in the
A(Li) = 1.0-1.5 and 1.5-2.0 panels at the luminosity $\log(L/{\rm
L_{\odot}})\sim 0.5$. The limit A(Li) $\geq$ 1.5 with representation across
four panels corresponds approximately to the Li abundance expected following
the first dredge-up, a value which has often been adopted in discussions of
Li-rich giants as the lower boundary for a Li-rich giant. (Note that lithium
post-FDU is further depleted in the lower mass giants – see below.) Li
abundances in these four panels include giants in which clearly Li has been
enriched. Inspection of each of these panels confirms that Li-enriched stars
appear tightly confined to the RC and also to a few stars more luminous and
generally cooler than the RC, i.e., to stars which may have exhausted He-
burning in their core and evolved off the RC. For each of these four samples,
there is a clear gap between the RC giants and the lower luminosity RGB
giants, i.e., all giants which are Li-enriched are RC giants. There is
potential confusion at luminosities above the RC between post-RC (early AGB)
giants and RGB stars up to the tip of the RGB, at $\log(L/{\rm
L_{\odot}})\approx 3.4$. (There may also be a possibility of confusion
provided by Li-rich giants at the luminosity bump on the RGB, an issue
discussed in Section 5). In the panel covering the lowest Li abundances, A(Li)
= $-0.5$ to $-0.0$, no stars are seen on the RC, and the majority are likely
RGB giants with a luminosity below that of the RGB tip. Since further
evolution to the RGB tip may result in even lower Li abundances, RC giants
emerging from the RGB tip without experiencing Li enrichment will have maximum
Li abundances A(Li) $\lesssim 0.0$. But there are very few RC giants with this
low Li abundance and, therefore, Li synthesis at some level from the He-core
flash may occur in all giants transitioning from the RGB to the RC, as
suggested by Kumar et al. (2020).
Consideration of the panels in Figure 4 provides additional information.
First, we explore the metallicity and mass dependence of Li-rich giants (see,
for example, Deepak et al., 2020). The metallicities (colour-coded) show that
Li-rich RC giants are predominantly of near-solar metallicities (Figure 5).
This is not a new result; see, for example, Figures 3 and 4 from Deepak et al.
(2020) and Figure 5 from Martell et al. (2021) but extends earlier results by
showing that the metallicity histograms are only weakly dependent on the Li
abundance. Histograms of the metallicity for RC and (presumed) evolved RC
stars in Figure 6 well illustrate that Li-rich giants are certainly
representatives of the Galactic thin disk and its spread in metallicity.
Another family of histograms (Figure 6, top-right panel) is provided for the
masses of the Li-rich RC giants. As Figure 6 shows, the Li-rich giants have a
narrow distribution peaking about 1.1 M⊙. The histograms extend earlier
results by showing that the mass distribution is independent of Li abundance.
There is perhaps a slight increase in mass from the cool to the warm end of
the RC. Distribution of Li abundances across the RBS by mass shows Li rich RC
giants from 0.8 M⊙ to 2.3 M⊙ with, of course, the most prominent sampling
around 1.1 M⊙ to 1.2 M⊙.
The histograms in the bottom panels of Figure 6 refers to the full sample
covered by the RBS at luminosities $\log(L/{\rm L_{\odot}})>1.5$. Note the
large increase in counts between this sample and the histograms compiled from
selected Li abundance intervals in the upper part of the figure. More
importantly, note the overall similarities in the two sets of histograms. To
first order, this suggests that the frequency of Li enriched RC giants is
approximately independent of mass and metallicity. The two sets of histograms
appear most similar for intermediate Li abundances inviting the idea that
levels of Li enrichment may vary with mass and metallicity.
Close scrutiny of Figure 4 suggests several insights into Li production at the
He-core flash. Most obviously, the RC’s location in the HR diagram is
independent of the Li abundance. For example, the mean luminosity of the RC’s
core at the mean effective temperature of 4775$\pm 100$ K is $\log(L/{\rm
L_{\odot}})$ = 1.73 to within $\pm 0.05$ across the bottom four panels in
Figure 4. The mild increase in luminosity with increasing effective
temperature is independent of Li abundance. Similarly, the RC’s width at a
given effective temperature appears to be constant across these panels with
the exception of a mild increase at the lowest Li abundances, which may be
related to the increase in the sample size at the lowest Li abundances. The
range of effective temperatures spanned by the RC in the panels is independent
of Li abundance above A(Li)$\geq 1.5$. Also, the proportion of giants at
luminosities above the RC is similar for the four panels. These
characteristics about the RC and the Li abundance show that the gross
properties of RC stars appear independent of the He-core flash’s ability to
synthesize Li.
However, stellar evolutionary calculations fail not only to account for Li
enhancement in RC giants but also do not account for their considerable spread
in effective temperature at approximately constant luminosity. Such
calculations predict a 1 M⊙ of about solar metallicity to have approximately
the observed luminosity at around 4500K. These are two failures of standard
calculations, but one supposes that while their resolutions may be related,
they cannot be tightly coupled because Li abundances are independent of a
star’s effective temperature.
It seems remarkable that the vast majority of Li enriched giants have very
similar characteristics, i.e., they are or were He-core burning giants of
about 1 M⊙ and near-solar metallicity. Figure 5 shows that among apparent
exceptions are subgiants ($\log(L/{\rm L_{\odot}})\sim 0.7$ with $T_{\rm
eff}\sim 5000-5600$ K) whose numbers decline with increasing A(Li). These are
in all likelihood normal stars of about 1 M⊙ and near-solar metallicity at
different stages of an incomplete FDU. Figure 5 also shows a few higher mass
stars with a luminosity spanning that of the RC. These too are stars observed
before completion of their FDU. Two Li enriched giants at $\sim$4800 K and at
a luminosity well below that of the RC in A(Li) $\geq 3.0$ panel in Figure 4
arouse our curiosity. Both of these giants are of solar mass. This pair “Gaia
DR2 5507464674226166144” and “Gaia DR2 3340856632069339776” with A(Li) = 3.09
$\pm$ 0.11 and 3.51 $\pm$ 0.11, respectively, deserve detailed study.
In essence, setting aside few stars, our selection of Li enriched stars from
the RBS is dominated by stars of about a solar mass and a solar metallicity.
Figure 7: HR diagram (odd-numbered rows) and A(Li) as a function of luminosity
(even-numbered rows) for the RBS stars selected with mass and metallicity
ranges as mentioned in panels’ titles. In luminosity-A(Li) panels, giants with
Li detections are shown in red, while giants with Li upper limits are shown in
blue. Figure 8: continued Figure 9: HR diagrams and A(Li) as a function of
luminosity for solar mass and solar metallicity models with different
$\alpha\rm_{ML}$ and $\alpha\rm_{th}$. Top and bottom panels show a model‘s
sensitivity to $\alpha\rm_{ML}$ and $\alpha\rm_{th}$, respectively. Dashed
lines show pre-main-sequence evolution, while solid lines show the evolution
from the zero-age main sequence (ZAMS). On each track, the LB’s position is
marked by the green coloured blob. Figure 10: Temporal evolution of Li for
solar mass and solar metallicity models with different $\alpha\rm_{ML}$ and
$\alpha\rm_{th}$. Time is shown relative to the onset of LB. Model’s
sensitivity to both $\alpha\rm_{ML}$ and $\alpha\rm_{th}$ is shown. The top
panel covers evolution from about sub-giant phase to the CHeB phase, while the
bottom panel covers evolution from near RGB on-wards. Evolution up to the RGB
tip is shown as solid lines, while dashed lines show the evolution in the CHeB
phase. On each track, the LB’s position is marked by the green coloured blob.
Timestep in each case varies from about 0.1 to 0.2 million years (Myr) during
the LB, and further decreases as the star ascend the RGB. Figure 11: Top
panel: Predicted evolution of A(Li) as a function of luminosity for solar
metallicity (Z = 0.0143) giants. Masses of 0.8 M⊙ to 2.0 M⊙ are shown, as
labelled. Bottom panel: Predicted evolution of A(Li) as a function of
luminosity for $1{\rm M_{\odot}}$ mass models with metallicities Z = 0.0045,
0.0080, 0.0143, and 0.0254, which are equivalent to [Fe/H] = -0.50, -0.25,
0.00, and 0.25 dex, respectively. For all models, we use $\alpha\rm_{ML}$ =
1.85, $\alpha\rm_{th}$ = 100 and initial surface rotation $\Omega_{\it
int}$/$\Omega_{\it crit}$ = 0. For each of the case, evolution is shown from
the ZAMS to the RGB tip. For all the masses, LBs are marked by green coloured
blobs. The abrupt change in A(Li) occurring at about $\log(L/{\rm
L_{\odot}})\sim 1.5$ is attributable to the onset of thermohaline mixing at
the LB. Figure 12: Observed and predicted HR diagram (top panel) and A(Li) as
a function of luminosity (middle and bottom panels) for RBS giants of masses
1.10 $\pm$ 0.02 M⊙ and metallicity 0.0 $\pm$ 0.1 dex. Predictions are shown
for best fit model with $\alpha_{\rm ML}$ = 1.85 along with $\alpha_{\rm th}$
= 50, 100, 150 and 200. The bottom panel shows luminosity-A(Li) trends after
adjustment to Li abundance level. In A(Li)-luminosity panels, giants with Li
detections are shown in red, while giants with Li upper limits are shown in
blue. Figure 13: Observed and predicted HR diagram (top panel and middle
panel) and A(Li) as a function of luminosity (bottom panel) for RBS giants of
masses 1.10 $\pm$ 0.02 M⊙ and metallicity 0.0 $\pm$ 0.1 dex. Predictions are
shown for best fit model with $\alpha_{\rm ML}$ = 1.85 and $\alpha_{\rm th}$ =
100 along with initial surface rotation $\Omega_{\rm int}/\Omega_{\rm crit}$ =
0.00, 0.05, and 0.10, which are equivalent to initial surface rotation of
about 0, 15, and 30 kms-1, respectively. In A(Li)-luminosity panels, giants
with Li detections are shown in red, while giants with Li upper limits are
shown in blue. Figure 14: Observed and predicted HR diagram (top panel) and
A(Li) as a function of luminosity (middle and bottom panel) for RBS giants of
masses 1.10 $\pm$ 0.02 M⊙ and metallicity -0.50 $\pm$ 0.10 dex. Prediction are
shown for best fit models with $\alpha_{\rm ML}$ = 1.80 along with
$\alpha_{\rm th}$ = 50, 100, 150 and 200. In A(Li)-luminosity panels, giants
with Li detections are shown in red, while giants with Li upper limits are
shown in blue. Figure 15: Observed and predicted HR diagram (top panel) and
A(Li) as a function of luminosity (bottom panel) for RBS giants of masses 1.50
$\pm$ 0.02 M⊙ and metallicity -0.25 $\pm$ 0.10 dex. Prediction are shown for
best fit model with $\alpha_{\rm ML}$ = 1.95 and $\alpha_{\rm th}$ = 100, 200,
300 and 400. The bottom panel shows luminosity-A(Li) trends after adjustment
to Li abundance level. In A(Li)-luminosity panels, giants with Li detections
are shown in red, while giants with Li upper limits are shown in blue.
## 4 MESA modelling for low-mass stars
To further our understanding of Li in giants, we use MESA models to predict
the Li abundance from the main sequence along the RGB to the RC for stars in
the mass and metallicity ranges covered by our RBS. Lithium dilution by the
FDU and lithium evolution to the RGB’s tip is modelled by MESA. The potential
episode of Li enrichment of the atmosphere resulting from the He-core flash is
not included. Also, AGB stars experiencing the third dredge-up and possible Li
enrichment are neither included in selection of giants from our RBS nor in our
use of the MESA modules.
Our initial interest in the MESA predictions for Li abundances focussed on
predicted Li abundances for red giants up to the RGB tip and their comparison
with the observations over the mass and metallicity ranges adequately sampled
by the RBS (Figure 7). Such a comparison is also a resource with which to
search for non-standard episodes of Li enrichment (or destruction) such as may
occur on the RGB or by engulfment of planetary material (Alexander, 1967).
### 4.1 Predicted Li abundances to the RGB tip
In pursuit of the predicted Li abundances in giants, we began with the MESA
model for a solar mass and solar metallicity model assembled by Schwab
(2020).555Schwab’s MESA model is publicly available on Zenodo at
https://doi.org/10.5281/zenodo.3960434. Based on our needs, we made some
changes to Schwab’s inlist. The prescription now includes phenomena like
rotation, semi-convection, etc., along with additional resolution
controls.666The source files for our solar mass and metallicity MESA model
along with the outputs for all the cases studied in this work are publicly
available on Zenodo at https://doi.org/10.5281/zenodo.5086893. For our
calculations, we use MESA version 15140. Convection in MESA is treated using
the mixing length theory (MLT) of convection, and for the current exploration
we use its default formalism from Cox & Giuli (1968). For our models, we use
the MESA’s available recipe for thermohaline mixing adapted from Kippenhahn et
al. (1980). We also use convective premixing (Paxton et al., 2019) which is
specified by the condition use_Ledoux_criterion = .true. and do_conv_premix =
.true. in the code. In MESA models, the strength of the thermohaline mixing is
set by a dimensionless efficiency parameter $\alpha\rm_{th}$ (Paxton et al.,
2013). Thermohaline mixing is initiated when the inversion of the mean
molecular weight creates the LB and the region is formally stable against
convection. Mixing in the interior at this time may result in Li synthesis.
This fresh Li will appear at the giant’s surface if the Li enters the deep
convective envelope. For cases where there is no overlap between the region of
Li synthesis and the convective envelope, the surface Li abundance will be
unaltered at its post-FDU value. Subsequent overlap with the convective
envelope is predicted to generally reduce the surface Li abundance. Another
parameter in MESA modelling is the mixing length ($\alpha\rm_{ML}$) for which
we generally adopt a value of about 1.85 times the pressure scale height.
Semi-convective mixing is also considered with efficiency $\alpha_{\rm sc}$ =
0.1 (Paxton et al., 2013). Our models use the standard MESA pp_and_cno_extras
nuclear network, which includes 25 species and the reactions covering the pp-
chains and CNO-cycles. Our adopted MESA nuclear reaction rates are the same as
used by Schwab (2020). GALAH’s abundances are scaled to the solar composition
from Asplund et al. (2009). We also use the same abundance mixture and related
opacity tables (a09, a09_co and lowT_fa05_a09p with condition
use_Type2_opacities = .true.) from MESA’s kap module. Mass loss is not
considered. A recipe for rotation is included in our models. However, in
general, most of our models are non-rotating, i.e., the initial surface
rotation is set at $\Omega_{\rm int}$/$\Omega_{\rm crit}$ = 0.0, and the
detailed effect of rotation is discussed only for a few cases (see Section
4.1.1). Also, unlike Schwab (2020), our models do not include the ad hoc
flash-induced diffusive mixing in the He core at the RGB tip. (However, if
required, it can be easily turned on in our final prescription also.)
Selection of timestep is very crucial in the evolution of models as it helps
to decide the next timestep and assist in the solution for the new structure
and composition (see Paxton et al., 2011; Lattanzio et al., 2015, for more
details). In MESA it is primarily controlled using parameter
varcontrol_target. In our initial test runs, we used the default minimum value
of varcontrol_target = 10-4 for evolution of solar metallicity models of
masses 0.8 to 2.2 M⊙ in 0.1 M⊙ interval. Thermohaline mixing was present in
all the models except for the 1.8 M⊙ case. We also noticed that adoption of a
slightly higher value (about three times than the default value) of
varcontrol_target resulted in no Li reduction (i.e., missing thermohaline
mixing) in the post-LB phase for masses higher than about 1.2 M⊙ until the
model has already evolved near to the tip of RGB. As a fix to this issue, we
forced smaller timesteps by adopting a lower value of the parameter
time_delta_coeff = 0.75 than its default value of unity. Adoption of this
lower timestep resulted in no significant effect on the Li trends for the
cases where Li reduction near the LB was possible with a larger timestep.
Models are begun in the pre-main sequence phase at the point where the core
first becomes radiative. Approach to zero-age main sequence (ZAMS) is shown by
dotted lines in Figures 9 and 11. Initial Li abundances dependent on
metallicity are consistent with the Li abundances in local stars (Lambert &
Reddy, 2004; Deepak & Reddy, 2020; Randich et al., 2020): A(Li) = 3.57, 3.30,
3.04 and 2.78 for [Fe/H] = $+0.25,0.00,-0.25$ and $-0.50$, respectively. Li
depletion dependent on $\alpha_{\rm ML}$ occurs on the approach to the ZAMS:
for the preferred value $\alpha_{\rm ML}\simeq 1.85$ (see below) depletion is
$\sim 0.3$ and $\sim 0.15$ dex for solar metallicity stars of masses 1.0 and
1.1 M⊙, respectively (Table 1). Evolution is followed along the RGB and
through the He-core flash at the culmination of the RGB and to He-burning in
the RC CHeB giant. We evolve all our models until the central mass fraction of
4He drops below 10-6 to conclude the CHeB phase.
We first study the sensitivity of the model predictions to $\alpha\rm_{ML}$
and $\alpha\rm_{th}$. We evolved 1 M⊙ and solar metallicity models with
$\alpha\rm_{ML}$ = 1.00, 1.85, and 2.70, while keeping $\alpha\rm_{th}$ = 100.
For $\alpha\rm_{th}$, we evolved 1 M⊙ and solar metallicity models with
$\alpha\rm_{th}$ = 50, 100, and 200, while keeping $\alpha\rm_{ML}$ fixed to
1.85. The HR diagram and A(Li) trends as a function of luminosity for all
these cases are shown in Figure 9 from the ZAMS through the RGB tip to the end
of core-He burning. The top panels show the effect of $\alpha\rm_{ML}$, and
the bottom panels illustrate sensitivity to $\alpha\rm_{th}$. An increase in
$\alpha\rm_{ML}$ leads to the expected evolutionary track shifts to higher
temperatures. Higher $\alpha\rm_{ML}$ also leads to increased depletion of Li
during the FDU. The temporal evolution of Li in our model stars is shown in
Figure 10 which suggests that adoption of higher $\alpha\rm_{ML}$ leads to
increased as well as prolonged depletion of Li as the star ascends the RGB. A
closer look at Figure 9 and 10 also suggests that for a given set of input
parameters, models with lower $\alpha_{\rm ML}$ experience slightly delayed Li
reduction at the LB in comparison to models with higher $\alpha_{\rm ML}$.
Solar mass and metallicity models with $\alpha_{\rm ML}$ = 1.00, 1.85, and
2.70 starts experiencing post-LB Li reduction after about 29, 17, and 9 Myrs,
respectively, from the onset of LB. However, for all three cases, the actual
timestep during LB’s evolution ranges from about 0.1 to 0.2 million years
(Myr) and further decreases as the star evolve towards the tip of RGB. These
timesteps are very small compared to the duration of LB and delays in Li
reduction. The delay in the post-LB Li reduction is sensitive to the timestep
and decreases with a decrease in the timestep. A test run with smaller
timestep by adopting time_delta_coeff = 0.50 and 0.25 (which resulted in a
timestep of about 0.1 and 0.05 Myr, respectively) for a solar mass model with
$\alpha_{\rm ML}$ = 1.00 resulted in delays of about 8 and 6 Myr,
respectively, along with almost tripling the computation time. It may be of
interest to note that the duration of the LB depends on the assumed value of
$\alpha_{\rm ML}$ and is longer for smaller values of $\alpha_{\rm ML}$.
Similar is the case for overall age of the stars i.e. stars with lower
$\alpha_{\rm ML}$ evolves more slowly (Figure 10). The evolutionary tracks are
insensitive to $\alpha_{\rm th}$, but Li depletion commencing at the LB is, as
anticipated, dependent on $\alpha_{\rm th}$.
To obtain model predictions for various masses, we evolve solar metallicity
models stars of masses 0.8 to 1.6 M⊙ in steps of 0.1 M⊙ and 1.6 to 2.0 in
steps of 0.2 M⊙ from the ZAMS to the RGB tip. The role of metallicity in the
evolution of A(Li) during the RGB was explored by 1 M⊙ models with
metallicities Z = 0.0045, 0.0080, 0.0143, and 0.0254, which are equivalent to
[Fe/H] = -0.50, -0.25, 0.00 and 0.25 dex, respectively. For all models, we
used $\alpha\rm_{ML}$ = 1.85, $\alpha\rm_{th}$ = 100 and initial surface
rotation $\Omega_{\rm int}$/$\Omega_{\rm crit}$ = 0. Changes in the Li
abundance are shown in the top panel of Figure 11 for the series of solar
metallicity models. Metallicity dependence of the Li abundance evolution is
shown in the bottom panel of Figure 11 for 1 M⊙ models. The initial
temperature, luminosity and $\it A$(Li) along with their values at ZAMS, at
the onset of LB and at RGB tip for models in Figure 11 are tabulated in Table
1.
Conditions promoting thermohaline mixing begin at a luminosity around the LB.
Surface Li abundance is affected by Li destruction and formation in the
interior and by the connection between the interior and the outer convective
envelope. Prescriptions for surface Li on the RGB predict in general an
appreciable loss of Li beginning around the LB’s luminosity, see examples in
Figures 9 and 11. A decline of Li on the ascent of the RGB results when the
convective envelope taps the region of Li synthesis. The predicted Li
abundance is necessarily sensitive to the detailed modelling of thermohaline
mixing, as well as stellar mass and composition. Lattanzio et al. (2015)
discuss Li abundance predictions for a 1.25 M⊙ solar metallicity model given
by five different codes. Across these codes, reduction of the Li abundance
post-LB varies considerably for comparable mixing coefficients. An increase in
A(Li) at luminosities just below the RGB tip is predicted under some
conditions for some codes. (Late (small) increases in Li abundance are seen in
Figure 9 for the 0.8 M⊙ model with [Fe/H] = 0 and the 1.0 M⊙ models with
[Fe/H] = 0.) Such late small increases of Li will not be detectable in DR3 and
similar surveys of K giants. In Figure 9 evolution of A(Li) is continued
beyond the He-core flash to the termination of He core burning. It is at the
He-core flash followed by a slight Li decrease as a RC giant. (Li synthesis at
the He-core flash is not included in our models.)
A goal of our study is to confront the MESA models with the HR diagrams and
luminosity-A(Li) trends displayed in Figure 7 where in all panels, we
distinguish between Li abundance measurements and upper limits. Within the
constraints of MESA models, it is apparent that the value of $\alpha_{\rm ML}$
may be determined independently of $\alpha_{\rm th}$ from an observed HR
diagram. Then, determination of the best fit to the luminosity-A(Li) trend for
a given mass-metallicity combination may proceed by adjustment of $\alpha_{\rm
th}$. But, perhaps, the most striking observation is that the luminosity-A(Li)
trend is very similar across all (M⊙,[Fe/H]) pairings well sampled by the RBS,
but the MESA predictions in Figure 9 and 11 show a diverse set of
luminosity-A(Li) trends. Ahead of detailed comparisons, it is apparent that
the fits of MESA Li predictions to observations in Figure 7 will range in
quality. Notably lacking among the observed trends in Figure 7 are examples
matching predictions where the Li abundance is effectively constant over the
RGB from completion of the FDU through the LB’s luminosity to the RGB’s
tip.777The mass and metallicity trends shown in Figure 11 are qualitatively
similar to those provided by Charbonnel & Lagarde (2010) from the STAREVOL-
GENEVA Code.
Investigation of mixing on the RGB is well worthy of study because
observations of Li on the RGB end at a luminosity of $\log(L/{\rm
L_{\odot}})\sim 2.2$ well short of the RGB tip at $\log(L/{\rm L_{\odot}})\sim
3.4$. Thus, inference of the Li abundance at the RGB’s tip involves a
substantial extrapolation from the known Li abundances. It is the Li abundance
at the tip which sets the low bound to the Li abundance among RC clump giants,
i.e., the RC giants which did not experience Li addition as a result of the
He-core flash.
#### 4.1.1 Constraining thermohaline mixing from Li predictions for red
giants
To investigate thermohaline mixing in red giants from the MESA models, we
start with the sample of giants with mass $M/{\rm M_{\odot}}$ = 1.10 $\pm$
0.02 and [Fe/H] = 0.00 $\pm$ 0.10. As discussed above, the evolutionary track
is sensitive to $\alpha_{\rm ML}$ and insensitive to $\alpha_{\rm th}$. A
model with $\alpha_{\rm ML}=1.85\pm 0.05$ provides a good match to the
observations (Figure 12). Li abundances for $\alpha_{\rm th}=50,100,150$ and
200 were predicted. Predictions and observations are compared in the bottom
panels of Figure 12.
Predicted A(Li) throughout the observed luminosity range is higher than the
observed value. This offset, apparently independent of luminosity, may be
negated by subtracting 0.6 dex from observed Li abundances, as achieved in the
bottom panel of Figure 12 showing a fair fit between predicted and observed Li
abundances. The short segment of the RGB above the LB with observed Li
abundances (after application of the offset) is fitted with $\alpha_{\rm
th}\sim 100\pm 50$. When predictions of Li abundances are extrapolated to the
RGB’s tip, a Li abundance A(Li) $\simeq-1.0\pm 0.3$ is obtained. Setting aside
the question of the $-0.6$ offset in Li abundance, the final match of
prediction to observation might seem acceptable. Two points deserve attention.
First, the decline of observed Li abundances on the RGB does not clearly show
the abrupt change predicted to occur at the LB’s luminosity. However, as
starkly shown by the bottom panel in the figure, the RGB at luminosities below
the LB is poorly represented by giants with Li detections and, thus, the true
slope may be ill determined.
Origins of the above Li abundance offset may be imagined. (Also, the presence
of Li upper limits among similar giants with Li abundances shows processes
actively depleting Li are common among these stars, as giants or in earlier
evolutionary phases.) Li in a main-sequence star survives in a very thin skin
at the surface. Should depletion occur via the skin’s base (prior to the FDU),
the Li abundance post-FDU will be reduced from its predicted value. Unless the
Li-containing skin is convective, the depletion of Li will not be seen in the
main sequence star yet revealed in the post-FDU giant. Similarly, mass loss
from the surface of the main-sequence star may not affect its Li surface
abundance but will be revealed by the FDU. There are examples of main-sequence
stars with reduced Li abundances. The Sun is a notable example with an
abundance of about 2.3 dex less than its initial abundance. Surely, the most
extraordinary examples belong to the so-called Li-dip discovered by Boesgaard
& Tripicco (1986) in the Hyades at a mass of $M\sim 1.4{\rm M_{\odot}}$. This
dip shifts to lower masses with decreasing metallicity (Balachandran, 1990).
Chen et al. (2001) map this decline: at [Fe/H] = 0.0 and -0.50, the mass
corresponding to the dip is $M\simeq$ 1.40 M⊙ and 1.26 M⊙, respectively,
suggesting that the Li-dip is not the primary origin for the Li offset
required in Figure 12. Severe, even moderate, reductions of Li abundance in
main sequence stars result in non-detection of Li in giants such that
reductions cannot be quantified. It is curious that when well-populated with
Li abundance measurements, the A(Li) - luminosity relations for the RGB in
Figure 7 have a smooth outer boundary suggesting uniform initial Li abundances
and effectively identical treatment of Li from the main sequence to the post-
FDU giant. The other boundary defined by Li abundance limits from non-
detection of the Li i 6707 Å line may be ragged and arises from stars which
experienced Li loss prior to the RGB.888 It may be worth remarking that Li
with its survival limited to a main-sequence star’s outer skin differs greatly
in its response to the FDU from the C, N and O elemental and isotopic
abundances traditionally identified as measures of the FDU.
To understand the effect of rotation, we evolved a 1.1 M⊙ solar metallicity
model with initial surface rotation $\Omega_{\rm int}$/$\Omega_{\rm crit}$ =
0.00, 0.05, and 0.10 which are equivalent to initial surface rotation of about
0, 15, and 30 kms-1, respectively. A comparison of predictions with the
observational data is shown in Figure 13. Predicted HRDs for both rotating and
non-rotating model are similar except for slight disagreements at the RC and
the RGB’s tip. (HRDs of the rotating and non-rotating models also disagree at
the main-sequence, but not shown here.) Rotating models show slightly lower
luminosity and slightly higher temperature for the RC star compared to non-
rotating models. This behaviour of rotating models at the RC is in contrast to
observations and suggests that giants post-He-Core-flash have small or no
rotation. The RGB’s tip for rotating models lies at a slightly lower
luminosity compared to non-rotating models (see the bottom panel of Figure
13). The figure suggests that rotating models experience smaller Li depletion
during the FDU compared to non-rotating models. However, MESA’s implementation
of rotation-induced mixing is but one realization of this complex physical
process - see Charbonnel & Lagarde (2010) for an alternative prescription and
different Li depletions.
Other samples in Figure 7 from the RBS were studied to compare predicted with
observed HR diagrams and the luminosity-A(Li) trends. Predictions comparable
to observations can be produced with small acceptable changes to $\alpha_{\rm
ML}$ and $\alpha_{\rm th}$. As a test of a fit to a metal-poor selection, we
consider the sample for ($M$,[Fe/H]) = ($1.10\pm 0.02{\rm M_{\odot}},-0.50\pm
0.10$) - see Figure 14. The HR diagram is well fit with a slightly lower
$\alpha_{\rm ML}=1.80\pm 0.05$. For this sample, the predicted luminosity of
the LB falls slightly above the predicted/observed luminosity of the RC. The
LB is not obvious from the observed HR diagram because it is merged with the
RC. The middle panel of Figure 14 shows a mildly satisfactory match between
observed and predicted A(Li)-luminosity trends. An additional offset of about
$-0.3$ to the predicted A(Li) is required to get a better fit. The predicted
Li-luminosity trend with a sharp break in slope at the LB’s luminosity seems a
poor match to the observed trend, as set by giants with Li abundance
detections. In contrast to Figure 12, Li detections span the RGB from its base
to the limiting luminosity. The short segment of the RGB above the LB with
observed Li abundances (after application of the offset) is fitted with a
slightly higher $\alpha_{th}$ $\sim$ 150 $\pm$ 50\. Extrapolation of predicted
Li trends to the RGB tip gives a Li abundance A(Li) $\simeq$ -0.8 $\pm$ 0.3 at
the RGB tip.
As a representative sample of higher mass giants, we consider the ($M$,[Fe/H])
= ($1.50\pm 0.02{\rm M_{\odot}},-0.25\pm 0.10$) selection - see Figure 15. The
rather sparse HRD is again well fit with $\alpha_{\rm ML}$ = 1.95 $\pm$ 0.05.
The LB is not prominent in the observed HR diagram. The predicted luminosity
of the LB falls slightly above the predicted/observed luminosity of the RC.
Due to the small sample size, especially very few giants with a Li detection,
the observed A(Li) trend is not very well defined by giants more luminous than
the LB. An offset of $-0.2$ to the Li abundance level is required to match the
luminosity-(Li) trend for giants less luminous than the RC. The predicted
trend Li abundance with luminosity again changes slope at the LB’s luminosity
and may not match the smoother observed trend. The segment of giants more
luminous than the RC suggests $\alpha_{\rm th}\simeq$ 300 $\pm$ 100, a value
almost three times higher than required for 1.1 M⊙ stars. Extrapolation of
predicted Li trends to the RGB’s tip gives a Li abundance A(Li) $\simeq$
$-0.1\pm 0.3$.
Final results from detailed modeling for the above three (M, [Fe/H]) cases are
summarized in Table 2.
Figure 16: Left panel: HR diagram showing tracks for RGB phase for solar
metallicity (Z = 0.0143) models of masses from 0.8 to 2.2 in steps of 0.2 M⊙.
Right panel: RGB tracks for solar mass models of metallicity Z = 0.0045,
0.0080, 0.0143, and 0.0254, which is equivalent to [Fe/H] = -0.50, -0.25, 0.00
and 0.25, respectively. Shown tracks are for models with $\alpha_{\rm ML}$ =
1.85 and $\alpha_{\rm th}$ = 100. The luminous end for each of the track marks
the position of RGB tip. Also, in each of the case, green blob mark the
position of LB.
### 4.2 MESA predictions for the red clump
Our MESA models include the He-core flash but not the possibility of
accompanying Li synthesis. In low-mass stars, there is not a single flash, but
a series of He-core flashes of declining severity as He-core burning is
established (Thomas, 1967; Despain, 1981; Bildsten et al., 2012; Deepak &
Lambert, 2021) and the star’s luminosity drops from its value at the RGB tip
to the value of the RC.999Iben (2013) provides a detailed discussion of a
solar metallicity 1 M⊙ model through its He-core flashes to the third dredge-
up on the AGB. This decline occurs so quickly that very few stars are expected
to be observed in this approach to the RC. During the RC phase, the luminosity
and effective temperature are little changed for about 100 Myr for a giant of
about 1 M⊙. As He-core burning transitions to He-shell burning with continuing
H-shell burning, luminosity increases and effective temperature decreases in
the transitions from the RC to the AGB. RC evolution according to MESA models
for 0.8, 1.0 and 1.2 M⊙ stars of solar metallicity is shown in Figure 17.
Initial lithium abundances for these RC models are assumed to be the low
values achieved at the tip of the RGB. A small additional decrease in Li
abundance is predicted on the RC and in the initial ascent of the AGB (Figure
10 and 17). An increase of the Li abundance resulting from the third dredge-up
is not expected until the AGB model reaches a luminosity exceeding that of the
RGB tip and an effective temperature of 3000 K or cooler.
Notably, observations and MESA predictions disagree sharply in two regards.
First, the luminosity and effective temperature of the post He-core flash RC
models are in good agreement with observations for the coolest examples in
Figure 5 but cannot account for the RC’s extension to higher temperatures
(Figure 3 and 4) at nearly a constant luminosity and nearly the same mass. For
example, the 1 M⊙ model in Figure 17 at 10 Myr has the luminosity $\log(L/{\rm
L_{\odot}})$ = 1.70 and $T\rm_{eff}$ = 4518 K which compares favourably with
the values for the coolest RC stars in Figure 5. Second, these standard MESA
models necessarily have the very low A(Li) of their RGB progenitors and cannot
account for the highly variable Li enrichment of all - ‘ubiquitous’ enrichment
(Kumar et al., 2020) – RC stars across their temperature range. One may
speculate that these two disagreements between observations and models may
have a single explanation or at least related explanations.
Lithium synthesis at the He core flash was recently investigated by Schwab
(2020) using a 1 M⊙ solar metallicity MESA model. He demonstrated that the
high Li abundance in RC stars may be explained by an event that occurs at the
time of the first and strongest He-core flash. Three versions of Li synthesis
and mixing events shown in Schwab’s Figure 4 give final Li abundances from
about 3.7 to 1.0. The mixing introduced by Schwab is diffusive and defined by
a mixing coefficient. Another parameter is the duration of the mixing event.
Li may also be destroyed in a subsequent weaker He-core flash. Schwab remarks
that excitation of internal gravity waves by turbulent convection during the
core flash may provide the basis for the mixing, which he describes by an ad
hoc mixing coefficient. These exploratory calculations do suggest that the He-
core flash and attendant Li synthesis may account for varying degrees of Li
enrichment seen in RC giants (Figure 4).
## 5 Lithium enrichment at the luminosity bump?
Lithium synthesis at the He-core flash results in an abundance increase
between a star at the RGB tip and the RC. In a parallel endeavour, one may
search for Li abundance increases in stars at or just following the LB. Theory
and observation, as discussed above, anticipate that Li abundance changes
resulting from the LB are expected to be revealed subsequently as the star
ascends the RGB. In general, the Li abundance post-LB is predicted to be
unchanged or to decline steadily up the RGB. Nonetheless, the history of Li
among giants suggests that one may usefully see if exceptional Li-rich giants
are to be found at about the luminosity of the LB. MESA predictions for the
LB’s location in the HR diagram are shown in Figure 16 for solar composition
models with $\alpha_{\rm ML}$ = 1.8. This figure shows that the LB’s
luminosity increases with stellar mass and crosses the RC’s luminosity at
$M\simeq 1.6{\rm M_{\odot}}$. The LB occurs at somewhat higher luminosities
and higher temperatures for metal-poor stars (Figure 16). The similarity in
the luminosity of RC and LB for low mass giants potentially complicates the
identity of Li enriched stars in a luminosity-A(Li) diagram.
Attribution of Li enrichment directly to the LB seems possible for those
(mass, metallicity) pairings providing a LB luminosity operationally
distinguishable by more than 0.2 dex from the RC’s luminosity. At [Fe/H] = 0,
this condition corresponds to $M\leq 1.2{\rm M}_{\odot}$ for a LB luminosity
less than that of the RC or $M\leq 1.0{\rm M}_{\odot}$ for a LB luminosity
greater than that of the RC. At [Fe/H] $=-0.25$ these limits are reduced by
about $0.2{\rm M_{\odot}}$. Figure 7 shows that only the search among the
samples with a luminosity less than that of the RC is practical for the DR3
survey; far too few giants with $M\geq 1.6{\rm M}_{\odot}$ are included in
DR3.
One suitable sample is shown in Figure 2. The distinct locations of the RC and
LB are well shown. Surely, the outstanding feature in the luminosity- A(Li)
diagram is the wide range of Li abundances at the RC’s luminosity. At the LB’s
luminosity, there are no giants with an obvious Li enrichment; most of the Li
abundances are upper limits. Several panels in Figure 7 confirm this result
for other (mass,metallicity) combinations.
Figure 7 shows that for all (mass, metallicity) combinations, the Li abundance
post-LB declines at a similar rate as the giant evolves beyond the LB. If
significant Li enrichment were to occur at the LB, it may vary even between
stars with identical properties, as it does among RC giants. Then, the width
of the A(Li) – $\log(L/{\rm L_{\odot}})$ trend on the RGB above the LB would
be increased relative to the width below the LB. In principle, this test of Li
enrichment at the LB seems applicable to all (mass, metallicity) combinations
provided that the RGB is well populated. Unfortunately, the presence of many
upper limits to the Li abundances impairs the test. In addition, at
luminosities above the RC and LB, Li enriched RGB stars are difficult to
isolate from evolved Li enriched RC giants.
Our conclusion that Li-rich giants do not occur on the RGB at the LB differs
from remarks made by Martell et al. (2021) from their analysis of Li-rich
giants in the GALAH DR3 survey. They conclude that lithium enrichment is not
limited to RC giants but is also present among an appreciable number of RGB
giants: “red clump stars are 2.5 times as likely to be lithium-rich as red
giant branch stars.” This disagreement with our analysis may arise from
Martell et al.’s use of Kiel diagrams ($T_{\rm eff},\log g)$ where giants are
assigned to either the RGB or the RC using the Bayesian probability estimates
provided in column is_redclump_bstep. Giants with is_redclump_bstep $\geq$ 0.5
and WISE $W_{\rm 2}$ absolute magnitude in the range $|W_{\rm 2}$ $+$ 1.63$|$
$\leq$ 0.80 are classified as RC giants, while giants with is_redclump_bstep
$<$ 0.5 or absolute magnitude outside the range $|W_{\rm 2}$ $+$ 1.63$|$
$\leq$ 0.80 are classified as RGB giants. This results in a classification of
all the luminous giants with $W_{\rm 2}$ $<$ -2.43 or $log(L/{\rm L_{\odot}})$
$\gtrapprox$ 2.1 as RGB giants, which in actuality are expected to be a
mixture of RGB, early CHeB, and post-RC giants. The use of Bayesian estimates,
which vary with luminosity, effective temperature and metallicity, appear to
have introduced some intermixing of the RGB and RC populations. We prefer to
assign evolutionary phases using a giant’s location in the HR diagram
($\log(L/{\rm L_{\odot}}),T_{\rm eff}$) with respect to the appropriate
evolutionary track. As noted above, there are regions in the HR diagram where
ambiguities may arise depending on the accuracy of the parameters used to
place a giant on the HR diagram, for example, at the confluence of the RC and
the RGB.
In summary, Li enrichment at the LB is not detected in the giants well
represented by DR3. An upper limit to the Li abundance increase of about 0.3
dex is suggested by visual inspection of Figure 7. This may provide a novel
challenge to the theoretical modelling of thermohaline mixing at the LB.
Figure 17: Variation of luminosity, Teff and A(Li) as a function of time (in
million years) during the CHeB phase for solar-metallicity model stars of
masses 0.8, 1.0 and 1.2 M⊙ with $\alpha_{\rm ML}$ = 1.85 and $\alpha_{\rm th}$
= 100. For the time axis, a base-10 log scale is used to highlight the fact
that CHeB giants spend almost their entire life in a relatively stable state
except for the brief He-flashing region, which lasts for about 1-2 Myr from
the He-core flash. In the CHeB phase, giants also experience additional Li
depletion. This Li depletion starts approximately about 5 Myr after the He-
core flash for a solar mass and solar metallicity star.
## 6 Compositions of red clump giants
The chemical composition of Li enriched giants has been studied by Deepak et
al. (2020) to a great extent based on the GALAH DR2 and then by Martell et al.
(2021) based on data from the GALAH DR3. The conclusion is clear: compositions
of Li-rich giants are not distinguishable from other giants. Here, attention
is returned specifically to the RC giants. All of these stars have been
affected by the He-core flash, which led to alterations of the Li abundance by
an amount varying from small to substantial. In examination of the [X/Fe]
versus [Fe/H] plots, the emphasis here is on elements affected by He-burning
and $s$-processing. The earlier conclusion about compositions of Li enriched
and normal giants is confirmed.
Unfortunately, elements provided by the GALAH survey are not well suited to
probing surface contamination by products of He burning mixed to the surface
during the short-lived episode of He-core flashes or the longer period of He-
core burning. Just 229 giants in the RC luminosity strip (with $\log(L/\rm
L_{\odot})$ of 1.6-1.9) have a determined C abundance. Nitrogen, which might
be affected via associated H-burning CN-cycling, is not provided in the GALAH
survey. Oxygen, which is well represented in this survey, shows the [O/Fe]
versus [Fe/H] relation expected for thin-disk giants with no alterations
dependent on Li abundance. This suggests that, although internal He-burning
may have altered surface Li abundances, oxygen which is possibly a direct
product of He-burning has not been altered.
Coupling of $s$-processing with He-burning is a theoretical possibility. GALAH
provides 11 elements from Rb to Sm as at least partial monitors of an
$s$-process and Eu as a measure of the $r$-process. For four elements - Rb,
Sr, Mo and Ru - abundances are sparsely reported. Among the RC giants, there
is no evidence that abundances of these 11 elements differ according to their
Li abundances or differ at all from the [X/Fe] versus [Fe/H] trends expected
for the thin disk.101010The [X/Fe] versus [Fe/H] for X = Sr, Mo and Ru show an
odd bifurcation. A majority of the measurements provide [X/Fe] $\simeq 0$, as
found for other $s$-process dominated elements, but a minority of giants
suggest a second trend of increasing [X/Fe] with decreasing [Fe/H] attaining
[X/Fe] $\sim+1$ at [Fe/H] $\sim-0.4$. Is this exceptional behaviour a
reflection of systematic errors, perhaps linked to misidentification of lines?
In brief, elements examined across the RBS sample of RC giants show no
abundance anomalies except for Li, but this conclusion can not yet be extended
to principal products of He-burning (C and N and their isotopes) for lack of
observations.
## 7 Concluding remarks
Since their discovery in 1982 by Wallerstein & Sneden, Li-rich giants have
occasioned considerable attention by both observers and theoreticians
searching for the evolutionary phases at which a stellar atmosphere’s Li
abundance may be increased by up to several dex. For red giants considered as
single stars, the search for internal lithium production has been directed at
two evolutionary phases in low mass giants (say, $M\lesssim 2{\rm
M_{\odot}}$): the luminosity bump on the RGB and the He-core flash. It is now
clear that an overwhelming majority of lithium-rich red giants are He-core
burning giants, members of the so-called red clump, featuring lithium
synthesized by the Cameron-Fowler mechanism during (presumably) the He-core
flashes occurring in the RC star’s predecessor at the RGB’s tip. Detectable Li
synthesis is not detectable at the luminosity bump. The preeminence of the
idea that the majority of lithium-rich giants are He-core burners ends a long
debate about the relative roles of the red clump and the luminosity bump. (The
role of binary stars in creating Li-rich giants has been examined. Casey et
al. (2019) proposed that Li was synthesized in a RGB giant when Li synthesis
was triggered by perturbations arising from a companion star. The observation
that the binary frequency for Li-rich giants resembles that for Li normal
giants Jorissen et al. (2020) is a challenge to the universal viability of
this binary hypothesis.)
The conclusion about the dominance of the He-core flash in creating Li-rich
giants was strengthened by Kumar et al.’s (2020) use of the GALAH DR2 survey.
Kumar et al. noted that the population of the Li-rich population was at the
luminosity of the red clump with Li abundances A(Li) spanning from about
$-0.5$ to $+3.7$. Red clump residents have evolved from the RGB and most
likely from a star at the RGB’s luminosity tip. Thus, the lithium abundance at
the RGB’s tip provides the baseline for a clump giant. Kumar et al. inferred
by extrapolation of the GALAH abundances for the RGB that the abundance at the
tip was A(Li) $\sim-0.9$. Thus, this baseline and the Li abundance range
$-0.5$ to $+3.7$ for RC giants led Kumar et al. to a conclusion that all RC
residents were enriched in lithium: across the range of Li abundances, the
enrichment is slight to dramatic. Kumar et al. referred to lithium enrichment
as ‘ubiquitous’ among red clump giants. Our probing of red clump giants in the
GALAH DR3 endorses Kumar et al.’s conclusions about the ubiquity of lithium
enrichment among RC giants.
MESA models do not fully account for the run of Li abundance along the RGB.
Across the sample examined here, the maximum predicted Li abundances of RGB
giants are about 0.2 - 0.6 dex greater than observed, which suggests a loss of
Li within the main sequence star prior to the RGB. Loss of Li is also
suggested by the absence of detectable Li in many RGB giants. All mass-
metallicity samples of RGB giants exhibit a smooth decrease of Li abundance
with increasing luminosity, but the MESA predictions provide for a steeper
decline of Li abundance at luminosities above that of the LB. A possible
resolution of this discrepancy is to invoke missing ‘physics’ in the models,
such as rotation-induced mixing.
Our analysis shows that the lithium enriched population peaks at a mass
$M\simeq$ 1.1 M⊙ and metallicity [Fe/H] $\simeq-0.2$. These values reflect the
mass and metallicity distributions of the GALAH population as a whole. The
lithium richest and lithium poorest stars in a mass-metallicity combination
may appear at the same effective temperature and at the same luminosity. The
ubiquity of Li enrichment among RC giants demands redefinition and restatement
of the frequency of Li-rich giants among a population, previously stated to be
at the few per cent level. If ubiquity is the correct description for RC
giants, all such giants with mass $M\simeq$ 1.1 M⊙ and metallicity [Fe/H]
$\simeq-0.2$ have experienced the addition of fresh Li to their atmosphere.
Future investigations exploring red giants beyond the masses and metallicities
well covered by the RBS are awaited with interest.
MESA models fail to account for observations of RC giants. (Li synthesis was
not included in our models.) There is an observed spread in effective
temperature (and Li abundance) for giants of a given mass and metallicity. But
models predict He-core burning to occur at essentially a single effective
temperature before the star evolves to the AGB phase at lower temperatures and
higher luminosity. This observed spread in effective temperature and an
uncorrelated spread in Li abundance suggests that the sequence of He-core
flashes initiating the RC phase at the RGB’s tip may result in internal
adjustments not considered by MESA (and similar models). The primary search
for the origin for the effective temperature spread among red clump giants
must be led by theoreticians. This issue, however, may be explored by
examining compositions of red clump giants across their full spread in
effective temperature. No abundance anomalies are shown by the GALAH
compositions, but key nuclides are very incompletely reported (as C) or not
reported at all (as N).
The luminosity bump has been offered as an opportunity to enhance the surface
lithium abundance. Guided by the MESA predictions on the bump’s location, we
searched for a direct association with Li enriched giants. Across the mass and
metallicity combinations providing HRDs with well-populated RGBs, we found no
evidence for the presence of Li enriched giants at the LB: an upper limit of
0.3 dex to the increase in Li abundance is set. This provides an interesting
limit with which to test calculations of lithium synthesis occurring at the
luminosity bump. Such synthesis does directly affect the Li abundance
development on the RGB above the LB, where without obvious exception, the Li
abundances are less than at pre-LB RGB values.
The saga of the origins of lithium enrichment in low mass red giants has now
transitioned to the challenge of providing a clear picture of Li synthesis at
the He-core flashes in luminous RGB giants. Clarification of the
identification of Li enriched giants is a notable achievement, but questions
concerning the quantitative accounting for giants’ lithium abundances remain,
most especially difficult questions about the He-core flashes and their effect
on the structure of red clump giants which may in many cases depart markedly
from the clump giants described by standard evolution calculations. Although
direct observations of Li abundance at the LB have not been found,
observations of Li and light nuclides (C, N and O) affected by thermohaline
and other forms of mixing along the full RGB deserve pursuit before the
understanding of RGB giants may be deemed complete. Undoubtedly, as the
history of lithium in observational astronomy reminds us, other puzzles about
the lithium abundances among giants as single and binary stars will occur
unexpectedly.
## Acknowledgments
We are indebted to the anonymous reviewer and to John Lattanzio for helpful
correspondence and, most especially, to the GALAH team and their publication
of the data release DR3. Without the catalogues associated with DR3, our
analysis would have been impossible. A similar debt is due to Bill Paxton and
his colleagues for developing the MESA suite of programmes. This work has also
made use of NASA’s Astrophysics Data System. The data for the GALAH survey are
acquired through the Australian Astronomical Observatory, under programs:
A/2013B/13 (The GALAH pilot survey); A/2014A/25, A/2015A/19, A2017A/18 (The
GALAH survey phase 1); A2018A/18 (Open clusters with HERMES); A2019A/1
(Hierarchical star formation in Ori OB1); A2019A/15 (The GALAH survey phase
2); A/2015B/19, A/2016A/22, A/2016B/10, A/2017B/16, A/2018B/15 (The HERMES-
TESS program); and A/2015A/3, A/2015B/1, A/2015B/19, A/2016A/22, A/2016B/12,
A/2017A/14 (The HERMES K2-follow-up program). We acknowledge the traditional
owners of the land on which the AAT stands, the Gamilaraay people, and pay our
respects to elders past and present. This paper includes data that has been
provided by AAO Data Central (https://datacentral.org.au/). We also
acknowledge using the Delphinus and Fornax server at IIA, Bengaluru, to evolve
our model stars.
## ORCID iDs
Deepak: https://orcid.org/0000-0003-2048-9870
David L. Lambert: https://orcid.org/0000-0003-1814-3379
## Data Availability
The observational data underlying this article are publicly available in the
Data Central at
https://cloud.datacentral.org.au/teamdata/GALAH/public/GALAH_DR3/. The source
files for our solar mass and metallicity MESA model along with the outputs for
all the cases studied in this work are publicly available on Zenodo at
https://doi.org/10.5281/zenodo.5086893.
## References
* Alexander (1967) Alexander J. B., 1967, The Observatory, 87, 238
* Asplund et al. (2009) Asplund M., Grevesse N., Sauval A. J., Scott P., 2009, ARA&A, 47, 481
* Balachandran (1990) Balachandran S., 1990, ApJ, 354, 310
* Bildsten et al. (2012) Bildsten L., Paxton B., Moore K., Macias P. J., 2012, ApJ, 744, L6
* Boesgaard & Tripicco (1986) Boesgaard A. M., Tripicco M. J., 1986, ApJ, 302, L49
* Brown et al. (1989) Brown J. A., Sneden C., Lambert D. L., Dutchover E. J., 1989, ApJS, 71, 293
* Buder et al. (2021) Buder S., et al., 2021, MNRAS, 506, 150
* Cameron (1955) Cameron A. G. W., 1955, ApJ, 121, 144
* Cameron & Fowler (1971) Cameron A. G. W., Fowler W. A., 1971, ApJ, 164, 111
* Casey et al. (2016) Casey A. R., et al., 2016, MNRAS, 461, 3336
* Casey et al. (2019) Casey A. R., et al., 2019, ApJ, 880, 125
* Charbonnel & Lagarde (2010) Charbonnel C., Lagarde N., 2010, A&A, 522, A10
* Chen et al. (2001) Chen Y. Q., Nissen P. E., Benoni T., Zhao G., 2001, A&A, 371, 943
* Cox & Giuli (1968) Cox J. P., Giuli R. T., 1968, Principles of stellar structure. Gordon & Breach, New York
* Deepak & Lambert (2021) Deepak ., Lambert D. L., 2021, MNRAS, 505, 642
* Deepak & Reddy (2019) Deepak ., Reddy B. E., 2019, MNRAS, 484, 2000
* Deepak & Reddy (2020) Deepak ., Reddy B. E., 2020, Mem. Soc. Astron. Italiana, 91, 134
* Deepak et al. (2020) Deepak ., Lambert D. L., Reddy B. E., 2020, MNRAS, 494, 1348
* Despain (1981) Despain K. H., 1981, ApJ, 251, 639
* Eggleton et al. (2008) Eggleton P. P., Dearborn D. S. P., Lattanzio J. C., 2008, ApJ, 677, 581
* Gaia Collaboration et al. (2016) Gaia Collaboration et al., 2016, A&A, 595, A1
* Gao et al. (2019) Gao Q., Shi J.-R., Yan H.-L., Yan T.-S., Xiang M.-S., Zhou Y.-T., Li C.-Q., Zhao G., 2019, ApJS, 245, 33
* Iben (2013) Iben I. J., 2013, Stellar Evolution Physics, Volume 2: Advanced Evolution of Single Stars. Cambridge University Press
* Jorissen et al. (2020) Jorissen A., Van Winckel H., Siess L., Escorza A., Pourbaix D., Van Eck S., 2020, A&A, 639, A7
* Karakas (2010) Karakas A. I., 2010, MNRAS, 403, 1413
* Kippenhahn et al. (1980) Kippenhahn R., Ruschenplatt G., Thomas H. C., 1980, A&A, 91, 175
* Kumar et al. (2020) Kumar Y. B., Reddy B. E., Campbell S. W., Maben S., Zhao G., Ting Y.-S., 2020, Nature Astronomy, 4, 1059
* Lagarde et al. (2011) Lagarde N., Charbonnel C., Decressin T., Hagelberg J., 2011, A&A, 536, A28
* Lambert & Reddy (2004) Lambert D. L., Reddy B. E., 2004, MNRAS, 349, 757
* Lattanzio et al. (2015) Lattanzio J. C., Siess L., Church R. P., Angelou G., Stancliffe R. J., Doherty C. L., Stephen T., Campbell S. W., 2015, MNRAS, 446, 2673
* Martell et al. (2021) Martell S. L., et al., 2021, MNRAS, 505, 5340
* McKellar (1940) McKellar A., 1940, PASP, 52, 407
* Paxton et al. (2011) Paxton B., Bildsten L., Dotter A., Herwig F., Lesaffre P., Timmes F., 2011, ApJS, 192, 3
* Paxton et al. (2013) Paxton B., et al., 2013, ApJS, 208, 4
* Paxton et al. (2015) Paxton B., et al., 2015, ApJS, 220, 15
* Paxton et al. (2018) Paxton B., et al., 2018, ApJS, 234, 34
* Paxton et al. (2019) Paxton B., et al., 2019, ApJS, 243, 10
* Randich et al. (2020) Randich S., et al., 2020, A&A, 640, L1
* Sanford (1950) Sanford R. F., 1950, ApJ, 111, 262
* Schwab (2020) Schwab J., 2020, ApJ, 901, L18
* Sharma et al. (2018) Sharma S., et al., 2018, MNRAS, 473, 2004
* Singh et al. (2019) Singh R., Reddy B. E., Bharat Kumar Y., Antia H. M., 2019, ApJ, 878, L21
* Singh et al. (2021) Singh R., Reddy B. E., Campbell S. W., Kumar Y. B., Vrard M., 2021, ApJ, 913, L4
* Smith & Lambert (1989) Smith V. V., Lambert D. L., 1989, ApJ, 345, L75
* Thomas (1967) Thomas H. C., 1967, Z. Astrophys., 67, 420
* Wallerstein & Sneden (1982) Wallerstein G., Sneden C., 1982, ApJ, 255, 577
* Yan et al. (2018) Yan H.-L., et al., 2018, Nature Astronomy, 2, 790
* Yan et al. (2021) Yan H.-L., et al., 2021, Nature Astronomy, 5, 86
## Appendix A Tables summarizing results from the MESA models
Table 1: The initial temperature, luminosity and $\it A$(Li) along with their values at ZAMS, at the onset of LB and at the tip of the RGB for the models shown in Figure 11 and 16. $M$/M⊙ | Z | Temperature (K) at | $\log$ (L/L⊙) at | A(Li) at
---|---|---|---|---
| | PMS | ZAMS | LB | RGB-tip | PMS | ZAMS | LB | RGB-tip | PMS | ZAMS | LB | RGB-tip
0.8 | 0.0143 | 4082 | 4894 | 4422 | 3028 | -0.19 | -0.56 | 1.32 | 3.40 | 3.30 | 2.07 | -3.03 | -4.67
0.9 | 0.0143 | 4243 | 5323 | 4430 | 3087 | -0.01 | -0.33 | 1.42 | 3.40 | 3.30 | 2.77 | -0.26 | -2.90
1.0 | 0.0045 | 4681 | 6165 | 4600 | 3454 | 0.36 | 0.07 | 1.76 | 3.36 | 2.78 | 2.77 | 1.28 | -0.50
1.0 | 0.0080 | 4533 | 5936 | 4537 | 3307 | 0.26 | -0.01 | 1.62 | 3.38 | 3.04 | 2.99 | 1.43 | -0.49
1.0 | 0.0143 | 4380 | 5678 | 4438 | 3141 | 0.15 | -0.11 | 1.51 | 3.40 | 3.30 | 3.02 | 0.68 | -1.35
1.0 | 0.0254 | 4262 | 5403 | 4351 | 2940 | 0.06 | -0.21 | 1.37 | 3.40 | 3.57 | 1.87 | -4.92 | -4.49
1.1 | 0.0143 | 4491 | 5966 | 4455 | 3189 | 0.29 | 0.09 | 1.57 | 3.40 | 3.30 | 3.15 | 1.13 | -0.45
1.2 | 0.0143 | 4580 | 6231 | 4469 | 3230 | 0.41 | 0.27 | 1.62 | 3.39 | 3.30 | 3.22 | 1.34 | 0.12
1.3 | 0.0143 | 4651 | 6510 | 4474 | 3267 | 0.53 | 0.43 | 1.68 | 3.39 | 3.30 | 3.26 | 1.44 | 0.50
1.4 | 0.0143 | 4708 | 6863 | 4477 | 3300 | 0.63 | 0.58 | 1.73 | 3.39 | 3.30 | 3.28 | 1.49 | 0.78
1.5 | 0.0143 | 4754 | 7305 | 4486 | 3331 | 0.73 | 0.71 | 1.78 | 3.39 | 3.30 | 3.29 | 1.51 | 0.98
1.6 | 0.0143 | 4789 | 7768 | 4493 | 3361 | 0.82 | 0.83 | 1.82 | 3.39 | 3.30 | 3.29 | 1.53 | 1.13
1.8 | 0.0143 | 4838 | 8628 | 4462 | 3433 | 0.99 | 1.05 | 1.97 | 3.36 | 3.30 | 3.30 | 1.53 | 1.32
2.0 | 0.0143 | 4866 | 9388 | 4494 | 3570 | 1.14 | 1.23 | 2.01 | 3.24 | 3.30 | 3.30 | 1.51 | 1.40
2.2 | 0.0143 | 4878 | 10091 | 4373 | 3886 | 1.28 | 1.39 | 2.26 | 2.89 | 3.30 | 3.30 | 1.50 | 1.46
Table 2: Input parameters for the best fit models along with lithium abundances at the start of the different evolutionary phases for the three (M, [Fe/H]) cases studied in details in Section 4.1.1. Mass | [Fe/H] | Model parameters | $\delta A$(Li) | A(Li) after correction for $\delta A$(Li) at
---|---|---|---|---
(M⊙) | (dex) | Z | $\alpha_{\rm ML}$ | $\alpha_{\rm th}$ | (dex) | PMS | ZAMS | LB | RGB-tip
1.1 | -0.50 | 0.0045 | 1.80 $\pm$ 0.05 | 150 $\pm$ 50 | -0.3 | 2.48 | 2.48 | 0.92 | -0.77 $\pm$ 0.3
1.1 | 0.00 | 0.0143 | 1.85 $\pm$ 0.05 | 100 $\pm$ 50 | -0.6 | 2.70 | 2.55 | 0.53 | -1.05 $\pm$ 0.3
1.5 | -0.25 | 0.0080 | 1.95 $\pm$ 0.05 | 300 $\pm$ 100 | -0.2 | 2.84 | 2.83 | 1.10 | -0.08 $\pm$ 0.3
|
# Does the Heisenberg uncertainty principle apply along the time dimension?
John Ashmead Visiting Scholar, University of Pennsylvania
<EMAIL_ADDRESS>
###### Abstract
Does the Heisenberg uncertainty principle (HUP) apply along the time dimension
in the same way it applies along the three space dimensions? Relativity says
it should; current practice says no. With recent advances in measurement at
the attosecond scale it is now possible to decide this question
experimentally.
The most direct test is to measure the time-of-arrival of a quantum particle:
if the HUP applies in time, then the dispersion in the time-of-arrival will be
measurably increased.
We develop an appropriate metric of time-of-arrival in the standard case;
extend this to include the case where there is uncertainty in time; then
compare. There is – as expected – increased uncertainty in the time-of-arrival
if the HUP applies along the time axis. The results are fully constrained by
Lorentz covariance, therefore uniquely defined, therefore falsifiable.
So we have an experimental question on our hands. Any definite resolution
would have significant implications with respect to the role of time in
quantum mechanics and relativity. A positive result would also have
significant practical applications in the areas of quantum communication,
attosecond physics (e.g. protein folding), and quantum computing.
## 1 Introduction
> “You can have as much junk in the guess as you like, provided that the
> consequences can be compared with experiment.” – Richard P. Feynman [1]
##### Heisenberg uncertainty principle in time
The Heisenberg uncertainty principle in space:
$\Delta x\Delta p\geq 1$ (1)
is a foundational principle of quantum mechanics. From special relativity and
the requirement of Lorentz covariance we expect that this should be extended
to time:
$\Delta t\Delta E\geq 1$ (2)
Both Bohr and Einstein regarded the uncertainty principle in time as essential
to the integrity of quantum mechanics.
At the sixth Conference of Solvay in 1930, Einstein devised the celebrated
Clock-in-a-Box experiment to refute quantum mechanics by showing the HUP in
time could be broken. Consider a box full of photons with a small fast door
controlled by a clock. Weigh the box. Open the door for a time $\Delta t$,
long enough for one photon to escape. Weigh the box again. From the
equivalence of mass and energy, the change in the weight of the box gives you
the _exact_ energy of the photon. If $\Delta E\to 0$, then you can make
$\Delta t\Delta E<1$.
Bohr famously refuted the refutation by looking at the specifics of the
experiment. If you weigh the box by adding and removing small weights till the
box’s weight is again balanced, the process will take time. During this
process the clock will move up and down in the gravitational field, and its
time rate will be affected by the gravitational redshift. Bohr showed that
resulting uncertainty is precisely what is required to restore the HUP in time
[2, 3]. The irony of employing Einstein’s own General Relativity (via the use
of the gravitational redshift) to refute Einstein’s argument was presumably
lost on neither Einstein nor Bohr.
In later work this symmetry between time and space was in general lost. This
is clear in the Schrödinger equation:
$\imath\frac{\partial}{\partial\tau}\psi_{\tau}\left(\vec{x}\right)=H\psi_{\tau}\left(\vec{x}\right)$
(3)
Here the wave function is indexed by time: if we know the wave function at
time $\tau$ we can use this equation to compute the wave function at time
$\tau+\epsilon$. The wave function has in general non-zero dispersion in
space, but is always treated as having zero dispersion in time. In quantum
mechanics “time is a parameter not an operator” (Hilgevoord [4, 5]). Or as
Busch [6] puts it “… different types of time energy uncertainty can indeed be
deduced in specific contexts, but … there is no unique universal relation that
could stand on equal footing with the position-momentum uncertainty relation.”
See also Pauli, Dirac, and Muga [7, 8, 9, 10].
So, we have a contradiction at the heart of quantum mechanics. Symmetry
requires that the uncertainty principle apply in time; experience to date says
this is not necessary. Both are strong arguments; neither is decisive on its
own. The goal of this work is to reduce this to an experimental question; to
show we can address the question with current technology.
##### Order of magnitude estimate
The relevant time scale is the time for a photon to cross an atom. This is of
order attoseconds. This is small enough to explain why such an effect has not
already been seen by chance. Therefore the argument from experience is not
decisive.
But recent advances in ultra-fast time measurements have been extraordinary;
we can now do measurements at even the sub-attosecond time scale [11].
Therefore we should now be able to measure the effects of uncertainty in time,
if they are present.
Further the principle of covariance strongly constrains the effects. In the
same way that the inhabitants of Abbott’s Flatland [12] can infer the
properties of a Sphere from the Sphere’s projection on Flatland, we should be
able to predict the effects of uncertainty in time from the known
uncertainties in space.
##### Operational meaning of uncertainty in time
In an earlier work [13] we used the path integral approach to do
this111Detailed references are provided in the earlier work. For here we note
that the initial ideas come from the work of Stueckelberg and Feynman [14, 15,
16, 17, 18, 19] as further developed by Horwitz, Fanchi, Piron, Land, Collins,
and others [20, 21, 22, 23, 24, 25, 26, 27].. We generalized the usual paths
in the three space dimensions to extend in time as well. No other change was
required. The results were manifestly covariant by construction, uniquely
defined, and consistent with existing results in the appropriate long time
limit.
While there are presumably other routes to the same end, the requirement of
Lorentz covariance is a strong one and implies – at a minimum – that the first
order corrections of any such route will be the same. We can therefore present
a well-defined and falsifiable target to the experimentalists.
##### Time-of-arrival measurements
The most obvious line of attack is to use a time-of-arrival detector. We emit
a particle at a known time, with known average velocity and dispersion in
velocity. We measure the dispersion in time at a detector. The necessary
uncertainty in space will be associated with uncertainty in time-of-arrival.
For instance if the wave function has an uncertainty in position of $\Delta x$
and an average velocity of $v$, there will be an uncertainty of time-of-
arrival of order $\frac{\Delta x}{v}$. We refer to this as the extrinsic
uncertainty.
If there is an intrinsic uncertainty in time associated with the particle we
expect to see an additional uncertainty in time-of-arrival from that. If the
intrinsic uncertainty in time is $\Delta t$, then we would expect a total
uncertainty in time-of-arrival of order $\Delta t+\frac{\Delta x}{v}$. It is
the difference between the two predictions that is the experimental target.
For brevity we will refer to standard quantum mechanics, without uncertainty
in time, as SQM. And we will refer to quantum mechanics with intrinsic
uncertainty in time as TQM.
##### Time-of-arrival in SQM
We clearly need a solid measure of the extrinsic uncertainty to serve as a
reference point. We look at a number of available measures but are forced to
conclude none of the established metrics are entirely satisfactory: they
impose arbitrary conditions only valid in classical mechanics, have free
parameters, or are physically unrealistic. We argue that the difficulties are
of a fundamental nature. In quantum mechanics there cannot really be a crisp
boundary between the wave function detected and the equipment doing the
detection: we must – as a Bohr would surely insist – consider both
simultaneously as part of a single system. However once this requirement is
recognized and accepted, we can work out the rules for a specific setup and
get reasonably solid predictions.
##### Comparison of time-of-arrival in SQM and TQM
With this preliminary question dealt with, the actual comparison of the
results with and without intrinsic uncertainty in time is straightforward. We
first look at a generic particle detector setup in SQM, then look at the
parallel results in TQM. By writing the latter as the direct product of a time
part and a space part, we get a clean comparison. As expected, if there is
intrinsic uncertainty in time then the uncertainty in time-of-arrival is
increased and by a specific amount.
In the non-relativistic limit, the relative increase in the uncertainty in
time-of-arrival is small. But by running the initial wave function through a
single slit in time, as a rapidly opening and closing camera shutter, we can
make the relative increase arbitrarily great.
_We therefore have falsifiability._
##### Implications
Since the predictions come directly from the principle of Lorentz covariance
and the basic rules for quantum mechanics, any definite result – negative or
positive – will have implications for foundational questions in relativity and
quantum mechanics.
If there is in fact intrinsic uncertainty in time, there will be practical
implications for quantum communication, attosecond physics (e.g. protein
folding), and quantum computing. This will open up new ways to carry
information, additional kinds of interaction between quantum particles (forces
of anticipation and regret, resulting from the extension in time), and a
deeper understanding of the measurement problem.
## 2 Time-of-arrival measurements in SQM
There does not appear to be a well-defined, generally accepted, and
appropriate metric for the time-of-arrival in the case of SQM.
The Kijowski metric [28] is perhaps the most popular metric for time-of-
arrival. It has a long history, back to 1974, has seen a great deal of use,
and has some experimental confirmation. However it is based on a deeply
embedded classical assumption which is unjustifiable from a quantum mechanical
viewpoint and which produces inconsistent results in some edge cases. We will
nevertheless find it useful as a starting point.
A conceptually sounder approach is supplied by Marchewka and Schuss, who use
Feynman path integrals to compute the time-of-arrival [29, 30, 31, 32].
Unfortunately their approach has free parameters, which rules it out for use
here.
The difficulties with the Marchewka and Schuss approach are partly a function
of an ad hoc treatment of the boundary between incoming wave function and the
plane of the detector. We are able to deal with this boundary in a systematic
way by using a discrete approach along the space coordinate, then taking the
continuum limit.
Unfortunately while this discrete approach solves the problem of how to handle
the discontinuity at the edge of the detector it does not address the more
fundamental problem: of assuming a crisp boundary between the quantum
mechanical and classical parts of the problem in the first place.
We address this problem by arguing that while the assumption of a crisp
boundary between the quantum mechanical and classical parts of a problem is
not in general justified it is also not in general necessary. If we work
bottom-up, starting from a quantum mechanical perspective, we can use various
ad hoc but reasonable heuristics to determine which parts may be described
with acceptable accuracy using classical mechanics, and where only a fully
quantum mechanical approach will suffice.
### 2.1 Kijowski time-of-arrival operator
> “A common theme is that classical mechanics, deterministic or stochastic, is
> always a fundamental reference on which all of these approaches [to the
> time-of-arrival operator] are based.” – Muga and Leavens [33]
#### 2.1.1 Definition
The most popular time-of-arrival operator seems to be one developed by
Kijowski in 1974 [28]. It and variations on it have seen much use since then
[34, 35, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45]. We used Kijowski as a
starting point in our previous paper: it gives reasonable results in simple
cases. Unfortunately its conceptual problems rule it out for use here.
The main problem is that classical requirements play an essential role in
Kijowski’s original derivation. In particular, Kijowski assumed that for the
part of the wave function to the left of the detector222Throughout we assume
that the particle is going left to right along the $x$ axis; that the detector
is at $x=0$; that the particle starts well to the left of the detector; and
that we do not need to consider the $y$ and $z$ axes. These assumptions are
conventional in the time-of-arrival literature. only components with $p>0$
will contribute; while for the part of the wave function to the right of the
detector only components with $p<0$ will contribute. This condition (which we
will refer to as the classical condition) is imposed on the basis of the
corresponding requirement in classical mechanics, as noted by Muga and Leavens
above.
Figure 1: The classical condition for time-of-arrival measurements
Focusing on the case of a single space dimension, with the detector at $x=0$,
the Kijowski metric is333Equation 122 in [33]. :
$\rho_{d}\left(\tau\right)={\left|{\int\limits_{0}^{\infty}dp\sqrt{\frac{p}{{2\pi
m}}}{e^{-\imath\frac{{{p^{2}}\tau}}{{2m}}}}{\varphi^{\left({left}\right)}}\left(p\right)}\right|^{2}}+{\left|{\int\limits_{-\infty}^{0}dp\sqrt{\frac{{-p}}{{2\pi
m}}}{e^{-\imath\frac{{{p^{2}}\tau}}{{2m}}}}{\varphi^{\left({right}\right)}}\left(p\right)}\right|^{2}}$
(4)
However while the restrictions on momentum may make sense
classically444Amusingly enough, this condition is sometimes violated even in
classical waves: Rayleigh surface waves in shallow water (e.g. tsunamis
approaching land) show retrograde motion of parts of the wave [46]., they are
more troubling in quantum mechanics. It is as if we were thinking of the wave
function as composed of a myriad of classical particles each of which travels
like a tiny billiard ball. This would appear to be a hidden variable
interpretation of quantum mechanics. This was shown to be inconsistent with
quantum mechanics by Bell in 1965 [47] experimentally confirmed [48, 49, 50,
51, 52].
However, Kijowski’s metric has produced reasonable results, see for instance
[53]. Therefore if we see problems with it, we need to explain the successes
as well.
#### 2.1.2 Application to Gaussian test functions
We will use Gaussian test functions (GTFs) to probe this metric. We define
GTFs as normalized Gaussian functions that are solutions to the free
Schrödinger equation. We will look at the results of applying Kijowski’s
metric to two different cases, which we refer to as the “bullet” and “wave”
variations, following Feynman’s terminology in his discussion of the double-
slit experiment [54]. By “bullet” we mean a wave function which is narrowly
focused around the central axis of motion; by “wave” we mean a wave function
which is widely spread around the central axis of motion.
##### Bullets
For the bullet version we will take a GTF which starts at time $\tau=0$,
centered at start position $x_{0}=-d$, with average momentum $p_{0}$, and
dispersion $\sigma_{p}^{2}\ll p_{0}$. These conditions are met in the Muga,
Baute, Damborenea, and Egusquiza [53] reference; they give:
$\delta
v=0.098{{cm}\mathord{\left/{\vphantom{{cm}s}}\right.\kern-1.2pt}s},{v_{0}}=10{{cm}\mathord{\left/{\vphantom{{cm}s}}\right.\kern-1.2pt}s}$
(5)
This is typical of high-energy particles, where the free particle wave
functions will have essentially no negative momentum component. The starting
wave function is:
${{\bar{\varphi}}_{0}}\left(x\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{x}^{2}}}}}{e^{\imath{p_{0}}x-\frac{1}{{2\sigma_{x}^{2}}}{{\left({x+d}\right)}^{2}}}}$
(6)
This is taken to satisfy the free Schrödinger equation:
$\imath\frac{\partial}{{\partial\tau}}{{\bar{\varphi}}_{\tau}}\left(x\right)=-\frac{1}{{2m}}\frac{{\partial^{2}}}{{\partial{x^{2}}}}{{\bar{\varphi}}_{\tau}}\left(x\right)$
(7)
so is given as a function of time by:
$\bar{\varphi}_{\tau}\left(x\right)=\sqrt[4]{\frac{1}{\pi\sigma_{x}^{2}}}\sqrt{\frac{1}{f_{\tau}^{\left(x\right)}}}e^{\imath
p_{0}x-\frac{1}{2\sigma_{x}^{2}f_{\tau}^{\left(x\right)}}\left(x+d-\frac{p_{0}}{m}\tau\right)^{2}-\imath\frac{p_{0}^{2}}{2m}\tau}$
(8)
with dispersion factor $f$ defined by:
$f_{\tau}^{\left(x\right)}\equiv 1+\imath\frac{\tau}{m\sigma_{x}^{2}}$ (9)
The momentum transform is:
${{\hat{\bar{\varphi}}}_{\tau}}\left(p\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{p}^{2}}}}}{e^{\imath
pd-\frac{{{\left({p-{p_{0}}}\right)}^{2}}}{{2\sigma_{p}^{2}}}-\imath\frac{{p^{2}}}{{2m}}\tau}}$
(10)
with $\sigma_{p}\equiv\frac{1}{\sigma_{x}}$. We will use this wave function as
a starting point throughout this work, referring to it as the “bullet wave
function”.
In Kijowski’s notation, there is no right hand wave function since the
particle started on the left. Further since the wave function is by
construction narrow in momentum space we can extend the lower limit in the
integral on the left from zero to negative infinity giving:
${\rho_{d}}\left(\tau\right)={\left|{\int\limits_{-\infty}^{\infty}dp\sqrt{\frac{p}{{2\pi
m}}}\sqrt[4]{{\frac{1}{{\pi\sigma_{p}^{2}}}}}{e^{\imath
pd-\frac{{{\left({p-{p_{0}}}\right)}^{2}}}{{2\sigma_{p}^{2}}}-\imath\frac{{p^{2}}}{{2m}}\tau}}}\right|^{2}}$
(11)
Since the wave function is narrowly focused, we can replace the square root in
the integral by the average:
$\sqrt{\frac{{{p_{0}}+\delta p}}{m}}\approx\sqrt{\frac{{{\text{p}}_{0}}}{m}}$
(12)
which we can then pull outside of the integral:
${\rho_{d}}\left(\tau\right)\approx\frac{{p_{0}}}{m}{\left|{\int\limits_{-\infty}^{\infty}{\frac{{dp}}{{\sqrt{2\pi}}}\sqrt[4]{{\frac{1}{{\pi\sigma_{p}^{2}}}}}{e^{\imath
pd-\frac{{{\left({p-{p_{0}}}\right)}^{2}}}{{2\sigma_{p}^{2}}}-\imath\frac{{p^{2}}}{{2m}}\tau}}}}\right|^{2}}$
(13)
We recognize the integral as simply the inverse Fourier transform of the
momentum space form:
${{\bar{\varphi}}_{\tau}}\left(d\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{x}^{2}}}}}\sqrt{\frac{1}{{f_{\tau}^{\left(x\right)}}}}{e^{\imath{p_{0}}d-\frac{1}{{2\sigma_{x}^{2}f_{\tau}^{\left(x\right)}}}{{\left({d-\frac{{p_{0}}}{m}\tau}\right)}^{2}}-\imath\frac{{p_{0}^{2}}}{{2m}}\tau}}$
(14)
so:
${\rho_{d}}\left(\tau\right)\approx\frac{{p_{0}}}{m}{\left|{{{\bar{\varphi}}_{\tau}}\left(d\right)}\right|^{2}}=\frac{{p_{0}}}{m}{{\bar{\rho}}_{\tau}}\left(d\right)$
(15)
with the probability density being:
${{\bar{\rho}}_{\tau}}\left(d\right)=\frac{{p_{0}}}{m}\sqrt{\frac{1}{{\pi\sigma_{x}^{2}\left({1+\frac{{\tau^{2}}}{{{m^{2}}\sigma_{x}^{4}}}}\right)}}}\exp\left({-\frac{{{\left({{\bar{x}}_{\tau}}\right)}^{2}}}{{\sigma_{x}^{2}\left({1+\frac{{\tau^{2}}}{{{m^{2}}\sigma_{x}^{4}}}}\right)}}}\right)$
(16)
and with the average location in space:
$\bar{x}_{\tau}\equiv-d+{v_{0}}\tau$ (17)
We have the probability density as a function of $x$; we need it as a function
of $\tau$. We therefore make the variable transformation:
$d\tau={v_{0}}dx=\frac{{p_{0}}}{m}dx$ (18)
And rewrite $\tau$ as the average value of $\tau$ plus an offset:
$\tau=\bar{\tau}+\delta\tau,\bar{\tau}\equiv\frac{d}{{v_{0}}}$ (19)
We rewrite the numerator in the density function in terms of $\delta\tau$:
$\bar{\rho}_{\tau}\left(d\right)\approx\sqrt{\frac{1}{\pi\sigma_{x}^{2}\left|f_{\tau}^{\left(x\right)}\right|^{2}}}e^{-\frac{\left(v_{0}\delta\tau\right)^{2}}{\sigma_{x}^{2}\left|f_{\tau}^{\left(x\right)}\right|^{2}}}$
(20)
We may estimate the uncertainty in time by expanding around the average time.
Since the numerator is already only of second order in $\delta\tau$ we need
only keep the zeroth order in $\delta\tau$ in the denominator:
$\sigma_{x}^{2}\left|f_{\tau}^{\left(x\right)}\right|^{2}=\sigma_{x}^{2}+\frac{\tau^{2}}{m^{2}\sigma_{x}^{2}}\approx\frac{\tau^{2}}{m^{2}\sigma_{x}^{2}}=\frac{\left(\bar{\tau}+\delta\tau\right)^{2}}{m^{2}\sigma_{x}^{2}}\approx\frac{\bar{\tau}^{2}}{m^{2}\sigma_{x}^{2}}$
(21)
giving:
$\bar{\rho}_{\delta\tau}\approx\sqrt{\frac{v_{0}^{2}m^{2}\sigma_{x}^{2}}{\pi\bar{\tau}^{2}}}e^{-\frac{v_{0}^{2}m^{2}\sigma_{x}^{2}}{\bar{\tau}^{2}}\left(\delta\tau\right)^{2}}$
(22)
We define an effective dispersion in time:
$\bar{\sigma}_{\tau}\equiv\frac{1}{mv_{0}\sigma_{x}}\bar{\tau}$ (23)
So we have the probability of detection as roughly:
$\bar{\rho}_{\delta\tau}\approx\sqrt{\frac{1}{\pi\bar{\sigma}_{\tau}^{2}}}e^{-\frac{\left(\delta\tau\right)^{2}}{\bar{\sigma}_{\tau}^{2}}}$
(24)
This is normalized to one, centered on $\tau=\bar{\tau}$, and with
uncertainty:
$\Delta\tau=\frac{1}{\sqrt{2}}\bar{\sigma}_{\tau}$ (25)
It is interesting that the classical condition has played no essential role in
this derivation. In spite of that, the result is about what we would expect
from a classical particle with probability distribution $\bar{\rho}$.
Therefore the implication of the experimental results in Muga and Leavens is
that they are confirmation of the reasonableness of the classical
approximation in general – rather than of Kijowski’s classical condition in
particular.
##### Waves
Now consider a more wave-like Gaussian test function: slow and wide. Take
$0<{p_{0}}\ll{\sigma_{p}}$ in equation 10, in fact make $p_{0}$ only very
slightly greater than 0.
Our GTF is still moving to the right but very slowly. If we wait long enough –
and if there were no detector in the way – it will eventually get completely
past the detector. Since there is a detector, the GTF will encounter this
instead. We will first assume a perfect detector with no reflection.
Use of the Kijowski distribution requires we drop the $p<0$ half of the wave
function because we started on the left:
${\varphi^{\left({wave}\right)}}\left(p\right)\Rightarrow\sqrt[4]{{\frac{1}{{\pi\sigma_{p}^{2}}}}}{e^{\imath
pd-\frac{{\left({p-{p_{0}}}\right)^{2}}}{{2\sigma_{p}^{2}}}}}\theta\left(p\right)$
(26)
We can no longer justify extending the lower bound on the left hand integral
from zero to negative infinity. We have instead:
${\rho_{d}}\left(\tau\right)={\left|{\int\limits_{0}^{\infty}dp\sqrt{\frac{p}{{2\pi
m}}}\sqrt[4]{{\frac{1}{{\pi\sigma_{p}^{2}}}}}{e^{\imath
pd-\frac{{{\left({p-{p_{0}}}\right)}^{2}}}{{2\sigma_{p}^{2}}}-\imath\frac{{p^{2}}}{{2m}}\tau}}}\right|^{2}}$
(27)
This means we have dropped nearly half of the wave function and therefore
expect a detection rate of only about 25%.
The integral over the momentum gives a sum over Bessel functions of fractional
order; the subsequent integrals over time are distinctly non-trivial. However
in the limiting case $d\to 0,p_{0}\to 0$ (the wave function starts at the
detector) the distribution simplifies drastically giving:
$\rho_{0}\left(\tau\right)={\left|{\frac{{\sqrt[4]{m}\sqrt{\sigma_{p}^{2}}\Gamma\left({\frac{3}{4}}\right)}}{{{{(2\pi)}^{3/4}}{{(m+\imath\sigma_{p}^{2}\tau)}^{3/4}}}}}\right|^{2}}$
(28)
And this in turn is simple enough to let us compute the norm analytically:
$\int\limits_{0}^{\infty}{d\tau\rho_{0}\left(\tau\right)}=\frac{1}{4}$ (29)
This is exactly as expected: if we throw out half the wave function, the
subsequent norms are the square of the wave function or one quarter.
This result is general. The GTF obeys the Schrödinger equation, which is norm
preserving. If we throw out fraction $F$ of the wave function, the resulting
wave function has norm ${\left({1-F}\right)^{2}}$. Assuming perfect detection,
the Kijowski distribution will under-count the actual detections by this
factor.
It is arguable that the problem is the assumption of perfect detection.
However if we drill down to the actual mechanism of detection, this will
involve some sort of interaction of the incoming particles with various atoms,
typically involving the calculation of matrix elements of the form (e.g.
Sakurai [55]):
$\left\langle{\varphi_{\tau}^{\left({n^{\prime}l^{\prime}m^{\prime}}\right)}\left({\vec{x}}\right)}\right|\sqrt{\frac{{n_{\vec{k},\alpha}}}{{2\omega
V}}}{{\vec{\varepsilon}}^{\left(\alpha\right)}}\exp\left({\imath\vec{k}\cdot\vec{x}-\imath\omega\tau}\right)\left|{\varphi_{\tau}^{\left({nlm}\right)}\left({\vec{x}}\right)}\right\rangle$
(30)
It is difficult to make sense of the classical condition in this context.
Would we have to apply it to the interaction with each atom in turn? And how
would the atoms know whether they are supposed to be detecting particles
coming from the left or the right? And therefore how would the unfortunate
atoms know whether they should drop the upper left and bottom right quadrants
on the diagram or vice versa?
#### 2.1.3 Requirements for a time-of-arrival metric
The classical condition was introduced without justification in terms of
quantum mechanics itself. As there are no cases where the rules of quantum
mechanics have failed to produce correct results, this is not acceptable.
Further it is unnecessary. The probability current, discussed below
(subsubsection 3.1.1), provides a conceptually sound and purely quantum
mechanical approach to the problem of detection.
We therefore have to dispense entirely with the classical condition. As this
is the foundation of the Kijowski class of metrics – per Muga and Leavens
above – we have to dispense with the class as well.
We have however learned a bit about our requirements: we are looking for a
time-of-arrival metric which satisfies the requirements:
1. 1.
Comes out of a completely quantum mechanical analysis, with no ad hoc
classical requirements imposed on it.
2. 2.
Includes an account of the interaction with the detector. As Bohr pointed out
during the EPR debate [56, 57], an analysis of a quantum mechanical detection
must include the specifics of the apparatus. There is no well-defined value of
a quantum parameter in the abstract, but only in the context of a specific
measurement.
3. 3.
Conserves probability.
4. 4.
And of course, is as simple as possible given the first three requirements.
### 2.2 Marchewka and Schuss path integral approach
> “According to our assumptions, trajectories that propagate into the
> absorbing boundary never return into the interval [a, b] so that the Feynman
> integral over these trajectories is supported outside the interval. On the
> other hand, the Feynman integral over the bounded trajectories in the
> interval is supported inside the interval. Thus the two integrals are
> orthogonal and give rise to no interference.” – Marchewka and Schuss [31]
We turn to an approach from Marchewka and Schuss [29, 30, 31, 32]. They use a
path integral approach plus the assumption of an absorbing boundary. To
compute the wave function, they use a recursive approach: computing the wave
function at each step, subtracting off the absorption at each step, and so to
the end.
Marchewka and Schuss start with the wave function defined over a general
interval from $a\to b$. We can simplify their treatment slightly by taking
$a=-\infty,b=0$. They break the time evolution up into steps of size
$\epsilon$. Given the wave function at step $n$, they compute the wave
function at step $n+1$ as:
${\psi_{n+1}}\left({x_{n+1}}\right)=\int\limits_{-\infty}^{0}{d{x_{n}}{K_{\varepsilon}}\left({{x_{n+1}};{x_{n}}}\right)}{\psi_{n}}\left({x_{n}}\right)$
(31)
The kernel can be quite general.
Marchewka and Schuss then introduce the assumption that the probability to be
absorbed at each endpoint is proportional to the value of the probability to
be at the end point times a characteristic length $\lambda$555They refer to
$\lambda$ as a “fudge parameter”. The value of $\lambda$ has to found by
running some reference experiments; after this is done the values are
determined for “nearby” experiments.:
${P_{n}}=\lambda{\left|{{\psi_{n}}\left(0\right)}\right|^{2}}$ (32)
Using the kernel they compute the probability of absorption as:
${P_{n}}=\frac{\varepsilon}{{2\pi{m}}}\left({\lambda\left|{\left.{{\partial_{x}}{\psi_{n}}\left(x\right)}\right|_{x=0}^{2}}\right|}\right)$
(33)
They then shrink the wave function at the next step by an appropriate factor,
the survival probability:
${\psi_{n+1}}\left(x\right)\to\sqrt{1-{P_{n}}}{\psi_{n+1}}\left(x\right)$ (34)
This guarantees the total probability is constant, where the total probability
is the probability to be detected plus the probability to survive. This
overcomes the most obvious problem with the Kijowski approach.
At the same time it represents an explicit loss of the phase information.
There is nothing to keep the “characteristic length” from being complex;
taking it as real is a choice. Therefore we are seeing the loss of information
normally associated with the act of measurement. So we have effectively broken
the total measurements at the end points down into a series of time dependent
“mini-measurements” or “mini-collapses” at each time step.
Computing the product of all these infinitesimal absorptions they get the
final corrected wave function as:
$\psi_{\tau}^{\left({corrected}\right)}\left(x\right)=\exp\left({-\frac{1}{{2\pi{m}}}\int\limits_{0}^{\tau}{d\tau^{\prime}\left({{\lambda}\left|{{\left.{{\partial_{x}}{\psi_{\tau}}\left(0\right)}\right|}^{2}}\right|}\right)}}\right){\psi_{\tau}}\left(x\right)$
(35)
They go on to apply this approach to a variety of cases, as GTFs going from
left to right.
The Marchewka and Schuss approach is a significant advance over the Kijowski
metric: it works at purely quantum mechanical level, it includes the detector
in the analysis, and it guarantees by construction overall conservation of
probability.
But at the same time the introduction of the characteristic length is a major
problem for falsifiability: how do we know whether the same length should be
used for SQM and for TQM?
And the trick for estimating the loss of probability at each step implicitly
assumes no interference in time. This is not a problem for SQM but a point we
need to keep in mind when including TQM.
We therefor further refine our requirements for a metric: we need to handle
any discontinuity at the detector in a well-defined way.
### 2.3 Discrete approach
Figure 2: Reflection principle
We can handle the discontinuity at the boundary in a well-defined way by
taking a discrete approach, by setting up a grid in $\tau,x$. We model quantum
mechanics as a random walk on the Feynman checkerboard [58]. Feynman used
steps taken at light speed; ours will travel a bit more slowly than that.
This lets us satisfy the previous requirements. And it lets us take advantage
of the extensive literature on random walks (e.g. [59, 60, 61, 62]).
We will start with a random walk model for detection, take the continuous
limit as the grid spacing goes to zero, Wick rotate in time, and then apply
the results to a GTF. This will give us a well-defined answer for the
detection rate for a GTF.
#### 2.3.1 Random Walk
We start with a grid, index by $n\in\left[{0,1,\ldots,\infty}\right]$ in the
time direction and by $m\in[-\infty,\ldots,-1,0,1,\ldots,\infty]$ in the space
direction. We start with a simple coin tossing picture. We will take the
probability to go left or right as equal, both $\frac{1}{2}$. $P_{nm}$ is the
probability to be at position $m$ at step $n.$ It is given recursively by:
${P_{n+1,m}}={P_{n,m-1}}/2+{P_{n,m+1}}/2$ (36)
The number of paths from $\left(0,0\right)\to\left(n,m\right)$ is given by the
binomial coefficient $\left({\begin{array}[]{c}n\\\
{\frac{{n+m}}{2}}\end{array}}\right)$. We will take this as zero if $n$ and
$m$ do not have the same parity, if they are not both even or both odd. If
$n>\left|{m}\right|$ then $P_{nm}=0$.
The total probability to get from $\left(0,0\right)\to\left(n,m\right)$ is
given by the number of paths times the probability $\frac{{1}}{2^{n}}$ for
each one. In our example:
${P_{nm}}=\left({\begin{array}[]{c}n\\\
{\frac{{n+m}}{2}}\end{array}}\right)\frac{1}{{2^{n}}}$ (37)
If we wish to start at a position other than zero, say $m^{\prime}$, we
replace $m$ with $m-m^{\prime}$.
##### First arrival
We can model detection as the first arrival of the path at the position of the
detector. We start the particle at step 0 at position $m=-d$ and position the
detector at $m=0$. We take the probability of the first arrival at step $n$
when starting at position $-d$ as ${F_{n}}$.
We define the function $G_{nm}$ as the probability to arrive at position $m$
at time $n$, _without_ having been previously detected. We can get this
directly from the raw probabilities using the reflection principle (e.g. [59])
as follows:
The number of paths that start at $\left(0,-d\right)$ and get to
$\left(n,m\right)$ is given by: $\left({\begin{array}[]{c}n\\\
{\frac{{n+{m+d}}}{2}}\end{array}}\right)$. Now consider all paths that get to
$m$ at step $n$ having first touched the detector at some step $f$. By
symmetry, the number of paths from $-d\to f$ is the same as the number of
paths from $d\to f$. The number of these paths is
$\left({\begin{array}[]{c}n\\\
{\frac{{n+\left|{m-d}\right|}}{2}}\end{array}}\right)$.
Therefore we have the number of paths that reach $m$ without first touching
the line $m=0$ as ([63]):
${G_{nm}}=\left({\left({\begin{array}[]{c}n\\\
{\frac{{n+m+d}}{2}}\end{array}}\right)-\left({\begin{array}[]{c}n\\\
{\frac{{n+m-d}}{2}}\end{array}}\right)}\right)\frac{1}{{2^{n}}}$ (38)
And of course, if $\left(n,m\right)$ do not have the same parity, $G_{nm}=0$.
To get to the detector at step $n$, the particle must have been one step to
the left at the previous step. It will then have a probability of
$\frac{{1}}{2}$ to get the detector. So we have:
${F_{n}}=\frac{{1}}{2}{G_{n-1,-1}}$ (39)
giving:
$F_{n}=\frac{d}{n}\left({\begin{array}[]{c}n\\\
{\frac{{n+d}}{2}}\end{array}}\right)\frac{{1}}{2^{n}}=\frac{d}{n}{P_{nd}}$
(40)
If the particle is already at position $0$ at step $0$ we declare that it has
arrived at time zero so we have:
$F_{0}={\delta_{d0}}$ (41)
#### 2.3.2 Diffusion equation
The passage to the continuum limit is well understood ([63, 62]). Define:
$\begin{gathered}\tau=n\Delta\tau\hfill\\\ x=m\Delta x\hfill\end{gathered}$
(42)
We get the diffusion equation if we take the limit so that both $\Delta x$ and
$\Delta\tau$ become infinitesimal while we keep fixed the ratio:
${D_{0}}=\mathop{\lim}\limits_{\Delta\tau,\Delta x\to 0}\frac{{{\left({\Delta
x}\right)}^{2}}}{{2\Delta\tau}}$ (43)
This gives the diffusion equation for the probability $P$:
$\frac{{\partial
P}}{{\partial\tau}}={D_{0}}\frac{{{\partial^{2}}P}}{{\partial{x^{2}}}}$ (44)
To further simplify we take $D_{0}=\frac{1}{2}$:
$\frac{{\partial{P_{\tau}}}}{{\partial\tau}}=\frac{1}{2}\frac{{\partial^{2}}}{{\partial{x^{2}}}}{P_{\tau}}$
(45)
If we start with the probability distribution
${P_{0}}\left(x-x^{\prime}\right)=\delta\left({x-x^{\prime}}\right)$ the
probability distribution as a function of time is then:
${P_{\tau}}\left({x;x^{\prime}}\right)=\frac{1}{{\sqrt{2\pi\tau}}}\exp\left({-\frac{{{\left({x-x^{\prime}}\right)}^{2}}}{{2\tau}}}\right)$
(46)
with the probability of a first arrival at $\tau$ being given by:
${F_{\tau}}\left({x;x^{\prime}}\right)=\frac{{\left|{x-x^{\prime}}\right|}}{{\sqrt{2\pi{\tau^{3}}}}}\exp\left({-\frac{{{\left({x-x^{\prime}}\right)}^{2}}}{{2\tau}}}\right)$
(47)
To include the mass, scale by $\tau\to\tau^{\prime}\equiv m\tau$ giving:
$\frac{{\partial{P_{\tau}}}}{{\partial\tau^{\prime}}}=\frac{1}{{2m}}\frac{{\partial^{2}}}{{\partial{x^{2}}}}{P_{\tau^{\prime}}}$
(48)
and:
${P_{\tau}}\left({x;x^{\prime}}\right)=\frac{{\sqrt{m}}}{{\sqrt{2\pi\tau}}}\exp\left({-m\frac{{{\left({x-x^{\prime}}\right)}^{2}}}{{2\tau}}}\right)$
(49)
To get the detection rate we have to handle the scaling of the time a bit
differently, since it is a probability distribution in time. The detection
rate only makes sense when integrated over $\tau$, so the substitution
implies:
$\int{d\tau{F_{\tau}}\to\int{\frac{{d\tau^{\prime}}}{m}{F_{\tau^{\prime}}}}}\to\int{d\tau^{\prime}{D_{\tau^{\prime}}}}\Rightarrow{D_{\tau}}=\frac{1}{m}{F_{\tau}}$
(50)
So we have:
${D_{\tau}}\left({x-x^{\prime}}\right)=\frac{{\left|{x-x^{\prime}}\right|}}{\tau}\frac{{\sqrt{m}}}{{\sqrt{2\pi\tau}}}\exp\left({-m\frac{{{\left({x-x^{\prime}}\right)}^{2}}}{{2\tau}}}\right)$
(51)
As a double check on this, we derive the detection rate directly from the
diffusion equation using, again, the method of images (see appendix A).
#### 2.3.3 Wick rotation
To go from the diffusion equation to the Schrödinger equation:
$\imath\frac{\partial}{{\partial\tau}}\psi=-\frac{1}{{2{m}}}\frac{{\partial^{2}}}{{\partial{x^{2}}}}\psi$
(52)
we use Wick rotation:
$\tau\to-\imath\tau,P\to\psi$ (53)
so we have the Wick-rotated kernel $K$:
${K_{\tau}}\left({x;x^{\prime}}\right)=\sqrt{\frac{m}{{2\pi\imath\tau}}}\exp\left({\imath
m\frac{{{\left({x-x^{\prime}}\right)}^{2}}}{{2\tau}}}\right)$ (54)
and for the kernel for first arrival $F$:
${F_{\tau}}\left({x;x^{\prime}}\right)=\frac{{\left|{x-x^{\prime}}\right|}}{\tau}{K_{\tau}}\left({x-x^{\prime}}\right)$
(55)
As a second double check, we derive $F$ directly using Laplace transforms (see
appendix B).
#### 2.3.4 Application to a Gaussian test function
We start with the “bullet” GTF from our analysis of the Kijowski metric
(equation 6):
${\varphi_{0}}\left({x}\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{x}^{2}}}}}{e^{\imath{p_{0}}x-\frac{1}{{2\sigma_{x}^{2}}}{{\left({x+d}\right)}^{2}}}}$
(56)
with:
$\left|{d}\right|\gg{\sigma_{x}}$ (57)
The wave function at the detector is then:
${\varphi_{\tau}}\left(0\right)=\int\limits_{-\infty}^{0}{dx^{\prime}{F_{\tau}}\left({0;x^{\prime}}\right)}{\varphi_{0}}\left({x^{\prime}}\right)=\int\limits_{-\infty}^{0}{dx^{\prime}\frac{{\left|{0-x^{\prime}}\right|}}{\tau}{K_{\tau}}\left({0;x^{\prime}}\right)}{\varphi_{0}}\left({x^{\prime}}\right)$
(58)
We can solve this explicitly in terms of error functions. However since
$\varphi_{0}$ is strongly centered on $-d$ we can use the same trick as we did
in the analysis of the Kijowski metric. We extend the upper limit of
integration to $\infty$:
${\varphi_{\tau}}\left(0\right)\approx-\frac{1}{\tau}\int\limits_{-\infty}^{\infty}{dx^{\prime}x^{\prime}{K_{\tau}}\left({0;x^{\prime}}\right)}{\varphi_{0}}\left({x^{\prime}}\right)$
(59)
We see by inspection we can pull down the $x^{\prime}$ by taking the
derivative of $\varphi_{0}$ with respect to $p_{0}$. This gives:
${\varphi_{\tau}}\left(0\right)\approx\imath\frac{1}{{\tau}}\frac{\partial}{{\partial{p_{0}}}}\int\limits_{-\infty}^{\infty}{dx^{\prime}{K_{\tau}}\left({0;x^{\prime}}\right)}{\varphi_{0}}\left({x^{\prime}}\right)$
(60)
We recognize the integral as the integral that propagates the free wave
function from $0\to\tau$:
${\varphi_{\tau}}\left(0\right)\approx\imath\frac{1}{{\tau}}\frac{\partial}{{\partial{p_{0}}}}\varphi_{\tau}^{\left({free}\right)}\left(0\right)$
(61)
We have:
$\varphi_{\tau}^{\left({free}\right)}\left(0\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{x}^{2}}}}}\sqrt{\frac{1}{{f_{\tau}^{\left(x\right)}}}}{e^{-\frac{1}{{2\sigma_{x}^{2}f_{\tau}^{\left(x\right)}}}{{\left({d-\frac{{p_{0}}}{m}\tau}\right)}^{2}}-\imath\frac{{p_{0}^{2}}}{{2m}}\tau}}$
(62)
so the wave function for a first arrival at $\tau$ is:
${\varphi_{\tau}}\left(0\right)\approx\left({\imath\frac{1}{{m}}\frac{{d-{v_{0}}\tau}}{{\sigma_{x}^{2}f_{\tau}^{\left(x\right)}}}+v_{0}}\right)\varphi_{\tau}^{\left({free}\right)}\left(0\right)$
(63)
#### 2.3.5 Metrics
We can compute the probability density to a sufficient order by expanding
around the average time of arrival:
$\bar{\tau}\equiv\frac{d}{{v_{0}}},\delta\tau\equiv\tau-\bar{\tau}$ (64)
The first term in the expression for the $\varphi_{\tau}$ scales as
$1/{\bar{\tau}}$; the second as a constant so at large $\tau$ we have:
${\rho_{\tau}}\left(0\right)\approx
v_{0}^{2}\rho_{\tau}^{\left({free}\right)}\left(0\right)$ (65)
The result is the same as with Kijowski, except for the overall factor of
$v_{0}^{2}$. Since the uncertainty is produced by normalizing this, the
uncertainty in time is the same as with the bullet calculation for Kijowski.
But the factor of $v_{0}^{2}$ is troubling, particularly when we recall we are
using natural units so that at non-relativistic speeds this is hard to
distinguish from zero.
The immediate problem is our somewhat thin-skinned approximation. The random
walk can only penetrate to position $m=0$ in the grid before it is absorbed.
And since the probability density is computed as the square, the result is
doubly small.
Given the taking of the limit, it is perhaps more surprising that we get a
finite detection rate at all. A realistic approximation would have the paths
penetrating some finite distance into the detector, with the absorption going
perhaps with some characteristic length – as with the Marchewka and Schuss
approach.
In return for a _more_ detailed examination of the boundary we have gotten a
_less_ physically realistic estimate. But the real problem with all three of
the approaches we have looked at is their assumption of a crisp boundary
between quantum and classical realms.
### 2.4 Implications for time-of-arrival measurements
> “What is it that we human beings ultimately depend on? We depend on our
> words. We are suspended in language. Our task is to communicate experience
> and ideas to others. We must strive continually to extend the scope of our
> description, but in such a way that our messages do not thereby lose their
> objective or unambiguous character.” – Niels Bohr as cited in [64]
In reality there can be no crisp boundary between quantum and classical
realms. Consider the lifespan of an atom built on classical principles, as in
say Bohr’s planetary model of the atom. A classical electron in orbit around
the nucleus undergoes centripetal acceleration from the nucleus; therefore
emits Larmor radiation; therefore decays in towards the nucleus. The estimated
lifespan is of order $10^{-11}$ seconds [65].
What keeps this from happening in the case of a quantum atom is the
uncertainty principle: as the electron spirals into the nucleus its
uncertainty in position is reduced so its uncertainty in momentum is
increased. A larger $\Delta p$ means a larger kinetic energy. In fact the
ground state energy can be estimated as the minimum total energy – kinetic
plus potential – consistent with the uncertainty principle.
Since everything material is built of atoms and since there are no classical
atoms there is no classical realm. And therefore it is incorrect to speak of a
transition to a classical end state. No such state exists.
However classical mechanics does work very well in practice. So what we have
is a problem in description. Which parts of our experimental setup – particle,
source, detector, observers – can be described to acceptable accuracy using
classical language; where must we use a fully quantum description?
This approach places us, with Bohr [66], firmly on the epistemological side of
the measurement debate. One possible starting point is the program of
decoherence [67, 68, 69, 70, 71, 72, 73, 74]. The key question for decoherence
is not when does a system go from quantum to classical but rather when does a
classical description become possible. Another is Quantum Trajectory Theory
(QTT) [75, 76, 77, 78]. QTT, as its name suggests, is a particularly good fit
with the path integral approach, the formalism we are primarily focused on
here.
For the moment we will explore a more informal approach, taking advantage of
the conceptual simplicity of the path integral approach. We can describe the
evolution of a quantum particle in terms of the various paths it takes. In the
case of the problem at hand, we visualize the paths as those that enter the
detector or do not, that are reflected or are not, that leave and do not
return (backflow) or return again and so on. Ultimately however each of the
paths will escape permanently or else enter an entropy sink of some kind, a
structure that is sufficiently macroscopic that it is no longer possible to
analyze the process in terms of the individual paths. Its effects will live
on, perhaps in a photo-electric cascade or a change of temperature. But the
specific path will be lost like a spy in a crowd.
The entropy sink is the one essential part needed for a measurement to take
place. In a quantum mechanical system information, per the no-cloning theorem,
is neither created nor destroyed. But the heart of a measurement is the
reduction of the complexity of the system under examination to a few numbers.
Information is necessarily lost in doing this. In general it is fairly clear
where this happens; typically in crossing the boundary from the microscopic to
the macroscopic scale.
Figure 3: Global character of the measurement problem
Typical paths may cross the boundary multiple times, ultimately either being
absorbed and registered (measured), lost in some way (detector inefficiency),
or reflected (backflow). There will be some time delay associated with all of
these cases666A typical representation could be done in terms of a Laplace
transform $\mathcal{D}\left(s\right)$ of the detection amplitude plus a
Laplace transform $\mathcal{R}\left(s\right)$ of the reflection amplitude. The
Marchewka and Schuss fudge parameter may be thought of as a useful first
approximation of $\mathcal{D}$..
This is not too far from conventional practice. A macroscopic detector
functions as the entropy sink, its efficiency is usually known, and
reflections are possible but minimized. In many cases the response time
associated with the process of detection may not require much attention – we
may only care that there was a detection. Our case is a bit trickier than
that. We have to care about the time delay associated with a detection,
especially if the time delay introduces some uncertainty in time in its own
right.
This picture is enough to show that there can not be, in general, a local
time-of-arrival operator. Suppose there is such. Consider the paths leaving it
on the right, continuing to an entropic sink of some kind. If we arrange a
quantum mirror of some kind (perhaps just a regular mirror), then some of the
paths will return to their source. The detection rate will be correspondingly
reduced. But this can only be predicted using knowledge of the global
situation, of the interposed mirror. An operator local to the boundary cannot
know about the mirror and therefore cannot give a correct prediction of the
time-of-arrival distribution. Therefore the time-of-arrival cannot in general
be given correctly by a purely local operator.
Since the fundamental laws are those of quantum mechanics, the analysis must
be carried out at a quantum mechanical level – except for those parts where we
can show the classical approximation suffices, usually the end states. In
fact, there is no guarantee that even the end states can be adequately
described in classical terms; while there are no cases – so far – of a
perdurable and macroscopic quantum system777But see the macroscopic wave
functions in [79, 80]., there is nothing to rule that out.
## 3 Effects of dispersion in time on time-of-arrival measurements
> “Clearly,” the Time Traveller proceeded, “any real body must have extension
> in four directions: it must have Length, Breadth, Thickness, and — Duration.
> But through a natural infirmity of the flesh, which I will explain to you in
> a moment, we incline to overlook this fact. There are really four
> dimensions, three which we call the three planes of Space, and a fourth,
> Time. There is, however, a tendency to draw an unreal distinction between
> the former three dimensions and the latter, because it happens that our
> consciousness moves intermittently in one direction along the latter from
> the beginning to the end of our lives.” – H. G. Wells [81]
We now return to our original problem, first estimating the dispersion in
time-of-arrival without dispersion in time, then with.
### 3.1 Time-of-arrival measurements without dispersion in time
Our efforts to correctly define a time of arrival operator have led to the
conclusion that the time-of–arrival – like measurement in general – is a
system level property; it is not something that can be correctly described by
a local operator.
We have therefore to a certain extent painted ourselves into a corner. We
would like to compute the results for the general case, but we have shown
there are only specific cases. We will deal with this by trading precision for
generality.
We will first assume a perfect detector, then justify the assumption.
#### 3.1.1 Probability current in SQM
We will first compute the detection rate by using the probability current per
Marchewka and Schuss [32]. We start with the Schrödinger equation:
$\imath\frac{\partial}{{\partial\tau}}{\psi_{\tau}}\left(x\right)=-\frac{1}{{2m}}\frac{{\partial^{2}}}{{\partial{x^{2}}}}{\psi_{\tau}}\left(x\right)$
(66)
We will assume we have in hand a wave function that includes the interaction
with the detector. We define the probability of detection at clock time $\tau$
as $D_{\tau}$.
The total probability of the wave function to be found to the left of the
detector at time $\tau$ is the integral of the probability density from
$-\infty\to 0$. The total probability to have been detected up to time $\tau$
is the integral of $D_{\tau}$ from $0\to\tau$. Therefore by conservation of
probability we have:
$1=\int\limits_{-\infty}^{0}{dx}\psi_{\tau}^{*}\left(x\right){\psi_{\tau}}\left(x\right)+\int\limits_{0}^{\tau}{d\tau^{\prime}{D_{\tau^{\prime}}}}$
(67)
We take the derivative with respect to $\tau$ to get:
$D_{\tau}=-\int\limits_{-\infty}^{0}{dx\frac{{\partial\psi_{\tau}^{*}\left(x\right)}}{{\partial\tau}}{\psi_{\tau}}\left(x\right)+\psi_{\tau}^{*}\left(x\right)\frac{{\partial{\psi_{\tau}}\left(x\right)}}{{\partial\tau}}}$
(68)
We use the Schrödinger equation:
${D_{\tau}}=\frac{\imath}{{2m}}\int\limits_{-\infty}^{0}{dx\frac{{{\partial^{2}}\psi_{\tau}^{*}\left(x\right)}}{{\partial{x^{2}}}}{\psi_{\tau}}\left(x\right)-\psi_{\tau}^{*}\left(x\right)\frac{{{\partial^{2}}{\psi_{\tau}}\left(x\right)}}{{\partial{x^{2}}}}}$
(69)
And integrate by parts to get the equation for the detection rate:
${D_{\tau}}={\left.{\frac{\imath}{{2m}}\left({\frac{{\partial\psi_{\tau}^{*}\left(x\right)}}{{\partial
x}}{\psi_{\tau}}\left(x\right)-\psi_{\tau}^{*}\left(x\right)\frac{{\partial{\psi_{\tau}}\left(x\right)}}{{\partial
x}}}\right)}\right|_{x=0}}$ (70)
We recognize the expression on the right as the probability current:
${J_{\tau}}\left(x\right)=-\frac{\imath}{{2}}\left({\psi_{\tau}^{*}\left(x\right)\frac{{\partial{\psi_{\tau}}\left(x\right)}}{{\partial
x}}-{\psi_{\tau}}\left(x\right)\frac{{\partial\psi_{\tau}^{*}\left(x\right)}}{{\partial
x}}}\right)$ (71)
or:
${J_{\tau}}=\psi_{\tau}^{*}\frac{p}{{2m}}{\psi_{\tau}}-{\psi_{\tau}}\frac{p}{{2m}}\psi_{\tau}^{*}$
(72)
so:
$D_{\tau}=J_{\tau}$ (73)
Note that $D_{\tau}$ can, in the general case, be negative. For instance, if
the detector includes – per the previous section – a mirror of some kind there
may be backflow, probability current going from right to left.
Ironically enough, this is in fact the first metric that Kijowski considered
in his paper, only to reject it because it violated the classical condition 2.
#### 3.1.2 Black box detector
Given the wave function we can compute the detection rate using the
probability current. The real problem is to compute the wave function in the
first place, given that this must in principle include the interaction of the
particle with the detector. To treat the general case, we would like a general
detector, one where the details of the interaction do not matter. We are
looking for something which is a perfect black.
We can use Einstein’s clock-in-a-box, but this time run it in reverse: have
the detector be a box that has a small window open for a fraction of a second
then check how many particles are in the box afterward by weighing the box.
The interior walls of the box provide the necessary entropy sink. We will
refer to this as a black box detector 888A “perfect black” is apparently not
quite as theoretical a concept as it sounds: the commercial material
Vantablack can absorb 99.965% of incoming light [82]. And Vantablack even uses
a similar mechanism: incoming light is trapped in vertically aligned nanotube
arrays..
In fact, this is a reasonable model of the elementary treatment of detection.
When, for instance, we compute the rate of scattering of particles into a bit
of solid angle, we assume that the particles will be absorbed with some
efficiency but we do not generally worry about subtleties as time delay till a
detection is registered or backflow.
#### 3.1.3 Application to a Gaussian test function
We will take a GTF at the plane as above:
${\varphi_{\tau}}\left(x\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{x}^{2}}}}}\sqrt{\frac{1}{{f_{\tau}^{\left(x\right)}}}}{e^{\imath{p_{0}}x-\frac{1}{{2\sigma_{x}^{2}f_{\tau}^{\left(x\right)}}}{{\left({x+d-\frac{{p_{0}}}{m}{\tau}}\right)}^{2}}-\imath\frac{{p_{0}^{2}}}{{2m}}{\tau}}}$
(74)
An elementary application of the probability current gives the detection rate
as the velocity times the probability density:
${D_{\tau}}=\frac{{p_{0}}}{m}{\rho_{\tau}}\left(0\right)$ (75)
This is the same as we had for the Kijowski bullet (equation 16). And it is,
as noted there, also the classical rate for detection of a probability
distribution impacting a wall.
Our use of the Einstein box in reverse is intended to provide a lowest common
denominator for quantum measurements; it is not surprising that the lowest
common denominator for quantum measurements should be the corresponding
classical distribution.
We have the uncertainty in time-of-arrival as before (equation 25):
$\Delta\tau=\frac{1}{{\sqrt{2}}}{{\bar{\sigma}}_{\tau}}=\frac{1}{{\sqrt{2}}}\frac{1}{{m{v_{0}}{\sigma_{x}}}}\bar{\tau}$
(76)
Since we have $\bar{p}=m{v_{0}}$ and – for a minimum uncertainty wave function
– ${\sigma_{p}}=\frac{1}{{\sigma_{x}}}$ we can also write:
$\frac{{\sigma_{\tau}}}{{\bar{\tau}}}=\frac{{{\hat{\sigma}}_{p}}}{{\bar{p}}}$
(77)
so the dispersion in momentum and in clock time are closely related, as one
might expect.
While the results are the same as with the Kijowski metric (for bullet wave
functions), the advantage is that we now have a clearer sense of what is
actually going on and therefore what corrections we might have to make in
practice. These may include (but will probably not be limited to):
1. 1.
Depletion, e.g. inefficiencies of the detector,
2. 2.
Backflow, e.g. a reflection coefficient,
3. 3.
Delay in the detection (not normally important, but significant here),
4. 4.
Self-interference of the wave function with itself,
5. 5.
Edge effects – one can imagine the incoming part of the wave function skimming
along the edge of the detector for instance,
6. 6.
And of course the increasingly complex effects that result from increasingly
complex models of the wave function-detector interaction.
### 3.2 Time-of-arrival measurements with dispersion in time
#### 3.2.1 Quantum mechanics with uncertainty in time
Figure 4: Paths in three and in four dimensions
Path integrals provide a natural way to extend quantum mechanics to include
time in a fully covariant way. In [13] we did this by first extending the
usual three-dimensional paths to four dimensions. The rest of the treatment
followed from this single assumption together with the twin requirements of
covariance and of consistency with known results. But as one might expect,
there are a fair number of questions to be addressed along the way – making
the full paper both rather long and rather technical. To make this treatment
self-contained we summarize the key points here.
##### Feynman path integrals in four dimensions
We start with clock time, defined operationally as what the laboratory clock
measures. This is the parameter $\tau$ in the Schrödinger equation, Klein-
Gordon equation, path integral sums, and so on.
The normal three dimensional paths are parameterized by clock time; we
generalize them to four dimensional paths, introducing the coordinate time $t$
to do so:
${\bar{\pi}_{\tau}}\left({x,y,z}\right)\to{\pi_{\tau}}\left({t,x,y,z}\right)$
(78)
It can be helpful to think of coordinate time as another space dimension. We
will review the relationship between clock time and coordinate time at the
end, once the formalism has been laid out. (This is a case where it is helpful
to let the formalism precede the intuition.)
In path integrals we get the amplitude to go from one point to another by
summing over all intervening paths, weighing each path by the corresponding
action. The path integral with four dimensional paths is formally identical to
the path integral with three dimensions, except that the paths take all
possible routes in four rather than three dimensions. We define the kernel as:
${K_{\tau}}\left({x^{\prime\prime};x^{\prime}}\right)=\mathop{\lim}\limits_{N\to\infty}\int\mathcal{D}\pi{e^{\imath\int\limits_{0}^{\tau}{d\tau^{\prime}\mathcal{L}\left[{\pi,\dot{\pi}}\right]}}}$
(79)
We use a Lagrangian which works for both the three and four dimensional cases
(see [83]):
$\mathcal{L}\left({{x^{\mu}},{{\dot{x}}^{\mu}}}\right)=-\frac{1}{2}m{{\dot{x}}^{\mu}}{{\dot{x}}_{\mu}}-q{{\dot{x}}^{\mu}}{A_{\mu}}\left(x\right)-\frac{m}{2}$
(80)
As usual with path integrals, to actually do the integrals we discretize clock
time $\tau=\epsilon n$ and replace the integral over clock time with a sum
over steps:
$\int\limits_{0}^{\tau}{d\tau^{\prime}\mathcal{L}}\to\sum\limits_{j=0}^{n}{\mathcal{L}_{j}}$
with:
$\mathcal{L}_{j}\equiv-m\frac{\left(x_{j}-x_{j-1}\right)^{2}}{2\varepsilon^{2}}-q\frac{x_{j}-x_{j-1}}{\epsilon}\frac{A\left(x_{j}\right)+A\left(x_{j-1}\right)}{2}-\frac{m}{2}$
(81)
And measure:
$\mathcal{D}\pi\equiv\left(-\imath\frac{m^{2}}{4\pi^{2}\varepsilon^{2}}\right)^{N}\prod\limits_{n=1}^{N-1}d^{4}x_{n}$
(82)
This gives us the ability to compute the amplitude to get from one point to
another in a way that includes paths that vary in time as well as space.
##### Schrödinger equation in four dimensions
Usually we derive the path integral expressions from the Schrödinger equation.
But here we derive the four dimensional Schrödinger equation from the short
time limit of the path integral kernel, running the usual derivation in
reverse and with one extra dimension. We get:
$\imath\frac{\partial\psi_{\tau}}{\partial\tau}=-\frac{1}{{2m}}\left({\left({{p_{\mu}}-q{A_{\mu}}}\right)\left({{p^{\mu}}-q{A^{\mu}}}\right)-m^{2}}\right){\psi_{\tau}}$
(83)
Note that here $p$ is an operator, $A$ is an external field, and $m$ is the
constant operator999In the extension to QED, $A$ becomes an operator as well..
This equation goes back to Stueckelberg [14, 15] with further development by
Feynman, Fanchi, Land, and Horwitz [16, 23, 24, 27].
We need only the free Schrödinger equation here. With the vector potential $A$
set to zero we have:
$\imath\frac{\partial}{{\partial\tau}}{\psi_{\tau}}=-\frac{{{E^{2}}-{\overrightarrow{p}^{2}}-m^{2}}}{{2m}}{\psi_{\tau}}$
(84)
Or spelled out:
$\imath\frac{\partial}{{\partial\tau}}{\psi_{\tau}}\left({t,\vec{x}}\right)=\frac{1}{{2m}}\left({\frac{{\partial^{2}}}{{\partial{t^{2}}}}-{\nabla^{2}}-{m^{2}}}\right){\psi_{\tau}}\left({t,\vec{x}}\right)$
(85)
If the left hand side were zero, the right hand side would be the Klein-Gordon
equation (with $t\to\tau$). Over longer times – femtoseconds or more – the
left side will in general average to zero, giving the Klein-Gordon equation as
the long time limit. More formally we expect that if we average over times
$\Delta\tau$ of femtoseconds or greater, we will have:
$\int\limits_{\tau}^{\tau+\Delta\tau}{d\tau^{\prime}\int{{d^{4}}x}}\psi_{\tau}^{*}\left(x\right)\frac{{{E^{2}}-{{\vec{p}}^{2}}-{m^{2}}}}{{2m}}{\psi_{\tau}}\left(x\right)\approx
0$ (86)
But at short times – attoseconds – we should see effects associated with
uncertainty in time. This is in general how we get from a fully four
dimensional approach to the usual three dimensional approach; the fluctuations
in coordinate time tend to average out over femtosecond and longer time
scales. But at shorter times – attoseconds – we will see the effects of
uncertainty in time.
It is much the same way that in SQM quantum effects in space average out over
longer time and distance scales to give classical mechanics. TQM is to SQM –
with respect to time – as SQM is to classical mechanics with respect to the
three space dimensions.
##### Wave functions in coordinate time
We need an estimate of the wave function at source, its evolution in clock
time, and the rules for detection – the birth, life, and death of the wave
function if you will.
###### Initial wave function
We start with the free Klein-Gordon equation and use the method of separation
of variables to break the wave function out into a space and a time part:
$\varphi_{0}\left(t,x\right)=\tilde{\varphi}_{0}\left(t\right)\bar{\varphi}_{0}\left(x\right)$
(87)
Or in energy/momentum space:
$\hat{\varphi}_{0}\left(E,p\right)=\hat{\tilde{\varphi}}_{0}\left(E\right)\hat{\bar{\varphi}}_{0}\left(p\right)$
(88)
We assume we already have the space part by standard methods. For instance, if
this is a GTF in momentum space it will look like:
${{\hat{\bar{\varphi}}}_{\tau}}\left(p\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{p}^{2}}}}}{e^{-\imath
p{x_{0}}-\frac{{{\left({p-{p_{0}}}\right)}^{2}}}{{2\sigma_{p}^{2}}}-\imath\frac{{p^{2}}}{{2m}}\tau}}$
(89)
This plus the Klein-Gordon equation give us constraints on the time part:
$\begin{array}[]{l}\left\langle 1\right\rangle=1\\\ \bar{E}\equiv\left\langle
E\right\rangle=\sqrt{m^{2}+\left\langle\vec{p}\right\rangle^{2}}\\\
\left\langle E^{2}\right\rangle=\left\langle
m^{2}+\vec{p}^{2}\right\rangle=m^{2}+\left\langle\vec{p}^{2}\right\rangle\end{array}$
(90)
To get a robust estimate of the shape of the time part we assume it is the
maximum entropy solution that satisfies the constraints. We used the method of
Lagrange multipliers to find this getting:
$\hat{\tilde{\varphi}}_{0}\left(E\right)=\sqrt[4]{\frac{1}{\pi\sigma_{E}^{2}}}e^{\imath\left(E-E_{0}\right)t_{0}-\frac{\left(E-E_{0}\right)^{2}}{2\sigma_{E}^{2}}}$
(91)
with values:
$\sigma_{E}^{2}=\sigma_{p}^{2}$ (92)
and:
$E_{0}=\sqrt{m^{2}+\bar{p}^{2}}$ (93)
To get the starting wave function in time we take the inverse Fourier
transform:
$\tilde{\varphi}_{0}\left(t\right)=\sqrt[4]{\frac{1}{\pi\sigma_{t}^{2}}}e^{-\imath
E_{0}\left(t-t_{0}\right)-\frac{t^{2}}{2\sigma_{t}^{2}}}$ (94)
$\sigma_{t}^{2}=\frac{1}{\sigma_{E}^{2}}$ (95)
The value of the $\sigma_{t}$ will normally be of order attoseconds. We set
$t_{0}=0$ as the overall phase is already supplied by the space/momentum part.
Since estimates from maximum entropy tend to be robust we expect this approach
will give estimates which are order-of-magnitude correct.
###### Propagation of a wave function
The TQM form of the Klein-Gordon equation is formally identical to the non-
relativistic Schrödinger equation with the additional coordinate time term. In
momentum space one can use this insight to write the kernel out by inspection:
${{\hat{K}}_{\tau}}\left({p^{\prime\prime};p^{\prime}}\right)={\delta^{\left(4\right)}}\left({p^{\prime\prime}-p^{\prime}}\right)\exp\left({\imath\frac{{{{p^{\prime}}^{2}}-{m^{2}}}}{{2m}}\tau}\right)\theta\left(\tau\right)$
(96)
The coordinate space form is:
${K_{\tau}}\left({x^{\prime\prime};x^{\prime}}\right)=-\imath\frac{{m^{2}}}{{4{\pi^{2}}{\tau^{2}}}}{e^{-\imath
m\frac{{{\left({x^{\prime\prime}-x^{\prime}}\right)}^{2}}}{{2\tau}}-\imath\frac{m}{2}\tau}}\theta\left(\tau\right)$
(97)
In both cases we have the product of a coordinate time part by the familiar
space part. Spelling this out in coordinate space101010We are using an over-
tilde and over-bar to distinguish between the coordinate time and the space
parts, when this is useful.:
$\begin{gathered}{K_{\tau}}\left({x^{\prime\prime};x^{\prime}}\right)={{\tilde{K}}_{\tau}}\left({{t^{\prime\prime}};{t^{\prime}}}\right){{\bar{K}}_{\tau}}\left({{{\vec{x}}^{\prime\prime}};{{\vec{x}}^{\prime}}}\right)\exp\left({-\imath\frac{m}{2}\tau}\right)\theta\left(\tau\right)\hfill\\\
{{\tilde{K}}_{\tau}}\left({{t^{\prime\prime}};{t^{\prime}}}\right)=\sqrt{\frac{{\imath
m}}{{2\pi\tau}}}\exp\left({-\imath
m\frac{{{\left({{t^{\prime\prime}}-{t^{\prime}}}\right)}^{2}}}{{2\tau}}}\right)\hfill\\\
{{\bar{K}}_{\tau}}\left({{{\vec{x}}^{\prime\prime}};{{\vec{x}}^{\prime}}}\right)={\sqrt{-\frac{\imath
m}{{2\pi\tau}}}^{3}}\exp\left({\imath
m\frac{{{\left({\vec{x}^{\prime\prime}-\vec{x}^{\prime}}\right)}^{2}}}{{2\tau}}}\right)\hfill\end{gathered}$
(98)
where the space part kernel is merely the familiar non-relativistic kernel
(e.g. Merzbacher [84]). The additional factor of
$\exp\left({-\imath\frac{m}{2}\tau}\right)$ only contributes an overall phase
which plays no role in single particle calculations.
If the initial wave function can be written as the product of a coordinate
time and a three-space part, then the propagation of the three-space part is
the same as in non-relativistic quantum mechanics. In general, if the free
wave function starts as a direct product in time and space it stays that way.
Of particular interest here is the behavior of a GTF in time. If at clock time
zero it is given by:
$\tilde{\varphi}_{0}\left(t\right)\equiv\sqrt[4]{\frac{1}{\pi\sigma_{t}^{2}}}e^{-\imath
E_{0}\left(t-t_{0}\right)-\frac{\left(t-t_{0}\right)^{2}}{2\sigma_{t}^{2}}}$
(99)
then as a function of clock time it is:
$\tilde{\varphi}_{\tau}\left(t\right)=\sqrt[4]{\frac{1}{\pi\sigma_{t}^{2}}}\sqrt{\frac{1}{f_{\tau}^{\left(t\right)}}}e^{-\imath
E_{0}t-\frac{1}{2\sigma_{t}^{2}f_{\tau}^{\left(t\right)}}\left(t-t_{0}-\frac{E_{0}}{m}\tau\right)^{2}+\imath\frac{E_{0}^{2}}{2m}\tau}$
(100)
with dispersion factor $f_{\tau}^{\left(t\right)}\equiv
1-\imath\frac{\tau}{m\sigma_{t}^{2}}$ and with expectation, probability
density, and uncertainty:
$\begin{array}[]{l}\left\langle{t_{\tau}}\right\rangle={t_{0}}+\frac{E}{m}\tau={t_{0}}+\gamma\tau\\\
{{\tilde{\rho}}_{\tau}}\left(t\right)=\sqrt{\frac{1}{{\pi\sigma_{t}^{2}}\left({1+\frac{{\tau^{2}}}{{{m^{2}}\sigma_{t}^{4}}}}\right)}}\exp\left({-\frac{{{\left({t-\left\langle{t_{\tau}}\right\rangle}\right)}^{2}}}{{\sigma_{t}^{2}\left({1+\frac{{\tau^{2}}}{{{m^{2}}\sigma_{t}^{4}}}}\right)}}}\right)\\\
{\left({\Delta t}\right)^{2}}\equiv\left\langle
t^{2}\right\rangle-{\left\langle
t\right\rangle^{2}}=\frac{{\sigma_{t}^{2}}}{2}\left|{1+\frac{{\tau^{2}}}{{{m^{2}}\sigma_{t}^{4}}}}\right|\end{array}$
(101)
The behavior of the time part is given by replacing $x\to t$ and taking the
complex conjugate. As noted, coordinate time behaves like a fourth space
dimension, albeit one that enters with the sign of $\imath$ flipped.
###### Detection of a wave function
To complete the birth, life, death cycle we look at the detection of a wave
function. This is obviously the place where things can get very tricky.
However the general rule that coordinate time acts as a fourth space
coordinate eliminates much of the ambiguity. If we were doing a measurement
along the $y$ dimension and we registered a click in a detector located at
$y_{detector}\pm\frac{\Delta y}{2}$ we would take this as meaning that we had
measured the particle as being at position $y_{detector}$ with uncertainty
$\Delta y$.
Because of our requirement of the strictest possible correspondence between
coordinate time and the space dimensions, the same rule necessarily applies in
coordinate time. If we have a clock in the detector with time resolution
$\Delta\tau$, then a click in the detector implies a measurement of the
particle at coordinate time $t=\tau$ with uncertainty $\Delta\tau.$
So the detector does an implicit conversion from coordinate time to clock
time, because we have assumed we know the position of the detector in clock
time. How do we justify this assumption?
The justification is that we take the clock time of the detector as being
itself the average of the coordinate time of the detector:
${\tau_{Detector}}\equiv\left\langle{t_{Detector}}\right\rangle$ (102)
The detector will in general be made up of a large number of individual
particles, each with a four-position in coordinate time and the three space
dimensions. While in TQM, the individual particles may be a bit in the future
or past, the average will behave as a classical variable, specifically as the
clock time. This drops out of the path integral formalism, which supports an
Ehrenfest’s theorem in time (again in [13]). If we take the coordinate time of
a macroscopic object as the sum of the coordinate times of its component
particles, the principle is macroscopically stronger. The motion in time of a
macroscopic object will no more display quantum fluctuations in time than its
motion in space displays quantum fluctuations in the $x,y,z$ directions.
This explains the differences in the properties of the clock time and
coordinate time. The clock time is something we think of as only going
forward; its corresponding energy operator is bounded below, usually by zero.
But if it is really an expectation of the coordinate time of a macroscopic
object, then it is not that the clock time cannot go backward, it is that it
is statistically extremely unlikely to do so. The principle is the same as the
argument that while the gas molecules in a room could suddenly pile up in one
half of a room and leave the other half a vacuum, they are statistically
extremely unlikely to do so.
Further the conjugate variable for coordinate time
${E^{\left({op}\right)}}\equiv\imath\frac{\partial}{{\partial t}}$ is not
subject to Pauli’s theorem [85]. The values of $E$ – the momentum conjugate to
the coordinate time $t$ – can go negative. If we look at a non-relativistic
GTF:
${{\hat{\tilde{\varphi}}}_{0}}\left(E\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{E}^{2}}}}}{e^{\imath
E{t_{0}}-\frac{{{\left({E-{E_{0}}}\right)}^{2}}}{{2\sigma_{E}^{2}}}}},{\sigma_{E}}=\frac{1}{{\sigma_{t}}},{E_{0}}\approx
m+\frac{{{\vec{p}}^{2}}}{{2m}}$ (103)
the value of $m$ for an electron is about 500,000 eV, while for $\sigma_{t}$
of order attoseconds the $\sigma_{E}$ will be of order 6000 eV. Therefore the
negative energy part is about $500000/6000$ or $83$ standard deviations away
from the average. And therefore the likelihood of detecting a negative energy
part of the wave function is zero to an excellent approximation. But not
_exactly_ zero.
So, clock time is the classical variable corresponding to the fully quantum
mechanical coordinate time. They are two perspectives on the same universe.
#### 3.2.2 Probability current in TQM
We now compute the probability current in TQM, working in parallel to the
Marchewka and Schuss derivation for SQM covered above (and with similar
caveats about its applicability).
By conservation of probability the sum of the detections as of clock time
$\tau$ plus the total probability remaining at $\tau$ must equal one:
$1=\int\limits_{0}^{\tau}{d\tau^{\prime}\int\limits_{-\infty}^{\infty}{dt{D_{\tau^{\prime}}}\left(t\right)}}+\int\limits_{-\infty}^{0}{dx}\int\limits_{-\infty}^{\infty}{dt}\psi_{\tau}^{*}\left({t,x}\right){\psi_{\tau}}\left({t,x}\right)$
(104)
Again we take the derivative with respect to clock time to get:
$\int\limits_{-\infty}^{\infty}{dt{D_{\tau}}\left(t\right)}=-\int\limits_{-\infty}^{0}{dx}\int\limits_{-\infty}^{\infty}{dt}\left({\dot{\psi}_{\tau}^{*}\left({t,x}\right){\psi_{\tau}}\left({t,x}\right)+\psi_{\tau}^{*}\left({t,x}\right){{\dot{\psi}}_{\tau}}\left({t,x}\right)}\right)$
(105)
From the Schrödinger equation:
$\begin{gathered}{{\dot{\psi}}_{\tau}}=-\imath\frac{1}{{2m}}\left({\frac{{\partial^{2}}}{{\partial{t^{2}}}}-\frac{{\partial^{2}}}{{\partial{x^{2}}}}}\right){\psi_{\tau}}\hfill\\\
\dot{\psi}_{\tau}^{*}=\imath\frac{1}{{2m}}\left({\frac{{\partial^{2}}}{{\partial{t^{2}}}}-\frac{{\partial^{2}}}{{\partial{x^{2}}}}}\right)\psi_{\tau}^{*}\hfill\end{gathered}$
(106)
We use integration by parts in coordinate time to show the terms in the second
derivative of coordinate time cancel:
$\imath\frac{1}{{2m}}\left({\frac{{\partial^{2}}}{{\partial{t^{2}}}}\psi_{\tau}^{*}}\right){\psi_{\tau}}-\imath\frac{1}{{2m}}\psi_{\tau}^{*}\frac{{\partial^{2}}}{{\partial{t^{2}}}}{\psi_{\tau}}\to-\imath\frac{1}{{2m}}\left({\frac{\partial}{{\partial
t}}\psi_{\tau}^{*}}\right)\frac{\partial}{{\partial
t}}{\psi_{\tau}}+\imath\frac{1}{{2m}}\left({\frac{\partial}{{\partial
t}}\psi_{\tau}^{*}}\right)\frac{\partial}{{\partial t}}{\psi_{\tau}}=0$ (107)
leaving:
$\mathop{\smallint}\limits_{-\infty}^{\infty}dt{D_{\tau}}\left(t\right)=\frac{\imath}{{2m}}\mathop{\smallint}\limits_{-\infty}^{\infty}dt\mathop{\smallint}\limits_{-\infty}^{0}dx\left({\left({\frac{{{\partial^{2}}\psi_{\tau}^{*}\left({t,x}\right)}}{{\partial{x^{2}}}}}\right){\psi_{\tau}}\left({t,x}\right)-\psi_{\tau}^{*}\left({t,x}\right)\left({\frac{{{\partial^{2}}{\psi_{\tau}}\left({t,x}\right)}}{{\partial{x^{2}}}}}\right)}\right)$
(108)
We have the probability current in the $x$ direction at each instant in
coordinate time for fixed clock time:
$j_{\tau}\left({t,x}\right)\equiv-\frac{\imath}{{2m}}\frac{{\partial\psi_{\tau}^{*}\left({t,x}\right)}}{{\partial
x}}{\psi_{\tau}}\left({t,x}\right)+\frac{\imath}{{2m}}\psi_{\tau}^{*}\left({t,x}\right)\frac{{\partial{\psi_{\tau}}\left({t,x}\right)}}{{\partial
x}}$ (109)
$j_{\tau}\left({t,x}\right)=\left({\frac{{p_{x}}}{{2m}}\psi_{\tau}^{*}\left({t,x}\right)}\right){\psi_{\tau}}\left({t,x}\right)+\psi_{\tau}^{*}\left({t,x}\right)\frac{{p_{x}}}{{2m}}{\psi_{\tau}}\left({t,x}\right)$
(110)
What we are after is the full detection rate at a specific coordinate time:
$D\left(t\right)=\int{d\tau{D_{\tau}}\left(t\right)}$ (111)
If we accept that the equality in equation 108 applies at each instant in
coordinate time (detailed balance) we have:
${D_{\tau}}\left(t\right)={\left.{{j_{\tau}}\left({t,x}\right)}\right|_{x=0}}$
(112)
giving:
$D\left(t\right)={\left.{\int{d\tau{j_{\tau}}\left({t,x}\right)}}\right|_{x=0}}$
(113)
Recall that the duration in clock time of a path represents the length of the
path – in the discrete case the number of steps. To get the sum of all paths
ending at a specific coordinate time, we need to sum over all possible path
lengths. That is this integral.
As in the SQM we are assuming that once the paths enter the detector they do
not return. And as with SQM, this means we have at best a reasonable first
approximation. However as the SQM approximation works well in practice and as
we are interested only in first corrections to SQM, this should be enough to
achieve falsifiability.
#### 3.2.3 Application to a Gaussian test function
If we construct the wave function as the direct product of time and space
parts:
${\varphi_{\tau}}\left({t,x}\right)={{\tilde{\varphi}}_{\tau}}\left(t\right){{\bar{\varphi}}_{\tau}}\left(x\right)$
(114)
then the expression for the probability current simplifies:
${j_{\tau}}\left({t,x}\right)=\left({\left({\frac{{p_{x}}}{{2m}}\bar{\psi}_{\tau}^{*}\left(x\right)}\right){{\bar{\psi}}_{\tau}}\left(x\right)+\bar{\psi}_{\tau}^{*}\left(x\right)\frac{{p_{x}}}{{2m}}{{\bar{\psi}}_{\tau}}\left(x\right)}\right)\tilde{\varphi}_{\tau}^{*}\left(t\right){{\tilde{\varphi}}_{\tau}}\left(t\right)$
(115)
or:
${D_{\tau}}\left(t\right)={{\bar{D}}_{\tau}}{{\tilde{\rho}}_{\tau}}\left(t\right)$
(116)
Using the time wave function from equation 100 we have for the non-
relativistic probability density in time:
$\tilde{\rho}_{\tau}\left(t\right)=\sqrt{\frac{1}{\pi\sigma_{t}^{2}\left|f_{\tau}^{\left(t\right)}\right|^{2}}}e^{-\frac{1}{\sigma_{t}^{2}\left|f_{\tau}^{\left(t\right)}\right|^{2}}\left(t-\tau\right)^{2}}$
(117)
To get the probability for detection at coordinate time $t$ we convolute clock
time with coordinate time. To solve we look at the limit in long time (as with
equation 21):
$\sigma_{t}^{2}\left|f_{\tau}^{\left(t\right)}\right|^{2}=\sigma_{t}^{2}+\frac{\tau^{2}}{m^{2}\sigma_{t}^{2}}=\sigma_{t}^{2}+\frac{\left(\bar{\tau}+\delta\tau\right)^{2}}{m^{2}\sigma_{t}^{2}}\approx\frac{\left(\bar{\tau}+\delta\tau\right)^{2}}{m^{2}\sigma_{t}^{2}}\approx\frac{\bar{\tau}^{2}}{m^{2}\sigma_{t}^{2}}$
(118)
This gives the probability density:
${{\tilde{\rho}}_{\tau}}\left(t\right)\approx\sqrt{\frac{1}{{\pi\tilde{\sigma}_{\tau}^{2}}}}{e^{-\frac{1}{{\tilde{\sigma}_{\tau}^{2}}}{{\left({t-\tau}\right)}^{2}}}},{{\tilde{\sigma}}_{\tau}}\equiv\frac{{\bar{\tau}}}{{m{\sigma_{t}}}}$
(119)
The convolution over $\tau$ is trivial giving:
${{\rho_{\bar{\tau}}}\left(t\right)=\sqrt{\frac{1}{{\pi\sigma_{\tau}^{2}}}}{e^{-\frac{{{\left({t-\bar{\tau}}\right)}^{2}}}{{\sigma_{\tau}^{2}}}}}}$
(120)
with the total dispersion in clock time being the sum of the dispersions in
coordinate time and in space:
${\sigma_{\tau}^{2}\equiv\tilde{\sigma}_{\tau}^{2}+\bar{\sigma}_{\tau}^{2}}$
(121)
This is reasonably simple. The uncertainty is:
$\Delta\tau=\frac{1}{{\sqrt{2}}}\sqrt{\tilde{\sigma}_{\tau}^{2}+\bar{\sigma}_{\tau}^{2}}$
(122)
We collect the definitions for the two dispersions:
$\begin{array}[]{c}\bar{\sigma}_{\tau}^{2}=\frac{{{\bar{\tau}}^{2}}}{{{m^{2}}{v^{2}}\sigma_{x}^{2}}}\hfill\\\
\tilde{\sigma_{t}}^{2}\approx\frac{{{\bar{\tau}}^{2}}}{{{m^{2}}\sigma_{t}^{2}}}\hfill\end{array}$
(123)
As noted, we expect the particle wave functions to have initial dispersions in
energy/time comparable to their dispersions in momentum/space:
$\sigma_{t}\sim\sigma_{x}$ (124)
In the non-relativistic case, $v\ll 1$ the total uncertainty will be dominated
by the space part. While we have looked specifically at wave functions
composed of a direct product of GTFs in time and space, the underlying
arguments are general. Therefore we expect that in the case of non-
relativistic particles, the uncertainty in time-of-arrival will show evidence
of dispersion in coordinate time, but will be dominated by the dispersion in
space, because this enters with a $\frac{1}{v}$ factor and in the non-
relativistic case $v$ is by definition small.
## 4 Applications
##### Single slit in time
At non-relativistic velocities, we expect that the uncertainty in time-of-
arrival will be dominated by the space part. To increase the effect of
uncertainty in time we can run the wave function through a single slit in
time, i.e. a camera shutter which opens and closes very rapidly. In SQM, the
wave function will merely be clipped and the uncertainty in time at the
detector correspondingly reduced. But in TQM the reduction in $\Delta t$ will
increase $\Delta E$. The increase in uncertainty in energy will cause the wave
function to be more spread out in time, making the uncertainty in time at the
detector arbitrarily great. The case is analogous to the corresponding single
slit in space, with $\Delta E,\Delta t$ substituting for $\Delta p,\Delta x$.
In our previous paper we examined this case, but our analysis took as its
starting point the Kijowski metric and associated approaches. In particular,
we shifted back and forth between $q$ and $p$ in phase space in a way that we
now recognize as suspect.
This obscured an important subtlety in the case of non-relativistic particles.
Consider a non-relativistic particle going through a time gate. If the gate is
open for only a short time, the particle must have come from an area close to
the time gate. Therefore its position in space is also fairly certain. If the
particle is traveling with non-relativistic velocity $v\ll 1$ then $\Delta
x\sim v\Delta t\Rightarrow\Delta x\ll\Delta t$. This in turn drives up the
uncertainty in momentum, $\Delta p\sim{1\mathord{\left/{\vphantom{1{\Delta
x}}}\right.\kern-1.2pt}{\Delta x}}$, leading to increased uncertainty in time
at the detector, even for SQM.
To avoid this problem, we replace the gate with a time-dependent source. We
let the particles propagate for a short time past the source, then compare the
expected uncertainties without and with dispersion in time. The basic
conclusion of the previous work is unchanged: we can make the difference
between the two cases arbitrarily large. We work out the details in appendix
C.
##### Double slit in time
The double slit experiment is often cited as _the_ key experiment for quantum
mechanics (mostly famously by Feynman [54]). And Lindner’s “Attosecond Double-
Slit Experiment” [86] provided a key inspiration for the current work, in that
it showed we could now explore interference effects at the necessary
attosecond scale. Building on this, Palacios et al [87] have proposed an
experiment using spin-correlated pairs of electrons which takes advantage of
the electrons being indistinguishable to look for interference in the time-
energy domain111111Unfortunately as far as we know this experiment has not yet
been performed..
Both experiments use the standard quantum theory to do the analysis of
interference in the time/energy dimension. Horwitz [25, 88, 27] notes that as
time is a parameter in standard quantum mechanics it is difficult to fully
justify this analysis. He provides an alternative analysis in terms of the
Stueckelberg equation (our eq 83) which is not subject to this objection. He
gets the same spacing for the fringes as Lindner found.
Unfortunately for this investigation since the fringe spacing is the same
using the Lindner’s analysis or the one here the fringe pattern does not let
us distinguish between the two approaches; it contributes nothing to
falsifiability.
We do expect, from the analysis of the single slit experiment, that the
individual fringes will be smoothed out from the additional dispersion in
time: the brights less bright, the darks less dark. However the contribution
of this effect to falsifiability is already covered by the analysis of the
single slit experiment.
##### Quantum electrodynamics
While the above is sufficient to establish that we can detect the effects of
uncertainty in time, it is clear that the strongest and most interesting
effects will be seen at high energies and relativistic speeds, where the
effects of quantum field theory (QFT) will need to be taken into account.
In the previous paper, we looked at the QFT approach to systems of spinless,
massive particles. We were able to extend TQM to QFT in a straightforward and
unambiguous way. We also showed that we recover the usual Feynman rules in the
appropriate limit. And that we will see various interesting effects from the
additional uncertainty in time.
One obvious concern was that the additional dimension in the loop integrals
might make the theory still more divergent than QFT already is, perhaps even
unrenormalizable. But in the spin 0 massive case we do not see the usual
ultraviolet (UV) divergences: the combination of uncertainty in time,
entanglement in time, and the use of finite initial wave functions (rather
than plane waves or $\delta$ functions) suppresses the UV divergences without
the need for regularization.
The extension to spin 1/2 particles and to photons appears to be
straightforward. We expect to see the various effects of time-as-observable
for spin 1/2 particles and photons: interference in time, entanglement in
time, correlations in time, and so on. We will also look at the implications
for spin correlations, symmetry/anti-symmetry under exchange of identical
particles, and the like.
One caveat: if we analyze the photon propagator using the familiar trick of
letting the photon mass $\mu\to 0$[89] we can see from inspection of the
$\frac{1}{2\mu}$ factor in the propagator in momentum space:
${K_{\tau}}\left({k;k^{\prime}}\right)=\exp\left({\imath\frac{{{k^{2}}-{\mu^{2}}}}{{2\mu}}\tau}\right){\delta^{4}}\left({k-k^{\prime}}\right)$
(125)
that excursions off-shell will be severely penalized. This in turn will limit
the size of the effects of TQM in experiments such as the single slit in time.
Such experiments are therefore better run with massive particles, the more
massive the better in fact.
In this paper we have estimated the initial wave functions in time using
dimensional and maximum entropy arguments. Once TQM has been extended to
include QED, we should be able to combine the standard QED results with these
to serve as the zeroth term in a perturbation expansion, then compute first
and higher corrections using the standard methods of time-dependent
perturbation theory.
This should open up a wide variety of interesting experiments.
##### General Relativity
As noted in the introduction, TQM is a part of the relativistic dynamics
program so can draw on the extensive relativistic dynamics literature. In
particular we can take advantage of Horwitz’s extension of relativistic
dynamics to General Relativity for single particles, the classical many body
problem, and the quantum many body problem [90, 91, 92].
There would appear to be an interesting complementarity between the work of
Horwitz and this work. First, TQM supplies an explicit source for the
invariant monotonic parameter $\tau$ so that we do not need to introduce this
as an additional assumption. Second, TQM avoids the UV divergences which have
created significant difficulties for work to extend QFT to General Relativity.
Therefore an appropriate combination of the two approaches might provide
useful insights into the extension of QFT to General Relativity.
## 5 Discussion
> “It is not surprising that our language should be incapable of describing
> the processes occurring within the atoms, for, as has been remarked, it was
> invented to describe the experience of daily life, and these consist only of
> processes involving exceedingly large numbers of atoms. Furthermore, it is
> very difficult to modify our language so that it will be able to describe
> these atomic processes, for words can only describe things of which we can
> form mental pictures, and this ability, too, is a result of daily
> experience. Fortunately, mathematics is not subject to this limitation, and
> it has been possible to invent a mathematical scheme — the quantum theory —
> which seems entirely adequate for the treatment of atomic processes…” –
> Werner Heisenberg [93]
###### Falsifiability
The Heisenberg uncertainty principle (HUP) in time follows directly from
quantum mechanics and relativity. This was clear to both Einstein and Bohr in
1930. However quantum mechanics since then has not in general included it.
Given that the relevant attosecond time scales have been too small to work
with – until recently – this is reasonable enough.
However it is now possible to look at the time at the relevant scales; it is
therefore necessary to do so, if we are understand either time or quantum
mechanics. The specific predictions made here are based on only covariance and
the requirement of consistency with existing results, they are by construction
without free parameters. They should therefore provide at a minimum a
reasonable order-of-magnitude target.
The time-of-arrival experiment we have focused on here provides merely the
most obvious line of attack. Essentially any experiment looking at time-
dependent phenomena is a potential candidate. For instance in Auletta [94]
there are about three hundred foundational experiments; most can be converted
into a test of uncertainty in time by replacing an $x$ with a $t.$ Examples
include the single slit in time, the double slit in time, and so on. One may
also look at variations intended to multiply normally small effects, as
experiments using resonance or diffraction effects.
###### Implications of negative result striking
From symmetry, one may argue that a positive result is the more likely, but
even a negative result would be interesting.
For instance, assume we have established there is no HUP in time in one frame.
Consider the same experiment from another frame moving at high velocity
relative to the first. Consider say a Gaussian test function (GTF) in $x$ in
the first; make a Lorentz boost of this into the second frame. The Lorentz
boost will transform $x\to\gamma\left(x-v\tau\right)$. This will turn the
uncertainty in space into a mixture of uncertainty in space and in time. We
then look for uncertainty in time in the boosted frame.
If the HUP in time is also rejected in the second, how do we maintain the
principle of covariance? If the HUP in time is present in the second frame,
then we can defined a preferred frame as the one in which HUP in time is
maximally falsified. Such a preferred frame is anathema to general relativity.
Therefore exploring the precise character of the negative result – uniform
across frames, more in some frames than others, and so on – would itself
represent an interesting research program.
###### Positive result would have significant practical applications
If on the other hand, the wave function extends in time, this would not only
be interesting from the point of view of fundamental physics, it would open up
a variety of practical applications. For instance, there would be an
additional channel for quantum communication systems to use. Memristors and
other time-dependent circuit elements would show interesting effects. In
attosecond chemistry and biochemistry we would expect to see forces of
anticipation and regret; if the wave function extends in time, it will cause
interactions to start earlier and end later than would otherwise be the case.
The mysteries of protein folding could be attacked from a fresh perspective
and perhaps unexpected temporal subtleties found.
The applications in quantum computing are particularly interesting. Quantum
computers will need to compensate for the effects of decoherence along the
time dimension. But they should also be able to take advantage of additional
computing opportunities along the time dimension. And if we have a deeper
understanding of the relationship between time and measurement, we may find
opportunities to “cut across the diagonal” in the design of quantum computers.
I thank my long time friend Jonathan Smith for invaluable encouragement,
guidance, and practical assistance.
I thank Ferne Cohen Welch for extraordinary moral and practical support.
I thank Martin Land, L. P. Horwitz, and the other organizers and participants
of the International Association for Relativistic Dynamics (IARD) 2020
Conference for encouragement, useful discussions, and hosting a talk on this
paper at the IARD 2020 conference.
I thank the reviewer who drew my attention to Horwitz’s [90].
I thank Steven Libby for several useful conversations.
I thank Larry Sorensen for many helpful references. I thank Ashley Fidler for
helpful references to the attosecond physics literature.
And I thank Avi Marchewka for an interesting and instructive conversation
about various approaches to time-of-arrival measurements.
And I note none of the above are in any way responsible for any errors of
commission or omission in this work.
.
## Appendix A Direct computation of the detection rate in diffusion
We use conservation of probability and the method of images to make a direct
computation of the detection rate in the case of diffusion. The approach is
similar to the one used by Marchewka and Schuss to compute the probability
current (subsection 3.1.1), although the context is classical rather than
quantum.
Define ${G_{\tau}}\left(x\right)$ as the probability to get from $d\to x$
without touching the detector at $0$. We get $G$ from the method of images by
the same logic as in the discrete case:
${G_{\tau}}\left(x\right)={P_{\tau}}\left({x;-d}\right)-{P_{\tau}}\left({x;d}\right)$
(126)
$G$ obeys the diffusion equation:
$\frac{{\partial{G_{\tau}}}}{{\partial\tau}}=\frac{1}{{2m}}\frac{{\partial^{2}}}{{\partial{x^{2}}}}{G_{\tau}}$
(127)
Take $D_{\tau}$ as the rate of detection at $\tau$. From conservation of
probability we have:
$1=\int\limits_{-\infty}^{0}{dx{G_{\tau}}\left(x\right)}+\int\limits_{0}^{\tau}{d\tau^{\prime}{D_{\tau^{\prime}}}}$
(128)
Take the derivative with respect to $\tau$ and apply the diffusion equation:
${D_{\tau}}=-\int\limits_{-\infty}^{0}{dx\frac{{\partial{G_{\tau}}}}{{\partial\tau}}\left(x\right)}=-\int\limits_{-\infty}^{0}{dx\frac{1}{{2m}}\frac{{\partial^{2}}}{{\partial{x^{2}}}}G}$
(129)
Since the term on the right is the integral of a derivative we have:
${D_{\tau}}=-\frac{1}{{2m}}{\left.{\frac{\partial}{{\partial
x}}G}\right|_{x=0}}$ (130)
By using the explicit form for $P$ (equation 49) we get:
${D_{\tau}}=\frac{d}{\tau}{P_{\tau}}\left(d\right)$ (131)
as above (equation 51).
## Appendix B Alternate derivation of the detection amplitude
Figure 5: Computation of amplitude for first arrival by using the Laplace
transform
We compute $F$ by using the Laplace transform.
To get to any specific point $x$ we have to get to the point the first time,
then return to it zero or more times. Therefore we can write the full kernel
$K$ as the convolution of the kernel $F$ to arrive for the first time and the
kernel $U$ to return zero or more times:
${K_{\tau}}\left({x;x^{\prime}}\right)=\int\limits_{0}^{\tau}{d\tau^{\prime}{U_{\tau^{\prime}}}\left(x\right){F_{\tau^{\prime}}}\left({x;x^{\prime}}\right)}$
(132)
Since the kernel is invariant under space translation:
$\begin{gathered}{K_{\tau}}\left({x;x^{\prime}}\right)={K_{\tau}}\left({x-x^{\prime}}\right)\hfill\\\
{F_{\tau^{\prime}}}\left({x;x^{\prime}}\right)={F_{\tau}}\left({x-x^{\prime}}\right)\hfill\end{gathered}$
(133)
and:
${U_{\tau}}\left(x\right)={U_{\tau}}\left(0\right)={K_{\tau}}\left(0\right)$
(134)
we can simplify the convolution to:
${K_{\tau}}\left({x-x^{\prime}}\right)=\int\limits_{0}^{\tau}{d\tau^{\prime}{U_{\tau^{\prime}}}F_{\tau^{\prime}}\left({x-x^{\prime}}\right)}$
(135)
The Laplace transform of $K$ is the product of the Laplace transform of $U$
and $F$:
$\mathcal{L}\left[K\right]=\mathcal{L}\left[U\right]\mathcal{L}\left[F\right]$
(136)
The free kernel is:
${K_{\tau}}\left(x\right)=\sqrt{\frac{m}{{2\pi\imath\tau}}}\exp\left({\imath
m\frac{{x^{2}}}{{2\tau}}}\right)$ (137)
with Laplace transform:
$\mathcal{L}\left[K\right]=-\left({-{1^{3/4}}}\right)\frac{{e^{\left({-1+\imath}\right)\sqrt{ms}\left|x\right|}}}{{\sqrt{2}\sqrt{s}}}\sqrt{m}$
(138)
To get the Laplace transform of $U$ take the $x=0$ case:
$\mathcal{L}\left[U\right]=-\frac{{\left({-{1^{3/4}}}\right)}}{{\sqrt{2}\sqrt{s}}}\sqrt{m}$
(139)
Therefore we have the Laplace transform for $F$:
$\mathcal{L}\left[F\right]={e^{\left({-1+\imath}\right)\sqrt{ms}\left|x\right|}}$
(140)
and we get $F_{\tau}$ from the inverse Laplace transform:
$\begin{gathered}\hfill\\\
{F_{\tau}}=\frac{\left|x\right|}{\tau}{K_{\tau}}\left(x\right)\hfill\end{gathered}$
(141)
## Appendix C Single slit in time
We compare the results of the single slit in time in SQM and TQM. We model the
single slit in time as a particle source located at $x=-d$, emitting particles
with momentum $p_{0}$ in the $x$-direction, velocity
${v_{0}}=\frac{{p_{0}}}{m}$.
In the SQM case, we assume that the wave function is emitted with probability
$\bar{G}_{\tau}$:
$\bar{G}\left(\tau\right)\equiv\frac{1}{{\sqrt{2\pi{W^{2}}}}}\exp\left({-\frac{{\tau^{2}}}{{2{W^{2}}}}}\right)$
(142)
This is normalized to one, with uncertainty in time $\Delta\tau=W$.
To extend to TQM we will replace this probability with an amplitude:
${{\tilde{\varphi}}_{0}}\left(t_{0}\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{t}^{2}}}}}\exp\left({-\imath
E_{0}t_{0}-\frac{{t_{0}^{2}}}{{2\sigma_{t}^{2}}}}\right)$ (143)
This has probability distribution and uncertainty:
$\begin{gathered}{{\tilde{\rho}}_{0}}\left({t_{0}}\right)=\sqrt{\frac{1}{{\pi\sigma_{t}^{2}}}}\exp\left({-\frac{{t_{0}^{2}}}{{\sigma_{t}^{2}}}}\right)\\\
\Delta t=\frac{{\sigma_{t}}}{{\sqrt{2}}}\end{gathered}$ (144)
We will take ${\sigma_{t}}=\sqrt{2}W$ so that the uncertainty from the gate is
equal to the uncertainty from the initial distribution in coordinate time, to
make the comparison as fair as possible.
The detector will be positioned at $x=0$. The average time of arrival is
$\bar{t}=\bar{\tau}=\frac{d}{{v_{0}}}$. We are interested in the uncertainty
in time at the detector as defined above for SQM and TQM.
### C.1 Single slit in time in SQM
Figure 6: Single slit in time in SQM
In SQM the wave function extends in only the three space dimensions. If we
break time up into slices, there is no interference across slices. At each
time slice, there is an amplitude for paths from that slice to get to the
detector. But if we look at the individual paths emitted during that time
slice, some get to the detector at one time, some at an earlier or a later
time. At the detector all paths that arrive at the same clock time interfere,
constructively or destructively, as may happen. This is the picture used by
Lindner et al in their analysis. The peaks in the incoming electric wavelet
ejected electrons at different times, but when the electrons ejected from
different times arrived at the detector at the same time they interfered.
To make this specific, we will use a simple model.
For convenience, we will assume that the source has an overall time dependency
of $\exp\left({-\imath\frac{{p_{0}^{2}}}{{2m}}{\tau_{G}}}\right)$ so that our
initial wave function is:
${{\bar{\varphi}}_{G}}\left({x_{G}}\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{x}^{2}}}}}{e^{\imath{p_{0}}{x_{G}}-\frac{1}{{2\sigma_{x}^{2}}}x_{G}^{2}-\frac{{p_{0}^{2}}}{{2m}}{\tau_{G}}}}$
(145)
This is normalized to one at the start:
$1=\int{d{x_{G}}\varphi_{G}^{*}\left({x_{G}}\right){\varphi_{G}}\left({x_{G}}\right)}$
(146)
So the particle will have a total probability of being emitted of one:
$1=\int\limits_{-\infty}^{\infty}{d\tau{{\bar{G}}_{\tau}}\int{d{x_{G}}\varphi_{G}^{*}\left({x_{G}}\right){\varphi_{G}}\left({x_{G}}\right)}}$
(147)
The amplitude at the detector from a single moment at the gate will be given
by:
${{\bar{\varphi}}_{DG}}\left(x\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{x}^{2}}}}}\sqrt{\frac{1}{{f_{DG}^{\left(x\right)}}}}{e^{\imath{p_{0}}x-\frac{1}{{2\sigma_{x}^{2}f_{DG}^{\left(x\right)}}}{{\left({x+d-{v_{0}}{\tau_{DG}}}\right)}^{2}}-\imath\frac{{p_{0}^{2}}}{{2m}}{\tau_{D}}}}$
(148)
with ancillary definitions:
$\begin{gathered}{\tau_{DG}}\equiv{\tau_{D}}-{\tau_{G}}\hfill\\\
f_{DG}^{\left(x\right)}=1+\imath\frac{{\tau_{DG}}}{{m\sigma_{x}^{2}}}\hfill\end{gathered}$
(149)
We have:
$x+d-{v_{0}}\left({{\tau_{D}}-{\tau_{G}}}\right)=-{v_{0}}\left({\delta\tau-{\tau_{G}}}\right)$
(150)
Both $\delta\tau$ and $\tau_{G}$ are expected small. We can therefore justify
taking:
$f_{DG}^{\left(x\right)}=1+\imath\frac{{\tau_{DG}}}{{m\sigma_{x}^{2}}}\approx
f_{\bar{\tau}}^{\left(x\right)}=1+\imath\frac{{\bar{\tau}}}{{m\sigma_{x}^{2}}}$
(151)
Giving:
${{\bar{\varphi}}_{DG}}\left(x\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{x}^{2}}}}}\sqrt{\frac{1}{{f_{\vec{\tau}}^{\left(x\right)}}}}{e^{\imath{p_{0}}x-\frac{{v_{0}^{2}}}{{2\sigma_{x}^{2}f_{\vec{\tau}}^{\left(x\right)}}}{{\left({\delta\tau-{\tau_{G}}}\right)}^{2}}-\imath\frac{{p_{0}^{2}}}{{2m}}{\tau_{D}}}}$
(152)
To get the full wave function at the detector we need to take the convolution
of this with the gate function:
${{\bar{\psi}}_{D}}\left(x\right)=\int\limits_{-\infty}^{\infty}{d{\tau_{G}}\bar{G}\left({\tau_{G}}\right){{\bar{\varphi}}_{DG}}\left(x\right)}$
(153)
giving:
${{\bar{\psi}}_{D}}\left(x\right)=\frac{{\sigma_{x}}}{{v_{0}}}\sqrt[4]{{\frac{1}{{\pi\sigma_{x}^{2}}}}}\frac{1}{{\sqrt{2\pi\left({\frac{{\sigma_{x}^{2}}}{{v_{0}^{2}}}+{W^{2}}+\imath\frac{{\bar{\tau}}}{{mv_{0}^{2}}}}\right)}}}\exp\left({\imath{p_{0}}x-\frac{{{\left({\delta\tau}\right)}^{2}}}{{2\left({\frac{{\sigma_{x}^{2}}}{{v_{0}^{2}}}+{W^{2}}+\imath\frac{{\bar{\tau}}}{{mv_{0}^{2}}}}\right)}}-\imath\frac{{p_{0}^{2}}}{{2m}}{\tau_{D}}}\right)$
(154)
We can see that the effect of the gate is to increase the effective dispersion
in space:
$\frac{{\sigma_{x}^{2}}}{{v_{0}^{2}}}\to\frac{{\sigma_{x}^{2}}}{{v_{0}^{2}}}+{W^{2}}\Rightarrow\sigma_{x}^{2}\to\Sigma_{x}^{2}\equiv\sigma_{x}^{2}+v_{0}^{2}{W^{2}}$
(155)
The gate effectively adds an uncertainty of $vW$ to the original uncertainty
in space. As $vW$ is about the distance a particle would cross while the gate
is open, this is reasonable.
We get then as the associated uncertainty in time (eq: 76):
$\Delta\tau=\frac{1}{{\sqrt{2}}}{{\bar{\Sigma}}_{\tau}}=\frac{1}{{\sqrt{2}}}\frac{1}{{m{v_{0}}{\Sigma_{x}}}}\bar{\tau}$
The longer the gate stays open the greater the resulting uncertainty in time
at the detector. The shorter the gate stays open the less the uncertainty in
time at the detector, with the minimum being the uncertainty for a free GTF
released at the time and location of the gate.
### C.2 Single slit in time in TQM
Figure 7: Single slit in time in TQM
In TQM the wave function extends in all four dimensions. There is interference
across time so it is no longer legitimate to break time up into separate
slices (except for purposes of analysis of course). We will take the source as
centered at a specific moment in coordinate time, $t=0$.
We can write the four dimensional wave function as the product of the time
(eq: 143) and space parts:
${\varphi_{0}}\left({{t_{0}},{{\vec{x}}_{0}}}\right)={\tilde{\varphi}_{0}}\left({t_{0}}\right){\bar{\varphi}_{0}}\left({{\vec{x}}_{0}}\right)$
(156)
The space part is as above for clock time $\tau=0$.
The time part at clock time $\tau$ is:
>
> ${{\tilde{\varphi}}_{\tau}}\left(t\right)=\sqrt[4]{{\frac{1}{{\pi\sigma_{t}^{2}}}}}\sqrt{\frac{1}{{f_{\tau}^{\left(t\right)}}}}{e^{-\imath{E_{0}}t-\frac{1}{{2\sigma_{t}^{2}f_{\tau}^{\left(t\right)}}}{{\left({t-\frac{{E_{0}}}{m}\tau}\right)}^{2}}}}$
> (157)
For a non-relativistic particle $\frac{E}{m}\approx 1$. The treatment in
subsection 3.2.3 applies giving the uncertainty in time at the detector as:
$\Delta\tau=\frac{1}{{\sqrt{2}}}\sqrt{\tilde{\sigma}_{\tau}^{2}+\bar{\sigma}_{\tau}^{2}}$
(158)
with contributions from the space and time parts of:
$\begin{gathered}\bar{\sigma}_{\tau}^{2}=\frac{{{\bar{\tau}}^{2}}}{{{m^{2}}{v^{2}}\sigma_{x}^{2}}}\hfill\\\
\tilde{\sigma}_{\tau}^{2}=\frac{{{\bar{\tau}}^{2}}}{{{m^{2}}\sigma_{t}^{2}}}=\frac{{{\bar{\tau}}^{2}}}{{2{m^{2}}{W^{2}}}}\hfill\end{gathered}$
(159)
So we have the scaled uncertainty in time as:
$\frac{{\Delta\tau}}{{\bar{\tau}}}=\frac{1}{{\sqrt{2}}}\frac{1}{m}\sqrt{\frac{1}{{{v^{2}}\sigma_{x}^{2}}}+\frac{1}{{2{W^{2}}}}}$
(160)
We can see that when $W\sim v\sigma_{x}$ the effects of dispersion in time
will be comparable to those from dispersion in space. And as $W\to 0$, the
uncertainty in time at the detector will be dominated by the width of the gate
in time, going as $\frac{1}{W}$.
So the intrinsic uncertainty in space of a GTF creates a corresponding minimum
uncertainty in time at the detector given by $\bar{\sigma}_{\tau}$. In SQM the
effects of the gate drop to zero as $W\to 0$ while in TQM they go to infinity.
In SQM the wave function is clipped in time; in TQM it is diffracted.
We therefore have the unambiguous signal – even in the non-relativistic case –
that we need to achieve practical falsifiability.
## References
* [1] Feynman R P 1965 The Character of Physical Law (Modern Library) ISBN 0 679 60127 9
* [2] Schilpp P A and Bohr N 1949 Discussion with Einstein on Epistemological Problems in Atomic Physics (Evanston: The Library of Living Philosophers) pp 200–241
* [3] Pais A 1982 Subtle is the Lord: The Science and Life of Albert Einstein (Oxford University Press)
* [4] Hilgevoord J 1996 American Journal of Physics 64 1451–1456
* [5] Hilgevoord J 1998 American Journal of Physics 66 396–402
* [6] Busch P 2001 The Time Energy Uncertainty Relation (Berlin: Springer-Verlag) pp 69–98 Lecture Notes in Physics
* [7] Pauli W 1980 General Principles of Quantum Mechanics (Springer-Verlag) URL http://www.springer.com/us/book/9783540098423
* [8] Dirac P A M 1958 General Principles of Quantum Mechanics 4th ed International series of monographs on physics (Oxford, Clarendon Press)
* [9] Muga J G, Sala Mayato R and Egusquiza I L 2002 Time in Quantum Mechanics (Berlin; New York: Springer) ISBN 3540432949
* [10] Muga J G, Sala Mayato R and Egusquiza I L 2008 Time in Quantum Mechanics - Vol 2 (Berlin; New York: Springer-Verlag) ISBN 3540734724 9783540734727 0075-8450 ;
* [11] Ossiander M, Siegrist F, Shirvanyan V, Pazourek R, Sommer A, Latka T, Guggenmos A, Nagele S, Feist J, Burgdörfer J, Kienberger R and Schultze M 2016 Nature Physics 13 280
* [12] Abbott E A 1884 Flatland: A Romance of Many Dimensions (Seeley and Co. of London)
* [13] Ashmead J 2019 Journal of Physics: Conference Series 1239 012015 URL https://iopscience.iop.org/article/10.1088/1742-6596/1239/1/012015
* [14] Stueckelberg E C G 1941 Helv. Phys. Acta. 14 51
* [15] Stueckelberg E C G 1941 Helv. Phys. Acta. 14 322–323
* [16] Feynman R P 1948 Rev. Mod. Phys. 20(2) 367–387
* [17] Feynman R P 1949 Phys Rev 76 749–759
* [18] Feynman R P 1949 Phys Rev 76 769–789
* [19] Feynman R P 1950 Physical Review 80 440–457
* [20] Horwitz L P and Piron C 1973 Helvetica Physica Acta 46
* [21] Fanchi J R and Collins R E 1978 Found Phys 8 851–877
* [22] Fanchi J R 1993 Found. Phys. 23
* [23] Fanchi J R 1993 Parameterized Relativistic Quantum Theory (Fundamental Theories of Physics vol 56) (Kluwer Academic Publishers)
* [24] Land M C and Horwitz L P 1996 ArXiv e-prints (Preprint hep-th/9601021v1) URL http://lanl.arxiv.org/abs/hep-th/9601021v1
* [25] Horwitz L P 2006 Physics Letters A 355 1–6 (Preprint quant-ph/0507044) URL http://arxiv.org/abs/quant-ph/0507044
* [26] Fanchi J R 2011 Found Phys 41 4–32
* [27] Horwitz L P 2015 Relativistic Quantum Mechanics Fundamental Theories of Physics (Springer Dordrecht Heidelberg New York London)
* [28] Kijowski J 1974 Reports on Mathematical Physics 6 361–386
* [29] Marchewka A and Schuss Z 1998 Phys.Lett. A240 177–184 (Preprint quant-ph/9708034) URL https://arxiv.org/abs/quant-ph/9708034
* [30] Marchewka A and Schuss Z 1999 Phys. Lett. A 65 042112 (Preprint quant-ph/9906078) URL https://arxiv.org/abs/quant-ph/9906078
* [31] Marchewka A and Schuss Z 1999 ArXiv e-prints (Preprint quant-ph/9906003) URL https://arxiv.org/abs/quant-ph/9906003
* [32] Marchewka A and Schuss Z 2000 Phys. Rev. A 61 052107 (Preprint quant-ph/9903076) URL https://arxiv.org/abs/quant-ph/9903076
* [33] Muga J G and Leavens C R 2000 Physics Reports 338 353–438
* [34] Baute A D, Mayato R S, Palao J P, Muga J G and Egusquiza I L 2000 Phys.Rev.A 61 022118 (Preprint quant-ph/9904055) URL https://arxiv.org/abs/quant-ph/9904055
* [35] Baute A D, Egusquiza I L, Muga J G and Sala-Mayato R 2000 Phys. Rev. A 61 052111 (Preprint quant-ph/9911088) URL https://arxiv.org/abs/quant-ph/9911088
* [36] Baute A D, Egusquiza I L and Muga J G 2001 Phys. Rev. A 64 012501 (Preprint quant-ph/0102005) URL https://arxiv.org/abs/quant-ph/0102005
* [37] Ruggenthaler M, Gruebl G and Kreidl S 2005 Journal of Physics A: Mathematical and General 38 (Preprint quant-ph/0504185) URL https://arxiv.org/abs/quant-ph/0504185
* [38] Anastopoulos C and Savvidou N 2006 J. Math. Phys. 47 122106 (Preprint quant-ph/0509020) URL https://arxiv.org/abs/quant-ph/0509020
* [39] Yearsley J M 2010 ArXiv e-prints (Preprint 1012.2575v1) URL https://arxiv.org/abs/1012.2575v1
* [40] Kiukas J, Ruschhaupt A, Schmidt P O and Werner R F 2012 Journal of Physics A: Mathematical and Theoretical 45 (Preprint 1109.5087) URL https://arxiv.org/abs/1109.5087
* [41] Yearsley J M, Downs D A, Halliwell J J and Hashagen A K 2011 Phys. Rev. A 84 022109 (Preprint 1106.4767) URL https://arxiv.org/abs/1106.4767
* [42] Halliwell J J, Evaeus J, London J and Malik Y 2015 Physics Letters A 379 2445 (Preprint 1504.02509) URL https://arxiv.org/abs/1504.02509
* [43] Kijowski J, Waluk P and Senger K 2015 ArXiv e-prints (Preprint 1512.02867) URL https://arxiv.org/abs/1512.02867
* [44] Das S 2018 ArXiv e-prints (Preprint 1802.01863) URL https://arxiv.org/abs/1802.01863
* [45] Yearsley J M 2011 ArXiv e-prints (Preprint 1110.5790) URL https://arxiv.org/abs/1110.5790
* [46] Rayleigh L 1885 Proceedings of the London Mathematical Society s1-17 4–11 ISSN 0024-6115 (Preprint https://academic.oup.com/plms/article-pdf/s1-17/1/4/4368144/s1-17-1-4.pdf)
* [47] Bell J S 1964 Physics 1 195–200
* [48] Clauser J F, Horne M A, Shimony A and Holt R A 1969 Phys. Rev. Lett. 23(15) 880–884 URL htt
* [49] Aspect A, Grangier P and Roger G 1981 Phys. Rev. Lett. 47(7) 460–463
* [50] Aspect A, Grangier P and Roger G 1982 Phys. Rev. Lett. 49(2) 91–94
* [51] Aspect A, Dalibard J and Roger G 1982 Phys. Rev. Lett. 49(25) 1804–1807
* [52] Hensen B, Bernien H, Dréau A E, Reiserer A, Kalb N, Blok M S, Ruitenberg J, Vermeulen R F L, Schouten R N, Abellán C, Amaya W, Pruneri V, Mitchell M W, Markham M, Twitchen D J, Elkouss D, Wehner S, Taminiau T H and Hanson R 2015 Nature 526 682–686
* [53] Muga J G, Baute A D, Damborenea J A and Egusquiza I L 2000 ArXiv e-prints (Preprint quant-ph/0009111) URL https://arxiv.org/abs/quant-ph/0009111
* [54] Feynman R P, Leighton R B and Sands M 1965 The Feynman Lectures on Physics vol Volumes I, II, and III (Addison-Wesley)
* [55] Sakurai J J 1967 Advanced Quantum Mechanics (Addison-Wesley) ISBN 0 201 06710 2
* [56] Einstein A, Podolsky B and Rosen N 1935 Phys. Rev. 47 777–780
* [57] Bohr N 1935 Physical Review 48 696–702
* [58] Feynman R P, Hibbs A R and Styer D F 2010 Quantum Mechanics and Path Integrals (Mineola, N.Y.: Dover Publications) ISBN 9780486477220 0486477223 URL http://www.loc.gov/catdir/enhancements/fy1006/2010004550-d.html
* [59] Feller W 1968 An Introduction to Probability Theory and Its Applications vol I (John Wiley and Sons, Inc.)
* [60] Biggs N L 1989 Discrete Mathematics (Oxford Science Publications)
* [61] Graham R L, Knuth D E and Patashnik O 1994 Concrete Mathematics: A Foundation for Computer Science (Addison-Wesley Professional) ISBN 0201558025
* [62] Ibe O C 2013 Elements of Random Walk and Diffusion Processes (Hoboken, N.J.: John Wiley & Sons, Inc.) ISBN 1-118-61793-2; 1-118-61805-X; 1-118-62985-X
* [63] Grimmett G R and Stirzaker D R 2001 Probability and Random Processes 3rd ed (Oxford University Press)
* [64] Petersen A 1963 The Bulletin of the Atomic Scientists 19
* [65] Olsen J D and McDonald K T 2017 URL reference (Preprint https://www.physics.princeton.edu/ mcdonald/examples/orbitdecay.pdf) URL https://www.physics.princeton.edu/ mcdonald/examples/orbitdecay.pdf
* [66] Camilleri K and Schlosshauer M 2015 Stud. Hist. Phil. Mod. Phys. 49, 73-83 (2015) (Preprint 1502.06547v1) URL http://arxiv.org/abs/1502.06547v1
* [67] Zurek W H 1991 Physics Today 44 33–44 URL https://arxiv.org/abs/quant-ph/0306072
* [68] Omnès R 1994 The Interpretation of Quantum Mechanics (Princeton: Princeton University Press) ISBN 0 691 03669 1
* [69] Giulini D, Joos E, Kiefer C, Kupsch J, Stamatescu I O and Zeh H D 1996 Decoherence and the Appearance of a Classical World in Quantum Theory (Berlin: Springer-Verlag)
* [70] Zeh H D 1996 ArXiv e-prints (Preprint quant-ph/9610014) URL https://arxiv.org/abs/quant-ph/9610014
* [71] Zeh H D 2000 Lect.Notes Phys 538 19–42 URL https://arxiv.org/abs/quant-ph/9905004
* [72] Kim Y S and Noz M E 2003 Optics and Spectroscopy 94 (Preprint quant-ph/0206146) URL https://arxiv.org/pdf/quant-ph/0206146.pdf
* [73] Joos E 2003 Decoherence and the Appearance of a Classical World in Quantum Theory 2nd ed (Berlin: Springer) ISBN 3540003908 (acid-free paper) URL http://www.loc.gov/catdir/toc/fy038/2003042813.html
* [74] Schlosshauer M A 2007 Decoherence and the Quantum-to-Classical Transition (Berlin: Springer) ISBN 9783540357735 (hbk.) URL http://www.loc.gov/catdir/enhancements/fy0814/2007930038-b.html
* [75] Carmichael H 1991 An Open Systems Approach to Quantum Optics: Lectures Presented at the Université Libre de Bruxelles, October 28 to November 4, 1991 (Springer)
* [76] Dalibard J, Castin Y and Mølmer K 1992 Phys. Rev. Lett. 68(5) 580–583
* [77] Dum R, Zoller P and Ritsch H 1992 Phys. Rev. A 45(7) 4879–4887
* [78] Hegerfeldt G C and Wilser T S 1991 Classical and Quantum Systems: Foundations and Symmetries ed HD Doebner; W Scherer; F Schroeck J (World Scientific)
* [79] Kovachy T, Asenbaum P, Overstreet C, Donnelly C A, Dickerson S M, Sugarbaker A, Hogan J M and Kasevich M A 2015 Nature 528 530–533 URL https://doi.org/10.1038/nature16155
* [80] Asenbaum P, Overstreet C, Kovachy T, Brown D D, Hogan J M and Kasevich M A 2017 Phys. Rev. Lett. 118(18) 183602 URL https://link.aps.org/doi/10.1103/PhysRevLett.118.183602
* [81] Wells H G 1935 The Time Machine (Everyman) URL http://www.fourmilab.ch/etexts/www/wells/timemach/html/
* [82] Jackson J J, Puretzky A A, More K L, Rouleau C M, Eres G and Geohegan D B 2010 ACS Nano 4 7573–7581 URL https://doi.org/10.1021/nn102029y
* [83] Goldstein H 1980 Classical Mechanics Second Edition (Reading, MA: Addison-Wesley)
* [84] Merzbacher E 1998 Quantum Mechanics (New York: John Wiley and Sons, Inc.)
* [85] Pashby T 2014 Time and the Foundations of Quantum Mechanics Ph.D. thesis Dietrich School of Arts and Sciences, University of Pittsburgh URL http://d-scholarship.pitt.edu/21183/1/Pashby2014TimeFoundationsQM.pdf
* [86] Lindner F, Schätzel M G, Walther H, Baltuška A, Goulielmakis E, Krausz F, Milošević D B, Bauer D, Becker W and Paulus G G 2005 Physical Review Letters 95 040401 URL http://link.aps.org/doi/10.1103/PhysRevLett.95.040401 http://lanl.arxiv.org/abs/quant-ph/0503165
* [87] Palacios A, Rescigno T N and McCurdy C W 2009 Phys. Rev. Lett. 103(25) 253001 URL https://link.aps.org/doi/10.1103/PhysRevLett.103.253001
* [88] Horwitz L P 2007 Foundations of Physics 37 734–746 URL https://doi.org/10.1007/s10701-007-9127-7
* [89] Zee A 2010 Quantum Field Theory in a Nutshell (Princeton, N.J.: Princeton University Press) ISBN 9780691140346 (hardcover alk. paper)
* [90] Horwitz L P 2019 The European Physical Journal Plus 134 313 (Preprint 1810.09248) URL https://arxiv.org/abs/1810.09248
* [91] Horwitz L P 2020 The European Physical Journal Plus 135 479 URL https://doi.org/10.1140/epjp/s13360-020-00446-0
* [92] Horwitz L P 2021 The European Physical Journal Plus 136 32 URL https://doi.org/10.1140/epjp/s13360-020-00967-8
* [93] Heisenberg W 1930 The Physical Principles of the Quantum Theory (Chicago: University of Chicago Press)
* [94] Auletta G 2000 Foundations and Interpretation of Quantum Mechanics: In the Light of a Critical-Historical Analysis of the Problems and of a Synthesis of the Results (Singapore: World Scientific Publishing Co. Pte. Ltd.)
|
# Helioseismic Properties of Dynamo Waves in the Variation of Solar
Differential Rotation
Krishnendu Mandal New Jersey Institute of Technology, Newark, NJ 07102, USA
Alexander G. Kosovichev New Jersey Institute of Technology, Newark, NJ 07102,
USA NASA Ames Research Center, Moffett Field, CA 94035, USA Valery V. Pipin
Institute of Solar-Terrestrial Physics, Russian Academy of Sciences, Irkutsk,
664033, Russia
(Received –; Revised –; Accepted –)
###### Abstract
Solar differential rotation exhibits a prominent feature: its cyclic
variations over the solar cycle, referred to as zonal flows or torsional
oscillations, are observed throughout the convection zone. Given the challenge
of measuring magnetic fields in subsurface layers, understanding deep
torsional oscillations becomes pivotal in deciphering the underlying solar
dynamo mechanism. In this study, we address the critical question of
identifying specific signatures within helioseismic frequency-splitting data
associated with the torsional oscillations. To achieve this, a comprehensive
forward modeling approach is employed to simulate the helioseismic data for a
dynamo model that, to some extent, reproduces solar-cycle variations of
magnetic fields and flows. We provide a comprehensive derivation of the
forward modeling process utilizing generalized spherical harmonics, as it
involves intricate algebraic computations. All estimated frequency-splitting
coefficients from the model display an $11$-year periodicity. Using the
simulated splitting coefficients and realistic noise, we show that it is
possible to identify the dynamo wave signal present in the solar zonal flow
from the tachocline to the solar surface. By analyzing observed data, we find
similar dynamo wave patterns in the observational data from MDI, HMI, and
GONG. This validates the earlier detection of dynamo waves and holds potential
implications for the solar dynamo theory models.
Sun: waves – Sun: oscillations – solar interior – solar convection zone –
solar differential rotation
## 1 Introduction
Parker (1955) showed that dynamo action involves cyclic transformations of the
poloidal and toroidal magnetic fields of the Sun. This scenario suggests that
the magnetic field of bipolar magnetic regions is formed from the large-scale
toroidal magnetic field that is generated from the axisymmetric poloidal
magnetic field, twisted by differential rotation deep in the convection zone.
As the solar dynamo revolves around comprehending the cyclic evolution of
toroidal and poloidal magnetic fields, there is a consensus on the generation
of the toroidal field. However, the generation process of the poloidal field
remains less understood. Parker (1955) suggested that the poloidal magnetic
field is generated from the toroidal field by electromotive force excited by
cyclonic convection which is known as “$\alpha-$effect”. Several alternative
mechanisms for the poloidal field generation can be found in the literature
(Babcock, 1961; Choudhuri & Dikpati, 1999; Charbonneau & Barlet, 2011). An
alternative scenario, known as the flux transport model proposed by Babcock
(1961); Leighton (1969), posits that the cyclical evolution of solar activity
is influenced by meridional circulation. This circulation is responsible for
transporting the magnetic field of decaying sunspots from low latitudes toward
the polar region, as outlined by Dikpati et al. (2009). The transported
magnetic field subsequently undergoes reconnection with the polar field of the
preceding solar cycle and migrates to the base of the convection zone. In the
tachocline, this magnetic field is carried equatorward by the meridional flow
at the base of the convection zone. Eventually, it surfaces due to buoyancy
instability, resulting in the formation of observable sunspots. This scenario
gains validation through the observed migration of sunspots, originating from
mid-latitudes at the onset of the solar cycle and progressively migrating
towards the equator by its culmination. Observational evidence further
indicates that the polar magnetic field attains its peak during sunspot
minima, with its magnitude correlating with the subsequent cycle’s sunspot
number (Schatten et al., 1978). This information enables forecasting the
magnitude of the upcoming solar cycle (Choudhuri et al., 2007). Using this
technique, the forecast window spans $5$ to $6$ years, half the duration of
the $11$-year solar cycle. This supports the propositions by Parker (1955) and
Babcock (1961) that the toroidal magnetic field is generated through the
stretching of the poloidal magnetic field in polar regions. Whether this
conveyor belt scenario, which depends on the structure of the meridional flow,
is responsible for flux transport dynamo was under scrutiny. Helioseismology
has found that it can either be single cell (Rajaguru & Antia, 2015; Mandal et
al., 2018; Gizon et al., 2020) or double cell meridional flow (Zhao et al.,
2013; Chen & Zhao, 2017). However, all the studies to detect the deep
structure of the meridional flow suffer from centre-to-limb systematics in the
time-distance helioseismology measurements (Zhao et al., 2012). The
contribution from the systematic and random errors is comparable in magnitude
to the signal itself (Stejko et al., 2022). Aside from this difficulty, both
the surface flux transport and dynamo-waves can reproduce the basic observed
features of magnetic field evolution observed on the solar surface, including
the correlation of the polar magnetic field during the magnetic cycle minimum
with the subsequent sunspot cycle magnitude. The surface observations are not
sufficient to distinguish between two dynamo scenarios. Currently, we can not
measure sub-surface magnetic fields. Thus, the information about the dynamo
process comes from the measurement of large-scale subsurface flows such as
differential rotation and meridional flow. It is expected that variations of
these flows over the scale of $11$ years are due to the solar-cycle evolution
of subsurface magnetic fields. It is to be noted that these variations are not
only due to magnetic torque. Other inertial forces can also contribute to this
variation. Nevertheless observed variations in subsurface flows provide an
important clue about the mechanism of solar dynamo.
The torsional oscillations were discovered by Howard & Labonte (1980). After
subtracting the mean rotation, the data revealed alternating zones of fast and
slow zonal flow bands, which originated at mid-latitudes and subsequently
migrated to the equator just like the magnetic butterfly diagram. It was then
suggested that these zone flows (called torsional oscillations because of
their cyclic variations) are due to the action of the magnetic field of active
regions. The observed zonal flow pattern is long-living and coherent over
essentially the whole solar circumference. Therefore, it can not be of
convective origin but may be induced by the dynamo-generated magnetic fields
and associated processes. Numerical 3D simulation by Guerrero et al. (2016)
suggested that torsional oscillation in the model is due to the magnetic
torque at the base of the convection zone. However, modulation of the
turbulent heat transport by the magnetic field in the convection zone results
in variations in the zonal and meridional flows, producing the ‘extended’
22-year cycle of the torsional oscillations (Pipin & Kosovichev, 2019).
Helioseismology can probe the subsurface flow profile of solar zonal flow
(Howe et al., 2018; Basu & Antia, 2019). Kosovichev & Schou (1997) found that
torsional oscillations are extended beneath the solar surface. Later Vorontsov
et al. (2002) found that torsional oscillations are present in the entire
convection zone with the phase propagating poleward and equatorward from mid-
latitude at all depths throughout the convective envelope. Pipin & Kosovichev
(2019) suggested that torsional oscillation can be helpful for advanced
prediction of solar cycles, $1-2$ sunspot cycles ahead. Kosovichev & Pipin
(2019) argued that the amplitude of zonal acceleration in the high-latitude
region of the base of the convection zone during solar maxima may give
information about the strength of following solar cycle maxima. If confirmed,
this relationship would give the solar activity forecast a full $11$-year
cycle ahead. Pipin & Kosovichev (2020) argued that the sign of the correlation
between the flow amplitude in the deep convection zone and the strength of the
subsequent solar cycle depends on the properties of long-term variations of
the dynamo cycles.
Recognizing the significance of conclusions drawn from the aforementioned
studies, we systematically revisit the problem. We conduct synthetic tests to
validate the detection of the dynamo wave patterns within the solar zonal
flow. Utilizing the variations of the zonal flow velocity from the dynamo
model proposed by Pipin & Kosovichev (2020), we calculate associated
coefficients of rotational splitting of frequencies of solar oscillations
corresponding to the p-modes observed by the MDI. HMI, and GONG instruments.
The flow variations depicted in Figure 1, feature two distinct branches of
faster-than-average rotational components akin to the observed solar zonal
flows. Our aim is to determine the precision with which we can reconstruct the
zonal flows in the solar convection zones, from the tachocline to the solar
surface by using helioseismic data. Through these tests, we demonstrate the
high accuracy achievable in capturing the zonal flow across this entire depth
range if the observational data are averaged over two-year periods. We then
implement this methodology for measuring the zonal flow across the convection
zone using the current helioseismic data from MDI, HMI, and GONG.
## 2 Data analysis and methods
### 2.1 Rotational Frequency Splitting Data
We utilize a comprehensive set of global helioseismology data products
obtained from various instruments, including space-based ones like the
Helioseismic Magnetic Imager (HMI) (HMI; Schou et al., 2012) on board the
Solar Dynamic Observatory (SDO) and Michelson Doppler Imager (MDI) (MDI;
Scherrer et al., 1995) on board the Solar and Heliospheric Observatory (SoHO),
as well as ground-based instrument Global Oscillation Network Group (GONG)
(GONG; Hill et al., 1996). Each acoustic mode is characterized by a triplet of
quantum numbers, denoted as $(n,\ell,m)$, where $n$ signifies the radial
order, $\ell$ represents the spherical harmonic degree, and $m$ designates the
azimuthal order. In a spherically symmetric model, the mode frequency
$\nu_{n\ell m}$ remains independent of the azimuthal order, $m$. However, the
presence of axisymmetric perturbations, such as differential rotation, breaks
this degeneracy in $m$ which can be expressed as
$\nu_{n\ell
m}=\nu_{n\ell}+\sum_{s}^{s_{\text{max}}}a_{s}(n,\ell)\mathcal{P}_{s}^{\ell}(m),$
(1)
where $\nu_{n\ell}$ is the central frequency of a frequency multiple of radial
order $n$ and angular degree $\ell$; $a_{s}$ are the rotational splitting
coefficients (called the a-coefficients). The odd-order coefficients,
$a_{1},a_{3},a_{5}$, etc., capture information about the solar rotation.
$\mathcal{P}_{s}^{\ell}$ are polynomials of degree $s$ (Ritzwoller & Lavely,
1991), and these polynomials have orthogonal properties. The MDI data
encompass the period from May $1996$ to February $2011$, while the SDO/HMI
data used in this study cover the period from April $2010$ to May $2023$. We
collect solar frequency data and their corresponding splitting information
from distinct, non-overlapping $72$-day time series. The time series data for
the MDI and HMI instruments are sourced from the SDO Joint Science Operations
Center (JSOC) website111JSOC: http://jsoc.stanford.edu/. To investigate the
temporal evolution of solar zonal flow across the entire duration of our
analysis, we utilize a consolidated dataset comprising both MDI and HMI
instruments. In the overlapping time frame of both instruments, we rely on the
data exclusively from HMI.
GONG’s data span from May $1995$ to January $2023$. The data are categorized
by GONG’s “months”, each spanning a duration of 36 days. Solar oscillation
frequencies and splitting coefficients are derived through the analysis of a
108-day period, which is equal to observation of 3 GONG’s months. There is an
overlap of $72-$ days between different data sets.
Figure 1: The zonal flow velocities, derived from the dynamo model of Pipin &
Kosovichev (2020), are depicted across various depths. Notably, these
velocities exhibit both equatorward and poleward migrating branches stemming
from the mid-latitude area. Employing this zonal flow velocity, forward
modeling is conducted to estimate splitting coefficients. The right panel
illustrates the temporal evolution of splitting coefficients measured in nHz.
### 2.2 Forward Modelling Approach
The solar rotational velocity can be expressed as the following
$v(r,\theta)=\sum_{s=1,3,5,\cdots}w_{s}(r)\partial_{\theta}Y_{s,0}(\theta,\phi)$
(2)
where $Y_{s,0}(\theta,\phi)$ is a spherical harmonic of angular degree $s$ and
zero azimuthal order, $w_{s}(r)$ are the radial coefficients in the velocity
expansion, $r$ is the radius from the solar center, $\theta$ is the co-
latitude, and $\phi$ is the azimuthal angle. Rotation leads to the splitting
of p and f-mode frequencies with respect to the azimuthal order, $m$, of the
oscillation eigenfunctions. This phenomenon can be expressed through the
following integral equation:
$a_{s}(n,\ell)=\int_{0}^{R}K_{s}(n,\ell;r)w_{s}(r)r^{2}dr,$ (3)
Here, $K_{s}(n,\ell)$ represents the sensitivity kernel for the mode
$(n,\ell),$ determining the extent to which this specific mode is influenced
by rotation, characterized by a spatial scale $s$. The precise expression of
this function can be found in the appendix (see Appendix A). By leveraging
vector spherical harmonics, we have formulated the kernel expression. A
similar approach was previously employed by Ritzwoller & Lavely (1991). The
beauty of using the generalized spherical harmonics lies in their capability
to enable expressing derivatives with respect to $\theta$ and $\phi$ using
vector spherical harmonics itself. Consequently, the entire derivation
transforms into an elegant algebraic exercise. The use of generalized
spherical harmonics has recently become prevalent in studies focusing on
normal mode coupling (see e.g. Hanasoge et al., 2017). For an in-depth
derivation, please refer to Section A. We present the entire derivation due to
the availability of the splitting a-coefficients using various normalizations.
This often necessitates converting these coefficients into a preferred form
for ease of use. Equation 3 allows us to estimate the a-coefficients provided
we have a known rotation profile, denoted as $v(r,\theta)$, which is
decomposed into a spherical harmonic basis to derive $w_{s}$ using Equation 2.
Figure 2: The splitting coefficients’ variation over time is depicted here.
Left panel: we compare average a-coefficients measured in nHz for all the
modes with the turning point radius of the p-modes between $0.9R_{\odot}$ and
$1R_{\odot}$ by using data from three instruments, SOHO/MDI (blue marker),
SDO/HMI (red marker), and GONG (black marker). Right panel: we limit our
analysis to a maximum harmonic degree of $120$ for MDI. This results in
consistency across all three instruments.
### 2.3 Inversion Method
In the preceding section, we discussed the application of forward modeling
using a predefined rotation or zonal flow profile. These artificially
generated a-coefficients serve as the basis for inversion, enabling an
assessment of our ability to accurately replicate the zonal flow. To enhance
the realism of the inversion process, random noise is introduced based on
observed data. Employing a regularized least squares formula, we aim to
minimize the misfit function, denoted as $\chi$, as the following
$\chi^{2}=\sum_{n,\ell}\frac{1}{\sigma_{n,\ell}^{2}}\left(a_{s}(n,\ell)-\int
K_{n,\ell,s}(r)w_{s}(r)r^{2}dr\right)^{2}+\lambda\int\left(\frac{\partial^{2}w_{s}}{\partial
r^{2}}\right)^{2}r^{2}dr,$ (4)
We expect our solution for $w_{s}(r)$ to manifest as a linear combination of
splitting coefficients $a_{s}$.
$\displaystyle w_{s}(r_{0})=\sum_{i\in M}c_{i}(r_{0})a_{s}(i)$
$\displaystyle=\sum_{i\in M}c_{i}(r_{0})\int K_{i,s}(r)w_{s}(r)r^{2}dr$
$\displaystyle=\int\mathcal{K}(r;r_{0})w_{s}(r)r^{2}dr.$ (5)
Equation 5 reveals that our inverted $w_{s}$ at depth $r_{0}$ represents a
convolution of the true $w_{s}(r)$ profile, with $\mathcal{K}$ representing
the averaging kernel, $M$ represents the set of measurements for all
accessible acoustic modes. In Section D, we derive the expression for the
averaging kernel $\mathcal{K}(r;r_{0})$ and demonstrate its behavior at
various depths. Figure 9 illustrates that the averaging kernel tends to be
narrower near the surface and widens with greater depths. This observation
underscores our higher resolution capability close to the surface compared to
the base of the convection zone. To attain a smooth solution for $w_{s}(r)$,
we employ second-order derivative smoothing. The function, $w_{s}$, is
expanded on the basis of B-spline functions with $50$ equally spaced knots
along the acoustic depth, ranging from the surface to the center. Additional
details regarding our inversion method are available in Appendix C. To compute
error bars, we utilize the Monte Carlo method. This involves perturbing the
measured splitting coefficients in accordance with observed noise, generating
a sample set of $100$ realizations. Subsequently, we conduct inversions with
each sample to derive profiles for $w_{s}(r)$. The variation among these
different profiles of $w_{s}(r)$ allows us to calculate the standard
deviation, which in turn quantifies the error in $w_{s}$. We then use Equation
2 to estimate the error in velocity, $v(r,\theta)$. Our inversion technique
follows a $1.5$D approach. Other inversion methods are also available; for
instance, refer to Schou & Brown (1994); Schou et al. (1998).
## 3 Results
### 3.1 Forward Modeling of Helioseismic Effects of Torsional Oscillations
In their research, Pipin & Kosovichev (2020) delved into the connection
between the amplitude of the “extended” mode of migrating zonal flows and
solar magnetic cycles using a nonlinear mean-field solar dynamo model. Their
findings unveiled a notable correlation between torsional oscillation
parameters, such as zonal acceleration, and the subsequent cycle magnitudes,
albeit with a time lag ranging from $11$ to $20$ years. This discovery holds
great significance, suggesting that helioseismic observations of torsional
oscillations can offer valuable insights for more advanced predictions of
solar cycles, potentially forecasting solar activity $1$ to $2$ sunspot cycles
in advance. Given the substantial importance of this finding, we employ zonal
flow data from their simulation and engage in a forward modeling approach.
This allows us to gain insight into the specific signatures we should be
seeking within observed a-coefficients. This result is depicted in Figure 1.
We consider modes with turning-point radii between 0.95-1R⊙. We calculate the
average values of the a-coefficients of those mode sets. We only compare the
variations of the a-coefficients with time. Upon examining Figure 1, it
becomes evident that periodicity is present among all the a-coefficients
across all harmonic degrees. To determine if similar observations can be made
in the actual observed data, we conduct identical tests. We focused on the
modes with the turning-point radii ranging from 0.9-1R⊙. The results for the
observed a-coefficients from all three instruments are shown in Figure 2.
Similar to the findings of Basu & Antia (2019), we see periodicity in all the
a-coefficients in the observed data except for $a_{3}$, which does not exhibit
periodicity during the available time period of data. It is possible that it
reflects a longer trend. While the splitting coefficients from GONG and HMI
exhibit consistency (refer to the left panel of Figure 2), those from MDI do
not align. Following the approach of Basu & Antia (2019); Antia et al. (2008),
we limit our analysis to a maximum harmonic degree of $120$ for MDI, which
results in consistency across all three instruments (see right panel of Figure
2). To unveil the signature of dynamo waves from the tachocline to the solar
surface, we must address the presence of outliers in our observed data. Given
the relatively small magnitude of the signal, we can either eliminate outliers
through data cleaning or extend the duration of the time series to enhance the
signal-to-noise ratio.
### 3.2 Inversion for synthetic tests
We use noise from the $72$-day observational data to perturb the measurement.
We use a model from Pipin & Kosovichev (2020) to generate artificial
a-coefficients and perform inversion of these coefficients. In Figure 3, we
present a comparison between our inversion results and the original profile.
We showcase results for both scenarios: without noise and with added noise.
It’s evident that without noise, the inversion performs exceptionally well.
Even with added noise, the inversion effectively captures all the features of
the dynamo wave pattern from the tachocline to the surface. We specifically
show two harmonic degrees, $s=1$ and $s=3$.
Figure 3: Zonal flow derived from the dynamo model undergoes spherical
harmonic decomposition, as described by Equation 2. The left and right panels
illustrate our inversion outcomes for the $a$-coefficients of degrees $s=1$
and $s=3$, respectively. The top panel displays the original profile of
$w_{s}$, utilized in forward modeling to derive the $a$-coefficients from
Equation 3. These $a$-coefficients serve as inputs for the inversion process,
showcased in the middle panel. Following the initial calculation of the
$a$-coefficients as previously mentioned, they are subsequently subjected to
perturbation using Gaussian random noise proportional to the observed values.
The inversion is then conducted based on this perturbed data, with the
resulting analysis depicted in the bottom panel.
In Figure 4 within the dynamo model, we observe a band originating from the
tachocline and migrating toward the surface. This migration takes longer near
low-latitude regions ($<30^{\circ}$) but considerably less time at higher
latitudes, around $60^{\circ}$. This identical profile has been decomposed
into spherical harmonics to derive $w_{s}(r)$ using Equation 2, facilitating
forward modeling via Equation 3. Having demonstrated in Figure 3 that we can
accurately recover the original profile using our inversion methods, we
combine components from all harmonic degrees to derive the velocity using
Equation 2. We compare our inverted profile against the original profile in
Figure 4.
Figure 4: Zonal flow from the dynamo model is plotted against radius over time
across various latitudes, indicated in each panel’s title. The left panel
illustrates the inverted profile, while the right panel displays the original
profile.
### 3.3 Accuracy near the tachocline
Kosovichev & Pipin (2019) discussed the potential for predicting the strength
of future solar cycles by measuring torsional oscillation strength at the base
of the convection zone, particularly at latitude $60^{\circ}$. To undertake a
similar analysis, validating our accuracy near the base of the convection zone
is imperative. To accomplish this, we assess the accuracy through synthetic
tests. In Figure 5, we compare our inverted values with the original ones at
depth $0.75R_{\odot}$ and latitude $60^{\circ}$. We consider two cases. In one
instance, we apply noise akin to a $72$-day measurement, while in another, we
simulate noise comparable to a $2$-year measurement. We observe consistent
accuracy in both scenarios near the surface layers, depth $0.98R_{\odot}$.
However, there is a notable shift in the results as we approach the base of
the convection zone. This variation aligns with expectations due to reduced
sensitivity in deeper layers. P-modes spend less time traversing these depths,
primarily because of the increasing sound speed from the surface towards the
center of the Sun. This leads to an increase in the inversion error bars from
the surface towards the center. As the error bars in splitting coefficients
decrease with the square root of the total time, the inversion error bars near
the tachocline tend to be smaller for longer time series measurements. This
characteristic aids in interpreting variations in torsional oscillations
around the tachocline. In Figure 5, we illustrate this trend. JSOC typically
provides the $a$-coefficient data for the time series spanning 72 days.
However, Korzennik (2023) has presented the splitting coefficients across
various time series, including longer periods such as $1$ or $2$ years. This
varied dataset might significantly reduce uncertainties near the base of the
convection zone, aiding in understanding how torsional oscillation strength
evolves over time. This understanding could be pivotal in forecasting the
traits of forthcoming solar cycles (Kosovichev & Pipin, 2019; Pipin &
Kosovichev, 2020). However, it is important to note that long-term averaged
data might blend various dynamical changes occurring at the tachocline.
Figure 5: We present our synthetic inversion results across varying noise
levels. The left panels depict outcomes using noise akin to a $72$-day time
series, while the right panel reflects results obtained from a $2$-year time
series. All panels depict results corresponding to a latitude of $60^{\circ}$.
Evidently, employing the $2$-year time series yields superior outcomes,
particularly near the tachocline. However, both cases demonstrate equally
commendable results near the surface at $0.98R_{\odot}$. To display the error
bar at a depth of $0.98R_{\odot}$, we scaled it by a factor of $6$.
### 3.4 Inversion with observed data
After showcasing our ability to retrieve dynamo waves across the entire
convection zone through the $72$-day measurement, we proceeded to conduct
inversions using both MDI and HMI data sets, and during the overlap period, we
used the HMI data. Kosovichev & Pipin (2019) demonstrated the presence of
dynamo waves in solar zonal flow by utilizing a-coefficients from the JSOC
pipeline. To validate their analysis, we cross-checked using additional
datasets from the NSO/GONG pipeline. Confirming the presence of the dynamo
wave pattern in these alternative datasets would substantiate Kosovichev &
Pipin (2019) earlier detection. The results from the analysis of datasets
collected by all three instruments are depicted in Figure 6 and 7. We employ a
Gaussian filter with a one-year width in time to smooth the solution. In
Figure 7, two branches of zonal flow are visible, both originating near mid-
latitudes. One migrates towards the equator, while the other moves poleward.
Zonal flow patterns persist across three solar cycles, encompassing cycles
$23$, $24$, and ongoing cycle $25$. From Figure 6, the radial migration of the
high-latitude branch is fast compared to the low-latitude branch, which takes
a longer time to migrate from the tachocline to the surface.
Figure 6: Zonal flow velocity is plotted as a function of time and radius for
four latitudes, indicated in the title of each panel. The MDI and HMI results
are shown in the left panel, and the GONG results are shown in the right
panel.
Figure 7: Zonal flow velocities plotted as a function of time and latitude for
four depths mentioned in the title of each panel. The left panel illustrates
the results derived from the combined MDI and HMI datasets, while the right
panel showcases the findings specifically from the GONG datasets. Figure 8:
Evolution of solar zonal flow in the solar convection zone from the analysis
of MDI and HMI combined data sets.
## 4 Discussion and conclusions
The detection of dynamo waves holds significant importance in discerning the
operational dynamo within the solar interior—whether it aligns with Babcock-
Leighton or Parker’s dynamo. This study aims to ascertain the feasibility of
detecting dynamo-like wave patterns within the solar zonal flow. However, this
detection poses a challenge due to its subtle nature compared to the
prevailing mean rotation, which has a significantly higher magnitude.
Understanding the dynamo-wave-like structure within the zonal flows assumes
crucial importance as it could elucidate the type of dynamo active in the Sun.
This comprehension, in turn, might aid in predicting the characteristics of
subsequent solar cycles, thereby contributing significantly to space weather
prediction. Through forward modeling, we observe that variations in $a_{s}$
exhibit an $11$-year periodicity, paralleling similar characteristics observed
in the a-coefficients depicted in Figure 2, barring the potential for longer
periodicity in $a_{3}$ coefficients. This forward modeling, along with the
inversions, is necessary to decipher any subtle structure like dynamo waves
present in the solar zonal flow measurements. After developing a forward
modeling approach, we perform the inversion of the synthetic data sets. We
show in Figure 4 that we can quite accurately recover the dynamo waves-like
pattern in the zonal flow inferences from the tachocline to the surface.
After this testing, we used the observed data from three instruments MDI, HMI,
and GONG, and found a similar dynamo-wave-like pattern in the inferred solar
zonal flows. The similarity of the zonal flows deduced from the observations
and the dynamo models allows us to link the observed variations to the dynamo
waves, confirming the interpretation of Kosovichev & Pipin (2019). We find
that at high latitudes, the radial migration of the dynamo waves from the
tachocline to the surface is very fast, whereas it is of the order of several
years in the low-latitude areas. It exhibits a distinct behavior within the
near-surface layers, where the zonal flows migrate from the surface to the
bottom of the near-surface shear layer in the opposite direction compared to
the rest of the convection zone, as illustrated in Figure 6. This phenomenon
mirrors a similar structure previously observed by Vorontsov et al. (2002)
(refer to Figure $3$ in their study). Its origin has to be studied further. It
can be connected with the magnetic field feedback on the on the angular
momentum fluxes and the heat-transport in the near-surface layers. In the
presented dynamo model, a similar feature can be noticed in the time-radius
diagram at 45∘ latitude in Figure 4, but it is much less pronounced.
We can speculate a plausible dynamo scenario, previously discussed by
Kosovichev & Pipin (2019), in which the generation of the poloidal magnetic
field occurs due to helical turbulence in this high-latitude zone around
$60^{\circ}$, near the base of the convection zone. The field swiftly migrates
to the surface within a time frame of $1$ to $2$ years. However, in low
latitudes, this migration process takes longer, reaching the surface after
being twisted by differential rotation, transforming from the poloidal to the
toroidal component. This aspect resembles Parker’s dynamo theory (Parker,
1955), and is reproduced in our test dynamo model (Pipin & Kosovichev, 2019).
Due to buoyancy instability, these magnetic fields emerge at the surface,
forming sunspots. As sunspots decay, the remnant magnetic fields are
transported by meridional flows to the polar regions, where they eventually
descend and likely contribute to the seed field for the subsequent solar
dynamo cycle. This facet of the dynamo process shares similarities with the
models proposed by Dikpati et al. (2009) and Pipin et al. (2023). Our results
show clear signs of the magnetic cycle effects on the near-surface dynamics of
the torsional oscillations.
Emerging dynamo models propose that the inflow toward active regions could
influence the characteristics of the solar dynamo (Teweldebirhan et al.,
2023). This prompts an investigation into the potential implications of these
inflows of the zonal flows and the corresponding observable in the frequency-
splitting data. Both forward modeling and inversion techniques are essential
to decipher the specific signature that should manifest in the frequency-
splitting data, providing valuable insights into the solar dynamo’s
characteristics.
In summary, in this work, we developed a methodology for forward modeling of
the effects of migrating zonal flows on the rotational frequency splitting
data obtained from global helioseismology measurements and inversion for
inferring the flows in the solar convection zone. Our inversion is based on
the regularized least-squares RLS method. We find that using our approach, we
can detect the migrating dynamo-like wave patterns in the solar zonal flows
using only 72-day measurements of p-modes of solar oscillations. However, the
error bars are notably large near the tachocline, and our study shows that
helioseismic analysis of the 2-year time series of solar oscillations should
be able to measure the zonal flows in the tachocline. We will address this
task in future work.
##
Acknowledgments: This work was partially supported by the NASA grants
80NSSC20K1320, 80NSSC20K0602, and 80NSSC22M0162. VP thanks the financial
support of the Ministry of Science and Higher Education (Subsidy
No.075-GZ/C3569/278).
## Appendix A Theory
### A.1 Properties of Generalized Spherical Harmonics
We use the following relations of generalized spherical harmonics
$\displaystyle Y_{\ell m}(\theta,\phi)$
$\displaystyle=\gamma_{\ell}Y_{\ell}^{0m}(\theta,\phi),$ (A1)
$\displaystyle\frac{\partial Y_{\ell}^{Nm}}{\partial\theta}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\Omega_{N}^{\ell}Y_{\ell}^{N-1,m}-\Omega_{N+1}^{\ell}Y_{\ell}^{N+1,m}\right),$
(A2) $\displaystyle-\frac{m}{\sin\theta}Y_{\ell}^{0m}$
$\displaystyle=\frac{\Omega_{0}^{\ell}}{\sqrt{2}}\left(Y_{\ell}^{-1,m}+Y_{\ell}^{1,m}\right)$
(A3) $\displaystyle\frac{N\cos\theta-m}{\sin\theta}Y_{\ell}^{Nm}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\Omega_{N}^{\ell}Y_{\ell}^{N-1,m}+\Omega_{N+1}^{\ell}Y_{\ell}^{N+1,m}\right)$
(A4)
where $\gamma_{\ell}=\sqrt{(2\ell+1)/(4\pi)}$ and
$\Omega_{N}^{\ell}=\sqrt{\frac{1}{2}(\ell+N)(\ell-N+1)}$.
### A.2 Properties of Wigner-3j symbol
We use the following symmetry properties of Wigner’s 3j symbol
$\begin{pmatrix}J_{1}&J_{2}&J_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}=\begin{pmatrix}J_{2}&J_{3}&J_{1}\\\
m_{2}&m_{3}&m_{1}\end{pmatrix}=\begin{pmatrix}J_{3}&J_{1}&J_{2}\\\
m_{3}&m_{1}&m_{2}\end{pmatrix}$ (A6)
For odd permutations we have
$\begin{pmatrix}J_{1}&J_{2}&J_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}=(-1)^{J_{1}+J_{2}+J_{3}}\begin{pmatrix}J_{2}&J_{1}&J_{3}\\\
m_{2}&m_{1}&m_{3}\end{pmatrix}$ (A7)
The following recurrence relation will be necessary for our future analysis.
$\displaystyle\left[(j_{2}-m_{2})(j_{2}+m_{2}+1)(j_{3}+m_{3})(j_{3}-m_{3}+1)\right]^{1/2}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}+1&m_{3}-1\end{pmatrix}$ (A8)
$\displaystyle+\left[(j_{2}+m_{2})(j_{2}-m_{2}+1)(j_{3}-m_{3})(j_{3}+m_{3}+1)\right]^{1/2}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}-1&m_{3}+1\end{pmatrix}$
$\displaystyle=\left(j_{1}(j_{1}+1)-j_{2}(j_{2}+1)-j_{3}(j_{3}+1)-2m_{2}m_{3}\right)\begin{pmatrix}j_{1}&j_{2}&j_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}$ (A9)
The integral of three generalized spherical harmonics obeys the following
relations
$\displaystyle\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\theta\sin\theta\left(Y_{\ell^{\prime}}^{N^{\prime}m^{\prime}}\right)^{*}Y_{\ell^{\prime\prime}}^{N^{\prime\prime}m^{\prime\prime}}Y_{\ell}^{Nm}=4\pi(-1)^{(N^{\prime}-m^{\prime})}\begin{pmatrix}\ell^{\prime}&\ell^{\prime\prime}&\ell\\\
-N^{\prime}&N^{\prime\prime}&N\end{pmatrix}\begin{pmatrix}\ell^{\prime}&\ell^{\prime\prime}&\ell\\\
-m^{\prime}&m^{\prime\prime}&m\end{pmatrix}$ (A10)
It is to be noted that Wigner-3j symbols in Equation A6 are non-zero when the
following relation is satisfied
$m1+m2+m3=0$
Here we define another term
$B^{(N)\pm}_{\ell^{\prime}\ell^{\prime\prime}\ell}=\frac{(-1)^{N}}{2}(1\pm(-1)^{\ell+\ell^{\prime}+\ell^{\prime\prime}})\left[\frac{(\ell^{\prime}+N)!(\ell+N)!}{(\ell-N)!(\ell-N)!}\right]\begin{pmatrix}\ell^{\prime}&\ell^{\prime\prime}&\ell\\\
-N&0&N\end{pmatrix}$ (A11)
One relation that shall be used to calculate the kernel is the following which
can be derived from Equation A9
$\displaystyle(\Omega^{\ell}_{0})^{2}B^{(1)-}_{s\ell\ell}+\Omega_{0}^{s}(\Omega^{\ell}_{0})^{2}\Omega^{\ell}_{2}(1-(-1)^{2\ell+s}\begin{pmatrix}\ell&s&\ell\\\
1&1&-2\end{pmatrix}=\frac{1}{2}[2\ell(\ell+1)-s(s+1)]B^{(1)-}_{\ell s\ell},$
(A12)
### A.3 Kernel calculations
Rotation frequency splitting is calculated using the perturbation theory based
on the variational principle:
$\delta\omega_{k}=-i\frac{\int_{V}\rho_{0}\boldsymbol{\xi}_{k}^{*}\cdot(\mathbf{v}_{0}\cdot\nabla)\boldsymbol{\xi}_{k}dV}{\int_{V}\rho_{0}|\delta\mathbf{r}|^{2}dV},$
(A13)
$\boldsymbol{\xi}_{k}$ is the displacement eigenfunction of a mode, and $k$
signifies the combination of the radial order, angular degree, and azimuthal
number $(n,\ell,m)$ of a mode. The eigenfunction of a spherically symmetric
Sun can be written as
$\boldsymbol{\xi}_{k}=U(r)Y_{\ell m}\hat{r}+V(r)\boldsymbol{\nabla}_{h}Y_{\ell
m},$ (A14)
where $U(r)$ and $V(r)$ are the radial and horizontal components of the
displacement vector. Reformulating Equation A14 using generalized spherical
harmonics and presenting them as components within a vector representation,
$\displaystyle\xi_{k,r}$ $\displaystyle=\gamma_{\ell}U(r)Y_{\ell}^{0,m}$
$\displaystyle\xi_{k,\theta}$
$\displaystyle=\gamma_{\ell}V(r)\partial_{\theta}Y_{\ell}^{0,m}=\frac{\gamma_{\ell}}{\sqrt{2}}V(r)\Omega^{\ell}_{0}(Y_{\ell}^{-1,m}-Y_{\ell}^{1,m})$
$\displaystyle\xi_{k,\phi}$
$\displaystyle=\gamma_{\ell}V(r)\partial_{\phi}Y_{\ell}^{0,m}=-i\frac{\gamma_{\ell}}{\sqrt{2}}V(r)\Omega^{\ell}_{0}(Y_{\ell}^{-1,m}+Y_{\ell}^{1,m})$
(A15)
The velocity of solar rotation can be written as
$\displaystyle\mathbf{v}_{0}(r,\theta)$
$\displaystyle=-\sum_{s=1,3,5}w_{s}^{0}(r)\partial_{\theta}Y_{s0}\hat{\phi}$
$\displaystyle\mathbf{v}_{0}(r,\theta)$
$\displaystyle=-\sum_{s=1,3,5}\gamma_{s}w_{s}^{0}(r)\partial_{\theta}Y_{s}^{0,0}\hat{\phi}=-\sum_{s=1,3,5}\frac{\gamma_{s}}{\sqrt{2}}w_{s}^{0}(r)\Omega_{0}^{s}(Y_{s}^{-1,0}-Y_{s}^{1,0})\hat{\phi}$
(A17)
We use Equation A2 within Equation A17 to denote the derivative with respect
to $\theta$. In order to calculate Equation A13, we need to define a tensor
called $T$.
$T=\nabla\boldsymbol{\xi}_{k}$ (A18)
The components of tensor $T$, which are required to calculate the sensitivity
kernels are shown below. Temporarily, we omit the subscript $k$ from the
eigenfunction $\boldsymbol{\xi}_{k}$ for the sake of brevity in notation and
use the $r$, $\theta$, and $\phi$ components of the eigenfunction
$\boldsymbol{\xi}_{k}$ from Equation A14.
$\displaystyle T_{\phi r}$
$\displaystyle=\frac{1}{r\sin\theta}(\partial_{\phi}\boldsymbol{\xi}).\hat{r}=\frac{1}{r\sin\theta}(\partial_{\phi}\xi_{r}-\xi_{\phi}\sin\theta)$
$\displaystyle=\gamma_{\ell}U\frac{im}{r\sin\theta}Y_{\ell}^{0m}-(-i)\frac{\gamma_{\ell}}{\sqrt{2}}V\Omega_{0}^{\ell}r^{-1}\left(Y_{\ell}^{-1,m}+Y_{\ell}^{1,m}\right)$
$\displaystyle=-ir^{-1}\gamma_{\ell}\frac{\Omega_{0}^{\ell}}{\sqrt{2}}(U-V)\left(Y_{\ell}^{-1,m}+Y_{\ell}^{1,m}\right),$
(A19) $\displaystyle T_{\phi\theta}$
$\displaystyle=\frac{1}{r\sin\theta}(\partial_{\phi}\xi_{\theta}-\cos\theta\xi_{\phi})$
$\displaystyle=r^{-1}\left(\frac{\gamma_{\ell}}{\sin\theta\sqrt{2}}V\Omega_{0}^{\ell}\partial_{\phi}(Y_{\ell}^{-1,m}-Y_{\ell}^{1,m})+\cot\theta
i\frac{\gamma_{\ell}}{\sqrt{2}}V\Omega_{0}^{\ell}(Y_{\ell}^{-1,m}+Y_{\ell}^{1,m})\right)$
$\displaystyle=ir^{-1}\frac{\gamma_{\ell}\Omega_{0}^{\ell}}{\sqrt{2}}V\left(\frac{m}{\sin\theta}(Y_{\ell}^{-1,m}-Y_{\ell}^{1,m})+\cot\theta(Y_{\ell}^{-1,m}+Y_{\ell}^{1,m})\right)$
$\displaystyle=ir^{-1}\frac{\gamma_{\ell}}{2}V\Omega_{0}^{\ell}\Omega_{2}^{\ell}\left(Y_{\ell}^{2m}-Y_{\ell}^{-2m}\right),$
(A20) $\displaystyle T_{\phi\phi}$
$\displaystyle=\frac{1}{r\sin\theta}\left[\xi_{r}(\partial_{\phi}\hat{r})\cdot\hat{\phi}+\xi_{\theta}(\partial_{\phi}\hat{\theta})\cdot\hat{\phi}+\partial_{\phi}\xi_{\phi}\right]$
$\displaystyle=\frac{1}{r\sin\theta}\left[\xi_{r}\sin\theta+\xi_{\theta}\cos\theta+\partial_{\phi}\xi_{\phi}\right]$
$\displaystyle=\frac{1}{r}\left[\gamma_{\ell}U(r)Y_{\ell}^{0m}+\cot\theta\frac{\gamma_{\ell}}{\sqrt{2}}V(r)\Omega_{0}^{\ell}(Y_{\ell}^{-1,m}-Y_{\ell}^{1,m})+\frac{m}{\sin\theta}\frac{\gamma_{\ell}}{\sqrt{2}}V(r)\Omega_{0}^{\ell}(Y_{\ell}^{-1,m}+Y_{\ell}^{1,m})\right]$
$\displaystyle=r^{-1}\left[\gamma_{\ell}U(r)Y_{\ell}^{0m}-\frac{\gamma_{\ell}}{2}V\Omega_{0}^{\ell}\left[\Omega_{2}^{\ell}\left(Y_{\ell}^{2m}+Y_{\ell}^{-2m}\right)+2\Omega_{0}^{\ell}Y_{\ell}^{0m}\right]\right]$
$\displaystyle=r^{-1}\gamma_{\ell}\left[(U-\Omega_{0}^{\ell}\Omega_{0}^{\ell}V)Y_{\ell}^{0m}-V\Omega_{0}^{\ell}\Omega_{2}^{\ell}\frac{Y_{\ell}^{2m}+Y_{\ell}^{-2m}}{2}\right],,$
(A21)
The numerator in Equation A13 can be calculated using the tensor components of
$T$ as following
$i\int\rho d\mathbf{r}\left[v_{\phi}(\xi_{k,r}^{*}T_{\phi
r}+\xi^{*}_{k,\theta}T_{\phi\theta}+\xi^{*}_{k,\phi}T_{\phi\phi})\right]=T1+T2+T3,$
(A22)
where
$\displaystyle T1=i\int\rho d\mathbf{r}v_{\phi}(\xi_{k,r}^{*}T_{\phi r}$
$\displaystyle T2=i\int\rho
d\mathbf{r}v_{\phi}\xi^{*}_{k,\theta}T_{\phi\theta}$ $\displaystyle
T3=i\int\rho d\mathbf{r}v_{\phi}\xi^{*}_{k,\phi}T_{\phi\phi}$ (A23)
We individually analyze each term in the above equation $T1$, $T2$, and $T3$
before combining them in the final step.
### A.4 Analyzing $T1$ term
Substituting $v_{\phi}$, $\xi_{k,r}$ and $T_{\phi r}$ from Equations A17,
LABEL:eq:xi_sph and A19 into Equation A23 for $T1$ term we get
$\displaystyle i\int\rho
r^{2}dr\left(-\frac{\gamma_{s}}{\sqrt{2}}w_{s}^{0}\Omega_{0}^{s}(Y_{s}^{-1,0}-Y_{s}^{1,0})\right)(-ir^{-1})\gamma_{\ell}\frac{\Omega_{0}^{\ell}}{\sqrt{2}}(U-V)\left(Y_{\ell}^{-1,m}+Y_{\ell}^{1,m}\right)\gamma_{\ell}UY_{\ell}^{0m*}$
$\displaystyle=-4\pi\frac{\gamma_{\ell}^{2}\gamma_{s}\Omega_{0}^{\ell}\Omega_{0}^{s}}{2}\int\rho
w_{s}^{0}d\mathbf{r}(U^{2}-UV)Y_{\ell}^{0m*}(Y_{s}^{-1,0}-Y_{s}^{1,0})\left(Y_{\ell}^{-1,m}+Y_{\ell}^{1,m}\right)$
$\displaystyle=-4\pi\frac{\gamma_{\ell}^{2}\gamma_{s}\Omega_{0}^{\ell}\Omega_{0}^{s}}{2}(-1)^{m}\begin{pmatrix}\ell&s&\ell\\\
-m&0&m\end{pmatrix}\left(1-(-1)^{2\ell+s}\right)\begin{pmatrix}\ell&s&\ell\\\
0&-1&1\end{pmatrix}\int\rho w_{s}^{0}r^{2}dr(U^{2}-UV)$
$\displaystyle=-4\pi\frac{\gamma_{\ell}^{2}\gamma_{s}}{2}(-1)^{m}\begin{pmatrix}\ell&s&\ell\\\
-m&0&m\end{pmatrix}\int\rho
w_{s}^{0}d\mathbf{r}(U^{2}-UV)B_{s\ell\ell}^{(1)-}$
In our case $m^{\prime}=m$ and selection rules of the 3j symbol dictates
$\displaystyle-N^{\prime}+N^{\prime\prime}+N$ $\displaystyle=0$
$\displaystyle-m^{\prime}+m^{\prime\prime}+m$
$\displaystyle=0=m^{\prime\prime}$
### A.5 Analyzing $T2$ term
Substituting $v_{\phi}$, $\xi_{k,\theta}$ and $T_{\phi\theta}$ from Equations
A17, LABEL:eq:xi_sph and A20 into Equation A23 for $T2$ term we get
$\displaystyle i\int\rho
r^{2}dr\left(-\frac{\gamma_{s}}{\sqrt{2}}w_{s}^{0}\Omega_{0}^{s}(Y_{s}^{-1,0}-Y_{s}^{1,0})\right)\left(ir^{-1}\frac{\gamma_{\ell}}{2}V\Omega_{0}^{\ell}\Omega_{2}^{\ell}\left(Y_{\ell}^{2m}-Y_{\ell}^{-2m}\right)\right)\left(\frac{\gamma_{\ell}}{\sqrt{2}}V(r)\Omega_{0}^{\ell}(Y_{\ell}^{-1,m}-Y_{\ell}^{1,m})\right)^{*}$
$\displaystyle=4\pi\frac{\gamma_{s}\gamma_{\ell}^{2}}{4}\Omega_{0}^{s}\left(\Omega_{0}^{\ell}\right)^{2}\Omega_{2}^{\ell}\int\rho
rV^{2}w_{s}^{0}dr\begin{pmatrix}\ell&s&\ell\\\
-m&0&m\end{pmatrix}\left((-1)^{(-1-m)}\begin{pmatrix}\ell&s&\ell\\\
1&1&-2\end{pmatrix}-(-1)^{1-m}\begin{pmatrix}\ell&s&\ell\\\
-1&-1&2\end{pmatrix}\right)$
$\displaystyle=-4\pi\frac{\gamma_{s}\gamma_{\ell}^{2}}{4}\Omega_{0}^{s}\left(\Omega_{0}^{\ell}\right)^{2}\Omega_{2}^{\ell}(-1)^{m}\begin{pmatrix}\ell&s&\ell\\\
-m&0&m\end{pmatrix}\int\rho
rV^{2}w_{s}^{0}dr(1-(-1)^{2\ell+s})\begin{pmatrix}\ell&s&\ell\\\
1&1&-2\end{pmatrix}$
The term that we need to simplify is
$\displaystyle(-1)^{(-1-m)}\begin{pmatrix}\ell&s&\ell\\\
1&1&-2\end{pmatrix}-(-1)^{1-m}\begin{pmatrix}\ell&s&\ell\\\
-1&-1&2\end{pmatrix}$
$\displaystyle=-(-1)^{m}(1-(-1)^{2\ell+s})\begin{pmatrix}\ell&s&\ell\\\
1&1&-2\end{pmatrix}$
where we have used the following identity
$\begin{pmatrix}J_{1}&J_{2}&J_{3}\\\
m_{1}&m_{2}&m_{3}\end{pmatrix}=(-1)^{J_{1}+J_{2}+J_{3}}\begin{pmatrix}J_{1}&J_{2}&J_{3}\\\
-m_{1}&-m_{2}&-m_{3}\end{pmatrix}$ (A24)
### A.6 Analyzing $T3$ term
Substituting $v_{\phi}$, $\xi_{k,\phi}$ and $T_{\phi\phi}$ from Equations A17,
LABEL:eq:xi_sph and A21 into Equation A23 for $T3$ term we get
$\displaystyle T3$ $\displaystyle=i\int\rho
d\mathbf{r}\left(-\frac{\gamma_{s}}{\sqrt{2}}w_{s}^{0}\Omega_{0}^{s}(Y_{s}^{-1,0}-Y_{s}^{1,0})\right)$
$\displaystyle\left\\{r^{-1}\gamma_{\ell}\left[(U-\Omega_{0}^{\ell}\Omega_{0}^{\ell}V)Y_{\ell}^{0m}-V\Omega_{0}^{\ell}\Omega_{2}^{\ell}\frac{Y_{\ell}^{2m}+Y_{\ell}^{-2m}}{2}\right]\right\\}$
$\displaystyle\left(-i\frac{\gamma_{\ell}}{\sqrt{2}}V(r)\Omega_{0}^{\ell}(Y_{\ell}^{-1,m}+Y_{\ell}^{1,m})\right)^{*}$
We can simplify this equation as following
$\displaystyle
T3=4\pi\frac{\gamma_{s}\gamma_{\ell}^{2}}{2}\Omega_{0}^{s}\Omega_{0}^{\ell}(-1)^{m}\int\rho
r^{-1}w_{s}^{0}r^{2}dr\begin{pmatrix}\ell&s&\ell\\\ -m&0&m\end{pmatrix}$
$\displaystyle\left(-(1-(-1)^{2\ell+s})\begin{pmatrix}s&\ell&\ell\\\
-1&0&1\end{pmatrix}\left(UV-\Omega_{0}^{\ell}\Omega_{0}^{\ell}V^{2}\right)-\frac{V^{2}\Omega_{0}^{\ell}\Omega_{2}^{\ell}}{2}[1-(-1)^{2\ell+s}]\begin{pmatrix}\ell&s&\ell\\\
1&1&-2\end{pmatrix}\right)$ (A25)
After using the formula for $B^{(1)-}_{s\ell\ell}$ above Equation can be
simplified further
$\displaystyle T3=4\pi\frac{\gamma_{s}\gamma_{\ell}^{2}}{2}(-1)^{m}\int\rho
r^{-1}w_{s}^{0}d\mathbf{r}\begin{pmatrix}\ell&s&\ell\\\ -m&0&m\end{pmatrix}$
$\displaystyle\left(\left(UV-\Omega_{0}^{\ell}\Omega_{0}^{\ell}V^{2}\right)B^{(1)-}_{s\ell\ell}-\frac{V^{2}\Omega_{0}^{s}(\Omega_{0}^{\ell})^{2}\Omega_{2}^{\ell}}{2}[1-(-1)^{2\ell+s}]\begin{pmatrix}\ell&s&\ell\\\
1&1&-2\end{pmatrix}\right)$ (A26)
### A.7 Summing over all the terms
After summing over $T1$, $T2$ and $T3$, coefficient associated with the term
$V^{2}$ is following
$-\Omega_{0}^{s}(\Omega^{\ell}_{0})^{2}\Omega^{\ell}_{2}(1-(-1)^{2\ell+s})\begin{pmatrix}\ell&s&\ell\\\
1&1&-2\end{pmatrix}-(\Omega^{\ell}_{0})^{2}B^{(1)-}_{s\ell\ell}=-\frac{1}{2}[2\ell(\ell+1)-s(s+1)]B^{(1)-}_{\ell
s\ell}$ (A27)
where we have used the identity Equation A12. We get the equation for the
kernel as follows
$M_{n\ell}=4\pi\frac{\gamma_{s}\gamma_{\ell}^{2}}{2}(-1)^{m}\int\rho
r^{-1}w_{s}^{0}d\mathbf{r}\begin{pmatrix}\ell&s&\ell\\\
-m&0&m\end{pmatrix}\left[-U^{2}+2UV-\frac{V^{2}}{2}(2\ell(\ell+1)-s(s+1))\right]B^{(1)-}_{\ell
s\ell},$ (A28)
where the form for $B^{(1)-}_{\ell s\ell}$ from Equation A11 is the following
$B^{(1)-}_{\ell
s\ell}=\frac{-1}{2}(1-(-1)^{2\ell+s})\ell(\ell+1)\begin{pmatrix}\ell&s&\ell\\\
-1&0&1\end{pmatrix}$ (A29)
We can rearrange all the terms in the Equation A28 and write it as follows
$M_{n\ell}=\sqrt{\frac{2s+1}{4\pi}}(2\ell+1)L^{2}H_{s}^{1}H_{s}^{m}F_{s}^{2}\int\rho
r^{-1}w_{s}^{0}d\mathbf{r}\left[-U^{2}+2UV-\frac{V^{2}}{2}(2\ell(\ell+1)-s(s+1))\right]$
(A30)
where we have used a similar notation as used by Ritzwoller & Lavely (1991)
(Equation $32-35$ of their paper).
## Appendix B Asymptotic relation for the $a$-coefficients
Matrix element due to frequency splitting is following (when $s<<l$)
$\displaystyle M_{n\ell}$ $\displaystyle=-i\int_{0}^{R_{\odot}}\rho
dV\bm{\xi}_{n\ell m}\cdot(\mathbf{u}\cdot\bm{\nabla}\bm{\xi}_{e\ell m}$
$\displaystyle=(-1)^{m+1}\left[\ell(\ell+1)(2\ell+1)\right]\sqrt{\frac{2s+1}{4\pi}}\sum_{s=1,2}\begin{pmatrix}s&\ell&\ell\\\
0&1&-1\end{pmatrix}\begin{pmatrix}s&\ell&\ell\\\
0&m&-m\end{pmatrix}\int_{0}^{R}\rho
r^{2}dr\frac{w_{s}(r)}{r}\left[U^{2}-2UV+\frac{2\ell(\ell+1)-s(s+1)}{2}V^{2}\right]$
$\displaystyle=(-1)^{m+1}\left[\ell(\ell+1)(2\ell+1)\right]\sqrt{\frac{2s+1}{4\pi}}\sum_{s=1,2}(-1)^{\frac{s+2m+1}{2}}\frac{(s!!)^{2}}{2\ell^{2}s!}P_{s}(m/\ell)\langle\frac{w_{s}(r)}{r}\rangle_{n\ell}$
$\displaystyle\approx\sum_{s}\left((-1)^{(s-1)/2}\sqrt{\frac{2s+1}{4\pi}}\frac{(s!!)^{2}}{s!}\langle\frac{w_{s}(r)}{r}\rangle_{n\ell}\right)\ell
P_{s}(m/\ell)$ $\displaystyle=\sum_{s}a_{s}\ell P_{s}(m/\ell)$
where the expression for $a_{s}$ is the following
$a_{s}(n,\ell)=(-1)^{(s-1)/2}\sqrt{\frac{2s+1}{4\pi}}\frac{(s!!)^{2}}{s!}\langle\frac{w_{s}(r)}{r}\rangle_{n\ell}$
(B1)
where we have used the following simplified relation (Equation $A4$ from
Vorontsov (2011) )
$\begin{pmatrix}s&\ell&\ell\\\
0&1&-1\end{pmatrix}\begin{pmatrix}s&\ell&\ell\\\
0&m&-m\end{pmatrix}=(-1)^{(s+2m+1)/2}\frac{(s!!)^{2}}{2\ell^{2}s!}P_{s}(m/\ell)$
(B2)
### B.1 Full expression of kernel
We can obtain relations for the $a$-coefficients as following from Equation
A30
$\displaystyle
a_{s}=4\ell(\ell+1)(2\ell+1)\frac{2s+1}{4\pi}\frac{\\{(2\ell-1)!\\}^{2}}{(2\ell-s)!(2\ell+s+1)!}\mathcal{P}_{s}^{\ell}(1)\langle\frac{w_{s}(r)}{r}\rangle$
$\displaystyle=\int_{0}^{R}K_{s}(n,\ell;r)w_{s}(r)rdr$ (B3)
where the expression for $K_{s}(n,\ell)$ is following
$K_{s}(n,\ell;r)=4\ell(\ell+1)(2\ell+1)\frac{2s+1}{4\pi}\frac{\\{(2\ell-1)!\\}^{2}}{(2\ell-s)!(2\ell+s+1)!}\mathcal{P}_{s}^{\ell}(1)\rho\left[U^{2}-2UV+\frac{2\ell(\ell+1)-s(s+1)}{2}V^{2}\right]$
(B4)
## Appendix C Inversion method
The frequency splitting is expressed in terms of the $a-$coefficients.
$\omega_{n\ell
m}/(2\pi)=\nu_{n\ell}+\sum_{s=1,3,5,\dots,s_{\text{max}}}a_{n,\ell,s}\mathcal{P}_{s}^{\ell}(m)$
(C1)
where splitting coefficients are related as follows
$a_{s}(n,\ell)=\int_{0}^{R_{\odot}}K_{n,\ell,s}(r)w_{s}(r)r^{2}dr,$ (C2)
where $w_{s}$ are used to express velocity profile
$v(r,\theta)=\sum_{s}w_{s}(r)\partial_{\theta}Y_{s0}(\theta,\phi)$ (C3)
We minimize the following equation to obtain $w_{s}(r)$
$\chi=\sum_{n,\ell}\frac{1}{\sigma_{n,\ell}^{2}}\left(a_{n,\ell,s}-\int
K_{n,\ell,s}(r)w_{s}(r)r^{2}dr\right)^{2}+\lambda\int\left(\frac{\partial^{2}w_{s}}{\partial
r^{2}}\right)^{2}r^{2}dr,$ (C4)
We express $w_{s}$ in terms of B-spline in radius as following
$w_{s}(r)=\sum_{i=1}^{N}b_{i}B_{i}(r),$ (C5)
$N$ corresponds to the total number of unknowns. If we substitute Equation C5
into Equation C4, we get
$\displaystyle\chi=\sum_{n,\ell}\frac{1}{\sigma_{n,\ell}^{2}}\left(a_{s}(n,\ell)-\sum_{i}\int
K_{n,\ell,s}(r)b_{i}B_{i}(r)r^{2}dr\right)^{2}+\lambda\int\left(\frac{\partial^{2}\sum_{i}b_{i}B_{i}(r)}{\partial
r^{2}}\right)^{2}r^{2}dr$ (C6)
We minimize the above equation with respect to unknown coefficients $b_{j}$ to
obtain
$\displaystyle\frac{\partial\chi}{\partial
b_{j}}=-\sum_{n,\ell}\frac{1}{\sigma_{n,\ell}^{2}}\left(a_{s}(n,\ell)-\sum_{i}\int
K_{n,\ell,s}(r)b_{i}B_{i}(r)r^{2}dr)\right)\int
K_{n,\ell,s}(r)B_{j}(r)r^{2}dr$
$\displaystyle+\lambda\int\sum_{i}b_{i}\frac{\partial^{2}B_{i}(r)}{\partial
r^{2}}r^{2}dr\int\frac{\partial^{2}B_{j}(r)}{\partial r^{2}}r^{2}dr$ (C7)
We define matrix $A$
$\displaystyle A_{M,i}$ $\displaystyle=\int K_{M}B_{i}(r)r^{2}dr$
$\displaystyle D_{ij}$ $\displaystyle=\int\frac{\partial^{2}B_{i}(r)}{\partial
r^{2}}r^{2}dr\int\frac{\partial^{2}B_{j}(r)}{\partial r^{2}}r^{2}dr$
where $M$ is a multi-index variable for $(n,\ell)$ and assume there are total
$N_{0}$ such measurements. We get
$\sum_{M\in
N_{0}}\frac{1}{\sigma_{M}^{2}}A_{Mi}A_{Mj}b_{i}+\lambda\sum_{i}D_{ij}b_{i}=\sum_{M\in
N_{0}}\frac{1}{\sigma_{M}^{2}}a_{M}A_{Mj}$ (C8)
Which we can rewrite as
$\sum_{i=1}^{N}\left[\sum_{M\in
N_{0}}(A^{T})_{jM}\Lambda_{MM}^{-1}A_{Mi}+\lambda
D_{ij}\right]b_{i}=\sum_{M\in N_{0}}A_{jM}^{T}\Lambda_{MM}^{-1}a_{M},$ (C9)
where $\Lambda$ is a diagonal matrix
$\Lambda^{-1}=\begin{pmatrix}1/\sigma_{1}^{2}&0&0&0&\dots\\\
0&1/\sigma_{2}^{2}&0&0&\dots\\\ 0&0&1/\sigma_{3}^{2}&0&\dots\\\
0&0&0&1/\sigma_{4}^{2}&\dots\\\ \dots&\dots&\dots&\dots&\dots\end{pmatrix}$
(C10)
$\Lambda^{-1}$ is a $N_{0}\times N_{0}$ matrix. The above equation can be
written in a matrix form as
$\displaystyle\left[(A^{T}\Lambda^{-1}A)_{ji}+\lambda D_{ji}\right]b_{i}$
$\displaystyle=(A^{T}\Lambda^{-1})_{jM}a_{M}$
$\displaystyle\left[A^{T}\Lambda^{-1}A+\lambda D\right]b$
$\displaystyle=A^{T}\Lambda^{-1}a$ (C11)
where matrix $b=(b_{1},b_{2},\dots,b_{N})$. Solving these equations provides
us with the values of $b$ which are used to reconstruct $w_{s}(r)$ from
equation C5.
## Appendix D Averaging kernels
The splitting coefficients, $a_{s}$ are related to the velocity coefficients,
$w_{s}$:
$a_{s}(n,\ell)=\int_{0}^{R_{\odot}}K_{s}(n,\ell;r)w_{s}(r)r^{2}dr,$ (D1)
We look for a linear combination of $a_{s}(n,\ell)$ such that we can get an
estimate of $w_{s}(r)$ at a target depth $r_{0}$ as following
$\bar{w}_{s}(r)=\sum_{M\in N_{0}}C_{M}a_{s}(M),$ (D2)
$C_{M}$ are the coefficients that need to be determined. Therefore, formula
for the averaging kernels is
$\mathcal{K}(r,r_{0})=\sum_{M}C_{M}K_{s}(M;r),$ (D3)
From Section C we know
$w_{s}(r_{0})=\sum_{i=1}^{N}b_{i}B_{i}(r_{0})$ (D4)
where $b_{i}$ is derived by performing the matrix operations outlined in
Equation C11.
$b=(A^{T}\Lambda^{-1}A+\lambda D)^{-1}(A^{T}\Lambda^{-1})a$ (D5)
Since $b$ and $w_{s}$ are related through $w_{s}(r)=\sum_{i}b_{i}B_{i}(r)$, we
can find the equation for coefficients $c_{M}$ using the following equations
$C_{M}=\sum_{j}B_{j}(r_{0})((A^{T}\Lambda^{-1}A+\lambda
D)^{-1}(A^{T}\Lambda^{-1}))_{jM}$ (D6)
or in matrix notation
$C=\mathbf{B}((A^{T}\Lambda^{-1}A+\lambda D)^{-1}(A^{T}\Lambda^{-1})),$ (D7)
where $B=[B_{1}(r_{0}),B_{2}(r_{0}),\cdots]$ represents a matrix created by
applying B-spline basis functions evaluated at $r_{0}$. After determining the
coefficients $C$, we substitute them into Equation D3 to derive the averaging
kernel at the target depth $r_{0}$. In Figure 9, we display averaging kernel
at two different depths $0.75R_{\odot}$ and $0.9R_{\odot}$.
Figure 9: The averaging kernels for two distinct depths, $r=0.75R_{\odot}$
(indicated by a black dashed line) and $r=0.9R_{\odot}$ (shown as a black
solid line). The target depths are highlighted by vertical dashed grey lines
and are explicitly labeled in the legend of the plot. It is evident that the
averaging kernel tends to have a narrower width closer to the surface, while
it becomes broader in deeper layers.
### D.1 Averaging Kernels in 2D
Solar rotation, $\Omega(r,\theta)$ can be expressed as a toroidal function
$\Omega(r,\theta)=\sum_{s}w_{s}(r)\Psi_{s}(\theta)$ (D8)
where $w_{s}$ is obtained from Equation D2 as a linear combination of
splitting coefficients $a_{s}$ as following
$w_{s}(r)=\sum_{n,\ell}C_{s}(r;n,\ell)a_{s}(n,\ell)$ (D9)
We can find a relationship between the $a-$coefficients and frequency
splitting by minimizing the following misfit function (Pijpers, 1997)
$\chi^{2}=\sum_{m=-\ell}^{\ell}\left(\nu_{n\ell
m}-\sum_{s=0}^{s_{\text{max}}}a_{s}(n,\ell)\mathcal{P}_{s}^{\ell}(m)\right)^{2}$
(D10)
with respect to $a_{s}$. This yields
$a_{s}(n,\ell)=\frac{\sum_{-\ell}^{\ell}\nu_{n\ell
m}\mathcal{P}_{s}^{\ell}(m)}{\sum_{m=-\ell}^{\ell}\mathcal{P}_{s}^{\ell}(m)^{2}}=\sum_{m=-\ell}^{\ell}A_{s}(n,\ell,m)\nu_{n\ell
m}$ (D11)
where $A_{s}(n,\ell,m$ is defined as
$A_{s}(n,\ell,m)=\frac{\mathcal{P}_{s}^{\ell}(m)}{\sum_{m^{\prime}=-\ell}^{\ell}\left(\mathcal{P}_{s}^{\ell}(m^{\prime})\right)^{2}}$
(D12)
Substituting Equation D11 and D9 into Equation D8 we get
$\displaystyle\Omega(r,\theta)=\sum_{s}\sum_{n,\ell}C_{s}(r;n,\ell)\left(\sum_{m=-\ell}^{\ell}A_{s}(n,\ell)\nu_{n\ell
m}\right)\psi(\theta)$ $\displaystyle=\sum_{n\ell
m}C_{s}(r;n,\ell)A_{s}(n,\ell)\psi(\theta)$ $\displaystyle=\sum_{n\ell
m}C_{n\ell m}^{2D}\nu_{n\ell m},$ (D13)
where
$C_{n\ell m}^{2D}=\sum_{s}c_{s}(r;n,\ell)A_{s}(n,\ell)\psi(\theta),$ (D14)
Therefore, we can write the equation for the averaging kernels as the
following
$\mathcal{K}(r,\theta,r_{0},\theta_{0})=\sum_{n\ell m}C_{n\ell
m}^{2D}(r_{0},\theta_{0})K_{n\ell m}(r,\theta),$ (D15)
where $K_{n\ell m}(r,\theta)$ is the $2-$D kernel defined as
$K_{n\ell m}(r,\theta)=\frac{R_{n\ell m}}{I_{n\ell m}}$ (D16)
where $R_{n\ell m}$ is defined as the following
$\displaystyle R_{n\ell m}$ $\displaystyle=m\rho r\sin\theta\Bigg{\\{}\lvert
U\rvert^{2}P_{\ell}^{m}(\cos\theta)^{2}+\lvert
V(r)\rvert^{2}\Bigg{[}\left(\frac{dP_{\ell}^{m}}{d\theta}\right)^{2}$
$\displaystyle\quad+\frac{m^{2}}{\sin^{2}\theta}P_{\ell}^{m}(\cos\theta)^{2}\Bigg{]}-P_{\ell}^{m}(\cos\theta)^{2}[U^{*}V+UV^{*}]$
$\displaystyle\quad-2P_{\ell}^{m}(\cos\theta)\frac{dP_{\ell}^{m}}{d\theta}\frac{\cos\theta}{\sin\theta}\lvert
V\rvert^{2}\Bigg{\\}}$ (D17)
where $I_{n\ell m}$ is
$I_{n\ell
m}=\frac{2}{2\ell+1}\frac{(\ell+|m|)!}{(\ell-|m|)!}\int_{0}^{R}\left[U^{2}+\ell(\ell+1)V^{2}\right]\rho
r^{2}dr$ (D18)
In Figure 10, we present the averaging kernels for various depths and
latitudes. The figure clearly illustrates a distinctive localization in both
latitude and radius.
Figure 10: Left panel: averaging kernel plotted at radius $0.8R_{\odot}$ and
latitude $20^{\circ}$. Right panel: averaging kernel plotted at radius
$0.75R_{\odot}$ and latitude $40^{\circ}$.
## References
* Antia et al. (2008) Antia, H. M., Basu, S., & Chitre, S. M. 2008, ApJ, 681, 680, doi: 10.1086/588523
* Babcock (1961) Babcock, H. W. 1961, ApJ, 133, 572, doi: 10.1086/147060
* Basu & Antia (2019) Basu, S., & Antia, H. M. 2019, ApJ, 883, 93, doi: 10.3847/1538-4357/ab3b57
* Charbonneau & Barlet (2011) Charbonneau, P., & Barlet, G. 2011, Journal of Atmospheric and Solar-Terrestrial Physics, 73, 198, doi: 10.1016/j.jastp.2009.12.020
* Chen & Zhao (2017) Chen, R., & Zhao, J. 2017, ApJ, 849, 144, doi: 10.3847/1538-4357/aa8eec
* Choudhuri et al. (2007) Choudhuri, A. R., Chatterjee, P., & Jiang, J. 2007, Phys. Rev. Lett., 98, 131103, doi: 10.1103/PhysRevLett.98.131103
* Choudhuri & Dikpati (1999) Choudhuri, A. R., & Dikpati, M. 1999, Sol. Phys., 184, 61, doi: 10.1023/A:1005092601436
* Dikpati et al. (2009) Dikpati, M., Gilman, P. A., Cally, P. S., & Miesch, M. S. 2009, ApJ, 692, 1421, doi: 10.1088/0004-637X/692/2/1421
* Gizon et al. (2020) Gizon, L., Cameron, R. H., Pourabdian, M., et al. 2020, Science, 368, 1469, doi: 10.1126/science.aaz7119
* Guerrero et al. (2016) Guerrero, G., Smolarkiewicz, P. K., de Gouveia Dal Pino, E. M., Kosovichev, A. G., & Mansour, N. N. 2016, ApJ, 828, L3, doi: 10.3847/2041-8205/828/1/L3
* Hanasoge et al. (2017) Hanasoge, S. M., Woodard, M., Antia, H. M., Gizon, L., & Sreenivasan, K. R. 2017, MNRAS, 470, 1404, doi: 10.1093/mnras/stx1298
* Hill et al. (1996) Hill, F., Stark, P. B., Stebbins, R. T., et al. 1996, Science, 272, 1292, doi: 10.1126/science.272.5266.1292
* Howard & Labonte (1980) Howard, R., & Labonte, B. J. 1980, ApJ, 239, L33, doi: 10.1086/183286
* Howe et al. (2018) Howe, R., Hill, F., Komm, R., et al. 2018, ApJ, 862, L5, doi: 10.3847/2041-8213/aad1ed
* Korzennik (2023) Korzennik, S. G. 2023, Frontiers in Astronomy and Space Sciences, 9, 1031313, doi: 10.3389/fspas.2022.1031313
* Kosovichev & Pipin (2019) Kosovichev, A. G., & Pipin, V. V. 2019, ApJ, 871, L20, doi: 10.3847/2041-8213/aafe82
* Kosovichev & Schou (1997) Kosovichev, A. G., & Schou, J. 1997, ApJ, 482, L207, doi: 10.1086/310708
* Leighton (1969) Leighton, R. B. 1969, ApJ, 156, 1, doi: 10.1086/149943
* Mandal et al. (2018) Mandal, K., Hanasoge, S. M., Rajaguru, S. P., & Antia, H. M. 2018, ApJ, 863, 39, doi: 10.3847/1538-4357/aacea2
* Parker (1955) Parker, E. N. 1955, ApJ, 122, 293, doi: 10.1086/146087
* Pijpers (1997) Pijpers, F. P. 1997, A&A, 326, 1235, doi: 10.48550/arXiv.astro-ph/9705234
* Pipin & Kosovichev (2019) Pipin, V. V., & Kosovichev, A. G. 2019, ApJ, 887, 215, doi: 10.3847/1538-4357/ab5952
* Pipin & Kosovichev (2020) —. 2020, ApJ, 900, 26, doi: 10.3847/1538-4357/aba4ad
* Pipin et al. (2023) Pipin, V. V., Kosovichev, A. G., & Tomin, V. E. 2023, The Astrophysical Journal, 949, 7, doi: 10.3847/1538-4357/acaf69
* Rajaguru & Antia (2015) Rajaguru, S. P., & Antia, H. M. 2015, ApJ, 813, 114, doi: 10.1088/0004-637X/813/2/114
* Ritzwoller & Lavely (1991) Ritzwoller, M. H., & Lavely, E. M. 1991, ApJ, 369, 557, doi: 10.1086/169785
* Schatten et al. (1978) Schatten, K. H., Scherrer, P. H., Svalgaard, L., & Wilcox, J. M. 1978, Geophys. Res. Lett., 5, 411, doi: 10.1029/GL005i005p00411
* Scherrer et al. (1995) Scherrer, P. H., Bogart, R. S., Bush, R. I., et al. 1995, Sol. Phys., 162, 129, doi: 10.1007/BF00733429
* Schou & Brown (1994) Schou, J., & Brown, T. M. 1994, A&AS, 107, 541
* Schou et al. (1998) Schou, J., Antia, H. M., Basu, S., et al. 1998, ApJ, 505, 390, doi: 10.1086/306146
* Schou et al. (2012) Schou, J., Scherrer, P. H., Bush, R. I., et al. 2012, Sol. Phys., 275, 229, doi: 10.1007/s11207-011-9842-2
* Stejko et al. (2022) Stejko, A. M., Kosovichev, A. G., Featherstone, N. A., et al. 2022, The Astrophysical Journal, 934, 161, doi: 10.3847/1538-4357/ac7a44
* Teweldebirhan et al. (2023) Teweldebirhan, K., Miesch, M., & Gibson, S. 2023, arXiv e-prints, arXiv:2310.00738, doi: 10.48550/arXiv.2310.00738
* Vorontsov (2011) Vorontsov, S. V. 2011, Monthly Notices of the Royal Astronomical Society, 418, 1146, doi: 10.1111/j.1365-2966.2011.19564.x
* Vorontsov et al. (2002) Vorontsov, S. V., Christensen-Dalsgaard, J., Schou, J., Strakhov, V. N., & Thompson, M. J. 2002, Science, 296, 101, doi: 10.1126/science.1069190
* Zhao et al. (2013) Zhao, J., Bogart, R. S., Kosovichev, A. G., Duvall, T. L., J., & Hartlep, T. 2013, ApJ, 774, L29, doi: 10.1088/2041-8205/774/2/L29
* Zhao et al. (2012) Zhao, J., Nagashima, K., Bogart, R. S., Kosovichev, A. G., & Duvall, T. L., J. 2012, ApJ, 749, L5, doi: 10.1088/2041-8205/749/1/L5
|
We present a microlocal analysis of two novel Radon transforms of interest in Compton Scattering Tomography (CST),
which map compactly supported
$L^2$ functions to their integrals over seven-dimensional sets of apple and lemon
surfaces. Specifically, we show that the apple and lemon transforms
are elliptic Fourier Integral Operators (FIO), which satisfy the
Bolker condition. After an analysis of the full seven-dimensional
case, we focus our attention on $n$-D subsets of apple and lemon surfaces
with fixed central axis, where $n<7$. Such subsets of surface integrals have
applications in airport baggage and security screening. When the data dimensionality is restricted, the
apple transform is shown to violate the Bolker condition, and there
are artifacts which occur on apple-cylinder intersections. The lemon transform is shown to satisfy the Bolker
condition, when the support of the function is restricted to the strip $\sparen{0<z<1}$.
[1]
L. Borg, J. Frikel, J. S. Jørgensen, and E. T. Quinto.
Analyzing reconstruction artifacts from arbitrary incomplete X-ray
CT data.
SIAM Journal on Imaging Sciences, 11(4):2786–2814, 2018.
[2]
J. Cebeiro, C. Tarpau, M. A. Morvidone, D. Rubio, and M. K. Nguyen.
On a three-dimensional compton scattering tomography system with
fixed source.
Inverse Problems, 37(5):054001, 2021.
[3]
J. J. Duistermaat.
Fourier integral operators, volume 130 of Progress in
Birkhäuser, Inc., Boston, MA, 1996.
[4]
J. J. Duistermaat and L. Hormander.
Fourier integral operators, volume 2.
Springer, 1996.
[5]
V. Guillemin and S. Sternberg.
Geometric Asymptotics.
American Mathematical Society, Providence, RI, 1977.
[6]
L. Hörmander.
The analysis of linear partial differential operators. I.
Classics in Mathematics. Springer-Verlag, Berlin, 2003.
Distribution theory and Fourier analysis, Reprint of the second
(1990) edition [Springer, Berlin].
[7]
L. Hörmander.
The analysis of linear partial differential operators. III.
Classics in Mathematics. Springer, Berlin, 2007.
Pseudo-differential operators, Reprint of the 1994 edition.
[8]
L. Hörmander.
The analysis of linear partial differential operators. IV.
Classics in Mathematics. Springer-Verlag, Berlin, 2009.
Fourier integral operators, Reprint of the 1994 edition.
[9]
C.-Y. Jung and S. Moon.
Inversion formulas for cone transforms arising in application of
Compton cameras.
Inverse Problems, 31(1):015006, 2015.
[10]
A. I. Katsevich.
Local tomography for the limited-angle problem.
J. Math. Anal. Appl., 213(1):160–182, 1997.
[11]
V. P. Krishnan and E. T. Quinto.
Microlocal analysis in tomography.
Handbook of mathematical methods in imaging, pages 1–50, 2014.
[12]
P. Kuchment and F. Terzioglu.
Three-dimensional image reconstruction from Compton camera data.
SIAM Journal on Imaging Sciences, 9(4):1708–1725, 2016.
[13]
S. Moon and M. Haltmeier.
Analytic inversion of a conical radon transform arising in
application of Compton cameras on the cylinder.
SIAM Journal on imaging sciences, 10(2):535–557, 2017.
[14]
M. K. Nguyen, T. T. Truong, and P. Grangeat.
Radon transforms on a class of cones with fixed axis direction.
Journal of Physics A: Mathematical and General, 38(37):8003,
[15]
E. T. Quinto.
The dependence of the generalized Radon transform on defining
Trans. Amer. Math. Soc., 257:331–346, 1980.
[16]
G. Rigaud.
3d compton scattering imaging with multiple scattering: Analysis by
fio and contour reconstruction.
Inverse Problems, 2021.
[17]
G. Rigaud and B. Hahn.
Reconstruction algorithm for 3d compton scattering imaging with
incomplete data.
Inverse Problems in Science and Engineering, 29(7):967–989,
[18]
G. Rigaud and B. N. Hahn.
3D Compton scattering imaging and contour reconstruction for a
class of Radon transforms.
Inverse Problems, 34(7):075004, 2018.
[19]
W. Rudin.
Functional analysis.
McGraw-Hill Book Co., New York, 1973.
McGraw-Hill Series in Higher Mathematics.
[20]
Sylvester's determinant theorem.
Accessed 11/30/2021.
[21]
J. J. Sylvester.
On the relation between the minor determinants of linearly
equivalent quadratic functions.
Philosophical Magazine, 1:295–305, 1851.
[22]
T. T. Truong, M. K. Nguyen, and H. Zaidi.
The mathematical foundations of 3D Compton scatter emission
International journal of biomedical imaging, 2007, 2007.
[23]
J. Webber and E. Miller.
Compton scattering tomography in translational geometries.
Technical report, Tufts University, 2019.
[24]
J. W. Webber and S. Holman.
Microlocal analysis of a spindle transform.
Inverse Problems & Imaging, 13(2):231–261, 2019.
[25]
J. W. Webber and W. R. Lionheart.
Three dimensional Compton scattering tomography.
Inverse Problems, 34(8):084001, 2018.
[26]
J. W. Webber and E. T. Quinto.
Microlocal analysis of generalized radon transforms from scattering
SIAM Journal on Imaging Sciences, 14(3):976–1003, 2021.
[27]
J. W. Webber, E. T. Quinto, and E. L. Miller.
A joint reconstruction and lambda tomography regularization technique
for energy-resolved x-ray imaging.
Inverse Problems, 36(7):074002, 2020.
|
# Realizing degree sequences as $Z_{3}$-connected graphs
Fan Yang1 , Xiangwen Li2 , Hong -Jian Lai3
1Department of Mathematics and Physics, Jiangsu University of Science and
Technology,
Zhenjiang 212003, China
2Department of Mathematics, Huazhong Normal University, Wuhan 430079, China
3Department of Mathematics, West Virginia University, Morgantown, WV 26506,
USA Supported by the Natural Science Foundation of China (11326215)Supported
by the Natural Science Foundation of China (11171129) and by Doctoral Fund of
Ministry of Education of China (20130144110001)Corresponding author:
<EMAIL_ADDRESS>
###### Abstract
An integer-valued sequence $\pi=(d_{1},\ldots,d_{n})$ is graphic if there is a
simple graph $G$ with degree sequence of $\pi$. We say the $\pi$ has a
realization $G$. Let $Z_{3}$ be a cyclic group of order three. A graph $G$ is
$Z_{3}$-connected if for every mapping $b:V(G)\to Z_{3}$ such that $\sum_{v\in
V(G)}b(v)=0$, there is an orientation of $G$ and a mapping $f:E(G)\to
Z_{3}-\\{0\\}$ such that for each vertex $v\in V(G)$, the sum of the values of
$f$ on all the edges leaving from $v$ minus the sum of the values of $f$ on
the all edges coming to $v$ is equal to $b(v)$. If an integer-valued sequence
$\pi$ has a realization $G$ which is $Z_{3}$-connected, then $\pi$ has a
$Z_{3}$-connected realization $G$. Let $\pi=(d_{1},\ldots,d_{n})$ be a graphic
sequence with $d_{1}\geq\ldots\geq d_{n}\geq 3$. We prove in this paper that
if $d_{1}\geq n-3$, then either $\pi$ has a $Z_{3}$-connected realization
unless the sequence is $(n-3,3^{n-1})$ or is $(k,3^{k})$ or $(k^{2},3^{k-1})$
where $k=n-1$ and $n$ is even; if $d_{n-5}\geq 4$, then either $\pi$ has a
$Z_{3}$-connected realization unless the sequence is $(5^{2},3^{4})$ or
$(5,3^{5})$.
## 1 Introduction
Graphs here are finite, and may have multiple edges without loops. We follow
the notation and terminology in [2] except otherwise stated.
For a given orientation of a graph $G$, if an edge $e\in E(G)$ is directed
from a vertex $u$ to a vertex $v$, then $u$ is the tail of $e$ and $v$ is the
head of $e$. For a vertex $v\in V(G)$, let $E^{+}(v)$ and $E^{-}(v)$ denote
the sets of all edges having tail $v$ or head $v$, respectively. A graph $G$
is $k$-flowable if all the edges of $G$ can be oriented and assigned nonzero
numbers with absolute value less than $k$ so that for every vertex $v\in
V(G)$, the sum of the values on all the edges in $E^{+}(v)$ equals that of the
values of all the edges in $E^{-}(v)$. If $G$ is $k$-flowable we also say that
$G$ admits a nowhere-zero $k$-flow.
Let $A$ be an abelian group with identity 0, and let $A^{*}=A-\\{0\\}$. Given
an orientation and a mapping $f:E(G)\rightarrow A$, the boundary of $f$ is a
function $\partial f:V(G)\rightarrow A$ defined by, for each vertex $v\in
V(G)$,
$\partial f(v)=\sum\limits_{e\in E^{+}(v)}f(e)-\sum\limits_{e\in
E^{-}(v)}f(e),$
where “$\sum$” refers to the addition in $A$.
A mapping $b:V(G)\rightarrow A$ is a zero-sum function if $\sum\limits_{v\in
V(G)}b(v)=0$. A graph $G$ is $A$-connected if for every zero-sum function
$b:V(G)\rightarrow A$, there exists an orientation of $G$ and a mapping
$f:E(G)\rightarrow A^{*}$ such that $\partial f(v)=b(v)$ for each $v\in V(G)$.
The concept of $k$-flowability was first introduced by Tutte [19], and this
theory provides an interesting way to investigate the coloring of planar
graphs in the sense that Tutte [19] proved a classical theorem: a planar graph
is $k$-colorable if and only if it is $k$-flowable. Jaeger et al. [10]
successfully generalized nowhere-zero flow problems to group connectivity. The
purpose of study in group connectivity is to characterize contractible
configurations for integer flow problems. Let $Z_{3}$ be a cyclic group of
order three. Obviously, if $G$ is $Z_{3}$-connected, then $G$ is $3$-flowable.
An integer-valued sequence $\pi=(d_{1},\ldots,d_{n})$ is graphic if there is a
simple graph $G$ with degree sequence $\pi$. We say $\pi$ has a realization
$G$, and we also say $G$ is a realization of $\pi$. If an integer-valued
sequence $\pi$ has a realization $G$ which is $A$-connected, then we say that
$G$ is a $A$-connected realization of $\pi$ for an abelian group $A$. In
particular, if $A=Z_{3}$, then $G$ is a $Z_{3}$-connected realization of
$\pi$, and we also say that $\pi$ has a $Z_{3}$-connected realization $G$. In
this paper, we write every degree sequence $(d_{1},\ldots,d_{n})$ is in
nonincrease order. For simplicity, we use exponents to denote degree
multiplicities, for example, we write $(6,5,4^{4},3)$ for $(6,5,4,4,4,4,3)$.
The problem of realizing degree sequences by graphs that have nowhere-zero
flows or are $A$-connected, where $A$ is an abelian group, has been studied.
Luo et al. [15] proved that every bipartite graphic sequence with least
element at least 2 has a 4-flowable realization. As a corollary, they
confirmed the simultaneous edge-coloring conjecture of Cameron [3]. Fan et al.
[7] proved that every degree sequence with least element at least 2 has a
realization which contains a spanning eulerian subgraph; such graphs is
4-flowable. Let $A$ be an abelian group with $|A|=4$. For a nonincreasing
$n$-element graphic sequence $\pi$ with least element at least 2 and sum at
least $3n-3$, Luo et al. [17] proved that $\pi$ has a realization that is
$A$-connected. Yin and Guo [20] determined the smallest degree sum that yields
graphic sequences with a $Z_{3}$-connected realization. For the literature for
this topic, the readers can see a survey [13]. In particular, Luo et al.[16]
completely answered the question of Archdeacon [1]: Characterize all graphic
sequences $\pi$ realizable by a 3-flowable graph. The natural group
connectivity version of Archdeacon’s problem is as follows.
###### Problem 1.1
Characterize all graphic sequences $\pi$ realizable by a $Z_{3}$-connected
graph.
On this problem, Luo et al. [16] obtained the next two results.
###### Theorem 1.2
Every nonincreasing graphic sequence $(d_{1},\ldots,d_{n})$ with $d_{1}=n-1$
and $d_{n}\geq 3$ has a $Z_{3}$-connected realization unless $n$ is even and
the sequence is $(k,3^{k})$ or $(k^{2},3^{k-1})$, where $k=n-1$.
###### Theorem 1.3
Every nonincreasing graphic sequence $(d_{1},\ldots,d_{n})$ with $d_{n}\geq 3$
and $d_{n-3}\geq 4$ has a $Z_{3}$-connected realization.
Motivated by Problem 1.1 and the results above, we present the following two
theorems in this paper. These results extend the results of [16] by extending
the characterizations to a large set of sequences.
###### Theorem 1.4
A nonincreasing graphic sequence $(d_{1},\ldots,d_{n})$ with $d_{1}\geq n-3$
and $d_{n}\geq 3$ has a $Z_{3}$-connected realization unless the sequence is
$(n-3,3^{n-1})$ for any $n$ or is $(k,3^{k})$ or $(k^{2},3^{k-1})$, where
$k=n-1$ and $n$ is even.
###### Theorem 1.5
A nonincreasing graphic sequence $(d_{1},\ldots,d_{n})$ with $d_{n}\geq 3$ and
$d_{n-5}\geq 4$ has a $Z_{3}$-connected realization unless the sequence is
$(5^{2},3^{4})$ or $(5,3^{5})$.
We end this section with some notation and terminology. A graph is trivial if
$E(G)=\emptyset$ and nontrivial otherwise. A $k$-vertex denotes a vertex of
degree $k$. Let $P_{n}$ denote the path on $n$ vertices and we call $P_{n}$ a
$n$-path. An $n$-cycle is a cycle on $n$ vertices. The wheel $W_{k}$ is the
graph obtained from a $k$-cycle by adding a new vertex, the center of the
wheel, and joining it to every vertex of the $k$-cycle. A wheel $W_{k}$ is an
odd (even) wheel if $k$ is odd (even). For simplicity, we say $W_{1}$ is a
triangle. For a graph $G$ and $X\subseteq V(G)$, denote by $G[X]$ the subgraph
of $G$ induced by $X$. For two vertex-disjoint subsets $V_{1},V_{2}$ of
$V(G)$, denote by $e(V_{1},V_{2})$ the number of edges with one endpoint in
$V_{1}$ and the other endpoint in $V_{2}$.
We organize this paper as follows. In Section 2, we state some results and
establish some lemmas that will be used in the following proofs. We will deal
with some special degree sequences, each of which has a $Z_{3}$-connected
realization in Section 3. In Sections 4 and 5, we will give the proofs of
Theorems 1.4 and 1.5.
## 2 Lemmas
Let $\pi=(d_{1},\ldots,d_{n})$ be a graphic sequence with $d_{1}\geq\ldots\geq
d_{n}$. Throughout this paper, we use $\bar{\pi}$ to represent the sequence
$(d_{1}-1,\ldots,d_{d_{n}}-1,d_{d_{n}+1},\ldots,d_{n-1})$, which is called the
_residual sequence_ obtained from $\pi$ by deleting $d_{n}$. The following
well-known result is due to Hakimi [8, 9] and Kleitman and Wang [11].
###### Theorem 2.1
A graphic sequence has even sum. Furthermore, a sequence $\pi$ is graphic if
and only if $\bar{\pi}$ is graphic.
Some results in [4, 5, 6, 10, 12] on group connectivity are summarized as
follows.
###### Lemma 2.2
Let $A$ be an abelian group with $|A|\geq 3$. The following results are known:
(1) $K_{1}$ is $A$-connected;
(2) $K_{n}$ and $K_{n}^{-}$ are $A$-connected if $n\geq 5$;
(3) An $n$-cycle is $A$-connected if and only if $|A|\geq n+1$;
(4) $K_{m,n}$ is $A$-connected if $m\geq n\geq 4$; neither $K_{2,t}$ $(t\geq
2)$ nor $K_{3,s}$ $(s\geq 3)$ is $Z_{3}$-connected;
(5) Each even wheel is $Z_{3}$-connected and each odd wheel is not;
(6) Let $H\subseteq G$ and $H$ be $A$-connected. $G$ is $A$-connected if and
only if $G/H$ is $A$-connected;
(7) If $G$ is not $A$-connected, then any spanning subgraph of $G$ is not
$A$-connected.
(8) Let $v$ be not a vertex of $G$. If $G$ is $A$-connected and $e(v,G)\geq
2$, then $G\cup\\{v\\}$ is $A$-connected.
Let $G$ be a graph having an induced path with three vertices $v,u,w$ in
order. Let $G_{[uv,uw]}$ be the graph by deleting $uv$ and $uw$ and adding a
new edge $vw$. The following lemma was first proved by Lai in [12] and
reformulated by Chen et al. in [4].
###### Lemma 2.3
Let $G$ be a graph with $u\in V(G)$, $uv,uw\in E(G)$ and $d(u)\geq 4$, and let
$A$ be an abelian group with $|A|\geq 3$. If $G_{[uv,uw]}$ is $A$-connected,
then so is $G$.
A graph $G$ is triangularly connected if for every edge $e,f\in E$ there
exists a sequence of cycles $C_{1},C_{2},\ldots,C_{k}$ such that $e\in
E(C_{1})$, $f\in E(C_{k})$, and $|E(C_{i})|\leq 3$ for $1\leq i\leq k$, and
$|E(C_{j})\cap E(C_{j+1})|\neq\emptyset$ for $1\leq j\leq k-1$.
###### Lemma 2.4
([5]) A triangularly connected graph $G$ is $Z_{3}$-connected if $G$ has
minimum degree at least 4 or has a nontrivial $Z_{3}$-connected subgraph.
An orientation $D$ of $G$ is a modular 3-orientation if
$|E^{+}(v)|-|E^{-}(v)|\equiv 0$ (mod 3) for every vertex $v\in V(G)$.
Steinberg and Younger [18] established the following relationship.
###### Lemma 2.5
A graph $G$ is 3-flowable if and only if $G$ admits a modular 3-orientation.
Let $v$ be a 3-vertex in a graph $G$, and let $N(v)=\\{v_{1},v_{2},v_{3}\\}$.
Denote by $G_{(v,v_{1})}$ the graph obtained from $G$ by deleting vertex $v$
and adding a new edge $v_{2}v_{3}$. The following lemma is due to Luo et al.
[14].
###### Lemma 2.6
Let $A$ be an abelian group with $|A|\geq 3$, and let $b:V(G)\mapsto A$ be a
zero-sum function with $b(v)\neq 0$. If $G_{(v,v_{1})}$ is $Z_{3}$-connected,
then there exists an orientation $D$ of $G$ and a nowhere-zero mapping
$f^{\prime}:E(G)\mapsto A$ such that $\partial f^{\prime}=b$ under the
orientation $D$ of $G$.
For any odd integer $k$, Luo et al. [16] proved that no realization of the
graphic sequence $(k,3^{k})$ and $(k^{2},3^{k-1})$ is 3-flowable. This yields
the following lemma.
###### Lemma 2.7
If $k$ is odd, then neither $(k,3^{k})$ nor $(k^{2},3^{k-1})$ has a
$Z_{3}$-connected realization.
Next we provide $Z_{3}$-connected realizations for some degree sequences.
$v_{1}$$v_{5}$$v_{3}$$v_{4}$$v_{6}$$v_{2}$(a) $\pi=(4^{2},3^{4})$
$v_{2}$$v_{1}$$v_{3}$$v_{4}$$v_{5}$$v_{6}$$v_{7}$(b) $\pi=(5,4,3^{5})$
$v_{5}$(120,25,130,25) (126,18,132,19) (130,25,132,19) (126,18,140,15)
(130,25,140,15) (120,25,126,18)
$v_{6}$$v_{7}$$v_{8}$$v_{1}$$v_{4}$$v_{3}$$v_{2}$(c) $\pi=(6,4,3^{6})$
$v_{5}$(120,25,130,25) (126,18,132,19) (130,25,132,19) (126,18,140,15)
(130,25,140,15) (120,25,126,18) (132,19,140,15)
$v_{6}$$v_{7}$$v_{8}$$v_{1}$$v_{4}$$v_{3}$$v_{2}$(d) $\pi=(5^{2},3^{6})$Fig.
1: Realizations of four degree sequences
###### Lemma 2.8
Each of the graphs in Fig. 1 is $Z_{3}$-connected.
Proof. If $G$ is the graph (a) in Fig. 1, then $G$ is $Z_{3}$-connected by
Lemma 2.2 of [14]. Thus, we may assume that $G$ is one of the graphs (b), (c)
and (d) shown in Fig 1.
We first prove here that the graph (b) in Fig. 1 is $Z_{3}$-connected. Assume
that $G$ is the graph (b) shown in Fig. 1. Let $b:V(G)\to Z_{3}$ be a zero-sum
function. If $b(v_{3})\neq 0$, then $G_{(v_{3},v_{4})}$ contains a 2-cycle
$(v_{1},v_{2})$. Contracting this 2-cycle and repeatedly contracting all
2-cycles generated in the process, we obtain $K_{1}$. By parts (1) and (6) of
Lemma 2.2, $G_{(v_{3},v_{4})}$ is $Z_{3}$-connected. It follows by Lemma 2.6
that there exists a nowhere-zero mapping $f:E(G)\to Z_{3}$ with $\partial
f=b$. Thus, we may assume that $b(v_{3})=0$. Similarly, we may assume that
$b(v_{4})=b(v_{5})=b(v_{6})=b(v_{7})=0$. This means that for such $b$, there
are only three possibilities to be considered:
$(b(v_{1}),b(v_{2}))\in\\{(0,0),(1,2),(2,1)\\}$.
If $(b(v_{1}),b(v_{2}))=(0,0)$, we show that $G$ is 3-flowable. Note that each
vertex of $v_{3},v_{4},v_{5},v_{6}$ and $v_{7}$ is of degree 3. The edges of
$G$ are oriented as follows: $|E^{+}(v_{3})|=3$, $|E^{-}(v_{4})|=3$,
$|E^{+}(v_{5})|=3$, $|E^{-}(v_{6})|=3$, $|E^{+}(v_{7})|=3$, and $v_{2}v_{1}$
is oriented from $v_{2}$ to $v_{1}$. It is easy to verify that
$|E^{+}(v)|-|E^{-}(v)|=0$ (mod 3) for each vertex $v\in V(G)$. By Lemma 2.5,
$G$ is 3-flowable. Thus, there is an $f:E(G)\to Z_{3}^{*}$ such that $\partial
f(v)=b(v)$ for each $v\in V(G)$.
If $(b(v_{1}),b(v_{2})=(1,2)$, note that $b(v)=0$ for each $v\in
V(G)-\\{v_{1},v_{2}\\}$. The edges of $G$ are oriented as follows:
$|E^{-}(v_{3})|=3$, $|E^{+}(v_{4})|=3$, $|E^{-}(v_{5})|=3$,
$|E^{+}(v_{6})|=3$, $|E^{-}(v_{7})|=3$ and edge $v_{2}v_{1}$ is oriented from
$v_{2}$ to $v_{1}$. If $(b(v_{1}),b(v_{2})=(2,1)$, then the edges of $G$ are
oriented as follows: $|E^{+}(v_{3})|=3$, $|E^{-}(v_{4})|=3$,
$|E^{+}(v_{5})|=3$, $|E^{-}(v_{6})|=3$, $|E^{+}(v_{7})|=3$ and edge
$v_{1}v_{2}$ is oriented from $v_{1}$ to $v_{2}$. In each case, for each $e\in
E(G)$, define $f(e)=1$. It is easy to see that for
$v\in\\{v_{3},v_{4},v_{5},v_{6},v_{7}\\}$, $\partial f(v)=0=b(v)$, $\partial
f(v_{1})=b(v_{1})$ and $\partial f(v_{2})=b(v_{2})$.
Thus, for any zero-sum function $b$, there exists an orientation of $G$ and a
nowhere-zero mapping $f:E(G)\to Z_{3}$ such that $\partial f=b$. Therefore,
$G$ is $Z_{3}$-connected.
Next, assume that $G$ is the graph (c) shown in Fig. 1. Let $b:V(G)\to Z_{3}$
be a zero-sum function. If $b(v_{3})\neq 0$, then $G_{(v_{3},v_{4})}$ contains
an even wheel $W_{4}$ induced by $\\{v_{1},v_{2},v_{6},v_{7},v_{8}\\}$ with
the center $v_{1}$. Contracting this $W_{4}$ and recursively contracting all
2-cycles generated in the process, we get $K_{1}$. By parts (1), (5) and (6)
of Lemma 2.2, $G_{(v_{3},v_{4})}$ is $Z_{3}$-connected. By Lemma 2.6, there
exists a nowhere-zero mapping $f:E(G)\to Z_{3}$ with $\partial f=b$. Then we
may assume $b(v_{3})=0$. Similarly, $G_{(v_{5},v_{4})}$, $G_{(v_{6},v_{7})}$
and $G_{(v_{8},v_{7})}$ are all $Z_{3}$-connected. Thus, we may assume
$b(v_{5})=b(v_{6})=b(v_{8})=0$.
If $b(v_{4})\neq 0$, then $G_{(v_{4},v_{3})}$ contains a 2-cycle
$(v_{1},v_{5})$. Contracting this 2-cycle, we get the graph containing an even
wheel $W_{4}$ induced by $\\{v_{1},v_{2},v_{6},v_{7},v_{8}\\}$ with the center
at $v_{1}$. Contracting this wheel $W_{4}$ and contracting the 2-cycle
generated in the process, we get $K_{1}$. By parts (1), (3), (5) and (6) of
Lemma 2.2, $G_{(v_{4},v_{3})}$ is $Z_{3}$-connected. Lemma 2.6 shows that
there exists a nowhere-zero mapping $f:E(G)\to Z_{3}$ with $\partial f=b$.
Thus, we may assume $b(v_{4})=0$. Similarly, $G_{(v_{7},v_{8})}$ is
$Z_{3}$-connected. Therefore, we may assume $b(v_{7})=0$.
Thus, we are left to consider the case that $b(v_{i})=0$ for $3\leq i\leq 8$.
This implies that for such function $b$, we need to consider three cases:
$b(v_{1})=b(v_{2})=0$; $b(v_{1})=1$ and $b(v_{2})=2$; $b(v_{1})=2$ and
$b(v_{2})=1$.
In the case that $b(v_{1})=b(v_{2})=0$, we have $b(v)=0$ for each $v\in V(G)$.
The edges of $G$ are oriented as follows: $|E^{+}(v_{3})|=3$,
$|E^{-}(v_{4})|=3$, $|E^{+}(v_{5})|=3$, $|E^{-}(v_{6})|=3$, $|E^{+}(v_{7})|=3$
and $|E^{-}(v_{8})|=3$. It is easy to verify that for each vertex $v\in V(G)$,
$|E^{+}(v)|-|E^{-}(v)|=0$ (mod 3). By Lemma 2.5, $G$ is 3-flowable. Thus,
there is an $f:E(G)\to Z_{3}^{*}$ such that $\partial f(v)=b(v)$ for each
$v\in V(G)$.
In the case that $b(v_{1})=1$ and $b(v_{2})=2$, we have $b(v)=0$ for each
$v\in V(G)-\\{v_{1},v_{2}\\}$. The edges of $G$ are oriented as follows:
$|E^{+}(v_{3})|=3$, $|E^{-}(v_{4})|=3$, $|E^{+}(v_{5})|=3$,
$|E^{+}(v_{6})|=3$, $|E^{-}(v_{7})|=3$, $|E^{+}(v_{8})|=3$. For each $e\in
E(G)$, define $f(e)=1$. It is easy to verify that $\partial f(v)=b(v)$ for
each $v\in V(G)$.
In the case that $b(v_{1})=2$ and $b(v_{2})=1$, we have $b(v)=0$ for each
$v\in V(G)-\\{v_{1},v_{2}\\}$. The edges of $G$ are oriented as follows:
$|E^{-}(v_{3})|=3$, $|E^{+}(v_{4})|=3$, $|E^{-}(v_{5})|=3$,
$|E^{-}(v_{6})|=3$, $|E^{+}(v_{7})|=3$, $|E^{-}(v_{8})|=3$. For each $e\in
E(G)$, define $f(e)=1$. It is easy to verify that $\partial f(v)=b(v)$ for
each $v\in V(G)$.
Thus, for every zero-sum function $b$, there exists an orientation of $G$ and
a mapping $f:E(G)\mapsto Z_{3}\setminus\\{0\\}$ such that $\partial f=b$.
Therefore, $G$ is $Z_{3}$-connected.
Finally, assume that $G$ is the graph (d) shown in Fig. 1. Let $b:V(G)\to
Z_{3}$ be a zero-sum function. As in the proof of the case when $G$ is the
graph (c) in Fig. 1, we may assume that that $b(v_{i})=0$ for $3\leq i\leq 8$.
This implies that for such $b$, we need to consider only three cases:
$b(v_{1})=b(v_{2})=0$; $b(v_{1})=1$ and $b(v_{2})=2$; $b(v_{1})=2$ and
$b(v_{2})=1$.
In the case that $b(v_{1})=b(v_{2})=0$, we have $b(v)=0$ for each $v\in V(G)$.
Assume that the edges of $G$ are oriented as follows: $|E^{+}(v_{3})|=3$,
$|E^{-}(v_{4})|=3$, $|E^{+}(v_{5})|=3$, $|E^{+}(v_{6})|=3$,
$|E^{-}(v_{7})|=3$, $|E^{+}(v_{8})|=3$. It is easy to verify that for each
vertex $v\in V(G)$, $|E^{+}(v)|-|E^{-}(v)|=0$ (mod 3). By Lemma 2.5, $G$ is
3-flowable. Thus, there is an $f:E(G)\to Z_{3}\setminus\\{0\\}$ such that
$\partial f(v)=b(v)$ for each $v\in V(G)$.
In the case that $b(v_{1})=1$ and $b(v_{2})=2$, we have $b(v)=0$ for each
$v\in V(G)-\\{v_{1},v_{2}\\}$. Assume that the edges of $G$ are oriented as
follows: $|E^{+}(v_{3})|=3$, $|E^{-}(v_{4})|=3$, $|E^{+}(v_{5})|=3$,
$|E^{-}(v_{6})|=3$, $|E^{+}(v_{7})|=3$, $|E^{-}(v_{8})|=3$. For each $e\in
E(G)$, define $f(e)=1$. It is easy to verify that $\partial f(v)=b(v)$ for
each $v\in V(G)$.
In the case that $b(v_{1})=2$ and $b(v_{2})=1$, we have $b(v)=0$ for each
$v\in V(G)-\\{v_{1},v_{2}\\}$. Assume that the edges of $G$ are oriented as
follows: $|E^{-}(v_{3})|=3$, $|E^{+}(v_{4})|=3$, $|E^{-}(v_{5})|=3$,
$|E^{+}(v_{6})|=3$, $|E^{-}(v_{7})|=3$, $|E^{+}(v_{8})|=3$. For each $e\in
E(G)$, define $f(e)=1$. It is easy to verify that $\partial f(v)=b(v)$ for
each $v\in V(G)$.
Thus, for every zero-sum function $b$, there exists an orientation of $G$ and
a mapping $f:E(G)\to Z_{3}\setminus\\{0\\}$ such that $\partial f=b$.
Therefore, $G$ is $Z_{3}$-connected.
###### Lemma 2.9
Each graph in Fig.2 is $Z_{3}$-connected.
Proof. We first prove here that the graph (a) in Fig. 2 is $Z_{3}$-connected.
Denote by $G$ the graph (a) in Fig. 2. We claim that $G$ is 3-flowable. Assume
that the edges of the graph are oriented as follows: $|E^{+}(v_{4})|=3$,
$|E^{+}(v_{5})|=0$, $|E^{+}(v_{6})|=3$, $|E^{+}(v_{7})|=0$ and $v_{2}v_{3}$
from $v_{2}$ to $v_{3}$. Define $f(e)=1$ for all $e\in E(G)$. It is easy to
verify that $\partial f(v)=0$ for each $v\in V(G)$. By Lemma 2.5, the graph
(a) is 3-flowable.
(10,35,25,28) (40,35,25,28) (30,25,25,28) (20,25,25,28) (20,25,40,15)
$v_{2}$$v_{1}$$v_{4}$$v_{5}$$v_{3}$$v_{6}$$v_{7}$(a) $\pi=(4^{3},3^{4})$
(75,20,70,15) $v_{2}$$v_{1}$$v_{7}$$v_{8}$$v_{6}$$v_{5}$$v_{3}$$v_{4}$(b)
$\pi=(5,4^{2},3^{5})$
$v_{3}$(125,30,110,15) (125,30,140,15) (125,30,110,35) (125,30,140,35)
(125,20,110,15) (125,20,140,15) (125,20,110,28) (125,20,140,28)
$v_{2}$$v_{7}$$v_{8}$$v_{1}$$v_{5}$$v_{6}$$v_{4}$(c) $\pi=(4^{4},3^{4})$Fig.
2: Realizations of three degree sequences
Let $b:V(G)\to Z_{3}$ be a zero-sum function. If $b(v_{4})\neq 0$, then
$G_{(v_{4},v_{2})}$ contains a 2-cycle $(v_{1},v_{5})$. Contracting the
2-cycle, we obtain an even wheel $W_{4}$ induced by
$\\{v_{1},v_{2},v_{3},v_{6},v_{7}\\}$ with the center at $v_{3}$. By parts
(3), (5) and (6) of Lemma 2.2, $G_{(v_{4},v_{2})}$ is $Z_{3}$-connected. By
Lemma 2.6, there exists a nowhere-zero mapping $f:E(G)\to Z_{3}$ with
$\partial f=b$. Thus, we assume $b(v_{4})=0$. By symmetry, we may assume
$b(v_{5})=0$. If $b(v_{6})\neq 0$, then $G_{(v_{6},v_{3})}$ is a graph
isomorphic to Fig 1 (a) which is $Z_{3}$-connected by Lemma 2.8. By Lemma 2.6,
there exists a nowhere-zero mapping $f:E(G)\to Z_{3}$ with $\partial f=b$. We
thus assume $b(v_{6})=0$. By symmetry, we assume $b(v_{7})=0$.
So far, we may assume $b(v_{4})=b(v_{5})=b(v_{6})=b(v_{7})=0$. We claim that
$b(v_{2})\not=0$. If $b(v_{2})=0$, then denote by $G(v_{2})$ the graph
obtained from $G$ by deleting $v_{2}$ and adding edges $v_{3}v_{7}$ and
$v_{4}v_{6}$. Contracting all 2-cycles, we finally get an even wheel $W_{4}$
with the center at $v_{1}$. By Lemma 2.2, $G(v_{2})$ is $Z_{3}$-connected.
Thus, there exists a nowhere-zero mapping $f:E(G)\to Z_{3}$ with $\partial
f=b$. By symmetry, we assume that $b(v_{3})\not=0$. Thus, we are left to
discuss three cases
$(b(v_{1}),b(v_{2}),b(v_{3}))\in\\{(1,1,1),(0,1,2),(2,2,2)\\}$.
If $(b(v_{1}),b(v_{2}),b(v_{3}))=(1,1,1)$, then we orient the edges of $G$ as
follows: $|E^{+}(v_{4})|=3$, $|E^{+}(v_{5})|=0$, $|E^{+}(v_{6})|=3$,
$|E^{+}(v_{7})|=3$, and $v_{2}v_{3}$ from $v_{2}$ to $v_{3}$; if
$(b(v_{1}),b(v_{2}),b(v_{3}))=(0,1,2)$, then we orient edges of $G$ as
follows: $|E^{+}(v_{4})|=3$, $|E^{+}(v_{5})|=0$, $|E^{+}(v_{6})|=3$,
$|E^{+}(v_{7})|=0$, and $v_{3}v_{2}$ from $v_{3}$ to $v_{2}$; if
$(b(v_{1}),b(v_{2}),b(v_{3}))=(2,2,2)$, then we orient edges of $G$ as
follows: $|E^{+}(v_{4})|=3$, $|E^{+}(v_{5})|=0$, $|E^{+}(v_{6})|=0$,
$|E^{+}(v_{7})|=0$, and $v_{2}v_{3}$ from $v_{2}$ to $v_{3}$. In each case,
for each $e\in E(G)$ define $f(e)=1$. It is easy to verify that $\partial
f(v)=b(v)$ for each $v\in V(G)$.
In each case, there exists an orientation of $G$ and a nowhere-zero mapping
$f:E(G)\to Z_{3}$ such that $\partial f=b$. Therefore, $G$ is
$Z_{3}$-connected.
Next we consider $G$ is the graph $(b)$ in Fig. 2. Let $b:V(G)\to Z_{3}$ be a
zero-sum function. If $b(v_{4})\neq 0$, then $G_{(v_{4},v_{3})}$ is isomorphic
the graph shown in Fig 1 (b). By Lemma 2.8, $G_{(v_{4},v_{3})}$ is
$Z_{3}$-connected. By Lemma 2.6, there exists a nowhere-zero mapping
$f:E(G)\to Z_{3}$ with $\partial f=b$. Thus, we may assume $b(v_{4})=0$.
Similarly, if $b(v_{5})\not=0$, we can prove that $G_{(v_{5},v_{3})}$ is
$Z_{3}$-connected. Thus we also assume $b(v_{5})=0$.
If $b(v_{6})\neq 0$, then $G_{(v_{6},v_{5})}$ contains a 2-cycle
$(v_{1},v_{8})$. Contracting 2-cycles obtained will result in a graph $K_{1}$.
By parts (1), (3) and (6) of Lemma 2.2, $G_{(v_{6},v_{5})}$ is
$Z_{3}$-connected. By Lemma 2.6, there exists a nowhere-zero mapping
$f:E(G)\to Z_{3}$ with $\partial f=b$. Thus, we may assume $b(v_{6})=0$.
Similarly, if $b(v_{7})\not=0$, we can prove that $G_{(v_{7},v_{8})}$,
$G_{(v_{8},v_{6})}$ is $Z_{3}$-connected. Thus, we may assume
$b(v_{7})=b(v_{8})=0$.
From now on we may assume $b(v_{4})=b(v_{5})=b(v_{6})=b(v_{7})=b(v_{8})=0$. We
claim that $b(v_{2})\not=0$. If $b(v_{2})=0$, then denote by $G(v_{2})$ the
graph obtained from $G$ by deleting $v_{2}$ and adding edges $v_{1}v_{7}$ and
$v_{3}v_{4}$. By contracting all 2-cycles, finally we get $K_{1}$. By Lemma
2.2, $G(v_{2})$ is $Z_{3}$-connected. Thus, for $b(v_{2})=0$, there exists a
nowhere-zero mapping $f:E(G)\to Z_{3}$ with $\partial f=b$. Similarly, we may
assume that $b(v_{3})\not=0$. Thus, it remains for us to discuss four cases
$(b(v_{1}),b(v_{2}),b(v_{3}))\in\\{(1,1,1),(0,1,2),(2,2,2),(0,2,1)\\}$.
If $(b(v_{1}),b(v_{2}),b(v_{3}))=(1,1,1)$, then we orient the edges of $G$ as
follows: $|E^{+}(v_{7})|=3$, $|E^{+}(v_{8})|=0$, $|E^{+}(v_{6})|=3$,
$|E^{+}(v_{5})|=0$, $|E^{+}(v_{4})|=3$ and $v_{1}v_{2}$ from $v_{1}$ to
$v_{2}$, $v_{2}v_{3}$ from $v_{2}$ to $v_{3}$, $v_{1}v_{3}$ from $v_{1}$ to
$v_{3}$; if $(b(v_{1}),b(v_{2}),b(v_{3}))=(0,1,2)$, then we orient the edges
of $G$ as follows: $|E^{+}(v_{7})|=3$, $|E^{+}(v_{8})|=0$, $|E^{+}(v_{6})|=3$,
$|E^{+}(v_{5})|=0$, $|E^{+}(v_{4})|=3$ and $v_{2}v_{1}$ from $v_{2}$ to
$v_{1}$, $v_{3}v_{2}$ from $v_{3}$ to $v_{2}$, $v_{3}v_{1}$ from $v_{3}$ to
$v_{1}$; if $(b(v_{1}),b(v_{2}),b(v_{3}))=(2,2,2)$, then we orient the edges
of $G$ as follows: $|E^{+}(v_{7})|=0$, $|E^{+}(v_{8})|=3$, $|E^{+}(v_{6})|=0$,
$|E^{+}(v_{5})|=3$, $|E^{+}(v_{4})|=0$ and $v_{2}v_{1}$ from $v_{2}$ to
$v_{1}$, $v_{3}v_{2}$ from $v_{3}$ to $v_{2}$, $v_{3}v_{1}$ from $v_{3}$ to
$v_{1}$; if $(b(v_{1}),b(v_{2}),b(v_{3}))=(0,2,1)$, then we orient the edges
of $G$ as follows: $|E^{+}(v_{7})|=0$, $|E^{+}(v_{8})|=3$, $|E^{+}(v_{6})|=0$,
$|E^{+}(v_{5})|=3$, $|E^{+}(v_{4})|=0$ and $v_{1}v_{2}$ from $v_{1}$ to
$v_{2}$, $v_{2}v_{3}$ from $v_{2}$ to $v_{3}$, $v_{1}v_{3}$ from $v_{1}$ to
$v_{3}$. In each case, for each $e\in E(G)$ define $f(e)=1$ in such
orientation. Clearly, we can verify that $\partial f(v)=b(v)$ for each $v\in
V(G)$. Thus, $G$ is $Z_{3}$-connected.
We are left to consider the case that $G$ is the graph $(c)$ in Fig. 2. Let
$b:V(G)\to Z_{3}$ be a zero-sum function. If $b(v_{5})\neq 0$, then
$G_{(v_{5},v_{6})}$ contains an even wheel $W_{4}$ with the center at $v_{3}$.
By parts (5) and (8) of Lemma 2.2, $G_{(v_{5},v_{6})}$ is $Z_{3}$-connected.
By Lemma 2.6, there exists a nowhere-zero mapping $f:E(G)\to Z_{3}$ with
$\partial f=b$. Thus, we assume that $b(v_{5})=0$. By symmetry, we may assume
that $b(v_{6})=0$.
If $b(v_{7})\neq 0$, then $G_{(v_{7},v_{5})}$ contains a 2-cycle
$(v_{2},v_{3})$. Contracting this 2-cycle and repeatedly contracting all
2-cycles generated in the process, we finally get $K_{1}$. By parts (1) and
(6) of Lemma 2.2, $G_{(v_{7},v_{5})}$ is $Z_{3}$-connected. By Lemma 2.6,
there exists a nowhere-zero mapping $f:E(G)\to Z_{3}$ with $\partial f=b$.
Thus, we may assume $b(v_{7})=0$. By symmetry, we may assume $b(v_{8})=0$.
If $b(v_{1})=0$, then denote by $G(v_{1})$ the graph obtained from $G$ by
deleting $v_{1}$ and adding edges $v_{5}v_{6}$ and $v_{3}v_{4}$. Contracting
all 2-cycles in the process, we finally get $K_{1}$. By Lemma 2.2, $G(v_{1})$
is $Z_{3}$-connected. Thus, for $b(v_{1})=0$, there exists a nowhere-zero
mapping $f:E(G)\to Z_{3}$ with $\partial f=b$. Thus, we may assume that
$b(v_{1})\not=0$. Similarly, we also assume that $b(v_{2})\not=0$,
$b(v_{3})\not=0$ and $b(v_{4})\not=0$. Thus, we only need to discuss four
cases
$(b(v_{1}),b(v_{2}),b(v_{3}),b(v_{4}))\in\\{(2,2,1,1),(2,1,2,1),(1,2,1,2),(1,1,2,2)\\}$
and $b(v_{5})=b(v_{6})=b(v_{7})=b(v_{8})=0$.
If $(b(v_{1}),b(v_{2}),b(v_{3}),b(v_{4}))=(2,2,1,1)$, then we orient the edges
of $G$ as follows: $|E^{+}(v_{7})|=0$, $|E^{+}(v_{5})|=3$, $|E^{+}(v_{6})|=0$,
$|E^{+}(v_{8})|=3$, and $v_{1}v_{3}$ from $v_{1}$ to $v_{3}$, $v_{1}v_{4}$
from $v_{1}$ to $v_{4}$, $v_{2}v_{3}$ from $v_{2}$ to $v_{3}$, $v_{2}v_{4}$
from $v_{2}$ to $v_{4}$, $v_{4}v_{3}$ from $v_{4}$ to $v_{3}$; if
$(b(v_{1}),b(v_{2}),b(v_{3}),b(v_{4}))=(2,1,2,1)$, then we orient the edges of
$G$ as follows: $|E^{+}(v_{7})|=0$, $|E^{+}(v_{5})|=3$, $|E^{+}(v_{6})|=0$,
$|E^{+}(v_{8})|=3$, and $v_{1}v_{3}$ from $v_{1}$ to $v_{3}$, $v_{1}v_{4}$
from $v_{1}$ to $v_{4}$, $v_{3}v_{2}$ from $v_{3}$ to $v_{2}$, $v_{4}v_{2}$
from $v_{4}$ to $v_{2}$, $v_{3}v_{4}$ from $v_{3}$ to $v_{4}$; if
$(b(v_{1}),b(v_{2}),b(v_{3}),b(v_{4}))=(1,2,1,2)$, then we orient the edges of
$G$ as follows: $|E^{+}(v_{7})|=3$, $|E^{+}(v_{5})|=0$, $|E^{+}(v_{6})|=3$,
$|E^{+}(v_{8})|=0$, and $v_{3}v_{1}$ from $v_{3}$ to $v_{1}$, $v_{4}v_{1}$
from $v_{4}$ to $v_{1}$, $v_{2}v_{3}$ from $v_{2}$ to $v_{3}$, $v_{2}v_{4}$
from $v_{2}$ to $v_{4}$, $v_{4}v_{3}$ from $v_{4}$ to $v_{3}$; if
$(b(v_{1}),b(v_{2}),b(v_{3}),b(v_{4}))=(1,1,2,2)$, then we orient the edges of
$G$ as follows: $|E^{+}(v_{7})|=0$, $|E^{+}(v_{5})|=3$, $|E^{+}(v_{6})|=0$,
$|E^{+}(v_{8})|=3$, and $v_{3}v_{1}$ from $v_{3}$ to $v_{1}$, $v_{4}v_{1}$
from $v_{4}$ to $v_{1}$, $v_{3}v_{2}$ from $v_{3}$ to $v_{2}$, $v_{4}v_{2}$
from $v_{4}$ to $v_{2}$, $v_{4}v_{3}$ from $v_{4}$ to $v_{3}$. In each case,
for each $e\in E(G)$ define $f(e)=1$ in such orientation. Clearly, we can
verify that $\partial f(v)=b(v)$ for each $v\in V(G)$. Thus, $G$ is
$Z_{3}$-connected.
## 3 Some special cases
Throughout this section, all sequences are graphic sequences. We provide
$Z_{3}$-connected realizations for some graphic sequences.
###### Lemma 3.1
Suppose that one of the following holds,
(i) $n\geq 6$ and $\pi=(n-2,4,3^{n-2})$;
(ii) $n\geq 5$ and $\pi=(4^{n-4},3^{4})$;
(iii) $n\geq 7$ and $\pi=(5,4^{n-6},3^{5})$.
Then $\pi$ has a $Z_{3}$-connected realization.
Proof. (i) If $n=6$, then by Lemma 2.8, $\pi$ has a $Z_{3}$-connected
realization $G$ in Fig. 1 (a). Thus, we assume that $n\geq 7$.
If $n=7,8$, then by Lemma 2.8, $\pi$ has a $Z_{3}$-connected realization $G$
in Fig. 1 (b) (c). Thus, we assume that $n\geq 9$.
Assume that $n$ is odd. Let $W_{n-5}$ be an even wheel with the center at
$v_{1}$ and $K_{4}^{-}$ on vertex set $\\{u_{1},u_{2},u_{3},u_{4}\\}$ with
$d_{K_{4}^{-}}(u_{1})=d_{K_{4}^{-}}(u_{3})=2$. Denote by $G$ the graph
obtained from $W_{n-5}$ and $K_{4}^{-}$ by adding edges $u_{i}v_{1}$ for each
$i\in\\{1,2,3\\}$. Obviously, the graph $G$ has a degree sequence
$(n-2,4,3^{n-2})$. By part (5) of Lemma 2.2, $W_{n-5}$ is $Z_{3}$-connected.
The graph $G/W_{n-5}$ is an even wheel $W_{4}$. By part (5) and (6) of Lemma
2.2, $G$ is $Z_{3}$-connected. This means that $\pi$ has a $Z_{3}$-connected
realization.
Assume that $n$ is even. Let $G_{0}$ be the graph in Fig. 1 (a) and $W_{n-6}$
be an even wheel with the center at $u_{1}$. Denote by $G$ the graph obtained
from $W_{n-6}$ and $G_{0}$ by identifying $u_{1}$ and $v_{1}$. Clearly, $G$
has a degree sequence $(n-2,4,3^{n-2})$. Since $n\geq 10$ is even, $W_{n-6}$
is $Z_{3}$-connected by (5) of Lemma 2.2. By Lemma 2.8, $G_{0}$ is
$Z_{3}$-connected. This shows that $G$ is $Z_{3}$-connected.
(ii) If $n=5$, then an even wheel $W_{4}$ is a $Z_{3}$-connected realization
of $\pi$; if $n=6$, then by Lemma 2.8, $\pi$ has a $Z_{3}$-connected
realization $G$ in Fig. 1 (a); if $n=7$, then by Lemma 2.9, $\pi$ has a
$Z_{3}$-connected realization shown in Fig. 2 (a); if $n=8$, then by Lemma
2.9, the graph (c) in Fig. 2 is $Z_{3}$-connected realization of $\pi$. If
$n=9$, then let $G_{1}$ be an even wheel $W_{4}$ induced by
$\\{u_{0},u_{1},u_{2},u_{3},u_{4}\\}$ with the center at $u_{0}$ and $G_{2}$
be a $K_{4}^{-}$ induced by $\\{v_{1},v_{2},v_{3},v_{4}\\}$ with
$d_{G_{2}}(v_{1})=d_{G_{2}}(v_{3})=3$. We construct a graph $G$ from $W_{4}$
and $K_{4}^{-}$ by adding three edges $u_{1}v_{2}$, $u_{2}v_{4}$ and
$u_{3}v_{1}$. Then $G$ is a $Z_{3}$-connected realization of $(4^{5},3^{4})$.
Thus, we assume that $n\geq 10$.
Assume that $n=2k$, where $k\geq 5$. By induction of hypothesis, let $G_{i}$
be a $Z_{3}$-connected realization of the degree sequence $(4^{k-4},3^{4})$
for $i\in\\{1,2\\}$. Assume that $n=2k+1$, where $k\geq 5$. By induction
hypothesis, let $G_{1}$ be a $Z_{3}$-connected realization of the degree
sequence $(4^{k-4},3^{4})$ and $G_{2}$ be a $Z_{3}$-connected realization of
the degree sequence $(4^{k-3},3^{4})$. In each case, we construct a graph $G$
from $G_{1}$ and $G_{2}$ by connecting a pair of 3-vertices of $G_{1}$ to a
pair of 3-vertices of $G_{2}$ one by one. It is easy to verify that $G$ is a
$Z_{3}$-connected realization of the degree sequence $(4^{n-4},3^{4})$.
(iii) If $n=7$, then by Lemma 2.8, the graph (b) in Fig. 1 is a
$Z_{3}$-connected realization of $\pi$; if $n=8$, then by Lemma 2.9, the graph
(b) in Fig. 2 is a $Z_{3}$-connected realization of $\pi$. If $n=9$, then
$\pi=(5,4^{3},3^{5})$. Let $G_{1}$ be an even wheel $W_{4}$ induced by
$\\{u_{0},u_{1},u_{2},u_{3},u_{4}\\}$ with the center at $u_{0}$ and $G_{2}$
be a $K_{4}^{-}$ induced by $\\{v_{1},v_{2},v_{3},v_{4}\\}$ with
$d_{G_{2}}(v_{1})=d_{G_{2}}(v_{3})=3$. We construct a graph $G$ from $W_{4}$
and $K_{4}^{-}$ by adding three edges $u_{0}v_{2}$, $u_{1}v_{1}$,
$u_{2}v_{4}$. We conclude that $G$ is a $Z_{3}$-connected realization of
degree sequence $(5,4^{3},3^{5})$. Thus, $n\geq 10$.
Assume that $n=2k$, where $k\geq 5$. By (ii), let $G_{1}$ and $G_{2}$ be
$Z_{3}$-connected realizations of degree sequence $(4^{k-4},3^{4})$. Assume
that $n=2k+1$, where $k\geq 5$. By (ii), let $G_{1}$ be a $Z_{3}$-connected
realization of degree sequence $(4^{k-4},3^{4})$ and $G_{2}$ be a
$Z_{3}$-connected realization of degree sequence $(4^{k-3},3^{4})$. In each
case, choose one 4-vertex $u_{1}$ and one 3-vertex $u_{2}$ of $G_{1}$; choose
two 3-vertices $v_{1},v_{2}$ of $G_{2}$. We construct a graph $G$ from $G_{1}$
and $G_{2}$ by adding $u_{1}v_{1}$ and $u_{2}v_{2}$. Thus, $G$ is a
$Z_{3}$-connected realization of degree sequence $(5,4^{n-6},3^{5})$.
###### Lemma 3.2
If $\pi=(n-3,3^{n-1})$, then $\pi$ has not a $Z_{3}$-connected realization.
Proof. Suppose otherwise that $G$ has a $Z_{3}$-connected realization of
degree sequence $(n-3,3^{n-1})$. Let
$V(G)=\\{u,u_{1},\ldots,u_{n-3},x_{1},x_{2}\\}$,
$N_{G}(u)=\\{u_{1},\ldots,u_{n-3}\\}$ ($N$ for short), and
$X=\\{x_{1},x_{2}\\}$. We now consider the following two cases.
Case 1. $x_{1}x_{2}\in E(G)$.
Since $G$ is $Z_{3}$-connected, $G$ is 3-flowable. By Lemma 2.5 and symmetry,
we may assume that $|E^{+}(x_{1})|=3$ and $|E^{-}(x_{2})|=3$. Since
$d(u_{i})=3$ for $i\in\\{1,\ldots,n-3\\}$, by Lemma 2.5, either
$|E^{+}(u_{i})|=3$ or $|E^{-}(u_{i})|=3$. This implies that there exists no
vertex $u_{i}$ in $N$ such that $u_{i}x_{1},u_{i}x_{2}\in E(G)$. Thus, $G[N]$
is the union of two paths $P_{1}$ and $P_{2}$. We relabel the vertices of $N$
such that $P_{1}=u_{1}\ldots u_{k}$ and $P_{2}=u_{k+1}\ldots u_{n-3}$.
Suppose first that $x_{1}u_{1},x_{1}u_{k}\in E(G)$ and
$x_{2}u_{k+1},x_{2}u_{n-3}\in E(G)$. Since $G$ is 3-flowable, by Lemma 2.5,
$P_{i}$ contains odd number of vertices for each $i\in\\{1,2\\}$. Define
$b(u)=b(x_{1})=b(x_{2})=1$ and $b(u_{i})=0$ for each $i\in\\{1,\ldots,n-3\\}$.
It is easy to verify that there exists no $f:E(G)\to Z_{3}^{*}$ such that
$\partial f(v)=b(v)$ for each $v\in V(G)$, contrary to that $G$ is
$Z_{3}$-connected.
Next, suppose that $x_{1}u_{1},x_{1}u_{k+1}\in E(G)$ and
$x_{2}u_{k},x_{2}u_{n-3}\in E(G)$. Since $G$ is 3-flowable, by Lemma 2.5
$P_{i}$ contains even number of vertices for each $i\in\\{1,2\\}$. Define
$b(u)=1$, $b(x_{2})=2$ and $b(u_{i})=b(x_{1})=0$ for each
$i\in\\{1,\ldots,n-3\\}$. It is easy to verify that there exists no $f:E(G)\to
Z_{3}^{*}$ such that $\partial f(v)=b(v)$ for each $v\in V(G)$, contrary to
that $G$ is $Z_{3}$-connected.
Case 2. $x_{1}x_{2}\notin E(G)$.
Since $d(x_{i})=3$ for each $i=1,2$, $0\leq|N(x_{1})\cap N(x_{2})|\leq 3$.
Assume first that $|N(x_{1})\cap N(x_{2})|=3$. We assume, without loss of
generality, that $u_{1},u_{2},u_{3}\in N(x_{1})\cap N(x_{2})$. The subgraph
induced by $\\{u,x_{1},x_{2},u_{1},u_{2},u_{3}\\}$ is $K_{3,3}$ which is not
$Z_{3}$-connected by part (4) of Lemma 2.2. By part (6) of Lemma 2.2, $G$ is
not $Z_{3}$-connected, a contradiction.
Assume that $|N(x_{1})\cap N(x_{2})|=2$. We assume, without loss of
generality, that $u_{1},u_{2}\in N(x_{1})\cap N(x_{2})$. Since $G$ is
3-flowable, the graph $H$ induced by $N\setminus\\{u_{1},u_{2}\\}$ consists of
even cycles and a path of length even. This means that $n$ is even. If $n=6$,
then this case cannot occur. Thus $n\geq 8$. Define $b(x_{1})=1$, $b(x_{2})=2$
and $b(u_{i})=b(u)=0$ for each $i\in\\{1,\ldots,n-3\\}$. It is easy to verify
that there exists no $f:E(G)\to Z_{3}^{*}$ such that $\partial f(v)=b(v)$ for
each $v\in V(G)$, contrary to that $G$ is $Z_{3}$-connected.
Next, assume that $|N(x_{1})\cap N(x_{2})|=1$. We assume, without loss of
generality, that $u_{1}\in N(x_{1})\cap N(x_{2})$. The graph induced by
$N\setminus\\{u_{1}\\}$ consists of even cycles and two paths $P_{1}$ and
$P_{2}$. Since $G$ is 3-flowable, $P_{i}$ contains odd vertices for each
$i\in\\{1,2\\}$. Then $n$ is even. If $n=6,8$, then this case cannot occur.
Thus, we assume that $n\geq 10$. Define $b(x_{1})=1$, $b(x_{2})=2$ and
$b(u_{i})=b(u)=0$ for each $i\in\\{1,\ldots,n-3\\}$. In this case, there
exists no $f:E(G)\to Z_{3}^{*}$ such that $\partial f(v)=b(v)$ for each $v\in
V(G)$, contrary to that $G$ is $Z_{3}$-connected.
Finally, assume that $|N(x_{1})\cap N(x_{2})|=0$. Then the graph induced by
the vertices of $N$ consists of three paths $P_{1}$, $P_{2}$ and $P_{3}$,
together with even cycles. We relabel the vertices of $N$ such that
$P_{1}=u_{1}\ldots u_{s}$, $P_{2}=u_{s+1}\ldots u_{t}$ and
$P_{3}=u_{t+1}\ldots u_{n-3}$. By symmetry, we consider two cases: $x_{1}$ is
adjacent to both the end vertices of some $P_{i}$; $x_{1}$ is adjacent to one
of each $P_{j}$ for $j\in\\{1,2,3\\}$.
In the former case, we may assume that $u_{1}x_{1},u_{s}x_{1}\in E(G)$ and
$x_{2}u_{s+1},x_{2}u_{t}$. Since $G$ is 3-flowable, by Lemma 2.5, both
$|V(P_{1})|$ and $|V(P_{2})|$ are odd. If $|V(P_{3})|$ is odd, then define
$b(x_{1})=1$, $b(x_{2})=2$ and $b(u_{i})=b(u)=0$ for each
$i\in\\{1,\ldots,n-3\\}$. If $|V(P_{3})|$ is even, then define $b(x_{1})=1$,
$b(x_{2})=1$, $b(u)=1$ and $b(u_{i})=0$ for each $i\in\\{1,\ldots,n-3\\}$. In
either case, there exists no $f:E(G)\to Z_{3}^{*}$ such that $\partial
f(v)=b(v)$ for each $v\in V(G)$, contrary to that $G$ is $Z_{3}$-connected.
In the latter case,
$x_{1}u_{1},x_{1}u_{s+1},x_{1}u_{t+1},x_{2}u_{s},x_{2}u_{t},x_{2}u_{n-3}\in
E(G)$. It follows that $|V(P_{1})|,|V(P_{2})|$ and $|V(P_{3})|$ have the same
parity. If each of $|V(P_{i})|$ for $i\in\\{1,2,3\\}$ is even, then define
$b(x_{1})=1$, $b(x_{2})=1$, $b(u)=1$ and $b(u_{i})=0$ for each
$i\in\\{1,\ldots,n-3\\}$. If each of $|V(P_{i})|$ for $i\in\\{1,2,3\\}$ is
odd, then define define $b(x_{1})=1$, $b(x_{2})=2$, $b(u)=b(u_{i})=0$ for each
$i\in\\{1,\ldots,n-3\\}$. In either case, there exists no $f:E(G)\to
Z_{3}^{*}$ such that $\partial f(v)=b(v)$ for each $v\in V(G)$, contrary to
that $G$ is $Z_{3}$-connected.
## 4 Proof of Theorem 1.4
In order to prove Theorem 1.4, we establish the following lemma.
###### Lemma 4.1
Suppose that $\pi=(d_{1},\ldots,d_{n})$ is a nonincreasing graphic sequence
with $d_{n}\geq 3$. If $d_{1}=n-2$, then $\pi$ has a $Z_{3}$-connected
realization.
Proof. Suppose, to the contrary, that $\pi=(d_{1},\ldots,d_{n})$ has no
$Z_{3}$-connected realization with $n$ minimized, where $d_{1}=n-2$. By
Theorem 1.3, we may assume that $d_{n-3}\leq 3$. In order to prove our lemma,
we need the following claim.
Claim 1. Each of the following holds.
(i) $d_{n-3}=d_{n-2}=d_{n-1}=d_{n}=3$;
(ii) $n\geq 6$.
Proof of Claim 1. (i) follows since $d_{n}\geq 3$.
(ii) Since $d_{n}=3$, $n\geq 4$. If $n=4$, then $d_{1}=3=n-1$, contrary to
that $d_{1}=n-2$. If $n=5$, then $\pi=(3^{5})$ is not graphic by Theorem 2.1.
This proves Claim 1.
If $n=6$, then $d_{1}=4$ and $d_{3}=d_{4}=d_{5}=d_{6}=3$. By Theorem 2.1,
$d_{2}=4$. By Lemma 2.8, $\pi=(4^{2},3^{4})$ has a $Z_{3}$-realization, a
contradiction. Thus, we may assume that $n\geq 7$.
Claim 2. $d_{3}=3$.
Proof of Claim 2. Suppose otherwise that $d_{3}\geq 4$ and $G$ is a
counterexample with $|V(G)|=n$ minimized. Then $d_{2}\geq d_{3}\geq 4$. Hence,
$\bar{\pi}=(n-3,d_{2}-1,d_{3}-1,d_{4},\ldots,d_{n-1})$=$(\bar{d_{1}},\ldots,\bar{d}_{n-1})$
with $\bar{d_{1}}\geq\ldots\geq\bar{d}_{n-1}$. This implies that
$\bar{d}_{n-1}\geq 3$ and $\bar{d_{1}}=(n-1)-2$ or $\bar{d_{1}}=d_{4}$. In the
former case, since $\bar{d_{1}}=n-3=(n-1)-2$, by the minimality of $n$,
$\bar{\pi}$ has a $Z_{3}$-connected realization $\bar{G}$. In the latter case,
$\bar{d_{1}}\neq n-3$ and hence $\bar{d_{1}}=d_{4}>n-3$. Since $d_{1}=n-2\geq
d_{4}$, $d_{4}=n-2$. This means that $\bar{d_{1}}=d_{4}=(n-1)-1$. By Theorem
1.2, either $\bar{\pi}$ has a $Z_{3}$-connected realization $\bar{G}$ or
$\bar{\pi}=(k,3^{k})$, $(k^{2},3^{k-1})$, where $k$ is odd. If
$\bar{\pi}=(k,3^{k})$, then $d_{1}=k+1=n-2$ and $d_{2}=d_{3}=4$. On the other
hand, $n=k+1+1=k+2$. This contradiction proves that $\pi\not=(k,3^{k})$.
Similarly, $\bar{\pi}\not=(k^{2},3^{k-1})$. If $\bar{\pi}$ has a
$Z_{3}$-connected realization $\bar{G}$, then $\pi$ has a realization $G$ of
$\pi$ from $\bar{G}$ by adding a new vertex $v$ and three edges joining $v$ to
the corresponding vertices of $\bar{G}$. By part (8) of Lemma 2.2, $G$ is
$Z_{3}$-connected, a contradiction. Thus $d_{3}\leq 3$. Clearly, $d_{3}\geq
3$. Then $d_{3}=3$. This proves Claim 2.
By Claims 1 and 2, $\pi=(n-2,d_{2},3^{n-2})$. Since $\pi$ is graphic, $d_{2}$
is even whenever $n$ is even or odd. Moreover, $d_{2}\geq 4$. Recall that
$n\geq 7$. In this case, $\pi=(n-2,4,3^{n-2})$. By (i) of Lemma 3.1, $\pi$ has
a $Z_{3}$-connected realization $G$, a contradiction. Thus, we may assume that
$d_{2}\geq 6$. Since $n-2=d_{1}\geq d_{2}\geq 6$, $n\geq d_{2}+2\geq 8$.
Consider the case that $n$ is even. Denote by $W_{n-d_{2}+2}$ an even wheel
with the center at $v_{1}$ and by $S$ a vertex set such that $|S|=d_{2}-4$ and
$V(W_{n-d_{2}+2})\cap S=\emptyset$. Note that $|S|$ is even. We construct a
graph $G$ from $W_{n-d_{2}+2}$ and $S$ as follows: First, pick two vertices
$s_{1},s_{2}$ of $S$ and add $(d_{2}-6)/2$ edges such that the subgraph
induced by $S\setminus\\{s_{1},s_{2}\\}$ is a perfect matching. Second, let
$v_{1}$ connect to each vertex of $S$. Third, pick a vertex $v_{2}$ in
$W_{n-d_{2}+2}$ and let $v_{2}$ join to each vertex of $S$. Finally, add one
new vertex $x$ adjacent to $v_{2}$, $s_{1}$ and $s_{2}$.
We claim that $G$ has a degree sequence $(n-2,d_{2},3^{n-2})$. Since
$d_{W_{n-d_{2}+2}}(v_{1})=n-d_{2}+2\geq 4$, $d(v_{1})=n-d_{2}+2+d_{2}-4=n-2$,
$d(v_{2})=3+d_{2}-4+1=d_{2}$, each vertex of $V(G)\setminus\\{v_{1},v_{2}\\}$
is a 3-vertex. Since $W_{n-d_{2}+2}$ is an even wheel, by part (5) of Lemma
2.2, this wheel is $Z_{3}$-connected. By part (8) of Lemma 2.2, $G$ is
$Z_{3}$-connected, a contradiction.
Consider the case that $n$ is odd. Denote by $W_{n-d_{2}+1}$ an even wheel
with the center at $v_{1}$ and by $S$ a vertex set with $|S|=d_{2}-3$ and
$V(W_{n-d_{2}+1})\cap S=\emptyset$. We construct a graph $G$ from
$W_{n-d_{2}+1}$ and $S$ as follows: First, let $v_{1}$ connect to each vertex
of $S$. Second, pick one vertex $v_{2}$ in $W_{n-d_{2}+1}$ and let $v_{2}$
join to each vertex of $S$. Third, add one vertex $x$ adjacent to three
vertices of $S$. Finally, add $(d_{2}-6)/2$ edges in $S$ so that the subgraph
induced by vertices of $S$, each of which is not adjacent to $x$, is a perfect
matching. We claim that $G$ is a realization of degree sequence
$(n-2,d_{2},3^{n-2})$. Since $d_{W_{n-d_{2}+1}}(v_{1})=n-d_{2}+1\geq 4$,
$d(v_{1})=n-d_{2}+1+d_{2}-3=n-2$. Note that $d(v_{2})=3+d_{2}-3=d_{2}$, each
vertex of $V(G)\setminus\\{v_{1},v_{2}\\}$ is a 3-vertex. Similarly, it can be
verified that $G$ is a $Z_{3}$-connected realization of $\pi$, a
contradiction.
Proof of Theorem 1.4. Assume that $\pi=(d_{1},\ldots,d_{n})$ is a
nonincreasing graphic sequence with $d_{1}\geq n-3$. If $\pi$ is one of
$(n-3,3^{n-1})$, $(k,3^{k})$ and $(k^{2},3^{k-1})$, then by Lemmas 2.7 and
3.2, $\pi$ has no $Z_{3}$-connected realization.
Conversely, assume that
$\pi\notin\\{(n-3,3^{n-1}),(k,3^{k}),(k^{2},3^{k-1})\\}$. Since $d_{1}\geq
n-3$ and $d_{n}\geq 3$, $n\geq 6$. In the case that $n=6$, by Theorem 1.2,
$d_{1}=3,4$. If $d_{1}=3$, then $\pi=(3^{6})$. Since $n=6$,
$(3^{6})=(n-3,3^{n-1})$, contrary to our assumption. If $d_{1}=4$, then by
(ii) of Lemma 3.1 $\pi$ has a $Z_{3}$-connected realization. In the case that
$n=7$, by Theorems 1.2 and 2.1, $4\leq d_{1}\leq 5$. If $d_{1}=5$, then any
realization of $\pi$ contains the graph (b) of fig. 1. By Lemma 2.8, $\pi$ has
a $Z_{3}$-connected realization. Assume that $d_{1}=4$. Since $n=7$,
$(4,3^{6})=(n-3,3^{n-1})$. Thus, by our assumption, $\pi\not=(4,3^{6})$. In
this case, any realization of $\pi$ contains the graph (a) in Fig. 2. By Lemma
2.9, $\pi$ has a $Z_{3}$-connected realization. Thus, assume that $n\geq 8$.
By Theorem 1.2 and by Lemmas 3.2 and 4.1, we are left to prove that if
$d_{1}=n-3,d_{n}\geq 3$ and $d_{2}\neq 3$, then $\pi$ has a $Z_{3}$-connected
realization. Suppose otherwise that $\pi=(d_{1},\ldots,d_{n})$ satisfying
$d_{1}=n-3$, $d_{2}\neq 3$, $d_{n}\geq 3$. (1)
Subject to (1),
$\pi$ has no $Z_{3}$-realization with $n$ minimized. (2)
We establish the following claim first.
Claim 1. (i) $d_{n-3}=d_{n-2}=d_{n-1}=d_{n}=3$.
(ii) $3\leq d_{3}\leq 4$.
_Proof of Claim 1._ By Theorem 1.3, $d_{n-3}\leq 3$. (i) follows since
$d_{n}\geq 3$.
(ii) Suppose otherwise that subject to (1) and (2), $\pi$ satisfies $d_{3}\geq
5$. Since $d_{2}\geq d_{3}$, $d_{2}\geq 5$. Define
$\bar{\pi}=(n-4,d_{2}-1,d_{3}-1,d_{4},\ldots,d_{n-1})$=$(\bar{d_{1}},\ldots,\bar{d}_{n-1})$
with $\bar{d_{1}}\geq\ldots\geq\bar{d}_{n-1}$. Since $d_{3}\geq 5$,
$d_{2}-1\geq d_{3}-1\geq 4$. This means that $\bar{d_{1}}\geq\bar{d_{2}}\geq
4$, and $\bar{d}_{n-1}\geq 3$. If $d_{1}>d_{4}$, then $\bar{d_{1}}=(n-1)-3$.
In this case, by the minimality of $n$, $\bar{\pi}$ has a $Z_{3}$-connected
realization $\bar{G}$. If $d_{1}=d_{4}$, then $\bar{d_{1}}=d_{4}$. It follows
that $d_{4}>n-4$. This implies that $d_{1}=d_{2}=d_{3}=d_{4}=n-3$. Thus,
$\bar{d_{1}}=d_{4}=n-3=(n-1)-2$. By Lemma 4.1, $\bar{\pi}$ has a
$Z_{3}$-connected realization $\bar{G}$. In either case, $\pi$ has a
realization $G$ obtained from $\bar{G}$ by adding a new vertex $v$ and three
edges joining $v$ to the corresponding vertices of $\bar{G}$. By part (8) of
Lemma 2.2, $G$ is $Z_{3}$-connected, a contradiction. Thus $d_{3}\leq 4$.
Since $d_{3}\geq 3$, $3\leq d_{3}\leq 4$. This proves Claim 1.
By Claim 1, we may assume that $\pi=(n-3,d_{2},d_{3},\ldots,d_{n-4},3^{4})$
with $d_{3}\in\\{3,4\\}$. We consider the following two cases.
Case 1. $d_{3}=3$.
In this case, $\pi=(n-3,d_{2},3^{n-2})$. Since $\pi$ is graphic, $d_{2}$ is
odd. Since $d_{2}\not=3$, $d_{2}\geq 5$. We first assume that $d_{2}=5$. In
this case, $\pi=(n-3,5,3^{n-2})$. If $n=8$, by Lemma 2.8, the graph (d) in
Fig. 1 is a $Z_{3}$-connected realization of $\pi=(5^{2},3^{6})$. Thus, assume
that $n\geq 9$.
Assume that $n$ is odd. Denote by $W_{n-5}$ an even wheel with the center at
$v_{1}$ and by $S$ a vertex set with $|S|=2$. We construct graph $G$ from
$W_{n-5}$ and $S$ as follows: First, connect $v_{1}$ to each vertex of $S$.
Second, choose one vertex $v_{2}$ in $W_{n-5}$ and add two vertices
$x_{1},x_{2}$ such that $x_{i}$ is adjacent to $v_{2}$ and each vertex of $S$
for each $i\in\\{1,2\\}$.
Since $d(v_{1})=n-5+2=n-3$, $d(v_{2})=3+2=5$ and each vertex of
$V(G)\setminus\\{v_{1},v_{2}\\}$ is a 3-vertex, this means that $G$ is a
realization of degree sequence $(n-3,5,3^{n-2})$. By part (5) of Lemma 2.2,
$W_{n-5}$ is $Z_{3}$-connected. Note that $G/W_{n-5}$ is an even wheel $W_{4}$
which is also $Z_{3}$-connected by Lemma 2.2. It follows by part (6) of Lemma
2.2 that $G$ is $Z_{3}$-connected, a contradiction.
Thus we may assume that $n$ is even. Denote by $W_{n-6}$ an even wheel with
the center at $v_{1}$ and let $S=\\{s_{1},s_{2},s_{3}\\}$ be a vertex set. We
construct graph $G$ from $W_{n-6}$ and $S$ as follows: First, connect $v_{1}$
to each vertex of $S$. Second, choose one vertex $v_{2}$ in $W_{n-6}$ and and
let $v_{2}$ be adjacent to $s_{1}$. Finally, add two vertices $x_{1},x_{2}$
such that $x_{1}$ is adjacent to $v_{2}$ and $s_{2}$, $s_{3}$; $x_{2}$ is
adjacent to each vertex of $S$.
It is easy to verify that $d(v_{1})=n-6+3=n-3$, $d(v_{2})=3+2=5$ and each
vertex of $V(G)\setminus\\{v_{1},v_{2}\\}$ is a 3-vertex. This means that $G$
is a realization of degree sequence $(n-3,5,3^{n-2})$. By (5) of Lemma 2.2,
$W_{n-6}$ is $Z_{3}$-connected. By part (8) of Lemma 2.2,
$W_{n-6}\cup\\{s_{1}\\}$ is $Z_{3}$-connected. Note that
$G/\\{W_{n-6}\cup\\{s_{1}\\}\\}$ is an even wheel $W_{4}$ which is
$Z_{3}$-connected. It follows by (6) of Lemma 2.2 that $G$ is
$Z_{3}$-connected, a contradiction.
From now on, we assume that $d_{2}\geq 7$. In this case, $n\geq d_{2}+3\geq
10$. Consider the case that $n$ is even. Denote by $W_{n-d_{2}+1}$ an even
wheel with the center at $v_{1}$ and by $S$ a vertex set with $|S|=d_{2}-4$.
We construct graph $G$ from $W_{n-d_{2}+1}$ and $S$ as follows: First, connect
$v_{1}$ to each vertex of $S$. Second, pick one vertex $s$ of $S$ and let
$S_{1}=S\setminus\\{s\\}$, pick one vertex $v_{2}$ in $W_{n-d_{2}+1}$ and
connect $v_{2}$ to each vertex of $S_{1}$. Third, pick two vertices
$s_{1},s_{2}$ of $S_{1}$, and add $(d_{2}-7)/2$ edges such that the induced
subgraph by $S_{1}\setminus\\{s_{1},s_{2}\\}$ is a perfect matching. Finally,
we add two vertices $x_{1}$ and $x_{2}$ such that $x_{i}$ is adjacent to
$v_{2}$, $x_{i}$ is adjacent to $s_{i}$ for $i=1,2$ and $s$ is adjacent to
each of $x_{1}$ and $x_{2}$.
Since $d(v_{1})=n-d_{2}+1+d_{2}-4=n-3$, $d(v_{2})=3+d_{2}-5+2=d_{2}$ and each
vertex of $V(G)-\\{v_{1},v_{2}\\}$ is a 3-vertex, this implies s that $G$ is a
realization of degree sequence $(n-3,d_{2},3^{n-2})$. By (5) of Lemma 2.2,
$W_{n-d_{2}+1}$ is $Z_{3}$-connected. Contracting this even wheel
$W_{n-d_{2}+1}$ and contracting all 2-cycles generated in the process, we get
$K_{1}$. By (8) of Lemma 2.2, $G$ is $Z_{3}$-connected, a contradiction.
Consider the case that $n$ is odd. Denote by $W_{n-d_{2}}$ an even wheel with
the center at $v_{1}$ and by $S$ a vertex set with $|S|=d_{2}-3$. We construct
a graph from $W_{n-d_{2}}$ and $S$ as follows. First, let $v_{1}$ be adjacent
to each vertex of $S$. Second, pick two vertices $s_{3}$ and $s_{4}$ of $S$,
define $S_{1}=S\setminus\\{s_{3},s_{4}\\}$ and pick one vertex $v_{2}$ in $W$
so that $v_{2}$ is adjacent to each vertex of $S_{1}$. Third, add two new
vertex $x_{1},x_{2}$ such that $x_{i}$ is adjacent to each of $v_{2}$, $s_{3}$
and $s_{4}$. Finally, add $(d_{2}-5)/2$ edges in $S_{1}$ so that the subgraph
induced by vertices of $S_{1}\setminus\\{s_{3},s_{4}\\}$ is a perfect
matching.
Since $d(v_{1})=n-d_{2}+d_{2}-3=n-3$, $d(v_{2})=3+d_{2}-5+2=d_{2}$, and each
vertex of $V(G)\setminus\\{v_{1},v_{2}\\}$ is a 3-vertex, $G$ is a realization
of degree sequence $(n-3,d_{2},3^{n-2})$. Similarly, by parts (5) and (8) of
Lemma 2.2, $G$ is $Z_{3}$-connected, a contradiction.
Case 2. $d_{3}=4$.
In this case, $d_{2}\geq 4$ and
$\pi=(n-3,d_{2},4,d_{4},\ldots,d_{n-4},3^{4})$. Define
$\bar{\pi}=(n-4,d_{2}-1,3,d_{4},\ldots,d_{n-4},\\\
3^{3})=(\bar{d}_{1},\ldots,\bar{d}_{n-1})$ with
$\bar{d}_{1}\geq\ldots\geq\bar{d}_{n-1}$. If $d_{1}=d_{4}$, then $n-3=4$ and
hence $n=7$, contrary to assumption that $n\geq 8$. Thus, $d_{1}>d_{4}$. In
this case, $\bar{d}_{1}=n-4$.
Claim 2. $d_{2}=4$.
Proof of Claim 2. Suppose otherwise that $d_{2}\geq 5$. Then $\bar{d_{2}}\geq
4$ and $\bar{\pi}$ satisfies (1). By the minimality of $n$, $\bar{\pi}$ has a
$Z_{3}$-connected realization $\bar{G}$. Thus, we conclude that $G$ is a
$Z_{3}$-connected realization of $\pi$ obtained from $\bar{G}$ by adding a new
vertex $v$ and three edges joining $v$ to the corresponding vertices of
$\bar{G}$. This contradiction proves Claim 2.
By Claim 2, $d_{2}=4$. Assume that $i\in\\{3,\ldots,n-4\\}$ such that
$d_{i}=4$ and $d_{i+1}=3$. Thus $\pi=(n-3,4^{i-1},3^{n-i})$.
Claim 3. $i=3$.
Proof of Claim 3. If $i$ is even, then $n-i$ is odd (even) when $n$ is odd
(even). No matter whether $n$ is odd or even, there are odd vertices of odd
degree, a contradiction. Thus, $i$ is odd. If $i\geq 5$, then
$\bar{\pi}=(n-4,4^{i-3},3^{n-i+1})$ satisfies (1). Recall that $n\geq 8$, by
the minimality of $n$, $\bar{\pi}$ has a $Z_{3}$-connected realization
$\bar{G}$. In this case, we can obtain a realization $G$ of $\pi$ from
$\bar{G}$ by adding a new vertex $v$ and three edges joining $v$ to the
corresponding vertices of $\bar{G}$. By (8) of Lemma 2.2, $G$ is
$Z_{3}$-connected, a contradiction. This proves Claim 3.
By Claim 3, $i=3$. This leads to that $\pi=(n-3,4^{2},3^{n-3})$. Recall that
$n\geq 8$. If $n=8$, then by Lemma 2.9, $\pi$ has a $Z_{3}$-connected
realization. Thus, we may assume that $n\geq 9$.
In the case that $n$ is odd, denote by $W_{n-5}$ the even wheel with the
center at $v_{1}$ and by $S$ a vertex set with $|S|=2$. We construct a graph
$G$ from $W_{n-5}$ and $S$ as follows. First, let $v_{1}$ be adjacent to each
vertex of $S$. Second, pick two vertices $v_{2},v_{3}$ in $W_{n-5}$ and add
two vertices $x_{1},x_{2}$ such that $x_{i}$ is adjacent to $v_{i+1}$ and each
vertex of $S$ for each $i\in\\{1,2\\}$.
It is easy to verify that $d(v_{1})=n-5+2=n-3$, $d(v_{i})=3+1=4$ for each
$i\in\\{2,3\\}$, and each vertex of $V(G)-\\{v_{1},v_{2},v_{3}\\}$ is a
3-vertex. Obviously, $G$ has a degree sequence $(n-3,4^{2},3^{n-3})$. By (5)
of Lemma 2.2, $W_{n-5}$ is $Z_{3}$-connected. $G/W_{n-5}$ is an even wheel
$W_{4}$ which is $Z_{3}$-connected. By (6) of Lemma 2.2, $G$ is
$Z_{3}$-connected, a contradiction.
In the case that $n$ is even, denote by $W_{n-6}$ the even wheel with the
center at $v_{1}$ and let $S=\\{s_{1},s_{2},s_{3}\\}$. We construct a graph
$G$ from $W_{n-6}$ and $S$ as follows. First, let $v_{1}$ be adjacent to each
vertex of $S$. Second, pick two vertices $v_{2},v_{3}$ in $W_{n-6}$ so that
$v_{2}$ is adjacent to $s_{1}$. Finally, we add two vertices $x_{1},x_{2}$
such that $x_{1}$ is adjacent to $v_{3}$ and $s_{2}$, $s_{3}$ and such that
$x_{2}$ is adjacent to each vertex of $S$.
It is easy to verify that $d(v_{1})=n-6+3=n-3$, $d(v_{i})=3+1=4$ for each
$i\in\\{2,3\\}$, and each vertex of $V(G)-\\{v_{1},v_{2},v_{3}\\}$ is a
3-vertex. Obviously, $G$ has a degree sequence $(n-3,4^{2},3^{n-3})$. By (5)
Lemma 2.2, $W_{n-6}$ is $Z_{3}$-connected. By (8) of Lemma 2.2,
$W_{n-6}\cup\\{s_{1}\\}$ is $Z_{3}$-connected.
$G/\\{W_{n-6}\cup\\{s_{1}\\}\\}$ is an even wheel $W_{4}$ which is
$Z_{3}$-connected. By (6) of Lemma 2.2, $G$ is $Z_{3}$-connected a
contradiction. $\blacksquare$
## 5 Proof of Theorem 1.5
We first establish the following lemma which is used in the proof of Theorem
1.5.
###### Lemma 5.1
Let $\pi=(d_{1},\ldots,d_{n})$ be a nonincreasing graphic sequence. If
$d_{n}\geq 3$ and $d_{n-4}\geq 4$, then either $\pi$ has a $Z_{3}$-connected
realization or $\pi=(5^{2},3^{4})$.
Proof. Since $d_{n-4}\geq 4$, $n\geq 5$. If $n=5$, then by Theorem 1.3,
$\pi=(4,3^{4})$. In this case, an even wheel $W_{4}$ is a $Z_{3}$-connected
realization of $\pi$. If $n=6$, then by Theorem 1.3, $\pi=(4^{2},3^{4})$ or
$(5^{2},3^{4})$. If $\pi=(4^{2},3^{4})$, then by Lemma 2.8, the graph (a)
shown in Fig.1 is a $Z_{3}$-connected realization of $\pi$. If $n=7$, then by
Theorem 1.3, $\pi=(5^{2},4,3^{4})$. Let $G$ be the graph (b) shown in Fig. 1
which has degree sequence $\pi=(5,4,3^{5})$. Denote by $G^{\prime}$ the graph
obtained from $G$ by adding an edge joining a vertex of degree 3 to a vertex
of degree 4. By Lemma 2.8, $G$ is a $Z_{3}$-connected realization of
$(5,4,3^{4})$ and so $G^{\prime}$ is a $Z_{3}$-connected realization of
$(5^{2},4,3^{4})$. Thus, assume that $n\geq 8$.
By Theorems 1.3 and 1.4, it is sufficient to prove that if $d_{n-3}=3$,
$d_{n-4}\geq 4$ and $d_{1}\leq n-4$, then $\pi$ has a $Z_{3}$-connected
realization. In this case, $\pi=(d_{1},\ldots,d_{n-4},3^{4})$. Suppose, to the
contrary, that $\pi=(d_{1},\ldots,d_{n})$ satisfies
$d_{n-3}=3$, $d_{n-4}\geq 4$ and $d_{1}\leq n-4$. (3)
Subject to (3),
$\pi$ has no $Z_{3}$-connected realization with $n$ minimized. (4)
Assume that $d_{n-4}\geq 5$. Define
$\bar{\pi}=(d_{1}-1,d_{2}-1,d_{3}-1,d_{4},\ldots,d_{n-4},3^{3})=(\bar{d_{1}},\ldots,\bar{d}_{n-1})$.
Since $d_{1},d_{2},d_{3}\geq 5$ and $n\geq 8$, $\bar{d}_{n-5}\geq 4$. This
implies that $\bar{\pi}$ satisfies (3), by the minimality of $n$, $\bar{\pi}$
has a $Z_{3}$-connected realization $\bar{G}$. Thus, we construct a
realization $G$ of $\pi$ from $\bar{G}$ by adding a new vertex $v$ and three
edges joining $v$ to the corresponding vertices of $\bar{G}$. It follows by
(8) of Lemma 2.2 that $G$ is $Z_{3}$-connected, a contradiction. Thus, we may
assume that $d_{n-4}=4$.
On the other hand, if $d_{1}=4$, then $\pi=(4^{n-4},3^{4})$. By Lemma 3.1,
$\pi$ has a $Z_{3}$-connected realization, a contradiction. Thus, assume
$d_{1}\geq 5$. Since $d_{n-4}=4$ and $n\geq 8$, $d_{2}\geq 5$ or $d_{2}=4$.
In the former case, $\bar{d}_{n-5}\geq 4$. In this case, $\bar{\pi}$ satisfies
(3), by the minimality of $n$, $\bar{\pi}$ has a $Z_{3}$-connected realization
$\bar{G}$. Thus, we can construct a realization $G$ of $\pi$ from $\bar{G}$ by
adding a new vertex $v$ and three edges joining $v$ to the corresponding
vertices of $\bar{G}$. By (8) of Lemma 2.2, $G$ is $Z_{3}$-connected, a
contradiction.
In the latter case, $\pi=(d_{1},4^{n-5},3^{4})$. Since $\pi$ is graphic,
$d_{1}$ is even. Since $d_{1}\leq n-4$, $n-d_{1}-1\geq 3$. In the case that
$n-d_{1}-1=3$, we have $d_{1}=n-4$ and $n\geq 10$ is even. Denote by $W_{n-4}$
an even wheel with the center at $v_{1}$. We construct a graph $G$ from
$W_{n-4}$ as follows. First, choose five vertices
$v_{2},v_{3},v_{4},v_{5},v_{6}$ of $W_{n-4}$. Second, add three vertices
$x_{1},x_{2},x_{3}$ and edges $x_{1}x_{2}$, $x_{2}x_{3}$. Finally, add edges
$v_{2}x_{1},v_{3}x_{1},v_{4}x_{2},v_{5}x_{3},v_{6}x_{3}$. In this case, for
each $i\in\\{1,2,3\\}$, $x_{i}$ is a 3-vertex, and for each
$j\in\\{2,\ldots,6\\}$ $v_{j}$ is a 4-vertex.
It is easy to see that $G$ is a $Z_{3}$-connected realization of degree
sequence $(n-4,4^{5},3^{n-6})$. Define
$S=V(W_{n-4})\setminus\\{v_{2},\ldots,v_{6}\\}=\\{v_{1},v_{7},\ldots,v_{n-3}\\}$.
Then $|S|=n-4-5=n-9\geq 1$. If $n=10$, then $G$ is a realization of
$(6,4^{5},3^{4})$. If $n\geq 12$, then define $G^{\prime}$ from $G$ by adding
$v_{j}v_{n-j+4}$ for $7\leq j\leq\frac{n}{2}+1$, that is, adding $(n-10)/2$
edges in $S$. Obviously, $G^{\prime}$ has a degree sequence
$(n-4,4^{n-5},3^{4})$. We conclude that $G^{\prime}$ is a $Z_{3}$-connected
realization of $\pi$.
In the case that $n-d_{1}-1=4$, we have $d_{1}=n-5$ and $n\geq 11$ is odd.
Denote by $W_{n-5}$ an even wheel with the center at $v_{1}$. We construct a
graph $G$ from $W_{n-5}$ as follows. First, choose six vertices
$v_{2},v_{3},v_{4},v_{5},v_{6},v_{7}$ of $W_{n-5}$. Second, add four vertices
$x_{1},x_{2},x_{3},x_{4}$ and edges $x_{1}x_{2}$, $x_{2}x_{3}$, $x_{3}x_{4}$.
Third, add edges
$x_{1}v_{2},x_{1}v_{3},x_{2}v_{4},x_{3}v_{5},x_{4}v_{6},x_{4}v_{7}$. In this
case, for each $i\in\\{1,2,3,4\\}$, $x_{i}$ is a 3-vertex, and for each
$j\in\\{2,3,\ldots,7\\}$ $v_{j}$ is a 4-vertex.
It is easy to see that $G$ is a $Z_{3}$-connected realization of degree
sequence $(n-5,4^{6},3^{n-7})$. Define
$S=V(W_{n-5})\setminus\\{v_{1},v_{2},\ldots,v_{7}\\}=\\{v_{8},\ldots,v_{n-4}\\}$.
Then $|S|=n-5-6=n-11\geq 0$. If $n=11$, then $G$ is a realization of
$(6,4^{6},3^{4})$. If $n\geq 13$, then define $G^{\prime}$ from $G$ by adding
edges $v_{j}v_{n-j+4}$ for $8\leq j\leq\frac{n+3}{2}$, that is, adding
$(n-11)/2$ edges in $S$. Obviously, $G^{\prime}$ is a $Z_{3}$-connected
realization of degree sequence $(n-5,4^{n-5},3^{4})$, a contradiction.
In the case that $n-d_{1}-1\geq 5$, denote by $W_{d_{1}}$ an even wheel with
the center at $v_{0}$. Let $V(W_{d_{1}})=\\{v_{0},v_{1},\ldots,v_{d_{1}}\\}$.
Let $C:u_{1}\ldots u_{n-d_{1}-1}u_{1}$ be a cycle of length $n-d_{1}-1$ and
and define a graph $H$ obtained from $C$ adding edges $u_{i}u_{i+2}$ for each
$i\in\\{1,\ldots,n-d_{1}-1\\}$, where the subscripts are taken modular
$n-d_{1}$. Clearly, $H$ is a 4-regular and is triangularly connected. Define
$H^{\prime}=H-\\{u_{2}u_{n-d_{1}-1}\\}$. Now we prove $H^{\prime}$ is
$Z_{3}$-connected. Clearly, $H^{\prime}_{[u_{1}u_{2},u_{1}u_{3}]}$ is
triangularly connected and contains a 2-circuit $u_{2}u_{3}u_{2}$. By Lemma
2.4 (a) and Lemma 2.2 (3), $H^{\prime}_{[u_{1}u_{2},u_{1}u_{3}]}$ is
$Z_{3}$-connected, and hence $H^{\prime}$ by Lemma 2.3. We construct a graph
$G$ from $W_{d_{1}}$ and $H^{\prime}$ as follows. If $d_{1}\geq 8$, then we
add two edges $v_{1}u_{n-d_{1}-1}$, $v_{2}u_{2}$ and add edges
$v_{j}v_{d_{1}-j+3}$ for $3\leq j\leq\frac{d_{1}}{2}-1$, that is, add
$(d_{1}-6)/2$ edges between vertices
$\\{v_{3},\ldots,v_{d_{1}-4}\\}\setminus\\{v_{\frac{d_{1}}{2}},v_{\frac{d_{1}}{2}+1},v_{\frac{d_{1}}{2}+2},v_{\frac{d_{1}}{2}+3}\\}$
such that $d(v_{i})=4$ for each vertex of
$\\{v_{3},\ldots,v_{d_{1}-4}\\}\setminus\\{v_{\frac{d_{1}}{2}},v_{\frac{d_{1}}{2}+1},v_{\frac{d_{1}}{2}+2},v_{\frac{d_{1}}{2}+3}\\}$
and the new graph is simple. If $d_{1}=6$, then we add two edges
$v_{1}u_{n-d_{1}-1}$, $v_{2}u_{2}$. In either case, $G$ is a $Z_{3}$-connected
realization of has a degree sequence $(d_{1},4^{n-5},3^{4})$, a contradiction
Proof of Theorem 1.5. If $\pi=(5^{2},3^{4})$ or $(5,3^{5})$, then by Lemma
2.7, $\pi$ has no $Z_{3}$-connected realization. Thus, assume that
$\pi\not=(5^{2},3^{4}),(5,3^{5})$.
Since $d_{n-5}\geq 4$, $n\geq 6$. If $n=6$, then by Theorem 1.3,
$\pi=(d_{1},d_{2},3^{4})$. By Lemma 5.1, $\pi=(d_{1},3^{5})$. By our
assumption that $d_{n-5}\geq 4$, $\pi=(5,3^{5})$, a contradiction. If $n=7$,
then by Theorem 1.3 and Lemma 5.1, $\pi=(d_{1},d_{2},3^{5})$. By our
assumption that $d_{n-5}\geq 4$, $\pi=(5,4,3^{5})$ or $(6,5,3^{5})$. In the
former, the graph (b) in Fig. 1 is a $Z_{3}$-connected realization of $\pi$.
In the latter case, Theorem 1.4 shows that $\pi$ has a $Z_{3}$-connected
realization. Thus, we may assume that $n\geq 8$.
By Theorems 1.3 and 1.4, and Lemma 5.1, it is sufficient to prove that if
$d_{n-4}=3$, $d_{n-5}\geq 4$ and $d_{1}\leq n-4$, then $\pi$ has a
$Z_{3}$-connected realization. Then $\pi=(d_{1},\ldots,d_{n-5},3^{5})$.
Suppose to the contrary that $\pi$ satisfies
$d_{n-4}=3$, $d_{n-5}\geq 4$ and $d_{1}\leq n-4$. (5)
Subject to (5),
$\pi$ has no $Z_{3}$-connected realization with $n$ minimized. (6)
We claim that $d_{3}=4$. Suppose otherwise that $d_{3}\geq 5$. Define
$\bar{\pi}=(d_{1}-1,d_{2}-1,d_{3}-1,d_{4},\ldots,d_{n-5},3^{4})=(\bar{d}_{1},\ldots,\bar{d}_{n-1})$.
Since $d_{1},d_{2},d_{3}\geq 5$, $\bar{d}_{n-6}\geq 4$. Thus $\bar{\pi}$
satisfies (5). By the minimality of $n$, $\bar{\pi}$ has a $Z_{3}$-connected
realization $\bar{G}$. Denote by $G$ the graph obtained from $\bar{G}$ by
adding a new vertex $v$ and three edges joining $v$ to the corresponding
vertices. It follows by (8) of Lemma 2.2 that $G$ is a $Z_{3}$-connected
realization of $\pi$, a contradiction. Thus, $d_{3}=4$ and
$\pi=(d_{1},d_{2},4^{n-7},3^{5})$.
We claim that $d_{2}=4$. Suppose otherwise that $d_{2}\geq 5$. In this case,
$\bar{\pi}=(d_{1}-1,d_{2}-1,4^{n-8},3^{6})=(\bar{d}_{1},\ldots,\bar{d}_{n-1})$.
Since $d_{2}\geq 5$ and $d_{3}=4$, $n\geq 3+5=8$. This implies that
$\bar{d}_{n-6}\geq 4$. Thus, $\bar{\pi}$ satisfies (5). By the minimality of
$n$, $\bar{\pi}$ has a $Z_{3}$-connected realization $\bar{G}$. Denote by $G$
the graph obtained from $\bar{G}$ by adding a new vertex $v$ and three edges
joining $v$ to the corresponding vertices of $\bar{G}$. It follows from (8) of
Lemma 2.2 that $G$ is a $Z_{3}$-connected realization of $\pi$, a
contradiction. Thus, $d_{2}=4$ and $\pi=(d_{1},4^{n-6},3^{5})$.
Since $\pi$ is graphic, $d_{1}$ is odd and $d_{1}\geq 5$. If $d_{1}=5$, then
by (iii) of Lemma 3.1, $\pi$ has a $Z_{3}$-connected realization.
We are left to the case that $d_{1}\geq 7$. Since $d_{1}\leq n-4$, $n\geq
d_{1}+4\geq 11$. By (iii) of Lemma 3.1, Let $G^{\prime}$ be a
$Z_{3}$-connected realization of degree sequence $(5,4^{n-6},3^{5})$. By the
construction of $G^{\prime}$ in (iii) of Lemma 3.1, $G^{\prime}$ has at least
$|E(G^{\prime})|-5-14=2n-21\geq 2(d_{1}+4)-21=2d_{1}-13\geq(d_{1}-5)/2$ edges
not incident with the any vertex of $N_{G^{\prime}}(u)\cup\\{u\\}$, where $u$
is a 5-vertex in $G^{\prime}$ since $G^{\prime}$ contains a pair of adjacent
neighbors of $u$. Choose $(d_{1}-5)/2$ such edges, say $u_{i}v_{i}$ for each
$i\in\\{1,\ldots,(d_{1}-5)/2)\\}$. Denote by the graph $G$ from $G^{\prime}$
by deleting edges $u_{i}v_{i}$ and adding edge $uu_{i}$, $uv_{i}$ for each
$i\in\\{1,\ldots,(d_{1}-5)/2\\}$. It follows by Lemma 2.3 that $G$ is a
$Z_{3}$-connected realization of degree sequence $(d_{1},4^{n-6},3^{5})$, a
contradiction. We complete our proof.
## References
* [1] D. Archdeacon, Realizing degree sequences with graphs having 3-flows, http://www.cems. uvm.edu/ archdeac/problems/seqflow. html.
* [2] J. A. Bondy, U. S. R. Murty, Graph Theory with Applications, North-Holland, New York, 1976.
* [3] P. J. Cameron, Problems from the 16th British Combinatorial Conference, Discrete Math., 197/198 (1999) 799-812.
* [4] J. Chen, E. Eschen, H. -J. Lai, Group connectivity of certain graphs, Ars Combin., 89 (2008) 141–158.
* [5] M. DeVos, R. Xu, G. Yu, Nowhere-zero $Z_{3}$-flows through $Z_{3}$-connectivity, Discrete Math., 306 (2006) 26–30.
* [6] G. Fan, H. -J. Lai, R. Xu, C. -Q. Zhang, C. Zhou, Nowhere-zero 3-flows in triangularly connected graphs, J. Combin. Theory, Ser. B 98 (2008) 1325–1336.
* [7] S. Fan, H. -J. Lai, Y. Shao, T. Zhang, J. Zhou, Degree sequence and supereulerian graphs, Discrete Math., 308 (2008) 6626–6631.
* [8] S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. I, SIAM J. Appl. Math., 10 (1962) 496–506.
* [9] S. L. Hakimi, On realizability of a set of integers as degrees of the vertices of a linear graph. H. Uniqueness, SIAM J. Appl. Math., 11 (1963) 135–147.
* [10] F. Jaeger, N. Linial, C. Payan, N. Tarsi, Group connectivity of graphs–a nonhomogeneous analogue of nowhere zero flow properties, J. Combin. Theory, Ser. B 56 (1992) 165–182.
* [11] D. J. Kleitman and D. L. Wang, Algorithm for constructing graphs and digraphs with given valences and factors, Discrete Math., 6 (1973) 79–88.
* [12] H. -J. Lai, Group connectivity of 3-edge-connected chordal graphs, Graphs Combin., 16 (2000) 165–176.
* [13] H.-J. Lai, X. Li, Y. Shao, M. Zhan, Group connectivity and group colorings of graphs-a survey, Acta Math. Sinica, 27 (2011) 405–434.
* [14] R. Luo, R. Xu, J. Yin, G. Yu, Ore-condition and $Z_{3}$-connectivity, European J. of Combin., 29 (2008) 1587–1595.
* [15] R. Luo, W. Zang, and C. -Q Zhang, Nowhere-zero 4-flows, simultaneous edge-colorings, and critical partial latin squares, Combinatorica, 24 (2004) 641–657.
* [16] R. Luo, R. Xu, W. Zang, and C. -Q Zhang, Realizing degree sequences with graphs having nowhere-zero 3-flows, SIAM J. Discrete Math., 22 (2008) 500–519.
* [17] R. Luo, R. Xu, X. Yu, An extremal problem on group connectivity of graphs, European J. Combin., 33 (2012) 1078–1085.
* [18] R. Steinberg and D. H. Younger, Grötzch’s theorem for the projectice plane, Ars Combin. 28 (1989) 15–31.
* [19] W. T. Tutte, A contribution on the theory of chromatic polynomial, Canad. J. Math., 6 (1954), 80–91.
* [20] J. Yin, G. Guo, The smallest degree sum that yields graphic sequences with a $Z_{3}$-connected realization, European J. Combin., 34 (2013) 806–811.
|
# On Vacuum Fluctuations in Quantum Gravity and Interferometer Arm
Fluctuations
Kathryn M. Zurek Walter Burke Institute for Theoretical Physics
California Institute of Technology, Pasadena, CA USA
###### Abstract
We propose a simple model of spacetime vacuum fluctuations motivated by
AdS/CFT, where the vacuum is described by a thermal density matrix,
$\rho=\frac{e^{-K}}{\mbox{Tr}(e^{-K})}$ with $K$ the modular Hamiltonian. In
AdS/CFT, both the expectation value of $K$ and its fluctuations $\langle\Delta
K^{2}\rangle$ have been calculated; both obey an area law identical to the
Bekenstein-Hawking area law of black hole mechanics: $\langle
K\rangle=\langle\Delta K^{2}\rangle=\frac{A}{4G_{N}}$, where $A$ is the area
of an (extremal) entangling surface. It has also been shown that $\Delta K$
gravitates in AdS, and hence generates metric fluctuations. These theoretical
results are intriguing, but it is not known how to precisely extend such ideas
about holographic quantum gravity to ordinary flat space. We take the approach
of considering whether experimental signatures in metric fluctuations could
determine properties of the vacuum of quantum gravity in flat space. In
particular, we propose a theoretical model motived by the AdS/CFT calculations
that reproduces the most important features of modular Hamiltonian
fluctuations; the model consists of a high occupation number bosonic degree of
freedom. We show that if this theory couples through ordinary gravitational
couplings to the mirrors in an interferometer with strain sensitivity similar
to what will be available for gravitational waves, vacuum fluctuations could
be observable.
Overview. One of the most important open problems in fundamental physics is
bridging the theoretical and observational chasm between quantum mechanics and
gravity, to understand quantum gravity. The divide can be understood at a
basic level by considering the length and time scales of gravity, which are
derived from the Newton constant, $l_{p}=\sqrt{8\pi G\hbar/c^{3}}\simeq
10^{-34}\mbox{ m},~{}t_{p}=l_{p}/c\simeq 10^{-43}\mbox{ s}$.111For the
remainer of this letter we will take $\hbar=c=1$. From the point of view of
quantum mechanics and Effective Field Theory (EFT), these numbers represent
the time and length scales of quantum fluctuations. They are far out of reach
of observational capabilities, and neatly encapsulate why the quantum effects
of gravity have never been probed.
On the other hand, theoretical progress, mostly in the context of AdS/CFT and
the black hole information paradox, suggests that non-locality and
entanglement plays an important role in the quantum theory of gravity (see
Ref. Van Raamsdonk (2016) for a pedagogical review and references therein).
The simplest way to see why quantum gravity may be highly non-local, and hence
why the naïve expectations of EFT may break down, is through holography.
Quantum Field Theory (QFT) dictates that the number of degrees of freedom in a
spacetime volume scales with the volume (in units of $l_{p}$). Holography, on
the other hand, says that the degrees-of-freedom of a spacetime volume scales
with the area of the surface bounding that volume. As a result, this implies
that QFT grossly over-counts the number of degrees-of-freedom when gravity is
involved, suggesting that in any EFT description of spacetime, there should be
long range correlations between the degrees-of-freedom. It is such long range
correlations that we seek to model in this letter, where we call each
spacetime degree-of-freedom a pixel.
Our ability to quantify spacetime fluctuations from pixels relies on
theoretical progress made recently in connecting the holographic notion of
spacetime to entanglement. Not surprisingly, the connection between geometry,
quantum information (QI) and holography has been formulated most clearly and
rigorously in the context of AdS/CFT Ryu and Takayanagi (2006a, b), though
there are reasons to think that the connection appears in generic spacetimes
Srednicki (1993); Callan and Wilczek (1994); Jacobson (1995, 2016). In the
context of AdS/CFT, the geometry-QI-holography connection is encapsulated most
prominently in the relation
$\displaystyle\beta\langle K\rangle=S_{\rm ent}=\frac{A(\Sigma)}{4G}.$ (1)
Let us discuss each part of this equality. $K$ is a measure of the total
energy in a spacetime volume called the modular Hamiltonian of the boundary
CFT ($K\equiv\int T_{\mu\nu}\xi^{\mu}dV^{\nu}$ with $T_{\mu\nu}$ the CFT
stress tensor, $\xi^{\mu}$ the conformal Killing vector on the boundary and
$dV^{\nu}$ an infinitesimal volume element); since energy directly sources the
metric through the Einstein equation, it represents geometry. The entanglement
entropy $S_{\rm ent}$ quantifies mixing of the vacuum state $\rho_{\beta}$
across an entangling boundary and represents the QI part of the
correspondence: $S_{\rm ent}=-\mbox{Tr}\rho_{\beta}\log\rho_{\beta}$, where
the entangling surface $\Sigma$ has area $A(\Sigma)$ and an associated inverse
temperature $\beta=2\pi R$, where $R$ is the AdS curvature.222We have
reintroduced the temperature; it can be reabsorbed in the definition of $K$ as
in the abstract, but we will find it convenient to leave it explicit. It was
remarkably shown Ryu and Takayanagi (2006b) that there is a special surface in
AdS, called the RT surface, where the entanglement entropy of the vacuum state
has precisely the form of the Bekenstein-Hawking area law for the entropy of
black hole horizons; this represents the holographic part of the connection.
Further, by calculating fluctuations of the modular Hamiltonian (and of the
entanglement entropy), one can compute fluctuations in the bulk (AdS)
geometry. The fluctuations of the modular Hamiltonian have been calculated in
the AdS bulk Verlinde and Zurek (2020), with the result
$\displaystyle\beta^{2}\langle\Delta K^{2}\rangle=\frac{A(\Sigma)}{4G}.$ (2)
With suitable identifications, this result agrees with an equivalent boundary
calculation Perlmutter (2014), and with the “capacity of entanglement”
Nakaguchi and Nishioka (2016); De Boer _et al._ (2019). This implies that,
when restricted to a finite part of the spacetime (defined by a causal diamond
with bifurcate horizon $\Sigma$), the vacuum has energy fluctuations $\Delta
K\neq 0$.
Eqs. (1), (2) together encourage an interpretation of spacetime as bits of
information, with the number ${\cal N}$ of degrees-of-freedom in a given
volume bounded by a surface of area $A$ given by the entanglement entropy
$\displaystyle S_{\rm ent}={\cal N}=\frac{A}{4G}.$ (3)
This result can be interpreted in light of the fact that the vacuum state of
any QFT, restricted to a causal diamond, is given by a thermal density matrix
Casini _et al._ (2011)
$\displaystyle\rho_{\beta}=\frac{e^{-\beta K}}{Z_{\beta}}.$ (4)
For example, in a high-temperature system with Maxwell-Boltzmann statistics,
the root-mean-square fluctuations of these ${\cal N}$ degrees-of-freedom is
given by
$\displaystyle\frac{\sqrt{\langle\Delta K^{2}\rangle}}{\langle
K\rangle}=\frac{1}{\sqrt{\cal N}},$ (5)
in agreement with Eqs. (1), (2). It was shown in Ref. Verlinde and Zurek
(2020) that the fluctuations in the modular Hamiltonian, $\Delta K$, gravitate
and hence, for certain observers, behave like a mass, sourcing metric
fluctuations.
It is not precisely known whether the vacuum state of quantum gravity in
ordinary flat space, when restricted to a causal diamond, can be described by
Eq. (4), and whether the results derived for AdS/CFT apply to the Universe we
observe. There are reasons to think that entropy, entanglement and their
connection to geometry are very generic concepts that apply to any spacetime,
including ours. Such ideas underly currents dating more than twenty years on
the entropic and holographic nature of spacetime Jacobson (1995); Verlinde
(2011); Banks (2011). For example, it was shown in Ref. Jacobson (2016) that
taking fixed volume variations of the first law of entanglement gives rise to
the Einstein Equations.
In this letter we approach quantum gravity in flat space in terms of
observational signatures derived from a model motivated by the known AdS/CFT
results. If vacuum fluctuations could be observed in an experiment consistent
with an entropic or thermal nature of the vacuum state, this would be a leap
forward in our understanding of quantum gravity. To this end, we propose a
simple model for the degrees-of-freedom in the density matrix, which we call
pixellons because the excitations are associated with each holographic pixel
of a volume bounded by an entangling surface of area $A$. More specifically,
we consider whether fluctuations in the spacetime degrees-of-freedom of the
density matrix Eq. (4) could be observable as fluctuations in the arm length
of an interferometer. The two arms of an interferometer mark out a
(spherically symmetric) volume of spacetime with the beamsplitter at the
center of the volume and the mirrors on the surface with area $A$; the
interferometer measures the geometric fluctuations in this volume.
The motivation for our pixellon Ansatz is as follows. Based on the discussion
of Eqs. (1)-(3), we interpret the spacetime volume bounded by a surface of
area $A$ as having ${\cal N}$ bits with total energy $\langle K\rangle$. The
energy per bit is then $\omega\sim\beta^{-1}\sim 1/L$. However, it was shown
in Ref. Verlinde and Zurek (2020) that $\langle K\rangle$ itself does not
gravitate; rather the fluctuations $\Delta K$ gravitate. From this point of
view, $\langle K\rangle$ should be treated as a chemical potential counting
the background degrees-of-freedom, and the energy per excitation is
$\beta\omega=\Delta K/K=1/\sqrt{\cal N}\ll 1$. We assume the low-energy
excitations will be bosonic; this is appropriate because, as we will see
below, they are associated with a gravitational potential. Because the energy
of these bosonic degrees-of-freedom is so low, $\beta\omega\ll 1$, they will
form a high-occupation-number bosonic state. We further show that such low-
energy bosonic excitations of the vacuum, when gravitationally coupled to test
masses, may give rise to observably large fluctuations of the mirror positions
in an interferometer.
The detailed outline of our proposal is as follows. In the next section we
introduce the bosonic excitations as vacuum fluctuations, and we suggest that
these bosonic fluctuations be associated with a scalar gravitational
potential. In the following section, we propose that these scalar degrees-of-
freedom have a Bose-Einstein density-of-states and a high occupation number
due to their low energy. Then, we gravitationally couple the pixellons to a
test mass whose position fluctuates due to the boson fluctuations. We utilize
the Feynman-Vernon influence functional–a path integral realization of the
fluctuation-dissipation theorem–to compute the size of those fluctuations. We
will conclude that vacuum fluctuations from a thermal density matrix can give
rise to interferometer mirror position fluctuations that appear as noise with
a peculiar angular correlation, a smoking gun signature. Note that while the
experimental system of interest for measuring metric fluctuations, an
interferometer, and the motivation derived from the discussion surrounding
Eqs. (1)-(5), is the same as Ref. Verlinde and Zurek (2020, 2019), the models
operationally share no overlap. We will nevertheless find a very similar
effect, perhaps suggesting dual languages to describe the vacuum state of
quantum gravity.
Pixellons and Vacuum Fluctuations. As outlined in the introduction,
excitations of the degrees-of-freedom, associated with the entanglement
entropy of a finite volume of space, we refer to as pixellons. We expect that
pixellons are complicated non-linear states of all the degrees-of-freedom
available in the complete theory. Our goal is to describe a consistent low-
energy theory based on the quantum information theoretic picture presented in
the introduction.
Our starting point is energy fluctuations given by Eq. (5), which we associate
with pixellons. The energy fluctuations in a volume of radius $L$ source a
potential
$\displaystyle 2\phi=\sqrt{\frac{\alpha}{\cal N}}=\frac{\bar{l}_{p}}{4\pi L},$
(6)
where $l_{p}^{2}\equiv 8\pi G_{N}$ and
$\bar{l}_{p}^{2}\equiv\frac{\alpha}{2}l_{p}^{2}$. Here we have allowed for an
${\cal O}(1)$ number $\alpha$ in the relation $\beta^{2}\langle\Delta
K^{2}\rangle=\alpha\beta\langle K\rangle$ ($\alpha=1$ in AdS/CFT) as a
generalization of Eq. (5). This gravitational potential should be interpreted
as the root-mean-square (RMS) fluctuations in the potential about the vacuum
state, for which $\langle\phi\rangle=0$. Note that this potential is in
agreement with the results of Ref. Verlinde and Zurek (2019), where $\alpha=2$
(in four dimensions) was derived by computing mass fluctuations
thermodynamically and then extracting the gravitational potential from a
topological black hole foliation of 4-d Minkowski space. Next we consider how
to represent this gravitational potential as a bosonic degree-of-freedom.
Thermal Representation of Vacuum. The effects of interest occur because of the
finite volume, and hence finite ${\cal N}$, under observation; in the limit
${\cal N}\rightarrow\infty$, fluctuations vanish. We divide our space into two
regions, $A$ (which is interrogated by an experiment), and $\tilde{A}$
(outside of the experimental apparatus). We assume that the vacuum state of
the quantized Einstein-Hilbert action, restricted to the subspace $A$, has a
density matrix given by Eq. (4), such that $\rho_{A}=\rho_{\beta}$. This
assumption is supported by Jacobson’s derivation Jacobson (2016) of the
Einstein Equation from the entanglement entropy of a diamond utilizing the
density matrix in Eq. (4).
Furthermore, we are seeking to describe the excitations associated with the
fluctuating scalar potential in Eq. (6). Accordingly, we assume that there is
an algebra of $N=\sqrt{\cal N}$ bosonic degrees-of-freedom to describe the
pixellon excitations, with thermal density matrix given by
$\displaystyle\mbox{Tr}(\rho_{\beta}a_{i}^{\dagger}a_{i})=\frac{1}{e^{\beta(\epsilon_{i}-\mu)}-1},$
(7)
where the bosonic creation and annihilation operators satisfy the usual
commutation relations
$\displaystyle[a_{i},a_{i^{\prime}}^{\dagger}]=\delta_{i,i^{\prime}},$ (8)
with $i,~{}i^{\prime}$ running from 1 to $N$. The Hilbert space of these
operators has been cut-off by the finite size of the causal diamond. If
$\epsilon_{i}-\mu$ represents the energy per degree-of-freedom of excitations
of the vacuum state, we have
$\displaystyle\beta(\epsilon_{i}-\mu)=\beta\frac{K-\langle K\rangle}{{\cal
N}}.$ (9)
Since $\beta^{2}\langle\Delta K^{2}\rangle=\alpha\cal N$, and $\beta\sim L$,
we immediately learn that the amplitude of a mass fluctuation is Planckian,
$\sqrt{\langle\Delta K^{2}\rangle}\sim 1/l_{p}$. This is a large mass
fluctuation, but it is divided over ${\cal N}$ pixels such that the typical
energy per degree-of-freedom, relative to the momentum $p\sim 1/\beta$, is
$\displaystyle\beta(\epsilon_{\rm pix}-\mu)\sim\frac{1}{\sqrt{\cal N}}\ll 1.$
(10)
To fully describe the distribution, we need the momentum dependence of
$\beta(\epsilon_{\rm pix}-\mu)$. To obtain this, we employ an Ansatz motivated
by the observation that we produce the scaling in Eq. (10) if we postulate
$\displaystyle\epsilon_{\rm pix}-\mu\equiv\frac{p^{2}}{2m_{\rm pix}},$ (11)
with $m_{\rm pix}\sim 1/l_{p}$. We thus have a pixellon density of states
given by
$\displaystyle\rho_{\rm pix}(p)=\frac{1}{e^{\beta\frac{p^{2}}{2m_{\rm
pix}}}-1}.$ (12)
Because of the extremely low energy of the bosonic degrees-of-freedom, the
occupation number of the excitations is high:
$\displaystyle\rho_{\rm pix}\approx\frac{2m_{\rm pix}}{\beta
p^{2}}\sim\sqrt{\cal N}.$ (13)
This high occupation number will amplify the effective gravitational coupling
of a massive object to the pixellons. Note that while Eq. (12) is the density
of states for the collective pixellon modes, one should not identify the
dispersion of an individual pixellon as $\omega_{p}=p^{2}/2m_{\rm pix}$; the
individual pixellon dispersion will be read off from the action Eq. (18)
below. We also note in passing that, while the context is different, our
hypothesis shares features in common with Ref. Dvali and Gomez (2013) for
black holes, a connection that might be interesting to explore in the future.
Length from Vacuum Fluctuations. For our purposes, the most important question
is whether such excitations are observable. The Ansatz of a pixellon
excitations, $\phi$, with density of states given by Eq. (12), coupled
gravitationally to the mirror system of an interferometer, is ideally suited
to the Feynman-Vernon (FV) influence functional Feynman and Vernon (1963).
Refs. Parikh _et al._ (2020a, b) have utilized the FV set-up to compute the
impact of graviton fluctuations coupled to an interferometer mirror; they find
that the mirror position fluctuations are Planckian (i.e. unobservably small)
for the vacuum state. In this section we utilize similar methods, but focus on
the pixellon vacuum fluctuations, and find the effects are large, amplified by
the high occupation number of the pixellon.
We will consider a single arm of a simple Michelson interferometer as the test
system, with the far mirror of mass $m_{M}$ at coordinate ${\bf M}$, with
equilibrium position a distance $L$ from a beam splitter at position ${\bf
B}$. Our starting point is the mode decomposition $h_{ij}(t,{\bf
x})=\int\frac{d^{3}k}{(2\pi)^{3}}\phi(t,{\bf k})e^{i{\bf k}\cdot{\bf
x}}\epsilon_{ij}^{s},$ where $\epsilon_{ij}^{s}$ is the graviton polarization
vector. We choose TT gauge where
$\epsilon_{ij}^{s}\epsilon^{ij}_{t}=2\delta^{s}_{t}$, and the metric
fluctuations are transverse to the interferometer arm. $\phi(t,{\bf k})$ will
be identified with the pixellon momentum modes. The action for the mirror and
metric is
$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\left[\frac{\partial_{\mu}h_{ij}\partial^{\mu}h^{ij}}{64\pi G}\right.$
(14) $\displaystyle+\left.\delta^{3}({\bf x}-{\bf
M})\frac{m_{M}(\delta_{ij}+h_{ij}(t,{\bf
x}))}{2}\dot{M}^{i}\dot{M}^{j}\right].$
The first term is identified as the usual Einstein-Hilbert action, and the
second as the mirror kinetic term. We have the freedom to fix the origin of
the coordinate system at the beam splitter, with displacements relative to the
beam splitter given by $M^{i}-B^{i}=X^{i}-\frac{h^{i}_{k}X^{k}}{2}$. We also
take the interferometer arm of interest aligned along the $x$ axis such that
$X^{x}\equiv X$. The action Eq. (14) then becomes
$\displaystyle S=$ $\displaystyle\int
dt\int\frac{d^{3}k}{(2\pi)^{3}}\frac{m_{\phi}}{2}\left(\dot{\tilde{\phi}}(t,{\bf
k})^{2}-k^{2}\tilde{\phi}(t,{\bf k})^{2}\right)$ $\displaystyle+\int
dt\frac{m_{M}}{2}\left(\dot{X}^{2}+g\int\frac{d^{3}k}{(2\pi)^{3}}\tilde{\phi}(t,{\bf
k})\frac{d}{dt}(\dot{X}X)\right),$
where $m_{\phi}=1/16\pi G^{2}$, $\tilde{\phi}=\sqrt{G}\phi$, and
$g=1/\sqrt{G}$. In this form, the action has been explicitly written as a non-
relativistic quantum mechanical system for which the FV action is ideally
suited.
Our goal is to compute the fluctuations in the mirror position $X$ driven by
the gravitational $\phi-X$ coupling, with the FV influence functional. The FV
influence functional is a path integral formulation for the transition
probability $P_{nm}$ of some test system $Q$ from a state $\psi_{n}$ at time
$t=\tau$ to $\psi_{m}$ at time $t=T$, when coupled to an interaction system
characterized by an influence functional ${\cal F}(Q,Q^{\prime})$:
$\displaystyle P_{nm}$ $\displaystyle=$
$\displaystyle\int\psi_{m}^{*}(Q_{T})\psi_{m}(Q^{\prime}_{T})e^{i(S_{0}(Q)-S_{0}(Q^{\prime}))}{\cal
F}(Q,Q^{\prime})$ (16) $\displaystyle\times$
$\displaystyle\psi_{n}^{*}(Q_{\tau})\psi_{n}(Q^{\prime}_{\tau}){\cal D}Q{\cal
D}Q^{\prime}dQ_{T}dQ^{\prime}_{T}dQ_{\tau}dQ^{\prime}_{\tau}.$
Here $S_{0}(Q)$ is the action in the absence of any coupling to an external
system, and the influences of the external system are contained in ${\cal
F}(Q,Q^{\prime})$, which is characterized by the influence phase
$\Phi(Q,Q^{\prime})$:
$\displaystyle{\cal F}(Q,Q^{\prime})=e^{i\Phi(Q,Q^{\prime})}.$ (17)
The influence phase is most simply derived for an oscillator $\tilde{\phi}$
coupled to the test system $Q$ through a linear interaction of the type (see
Eq. (4.6) of Ref. Feynman and Vernon (1963)),
$\displaystyle
S_{k}=\int_{\tau}^{T}dt\left[\frac{m_{\phi}}{2}(\dot{\tilde{\phi}}^{2}-k^{2}\tilde{\phi}^{2})+\frac{gm_{M}}{2}\tilde{\phi}Q\right]+S_{0}^{Q},$
(18)
where $S_{0}^{Q}$ is the uncoupled action of the test system, and we have left
a subscript $k$ explicit to denote that we have chosen a single $k$ mode. For
this action FV showed the influence functional is
$\displaystyle i\Phi_{k}(Q,Q^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{g^{2}m_{M}^{2}}{8m_{\phi}k}\int_{\tau}^{T}dt\int_{\tau}^{t}ds(Q_{t}-Q^{\prime}_{t})$
(19)
$\displaystyle\times\left(Q_{s}e^{-ik(t-s)}-Q_{s}^{\prime}e^{ik(t-s)}\right).$
For a distribution of oscillators $\rho(k)$ of momentum $k$ acting on the test
system, we have
$\displaystyle\Phi(Q,Q^{\prime})=\int\frac{d^{3}k}{(2\pi)^{3}}\rho(k)\Phi_{k}(Q,Q^{\prime}).$
(20)
Now we decompose the influence functional into real and imaginary parts,
according to
$\displaystyle f(\Delta t)$ $\displaystyle\equiv$
$\displaystyle\frac{g^{2}m_{M}^{2}}{4m_{\phi}}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\rho(k)}{k}e^{ik\Delta
t}$ (21) $\displaystyle\equiv$ $\displaystyle A_{FV}(\Delta t)+iB_{FV}(\Delta
t).$
The real and imaginary parts of the influence phase are fixed relative to each
other by the oscillator density of states $\rho(k)$, which is the path
integral realization of the Fluctuation-Dissipation Theorem. The influence
phase can be written in terms of this function as
$\displaystyle i\Phi(Q,Q^{\prime})$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\int_{\tau}^{T}dt\int_{\tau}^{t}ds(Q_{t}-Q^{\prime}_{t})$
(22) $\displaystyle\times\left(Q_{s}f^{*}(t-s)-Q_{s}^{\prime}f(t-s)\right).$
One can see that $A_{FV}(t-s)$ represents the variance in $Q(t)$, i.e. the
amplitude of the noise in the variable $Q$. This is most clearly seen by
rewriting
$\displaystyle\mbox{exp}\left[\mbox{Re}\left[i\Phi(Q,Q^{\prime})\right]\right]$
$\displaystyle=$
$\displaystyle\mbox{exp}\left[-\frac{1}{2}\int_{0}^{T}dt\int_{0}^{t}ds~{}A_{FV}(t-s)(Q_{t}-Q^{\prime}_{t})(Q_{s}-Q^{\prime}_{s})\right]$
$\displaystyle=$ $\displaystyle\int{\cal
D}F_{Q}\mbox{exp}\left[-\frac{1}{2}\int_{0}^{T}dt\int_{0}^{t}ds~{}A_{FV}^{-1}(t-s)F_{Q}(t)F_{Q}(s)\right.$
$\displaystyle\left.+i\int_{0}^{T}dtF_{Q}(t)(Q_{t}-Q^{\prime}_{t})\right],$
from which one reads
$\displaystyle\langle F_{Q}^{2}\rangle=A_{FV}(\Delta t).$ (24)
The only step that remains is to relate this noise to the observable,
fluctuations in the position of the mirror, $X(t)$. The influence phase has
allowed us to effectively integrate out the pixellon oscillators $\phi$ and
write their effect only in terms of the degree-of-freedom $F_{Q}$ which
describes the distribution of the mirror displacements according to
$Q\equiv\frac{d}{dt}(\dot{X}X)$. The noise $F_{Q}$ accordingly becomes a term
in the effective action $S_{\rm eff}$ replacing the pixellon:
$\displaystyle S_{\rm eff}=\int
dt\left[\frac{m_{M}}{2}\dot{X}^{2}+F_{Q}(t)\frac{d}{dt}(\dot{X}X)\right].$
(25)
The equation of motion is thus
$\displaystyle m_{M}\ddot{X}(t)=\ddot{F}_{Q}(t)X(t),$ (26)
where we have integrated the right-hand-side by parts as necessary to move all
time derivatives onto $F_{Q}$. This result agrees with Ref. Parikh _et al._
(2020b) in the relevant limit (up to conventions), though we have arrived at
it more compactly by utilizing the FV results directly. Identifying the time-
dependent change in $X(t)$ with the mirror displacement $\delta L(t)$ observed
in an interferometer, we have
$\displaystyle\frac{\delta L(t)}{L}=\frac{F_{Q}(t)}{m_{M}},$ (27)
or equivalently
$\displaystyle\langle\frac{\delta L^{2}(\Delta t)}{L^{2}}\rangle$
$\displaystyle=$ $\displaystyle\frac{A_{FV}(\Delta t)}{m_{M}^{2}}$
$\displaystyle=$ $\displaystyle 4\pi
G\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\rho(k)}{k}\cos(k\Delta t)dk.$
Observational Consequences of the Pixellon. We now identify the pixellon with
the degrees-of-freedom having density of states $\rho(k)$ in Eq. (20). Taking
$p=k$ in Eq. (13), we have $\rho_{\rm pix}(k)\approx\frac{2m_{\rm pix}}{\beta
k^{2}}$ such that
$\displaystyle\langle\frac{\delta L^{2}(\Delta
t)}{L^{2}}\rangle\approx\frac{4\pi Gm_{\rm pix}}{\pi^{2}\beta}\int_{k_{\rm
min}}^{k_{\rm max}}\frac{dk}{k}\cos(k\Delta t),$ (29)
where ${k_{\rm min}},~{}{k_{\rm max}}$ are the maximum and minimum frequencies
detectable by the apparatus measuring the metric fluctuations. We previously
argued that $m_{\rm pix}$ is near the Planck scale; ultimately it should be
experimentally determined. Taking, for example, $2m_{\rm pix}/\beta=a/l_{p}L$,
and in the limit $\Delta t\rightarrow 0$, we recover
$\displaystyle\langle\frac{\delta
L^{2}}{L^{2}}\rangle\approx\frac{al_{p}}{4\pi^{2}L}\log\left(\frac{k_{\rm
max}}{k_{\rm min}}\right).$ (30)
Now $k$ is the momentum conjugate to the spacetime position $x$. Thus we can
identify $~{}k_{\rm min}$ with the overall apparatus size $k_{\rm min}\sim
1/L$, and the spatial separation between the two arms of the interferometer
$k_{\rm max}\sim 1/|{\bf r}_{1}-{\bf r}_{2}|$. Note that this is remarkably
similar to the result obtained in Ref. Verlinde and Zurek (2019) despite the
distinct Ansatz and calculational techniques.
The amplitude of the effect is within reach. For example, the interferometer
of Ref. Chou _et al._ (2017) was able to constrain a power spectral density
$S_{h}(\omega)\equiv\int_{-\infty}^{\infty}d\Delta t\langle\frac{\delta
L^{2}(\Delta t)}{L^{2}}\rangle e^{-i\omega\Delta t}\lesssim
0.15~{}l_{p}/\sqrt{8\pi}(\operatorname{sinc}^{2}(\omega
L/2)-2\operatorname{sinc}^{2}(\omega L))$. While a precise phenomenological
analysis comparing our model to the data is not the purpose of this present
work, we can obtain an estimate by integrating Eq. (29) times
$e^{-i\omega\Delta t}$ over $-L$ to $L$, taking $\omega\rightarrow 0$. With
this naive estimate, we find that the data is roughly consistent with
$a\lesssim 2$. This motivates future work to make precise phenomenological
predictions and to search for the unique signatures of this model. For
example, Ref. Verlinde and Zurek (2019) went one step further and used the
spherical symmetry of the entangling surface to argue that length fluctuations
correlated between two interferometer arms should obey the Green function of
the 2-d Laplacian on the sphere. While we have not repeated this calculation
here, the general form of Eq. (30) suggests similar behavior holds for this
model. The observation of these angular correlations would be a spectacular
confirmation of the idea of low momentum vacuum fluctuations in quantum
gravity, and could be tested with a suitable modification of a planned table
top interferometer experiment Vermeulen _et al._ (2020). We anticipate a
vibrant experimental program along these directions.
Acknowledgments. I especially thank Tom Banks, Cindy Keeler, Michele Papucci
and Julio Parra-Martinez for detailed comments on the draft. I thank many
others for discussion and criticism, including Dan Carney, Yanbei Chen,
Clifford Cheung, Patrick Draper, David B. Kaplan, Dongjun Li, Juan Maldacena,
Maulik Parikh, and Erik Verlinde.
## References
* Van Raamsdonk (2016) M. Van Raamsdonk, in _Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings_ (2016) arXiv:1609.00026 [hep-th] .
* Ryu and Takayanagi (2006a) S. Ryu and T. Takayanagi, JHEP 08, 045 (2006a), arXiv:hep-th/0605073 .
* Ryu and Takayanagi (2006b) S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006b), arXiv:hep-th/0603001 .
* Srednicki (1993) M. Srednicki, Phys. Rev. Lett. 71, 666 (1993), arXiv:hep-th/9303048 .
* Callan and Wilczek (1994) C. G. Callan, Jr. and F. Wilczek, Phys. Lett. B 333, 55 (1994), arXiv:hep-th/9401072 .
* Jacobson (1995) T. Jacobson, Phys. Rev. Lett. 75, 1260 (1995), arXiv:gr-qc/9504004 .
* Jacobson (2016) T. Jacobson, Phys. Rev. Lett. 116, 201101 (2016), arXiv:1505.04753 [gr-qc] .
* Verlinde and Zurek (2020) E. Verlinde and K. M. Zurek, JHEP 04, 209 (2020), arXiv:1911.02018 [hep-th] .
* Perlmutter (2014) E. Perlmutter, JHEP 03, 117 (2014), arXiv:1308.1083 [hep-th] .
* Nakaguchi and Nishioka (2016) Y. Nakaguchi and T. Nishioka, JHEP 12, 129 (2016), arXiv:1606.08443 [hep-th] .
* De Boer _et al._ (2019) J. De Boer, J. Järvelä, and E. Keski-Vakkuri, Phys. Rev. D 99, 066012 (2019), arXiv:1807.07357 [hep-th] .
* Casini _et al._ (2011) H. Casini, M. Huerta, and R. C. Myers, JHEP 05, 036 (2011), arXiv:1102.0440 [hep-th] .
* Verlinde (2011) E. P. Verlinde, JHEP 04, 029 (2011), arXiv:1001.0785 [hep-th] .
* Banks (2011) T. Banks, (2011), arXiv:1109.2435 [hep-th] .
* Verlinde and Zurek (2019) E. P. Verlinde and K. M. Zurek, (2019), arXiv:1902.08207 [gr-qc] .
* Dvali and Gomez (2013) G. Dvali and C. Gomez, Fortsch. Phys. 61, 742 (2013), arXiv:1112.3359 [hep-th] .
* Feynman and Vernon (1963) R. Feynman and J. Vernon, F.L., Annals Phys. 24, 118 (1963).
* Parikh _et al._ (2020a) M. Parikh, F. Wilczek, and G. Zahariade, (2020a), arXiv:2005.07211 [hep-th] .
* Parikh _et al._ (2020b) M. Parikh, F. Wilczek, and G. Zahariade, (2020b), arXiv:2010.08208 [hep-th] .
* Chou _et al._ (2017) A. Chou _et al._ (Holometer), Class. Quant. Grav. 34, 165005 (2017), arXiv:1703.08503 [gr-qc] .
* Vermeulen _et al._ (2020) S. Vermeulen, L. Aiello, A. Ejlli, W. Griffiths, A. James, K. Dooley, and H. Grote, (2020), arXiv:2008.04957 [gr-qc] .
|
# The Prime Geodesic Theorem in Arithmetic Progressions
Dimitrios Chatzakos Department of Mathematics, University of Patras, 26 110,
Patras, Greece https://dchatzakos.math.upatras.gr<EMAIL_ADDRESS>, Gergely Harcos Alfréd Rényi Institute of Mathematics, POB 127, Budapest
H-1364, Hungary https://users.renyi.hu/ gharcos/<EMAIL_ADDRESS>and Ikuya
Kaneko The Division of Physics, Mathematics and Astronomy, California
Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA
https://sites.google.com/view/ikuyakaneko/<EMAIL_ADDRESS>
###### Abstract.
We address the prime geodesic theorem in arithmetic progressions, and resolve
conjectures of Golovchanskiĭ–Smotrov (1999). In particular, we prove that the
traces of closed geodesics on the modular surface do not equidistribute in the
reduced residue classes of a given modulus.
###### Key words and phrases:
prime geodesic theorem, arithmetic progressions, Kuznetsov–Bykovskiĭ formula
###### 2020 Mathematics Subject Classification:
11F72 (primary); 11M36 (secondary)
The first author acknowledges the financial support from the ELKE of the
University of Patras (MEDIKOS Program no. 82043). The second author was
supported by the Rényi Intézet Lendület Automorphic Research Group and NKFIH
(National Research, Development and Innovation Office) grant K 143876. The
third author acknowledges the support of the Masason Foundation.
## 1\. Introduction
### 1.1. Historical prelude
The prime geodesic theorem asks for an asymptotic evaluation of the counting
function of oriented primitive closed geodesics on hyperbolic manifolds. If
the underlying group is a cofinite Fuchsian group
$\Gamma\subset\mathrm{PSL}_{2}(\mathbb{R})$, then this problem has received
distinguished attention among number theorists; see for instance [Hej76a,
Hej76b, Hej83, Hub61a, Hub61b, Kuz78, Sar80, Sel89, Ven90] and references
therein for classical triumphs.
A closed geodesic $P$ on $\Gamma\backslash\mathbb{H}$ corresponds bijectively
to a hyperbolic conjugacy class in $\Gamma$ (cf. [Hub59]). If $\mathrm{N}(P)$
denotes the norm of this conjugacy class, then the hyperbolic length of $P$
equals $\log\mathrm{N}(P)$. As usual, we set
$\Lambda_{\Gamma}(P)=\log\mathrm{N}(P_{0})$, where $P_{0}$ is the primitive
closed geodesic underlying $P$, and we introduce the Chebyshev-like counting
function
$\Psi_{\Gamma}(x)\coloneqq\sum_{\mathrm{N}(P)\leq x}\Lambda_{\Gamma}(P).$
Because the prime geodesic theorem is reminiscent of the prime number theorem,
the norms are sometimes called pseudoprimes. Selberg [Sel89] established an
asymptotic formula of the shape
$\Psi_{\Gamma}(x)=\sum_{\frac{1}{2}<s_{j}\leq
1}\frac{x^{s_{j}}}{s_{j}}+\mathcal{E}_{\Gamma}(x),$ (1.1)
where the main term emerges from the small eigenvalues
$\lambda_{j}=s_{j}(1-s_{j})<\frac{1}{4}$ of the Laplacian on
$\Gamma\backslash\mathbb{H}$ for the upper half-plane $\mathbb{H}$, and
$\mathcal{E}_{\Gamma}(x)$ is an error term. It is known that
$\mathcal{E}_{\Gamma}(x)\ll_{\Gamma,\varepsilon}x^{\frac{3}{4}+\varepsilon}$;
see the explicit formulæ in [Iwa84b, KK22]. This barrier is often termed the
trivial bound. Given an analogue of the Riemann Hypothesis for Selberg zeta
functions apart from a finite number of the exceptional zeros, one should
expect the best possible estimate to be
$\mathcal{E}_{\Gamma}(x)\ll_{\Gamma,\varepsilon}x^{\frac{1}{2}+\varepsilon}$.
This remains unresolved due to the abundance of Laplace eigenvalues.
If $\Gamma$ is arithmetic, then an improvement over the barrier was achieved
by Iwaniec [Iwa84b], who proved that
$\mathcal{E}_{\Gamma}(x)\ll_{\varepsilon}x^{\frac{35}{48}+\varepsilon}$ for
the full modular group $\Gamma=\mathrm{PSL}_{2}(\mathbb{Z})$. Iwaniec [Iwa84a]
stated that the stronger exponent $\frac{2}{3}+\varepsilon$ follows from the
Generalised Lindelöf Hypothesis for quadratic Dirichlet $L$-functions.
Furthermore, under the Generalised Lindelöf Hypothesis for Rankin–Selberg
$L$-functions, the same exponent follows from the Weil bound for Kloosterman
sums. Luo–Sarnak [LS95] then strengthened the machinery of Iwaniec to obtain
the exponent $\frac{7}{10}+\varepsilon$; see also [Koy98, LRS95]. As a further
refinement, Cai [Cai02] derived the exponent $\frac{71}{102}+\varepsilon$. The
crucial gist in all these works is to estimate nontrivially a certain spectral
exponential sum via the Kuznetsov formula. On the other hand, the subsequent
work of Soundararajan–Young [SY13] demonstrated that
$\mathcal{E}_{\Gamma}(x)\ll_{\varepsilon}x^{\frac{2}{3}+\frac{\vartheta}{6}+\varepsilon},$
where $\vartheta$ is a subconvex exponent for quadratic Dirichlet
$L$-functions. The current record $\vartheta=\frac{1}{6}$ by Conrey–Iwaniec
[CI00] implies the best known exponent $\frac{25}{36}+\varepsilon$. The proof
utilises the Kuznetsov–Bykovskiĭ formula (see [Byk94, Kuz78, SY13]), leaving
the theory of Selberg zeta functions aside. For recent progress on the prime
geodesic theorem and its generalisations, we direct the reader to [BBCL22,
BBHM19, BCC+19, BF19, BF20, BFR22, CCL22, CG18, CWZ22, DM23, Kan20, Kan22,
Kan23, Koy01, PR17].
### 1.2. Statement of main results
By a classical theorem of Dirichlet, the primes equidistribute in the reduced
residue classes of a given modulus. As we shall see, the prime geodesic
analogue of this phenomenon breaks down, and the corresponding non-uniform
distribution can be determined explicitly.
We fix $\Gamma=\mathrm{PSL}_{2}(\mathbb{Z})$. By definition, an element
$P\in\Gamma$ is said to be hyperbolic if as a Möbius transformation
$Pz=\frac{az+b}{cz+d},$
it possesses two distinct real fixed points. By conjugation, any hyperbolic
element $P\in\Gamma$ may be expressed as $P=\sigma^{-1}\widetilde{P}\sigma$,
where $\sigma\in\mathrm{PSL}_{2}(\mathbb{R})$ and
$\widetilde{P}=\begin{pmatrix}\lambda&0\\\ 0&\lambda^{-1}\end{pmatrix}$ with
$\lambda>1$. Here $\widetilde{P}$ acts as multiplication by $\lambda^{2}$,
namely $\widetilde{P}z=\lambda^{2}z$. The factor $\lambda^{2}$ is called the
norm of $P$, which depends only on the
$\mathrm{PSL}_{2}(\mathbb{R})$-conjugacy class of $P$. We note that the
positive trace
$t=\mathrm{tr}(P)=\mathrm{tr}(\widetilde{P})=\lambda+\lambda^{-1}$
is an integer exceeding $2$, and hence the norm takes the form
$\mathrm{N}(P)=\left(\frac{t+\sqrt{t^{2}-4}}{2}\right)^{2}=t^{2}-2+O(t^{-2}).$
Given a prime $p\geq 2$, we define the Chebyshev-like counting function in
arithmetic progressions modulo $p$ by
$\Psi_{\Gamma}(x;p,a)\coloneqq\sum_{\begin{subarray}{c}\mathrm{N}(P)\leq x\\\
\mathrm{tr}(P)\equiv
a\,(\mathrm{mod}{\,p})\end{subarray}}\Lambda_{\Gamma}(P).$
The following result shows that the main term in an asymptotic for
$\Psi_{\Gamma}(x;p,a)$ depends on the residue class $a\,(\mathrm{mod}{\,p})$
unlike for primes in arithmetic progressions.
###### Theorem 1.1.
Let $\Gamma=\mathrm{PSL}_{2}(\mathbb{Z})$, and let $p\geq 3$ be a prime. Then
we have that
$\Psi_{\Gamma}(x;p,a)=\begin{dcases}\frac{1}{p-1}\cdot
x+O_{\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon})&\text{if
$\left(\frac{a^{2}-4}{p}\right)=1$,}\\\ \frac{1}{p+1}\cdot
x+O_{\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon})&\text{if
$\left(\frac{a^{2}-4}{p}\right)=-1$,}\\\ \frac{p}{p^{2}-1}\cdot
x+O_{p,\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon})&\text{if
$\left(\frac{a^{2}-4}{p}\right)=0$,}\end{dcases}$ (1.2)
where $\vartheta$ is a subconvex exponent for quadratic Dirichlet
$L$-functions.
###### Remark 1.
The implied constant in the error term is independent of $p$ when
$a\not\equiv\pm 2\,(\mathrm{mod}{\,p})$.
###### Remark 2.
When $p=3$, the first case of (1.2) is void, while the second case is covered
with a stronger error term by [GS99, Theorem $1$].
###### Remark 3.
Apart from the size of the error term, Theorem 1.1 resolves [GS99, Conjecture
$2$] for level $N=1$. In fact, we expect that our method works for
$\Gamma=\Gamma_{0}(N)$ when $(N,p)=1$, but we restrict to
$\Gamma=\mathrm{PSL}_{2}(\mathbb{Z})$ for simplicity. Furthermore, we expect
that the error term can be improved significantly by a more careful analysis
(e.g. by combining the Kuznetsov–Bykovskiĭ formula with an adèlic trace
formula), but we solely focused on determining the main term according to the
sign of the Legendre symbol. We leave such pursuits for future work.
The method of Golovchanskiĭ–Smotrov [GS99, Theorem $1$] is different from
ours, and they delve into properties of the norms and traces, expressing
$\Psi_{\Gamma}(x;p,a)$ as a linear combination of
$\Psi_{\Gamma_{0}(2^{k})}(x)$ for some $k\geq 0$, for which an asymptotic
formula is already known as in (1.1). For example, they derived a general
linear combination of the shape
$3\Psi_{\Gamma_{0}(N)}(x)-3\Psi_{\Gamma_{0}(2N)}(x)+\Psi_{\Gamma_{0}(4N)}(x)=3\Psi_{\Gamma_{0}(N)}(x;2,1),$
from which it follows that111We emphasise that the case of $p=2$ is not
contained in Theorem 1.1.
$\Psi_{\Gamma_{0}(N)}(x;2,1)=\frac{1}{3}\cdot
x+\mathcal{E}_{\Gamma_{0}(N)}(x),\qquad\Psi_{\Gamma_{0}(N)}(x;2,0)=\frac{2}{3}\cdot
x+\mathcal{E}_{\Gamma_{0}(N)}(x).$
The level structure that they developed is delicate, and it appears that their
idea only works for some specific values of $p$ and $a$. Hence, some new ideas
are needed to prove Theorem 1.1.
An elementary counting argument implies the following result.
###### Corollary 1.2.
Let $\Gamma=\mathrm{PSL}_{2}(\mathbb{Z})$, and let $p\geq 3$ be a prime. Then
we have that
$\displaystyle\sum_{\begin{subarray}{c}a\,(\mathrm{mod}{\,p})\\\
\left(\frac{a^{2}-4}{p}\right)=1\end{subarray}}\Psi_{\Gamma}(x;p,a)$
$\displaystyle=\frac{p-3}{2(p-1)}\cdot
x+O_{p,\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon}),$
$\displaystyle\sum_{\begin{subarray}{c}a\,(\mathrm{mod}{\,p})\\\
\left(\frac{a^{2}-4}{p}\right)=-1\end{subarray}}\Psi_{\Gamma}(x;p,a)$
$\displaystyle=\frac{p-1}{2(p+1)}\cdot
x+O_{p,\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon}),$
$\displaystyle\sum_{\begin{subarray}{c}a\,(\mathrm{mod}{\,p})\\\
\left(\frac{a^{2}-4}{p}\right)=0\end{subarray}}\Psi_{\Gamma}(x;p,a)$
$\displaystyle=\frac{2p}{p^{2}-1}\cdot
x+O_{p,\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon}),$
where $\vartheta$ is a subconvex exponent for quadratic Dirichlet
$L$-functions.
###### Remark 4.
Apart from the size of the error term, Corollary 1.2 resolves [GS99,
Conjecture $1$] in the case of full level $N=1$, and again the method should
work for $\Gamma=\Gamma_{0}(N)$ when $(N,p)=1$.
### Acknowledgements
The authors thank Mikhail Nikolaevich Smotrov for sending us the preprint
[GS99].
## 2\. Key propositions
This section prepares for the proof of Theorem 1.1. Throughout, we follow
[SY13, Section 2] closely.
Let $\Gamma=\mathrm{PSL}_{2}(\mathbb{Z})$ as before. Sarnak [Sar82,
Proposition 1.4] showed that the primitive hyperbolic conjugacy classes in
$\Gamma$ correspond bijectively to the $\Gamma$-equivalence classes of
primitive indefinite binary quadratic forms. We recall this correspondence
briefly. For a given primitive quadratic form $ax^{2}+bxy+cy^{2}$ of
discriminant $d>0$, the automorphs are the elements
$P(t,u)=\begin{pmatrix}\dfrac{t-bu}{2}&-cu\\\
au&\dfrac{t+bu}{2}\end{pmatrix}\in\Gamma,$
with $t^{2}-du^{2}=4$ being a solution of the Pell equation. For $u$ nonzero,
$P(t,u)$ is hyperbolic with norm $(t+u\sqrt{d})^{2}/4$ and trace $t$. Because
$P(-t,-u)=P(t,u)$ holds in $\Gamma$, we shall restrict to $t>0$ without loss
of generality. This is in harmony with our convention in Section 1.2 that
$\mathrm{tr}(P)>2$ for a hyperbolic element $P\in\Gamma$. If $(t_{d},u_{d})$
denotes the fundamental solution of the Pell equation, then $P(t_{d},u_{d})$
is a primitive hyperbolic matrix of norm $\varepsilon_{d}^{2}$ and trace
$t_{d}$. Moreover, every automorph $P(t,u)$ with $u>0$ (resp. $u<0$) is a
unique positive (resp. negative) integral power of $P(t_{d},u_{d})$. Sarnak’s
bijection sends the quadratic form $ax^{2}+bxy+cy^{2}$ to the conjugacy class
of $P(t_{d},u_{d})$ in $\Gamma$. Thus, for a given discriminant $d>0$, there
are $h(d)$ primitive hyperbolic conjugacy classes in $\Gamma$, each of norm
$\varepsilon_{d}^{2}$ and trace $t_{d}$.
Now every hyperbolic conjugacy class $\\{P\\}$ can be written uniquely as
$\\{P_{0}^{n}\\}$ for $n\geq 1$ and a primitive hyperbolic conjugacy class
$\\{P_{0}\\}$ (cf. [Hub59]). Combining this with Sarnak’s bijection described
above, we obtain
$\Psi_{\Gamma}(x;p,a)=2\sum_{\begin{subarray}{c}3\leq t\leq X\\\ t\equiv
a\,(\mathrm{mod}{\,p})\end{subarray}}\sum_{t^{2}-du^{2}=4}h(d)\log\varepsilon_{d},$
where $X$ abbreviates $\sqrt{x}+\frac{1}{\sqrt{x}}$, and $d>0$ (resp. $u>0$)
runs through discriminants (resp. integers). The class number formula
$h(d)\log\varepsilon_{d}=\sqrt{d}L(1,\chi_{d})$, where $\chi_{d}$ is the not
necessarily primitive quadratic Dirichlet character associated to the
discriminant $d$, allows us to write
$\Psi_{\Gamma}(x;p,a)=2\sum_{\begin{subarray}{c}3\leq t\leq X\\\ t\equiv
a\,(\mathrm{mod}{\,p})\end{subarray}}\sum_{t^{2}-du^{2}=4}\sqrt{d}L(1,\chi_{d}).$
(2.1)
For an arbitrary discriminant $\delta>0$, we define Zagier’s $L$-series by
(cf. [SY13, (6) & (3)])
$L(s,\delta)\coloneqq\sum_{du^{2}=\delta}L(s,\chi_{d})u^{1-2s}=\sum_{q=1}^{\infty}\frac{\lambda_{q}(\delta)}{q^{s}},$
where $d>0$ (resp. $u>0$) runs through discriminants (resp. integers). This
series admits a transparent Euler product expansion (to be discussed below),
while it simplifies (2.1) as
$\Psi_{\Gamma}(x;p,a)=2\sum_{\begin{subarray}{c}3\leq t\leq X\\\ t\equiv
a\,(\mathrm{mod}{\,p})\end{subarray}}\sqrt{t^{2}-4}L(1,t^{2}-4).$ (2.2)
If $\delta=Dl^{2}$, where $D>0$ is a fundamental discriminant and $l>0$ is an
integer, then we obtain the following Euler product expansion of Zagier’s
$L$-series (cf. [SY13, (2)]):
$\displaystyle L(s,\delta)$ $\displaystyle=\sum_{u\mid
l}L(s,\chi_{Dl^{2}/u^{2}})u^{1-2s}$ $\displaystyle=L(s,\chi_{D})\sum_{u\mid
l}u^{1-2s}\prod_{\bm{p}\mid\frac{l}{u}}(1-\chi_{D}(\bm{p}))$
$\displaystyle=\prod_{\bm{p}}\left(\sum_{0\leq
m<v_{\bm{p}}(l)}\bm{p}^{m(1-2s)}+\frac{\bm{p}^{v_{\bm{p}}(l)(1-2s)}}{1-\chi_{D}(\bm{p})\bm{p}^{-s}}\right).$
(2.3)
In particular, for fixed $\delta$, the arithmetic function
$q\mapsto\lambda_{q}(\delta)$ is multiplicative. The idea of the proof of
Theorem 1.1 is to group together certain values of $t$ in (2.2) such that the
corresponding Zagier $L$-series $L(s,t^{2}-4)$ has a constant Euler factor at
$\bm{p}=p$. Thus we are led to consider $L(s,\delta)$ with its Euler factor at
$\bm{p}=p$ removed:
$L^{p}(s,\delta)\coloneqq\sum_{\begin{subarray}{c}q\geq 1\\\
(q,p)=1\end{subarray}}\frac{\lambda_{q}(\delta)}{q^{s}}.$
###### Proposition 2.1.
Let $p\geq 3$ be a prime, and let $n\geq 1$ be an integer. Let
$r\,(\mathrm{mod}{\,p^{n}})$ be an arbitrary residue class. If $(q,p)=1$ and
$b$ denotes the squarefree part of $q$, then
$\sum_{\begin{subarray}{c}3\leq t\leq X\\\ t\equiv
r\,(\mathrm{mod}{\,p^{n}})\end{subarray}}\lambda_{q}(t^{2}-4)=\frac{X}{p^{n}}\cdot\frac{\mu(b)}{b}+O_{\varepsilon}(q^{\frac{1}{2}+\varepsilon}).$
###### Proof.
It follows from [SY13, Lemma 2.3] that
$\lambda_{q}(t^{2}-4)=\sum_{q_{1}^{2}q_{2}=q}\frac{1}{q_{2}}\sum_{k\,(\mathrm{mod}{\,q_{2}})}e\left(\frac{kt}{q_{2}}\right)S(k^{2},1;q_{2}).$
This leads to
$\sum_{\begin{subarray}{c}3\leq t\leq X\\\ t\equiv
r\,(\mathrm{mod}{\,p^{n}})\end{subarray}}\lambda_{q}(t^{2}-4)=\sum_{q_{1}^{2}q_{2}=q}\frac{1}{q_{2}}\sum_{k\,(\mathrm{mod}{\,q_{2}})}S(k^{2},1;q_{2})\sum_{\begin{subarray}{c}3\leq
t\leq X\\\ t\equiv
r\,(\mathrm{mod}{\,p^{n}})\end{subarray}}e\left(\frac{kt}{q_{2}}\right).$
When $k\equiv 0\,(\mathrm{mod}{\,q_{2}})$, the inner sum over $t$ is
$\frac{X}{p^{n}}+O(1)$, and $S(0,1;q_{2})=\mu(q_{2})$, yielding the expected
main term. When $k\not\equiv 0\,(\mathrm{mod}{\,q_{2}})$, we apply the Weil
bound for Kloosterman sums and the estimate
$\sum_{\begin{subarray}{c}3\leq t\leq X\\\ t\equiv
r\,(\mathrm{mod}{\,p^{n}})\end{subarray}}e\left(\frac{kt}{q_{2}}\right)\ll\left\lVert\frac{kp^{n}}{q_{2}}\right\rVert^{-1}\leq
q_{2},$
where $\left\lVert\cdot\right\rVert$ is the distance to the nearest integer.
The proof is complete. ∎
Guided by Proposition 2.1 and (2.2), we consider the sum
$\Psi_{\Gamma}^{\star}(x;p^{n},r)\coloneqq 2\sum_{\begin{subarray}{c}3\leq
t\leq X\\\ t\equiv
r\,(\mathrm{mod}{\,p^{n}})\end{subarray}}\sqrt{t^{2}-4}L^{p}(1,t^{2}-4).$
(2.4)
We shall deduce Theorem 1.1 from the following analogue of [SY13, Theorem
3.2]:
###### Proposition 2.2.
Let $p\geq 3$ be a prime, and let $n\geq 1$ be an integer. Let
$r\,(\mathrm{mod}{\,p^{n}})$ be an arbitrary residue class. Then
$\Psi_{\Gamma}^{\star}(x+u;p^{n},r)-\Psi_{\Gamma}^{\star}(x;p^{n},r)=\frac{u}{p^{n}}+O_{\varepsilon}(u^{\frac{1}{2}}x^{\frac{1}{4}+\frac{\vartheta}{2}+\varepsilon}),\qquad\sqrt{x}\leq
u\leq x.$ (2.5)
###### Proof.
Let $\sqrt{x}\leq u\leq x$, and set
$X\coloneqq\sqrt{x}+\frac{1}{\sqrt{x}}\qquad\text{and}\qquad
X^{\prime}\coloneqq\sqrt{x+u}+\frac{1}{\sqrt{x+u}}.$
From the definition (2.4), it is clear that
$\displaystyle\Psi_{\Gamma}^{\star}(x+u;p^{n},r)-\Psi_{\Gamma}^{\star}(x;p^{n},r)$
$\displaystyle=2\sum_{\begin{subarray}{c}X<t\leq X^{\prime}\\\ t\equiv
r\,(\mathrm{mod}{\,p^{n}})\end{subarray}}\sqrt{t^{2}-4}L^{p}(1,t^{2}-4)$
$\displaystyle=\left(2+O\left(x^{-1}\right)\right)\sum_{\begin{subarray}{c}X<t\leq
X^{\prime}\\\ t\equiv
r\,(\mathrm{mod}{\,p^{n}})\end{subarray}}tL^{p}(1,t^{2}-4),$
because $\sqrt{t^{2}-4}=t(1+O(t^{-2}))$. We shall approximate
$L^{p}(1,t^{2}-4)$ in terms of a suitable Dirichlet series. Let $V\geq 1$ be a
parameter to be chosen later, and let
$\delta=t^{2}-4=Dl^{2},$
where $D>0$ is a fundamental discriminant and $l>0$ is an integer. Consider
$\Lambda_{V}^{p}(\delta)\coloneqq\sum_{\begin{subarray}{c}q\geq 1\\\
(q,p)=1\end{subarray}}\frac{\lambda_{q}(\delta)}{q}e^{-\frac{q}{V}}.$
Shifting the contour yields the expression
$\Lambda_{V}^{p}(\delta)=\int_{(1)}L^{p}(1+s,\delta)V^{s}\Gamma(s)\frac{ds}{2\pi
i}=L^{p}(1,\delta)+\int_{(-\frac{1}{2})}L^{p}(1+s,\delta)V^{s}\Gamma(s)\frac{ds}{2\pi
i}.$
On the right-hand side, for some $A>0$,
$L^{p}(1+s,\delta)\ll|L(1+s,\delta)|\ll_{\varepsilon}|L(1+s,\chi_{D})|l^{\varepsilon}\ll_{\varepsilon}\delta^{\vartheta+\varepsilon}|s|^{A},\qquad\mathrm{Re}s=-\frac{1}{2},$
while $\Gamma(s)$ decays exponentially. It follows that
$L^{p}(1,\delta)=\Lambda_{V}^{p}(\delta)+O_{\varepsilon}(\delta^{\vartheta+\varepsilon}V^{-\frac{1}{2}}),$
and hence
$\Psi_{\Gamma}^{\star}(x+u;p^{n},r)-\Psi_{\Gamma}^{\star}(x;p^{n},r)=(2+O(x^{-1}))\sum_{\begin{subarray}{c}X<t\leq
X^{\prime}\\\ t\equiv
r\,(\mathrm{mod}{\,p^{n}})\end{subarray}}t\Lambda_{V}^{p}(t^{2}-4)+O_{\varepsilon}(ux^{\vartheta+\varepsilon}V^{-\frac{1}{2}}).$
If $(q,p)=1$ and $q=bc^{2}$ with $b$ squarefree, then Proposition 2.1 along
with partial summation leads to
$2\sum_{\begin{subarray}{c}X<t\leq X^{\prime}\\\ t\equiv
r\,(\mathrm{mod}{\,p^{n}})\end{subarray}}t\lambda_{q}(t^{2}-4)=\frac{u}{p^{n}}\cdot\frac{\mu(b)}{b}+O_{\varepsilon}(Xq^{\frac{1}{2}+\varepsilon}).$
It thus follows that
$2\sum_{\begin{subarray}{c}X<t\leq X^{\prime}\\\ t\equiv
r\,(\mathrm{mod}{\,p^{n}})\end{subarray}}t\Lambda_{V}^{p}(t^{2}-4)=\frac{u}{p^{n}}\sum_{\begin{subarray}{c}b,c\geq
1\\\
(bc,p)=1\end{subarray}}\frac{\mu(b)}{b^{2}c^{2}}e^{-\frac{bc^{2}}{V}}+O_{\varepsilon}(XV^{\frac{1}{2}+\varepsilon}).$
A standard contour shift argument gives
$\sum_{\begin{subarray}{c}b,c\geq 1\\\
(bc,p)=1\end{subarray}}\frac{\mu(b)}{b^{2}c^{2}}e^{-\frac{bc^{2}}{V}}=\int_{(1)}V^{s}\Gamma(s)\frac{\zeta^{p}(2+2s)}{\zeta^{p}(2+s)}\frac{ds}{2\pi
i}=1+O(V^{-\frac{1}{2}}),$
where $\zeta^{p}(s)$ is the Riemann zeta function with the Euler factor at $p$
removed. As a result,
$\Psi_{\Gamma}^{\star}(x+u;p^{n},r)-\Psi_{\Gamma}^{\star}(x;p^{n},r)=\frac{u}{p^{n}}+O_{\varepsilon}(x^{\frac{1}{2}}V^{\frac{1}{2}+\varepsilon}+ux^{\vartheta+\varepsilon}V^{-\frac{1}{2}}).$
Setting $V=ux^{-\frac{1}{2}+\vartheta}$ yields (2.5). ∎
###### Corollary 2.3.
Let $p\geq 3$ be a prime, and let $n\geq 1$ be an integer. Let
$r\,(\mathrm{mod}{\,p^{n}})$ be an arbitrary residue class. Then
$\Psi_{\Gamma}^{\star}(x;p^{n},r)=\frac{x}{p^{n}}+O_{\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon}).$
(2.6)
###### Proof.
Setting $u=x$ in (2.5) yields
$\Psi_{\Gamma}^{\star}(2x;p^{n},r)-\Psi_{\Gamma}^{\star}(x;p^{n},r)=\frac{x}{p^{n}}+O_{\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon}).$
Now (2.6) follows readily by a dyadic subdivision. ∎
## 3\. Proof of Theorem 1.1
We shall approximate the $t$-sum in (2.2). As before, we write
$\delta=t^{2}-4=Dl^{2},$
where $D>0$ is a fundamental discriminant and $l>0$ is an integer.
If $a\not\equiv\pm 2\,(\mathrm{mod}{\,p})$, then for every $t$ participating
in (2.2), we have that $p\nmid l$ and
$\chi_{D}(p)=\left(\frac{D}{p}\right)=\left(\frac{Dl^{2}}{p}\right)=\left(\frac{t^{2}-4}{p}\right)=\left(\frac{a^{2}-4}{p}\right).$
By (2.3), the corresponding Zagier $L$-series $L(s,t^{2}-4)$ factorises as
$L(s,t^{2}-4)=\left(1-\left(\frac{a^{2}-4}{p}\right)p^{-s}\right)^{-1}L^{p}(s,t^{2}-4),$
hence by (2.2) and (2.4), it also follows that
$\Psi_{\Gamma}(x;p,a)=\left(1-\left(\frac{a^{2}-4}{p}\right)p^{-1}\right)^{-1}\Psi_{\Gamma}^{\star}(x;p,a).$
Applying (2.6), we obtain the first two cases of (1.2).
If $a\equiv\pm 2\,(\mathrm{mod}{\,p})$, then we shall assume (without loss of
generality) that $a=\pm 2$. We subdivide the $t$-sum in (2.2) according to the
exponent of $p$ in the positive integer $t-a$:
$\Psi_{\Gamma}(x;p,a)=\sum_{k=1}^{\infty}\Psi_{\Gamma}(x;p,a;k),$
where
$\Psi_{\Gamma}(x;p,a;k)\coloneqq 2\sum_{\begin{subarray}{c}3\leq t\leq X\\\
v_{p}(t-a)=k\end{subarray}}\sqrt{t^{2}-4}L(1,t^{2}-4).$
We shall approximate these pieces individually. Note that
$\Psi_{\Gamma}(x;p,a;k)=0$ for $p^{k}>t-a$. Moreover, the condition
$v_{p}(t-a)=k$ constrains $t$ to $p-1$ residue classes modulo $p^{k+1}$, and
it also yields $v_{p}(t^{2}-4)=k$.
If $k=2n-1$ is odd, then $p\mid D$ and $v_{p}(l)=n-1$, hence by (2.3),
$L(s,t^{2}-4)=\frac{1-p^{n(1-2s)}}{1-p^{1-2s}}L^{p}(s,t^{2}-4).$
Using also (2.4) and (2.6), we obtain
$\displaystyle\Psi_{\Gamma}(x;p,a;2n-1)$
$\displaystyle=\frac{p-1}{p^{2n}}\cdot\frac{1-p^{-n}}{1-p^{-1}}\cdot
x+O_{p,\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon})$
$\displaystyle=(p^{1-2n}-p^{1-3n})x+O_{p,\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon}).$
(3.1)
It is important that the implied constant in the error term is independent of
$n$.
If $k=2n$ is even, then $p\nmid D$ and $v_{p}(l)=n$, hence by (2.3),
$L(s,t^{2}-4)=\left(\frac{1-p^{n(1-2s)}}{1-p^{1-2s}}+\frac{p^{n(1-2s)}}{1-\chi_{D}(p)p^{-s}}\right)L^{p}(s,t^{2}-4).$
We can understand $\chi_{D}(p)$ by writing $t=a+p^{2n}r$. Indeed, then
$t^{2}-4=2ap^{2n}r+p^{4n}r^{2}$, hence
$\chi_{D}(p)=\left(\frac{D}{p}\right)=\left(\frac{Dl^{2}p^{-2n}}{p}\right)=\left(\frac{2ar}{p}\right).$
This means that among the $p-1$ choices for $t\,(\mathrm{mod}{\,p^{2n+1}})$,
half the time $\chi_{D}(p)$ equals $1$ and half the time it equals $-1$. Using
also (2.4) and (2.6), we obtain
$\displaystyle\Psi_{\Gamma}(x;p,a;2n)$
$\displaystyle=\frac{p-1}{p^{2n+1}}\left(\frac{1-p^{-n}}{1-p^{-1}}+\frac{(1/2)p^{-n}}{1-p^{-1}}+\frac{(1/2)p^{-n}}{1+p^{-1}}\right)x+O_{p,\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon})$
$\displaystyle=\left(p^{-2n}-\frac{p^{-3n}}{p+1}\right)x+O_{p,\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon}).$
(3.2)
Again, the implied constant in the error term is independent of $n$.
Summing up the pieces $\Psi_{\Gamma}(x;p,a;2n-1)$ and
$\Psi_{\Gamma}(x;p,a;2n)$ for $1\leq n\leq\log(X+2)$, and inserting the
approximations (3.1)–(3.2), we deduce the asymptotic formula
$\Psi_{\Gamma}(x;p,\pm
2)=c_{p}x+O_{p,\varepsilon}(x^{\frac{3}{4}+\frac{\vartheta}{2}+\varepsilon}),$
where
$c_{p}\coloneqq\sum_{n=1}^{\infty}\left(p^{1-2n}-p^{1-3n}+p^{-2n}-\frac{p^{-3n}}{p+1}\right)=\frac{p}{p^{2}-1}.$
This is the third case of (1.2), and the proof of Theorem 1.1 is complete.
## 4\. Proof of Corollary 1.2
There are $\frac{p-3}{2}$ residues $a\,(\mathrm{mod}{\,p})$ such that
$a^{2}-4$ is a nonzero quadratic residue. Indeed, this occurs if and only if
$a^{2}-4\equiv b^{2}\,(\mathrm{mod}{\,p})$ for some $b\not\equiv
0\,(\mathrm{mod}{\,p})$, which can be written as $(a+b)(a-b)\equiv
4\,(\mathrm{mod}{\,p})$. Making the change of variables $a+b\equiv
2x\,(\mathrm{mod}{\,p})$ and $a-b\equiv 2x^{-1}\,(\mathrm{mod}{\,p})$, we
obtain $a\equiv x+x^{-1}\,(\mathrm{mod}{\,p})$ and $b\equiv
x-x^{-1}\,(\mathrm{mod}{\,p})$ with the condition
$x\not\equiv-1,0,1\,(\mathrm{mod}{\,p})$. This restriction leaves
$\frac{p-3}{2}$ different ways to choose $a\,(\mathrm{mod}{\,p})$. Since there
are two solutions to $(\frac{a^{2}-4}{p})=0$, we conclude that there are
$\frac{p-1}{2}$ residues $a\,(\mathrm{mod}{\,p})$ such that $a^{2}-4$ is a
nonzero quadratic nonresidue. Now Corollary 1.2 is immediate from Theorem 1.1.
## References
* [BBCL22] A. Balog, A. Biró, G. Cherubini, and N. Laaksonen, _Bykovskii-type theorem for the Picard manifold_ , Int. Math. Res. Not. IMRN 2022 (2022), no. 3, 1893–1921.
* [BBHM19] A. Balog, A. Biró, G. Harcos, and P. Maga, _The prime geodesic theorem in square mean_ , J. Number Theory 198 (2019), 239–249, Errata: https://users.renyi.hu/~gharcos/primegeodesic_errata.txt.
* [BCC+19] O. Balkanova, D. Chatzakos, G. Cherubini, D. Frolenkov, and N. Laaksonen, _Prime geodesic theorem in the 3-dimensional hyperbolic space_ , Trans. Amer. Math. Soc. 372 (2019), no. 8, 5355–5374.
* [BF19] O. Balkanova and D. Frolenkov, _Sums of Kloosterman sums in the prime geodesic theorem_ , Quart. J. Math. 70 (2019), no. 2, 649–674.
* [BF20] by same author, _Prime geodesic theorem for the Picard manifold_ , Adv. Math. 375 (2020), Paper No. 107377, 42 pages.
* [BFR22] O. Balkanova, D. Frolenkov, and M. S. Risager, _Prime geodesics and averages of the Zagier $L$-series_, Math. Proc. Cambridge Philos. Soc. 172 (2022), no. 3, 705–728.
* [Byk94] V. A. Bykovskiĭ, _Density theorems and the mean value of arithmetic functions in short intervals (Russian)_ , Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 212 (1994), no. Anal. Teor. Chisel i Teor. Funktsiĭ. 12, 56–70, 196, translation in J. Math. Sci. (N.Y.) 83 (1997), no. 6, 720–730.
* [Cai02] Y. Cai, _Prime geodesic theorem_ , J. Théor. Nombres Bordeaux 14 (2002), no. 2, 59–72.
* [CCL22] D. Chatzakos, G. Cherubini, and N. Laaksonen, _Second moment of the prime geodesic theorem for $\mathrm{PSL}(2,\mathbb{Z}[i])$_, Math. Z. 300 (2022), no. 1, 791–806.
* [CG18] G. Cherubini and J. Guerreiro, _Mean square in the prime geodesic theorem_ , Algebra & Number Theory 12 (2018), no. 3, 571–597.
* [CI00] J. B. Conrey and H. Iwaniec, _The cubic moment of central values of automorphic $L$-functions_, Ann. of Math. (2) 151 (2000), no. 3, 1175–1216.
* [CWZ22] G. Cherubini, H. Wu, and G. Zábrádi, _On Kuznetsov–Bykovskii’s formula of counting prime geodesics_ , Math. Z. 300 (2022), no. 1, 881–928.
* [DM23] L. Dever and D. Milićević, _Ambient prime geodesic theorems on hyperbolic $3$-manifolds_, Int. Math. Res. Not. IMRN 2023 (2023), no. 1, 588–635.
* [GS99] V. V. Golovchanskiĭ and M. N. Smotrov, _The distribution of the number classes of the primitive hyperbolic elements of the group $\Gamma_{0}(N)$ in arithmetic progressions (Russian)_, Preprint FEB RAS Khabarovsk Division of the Institute for Applied Mathematics (1999), no. 15, 15 pages, https://drive.google.com/file/d/1MxrJbbhLTGKzh4-1lCWa4Up_EDGeaqHc/view.
* [Hej76a] D. Hejhal, _The Selberg trace formula and the Riemann zeta function_ , Duke Math. J. 43 (1976), no. 3, 441–482.
* [Hej76b] by same author, _The Selberg trace formula for $\mathrm{PSL}(2,\mathbb{R})$ I_, Lecture Notes in Mathematics, vol. 548, Springer-Verlag, Berlin Heidelberg, 1976.
* [Hej83] by same author, _The Selberg trace formula for $\mathrm{PSL}(2,\mathbb{R})$ II_, Lecture Notes in Mathematics, vol. 1001, Springer-Verlag, Berlin Heidelberg, 1983.
* [Hub59] H. Huber, _Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen_ , Math. Ann. 138 (1959), no. 1, 1–26.
* [Hub61a] by same author, _Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II_ , Math. Ann. 142 (1961), no. 4, 385–398.
* [Hub61b] by same author, _Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II. Nachtrag zu Math. Ann. 142, 385–398 (1961)_, Math. Ann. 143 (1961), no. 5, 463–464.
* [Iwa84a] H. Iwaniec, _Non-holomorphic modular forms and their applications_ , Modular forms (Durham, 1983) (R. A. Rankin, ed.), Ellis Horwood Ser. Math. Appl., Statist. Oper. Res., Horwood, Chichester, 1984, pp. 157–196.
* [Iwa84b] by same author, _Prime geodesic theorem_ , J. Reine Angew. Math. 349 (1984), 136–159.
* [Kan20] I. Kaneko, _The second moment for counting prime geodesics_ , Proc. Japan Acad. Ser. A Math. Sci. 96 (2020), no. 1, 7–12.
* [Kan22] by same author, _The prime geodesic theorem for $\mathrm{PSL}_{2}(\mathbb{Z}[i])$ and spectral exponential sums_, Algebra Number Theory 16 (2022), no. 8, 1845–1887.
* [Kan23] by same author, _Spectral exponential sums on hyperbolic surfaces_ , Ramanujan J. 60 (2023), no. 3, 837–846.
* [KK22] I. Kaneko and S. Koyama, _Euler products of Selberg zeta functions in the critical strip_ , Ramanujan J. 59 (2022), no. 2, 437–458.
* [Koy98] S. Koyama, _Prime geodesic theorem for arithmetic compact surfaces_ , Int. Math. Res. Not. IMRN 1998 (1998), no. 8, 383–388.
* [Koy01] by same author, _Prime geodesic theorem for the Picard manifold under the mean-Lindelöf hypothesis_ , Forum Math. 13 (2001), no. 6, 781–793.
* [Kuz78] N. V. Kuznetsov, _The arithmetic form of Selberg’s trace formula and the distribution of norms of the primitive hyperbolic classes of the modular group_ , Preprint (Khabarovsk), 1978.
* [LRS95] W. Luo, Z. Rudnick, and P. Sarnak, _On Selberg’s eigenvalue conjecture_ , Geom. Funct. Anal. 5 (1995), no. 2, 387–401.
* [LS95] W. Luo and P. Sarnak, _Quantum ergodicity of eigenfunctions on $\mathrm{PSL}_{2}(\mathbf{Z})\backslash\mathbf{H}^{2}$_, Publ. Math. Inst. Hautes Études Sci. 81 (1995), 207–237.
* [PR17] Y. N. Petridis and M. S. Risager, _Local average in hyperbolic lattice point counting, with an appendix by Niko Laaksonen_ , Math. Z. 285 (2017), no. 3–4, 1319–1344.
* [Sar80] P. Sarnak, _Prime geodesic theorems_ , Ph.D. thesis, Stanford University, August 1980.
* [Sar82] by same author, _Class numbers of indefinite binary quadratic forms_ , J. Number Theory 15 (1982), no. 2, 229–247.
* [Sel89] A. Selberg, _Collected papers_ , Springer Collected Works in Mathematics, vol. 1, Springer-Verlag Berlin Heidelberg, 1989.
* [SY13] K. Soundararajan and M. P. Young, _The prime geodesic theorem_ , J. Reine Angew. Math. 676 (2013), 105–120, Errata: https://www.math.tamu.edu/%7Emyoung/PrimeGeodesicCorrections.pdf.
* [Ven90] A. B. Venkov, _Spectral theory of automorphic functions and its applications_ , Mathematics and its Applications (Soviet Series), vol. 51, Kluwer Academic Publishers Group, 1990, Translated from the Russian by N. B. Lebedinskaya.
|
††thanks<EMAIL_ADDRESS>
# Differentiable Predictions for Large Scale Structure with SHAMNet
Andrew P. Hearin1,⋆ Nesar Ramachandra2,1 Matthew R. Becker1 Joseph DeRose3
1High Energy Physics Division, Argonne National Laboratory, 9700 South Cass
Avenue, Lemont, IL 60439, USA 2Computational Science Division, Argonne
National Laboratory, 9700 South Cass Avenue, Lemont, IL 60439, USA 3Lawrence
Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA, 94720 USA
###### Abstract
In simulation-based models of the galaxy–halo connection, theoretical
predictions for galaxy clustering and lensing are typically made based on
Monte Carlo realizations of a mock universe. In this paper, we use Subhalo
Abundance Matching (SHAM) as a toy model to introduce an alternative to
stochastic predictions based on mock population, demonstrating how to make
simulation-based predictions for clustering and lensing that are both exact
and differentiable with respect to the parameters of the model. Conventional
implementations of SHAM are based on iterative algorithms such as Richardson-
Lucy deconvolution; here we use the JAX library for automatic differentiation
to train SHAMNet, a neural network that accurately approximates the stellar-
to-halo mass relation (SMHM) defined by abundance matching. In our approach to
making differentiable predictions of large scale structure, we map
parameterized PDFs onto each simulated halo, and calculate gradients of
summary statistics of the galaxy distribution by using autodiff to propagate
the gradients of the SMHM through the statistical estimators used to measure
one- and two-point functions. Our techniques are quite general, and we
conclude with an overview of how they can be applied in tandem with more
complex, higher-dimensional models, creating the capability to make
differentiable predictions for the multi-wavelength universe of galaxies.
###### keywords:
Cosmology: large-scale structure of Universe; methods: N-body simulations
## 1 Introduction
Numerical simulations of cosmological structure formation can be viewed as
prediction engines for the density field, and for the abundance and spatial
distribution of dark matter halos. Of course, neither dark matter nor
gravitationally self-bound halos are directly observed in the sky, and so
additional modeling is required in order to transform a cosmological
simulation into a prediction that is commensurable with astronomical
measurements of galaxies. Contemporary simulations have by now achieved
percent-level precision in the ability to characterize the density field, halo
abundance and halo clustering in the nonlinear regime, whereas models
connecting observed galaxies to the fundamental quantities in simulations lag
far behind this level of accuracy (see Wechsler & Tinker, 2018, for a recent
review). Improving theoretical techniques for transforming cosmological
simulations into predictions for the galaxy density field is thus a critical
component of the precision-cosmology program.
In many of the structure formation models that are used for cosmological
inference, predictions for the galaxy distribution are made by modeling
galaxies as biased tracers of the underlying density field of matter. There
are a wide range of techniques that can be used for this purpose, including
methods based on Lagrangian perturbation theory (LPT, Bernardeau et al., 2002;
Desjacques et al., 2018), effective field theory (Carrasco et al., 2012), the
Zel’dovich approximation (White, 2014), and hybrid techniques that blend LPT
with simulations (Kokron et al., 2021; DeRose et al., 2021b). This approach to
generating cosmological predictions has now been used by numerous galaxy
surveys to derive constraints on the fundamental parameters of cosmology
(e.g., DES Collaboration et al., 2021; Joudaki et al., 2018).
An alternative approach is to model the connection between dark matter halos
and the galaxies residing within them. The conventional “halo occupation
model” approach proceeds with an initial step in which a fitting function
and/or a machine learning algorithm is calibrated to capture the cosmology-
dependence of various summary statistics of dark matter halos, such as the
halo mass function (e.g., Jenkins et al., 2001; McClintock et al., 2019a;
Bocquet et al., 2020) and halo bias (e.g, Tinker et al., 2010; McClintock et
al., 2019b); once the approximations for these quantities are specified,
models for the galaxy–halo connection determine the prediction for the
clustering and lensing of galaxies, enabling the derivation of constraints on
cosmological parameters (Cacciato et al., 2013; Reddick et al., 2014; Miyatake
et al., 2021).
As an alternative to traditional implementations of halo occupation modeling,
it has become increasingly common to directly populate simulated halos with a
Monte Carlo realization of the galaxy population; predictions for summary
statistics of large scale structure are then made directly from the synthetic
galaxy distribution using the same point estimators used to analyze
observational data. In the conventional approach to halo occupation modeling,
phenomena such as halo exclusion (García et al., 2021), satellite anisotropy
(Sgró et al., 2013), and galaxy assembly bias (Zentner et al., 2014)
constitute a major technical challenge, particularly in the one-to-two-halo
regime (van den Bosch et al., 2013); one of the key advantages of the mock-
population approach is that the numerical treatment of such effects is exact,
with a precision limited only by the resolution and finite size of the
simulation.
One of the principal sources of motivation for using mock-population
techniques lies in the cosmological information content of the nonlinear
regime. The potential to substantially improve cosmological constraints by
incorporating smaller-scale information has been known for many years (Zentner
et al., 2013; Reid et al., 2014; Krause & Eifler, 2017). Recent studies of the
clustering of galaxies in the Baryon Oscillation Spectroscopic Survey (BOSS)
have confirmed the long-forecasted constraining power of the nonlinear regime.
In an analysis of the redshift-space clustering of the LOWZ galaxy sample
(Lange et al., 2021), the authors derived better-than 5% constraints on the
cosmological growth of structure, $f\sigma_{8},$ a full factor of two stronger
than any previous BOSS analysis that restricted attention to larger-scale
measurements. Comparable gains in cosmological constraining power from the
nonlinear regime of BOSS LOWZ galaxies were also found in Wibking et al.
(2020). In closely related work analyzing the clustering of the Luminous Red
Galaxy sample in eBOSS (Chapman et al., 2021), it was found that constraints
on $f\sigma_{8}$ are improved by 70% when including information from nonlinear
scales.
Even when simulation-based predictions are made based on simple empirical
models implemented in specialized libraries such as Halotools (Hearin et al.,
2017), Corrfunc (Sinha & Garrison, 2020), or AbacusHOD (Yuan et al., 2021),
the computational demands of conducting Bayesian inference with survey-scale
simulations are considerable. This challenge is tailor-made for a set of
methods generally referred to as “emulation”, which is a machine learning
technique that is specifically designed for situations in which the behavior
of some parametric function, $\mathcal{P}(\theta),$ is calculable, but
expensive to evaluate. When confronted with such a situation, a natural way to
proceed is to pre-compute $\mathcal{P}(\theta)$ for a finite collection of
$\theta_{\rm i}$ that spans the domain of interest, and to use
$\mathcal{P}(\theta_{\rm i})$ to train a machine learning algorithm to serve
as an “emulator”, or surrogate function, $\mathcal{F}(\theta),$ that
approximates $\mathcal{P}(\theta).$ Once the emulator has been trained, one
then proceeds to carry out performance-critical analyses such as MCMCs using
$\mathcal{F}(\theta)$ rather than $\mathcal{P}(\theta),$ since the surrogate
function is typically inexpensive to evaluate. Since the introduction of
emulation methods to computational cosmology (Heitmann et al., 2006), these
techniques have become a widely-used tool throughout the field (e.g., Kwan et
al., 2015; Nishimichi et al., 2019; Euclid Collaboration et al., 2020).
The conventional approach to emulation is based on Gaussian Process (GP)
regression. Briefly, in GP regression, it is assumed that the values
$\mathcal{P}(\theta_{\rm i})$ are drawn from a multi-dimensional Gaussian
distribution, $\mathcal{N}(\mathcal{P}(\theta_{\rm i}),\Sigma(\theta_{\rm
i})),$ where $\Sigma(\theta_{\rm i})$ is a correlation matrix with hyper-
parameters that are optimized during training (see Rasmussen & Williams, 2006,
for a contemporary review). For example, in Lawrence et al. (2017), the
abscissa $\theta_{\rm i}$ are cosmological parameters, and
$\mathcal{P}(\theta_{\rm i})$ are values of the matter power spectrum; as
another example, in Zhai et al. (2019), the abscissa reside in the joint space
of cosmology and parameters of the Halo Occupation Distribution (HOD, Berlind
& Weinberg, 2002; Zheng et al., 2005), and the ordinates are values of the
redshift-space galaxy correlation function.
Although the GP approach to emulation has thus far been quite successful, this
technique has limitations that create a significant impediment to applying it
in tandem with the realistically complex models that will be required by near-
future cosmological datasets. In Zhai et al. (2019), the authors found that
the errors of their emulator were comparable to present-day measurement
uncertainties on their predicted data vector, resulting in a nearly $50\%$
degradation of the cosmological constraints due to emulator noise. In
principle, these inaccuracies could be remedied by increasing the number of
training points $\theta_{\rm i};$ in practice, however, the size of the
training data cannot be too large or the hyper-parameter optimization becomes
computationally intractable due to the need to invert $\Sigma(\theta_{\rm
i}).$ We point out that the effort in Zhai et al. (2019) is one of the most
ambitious GP applications in cosmology to date in the sense that the emulated
parameter space includes parameters that jointly encode variations in
cosmology as well as the galaxy–halo connection. And yet, the HOD model
emulated in Zhai et al. (2019) is essentially the simplest, lowest-dimensional
model that can plausibly be used to interpret galaxy clustering data with
present-day levels of uncertainty. Thus these shortcomings will only become
more severe as observational cosmology progresses further into the 2020s: not
only will the precision of cosmological measurements improve dramatically, but
perhaps even more importantly, the dimension of the emulated model will need
to increase substantially in order for the theoretical predictions to match
the quality and richness of the data.
In the present work, we introduce a new theoretical framework for the
galaxy–halo connection that is designed to address these issues. Our formalism
differs from the typical simulation-based methodology outlined above in
several respects. First, our simulation-based predictions for large scale
structure observables are not based on stochastic Monte Carlo realizations;
instead, we analytically propagate probability distributions from the
galaxy–halo connection through to the corresponding summary statistics, and so
the only source of stochasticity in our predictions derives from the finite
size and resolution of the simulation. Second, instead of relying upon
classical machine learning techniques such as Gaussian Process emulation, we
instead rely on Artificial Intelligence (AI) algorithms for our surrogate
functions. Since the neural networks implemented in contemporary deep learning
libraries are routinely used in industry applications to approximate the
behavior of million-parameter systems, our use of AI-based surrogate functions
improves upon the problems associated with emulator accuracy and complexity
outlined above. As a result of these techniques, our predictions for large
scale structure observables are fully differentiable, end-to-end, enabling us
to use gradient-based optimization and inference algorithms that exhibit much
better scaling to the problem sizes that will characterize cosmological
modeling in the 2020s.
In the present paper, for our model of the galaxy–halo connection, we use
subhalo abundance matching (SHAM, Kravtsov et al., 2004; Conroy et al., 2006)
as a toy model to demonstrate our framework; in §2, we provide a self-
contained overview of SHAM. We train a neural network, SHAMNet, that serves as
a surrogate function approximating the SHAM mapping between stellar mass and
halo mass; we describe our implementation of SHAMNet in §3, relegating
technical details to the appendices. Although any suitably formulated neural
network is naturally differentiable, this is not the case for typical
pipelines that make downstream predictions for the $n-$point functions of
large scale structure; in §4, we describe our framework for constructing
differentiable point estimators of the stellar mass function, galaxy-galaxy
lensing, and galaxy clustering. We discuss our results in the broader context
of related efforts in the literature in §5, and we conclude by summarizing our
primary findings in §6.
Throughout the paper, values of halo mass and distance are quoted assuming
$h=1.$ For example, when writing $M_{\rm peak}=10^{12}M_{\odot},$ we suppress
the $M_{\odot}/h$ notation and write the units as $M_{\odot}.$ Values of
stellar mass are quoted assuming $h=0.7.$
## 2 Abundance Matching Primer
The $M_{\star}-M_{\rm halo}$ relation in traditional abundance matching is the
unique, non-parametric mapping defined by equating the abundance of galaxies
to the abundance of halos at the same redshift,
$\displaystyle\Phi_{\rm g}(>M_{\star}|z)=\Phi_{\rm h}(>M_{\rm halo}|z),$ (1)
where $\Phi_{\rm g}(>M_{\star})$ is the cumulative number density of galaxies,
and $\Phi_{\rm h}(>M_{\rm halo})$ is the cumulative number density of
subhalos. Contemporary forms of SHAM generalize Eq.1 to define non-parametric
mappings between some subhalo property, $x_{\rm h},$ and some observed galaxy
property, $y_{\rm g}.$ Application of the defining abundance matching equation
guarantees that no matter the choice for $x_{\rm h}$ and $y_{\rm g},$ the
observed and predicted number density of galaxies will be in exact
correspondence, by construction. We restrict the present investigation to
$x_{\rm h}=M_{\rm peak}$ and $y_{\rm g}=M_{\star},$ and refer the reader to §5
for discussion of how our methodology could be extended to generalized
galaxy/halo properties.
Even in the early literature on abundance matching (e.g., Tasitsiomi et al.,
2004), the importance of stochasticity in the $M_{\star}-M_{\rm halo}$
relation was recognized to play an important role in the predictions of the
model. Stochasticity in abundance matching is typically treated by assuming a
model for the probability density $P(M_{\star}|M_{\rm peak})$ such as a log-
normal, and then equating the number density of galaxies to the convolution of
the subhalo mass function against the assumed PDF :
$\displaystyle\phi_{\rm g}(M_{\star})=\int_{0}^{\infty}dM_{\rm peak}\phi_{\rm
h}(M_{\rm peak})P(M_{\star}|M_{\rm peak}),$ (2)
where $\phi_{\rm g}(M_{\star})$ is the differential number density galaxies as
a function of stellar mass, i.e. the stellar mass function (SMF), and
$\phi_{\rm h}(M_{\rm peak})$ is the differential number density of subhalos as
a function of mass, i.e. the subhalo mass function (SHMF).
In general, the $M_{\star}-M_{\rm peak}$ relation defined by Equation 2 has no
closed-form analytic solution, and so one must rely on approximations and
numerical techniques to determine the scaling relation $\langle
M_{\star}|M_{\rm peak}\rangle_{\rm med}$ that gives rise to the observed SMF
when applied to the SHMF of a subhalo population. In §2.1, we discuss the
conventional, non-parametric approach to determining the $M_{\star}-M_{\rm
peak}$ relation predicted by SHAM, and in §2.2 we describe the commonly-used
alternative approach based on a parameterized approximation to SHAM, in both
cases highlighting the close relationship between the SMF, the median relation
$\langle M_{\star}|M_{\rm peak}\rangle_{\rm med},$ and scatter in stellar mass
at fixed halo mass.
### 2.1 Non-parametric SHAM with scatter
In non-parametric approaches to SHAM, the starting point is typically a
catalog of simulated subhalos that defines the SHMF, and a volume-limited
galaxy sample that defines the SMF. A particularly simple way to measure the
cumulative abundance of galaxies or halos is simply to divide each object’s
rank-order by the volume containing the sample, although one may instead use a
fitting function approximation to one or both abundance functions. However
$\phi_{\rm g}$ and $\phi_{\rm h}$ are characterized, in non-parametric SHAM
the SMF and SHMF are held fixed, and one numerically solves for the quantity
$P(M_{\star}|M_{\rm peak})$ that is constrained by Eq. 2. Thus when generating
a synthetic galaxy population with non-parametric SHAM, the stellar mass
function of the mock is guaranteed to match the observed SMF, by construction,
with an accuracy limited only by the convergence and robustness of the
computational technique used to numerically solve Eq. 2. In §2.1.1 we describe
two commonly used approaches to obtaining such a solution, and in §2.1.2 we
highlight the basic features of the stellar-to-halo mass relation derived from
the solution.
#### 2.1.1 Numerical Methods
The most widely used numerical approach to solving Eq. 2 is based on
Richardson-Lucy deconvolution (Richardson, 1972; Lucy, 1974), an iterative
algorithm originally developed to recover a true underlying image that has
been blurred by a point-spread function (PSF). In the context of abundance
matching, we can think of the “noisy image” as $\phi_{\rm g}(M_{\star}),$
i.e., the SMF that we measure in our observed galaxy sample; the PSF blurring
the image is typically assumed to be a log-normal distribution in stellar mass
at fixed halo mass; the RL deconvolution algorithm determines the “true
image”, $\phi^{\rm true}_{\rm g}(M_{\star}),$ which in this case is the SMF
that one would measure in the absence of any scatter in the $M_{\star}-M_{\rm
halo}$ relation. Once $\phi^{\rm true}_{\rm g}(M_{\star})$ is determined, then
Eq. 1 is used to define $\langle M_{\star}|M_{\rm peak}\rangle_{\rm med};$
together with the assumed level of log-normal scatter, this scaling relation
then defines the quantity $P(M_{\star}|M_{\rm peak})$ that is used to map
stellar mass onto a simulated subhalo population. In the conventional Monte
Carlo based implementation, a synthetic galaxy population is generated by
randomly drawing a value of $M_{\star}$ from $P(M_{\star}|M_{\rm peak})$ for
each subhalo; by construction, the resulting SMF of the synthetic population
will agree with the SMF of the observed galaxy sample, $\phi_{\rm
g}(M_{\star}),$ again with accuracy limited only by the RL deconvoluation-
based estimation of $\phi^{\rm true}_{\rm g}(M_{\star})$ (see Behroozi et al.,
2010, Section 3, for further details).
An alternative approach presented in Kravtsov et al. (2018) is instead based
on the assumption that the $M_{\star}-M_{\rm peak}$ relation can be locally
approximated as a power law. Under this assumption, for any particular value
of $M_{\rm peak}$ and level of scatter, it is straightforward to solve for the
power-law normalization and index that produces the observed $\phi_{\rm
g}(M_{\star})$ from the simulated $\phi_{\rm h}(M_{\rm peak}).$ By repeating
this exercise at a finite set of control points in $M_{\rm peak}$ that densely
spans the relevant range, one can use the results of the computation as an
interpolation table that defines the $\langle M_{\star}|M_{\rm
peak}\rangle_{\rm med}$ scaling relation (see Kravtsov et al., 2018, Appendix
A, for further details).
#### 2.1.2 Stellar-to-Halo Mass Relation of Non-Parameteric SHAM
Figure 1 gives a simple demonstration of the role of scatter in non-parametric
abundance matching. Each curve shows the abundance matching prediction for
$\langle M_{\star}|M_{\rm peak}\rangle_{\rm med}$ using the same stellar mass
function, and the same subhalo mass function, but with different levels of
scatter as indicated in the legend. For our SMF, $\phi_{\rm fid},$ we use a
Schechter function with best-fit parameters taken from Panter et al. (2007),
and for the subhalo mass function we use the fitting function presented in
Appendix A. We calculate $\langle M_{\star}|M_{\rm peak}\rangle_{\rm med}$
using a publicly available python
wrapper111https://bitbucket.org/yymao/abundancematching/src of the RL
deconvolution implementation originally developed in Behroozi et al. (2010).
All curves in Figure 1 present the same qualitative shape of a double power-
law, with stronger levels of scatter producing shallower power-law relations
at the high-mass end. We can understand this characteristic flattening in
terms of Eddington bias (Eddington, 1913): stochasticity in the
$M_{\star}-M_{\rm peak}$ relation will naturally result in proportionally more
up-scatter above a threshold relative to down-scatter below the threshold,
simply because low-mass halos are more abundant than high-mass halos. Thus it
is sensible that we see stronger effects of scatter at high mass, where the
slope of the mass function is rapidly steepening. We return to this issue in
the subsequent section on parametric approximations to abundance matching.
Figure 1: Role of scatter in non-parametric SHAM. Each curve shows the
$M_{\star}-M_{\rm halo}$ relation of a SHAM model that predicts the same
stellar mass function, $\phi_{\rm fid}(M_{\star}),$ but with different levels
of scatter in $M_{\star}$ at fixed halo mass, as indicated in the legend. For
each model, the median relation $\langle M_{\star}|M_{\rm peak}\rangle_{\rm
med}$ is plotted on the vertical axis as a function of halo mass, $M_{\rm
peak}.$ Figure 2: Role of scatter in parameterized abundance matching. Each
curve shows the stellar mass function predicted by a parameterized abundance
matching model with the same $\langle M_{\star}|M_{\rm peak}\rangle_{\rm
med},$ but with different levels of scatter in $M_{\star}$ at fixed halo mass.
For each model, the number density of galaxies, $\phi(M_{\star}),$ is plotted
on the vertical axis as a function of stellar mass.
### 2.2 Parametric Approximations to SHAM with scatter
In studies of the galaxy-halo connection connecting stellar mass to halo mass,
numerous models have directly parameterized the scaling relation $\langle
M_{\star}|M_{\rm peak}\rangle_{\rm med},$ rather than using numerical methods
of solution to Eq. 2 (e.g., Moster et al., 2010, 2013; Behroozi et al., 2013a;
Rodríguez-Puebla et al., 2015). In such models, the assumed functional forms
for $\langle M_{\star}|M_{\rm peak}\rangle_{\rm med}$ all have the same
general double power-law shape shown in Figure 1; the free parameters of these
models regulate the low- and high-mass slopes, as well as the normalization
and shape of the transition between the two regimes.
It has been shown explicitly that the $\langle M_{\star}|M_{\rm
peak}\rangle_{\rm med}$ predicted via RL deconvolution can indeed be
accurately approximated with the functional forms used in Behroozi et al.
(2010) and related studies, and so it is common for such parametric models to
be informally referred to as “abundance matching.” However, we point out a
clear distinction in the way these two models utilize information from
observations: in parameterized models of $\langle M_{\star}|M_{\rm
peak}\rangle_{\rm med},$ there is no guarantee that the observed stellar mass
function will be correctly predicted, as is the case with non-parametric
abundance matching. Instead, in parameterized models, observations of the SMF
are treated as constraining data that can be used to derive confidence
intervals on the parameters controlling the stellar-to-halo mass relation.
Thus parametric vs. non-parametric SHAM models use information from
observations in quite a different manner when evaluating the likelihood of a
proposed point in parameter space.
Figure 2 illustrates the role of scatter in predictions for the stellar mass
function made by parameterized abundance matching. The axes show the number
density of galaxies as a function of stellar mass, $\phi(M_{\star}).$ Each
curve shows results for models with different levels of scatter about the same
$M_{\star}-M_{\rm halo}$ relation. Again we see the proportionally larger
influence of Eddington bias at higher mass, due to up-scatter from lower-mass
halos that outnumber higher-mass halos.
## 3 SHAMNet
In this section, we describe a new approach to abundance matching based on
SHAMNet: a neural network approximation to the numerical solution to Eq. 2. We
begin in §3.1 with a high-level overview of how SHAMNet is defined. In §3.2 we
describe the key ingredients we use in our implementation, and in §3.3 we
describe how we have used these ingredients to train SHAMNet to approximate
the abundance matching prediction for the SMHM scaling relation, $\langle
M_{\star}|M_{\rm peak}\rangle_{\rm med}.$ We give a detailed account of the
architecture and training of SHAMNet in the appendices; our source code is
publicly available at https://github.com/ArgonneCPAC/shamnet, and is available
for installation with pip.
### 3.1 SHAMNet Overview
As discussed in §2, the SHAM technique defines the probability distribution
$P(M_{\star}|M_{\rm peak})$ such that when this PDF is convolved against the
subhalo mass function, $\phi_{\rm h}(M_{\rm peak}),$ the galaxy stellar mass
function, $\phi_{\rm g}(M_{\star}),$ is recovered. Under the assumption of
log-normal scatter, this PDF is fully described by its first and second
moments, $\langle M_{\star}|M_{\rm peak}\rangle_{\rm med}$ and $\sigma(M_{\rm
peak}),$ respectively. Thus the goal of any SHAM implementation is to accept
ingredients for $\phi_{\rm g}(M_{\star}),$ $\phi_{\rm h}(M_{\rm peak}),$ and
$\sigma(M_{\rm peak})$ as inputs, and to return the scaling relation $\langle
M_{\star}|M_{\rm peak}\rangle_{\rm med}$ that provides a solution to Eq. 2.
SHAMNet is simply a neural network that provides a mapping from these
ingredients to the desired stellar-to-halo mass relation.
In building SHAMNet, we assume that the observed SMF can be characterized with
sufficient precision using some parametrized functional form, $\phi_{\rm
g}(M_{\star}|\theta_{1});$ we similarly assume that the subhalo mass function
can be parametrically described as $\phi_{\rm h}(M_{\rm peak}|\theta_{2});$
finally, we assume log-normal scatter in stellar mass at fixed halo mass,
allowing the level of scatter, $\sigma,$ to be a parametrized function of halo
mass, $\sigma(M_{\rm peak}|\theta_{3}).$ For notational convenience, we will
use the generic variable $\theta$ to refer to the collection of these
parameters, so that $\theta$ fully specifies $\phi_{\rm g}(M_{\star}),$
$\phi_{\rm h}(M_{\rm peak}),$ and $\sigma(M_{\rm peak}).$ SHAMNet is defined
to be a neural network that accepts $\theta$ and $M_{\rm peak}$ as input, and
returns $M_{\star}$ as output; the network parameters of SHAMNet are trained
so that Eq. 2 is satisfied by the mapping.
### 3.2 SHAMNet ingredients
Our formulation of SHAMNet requires a parametric description of $\phi_{\rm
g}(M_{\star}),$ $\phi_{\rm h}(M_{\rm peak}),$ and $\sigma(M_{\rm peak}).$ In
this section, we describe our models for each of these ingredients in turn.
Our parameterization of $\phi_{\rm g},$ the differential number density of
galaxies, is based on the Schechter function, $\phi_{\rm S},$ defined as
(3) $\displaystyle\phi_{\rm S}(x|x_{\ast},\phi_{\ast},\alpha)={\rm
ln}(10)\phi_{\ast}10^{(x-x_{\ast})\cdot(\alpha+1)}\exp(-10^{x-x_{\ast}}),$
where $x=\log_{10}M_{\star}$ in units of $M_{\odot}.$ Our galaxy SMF is thus
characterized by $x_{\ast},\phi_{\ast},$ and $\alpha,$ and we define SHAMNet
with a fixed value $x_{\ast}=10.85,$ closely mimicking the SMF in the low-
redshift universe (Li & White, 2009).
We parameterize the subhalo mass function in terms of the cumulative number
density of subhalos, $\Phi_{\rm h}(>M_{\rm peak}|z),$ using the fitting
function described in Appendix A. Briefly, $\Phi_{\rm h}(>M_{\rm peak}|z)$
behaves like a power-law at low mass, with a normalization $A_{\rm h},$ and an
index $\beta_{\rm h},$ with an exponential cutoff the high-mass end
characterized by a cutoff mass, $x_{\rm h},$ and cutoff speed, $k_{\rm h}.$ In
defining SHAMNet, we hold the subhalo mass function parameters fixed to the
values supplied in Appendix A that have been tuned to closely match the SHMF
in the Bolshoi-Planck simulation (BPL, Klypin et al., 2011).
We model the scatter in $M_{\star}$ at fixed $M_{\rm peak}$ as a log-normal
distribution with a width that is allowed to vary as a function of halo mass,
$\sigma(M_{\rm peak}).$ For the mass-dependence of the scatter, we use the
sigmoid function given in 21, with the parameters $y^{\sigma}_{\rm lo}$ and
$y^{\sigma}_{\rm hi}$ specifying the level of scatter at the low- and high-
mass end, respectively, and with $k^{\sigma}=1$ and $x^{\sigma}_{0}=12$ held
fixed:
$\displaystyle\sigma(x|y_{\rm lo}^{\sigma},y_{\rm
hi}^{\sigma})=\mathcal{S}(x,12,1,y_{\rm lo}^{\sigma},y_{\rm hi}^{\sigma}),$
(4)
where $x=\log_{10}M_{\rm peak}.$
### 3.3 SHAMNet Definition
With the ingredients defined in §3.2, the parameters $\theta$ collectively
describe our model for the SMF, $\phi_{\rm g}(M_{\star}|\theta),$ the subhalo
mass function, $\phi_{\rm h}(M_{\rm peak}|\theta),$ and the level of log-
normal scatter, $\sigma(M_{\rm peak}|\theta).$ For any particular value of
$\theta,$ we seek to identify the stellar-to-halo mass relation, $\langle
M_{\star}|M_{\rm peak};\theta\rangle_{\rm med},$ that provides the self-
consistent relationship between these quantities (as defined by Eq. 2). We
define SHAMNet, $\mathcal{F}_{\rm SHAM},$ to be a neural network that has been
trained to supply the appropriate scaling relation:
$\displaystyle\mathcal{F}_{\rm SHAM}(M_{\rm peak},\theta|\psi)\equiv\langle
M_{\star}|M_{\rm peak};\theta\rangle_{\rm med},$ (5)
where the variables $\psi$ are the weights and biases of the neural network.
Thus for any particular values of $\psi,$ the SHAMNet function
$\mathcal{F}_{\rm SHAM}$ accepts $M_{\rm peak}$ and $\theta$ as input, and
returns $M_{\star}$ as output.
The objective of training SHAMNet is to optimize the weights and biases $\psi$
such that the resulting mapping in Eq. 5 supplies a stellar-to-halo mass
relation that satisfies Eq. 2 to the required precision. Once trained, the
parameters $\psi$ are thereafter held fixed, and $\mathcal{F}_{\rm
SHAM}(M_{\rm peak},\theta)=M_{\star}$ defines the stellar-to-halo mass
relation for any input combination of SMF, subhalo mass function, and scatter.
As shown in Figure 3, accurately training SHAMNet does not present a
significant challenge for a simply-connected network with only a few short
layers. On the vertical axis, we show the logarithmic difference between
target SMF defined by $\theta,$ and the SMF that results from the SHAMNet
prediction for $\langle M_{\star}|M_{\rm peak};\theta\rangle_{\rm med}.$ The
level of success shown in Fig. 3 should not be surprising, since neural
networks are routinely called upon to approximate far more complex functions
than abundance matching. We refer the reader to Appendix D for detailed
information on how we trained SHAMNet.
Figure 3: Accuracy of SHAMNet. The vertical axis shows the logarithmic
difference between the target stellar mass function and the SMF that results
from the SHAMNet prediction for the stellar-to-halo-mass relation. The red
band shows the variance in the residual error amongst 100 randomly selected
points in the SMF parameter space; the blue curves show the particular
residuals for 10 of these points.
Beyond its accuracy and convenience, the function $\mathcal{F}_{\rm
SHAM}(M_{\rm peak},\theta)$ is an analytically differentiable function of the
parameters $\theta,$ because the behavior of the function is simply the
composition of a chain of $C^{\infty}$ functions. As described in Appendix D,
our implementation of SHAMNet in the JAX library makes it straightforward and
efficient to compute these gradients via automatic differentiation. In §4
below, we describe how these SHAMNet gradients can be propagated to permit
calculation of derivatives of large scale structure observables such as two-
point clustering and lensing.
## 4 Differentiable Predictions for Large Scale Structure
In §3, we described how SHAMNet, $\mathcal{F}_{\rm SHAM},$ serves as a
surrogate function that approximates the stellar-to-halo mass relation of
abundance matching, $\mathcal{F}_{\rm SHAM}(M_{\rm peak}|\theta)\equiv\langle
M_{\star}|M_{\rm peak};\theta\rangle_{\rm med}.$ We remind the reader that the
parameters $\theta$ control the behavior of the galaxy stellar mass function,
$\phi_{\rm g}(M_{\star}|\theta),$ the subhalo mass function, $\phi_{\rm
h}(M_{\rm peak}|\theta),$ and halo mass-dependent scatter, $\sigma(M_{\rm
peak}|\theta).$ For the remainder of the paper, our principal focus will be on
leveraging the differentiability of $\mathcal{F}_{\rm SHAM}(M_{\rm
peak}|\theta)$ with respect to the parameters $\theta.$ Physically, the
gradients $\partial\mathcal{F}_{\rm SHAM}/\partial\theta$ encode how the
stellar-to-halo mass relation changes in response to changes in $\phi_{\rm
g}(M_{\star}),$ $\phi_{\rm h}(M_{\rm peak}),$ and $\sigma(M_{\rm peak}).$
In conventional SHAM implementations, the scaling relation $\langle
M_{\star}|M_{\rm peak};\theta\rangle_{\rm med}$ is computed numerically via an
iterative algorithm such as RL deconvolution, and so one must rely on finite
differencing methods to estimate the gradients of the scaling relation with
respect to the parameters $\theta.$ By contrast, our JAX implementation of
$\mathcal{F}_{\rm SHAM}$ allows us to calculate $\partial\langle
M_{\star}|M_{\rm peak};\theta\rangle_{\rm med}/\partial\theta$ to machine
precision with high efficiency. In this section, we describe a set of
techniques that will allow us to leverage the availability of these gradients
to directly differentiate simulation-based forward-modeling predictions for
large scale structure observables.
### 4.1 Stellar Mass Function
Simulation-based predictions for large scale structure observables are
typically based on Monte Carlo realizations of a model applied to a (sub)halo
catalog. Using such Monte Carlo methods, the SHAM prediction for the stellar
mass function proceeds in two steps:
1. 1.
Assign a value $M_{\star}$ to every subhalo in the catalog by randomly drawing
from a log-normal PDF centered at $\langle M_{\star}|M_{\rm peak}\rangle_{\rm
med}$ with scatter $\sigma.$
2. 2.
For the $i^{\rm th}$ bin of the SMF, $\phi_{\rm g}(M_{\star}^{\rm i}),$ sum
the number of synthetic galaxies with $M_{\star}^{\rm
i}<=M_{\star}<M_{\star}^{\rm i+1},$ and divide by the normalization factor
appropriate for the population volume and bin size, $V_{\rm i}.$
In order to calculate gradients of the predicted SMF with respect to model
parameters $\theta,$ the above procedure is repeated for parameter values
$\theta$ that have been perturbed about some fiducial value, and the gradient
is estimated via finite-differencing methods, e.g., $\Delta\phi_{\rm
g}(M_{\star})/\Delta\theta.$
Here we consider an alternative approach to predicting $\phi_{\rm
g}(M_{\star})$ that does not rely on a Monte Carlo realization. For the
$\alpha^{\rm th}$ subhalo in the catalog with halo mass $M_{\rm
peak}^{\alpha},$ we can analytically calculate the probability that the galaxy
mapped onto the subhalo will fall within the ${\rm i^{th}}$ bin:
$\displaystyle w_{\rm i}^{\alpha}$ $\displaystyle=$
$\displaystyle\int_{M_{\star}^{\rm i}}^{M_{\star}^{\rm i+1}}{\rm
dM_{\star}}P(M_{\star}|M_{\rm peak}^{\alpha}).$ (6)
Once all the weights $w_{\rm i}^{\alpha}$ have been calculated for each
subhalo in the catalog, then predicting the SMF is simply a matter of summing
the weights in each bin and normalizing the result:
$\displaystyle\phi_{\rm g}(M_{\star}^{\rm i})$ $\displaystyle=$
$\displaystyle\sum_{\alpha}w_{\rm i}^{\alpha}/V_{\rm i}.$ (7)
Figure 4: Gradients of simulation-based SMF predictions. In the top panel we
show three different SMFs, with different values of $\alpha$ as indicated in
the legend. In the bottom panel we show the logarithmic derivative of the SMF
with respect to $\alpha,$ calculating the result with both finite-differencing
methods and automatic differentiation.
Operationally, the only difference between these two approaches is that we
compute the integrals in Eq. 6 analytically rather than via Monte Carlo
integration. While Monte Carlo-based predictions will be statistically
commensurable with the SMF computed via Eq. 7, the analytical approach can be
more computationally efficient because numerical convergence may require a
large number of realizations, particularly at the high-mass end due to the
rarity of massive galaxies. Beyond this computational advantage, from
Equations 6-7 we can readily see that the gradient $\partial\phi_{\rm
g}(M_{\star})/\partial\theta$ can be calculated exactly, since it is simply a
sum of $C^{\infty}$ functions:
$\displaystyle\frac{\partial}{\partial\theta}\phi_{\rm g}(M_{\star}^{\rm
i})=\frac{1}{V_{\rm i}}\sum_{\alpha}\frac{\partial w_{\rm
i}^{\alpha}}{\partial\theta}$ (8)
In Figure 4, we show how the stellar mass function responds to changes with
respect to $\alpha,$ the parameter controlling the power-law slope of the
Schechter function defined in Eq. 3. In the top panel we show three different
SMFs, with different values of $\alpha$ as indicated in the legend. To compute
$\phi_{\rm g}(M_{\star}),$ we used Eq. 7 to calculate the weight attached to
every subhalo of the BPL simulation at $z=0,$ using 50 logarithmically-spaced
bins spanning $10^{9}M_{\odot}<M_{\star}<10^{11.5}M_{\odot}.$ In the bottom
panel of Figure 4, we show the logarithmic derivative of the SMF with respect
to $\alpha.$ For the curve labeled “finite differencing”, we calculated the
gradient numerically by repeatedly calculating the result of Eq. 7 for low and
high values of $\alpha.$ For the curve labeled “autodiff”, we use JAX to
calculate $\partial w_{\alpha}/\partial\theta$ for every subhalo, and then we
use numba to propagate these derivatives through to the computation of the SMF
according to Eq. 8.
### 4.2 Galaxy Lensing
In this section, we adapt the methods described in §4.1 to make differentiable
predictions for galaxy lensing, $\Delta\Sigma(R),$ the excess surface mass
density at a projected distance $R$ from the center of a stacked sample of
galaxies, defined as
$\displaystyle\Delta\Sigma(R)\equiv\bar{\Sigma}(<R)-\Sigma(R).$ (9)
In Eq. 9, $\Sigma(R)$ is the surface mass density projected along the line-of-
sight to the stack, and $\bar{\Sigma}(<R)$ is the value of $\Sigma$ averaged
over the cylinder interior to $R.$
Using the Halotools implementation of the technique presented in Lange et al.
(2019) and reviewed in Appendix F, the profiles $\Delta\Sigma(R)$ can be
computed in advance on a per-object basis for every subhalo in a simulated
snapshot. Once tabulated, the lensing signal produced by a stack of galaxies
can be computed as the average signal produced by each of the subhalos in the
sample,
$\displaystyle\langle\Delta\Sigma(R)\rangle=\langle\Delta\Sigma^{\alpha}(R)\rangle_{\alpha\in\
{\rm sample}},$ (10)
where $\Delta\Sigma^{\alpha}$ is the lensing of the $\alpha^{\rm th}$ subhalo
in the catalog.
Traditional SHAM predictions for galaxy lensing are straightforward to compute
using standard Monte Carlo methods. First, a value of $M_{\star}^{\alpha}$ is
mapped onto every subhalo in the snapshot according to a random draw from the
appropriate log-normal. The SHAM prediction for the lensing signal produced by
a stack of galaxies is then given simply by Eq. 10, taking the average over
only those galaxies with randomly drawn values of $M_{\star}^{\alpha}$ that
satisfy the desired selection criteria, e.g., $M_{\star}^{\rm
lo}<M_{\star}^{\alpha}<M_{\star}^{\rm hi}.$
Figure 5: Gradients of simulation-based $\Delta\Sigma$ predictions. In the top
panel, we show $\Delta\Sigma$ for galaxies with $M_{\star}>10^{11}M_{\odot}$
residing in subhalos in the BPL simulation, with different curves
corresponding to values of the parameters controlling mass-dependent scatter,
as indicated in the legend. In the bottom panel we show the logarithmic
derivative of $\Delta\Sigma,$ calculating the result with both finite-
differencing methods and automatic differentiation.
As a differentiable alternative to Monte Carlo-based predictions, we can
instead compute the left-hand side of Eq. 10 as a weighted average, where the
weight attached to each subhalo, $w_{\alpha},$ is the probability that the
galaxy passes the selection function, calculated according to Eq. 6:
$\displaystyle\langle\Delta\Sigma(R)\rangle=\frac{\sum_{\alpha}w_{\alpha}\cdot\Delta\Sigma^{\alpha}(R)}{\sum_{\alpha}w_{\alpha}},$
(11)
where the summations in Eq. 11 are performed over all subhalos with non-zero
probability of residing in the stellar mass bin. In the limit of an infinite
simulated volume, Eq. 10 converges to Eq. 11, but the latter method has the
advantage of being differentiable with respect to parameters $\theta$ of the
galaxy–halo connection:
$\displaystyle\frac{\partial}{\partial\theta}\langle\Delta\Sigma(R)\rangle=\sum_{\alpha}\frac{\partial}{\partial\theta}\left(\frac{w_{\alpha}}{\sum_{\alpha}w_{\alpha}}\right)\Delta\Sigma^{\alpha}(R).$
(12)
To compute the left-hand side of Eq. 12, we use JAX to calculate the quantity
$\frac{\partial}{\partial\theta}(w_{\alpha}/\sum_{\alpha}w_{\alpha})$ for each
simulated subhalo, and we then we use numba to perform the weighted-sum on the
right-hand-side of Eq. 12 using the precomputed lensing profile of each
individual subhalo.
In the top panel of Figure 5, we show a few examples of computations of
$\Delta\Sigma$ for galaxies with $M_{\star}>10^{11}M_{\odot}$ based on the BPL
simulation, with different curves corresponding to different values of the
parameters controlling mass-dependent scatter. For the fiducial model shown
with the black curve, we use $(\sigma_{\rm lo},\sigma_{\rm hi})=(0.4,0.25);$
results based on independent perturbations to $\sigma_{\rm lo}$ and
$\sigma_{\rm hi}$ are color-coded as indicated in the legend. In the bottom
panel of Figure 5, we show gradients of $\Delta\Sigma$ with respect to
$\sigma_{\rm lo}$ and $\sigma_{\rm hi},$ again comparing the results of the
computation based on autodiff vs. finite-differencing methods. Due to the
power-law shape of the subhalo mass function, larger scatter in the SMHM at
any mass corresponds to a lower amplitude of $\Delta\Sigma,$ since increasing
the scatter results in a higher proportion of low-mass, weakly-clustered
subhalos that up-scatter into the sample. The effect on $\Delta\Sigma$ for
this relatively massive galaxy sample is more pronounced for $\sigma_{\rm
hi};$ this is sensible, since at high mass the slope of the mass function is
falling off rapidly, and the halo bias function is rapidly steepening.
### 4.3 Galaxy Clustering
The computation of differentiable two-point functions proceeds in much the
same way as that for galaxy lensing. Each object again has a probability
$w_{\alpha}$ of being in the sample. The product of the weights,
$w_{\alpha}\cdot w_{\beta}$, is interpreted as the probability that a given
pair of objects would be counted when accumulating the number of pairs at a
given separation for, e.g., computing a two-point function. Formally, we are
assuming that when conditioned on the values of the weights themselves, the
presence of any given object in the sample is independent of all of the
others. Finally, there exists some ambiguity in how exactly to define an
unclustered or “random” sample given a set of objects with weights. We define
the random sample as a set of points with a random spatial distribution, each
with a weight that has been randomly assigned from the original sample. We
show below that this definition properly reproduces zero clustering signal for
samples where points are distributed randomly, and only kept with a
probability equal to the weight.
With these assumptions, we use the following estimator for the two-point
correlation function
$\displaystyle\hat{\xi}_{i}=\frac{DD_{i}}{RR_{i}}-1$ (13)
where
$\displaystyle
DD_{i}=\sum_{\alpha}\sum_{\beta\neq\alpha}w_{\alpha}w_{\beta}{\cal
S}_{i}(x_{\alpha},x_{\beta})$ (14)
and
$\displaystyle
RR_{i}=\left(\sum_{\alpha}w_{\alpha}\right)^{2}\left(1-\frac{1}{N_{\rm
eff}}\right)\frac{V_{i}}{V}\ .$ (15)
Here ${\cal S}_{i}(x_{\alpha},x_{\beta})$ is the bin selection function, which
is unity if points $\alpha$ and $\beta$ are separated such that their
separation falls in bin $i$, and zero otherwise; $N$ is the total number of
points in the sample, $V_{i}$ is the volume of the bin, and $V$ is the total
volume of the domain; finally, the quantity $N_{\rm eff}$ is defined as:
$\displaystyle N_{\rm
eff}\equiv\frac{\left(\sum_{\alpha}w_{\alpha}\right)^{2}}{\sum_{\alpha}w_{\alpha}^{2}},$
(16)
so that $N_{\rm eff}$ is interpreted as the effective sample size.
To better understand the role of the denominator, $RR_{i},$ we can rewrite the
estimator as
$\displaystyle
RR_{i}=\left(\sum_{\alpha}w_{\alpha}\left(\sum_{\beta}w_{\beta}-w_{\alpha}\right)\right)\times\frac{V_{i}}{V}$
(17)
In this form, we can see that this estimator directly computes the expected
pair counts, excluding self-pairs, if we have randomly assigned the weights to
the random sample. This estimator is equivalent to the
$\mathcal{O}(\frac{1}{N})$ corrections for finite samples described in the
Corrfunc documentation (Sinha & Garrison, 2020), but with weights that
represent membership probabilities. Finally, the factor $V_{i}/V$ is the
fraction of the total volume occupied by the $i^{\rm th}$ bin.
Figure 6: Gradients of simulation-based $w_{\rm p}$ predictions. Same as Fig.
5, but for $w_{\rm p}({\rm r_{p}}).$ In the top panel, different curves show
$w_{\rm p}({\rm r_{p}})$ for different values of the parameters controlling
mass-dependent scatter, as indicated in the legend. In the bottom panel we
show the logarithmic derivative of $w_{\rm p}({\rm r_{p}}),$ calculating the
result with both finite-differencing methods and automatic differentiation.
From this definition of the two-point clustering estimator, we can use the
chain rule to compute the derivative,
$\displaystyle\frac{\partial\hat{\xi}_{i}}{\partial\theta}=\frac{1}{RR_{i}}\frac{\partial
DD_{i}}{\partial\theta}-\frac{DD_{i}}{RR_{i}^{2}}\frac{\partial
RR_{i}}{\partial\theta}$ (18)
where
(19) $\displaystyle\frac{\partial
DD_{i}}{\partial\theta}=\sum_{\alpha}\sum_{\beta\neq\alpha}\left[\frac{\partial
w_{\alpha}}{\partial\theta}w_{\beta}+w_{\alpha}\frac{\partial
w_{\beta}}{\partial\theta}\right]{\cal S}_{i}(x_{\alpha},x_{\beta}),$
and
$\displaystyle\frac{\partial RR_{i}}{\partial\theta}$ $\displaystyle=$
$\displaystyle\left(\sum_{\alpha}\left[\frac{\partial
w_{\alpha}}{\partial\theta}\left(\sum_{\beta}w_{\beta}-w_{\alpha}\right)\right.\right.$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \
\left.\left.+w_{\alpha}\left(\sum_{\beta}\frac{\partial
w_{\beta}}{\partial\theta}-\frac{\partial
w_{\alpha}}{\partial\theta}\right)\right]\right)\times\frac{V_{i}}{V}$
We assume here that the parameters $\theta$ exclusively impact the value of
the weights mapped onto each object. Should the object positions also depend
on $\theta$, additional terms proportional to the spatial derivative of the
bin selection function ${\cal S}_{i}(x_{\alpha},x_{\beta})$ would arise. We
leave the treatment of such contributions to the derivative to future work
exploring models that parameterize changes to galaxy position.
We note that the expressions derived in this section can be quite useful in
cases where one has existing weighted pair-counting code to efficiently
compute two-point functions. In this case, an autodiff library such as JAX can
be used to calculate the per-object weights, $w_{\alpha},$ and the gradients
$\partial w_{\alpha}/\partial\theta;$ subsequently, the pair-counting code can
proceed with its usual computation, using the equations derived in this
section to transform the results into a computation of a two-point function
and its exact derivatives. In carrying out the calculations in this section,
we use used the corrfunc code for this purpose.
In the top panel of Figure 6, we show $w_{\rm p}({\rm r_{p}})$ for galaxies
with $M_{\star}>10^{11}M_{\odot}$ based on the BPL simulation, with the same
fiducial scatter model and perturbations used in Fig. 5. In the bottom panel
of Figure 6, we show gradients of $w_{\rm p}({\rm r_{p}})$ with respect to
$\sigma_{\rm lo}$ and $\sigma_{\rm hi}.$ We see the same trends in Figure 6 as
we saw in Fig. 5: increasing scatter in the SMHM decreases clustering
strength, with more pronounced effects from scatter at high mass.
## 5 Discussion & Future Work
We have presented a new approach to making simulation-based predictions of the
galaxy–halo connection. In the conventional methodology, synthetic galaxy
populations are generated from a stochastic Monte Carlo realization of some
probability distribution defined by the underlying model. For example, in most
abundance matching studies, stellar masses are drawn from a realization of a
log-normal PDF; the synthetic galaxy at the center of each subhalo is then
assigned to a unique bin of stellar mass (or otherwise discarded from the
sample), and predictions for clustering and lensing are made from the
resulting bins of point data (as in, e.g., Reddick et al., 2013; Hearin et
al., 2013). By contrast, the approach taken here is not based on Monte Carlo
realizations. Instead, every subhalo makes a PDF-weighted contribution to each
stellar mass bin, and the statistical estimators used to calculate summary
statistics are defined in terms of the weighted point data.
Previous implementations of the galaxy–halo connection have also opted in
favor of weighted summary statistics over hard-edged bins filled with noisy
Monte Carlo realizations. For example, a variation on this technique was used
in Reid et al. (2014) in their treatment of fiber collisions; the
UniverseMachine source code used PDF-weighting methods in the computation of
its DR1 summary statistics (Behroozi et al., 2019); these same techniques form
the basis of the TabCorr source code222https://github.com/johannesulf/TabCorr,
as well as a variety of previous works on the galaxy–halo connection that
utilize pre-computation methods (e.g., Neistein & Khochfar, 2012; Zheng & Guo,
2016).
Although implementations of the galaxy–halo connection utilizing pre-
computation methods also enjoy the same reduction of stochasticity as the
approach presented here, our framework has two distinct advantages that make
it more extensible to the problem sizes of cosmological modeling in the 2020s.
First, conventional pre-computation methods are implemented by tabulating a
set of results over a grid of (sub)halo properties, and then linearly
interpolating from the resulting lookup table. While this methodology has
proven to be quite effective in deriving constraints on cosmological
parameters (e.g., Lange et al., 2021), extending these techniques to models
that depend upon more than one or two (sub)halo properties would be quite
challenging, since implementations based on lookup-table interpolation have
memory demands that rise sharply with the dimension of the table. Our use of
AI resolves this issue of excessive memory demands, in essence by storing the
lookup-table information in the weights and biases of a neural network.
Second, our predictions for large scale structure are differentiable. Although
we achieve this property in part due to our use of PDF-weighted summary
statistics, as discussed above this is not the novel feature of our
methodology. As described in §4, our predictions for the stellar mass
function, galaxy–galaxy lensing, and galaxy clustering are differentiable as a
result of the analytical propagation of the gradients of each synthetic
galaxy’s weight to the point estimators of the summary statistics. The
availability of gradient information is the central feature that enables
modern Bayesian inference techniques such as Hamiltonian Monte Carlo to derive
converged posteriors for models of hundreds of parameters (Hoffman & Gelman,
2014), and the extensibility of our framework to physical models of higher
dimension is one of the principal motivations underlying this paper.
Our paper focuses on the most widely-used 1- and 2-point functions in
cosmology, but our framework could naturally extend to higher-order summary
statistics. In some cases, virtually no adaptation of the computations
detailed in §4 would be required. For example, both the under-density
probability function (UPF) and counts-in-cells (CIC) summary statistics have
been shown to be sensitive probes of galaxy–halo information that is not
contained in galaxy clustering or lensing (Tinker et al., 2008; Wang et al.,
2019). Both of these summary statistics are defined in terms of number counts
of galaxies residing within some enclosing volume. If the UPF and CIC
estimators are instead defined in terms of the total sum of galaxy weights
residing in the enclosing volume, the predictions for these statistics become
differentiable using the same techniques in §4. Recent work has shown that the
information content of $n-$point functions of arbitrarily high order is
formally contained in summary statistics based on $k$ nearest-neighbor ($k$NN)
computations (Banerjee & Abel, 2021a, b); this approach to extracting
information from the density field is an attractive alternative to using
higher-order $n-$point functions directly, since the computation of $k$NN-
based summary statistics scales like $\mathcal{O}(N\log N);$ moreover, it has
also recently been shown that $k$NN-based measurements are sensitive probes of
the galaxy–halo connection (Behroozi et al., 2021). In the standard
computation of these summary statistics, the estimator is defined in terms of
the spatial distance to the $N^{\rm th}-$nearest member of a galaxy sample
defined by some hard-edged bin or threshold; in the corresponding
differentiable version, one would instead use the smallest distance for which
the sum of galaxy weights exceeds some chosen value. We relegate the extension
of our framework to these and other higher-order summary statistics to future
work.
Our work is closely related to Horowitz et al. (2021), who have implemented
DiffHOD, a differentiable form of the Halo Occupation Distribution (HOD).
Whereas the HOD populates host dark matter halos with variable numbers of
satellites, SHAM populates each simulated subhalo with a single synthetic
galaxy. Nonetheless, the theoretical roots of these two empirical models are
tightly connected (Kravtsov et al., 2004), and DiffHOD and SHAMNet are part of
a growing trend of differentiable formulations of galaxy–halo connection
models. In these two examples, the differentiability of the model predictions
is achieved through a different methodology. In SHAMNet, the derivatives of
the galaxy–halo connection model are propagated through analytical PDFs of the
galaxy residing in each halo, so that SHAMNet predictions have no
stochasticity; by contrast, DiffHOD uses a differentiable form of stochastic
sampling based on a Gumbel-Softmax distribution, coupled with an annealing
technique (Jang et al., 2016).
The GalaxyNet model (Moster et al., 2020) is also a differentiable form of the
galaxy–halo connection, and so the techniques presented here could naturally
be used to transform GalaxyNet predictions for clustering and lensing into
differentiable computations. Whereas DiffHOD and SHAMNet use AI-based methods
in a manner that mimics the scaling relations of simple empirical models,
GalaxyNet builds a highly nonlinear galaxy–halo connection with its neural
network, and so is a far more expansive application of AI. Our two-phase
training of SHAMNet is an adaption of the same technique used to train
GalaxyNet. As detailed in Appendix D, we trained SHAMNet by first carrying out
an initialization phase in which the weights and biases of the network were
tuned to reproduce an existing model; training then proceeded with a
subsequent phase in which the cost function was defined directly in terms of
target summary statistics of the galaxy distribution. The initialization phase
of SHAMNet was based on a flexible parametric form of the stellar-to-halo-mass
relation (see Appendix C), whereas GalaxyNet was initialized based on the
EMERGE model of star formation history (Moster et al., 2018).
Our primary interest in the differentiable techniques introduced here is their
application to more physically complex and higher-dimensional models than
abundance matching. In particular, in closely related work (Hearin et al.,
2021b), we have introduced diffmah, a differentiable model for the assembly of
dark matter halos; the diffmah model is the basis of a new approach to the
galaxy–halo connection, diffstar, in which parameterized star formation
histories (SFH) are statistically mapped onto the merger trees of dark matter
halos (Alarcon et al., 2021). Formulating the galaxy–halo connection in terms
of parameterized SFHs allows us to additionally forward model galaxy SEDs in a
differentiable fashion through the use of DSPS, a JAX-based implementation of
stellar population synthesis (Hearin et al., 2021a). Our focus on the benefits
of gradient information in likelihood analyses anticipates the expansion of
the dimension of the parameter space required by the complexity of these
models.
In the present work, we have used SHAM primarily as a toy model to demonstrate
how gradients of parameters of a galaxy–halo connection model propagate
through to derivatives of predictions for the summary statistics of large
scale structure. However, analyzing the galaxy distribution with SHAM is an
active area of research unto itself, and SHAMNet offers some improvements upon
conventional deconvolution-based abundance matching that may be useful for
such purposes. For example, the RL deconvolution algorithm outlined in §2 has
notoriously finicky convergence properties at the high-mass end, whereas
SHAMNet provides a robust solution to Eq. 2 even at very high mass (see Figure
3). More importantly, SHAMNet naturally incorporates mass-dependent scatter in
the stellar-to-halo mass relation, whereas all previous deconvolution-based
SHAM constraints have been derived under the assumption of constant scatter,
at least in part due to the technical challenge of incorporating this feature
into the RL-deconvolution algorithm.
With only modest extensions to SHAMNet, differentiable versions of
contemporary analyses could be applied to observational data. The present work
is based on $M_{\rm peak}-$based abundance matching, and so in order to
support abundance matching based on alternative mass proxies such as $V_{\rm
peak}$ (Reddick et al., 2013), $V_{\rm relax}$ (Chaves-Montero et al., 2016),
or more generalized proxies (Lehmann et al., 2017), it would be necessary to
retrain SHAMNet based on an alternative parameterization of the subhalo
abundance function. Although in §4 we focused on the gradients of the mass-
dependent scatter, we trained SHAMNet to additionally capture the dependence
of $\langle M_{\star}|M_{\rm peak}\rangle_{\rm med}$ upon parameters
specifying the SMF, and so our approach could also be used in abundance
matching formulations such as Saito et al. (2016) that incorporate uncertainty
in the SMF. In order to incorporate cosmology-dependence into abundance
matching, as in Contreras et al. (2021), an additional model providing a
mapping from cosmological parameters to the parameterized subhalo abundance
function would need to be developed. If an analysis required additional
marginalization over uncertainty in orphan abundance, as in DeRose et al.
(2021a), it would be necessary to develop an additional ingredient for how the
parameters of the orphan prescription change the parameters of the subhalo
abundance function.
With comparably modest adaptations, our framework could also be used to
conduct differentiable analyses of conditional abundance matching (CAM, Hearin
et al., 2014). While SHAM supplies a mapping from e.g., halo mass to stellar
mass, the CAM framework supplies a mapping from some secondary subhalo
property (such as a halo formation time proxy, as in Hearin & Watson, 2013;
Masaki et al., 2013), to some secondary galaxy property (such as specific star
formation rate, as in Watson et al., 2015). In order to make CAM
differentiable, rather than parameterizing the SMF and SHMF, one would instead
need to calibrate parameterized models for the conditional abundance of the
secondary halo and galaxy properties, and then use, e.g., the
conditional_abunmatch function in halotools to generate training data for
CAMNet. Although the version of SHAMNet that we trained in the present work
does not directly support these features, we have made our source code
publicly available to facilitate these and other adaptations in future work.
## 6 Summary
1. 1.
We have devised a new, differentiable framework for making simulation-based
predictions for large scale structure. Our approach is not based on stochastic
Monte Carlo realizations, but is instead formulated in terms of parameterized
PDFs that are mapped onto each simulated halo. We then use automatic
differentiation to propagate gradients of the galaxy–halo connection through
the point estimators used to measure summary statistics of the galaxy
distribution.
2. 2.
We have used a neural network, SHAMNet, to approximate the stellar-to-halo
mass relationship (SMHM) of abundance matching. Our JAX-based implementation
of SHAMNet is available for installation with pip. In addition to being
differentiable, SHAMNet captures variable levels of $M_{\rm halo}$-dependent
scatter in the SMHM, which is challenging to incorporate into conventional
deconvolution-based implementations of abundance matching.
3. 3.
Our source code is publicly available at
https://github.com/ArgonneCPAC/shamnet, and provides a set of recipes that can
be adapted to formulate other differentiable, AI-accelerated models of the
galaxy–halo connection.
## 7 Acknowledgements
We thank Peter Behroozi for useful discussions. APH thanks José Feliciano
again for Feliz Navidad.
We thank the developers of NumPy (Van Der Walt et al., 2011), SciPy (Jones et
al., 2001-2016), Jupyter (Ragan-Kelley et al., 2014), IPython (Pérez &
Granger, 2007), scikit-learn (Pedregosa et al., 2011), JAX (Bradbury et al.,
2018), numba (Lam et al., 2015), conda-forge (conda-forge community, 2015),
and Matplotlib (Hunter, 2007) for their extremely useful free software. While
writing this paper we made extensive use of the Astrophysics Data Service
(ADS) and arXiv preprint repository. The Bolshoi simulations used in this work
have been performed within the Bolshoi project of the University of California
High-Performance AstroComputing Center (UC-HiPACC) and were run at the NASA
Ames Research Center.
Work done at Argonne National Laboratory was supported under the DOE contract
DE-AC02-06CH11357. JD is supported by the Chamberlain Fellowship at Lawrence
Berkeley National Laboratory. We gratefully acknowledge use of the Bebop
cluster in the Laboratory Computing Resource Center at Argonne National
Laboratory.
## References
* Alarcon et al. (2021) Alarcon, A., Hearin, A., Becker, M., & Chaves-Montero, J. 2021, in prep
* Banerjee & Abel (2021a) Banerjee, A., & Abel, T. 2021a, MNRAS, 500, 5479, doi: 10.1093/mnras/staa3604
* Banerjee & Abel (2021b) —. 2021b, MNRAS, 504, 2911, doi: 10.1093/mnras/stab961
* Behroozi et al. (2021) Behroozi, P., Hearin, A., & Moster, B. P. 2021, arXiv:2101.05280, arXiv:2101.05280. https://arxiv.org/abs/2101.05280
* Behroozi et al. (2019) Behroozi, P., Wechsler, R. H., Hearin, A. P., & Conroy, C. 2019, MNRAS, 488, 3143, doi: 10.1093/mnras/stz1182
* Behroozi et al. (2010) Behroozi, P. S., Conroy, C., & Wechsler, R. H. 2010, ApJ, 717, 379, doi: 10.1088/0004-637X/717/1/379
* Behroozi et al. (2013a) Behroozi, P. S., Wechsler, R. H., & Conroy, C. 2013a, ApJ, 770, 57, doi: 10.1088/0004-637X/770/1/57
* Behroozi et al. (2013b) Behroozi, P. S., Wechsler, R. H., & Wu, H.-Y. 2013b, ApJ, 762, 109, doi: 10.1088/0004-637X/762/2/109
* Behroozi et al. (2013c) Behroozi, P. S., Wechsler, R. H., Wu, H.-Y., et al. 2013c, ApJ, 763, 18, doi: 10.1088/0004-637X/763/1/18
* Berlind & Weinberg (2002) Berlind, A. A., & Weinberg, D. H. 2002, ApJ, 575, 587, doi: 10.1086/341469
* Bernardeau et al. (2002) Bernardeau, F., Colombi, S., Gaztañaga, E., & Scoccimarro, R. 2002, Phys. Rep., 367, 1, doi: 10.1016/S0370-1573(02)00135-7
* Bocquet et al. (2020) Bocquet, S., Heitmann, K., Habib, S., et al. 2020, ApJ, 901, 5, doi: 10.3847/1538-4357/abac5c
* Bradbury et al. (2018) Bradbury, J., Frostig, R., Hawkins, P., et al. 2018, JAX: composable transformations of Python+NumPy programs, 0.2.10. http://github.com/google/jax
* Cacciato et al. (2013) Cacciato, M., van den Bosch, F. C., More, S., Mo, H., & Yang, X. 2013, MNRAS, 430, 767, doi: 10.1093/mnras/sts525
* Carrasco et al. (2012) Carrasco, J. J. M., Hertzberg, M. P., & Senatore, L. 2012, Journal of High Energy Physics, 2012, 82, doi: 10.1007/JHEP09(2012)082
* Chapman et al. (2021) Chapman, M. J., Mohammad, F. G., Zhai, Z., et al. 2021, arXiv:2106.14961, arXiv:2106.14961. https://arxiv.org/abs/2106.14961
* Chaves-Montero et al. (2016) Chaves-Montero, J., Angulo, R. E., Schaye, J., et al. 2016, MNRAS, 460, 3100, doi: 10.1093/mnras/stw1225
* conda-forge community (2015) conda-forge community. 2015, The conda-forge Project: Community-based Software Distribution Built on the conda Package Format and Ecosystem, Zenodo, doi: 10.5281/zenodo.4774216
* Conroy et al. (2006) Conroy, C., Wechsler, R. H., & Kravtsov, A. V. 2006, ApJ, 647, 201, doi: 10.1086/503602
* Contreras et al. (2021) Contreras, S., Chaves-Montero, J., Zennaro, M., & Angulo, R. E. 2021, arXiv e-prints, arXiv:2105.05854. https://arxiv.org/abs/2105.05854
* DeRose et al. (2021a) DeRose, J., Becker, M. R., & Wechsler, R. H. 2021a, arXiv e-prints, arXiv:2105.12104. https://arxiv.org/abs/2105.12104
* DeRose et al. (2021b) DeRose, J., Chen, S.-F., White, M., & Kokron, N. 2021b, arXiv:2112.05889, arXiv:2112.05889. https://arxiv.org/abs/2112.05889
* DES Collaboration et al. (2021) DES Collaboration, Abbott, T. M. C., Aguena, M., et al. 2021, arXiv e-prints, arXiv:2105.13549. https://arxiv.org/abs/2105.13549
* Desjacques et al. (2018) Desjacques, V., Jeong, D., & Schmidt, F. 2018, Phys. Rep., 733, 1, doi: 10.1016/j.physrep.2017.12.002
* Eddington (1913) Eddington, A. S. 1913, MNRAS, 73, 359, doi: 10.1093/mnras/73.5.359
* Euclid Collaboration et al. (2020) Euclid Collaboration, Knabenhans, M., Stadel, J., et al. 2020, arXiv:2010.11288, arXiv:2010.11288. https://arxiv.org/abs/2010.11288
* Foreman-Mackey et al. (2019) Foreman-Mackey, D., Farr, W., Sinha, M., et al. 2019, The Journal of Open Source Software, 4, 1864, doi: 10.21105/joss.01864
* García et al. (2021) García, R., Rozo, E., Becker, M. R., & More, S. 2021, MNRAS, doi: 10.1093/mnras/stab1317
* Hearin et al. (2021a) Hearin, A. P., Chaves-Montero, J., Alarcon, A., Becker, M. R., & Benson, A. 2021a, arXiv:2112.06830, arXiv:2112.06830. https://arxiv.org/abs/2112.06830
* Hearin et al. (2021b) Hearin, A. P., Chaves-Montero, J., Becker, M. R., & Alarcon, A. 2021b, arXiv e-prints, arXiv:2105.05859. https://arxiv.org/abs/2105.05859
* Hearin & Watson (2013) Hearin, A. P., & Watson, D. F. 2013, MNRAS, 435, 1313, doi: 10.1093/mnras/stt1374
* Hearin et al. (2014) Hearin, A. P., Watson, D. F., Becker, M. R., et al. 2014, MNRAS, 444, 729, doi: 10.1093/mnras/stu1443
* Hearin et al. (2013) Hearin, A. P., Zentner, A. R., Berlind, A. A., & Newman, J. A. 2013, MNRAS, 433, 659, doi: 10.1093/mnras/stt755
* Hearin et al. (2017) Hearin, A. P., Campbell, D., Tollerud, E., et al. 2017, AJ, 154, 190, doi: 10.3847/1538-3881/aa859f
* Heitmann et al. (2006) Heitmann, K., Higdon, D., Nakhleh, C., & Habib, S. 2006, ApJ, 646, L1, doi: 10.1086/506448
* Hoffman & Gelman (2014) Hoffman, M., & Gelman, A. 2014, Journal of Machine Learning Research, 15, 1593\. http://jmlr.org/papers/v15/hoffman14a.html
* Horowitz et al. (2021) Horowitz, B., Hahn, C., Lanusse, F., Modi, C., & Ferraro, S. 2021, in prep
* Hunter (2007) Hunter, J. D. 2007, Computing In Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
* Iman et al. (1981) Iman, R., Helton, J., & Campbell, J. 1981, Journal of Quality Technology, 13, 174, doi: 10.1080/00224065.1981.11978748
* Jang et al. (2016) Jang, E., Gu, S., & Poole, B. 2016, arXiv:1611.01144, arXiv:1611.01144. https://arxiv.org/abs/1611.01144
* Jenkins et al. (2001) Jenkins, A., Frenk, C. S., White, S. D. M., et al. 2001, MNRAS, 321, 372, doi: 10.1046/j.1365-8711.2001.04029.x
* Jones et al. (2001-2016) Jones, E., Oliphant, T., Peterson, P., et al. 2001-2016, http://www.scipy.org
* Joudaki et al. (2018) Joudaki, S., Blake, C., Johnson, A., et al. 2018, MNRAS, 474, 4894, doi: 10.1093/mnras/stx2820
* Kingma & Ba (2014) Kingma, D. P., & Ba, J. 2014, arXiv:1412.6980, arXiv:1412.6980. https://arxiv.org/abs/1412.6980
* Klambauer et al. (2017) Klambauer, G., Unterthiner, T., Mayr, A., & Hochreiter, S. 2017, arXiv:1706.02515, arXiv:1706.02515. https://arxiv.org/abs/1706.02515
* Klypin et al. (2011) Klypin, A. A., Trujillo-Gomez, S., & Primack, J. 2011, ApJ, 740, 102, doi: 10.1088/0004-637X/740/2/102
* Kokron et al. (2021) Kokron, N., DeRose, J., Chen, S.-F., White, M., & Wechsler, R. H. 2021, MNRAS, 505, 1422, doi: 10.1093/mnras/stab1358
* Krause & Eifler (2017) Krause, E., & Eifler, T. 2017, MNRAS, 470, 2100, doi: 10.1093/mnras/stx1261
* Kravtsov et al. (2004) Kravtsov, A. V., Berlind, A. A., Wechsler, R. H., et al. 2004, ApJ, 609, 35, doi: 10.1086/420959
* Kravtsov et al. (1997) Kravtsov, A. V., Klypin, A. A., & Khokhlov, A. M. 1997, ApJS, 111, 73, doi: 10.1086/313015
* Kravtsov et al. (2018) Kravtsov, A. V., Vikhlinin, A. A., & Meshcheryakov, A. V. 2018, Astronomy Letters, 44, 8, doi: 10.1134/S1063773717120015
* Kwan et al. (2015) Kwan, J., Heitmann, K., Habib, S., et al. 2015, ApJ, 810, 35, doi: 10.1088/0004-637X/810/1/35
* Lam et al. (2015) Lam, S. K., Pitrou, A., & Seibert, S. 2015, in Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC, 1–6
* Lange et al. (2021) Lange, J. U., Hearin, A. P., Leauthaud, A., et al. 2021, arXiv:2101.12261, arXiv:2101.12261. https://arxiv.org/abs/2101.12261
* Lange et al. (2019) Lange, J. U., Yang, X., Guo, H., Luo, W., & van den Bosch, F. C. 2019, MNRAS, 488, 5771, doi: 10.1093/mnras/stz2124
* Lawrence et al. (2017) Lawrence, E., Heitmann, K., Kwan, J., et al. 2017, ApJ, 847, 50, doi: 10.3847/1538-4357/aa86a9
* Lehmann et al. (2017) Lehmann, B. V., Mao, Y.-Y., Becker, M. R., Skillman, S. W., & Wechsler, R. H. 2017, ApJ, 834, 37, doi: 10.3847/1538-4357/834/1/37
* Li & White (2009) Li, C., & White, S. D. M. 2009, MNRAS, 398, 2177, doi: 10.1111/j.1365-2966.2009.15268.x
* Lucy (1974) Lucy, L. B. 1974, AJ, 79, 745, doi: 10.1086/111605
* Lupton et al. (1999) Lupton, R. H., Gunn, J. E., & Szalay, A. S. 1999, AJ, 118, 1406, doi: 10.1086/301004
* Masaki et al. (2013) Masaki, S., Lin, Y.-T., & Yoshida, N. 2013, MNRAS, 436, 2286, doi: 10.1093/mnras/stt1729
* McClintock et al. (2019a) McClintock, T., Rozo, E., Becker, M. R., et al. 2019a, ApJ, 872, 53, doi: 10.3847/1538-4357/aaf568
* McClintock et al. (2019b) McClintock, T., Rozo, E., Banerjee, A., et al. 2019b, arXiv:1907.13167, arXiv:1907.13167. https://arxiv.org/abs/1907.13167
* McKay et al. (1979) McKay, M., Beckman, R., & Conover, W. 1979, Technometrics, 21, 239, doi: 10.1080/00401706.1979.10489755
* Miyatake et al. (2021) Miyatake, H., Sugiyama, S., Takada, M., et al. 2021, arXiv:2111.02419, arXiv:2111.02419. https://arxiv.org/abs/2111.02419
* Moster et al. (2020) Moster, B. P., Naab, T., Lindström, M., & O’Leary, J. A. 2020, arXiv:2005.12276, arXiv:2005.12276. https://arxiv.org/abs/2005.12276
* Moster et al. (2013) Moster, B. P., Naab, T., & White, S. D. M. 2013, MNRAS, 428, 3121, doi: 10.1093/mnras/sts261
* Moster et al. (2018) —. 2018, MNRAS, 477, 1822, doi: 10.1093/mnras/sty655
* Moster et al. (2010) Moster, B. P., Somerville, R. S., Maulbetsch, C., et al. 2010, ApJ, 710, 903, doi: 10.1088/0004-637X/710/2/903
* Neistein & Khochfar (2012) Neistein, E., & Khochfar, S. 2012, arXiv:1209.0463, arXiv:1209.0463. https://arxiv.org/abs/1209.0463
* Nishimichi et al. (2019) Nishimichi, T., Takada, M., Takahashi, R., et al. 2019, ApJ, 884, 29, doi: 10.3847/1538-4357/ab3719
* Panter et al. (2007) Panter, B., Jimenez, R., Heavens, A. F., & Charlot, S. 2007, MNRAS, 378, 1550, doi: 10.1111/j.1365-2966.2007.11909.x
* Pedregosa et al. (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., et al. 2011, Journal of Machine Learning Research, 12, 2825
* Pérez & Granger (2007) Pérez, F., & Granger, B. E. 2007, Computing in Science and Engineering, 9, 21, doi: 10.1109/MCSE.2007.53
* Planck Collaboration et al. (2014) Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2014, A&A, 571, A16, doi: 10.1051/0004-6361/201321591
* Press & Schechter (1974) Press, W. H., & Schechter, P. 1974, ApJ, 187, 425, doi: 10.1086/152650
* Ragan-Kelley et al. (2014) Ragan-Kelley, M., Perez, F., Granger, B., et al. 2014, in American Geophysical Union Fall Meeting Abstracts, Vol. D7
* Rasmussen & Williams (2006) Rasmussen, C. E., & Williams, C. K. I. 2006, Gaussian Processes for Machine Learning
* Reddick et al. (2014) Reddick, R. M., Tinker, J. L., Wechsler, R. H., & Lu, Y. 2014, ApJ, 783, 118, doi: 10.1088/0004-637X/783/2/118
* Reddick et al. (2013) Reddick, R. M., Wechsler, R. H., Tinker, J. L., & Behroozi, P. S. 2013, ApJ, 771, 30, doi: 10.1088/0004-637X/771/1/30
* Reid et al. (2014) Reid, B. A., Seo, H.-J., Leauthaud, A., Tinker, J. L., & White, M. 2014, MNRAS, 444, 476, doi: 10.1093/mnras/stu1391
* Richardson (1972) Richardson, W. H. 1972, J. Opt. Soc. Am., 62, 55, doi: 10.1364/JOSA.62.000055
* Rodríguez-Puebla et al. (2015) Rodríguez-Puebla, A., Avila-Reese, V., Yang, X., et al. 2015, ApJ, 799, 130, doi: 10.1088/0004-637X/799/2/130
* Rodríguez-Puebla et al. (2016) Rodríguez-Puebla, A., Behroozi, P., Primack, J., et al. 2016, MNRAS, 462, 893, doi: 10.1093/mnras/stw1705
* Saito et al. (2016) Saito, S., Leauthaud, A., Hearin, A. P., et al. 2016, MNRAS, 460, 1457, doi: 10.1093/mnras/stw1080
* Sgró et al. (2013) Sgró, M. A., Paz, D. J., & Merchán, M. 2013, MNRAS, 433, 787, doi: 10.1093/mnras/stt773
* Sinha & Garrison (2020) Sinha, M., & Garrison, L. H. 2020, MNRAS, 491, 3022, doi: 10.1093/mnras/stz3157
* Tasitsiomi et al. (2004) Tasitsiomi, A., Kravtsov, A. V., Wechsler, R. H., & Primack, J. R. 2004, ApJ, 614, 533, doi: 10.1086/423784
* Tinker et al. (2008) Tinker, J. L., Conroy, C., Norberg, P., et al. 2008, ApJ, 686, 53, doi: 10.1086/589983
* Tinker et al. (2010) Tinker, J. L., Robertson, B. E., Kravtsov, A. V., et al. 2010, ApJ, 724, 878, doi: 10.1088/0004-637X/724/2/878
* van den Bosch et al. (2013) van den Bosch, F. C., More, S., Cacciato, M., Mo, H., & Yang, X. 2013, MNRAS, 430, 725, doi: 10.1093/mnras/sts006
* Van Der Walt et al. (2011) Van Der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, ArXiv:1102.1523
* Wang et al. (2019) Wang, K., Mao, Y.-Y., Zentner, A. R., et al. 2019, MNRAS, 488, 3541, doi: 10.1093/mnras/stz1733
* Watson et al. (2015) Watson, D. F., Hearin, A. P., Berlind, A. A., et al. 2015, MNRAS, 446, 651, doi: 10.1093/mnras/stu2065
* Wechsler & Tinker (2018) Wechsler, R. H., & Tinker, J. L. 2018, ARA&A, 56, 435, doi: 10.1146/annurev-astro-081817-051756
* White (2014) White, M. 2014, MNRAS, 439, 3630, doi: 10.1093/mnras/stu209
* Wibking et al. (2020) Wibking, B. D., Weinberg, D. H., Salcedo, A. N., et al. 2020, MNRAS, 492, 2872, doi: 10.1093/mnras/stz3423
* Yuan et al. (2021) Yuan, S., Garrison, L. H., Hadzhiyska, B., Bose, S., & Eisenstein, D. J. 2021, MNRAS, doi: 10.1093/mnras/stab3355
* Zentner (2007) Zentner, A. R. 2007, International Journal of Modern Physics D, 16, 763, doi: 10.1142/S0218271807010511
* Zentner et al. (2014) Zentner, A. R., Hearin, A. P., & van den Bosch, F. C. 2014, MNRAS, 443, 3044, doi: 10.1093/mnras/stu1383
* Zentner et al. (2013) Zentner, A. R., Semboloni, E., Dodelson, S., et al. 2013, Phys. Rev. D, 87, 043509, doi: 10.1103/PhysRevD.87.043509
* Zhai et al. (2019) Zhai, Z., Tinker, J. L., Becker, M. R., et al. 2019, ApJ, 874, 95, doi: 10.3847/1538-4357/ab0d7b
* Zheng & Guo (2016) Zheng, Z., & Guo, H. 2016, MNRAS, 458, 4015, doi: 10.1093/mnras/stw523
* Zheng et al. (2005) Zheng, Z., Berlind, A. A., Weinberg, D. H., et al. 2005, ApJ, 633, 791, doi: 10.1086/466510
## Appendix A Parameterized Subhalo Mass Function
In this section, we describe the fitting function we use to approximate the
subhalo mass function in the Bolshoi-Planck simulation (BPL, Klypin et al.,
2011). The BPL simulation was run with cosmological parameters closely
matching Planck Collaboration et al. (2014), and was carried out using the ART
code (Kravtsov et al., 1997) by evolving $2048^{3}$ dark-matter particles of
mass $m_{\rm p}=1.55\times 10^{8}M_{\odot}$ on a simulation box of $250\,{\rm
Mpc}$ on a side.
For notational convenience, throughout this section we will refer to subhalo
mass using the variable $m_{\rm h}\equiv\log_{10}M_{\rm peak}.$ The basic
quantity that we model is the cumulative mass function, $\rm dN_{\rm
h}(>m_{\rm h})/\rm dV,$ where the quantity $V$ is the comoving volume. The
relationship the cumulative mass function and the differential mass function,
$\rm dn_{\rm h}(m_{\rm h})/\rm dm_{\rm h}\rm dV,$ is given by the following
equation:
$\rm dN_{\rm h}(>m_{\rm h})/\rm dV=\int_{m_{\rm h}}^{\infty}\rm dm_{\rm
h}^{\prime}\rm dn_{\rm h}/\rm dm_{\rm h}^{\prime}\rm dV.$
Figure 7: Fitting function approximation of the subhalo mass function in the
Bolshoi-Planck simulation. Values for $\rm dN_{\rm h}(>x)/\rm dV$ plotted on
the vertical axis are offset by an order of magnitude at different redshifts
for visual clarity.
We parameterize the cumulative mass function to behave like a power-law with
constant index at the low-mass end, and like a power-law with a steeply
declining power-law index at high mass. We capture this behavior by modeling
$y_{\rm h}\equiv\log_{10}N_{\rm h}$ to behave like a linear function of
$m_{\rm h}$ at low mass, and to exhibit a rapidly declining slope above some
critical value of subhalo mass, $m_{\rm c}.$ To accomplish this behavior, we
utilize a sigmoid function:
(21) $\displaystyle\mathcal{S}(x|x_{\rm c},k,y_{\rm lo},y_{\rm hi})\equiv
y_{\rm lo}+\frac{y_{\rm hi}-y_{\rm lo}}{1+\exp(-k\cdot(x-x_{\rm c}))}.$
Using the sigmoid function defined in Eq. 21, we model the cumulative number
density as follows:
(22) $\displaystyle y_{\rm h}(m_{\rm h})=y_{0}+S(m_{\rm h},m_{\rm c},k,y_{\rm
lo},y_{\rm hi})\cdot(m_{\rm h}-m_{\rm c})$
In Eq. 22, the parameter $y_{0}$ controls the normalization of the subhalo
mass function at the critical mass $m_{\rm c};$ the power-law index at the
low-mass end is controlled by $y_{\rm lo},$ and the shape of the high-mass
cutoff is characterized by both $k$ and $y_{\rm hi}.$ We note that basic
theoretical considerations lead one to expect the true halo mass function to
have an exponential cutoff at the high-mass end (see, e.g., Press & Schechter,
1974; Zentner, 2007), whereas even in the limit of infinite mass, Eq. 22 never
attains exponentially declining behavior. For our purposes, however, this
shortcoming is immaterial, since our fitting function closely mimics an
exponential decline for all halo masses relevant for the present study, and we
find that Eq. 22 has more numerically stable behavior in applications
requiring automatic differentiation at the high-mass end.
We calibrate the values of our fitting function parameters using publicly
available333https://www.peterbehroozi.com/data.html subhalo catalog identified
with Rockstar and ConsistentTrees (Behroozi et al., 2013b, c; Rodríguez-Puebla
et al., 2016). While fitting the free parameters, we hold fixed $y_{\rm
lo}=-1.4$, $k=0.9,$ and $y_{\rm hi}=-5.25.$ The remaining two parameters,
$y_{0}$ and $m_{\rm c},$ exhibit redshift dependence that is well approximated
with the sigmoid function defined by Eq. 21. That is, we model $m_{\rm c}(z)$
as:
$\displaystyle m_{\rm c}(z)$ $\displaystyle=$
$\displaystyle\mathcal{S}(z,z_{\rm c}^{m_{\rm c}},k_{\rm c}^{m_{\rm c}},y_{\rm
lo}^{m_{\rm c}},y_{\rm hi}^{m_{\rm c}}),$
and similarly for $y_{0}(z).$ For our best-fitting parameters controlling
$m_{\rm c}(z),$ we have $z_{\rm c}^{m_{\rm c}}=0.75,k_{\rm c}^{m_{\rm
c}}=1,y_{\rm lo}^{m_{\rm c}}=16.175,$ and $y_{\rm hi}^{m_{\rm c}}=12.925.$ For
$y_{0}(z),$ we have $z_{\rm c}^{y_{0}}=0.5,k_{\rm c}^{y_{0}}=1.35,y_{\rm
lo}^{y_{0}}=-8.1,$ and $y_{\rm hi}^{y_{0}}=-5.9.$ In Figure 7 we show the
fidelity with which our calibration approximates the cumulative mass function
of subhalos in BPL. Throughout this paper, we define SHAMNet according to the
$z=0$ mass function, and we supply these parameters as a convenience for
future applications.
## Appendix B Analytical SMF predictions
In §4.1, we outlined a technique to make a differentiable prediction for the
SMF based on an input subhalo catalog. In this section, we describe a
variation on this method that allows one to predict the SMF based an
analytical subhalo mass function (SHMF), such as the one supplied in Appendix
A.
In the absence of scatter between stellar mass, and halo mass, it is
straightforward to predict the SMF, $\phi_{\rm g}(M_{\star}),$ from knowledge
of the SHMF, $\phi_{\rm h}(M_{\rm peak}),$ and the stellar-to-halo mass
relation, $\langle M_{\star}|M_{\rm peak}\rangle,$
$\displaystyle\phi_{\rm g}(M_{\star})=\phi_{\rm h}(\langle M_{\star}|M_{\rm
peak}\rangle)\frac{\rm dM_{\rm peak}}{\rm dM_{\star}},$ (23)
where ${\rm dM_{\rm peak}}/{\rm dM_{\star}}$ is the Jacobian of $\langle
M_{\star}|M_{\rm peak}\rangle.$
Equation 23 is simply the differential version of Equation 1, and so the
calculation of $\phi_{\rm g}(M_{\star})$ in the presence of scatter is given
by Equation 2, repeated here for convenience:
$\displaystyle\phi_{\rm g}(M_{\star})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dM_{\rm peak}\phi_{\rm h}(M_{\rm
peak})P(M_{\star}|M_{\rm peak})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\rm dM_{\star}^{\prime}\frac{\rm dM_{\rm
peak}}{\rm dM_{\star}^{\prime}}\phi_{\rm h}(\langle M_{\star}^{\prime}|M_{\rm
peak}\rangle)P(M_{\star}^{\prime}|M_{\rm peak})$
where scatter in the stellar-to-halo mass relation is encoded by
$P(M_{\star}|M_{\rm peak}).$ To differentiably calculate the left-hand side of
Equation B, one can simply tabulate the integrand on the right-hand side using
a grid of $M_{\star}^{\prime}$ that is sufficiently broad to cover the support
of convolution, and sufficiently dense to achieve the desired precision:
$\displaystyle\phi_{\rm g}(M_{\star})=\sum_{i}\Delta M_{\star,i}^{\prime}$
$\displaystyle\frac{\rm dM_{\rm peak}}{\rm dM_{\star,i}^{\prime}}\phi_{\rm
h}(\langle M_{\star,i}^{\prime}|M_{\rm peak}\rangle)$ (25)
$\displaystyle\times P(M_{\star,i}^{\prime}|M_{\rm peak}).$
The principal difference between Equation 25 and Equation 2 in the main body
of the paper is the presence of the Jacobian, ${\rm dM_{\rm peak}}/{\rm
dM_{\star}};$ this factor does not appear in the simulation-based formulation
because its effect is accounted for by the relative abundance of simulated
halos as a function of $M_{\rm peak}.$
Equation 25 makes it plain to see how to calculate derivatives of $\phi_{\rm
g}(M_{\star}|\theta)$ with respect to the model SHAMNet parameters, $\theta$:
the gradient operator passes through the summation, and $\partial\phi_{\rm
g}(M_{\star})/\partial\theta$ can be calculated by simply summing each term.
For the analytical approximations we use throughout this paper, each of these
terms can in principle be calculated symbolically; we refer the reader to the
smf_scatter_convolution.py module in our source code for our JAX-based
computation of the gradients of Equation 25 based on automatic
differentiation.
## Appendix C Three-roll stellar-to-halo mass relation
As discussed in Appendix §D, in order to train SHAMNet we used an analytical
parameterization of the stellar-to-halo mass relation to support the
initialization of the network. In §C.1, we describe the functional form we use
for this purpose, and in §C.2 we detail our procedure for optimizing the
parameters of this model as a function of the variables $\theta$ that define
the abundance matching problem space reviewed in §3.2.
### C.1 Basic behavior of the three-roll SMHM
In this section, we describe a new functional form we developed to capture the
principal scaling relation of abundance matching, the stellar-to-halo-mass
relation (SMHM), i.e., $\langle M_{\star}|M_{\rm peak}\rangle_{\rm med}.$ In
its general shape, the SMHM relation exhibits a characteristic peak at $M_{\rm
halo}\approx 10^{12}M_{\odot},$ and declines roughly like a power-law at both
the low- and high-mass ends. We parametrically capture this behavior in terms
of the relationship between $m_{\rm h}\equiv\log_{10}M_{\rm peak},$ and
$m_{\star}\equiv\log_{10}M_{\star}:$
$\displaystyle m_{\star}(m_{\rm h})=m_{0}+\alpha(m_{\rm h})\cdot(m_{\rm
h}-m_{\rm crit}).$ (26)
From Eq. 26 we see that $M_{\star}$ scales like a power-law with $M_{\rm
halo},$ with rolling index $\alpha,$ and normalization $m_{0}$ defined by the
stellar mass at halo mass $m_{\rm crit}.$ For the function $\alpha(x),$ we use
the sigmoid function $\mathcal{S}(x)$ defined in Eq. 21 to control the
transition between the low-mass slope, $\alpha_{\rm lo}=\alpha(x\rightarrow
0),$ and the high-mass slope, $\alpha_{\rm hi}=\alpha(x\rightarrow\infty).$
That is, we have
$\alpha(m_{\rm h})=\mathcal{S}(m_{\rm h},m_{\rm crit},k_{\rm crit},\alpha_{\rm
lo},\alpha_{\rm hi}).$
We further allow the power-law indices at low- and high-mass to be functions
of halo mass, $\alpha_{\rm lo}\rightarrow\alpha_{\rm lo}(m_{\rm h}),$ and
$\alpha_{\rm hi}\rightarrow\alpha_{\rm hi}(m_{\rm h}),$ where the mass-
dependence is again controlled by a sigmoid as defined by Eq. 21. The sigmoid-
dependence to $\alpha_{\rm lo}(m_{\rm h})$ and $\alpha_{\rm hi}(m_{\rm h})$
gives the model freedom for an additional roll in the power-law index at both
low- and high-mass ends.
The above formulation gives our SMHM relation a total of eleven parameters:
$m_{0},m_{\rm crit},k_{\rm crit},$ plus four parameters for each of the two
sigmoid functions, $\alpha_{\rm lo}(m_{\rm h}),$ and $\alpha_{\rm hi}(m_{\rm
h}).$ We find that none of the three $k$ parameters provides a physically
useful degree of freedom, and so in all of our applications of this SMHM
relation, we hold these fixed to unity, giving us an eight-dimensional model
for this three-roll SMHM relation.
Figure 8: Flexible model for the stellar-to-halo-mass relation (SMHM). Using
the model described in Appendix §C, different curves show the median stellar
mass as a function of halo mass for different values of $\omega$ in our eight-
dimensional parameter space.
### C.2 Parameterized SHAM with the three-roll SMHM
In this section, we describe how we use the functional form defined in §C.1 to
identify particular solutions to the stellar-to-halo mass relation that
defines abundance matching (see Equation 2). In particular, here describe how
we identify an optimal choice of parameters, $\omega,$ such that the
functional form $m_{\star}(m_{\rm h}|\omega)$ defined in Equation 26 supplies
a solution to Equation 2 for an input $\theta$ that defines the SMF,
$\phi_{\rm g}(M_{\star}|\theta),$ the SHMF, $\phi_{\rm h}(M_{\rm
peak}|\theta),$ and log-normal scatter, $\sigma(M_{\star}|M_{\rm peak}),$
where the $\theta$-dependence of these quantities is specified in §3.
Since the goal of solving Equation 2 is to define a scaling relation that
gives rise to some desired SMF, then in order to solve for $\omega$ we must
quantify our choice for the closeness of two stellar mass functions,
$\phi_{1}(M_{\star})$ and $\phi_{2}(M_{\star}).$ Due to the exponentially
declining nature of the Schechter function, we have found that minimizing the
simple least-squares difference between two SMFs can lead to numerically
unstable results during gradient descent with automatic differentiation,
particularly when seeking high-quality fits at the very massive end. Computing
the logarithmic difference between the two mass functions improves the
numerical instability, but does not resolve it; we also find instabilities
when implementing a hard-edged clip, $\phi_{\rm g}(M_{\star})\rightarrow{\rm
max}\left\\{\phi_{\rm g}(M_{\star}),\phi_{\rm min}\right\\},$ due to the
discontinuity of the derivative at the tiny value of the clip. To resolve
these issues, we have found it beneficial to use a differentiable clipping
function, $\mathcal{R}(x;y_{\rm min}),$ defined as follows
(27) $\displaystyle\mathcal{R}(x;y_{\rm min})\equiv\frac{1}{\ln(10)}\left({\rm
sinh}^{-1}(x/2y_{\rm min})+\ln(y_{\rm min})\right),$
which largely behaves like the base-10 logarithm, but smoothly asymptotes to
$y_{\rm min}$ rather than falling below this value (note that this is the same
transformation used to define magnitudes of SDSS galaxies in Lupton et al.,
1999). Thus when minimizing the difference between two stellar mass functions,
in practice we minimize the difference between
$\mathcal{R}(\phi_{1}(M_{\star}),\phi_{\rm min})$ and
$\mathcal{R}(\phi_{2}(M_{\star}),\phi_{\rm min}),$ using $\phi_{\rm
min}=10^{-15}{\rm Mpc^{-3}M_{\odot}^{-1}}$ to protect against numerical
instabilities in the gradient evaluations.
For each choice of $\theta,$ we searched our model parameter space for an
optimum value of $\omega$ by minimizing the quantity $\mathcal{L}_{\rm MSE},$
defined as
$\displaystyle\mathcal{L}_{\rm MSE}(\theta,\omega)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{N}\sum_{i}\left(y^{\rm i}(\theta)-x^{\rm
i}(\theta,\omega)\right)^{2}$ $\displaystyle y^{\rm i}(\theta)$
$\displaystyle=$ $\displaystyle\mathcal{R}(\phi_{\rm
g}(M_{\star,i}|\phi_{\ast},\alpha))$ (28) $\displaystyle x^{\rm
i}(\theta,\omega)$ $\displaystyle=$ $\displaystyle\mathcal{R}(\phi_{\rm
g}(M_{\star,i}|\theta,\omega))$
where $y^{\rm i}(\theta)$ is the differentiably-clipped logarithm of target
double-Schechter function defined only by $\\{\phi_{\ast},\alpha\\},$ and
$x^{\rm i}(\theta,\omega)$ is the theoretical prediction for the SMF, computed
by the method described in Appendix B. We calculate $\mathcal{L}_{\rm MSE}$ by
evaluating the predicted and target mass functions at a set of $N$ control
points, $M_{\star}^{\rm i},$ using $N=100$ logarithmically-spaced values
spanning the range $10^{9}M_{\odot}<M_{\star}<10^{12}M_{\odot}.$ To minimize
$\mathcal{L}_{\rm MSE}$, we use the JAX implementation of the Adam algorithm
(Kingma & Ba, 2014), which is a gradient descent technique with an adaptive
learning rate, where we also use JAX to compute the gradients. We use 2
successive burn-in cycles with a step-size parameter $s=0.05$ for $\sim 50$
updates, followed by $1000$ updates with $s=0.01.$
In this minimization calculation, the eight parameters $\omega$ defined in
§C.1 are not the actual variables we use to calculate our predicted values for
the SMF; instead we minimize $\omega^{\prime},$ defined by:
$\omega^{\prime}\equiv\mathcal{S}(\omega|\omega_{0},k_{\omega},\omega_{\rm
lo},\omega_{\rm hi}),$
where $\mathcal{S}(x)$ is the sigmoid function defined by Eq. 21. We tailor
the lower and upper bounds of each dimension of $\omega$ according to
reasonable expectations for physically plausible SMHMs, and we hold
$k_{\omega}$ fixed to a constant value of $0.1,$ and $\omega_{0}$ fixed to the
halfway point between $\omega_{\rm lo}$ and $\omega_{\rm hi}.$ We refer the
reader to the relevant section of our source code for additional details of
our implementation.
## Appendix D SHAMNet Definition and Training
In this appendix, we describe our procedure for training SHAMNet,
$\mathcal{F}_{\rm SHAM}(M_{\rm peak},\theta|\psi).$ We remind the reader that
in this notation, the parameters $\psi$ refer to the weights and biases of the
neural network described in detail in this section, and as described in §3.2,
the parameters $\theta$ control the behavior of the three ingredients needed
to define SHAM:
1. 1.
$\phi_{\rm g}(M_{\star}|\theta),$ the galaxy stellar mass function,
2. 2.
$\phi_{\rm h}(M_{\rm peak}|\theta),$ the subhalo mass function described in
§A,
3. 3.
$\sigma(M_{\rm peak}|\theta),$ the halo mass-dependent scatter.
As outlined in §2, for a particular choice of $\theta,$ SHAM is defined to be
the scaling relation, $\langle M_{\star}|M_{\rm peak};\theta\rangle_{\rm
med},$ that gives rise to the target stellar mass function, $\phi_{\rm
g}(M_{\star}|\theta),$ when applied to the subhalo population, $\phi_{\rm
h}(M_{\rm peak}|\theta),$ in the presence of (log-normal) scatter,
$\sigma(M_{\rm peak}|\theta).$ The goal of training SHAMNet is to identify the
parameters $\psi$ such that for all physically relevant values of $\theta,$
the function $\mathcal{F}_{\rm SHAM}(M_{\rm peak},\theta|\psi)=\langle
M_{\star}|M_{\rm peak};\theta\rangle_{\rm med}$ satisfies the SHAM-defined
relationship between $\phi_{\rm g},$ $\phi_{\rm h},$ and $\sigma.$ The nature
of this objective makes training $\mathcal{F}_{\rm SHAM}(M_{\rm
peak},\theta|\psi)$ different from a standard problem in neural network
regression, since there does not exist a target function whose output defines
the training data; instead, the parameters $\psi$ will be considered optimal
when the SMF that emerges from $\langle M_{\star}|M_{\rm
peak};\theta\rangle_{\rm med}$ agrees with $\phi_{\rm g}(M_{\star}|\theta)$ to
the desired precision.
We begin in §D.1 with a description of the design of the neural network we use
for $\mathcal{F}_{\rm SHAM}.$ Our training then proceeds in two phases. In the
first phase described in §D.2, we tune the network parameters $\psi$ so that
$\mathcal{F}_{\rm SHAM}(M_{\rm peak},\theta|\psi)$ closely agrees with a
parametric form for the stellar-to-halo mass relation that has been tuned in
advance to approximately solve Eq. 2. After this initial phase, we optimize
the parameters $\psi$ using the training procedure described in §D.3. We will
provide a reasonably comprehensive description of this two-phase procedure in
this appendix, but we refer the reader to
https://github.com/ArgonneCPAC/shamnet for all the quotidian details.
### D.1 Architecture
In this section, we describe the architecture of the neural network we use for
SHAMNet, which provides a mapping
$\mathcal{F}_{\rm SHAM}:\\{\theta,M_{\rm peak}\\}\rightarrow M_{\star}$
based on a simply-connected multi-layer perceptron (MLP). For the input
variables $\theta,$ we use a $2$-parameter Schechter function to capture
variations in the galaxy stellar mass function, and a 2-parameter sigmoid to
describe the halo mass dependence of scatter, adopting the forms described in
§3.2. For the subhalo mass function, we use the parametric model defined in
§A, but in this case we hold these parameters fixed to the values calibrated
to approximate the $z=0$ mass function in the Bolshoi simulation. Thus in
addition to $M_{\rm peak},$ SHAMNet accepts a 5-dimensional parameter,
$\theta,$ and returns stellar mass, $M_{\star},$ so that $\mathcal{F}_{\rm
SHAM}:\mathbb{R}^{5}\rightarrow\mathbb{R}^{1}.$
Our neural network $\mathcal{F}_{\rm SHAM}$ thus has an input layer of 5
nodes, one for each dimension of $\\{\theta,M_{\rm peak}\\},$ and an output
layer of a single node, $M_{\star}.$ Between the inputs and outputs, we use 4
hidden layers composed of $64,32,8,$ and $4$ nodes, respectively. Each node in
a layer is connected to every node in the previous layer, so that all 4 of our
hidden layers are dense. As with all simply-connected MLPs, the computations
performed by the $i^{\rm th}$ node of a layer, $y_{\rm i},$ consists of two
successive operations performed on the outputs of all the nodes of the
previous layer, $x_{\rm j}.$ The first set of operations consists of a linear
transformation,
$\displaystyle y_{\rm i}=\sum_{\rm j}W_{\rm ij}x_{\rm j}+b_{\rm i},$ (29)
and the second operation is a nonlinear transformation, $\Upsilon_{\rm act},$
that is independently applied to the each node $y_{\rm i}$ in the layer; the
result $\Upsilon_{\rm act}(y_{\rm i})$ is then fed as an input to each node in
the next layer. For the activation function $\Upsilon_{\rm act}$ in every node
of each dense hidden layer, we use a SELU function,
$\displaystyle\Upsilon_{\rm act}(z)=\lambda\begin{cases}z&\mbox{if }z>0\\\
\alpha e^{z}-\alpha&\mbox{if }z\leq 0\end{cases},$ (30)
where $\lambda=1.05$ and $\alpha=1.67$ are chosen so that the mean and
variance of the inputs are preserved between two consecutive layers (Klambauer
et al., 2017). For the activation function applied to the output layer, we use
the sigmoid function defined in Eq. 21 to enforce that the returned stellar
mass is bounded within the range
$10^{-5}M_{\odot}<M_{\star}<10^{25}M_{\odot}.$ Our MLP is implemented in the
stax subpackage of the JAX library.
### D.2 Initialization
Our goal for training SHAMNet is to identify the weights, $W_{\rm ij},$ and
biases, $b_{\rm j},$ collectively represented with the variable $\psi,$ such
that $\mathcal{F}_{\rm SHAM}$ provides the desired mapping from $M_{\rm
peak}\rightarrow M_{\star}$ for any physically relevant parameters, $\theta.$
We begin our procedure for optimizing $\psi$ by generating approximations to
the desired mapping, $\langle M_{\star}|M_{\rm peak};\theta\rangle_{\rm med},$
for a large collection of $\theta;$ the collection of $\theta$ and associated
mappings will serve as training data for the first phase of optimizing
$\mathcal{F}_{\rm SHAM}.$ Thus at the end of this phase, we will have a
collection of weights and biases, $\psi_{\rm init},$ that roughly provides the
desired scaling relation, $\langle M_{\star}|M_{\rm peak};\theta\rangle_{\rm
med},$ for each $\theta$ in the training set. The point in parameter space
$\psi_{\rm init}$ will be used as the starting point for the optimization
procedure described in §D.3, and so we refer to calculations described in the
present section as the initialization phase of our training.
As described in §3, the parameters $\theta$ define a specific combination of
$\phi_{\rm g}(M_{\star}|\theta),\phi_{\rm h}(M_{\rm peak}|\theta),$ and
$\sigma(M_{\rm peak}|\theta);$ for each such combination, we seek to determine
the abundance matching relation, $\langle M_{\star}|M_{\rm peak}\rangle_{\rm
med},$ that provides a solution to Eq. 2. In the initialization phase of our
training, we assume that the parametric form $m_{\star}(m_{\rm h}|\omega),$ as
defined by Eq. 26, is sufficiently flexible to supply an adequate stellar-to-
halo mass relation, and we train SHAMNet to reproduce this parameterized
relation. Thus in order to generate training data for the initialization phase
of our training, we need to generate a large collection of pairs,
$\\{\theta,\omega_{\theta}\\},$ where the values of $\theta$ span the
physically relevant range, and the accompanying value $\omega_{\theta}$
represents an approximate solution to Eq. 2. To generate the collection of
pairs, $\\{\theta,\omega_{\theta}\\},$ we randomly draw a value of $\theta$
from an observationally relevant range, and for each random draw, we use the
gradient descent technique described in Appendix C to find a point
$\omega_{\theta}$ that optimally solves Eq. 2.
Since $\theta=\\{\phi_{\ast},\alpha,y^{\sigma}_{\rm lo},y^{\sigma}_{\rm
hi}\\},$ then we have four dimensions for which we need to define an
observationally relevant range. The last two dimensions control the level of
$M_{\rm halo}$-dependent scatter, which we allow to vary independently within
the open interval, (0.1, 0.5). The first two variables define the variations
in the SMF, which for purposes of this paper, we wish to be reasonably close
to SDSS measurements of the low-redshift universe (e.g., Li & White, 2009). To
determine the relevant range of the SMF parameters, we define an SDSS-like
SMF, $\phi_{\rm SDSS}(M_{\star}),$ by selecting fiducial values
$\phi_{\ast}=0.005\ {\rm Mpc^{-3}M_{\odot}^{-1}},$ and $\alpha=-1.06.$ Using
emcee (Foreman-Mackey et al., 2019), we run an MCMC to determine posteriors on
the parameters $\phi_{\ast}$ and $\alpha,$ assuming a Gaussian likelihood with
a diagonal covariance matrix defined by $M_{\star}$-independent uncertainty of
$0.1$ dex on $\phi_{\rm SDSS}(M_{\star}).$ The results of this MCMC supply a
Gaussian-like distribution of points in our SMF parameter space, and we fit
this posterior distribution with a two-dimensional Gaussian distribution,
$\mathcal{N}(\mu,{\rm Cov(}\phi_{\ast},\alpha)).$ When generating training
data for SHAMNet, we sample SMF parameters by randomly draw pairs
$\\{\phi_{\ast},\alpha\\}$ based on a Latin Hypercube, where the axes of the
Latin Hypercube are aligned with the eigenvectors of ${\rm
Cov(}\phi_{\ast},\alpha),$ and span a $5\sigma$ length in each direction; any
values with $\alpha<1$ correspond to non-monotonic SMFs, and are
discarded.555We use the pyDOE2 package444https://github.com/clicumu/pyDOE2 in
our random sampling based on a Latin Hypercube (McKay et al., 1979; Iman et
al., 1981).
For each value of $\theta$ generated by this sampling method, we identify a
best-fitting stellar-to-halo mass relation, $\omega_{\theta},$ using the
optimization techniques described in §C.2; the quality of each fit is
quantified by $\mathcal{L}_{\rm MSE}(\omega,\theta),$ as defined in Eq. C.2.
For some points $\theta,$ the best-fitting value $\omega_{\theta}$ produces
only a rough recovery of the target stellar mass function, $\phi_{\rm
g}(M_{\star}|\theta).$ In the initialization phase of training SHAMNet, we
discard all pairs $\\{\theta,\omega_{\theta}\\}$ with $\mathcal{L}_{\rm
MSE}>0.05,$ which we find is about 5% of the sampling data. While this
rejection rate implies that our training set will not fairly sample the full,
observationally relevant region of parameter space as defined above, this is
rather harmless because the primary purpose of this procedure is just to
generate some collection $\\{\theta,\omega_{\theta}\\}$ used to identify
$\psi_{\rm init}.$ As described in §D.1, the design of SHAMNet has no reliance
upon the functional form of the stellar-to-halo mass relation defined in
Appendix C, $m_{\star}(m_{\rm h}|\omega)$, and so rejected points
$\omega_{\theta}$ simply represent an inflexibility of the three-roll SMHM
functional form, but this will not impact SHAMNet as we will not reject such
pairs $\\{\theta,\omega_{\theta}\\}$ after $\psi_{\rm init}$ has been
identified.
The goal of the initialization phase of our training is to identify a set of
weights and biases, $\psi_{\rm init},$ such that the behavior of our neural
network, $\mathcal{F}_{\rm SHAM},$ gives an accurate approximation of the
parametric function defined in Appendix C. To achieve this goal, we seek to
identify the point $\psi_{\rm init}$ that minimizes the following cost
function:
$\displaystyle\mathcal{L}_{\rm MSE}(\theta,\omega)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{N}\sum_{i}\left(y^{\rm i}(\theta)-x^{\rm
i}(\theta,\omega)\right)^{2}$ $\displaystyle y^{\rm i}(\theta)$
$\displaystyle=$ $\displaystyle m_{\star}(m_{\rm h}|\omega_{\theta})$ (31)
$\displaystyle x^{\rm i}(\theta,\omega)$ $\displaystyle=$
$\displaystyle\mathcal{F}_{\rm SHAM}(M_{\rm peak},\theta|\phi)$
In evaluating $\mathcal{L}_{\rm MSE}(\theta,\omega),$ we used batch sizes of
$N=50$ pairs of points, $\\{\theta,\omega_{\theta}\\},$ that randomly sample
the parameter space of $\theta$ using the procedure outlined above. We used
the implementation of the Adam optimizer in JAX with a step-size parameter of
0.001 to identify $\psi_{\rm init}$ through 1500 gradient evaluations, which
we found through experimentation to be sufficient for the network to achieve
the level of accuracy required for this initializing phase of training the
network.
### D.3 SHAMNet Training
The goal of the final phase of our training is to identify a set of weights
and biases, $\psi_{\rm SHAM},$ such that the behavior of our neural network,
$\mathcal{F}_{\rm SHAM},$ supplies a stellar-to-halo mass relation that
accurately solves the defining equation of abundance matching, Eq. 2. We note
that if the parametric function $m_{\star}(m_{\rm h}|\omega)$ defined in
Appendix C were sufficiently flexible, then this goal would already be
achieved via the identification of $\psi_{\rm init}.$ However, as described in
Appendix D.2, roughly 5% of points in the observationally relevant range have
SMFs that cannot be described to high accuracy using a stellar-to-halo mass
relation defined by this functional form. Thus in the final training phase of
SHAMNet, we no longer rely on training data based on the parametric function
$m_{\star}(m_{\rm h}|\omega).$ Instead, we seek to identify the point
$\psi_{\rm SHAM}$ that directly minimizes the difference between the predicted
and target SMF, and so we minimize a cost function that directly quantifies
this difference:
$\displaystyle\mathcal{L}_{\rm MSE}(\theta)$ $\displaystyle\equiv$
$\displaystyle\frac{1}{N}\sum_{i}\left(y^{\rm i}(\theta)-x^{\rm
i}(\theta)\right)^{2}$ $\displaystyle y^{\rm i}(\theta)$ $\displaystyle=$
$\displaystyle\mathcal{R}(\phi_{\rm g}(M_{\star,i}|\phi_{\ast},\alpha))$ (32)
$\displaystyle x^{\rm i}(\theta,\phi)$ $\displaystyle=$
$\displaystyle\mathcal{R}(\phi_{\rm g}(M_{\star,i}|\theta,\phi)).$
Equation D.3 is very similar to Eq. C.2, where the latter defined the cost
function that we minimized in order to identify the parametric function
$m_{\star}(m_{\rm h}|\omega_{\theta})$ that solves the defining equation of
abundance matching for a particular value of $\theta.$ In fact, in both
equations the values of $y^{\rm i}(\theta)$ that specify the target values of
the SMF are the same, and in both cases, the predicted values of the SMF,
$x^{\rm i}(\theta),$ are derived using the same techniques detailed in
Appendix B for calculating the SMF in the presence of scatter via an
analytical expression for the stellar-to-halo mass relation. But in evaluating
the cost function defined by Eq. D.3, here our predicted values $x^{\rm
i}(\theta)$ come directly from $\mathcal{F}_{\rm SHAM}(M_{\rm
peak},\theta|\psi).$
We use the Adam optimizer with a step size parameter of 0.001 to minimize
$\mathcal{L}_{\rm MSE}(\theta),$ where each of our $10^{5}$ cost function
evaluations used a batch size of $N=50$ points that span the parameter space
of $\theta,$ again using the same sampling method detailed in §D.2. The end
result of this procedure is the identification of $\psi_{\rm SHAM},$ which
defines the behavior of the $\mathcal{F}_{\rm SHAM}$ function whose
performance is illustrated in Figure 3.
## Appendix E Triweight Kernel Convolutions
Many of the calculations in this paper involve convolutions of log-normal
distributions such as the one appearing in Eq. 2. In this section, we review
an alternative to Gaussian convolution based on the triweight kernel,
$\mathcal{K},$ defined as:
$\displaystyle\mathcal{K}(z)\equiv\begin{cases}\frac{35}{96}\left[1-(z/3)^{2}\right]^{3},&-3\sigma\leq
x\leq 3\sigma\\\ 0,&\text{otherwise}\end{cases}$ (33)
where $z\equiv(x-\mu)/\sigma.$ The comparison to a Gaussian is shown in Figure
9. The two distributions have the same first and second moments, $\mu$ and
$\sigma,$ but differ in their higher-order moments.
Figure 9: Triweight convolution kernel. The blue curve shows the probability
density function of a typical log-normal distribution in stellar mass,
$\mathcal{N}(\mu=10,\sigma=0.25);$ the orange curve shows the triweight
kernel, $\mathcal{K},$ defined in Eq. 33. The triweight function
$\mathcal{K}(x)$ vanishes at points beyond $\mu\pm 3\sigma,$ is differentiable
for all $x,$ and is highly performant on GPUs.
The function $\mathcal{K}(x)$ has several properties that make it convenient
for our purposes. As described in §4, differentiable predictions for
cosmological summary statistics involve weights, $w_{\rm halo},$ computed by
integrating some assumed PDF (typically a log-normal) across some bin(s) of
stellar mass. Once $w_{\rm halo}$ has been computed for every simulated
subhalo, various point-estimators are applied to the population of synthetic
galaxies in order to make predictions for summary statistics.
Because Gaussians are everywhere non-zero, computing weighted two-point
summary statistics using exact log-normals can be expensive due to the need to
keep track of a large number of tiny contributions. One simple technique to
mitigate this computational expense is to impose a clip at some multiple of
$\sigma.$ This gains back performance at the cost of some (controllably small)
roundoff error, but results in discontinuity in the weights at $x=\mu\pm
3\sigma.$ On the other hand, the triweight kernel is a $C^{\infty}$ function
on the real line, and points with $|x-\mu|>\pm 3\sigma$ contribute formally
zero weight, and so can be neglected from two-point calculations without
consequence. Moreover, computations of predictions based on Gaussians require
special-function evaluations that can be far slower on GPU accelerator devices
in comparison to the small number of elementary arithmetical operations
required to evaluate Eq. 33.
In computing predictions based on $\mathcal{K}$ rather than $\mathcal{N},$ we
are, in effect, proposing a slightly different probability distribution
$P(M_{\star}|M_{\rm halo})$ than the traditional log-normal. However, the
difference between models distinguished only by $\mathcal{K}$ and
$\mathcal{N}$ is likely to be observationally immaterial, as it is already
challenging to obtain tight constraints on the second moment
$P(M_{\star}|M_{\rm halo}),$ which is the same in the two distributions.
## Appendix F Simulation-Based $\Delta\Sigma$ Predictions
In this section, we derive the equations associated with our calculation of
$\Delta\Sigma(R),$ the excess surface mass density profile of a sample of
points in a cosmological simulation. We highlight that the computation
described below is exact, can be used to calculate the lensing of either
galaxies or halos simulated with or without hydrodynamics, and applies on all
spatial scales resolved by the simulation, including the deeply nonlinear
regime. Our approach also facilitates a highly convenient pre-computation of
the lensing profile on a per-object basis, so that galaxy-halo model
predictions for the lensing of a stacked sample can be calculated simply as a
masked or weighted sum over the pre-computed profiles.
We begin by considering the gravitational lensing at a projected distance $R$
from a single point mass, ${\rm m_{p}}.$ The surface density profile,
$\Sigma(R),$ satisfies:
$\displaystyle\int_{R_{\rm min}}^{R_{\rm max}}\Sigma(R)2\pi R{\rm
d}R=\begin{cases}{\rm m_{p}},&\text{if}\ R_{\rm min}<R<R_{\rm max}\\\
0,&\text{otherwise}\end{cases}$
Using this expression, we can calculate $\langle\Sigma(R)\rangle,$ the average
surface density in an annulus with inner radius $R_{\rm min}$ and outer radius
$R_{\rm max}:$
$\displaystyle\langle\Sigma(R)\rangle$ $\displaystyle\equiv$
$\displaystyle\frac{\int_{R_{\rm min}}^{R_{\rm max}}\Sigma(R)2\pi R{\rm
d}R}{\int_{R_{\rm min}}^{R_{\rm max}}2\pi R{\rm d}R}$ (34) $\displaystyle=$
$\displaystyle\begin{cases}{\rm m_{p}}/A_{\rm ann},&\text{if}\ R_{\rm
min}<R<R_{\rm max}\\\ 0,&\text{otherwise}\end{cases}$
where $A_{\rm ann}=\pi(R_{\rm max}^{2}-R_{\rm min}^{2})$ is the area of the
annulus.
We can similarly compute $\bar{\Sigma}(<R)={\rm m_{p}}/\pi R^{2}$ averaged
over the same annulus:
$\displaystyle\langle\bar{\Sigma}(<R)\rangle$ $\displaystyle\equiv$
$\displaystyle\frac{1}{A_{\rm ann}}\int_{R_{\rm min}}^{R_{\rm
max}}\bar{\Sigma}(<R)2\pi R{\rm d}R,$
which reduces to
$\displaystyle\langle\bar{\Sigma}(<R)\rangle=\frac{2{\rm m_{p}}}{A_{\rm
ann}}\ln(R_{\rm max}/a),$ (35)
where the value of $a$ depends on whether the point mass is located inside the
annulus:
$\displaystyle a=\begin{cases}R,&R>R_{\rm min}\\\ R_{\rm min},&R<R_{\rm
min}\end{cases}$ (36)
Using Eqs. 34 & 35 together with Eq. 9, we can calculate the value of
$\Delta\Sigma$ averaged over the annulus:
$\displaystyle\langle\Delta\Sigma(R)\rangle$ $\displaystyle\equiv$
$\displaystyle\langle\bar{\Sigma}(<R)-\Sigma(R)\rangle$ $\displaystyle=$
$\displaystyle\frac{{\rm m_{p}}}{A_{\rm ann}}G(R,R_{\rm min},R_{\rm max}),$
where
$\displaystyle G$ $\displaystyle=$ $\displaystyle\begin{cases}2\ln(R_{\rm
max}/R_{\rm min}),&R<R_{\rm min}\\\ 2\ln(R_{\rm max}/R)-1,&R_{\rm
min}<R<R_{\rm max}\\\ 0,&R_{\rm max}<=R.\end{cases}$
In order to calculate the average lensing profile
$\langle\Delta\Sigma(R)\rangle$ about an individual point in a N-body
simulation, Equation F generalizes in the obvious way as a sum over the point-
masses used to trace the simulated density field. For a simulated snapshot at
redshift $z_{\rm snap},$ if the cosmological distance $D(z_{\rm snap})$
satisfies $D(z_{\rm snap})\gg R_{\rm max},$ then we can use any of the
Cartesian axes of the snapshot as the line-of-sight, and the remaining two
axes to define $R.$
For a fixed choice of $R_{\rm min}$ and $R_{\rm max},$ the average lensing
profile $\langle\Delta\Sigma\rangle$ can be computed once and for all about
every subhalo in a simulated snapshot. Once tabulated, predicting the lensing
produced by a stack of subhalos can be computed simply by averaging over the
pre-computed values of $\langle\Delta\Sigma\rangle$ for each member of the
stack (or alternatively, by calculating an average that has been weighted by a
continuously-valued sample selection function, as in §4.2).
|
# Optimal Learning Rates for Regularized Least-Squares with a Fourier Capacity
Condition ††thanks: Citation: Authors. Title. Pages…. DOI:000000/11111.
Prem Talwai
Operations Research Center
MIT David Simchi-Levi
Institute for Data, Systems, and Society
MIT
###### Abstract
We derive minimax adaptive rates for a new, broad class of Tikhonov-
regularized learning problems in Hilbert scales under general source
conditions. Our analysis does not require the regression function to be
contained in the hypothesis class, and most notably does not employ the
conventional a priori assumptions on kernel eigendecay. Using the theory of
interpolation, we demonstrate that the spectrum of the Mercer operator can be
inferred in the presence of “tight” $L^{\infty}$ embeddings of suitable
Hilbert scales. Our analysis utilizes a new Fourier capacity condition, which
characterizes the optimal Lorentz range space of a modified Mercer operator in
certain parameter regimes.
## 1 Introduction
Consider the classical learning problem:
$Y=f^{*}(X)+\epsilon$ (1)
where we wish to estimate the regression function $f^{*}$ in the presence of
additive noise $\epsilon$ using a dataset
$\mathcal{D}=\\{(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots(X_{n},Y_{n})\\}$ sampled
i.i.d from a distribution $P$ over $\mathcal{X}\times\mathcal{Y}$. Here, we
focus on kernel-based regularized least-squares (ridge) regression, i.e. we
wish to solve the optimization problem
$f_{\lambda,D}=\text{arg}\min_{f\in\mathcal{H}_{K}}\frac{1}{n}\sum_{i=1}^{n}(Y_{i}-f(X_{i}))^{2}+\lambda||f||^{2}_{\mathcal{K}}$
(2)
where $\mathcal{H}_{K}\subset L^{2}(P_{\mathcal{X}})$ is a reproducing kernel
Hilbert space (RKHS) with kernel $K(\cdot,\cdot)$.
The performance of a regularization scheme is typically quantified using the
expected risk:
$R(f_{D,\lambda})=||f_{D,\lambda}-f^{*}||^{2}_{L^{2}(P)}$ (3)
This problem has been studied extensively in the statistics and machine
learning literature [17, 8, 4, 36]. These works primarily focus on
establishing optimal convergence rates for $R(f_{D,\lambda})$ under light-
tailed noise (typically subgaussian or subexponential) and various source
conditions, which characterize the regularity/smoothness of the learning
target $f^{*}$. The two main approaches for this task have included the
integral operator technique (e.g. [17, 4, 8]) and empirical process technique
[41, 31]. While the latter approach enjoys easy adaptability to general convex
losses, the former technique more directly exploits the Hilbertian structure
of the hypothesis space, and aligns seamlessly with the method of real
interpolation between $\mathcal{H}_{K}$ and $L^{2}(P_{\mathcal{X}})$ (see
section 2.1).
We consider the hard-learning scenario, where $f^{*}\not\in\mathcal{H}$. This
setting has been treated in previous works [17, 24, 4, 37, 36] both for the
Hölder source condition [17, 4] where $f^{*}$ lies in some interpolation space
$[L^{2}(P_{\mathcal{X}}),\mathcal{H}_{K}]_{\theta,2}$ (for $\theta\in(0,1)$)
and the general source condition [37] where $f^{*}$ lies in a suitable Hilbert
scale (defined in section 2.1), which correspond to more general interpolation
spaces between $L^{2}(P_{\mathcal{X}})$ and $\mathcal{H}_{K}$ obtained using a
function parameter. While these works treat the problem of misspecification,
their analysis hinges on a kernel eigendecay assumption, which characterizes
the capacity of the hypothesis space $\mathcal{H}_{K}$ by the decay rate of
the eigenvalues of the kernel Mercer operator. While this spectral assumption
has been ubiquitous in the statistical analysis of kernel ridge regression
[26, 4, 8], it depends crucially on the data-generating measure $P$, which is
typically unknown in practice.
In this paper, we replace the classical kernel eigendecay condition with a new
Fourier capacity condition which characterizes the decay of the Fourier power
spectrum of the kernel ($P$-independent) as opposed to its Mercer spectrum
($P$-dependent). Indeed, our capacity assumption relates to $P$ only through
the eigenbasis of the Mercer operator, which may be shared by several
different kernels and is often independent of the specific interplay between
$\mathcal{H}_{K}$ and $P$ (see discussion in section 2.4). Fourier analytic
approaches have been applied previously to study the capacity of the
hypothesis class [51, 49, 40]. However, these works typically either assume a
uniform (Lebesgue-equivalent) data-generating measure or particular RKHS
structure (i.e a Sobolev space) in order to apply Fourier transform
techniques. In this paper, we demonstrate that Fourier techniques remain
powerful over general probability spaces, in the presence of the embedding of
a suitable Hilbert scale in $L^{\infty}(\mathcal{X})$. Indeed, we demonstrate
that if the embedding of an intermediate Hilbert scale is sufficiently “sharp”
(characterized by the optimality of its index function in “covering” the range
of a modified Mercer operator), then a Gagliardo-Nirenberg type inequality
between the scales enables us to estimate kernel eigendecay via the Bernstein
widths of the RKHS in $L^{\infty}(\mathcal{X})$. To the best of our knowledge,
this is the first work to establish minimax learning rates under general
source conditions without appealing to any direct assumptions on the
eigendecay or effective dimension of the Mercer operator.
## 2 Preliminaries
We recall our learning problem:
$f_{\lambda,D}=\text{arg}\min_{f\in\mathcal{H}_{K}}\frac{1}{n}\sum_{i=1}^{n}(Y_{i}-f(X_{i}))^{2}+\lambda||f||^{2}_{\mathcal{K}}$
where $\mathcal{H}_{K}$ be a separable reproducing kernel Hilbert space (RKHS)
on $\mathcal{X}$ (see e.g. [41] for a definition). A solution to this problem
can be computed in closed form:
$f_{\lambda,D}=(C_{D}+\lambda)^{-1}g_{D}$
where $C_{D}=\mathbb{E}_{\mathcal{D}}[k(X,\cdot)\otimes k(X,\cdot)]$ is the
empirical covariance, and $g_{D}=\sum_{i=1}^{n}y_{i}k(x_{i},\cdot)$. Before
discussing our model framework and results, we begin with an overview of some
mathematical preliminaries required for this section.
### 2.1 Hilbert Scales
Suppose the imbedding $I_{\nu}:\mathcal{H}_{K}\to L^{2}(\nu)$ of
$\mathcal{H}_{K}$ into $\mathcal{L}^{2}(\nu)$ is injective (here $\nu=P_{X}$
is the marginal on $\mathcal{X}$). Let $S_{\nu}=I^{*}_{\nu}$ be its adjoint.
Then, it can be shown that $S_{\nu}$ is an integral operator given by:
$S_{\nu}f(x)=\int_{\mathcal{X}}k(x,\cdot)f(y)d\nu(y)$ (4)
Using $S_{\nu}$ and $I_{\nu}$, we construct the following positive self-
adjoint operators on $\mathcal{H}_{K}$ and $\mathcal{L}^{2}(\nu)$,
respectively:
$\displaystyle C_{\nu}$ $\displaystyle=S_{\nu}I_{\nu}=I^{*}_{\nu}I_{\nu}$
$\displaystyle T_{\nu}$ $\displaystyle=I_{\nu}S_{\nu}=I_{\nu}I^{*}_{\nu}$
We observe that $C_{\nu}$ and $T_{\nu}$ are nuclear (see Lemma 2.2/2.3 in
[42]) Since, $T_{\nu}$ is nuclear and self-adjoint, it admits a spectral
representation:
$T_{\nu}=\sum_{j=1}^{\infty}\mu_{j}e_{j}\langle
e_{j},\cdot\rangle_{L^{2}(\nu)}$
where $\\{\mu_{j}\\}_{j=1}^{\infty}\in(0,\infty)$ are nonzero eigenvalues of
$T_{\nu}$ (ordered nonincreasingly) and $\\{e_{j}\\}_{j=1}^{\infty}\subset
L^{2}(\nu)$ form an orthonormal system of corresponding eigenfunctions. Note
that formally, the elements $e_{j}$ of $L^{2}(\nu)$ are equivalence classes
$[e_{j}]_{\nu}$ whose members only differ on a set of $\nu$-measure zero—
notationally, we consider this formalism to be understood here and simply
write $e_{j}$ to refer to elements in both $\mathcal{H}_{K}$, $L^{2}(\nu)$,
and their interpolation spaces (with the residence of $e_{j}$ understood from
context). Given a nonincreasing index function
$\phi:(0,||T_{\nu}||)\to\mathbb{R}_{+}$, we define the Hilbert scales
$\mathcal{H}^{\phi}_{K}$ as [30]:
###### Definition 2.1.
Let $\phi:(0,||T_{\nu}||)\to\mathbb{R}_{+}$ be nondecreasing, continuous, with
$\phi(0)=0$. Then, the Hilbert scale $\mathcal{H}^{\phi}_{K}$ is the
completion of the space:
$\mathcal{H}^{\phi}_{K}=\Big{\\{}f\in
L^{2}(\nu):\sum_{i=1}^{\infty}\frac{\langle
f,e_{i}\rangle_{L^{2}(\nu)}^{2}}{\phi(\mu_{i})}<\infty\Big{\\}}$
with respect to the inner product $\langle
f,g\rangle_{\phi}=\sum_{i}\frac{\langle f,e_{i}\rangle\langle
g,e_{i}\rangle}{\phi(\mu_{i})}$.
It is easy to see that
$\mathcal{H}^{\phi}_{K}\cong\text{ran}(\phi^{\frac{1}{2}}(T_{\nu}))$. The
subscript $K$ in $\mathcal{H}^{\phi}_{K}$ reflects that $\phi$ is acting on
the spectrum of $K$, when the kernel is fixed and understood from context, we
will omit this subscript and simply denote the Hilbert scale
$\mathcal{H}^{\phi}$.
We define:
$\|k^{\phi}\|^{2}_{\infty}=\sup_{x\in\mathcal{X}}\sum_{i=1}^{\infty}\phi(\mu_{i})e_{i}^{2}(x)$
where we allow $\|k^{\phi}\|_{\infty}=\infty$. If
$\|k^{\phi}\|_{\infty}<\infty$, then it is easy to show that
$\mathcal{H}^{\phi}_{K}$ is continuously embedded in $L^{\infty}(\mathcal{X})$
with norm $\|k^{\phi}\|_{\infty}$ (see e.g. Theorem 9 in [17] for the case of
Holder source conditions).
The Hilbert scale $\mathcal{H}^{\phi}_{K}$ generalizes the notion of RKHS
interpolation (power) spaces discussed in [42], which result from the specific
choice of $\phi(t)=t^{\alpha}$ for some $\alpha\in(0,1)$. Intuitively, the
Hilbert scales can be viewed as a nonlinear transformation of the infinite-
dimensional RKHS ellipsoid $B(\mathcal{H}_{K})$, realized by transforming the
axes lengths with the index function $\phi$. As we will see later, the growth
properties of $\phi$ play a fundamental role in specifying the smoothness of
the elements of $\mathcal{H}^{\phi}_{K}$.
### 2.2 Rearrangement Invariant Spaces
Let $(R,\mu)$ be a $\sigma$-finite measure space. For any positive increasing
function $\phi:(0,\infty)\to(0,\infty)$, we define the dilation function
$d_{\phi}:(0,\infty)\to(0,\infty)$ as:
$d_{\phi}(t)=\sup_{s\in(0,\infty)}\frac{\phi(st)}{\phi(s)}$ (5)
We define the extension indices:
$\displaystyle\alpha_{\phi}$ $\displaystyle\equiv\lim_{t\to 0}\frac{\log
d_{\phi}(t)}{\log t}$ $\displaystyle\beta_{\phi}$
$\displaystyle\equiv\lim_{t\to\infty}\frac{\log d_{\phi}(t)}{\log t}$
If $\phi$ has finite extension indices (i.e.
$-\infty<\alpha_{\phi}<\beta_{\phi}<\infty$), we define the Lorentz endpoint
space $\Lambda_{\phi}$ and Marcinkiewicz endpoint space $M_{\phi}$ via the
(quasi)-norms:
$||g||_{M_{\phi}}=\sup_{t\in(0,\mu(R))}\phi(t)g^{*}(t)\hskip
11.38109pt||g||_{\Lambda_{\phi}}=\int_{0}^{\mu(R)}g^{*}(t)d\phi(t)$
where $g^{*}$ denotes the decreasing rearrangement of $g$ (see e.g. [34]) for
a definition). Note, in the definition of $M_{\phi}$, we follow [35], rather
than the more standard convention in [34] (which replaces $g^{*}$ by the
maximal decreasing rearrangement), in order to allow for quasi-normed spaces
(under suitable growth conditions on $\phi$, which we demonstrate later on,
these two definitions are equivalent by Proposition 7.10.5 in [34]).
Every rearrangement invariant space $X$ with fundamental function $\phi$ lies
between $\Lambda_{\phi}$ and $M_{\phi}$, i.e. $\Lambda_{\phi}\subset X\subset
M_{\phi}$ (Propositions 7.10.6 and 7.10.15 in [34]). For $p\geq 1$, we hence
define the $p$-Lorentz spaces $\Lambda^{p}_{\phi}$ as:
$\Lambda^{p}_{\phi}=\Big{\\{}g:\int_{0}^{\mu(R)}(g^{*}(t))^{p}d\phi^{p}(t)<\infty\Big{\\}}$
(6)
A prototypical example of this scale are the “classical” Lorentz spaces:
$\Lambda_{t^{\frac{1}{p}}}(\mathbb{R}^{d})\equiv
L^{p,1}(\mathbb{R}^{d})\subset L^{p,q}(\mathbb{R}^{d})\subset
L^{p,\infty}(\mathbb{R}^{d})\equiv M_{t^{\frac{1}{p}}}$ for $q\in(1,\infty)$.
For a continuous, increasing function $\Phi$, we may further define the
Orlicz-Lorentz spaces $L^{\Phi,w}$ [19] containing measurable functions for
which the quantity:
$||f||_{\Phi,w}=\inf\Big{\\{}\lambda>0:\int_{0}^{\infty}\Phi\Big{(}\frac{f^{*}(t)}{\lambda}\Big{)}w(t)dt\leq
1\Big{\\}}$ (7)
is finite (where $w(t)$ is a decreasing weight function). Note that
$L^{\Phi,w}$ in (7) (following [19]) are sometimes known as weighted Orlicz
spaces, with Orlicz-Lorentz spaces instead being defined with respect to the
homogeneous measure $\frac{dt}{t}$ (e.g. [2]). This vernacular difference
significantly impacts the interpolation properties of the Lorentz-Orlicz space
(as elaborated below). More precisely, the latter definition enables the space
to be characterized as an “ultrasymmetric” space intermediate between Lorentz
and Marcinkiewicz endpoint spaces, while this characterization is rarely
possible with the formulation in (7) (see [2]). We use the formulation in (7)
for the added generality provided by the nonincreasing weight function $w(t)$.
When, $w=1$, we simply write $L^{\Phi}$.
It is also important to note that we have again not restricted $\Phi$ to be an
Orlicz or Young function. Hence, (7) only defines a quasi-norm. In Lemma 2 in
the appendix, we derive a fundamental function for for $L^{\Phi,w}$ (up to
equivalence) when the extension indices $\alpha_{\Phi}$ and $\beta_{\Phi}$
coincide.
Finally, we say that a function $\xi$ satisfies $\Delta_{2}$ condition if
there exists constants $D_{1},D_{2}>0$, such that:
$D_{1}\xi(\lambda)\leq\xi(2\lambda)\leq D_{2}\xi(\lambda)\hskip
5.69054pt\forall\lambda>0$ (8)
### 2.3 Approximation Widths
We review some standard widths in approximation theory that describe the
compactness of spaces in various norms. The $n^{\text{th}}$ entropy number
$\epsilon_{n}(\mathcal{X})$ of a compact set $\mathcal{X}$, is informally the
smallest radius $\epsilon$ such that there exists an $\epsilon$-covering of
$\mathcal{X}$ of at most $n$ balls. Precisely, we have:
$\epsilon_{n}(\mathcal{X})=\inf\\{\epsilon>0:\exists\hskip
1.42262pt\\{x_{i}\\}_{i=1}^{n}\subset\mathcal{X};\text{s.t.}\hskip
5.69054pt\mathcal{X}\subset\cup_{i=1}^{n}B(x_{i},\epsilon)\\}$ (9)
We also define the $n$-Bernstein width of a Banach space $X$ embedded in
another Banach space $Y$:
$b_{n}(X,Y)=\sup_{Z\subset X,\text{dim}(Z)\geq n+1}\inf_{z\in
Z}\frac{||z||_{Y}}{||z||_{X}}$ (10)
Intuitively, the Bernstein width gives the radius of the largest $Y$-ball that
may be inscribed in a unit $X$-ball in subspaces of dimension greater than
$n$. When $X$ and $Y$ are both Hilbert spaces, then $b_{n}$ coincides with the
$n+1^{\text{st}}$ singular number of the embedding $X\hookrightarrow Y$.
Moreover, if $X\hookrightarrow Y$ is compact, then $b_{n}(X,Y)\to 0$. For a
detailed discussion of the relationships between various approximation widths
and function space embeddings, we direct the reader to [9].
###### Remark (Notation).
For any two Banach spaces $\mathcal{A}$ and $\mathcal{B}$, we write
$\mathcal{A}\hookrightarrow\mathcal{B}$, if $\mathcal{A}$ is continuously
embedded in $\mathcal{B}$, and $\mathcal{A}\stackrel{{\scriptstyle
c}}{{\hookrightarrow}}\mathcal{B}$ if this embedding is further compact. We
denote by $B(\mathcal{A})$ the unit ball in $\mathcal{A}$.
$\mathbb{E}_{\mathcal{D}}[\cdot]$ denotes the sample expectation, and
$C_{D}=\mathbb{E}_{\mathcal{D}}[k(X,\cdot)\otimes k(X,\cdot)]$ denotes the
sample covariance. We will denote by $f_{D,\lambda}$ the solution to (2) and
$f_{\lambda}=(C_{\nu}+\lambda)^{-1}S_{\nu}f^{*}$, the regularized population
solution. $\mathcal{F}$ denotes the Fourier transform, while $\mathcal{F}_{d}$
denotes the $d$-dimensional Fourier transform of a univariate function (see
e.g. Theorem 5.26 in [47]). $L^{p}(\mathcal{X},\mu)$ denotes the $L^{p}$ space
on the measure space $(\mathcal{X},\mu)$ — the domain will be omitted in the
notation if it is understood from context; likewise the measure will be
omitted if it is equivalent to Lebesgue measure. $||\cdot||_{2}$ or
$||\cdot||_{L^{2}(\nu)}$ are used to for norms in $L^{2}(\nu)$ (the latter if
we want to make the space explicit); $||\cdot||_{\phi}$ for the norm in
Hilbert scale $\mathcal{H}^{\phi}$, and $||\cdot||_{K}$ for the RKHS
$\mathcal{H}_{K}$. Typically, we will take $\nu=P_{X}=P|_{\mathcal{X}}$ where
$P$ is the data-generating measure, and $|_{\mathcal{Z}}$ denotes the
“marginal on” the space $\mathcal{Z}$. Finally, we will write $a(x)\preceq
b(x)$ ($a(x)\succeq b(x)$) if, $a(x)\leq Kb(x)$ (resp. $b(x)\leq Ka(x)$), for
some $K>0$ that depends only on problem parameters (and independent of $x$ or
the confidence level in the statement). If $a\preceq b$ and $b\preceq a$, then
we write $a\asymp b$.
### 2.4 Assumptions
Let $\phi$ and $\psi$ be two admissible index functions (i.e. $\phi$ and
$\psi$ are nondecreasing, nonnegative, and continuous, with
$\phi(0)=\psi(0)=0$)
###### Assumption 1.
$H_{K}\stackrel{{\scriptstyle
c}}{{\hookrightarrow}}H^{\phi}\stackrel{{\scriptstyle
c}}{{\hookrightarrow}}H^{\psi}\hookrightarrow L^{\infty}$
###### Assumption 2.
(a) $\frac{t}{\phi(t)}$ and $\frac{\phi(t)}{\psi(t)}$ are nondecreasing, and
$\frac{t}{\psi(t)}$ is concave; (b) $\phi,\psi$ satisfy the $\Delta_{2}$
condition (8) for some $D^{\phi}_{1},D^{\phi}_{2},D^{\psi}_{1},D^{\psi}_{2}>1$
###### Assumption 3.
$f^{*}\in\mathcal{H}^{\phi}$
###### Assumption 4.
$\sup\Big{\\{}\Big{\|}\sum_{i=1}^{n}c_{i}e_{i}(\cdot)\Big{\|}_{\infty}:\sum_{i=1}^{n}c^{2}_{i}=1\Big{\\}}\asymp\sqrt{s(n)}$
for some increasing functions $s$, where
$\sum_{n=1}^{\infty}\frac{1}{s(n)}<\infty$ and $s$ satisfies (8) for some
$D^{s}_{1},D^{s}_{2}>1$
###### Assumption 5.
There are constants $\sigma,L>0$ such that:
$\int_{\mathbb{R}}|y-f^{*}(x)|^{m}dP(y|x)\leq\frac{m!\sigma^{2}L^{m-2}}{2}$
for $\nu$-almost all $x\in\mathcal{X}$
###### Assumption 6*.
$k$ is radial, i.e. $k(x,y)=\kappa(||x-y||)$ for some positive definite
function $\kappa\in L^{1}(\mathbb{R})$, such that either
$d_{\kappa}(t)<\infty$ or $d_{\mathcal{F}_{d}\kappa}(t)<\infty$ for all $t>0$.
###### Assumption 7*.
$\mathcal{X}\subset\mathbb{R}^{d}$ is compact with
$\epsilon_{n}(\mathcal{X})\asymp n^{-\frac{1}{d}}$ for $n\in\mathbb{N}$.
###### Assumption 8*.
$\psi\Big{(}\frac{\mathcal{F}_{d}\kappa(t^{\frac{1}{d}})}{s(t)}\Big{)}\asymp\frac{1}{s(t)}$
as $t\to\infty$
Assumptions 1 and 2 dictates the relative growth rates of the index functions
$\phi$ and $\psi$. Indeed, the compactness of the embedding
$H^{\phi}\stackrel{{\scriptstyle c}}{{\hookrightarrow}}H^{\psi}$ ensures that
$\frac{\phi(t)}{\psi(t)}\to 0$ as $t\to 0$. Moreover, the first embedding
$H_{K}\stackrel{{\scriptstyle c}}{{\hookrightarrow}}H^{\phi}$ ensures that
both $\frac{t}{\phi(t)}\to 0$ and $\frac{t}{\psi(t)}\to 0$ as $t\to 0$.
Intuitively, this condition implies that the norms in $\mathcal{H}_{\psi}$ and
$\mathcal{H}_{\phi}$ are weaker than those in our hypothesis class
$\mathcal{H}_{K}$, allowing us to study convergence in these larger spaces of
sample estimators constructed in $\mathcal{H}_{K}$. The second part of the
assumption requiring $\frac{t}{\phi(t)}$ and $\frac{\phi(t)}{\psi(t)}$ to be
nondecreasing extends this behavior as $t\to\infty$; collectively the two
growth conditions characterize $\phi^{-1}$ and $\psi^{-1}$ as resembling
Young’s functions; a requirement commonly imposed in the study of Orlicz
spaces [1]. The requirement that $\frac{t}{\phi(t)}$ is additionally concave
enables the application of a Gagliardo-Nirenberg type interpolation inequality
[29] that relates $L^{\infty}$ norms to those in $\mathcal{H}_{K}$ and
$L^{2}(\nu)$, which will be crucial in our analysis of uniform error rates.
Finally, the condition that index functions are $\Delta_{2}$ is found quite
commonly in the literature on statistical inverse problems [30, 39], where it
has been used to demonstrate adaptivity of balancing strategies to unknown
source conditions. It should be noted that all the growth conditions in
Assumption 2 are satisfied when $\phi(t)=t^{\beta}$ and $\psi(t)=t^{\alpha}$
are power functions with $0<\alpha<\beta<1$ (the so-called Hölder source
conditions).
Assumption 3 characterizes the smoothness of the learning target $f^{*}$. Note
a distinctive feature of our analysis is that we allow $f^{*}$ to lie outside
the hypothesis class $\mathcal{H}_{K}$ (in light of Assumption 1), while still
in $L^{\infty}(\mathcal{X})$ (see Assumption 1). Assumption 5 is standard
subexponential noise condition in kernel regression [17, 43].
Assumption 4 characterizes the embedding of $L^{\infty}$ in $L^{2}(\nu)$ via
the eigenspaces of $T_{\nu}$. The “Lebesgue” function
$M_{n}(x)=\sum_{i=1}^{n}e_{i}(x)$ was first considered by Littlewood [18] for
the Fourier basis; later its $L^{p}$-norms were studied in several works [21,
6, 7], primarily in the context of finding roots for various generalized
polynomials. [44] and [20] later explored the intimate connection between such
Lebesgue-type inequalities and the greedy approximation properties of the
basis $\\{e_{i}\\}_{i=1}^{\infty}$. Indeed, it can be seen directly from the
definition (10) that assumption 4 implies a lower bound of $s(n)^{-1}$ on the
Bernstein widths $b^{2}_{n-1}(L^{\infty},L^{2}(\nu))$ of $L^{\infty}$ in
$L^{2}(\nu)$. Note, that although the orthonormality of
$\\{e_{i}\\}_{i=1}^{\infty}$ in $L^{2}(\nu)$ is $\nu$-dependent, this system
is always an orthonormal basis in $\mathcal{H}_{K}$ by separability and the
injectivity of $I_{\nu}$ (see Theorem 3.3 in [42]). Moreover, Assumption 4 is
independent of the Mercer eigenvalues $\\{\mu_{i}\\}_{i=1}^{\infty}$ and is
hence applicable to any RKHS with a common eigenbasis (e.g. the popular
Fourier basis in $L^{2}([-\pi,\pi]^{d})$). These situations arise commonly in
the study of stochastic differential equations [13, 14], where $\nu$ is a
Gaussian measure, $\\{e_{i}\\}_{i=1}^{\infty}$ is a fixed basis of
multivariate Hermite polynomials, and $\\{\mu^{-1}_{i}\\}_{i=1}^{\infty}$ are
the (possibly unknown) eigenvalues of a general operator drift coefficient.
The square of the left-hand side in Assumption 4 is known as the Christoffel
function, whose asymptotic growth (and its interaction with the measure $\mu$)
is an active area of research [28, 48, 3] with diverse applications to
quadrature/interpolation and random matrices [45] and [3]. Moreover, it is
important to note that Assumption 3 is significantly more general than the
more typical assumption of uniformly bounded bases, which has been shown to be
violated by several common kernels (see discussion in [50, 42]).
While Assumptions 6*-8* are not necessary for establishing the upper bound in
Theorem 2, they are required to demonstrate its optimality. Namely, the
radiality assumption on the kernel $k$ in Assumption 6* enables the tight
estimation of the approximation widths of $\mathcal{H}_{K}$ in $L^{\infty}$
via the compactness properties of the domain $\mathcal{X}$, the latter of
which are characterized by the entropy numbers specified in Assumption 7*. The
dilation condition can be satisfied either by the kernel or its
$d$-dimensional Fourier transform; an example of the latter being the popular
Matern kernel. This dilation condition is primarily intended for streamlining
the analysis, it could be removed at the expense of requiring Assumption 8* be
satisfied instead by a dilation of the Fourier transform
$\mathcal{F}_{d}\kappa$ (this tradeoff is highlighted in Remark Remark in the
Appendix). Similarly, the precise asymptotic behavior of the entropy numbers
of $\mathcal{X}$ is only included for simplicity; the general case would
require replacing $\mathcal{F}_{d}\kappa(t^{\frac{1}{d}})$ with
$\mathcal{F}_{d}\kappa(\epsilon_{t}(\mathcal{X})^{-1})$ in Assumption 8*.
Assumption 8* ensures that the index function $\psi$ is indeed the “optimal”
choice for characterizing the embedding of our hypothesis class
$\mathcal{H}_{K}$ in $L^{\infty}$. It is important to note that while
Assumptions 1 and 3 implicitly depend on the ambient measure $\nu$ (as the
index functions act spectrally on the Mercer operator $T_{\nu}$), the
optimality condition is independent of this measure (which is typically
unknown in practice and determined by the data-generating process) given the
orthonormal basis $\\{e_{i}\\}_{i=1}^{\infty}$ of $\mathcal{H}_{K}$ in
Assumption 4. This contrasts with the exact decay rates of the eigenvalues
$\mu_{i}(T_{\nu})$ or the effective dimension [17, 4] traditionally required
to establish minimax optimal learning rates. Indeed, as is shown in Appendix
C, the combination of Assumptions 1, 2, 4, and 8* enable us to infer that
$\mu_{i}\asymp\psi^{-1}(s(i)^{-1})$ via a comparison of
$b^{2}_{n}(\mathcal{H}_{K},L^{\infty})$ (estimated by
$\mathcal{F}_{d}\kappa(n^{\frac{1}{d}}))$) and
$b^{2}_{n}(L^{\infty},L^{2}(\nu))$ (estimated by $s(n)^{-1}$). In the
following lemma, we demonstrate Assumption 8* is satisfied when $\psi$
characterizes the “optimal” scale of range spaces for a modified Mercer
operator.
###### Lemma 1.
Suppose $s(n)=n^{\alpha}$ for some $\alpha>1$ and
$\alpha_{\psi}=\beta_{\psi}\equiv\rho$. Define the Fourier multiplier operator
$T$ by
$\mathcal{F}(Tf)=\mathcal{F}k(||\cdot||^{\frac{1}{\alpha}})\mathcal{F}f$. Let
$\Lambda_{\text{rad}}=\\{f=\tilde{f}(||x||):\tilde{f}(r)r^{\frac{1}{2}}\in
L^{1}(0,\infty),\hskip 2.84526pt\tilde{f}(r)r^{d-1}\hskip
2.84526pt\text{decreasing}\\}$ be a space of radially decreasing functions on
$\mathbb{R}^{d}$. Then, if $\rho>\frac{p}{p+1}$ and Assumption 8* holds, $T$
is bounded from $L^{p,q}(\mathbb{R}^{d})\cap\Lambda_{\text{rad}}$ to
$\Lambda^{q}_{\Psi}$ where $\Psi$ is the fundamental function of
$L^{\psi,t^{\bar{p}}}$ with $\bar{p}=\rho\Big{(}1+\frac{1}{p}\Big{)}-2$.
Moreover, $\Lambda^{q}_{\Psi}$ is the smallest $q$-Lorentz space satisfying
this conclusion.
The choice of polynomial growth $s(n)=n^{\alpha}$ is inspired by Lebesgue-type
inequalities for common greedy bases (see e.g. Chapter 2 of [44]). Note, that
in the limiting case when $\alpha=1$, $T$ is simply the Mercer operator
$T_{m}$ with respect to the Lebesgue measure $m$. Hence, intuitively Lemma 1
states that when $\alpha_{\psi}=\beta_{\psi}$ the interpolation scale
$(\Lambda_{\Psi},M_{\Psi})$ characterizes the optimal range space for the
kernel embedding.
The requirement $\alpha_{\psi}=\beta_{\psi}$ is crucial in Lemma 1 in order to
identify the couple $(\Lambda_{\Psi},M_{\Psi})$ with an Orlicz-Lorentz space.
This is unsurprising, when considered in analogy with the Banach space case
(i.e. when $\psi$ is a Young function), for which stronger results hold [25].
Indeed, there it is known that $L^{\psi}$ is only a Lorentz endpoint space
when $\psi(t)=t^{\frac{1}{r}}v(t)$ for some $r\in(1,\infty)$ and function $v$
that satisfies the Lorentz condition (Theorem 3 in [25]) which implies sub-
power growth. Hence, intuitively, Lemma 1 suggests that we may only associate
the optimal range scale for $T$ with an Orlicz-Lorentz space when the latter
space itself also “belongs” to this interpolation scale (see Remark after the
proof of Lemma 1 for further discussion).
We observe $\alpha_{\psi}=\beta_{\psi}\equiv\rho$ most notably when
$\psi(t)=t^{\rho}$ (e.g. Assumption EMB in [17]). Here, Lemma 1 states that
$T$ is bounded from $L^{p,q}(\mathbb{R}^{d})\cap\Lambda_{\text{rad}}$ to
$L^{\frac{p\rho^{*}}{\rho^{*}+p},q}(\mathbb{R}^{d})$, where $\rho^{*}<0$ is
the dual exponent of $\rho\in(0,1)$. Observe that
$\frac{p\rho^{*}}{\rho^{*}+p}>1$ as $\rho<p$ by Assumption 2.
In the general case of $\alpha_{\psi}<\beta_{\psi}$, we only obtain the
following Boas-type result (see [23]):
###### Corollary 1.1.
Suppose $s(n)=n^{\alpha}$ for some $\alpha>1$. Define the Fourier multiplier
operator $T$ as in Lemma 1. Then, if $\beta_{\psi}>\frac{p}{p+1}$ and
Assumption 8* holds, $T$ is bounded from
$L^{p,q}(\mathbb{R}^{d})\cap\Lambda_{\text{rad}}$ to $\Lambda^{q}_{\Psi}$
where $\Psi(t)=\frac{t^{1+\frac{1}{p}}}{\psi^{-1}(t)}$. Moreover,
$\Lambda^{q}_{\Psi}$ is the smallest $q$-Lorentz space satisfying this
conclusion.
#### 2.4.1 Related Work
Assumptions 1 and 3 can be viewed as a generalization of those in [17] to
Hilbert scales. Namely, these assumptions reduce to the corresponding source
and embedding conditions in [17] with the choice of $\phi(t)=t^{\beta}$ and
$\psi(t)=t^{\alpha}$ (using their notation). The additional embedding
$H^{\phi}\hookrightarrow H^{\psi}$ in Assumption 1 is not strictly necessary
for the derivation of minimax rates, but helps simplify the analysis.
Intuitively, this embedding ensures that $f^{*}$ is bounded.
We emphasize that a distinctive feature of our analysis is that we do not make
any assumption on the eigendecay of $T_{\nu}$ or its effective dimension.
Indeed, while [4, 5] established learning rates in the stronger
$\|\cdot\|_{K}$ norm for more general regularization schemes, their analysis
only considered Holder-type source conditions and hinged on a particular
asymptotic eigendecay of $T_{\nu}$ (which depends on the unknown measure
$\nu=P|_{\mathcal{X}}$). [36] and [37] consider general source conditions
similar to this paper (and for more general regularization schemes), although
they too place conditions on the decay of the effective dimension of $T_{\nu}$
(which is strongly related to its eigendecay). Recently, [26] considered
Lepski-based parameter selection strategy for general regularization schemes,
and demonstrated that known estimates on the effective dimension of $T_{\nu}$
can be effectively leveraged for adapting to misspecification.
To the best of our knowledge, this is the first work to establish minimax
learning rates under general source conditions without appealing to any direct
assumptions on the eigendecay or effective dimension of the Mercer operator
$T_{\nu}$. Instead, our approach leverages the $L^{\infty}$ embedding of the
Hilbert scale $\mathcal{H}^{\psi}$, the distance between $L^{\infty}$ and
$L^{2}(\nu)$ (Assumption 4), and a compatibility condition between $\psi$ and
the Fourier transform of the kernel (Assumption 8*) to infer spectral
information on $T_{\nu}$. In a sense, our analysis demonstrates that, via an
interpolation inequality, evaluating the tightness of the embedding
$\mathcal{H}^{\psi}\hookrightarrow L^{\infty}$ is in a sense equivalent to
evaluating the optimality of $\psi$ in characterizing the range of a Mercer-
type operator (via Lemma 1). When $\psi$ is indeed optimal, the interpolation
inequality (Lemma 1) provides the correct “change of measure”. However, this
delicate relationship hinges on the radiality of the kernel, and does not
obviously extend to more general kernel families.
### 2.5 Results
We first present our upper bound on the mean-square error in Theorem 2.
###### Theorem 2.
Suppose Assumptions 1-8* hold. Let $\tilde{s}(t)=s^{-1}(t^{-1})$. Choosing
$\lambda_{n}\asymp(\frac{\phi}{\tilde{s}\circ\psi})^{-1}(n^{-1})$, we obtain,
with probability $1-2\delta$:
$||f_{D,\lambda_{n}}-f^{*}||_{L^{2}(\nu)}\preceq\log(\delta^{-1})\sqrt{\phi\Big{(}\Big{(}\frac{\phi}{\tilde{s}\circ\psi}\Big{)}^{-1}(n^{-1})\Big{)}}$
(11)
Intuitively, the function
$\phi\Big{(}\Big{(}\frac{\phi}{\tilde{s}\circ\psi}\Big{)}^{-1}\Big{)}$
appearing in (11) captures the distance between $\mathcal{H}^{\phi}$ and the
hypothesis class $\mathcal{H}_{K}$, i.e. the “degree of misspecification”.
Indeed, since $\mu_{i}\asymp\psi^{-1}(s(i)^{-1})$ (as shown in Appendix C),
the upper rate effectively compares $\phi(\mu_{i})$ to $\mu_{i}$. For example,
when $\phi(t)=t^{\beta}$ and $\mu_{i}\asymp i^{-\frac{1}{p}}$ (for
$\beta,p\in(0,1)$), then
$\phi\Big{(}\Big{(}\frac{\phi}{\tilde{s}\circ\psi}\Big{)}^{-1}\Big{)}(n^{-1})\asymp
n^{-\frac{\beta}{p+\beta}}$. If $\frac{\beta}{p}$ is close to $1$, then
$\mathcal{H}^{\phi}$ is weakly embedded in $L^{2}(\nu)$ (i.e. “far” from
$\mathcal{H}_{K}$) and the setting is poorly specified, producing the slow
rate of $n^{-\frac{1}{4}}$. If $\beta\gg p$ is large, i.e $\mathcal{H}^{\phi}$
is close to $\mathcal{H}_{K}$, our setting is well-specified, and we approach
the fast rate of $n^{-\frac{1}{2}}$.
The proof of Theorem 2 hinges on the following bias-variance decomposition,
which illustrates that, up to a constant, both components of the error can be
controlled purely in terms of $\phi,\psi$, and $s$:
###### Lemma 3.
Suppose Assumptions 1-8* hold. Then:
$||f^{*}-f_{D,\lambda}||_{L^{2}(\nu)}\preceq||f^{*}||_{\phi}\sqrt{\phi(\lambda)}+||k_{\psi}||_{\infty}||f^{*}||_{\phi}\sqrt{\frac{\log(2\delta^{-1})}{n}\Big{(}\frac{1}{n\psi(\lambda)}+\frac{\phi(\lambda)}{\psi(\lambda)}+s^{-1}(\psi(\lambda)^{-1})\Big{)}}$
(12)
with probability $1-3\delta$.
As noted above, we observe here that the kernel only appears in (12) via the
constant term $||k_{\psi}||_{\infty}$, and hence $k$ does not influence the
asymptotic behavior of the upper bound as $\lambda\to 0$. This surprising
absence of the kernel (beyond the eigenbasis $\\{e_{i}\\}_{i=1}^{\infty}$),
can be attributed to the embedding conditions in Assumption 1, which ensure
that (via a Gagliardo-Nirenberg type inequality) uniform bounds on the bias
depend only on the ratios between norms in $\mathcal{H}^{\psi}$, and
$\mathcal{H}^{\phi}$ and $\mathcal{H}_{K}$, respectively. Moreover, due to
relative growth rates of the index functions implied by compactness, these
capacity ratios depend solely on $\phi$, $\psi$, and $s$.
We now discuss the optimality of the upper bound in Theorem 2. The derivation
of a minimax lower bound matching (11) is significantly more involved and
hinges crucially on the sharpness of the embedding
$\mathcal{H}^{\psi}\hookrightarrow L^{\infty}$, which in turn depends on the
compactness properties of the domain $\mathcal{X}\subset\mathbb{R}^{d}$
(captured by the entropy condition in Assumption 7*) and the smoothness of the
kernel $k$. Unlike the derivation of the upper bound, here the analysis does
not reduce simply to a comparison of the index functions, as we must
demonstrate the maximality of the spectral function $\psi$ in “stretching” the
kernel class $\mathcal{H}_{K}$. Informally, we need to ensure that the
enlarged (infinite-dimensional) ellipsoid $B(\psi(\mathcal{H}_{K}))$ is
“maximal” in $L^{\infty}$, otherwise our use of $\mathcal{H}^{\psi}$ as a
proxy for $L^{\infty}$ in the derivation of (11) is suboptimal (i.e.
$||\cdot||_{\mathcal{H}^{\psi}}$ may provide a very coarse approximation of
$||\cdot||_{\infty}$). We quantify this maximality via estimating the
Bernstein widths of $\mathcal{H}_{K}$ in $L^{\infty}$, which can then be
compared to the eigenvalues $\mu_{i}$ (the Bernstein widths of
$\mathcal{H}_{K}$ in $L^{2}(\nu)$; see e.g. Prop. 5 in [29]) via the Gagliardo
interpolation inequality. The former widths are independent of the measure
$\nu$ and depend only on the smoothness of $\kappa$ (captured by its Fourier
decay) and the compactness of $\mathcal{X}$. Intuitively, the embedding
$\mathcal{H}^{\psi}\hookrightarrow L^{\infty}$ is sharp (and our learning rate
in (11) is optimal) if the index function $\psi$ provides the correct “change-
of-measure” from $\nu$ to the $d$-dimensional Lebesgue measure.
###### Theorem 4.
There exists a distribution $P$ on $\mathcal{X}\times\mathbb{R}$ satisfying
Assumption 5 with $P|_{\mathcal{X}}=\nu$, $||f^{*}_{P}||_{\phi}\leq B_{\phi}$,
$||f^{*}_{P}||_{\infty}\leq B_{\infty}$, such that, for any learning algorithm
$\mathcal{D}\mapsto f_{D}$, we have with $P^{n}$ probability not less than
$1-\frac{432\tau K^{\log\tau}}{U_{d,\infty,\phi}\sigma^{2}\log 2}$:
$||f_{D}-f^{*}_{P}||_{L^{2}(\nu)}\succeq\sqrt{\tau\phi\Big{(}\Big{(}\frac{\phi}{\tilde{s}\circ\psi}\Big{)}^{-1}(n^{-1})\Big{)}}$
where $U_{d,\infty,\phi}$ does not depend on $n$ or $\tau$ and $K>1$ is an
absolute constant.
## 3 Conclusion
In this paper, we derive novel minimax learning rates for ridge regression
over Hilbert scales with general source conditions. A notable feature of our
analysis is that we do not impose the conventional conditions on eigendecay
rates, and instead present a new approach based on estimating the Bernstein
widths of the kernel class in $L^{\infty}$ to demonstrate minimax optimality.
We introduce a new Fourier capacity condition, which characterizes the
complexity of the kernel via the optimal Lorentz range space of a modified
Mercer operator. Future work involves extending our convergence analysis to
stronger interpolation norms and online regression settings.
## References
* [1] Waldo Arriagada “Matuszewska–orlicz indices of the sobolev conjugate young function” In _Partial Differential Equations in Applied Mathematics_ 3 Elsevier, 2021, pp. 100029
* [2] Sergei V Astashkin and Lech Maligranda “Ultrasymmetric Orlicz spaces” In _Journal of mathematical analysis and applications_ 347.1 Elsevier, 2008, pp. 273–285
* [3] P Nevai AT983168RF14 and Géza Freud “Orthogonal polynomials and Christoffel functions: A case study” In _J. Approx. Theory_ 48, 1986, pp. 3–167
* [4] Gilles Blanchard and Nicole Mücke “Optimal rates for regularization of statistical inverse learning problems” In _Foundations of Computational Mathematics_ 18.4 Springer, 2018, pp. 971–1013
* [5] Gilles Blanchard and Nicole Mücke “Kernel regression, minimax rates and effective dimensionality: Beyond the regular case” In _Analysis and Applications_ 18.04 World Scientific, 2020, pp. 683–696
* [6] Sergei Viktorovich Bochkarev “Multiplicative inequalities for the L 1 norm: Applications in analysis and number theory” In _Proceedings of the Steklov Institute of Mathematics_ 255.1 Springer, 2006, pp. 49–64
* [7] Peter Borwein and Richard Lockhart “The expected $L_{p}$ norm of random polynomials” In _Proceedings of the American Mathematical Society_ 129.5, 2001, pp. 1463–1472
* [8] Andrea Caponnetto and Ernesto De Vito “Optimal rates for the regularized least-squares algorithm” In _Foundations of Computational Mathematics_ 7.3 Springer, 2007, pp. 331–368
* [9] Bernd Carl and Irmtraud Stephani “Entropy, Compactness and the Approximation of Operators”, Cambridge Tracts in Mathematics Cambridge University Press, 1990 DOI: 10.1017/CBO9780511897467
* [10] M Carro, Luboš Pick, Javier Soria and Vladimir D Stepanov “On embeddings between classical Lorentz spaces” In _Mathematical Inequalities and Applications_ 4 ELEMENT, 2001, pp. 397–428
* [11] C Carton-Lebrun and HP Heinig “Weighted Fourier transform inequalities for radially decreasing functions” In _SIAM journal on mathematical analysis_ 23.3 SIAM, 1992, pp. 785–798
* [12] Marianna Chatzakou and Vishvesh Kumar “$L^{p}L^{q}$ boundedness of Fourier multipliers associated with the anharmonic Oscillator” In _arXiv preprint arXiv:2004.07801_ , 2020
* [13] Pao-Liu Chow “Infinite-dimensional Kolmogorov equations in Gauss-Sobolev spaces” In _Stochastic Analysis and Applications_ 14.3 Taylor & Francis, 1996, pp. 257–282
* [14] Pao-Liu Chow “Solutions of linear elliptic equations in Gauss-Sobolev spaces” In _Communications on Stochastic Analysis_ 6.3, 2012, pp. 8
* [15] A Debernardi “Uniform convergence of Hankel transforms” In _Journal of Mathematical Analysis and Applications_ 468.2 Elsevier, 2018, pp. 1179–1206
* [16] Alberto Debernardi “The Boas problem on Hankel transforms” In _Journal of Fourier Analysis and Applications_ 25.6 Springer, 2019, pp. 3310–3341
* [17] Simon Fischer and Ingo Steinwart “Sobolev Norm Learning Rates for Regularized Least-Squares Algorithms.” In _J. Mach. Learn. Res._ 21, 2020, pp. 205–1
* [18] Godfrey Harold Hardy and John Edensor Littlewood “A new proof of a theorem on rearrangements” In _Journal of the London Mathematical Society_ 1.3 Wiley Online Library, 1948, pp. 163–168
* [19] Anna Kamińska “Some remarks on Orlicz-Lorentz spaces” In _Mathematische Nachrichten_ 147.1 Wiley Online Library, 1990, pp. 29–38
* [20] Boris Sergeevic Kashin and Artur Artushovich Saakyan “Orthogonal series” American Mathematical Soc., 2005
* [21] Sergei Vladimirovich Konyagin “On a problem of Littlewood” In _Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya_ 45.2 Russian Academy of Sciences, Steklov Mathematical Institute of Russian …, 1981, pp. 243–265
* [22] Aigerim Kopezhanova “Summability of Fourier transforms of functions from Lorentz spaces”, 2017
* [23] Aigerim Kopezhanova, Erlan Nursultanov and Lars Erik Persson “A new generalization of Boas theorem for some Lorentz spaces $\Lambda$q ($\omega$)” Ele-Math, 2018
* [24] Junhong Lin, Alessandro Rudi, Lorenzo Rosasco and Volkan Cevher “Optimal rates for spectral algorithms with least-squares regression over Hilbert spaces” In _Applied and Computational Harmonic Analysis_ 48.3 Elsevier, 2020, pp. 868–890
* [25] GG Lorentz “Relations between function spaces” Mathematical Science Directorate, Office of Scientific Research, US Air Force, 1959
* [26] Shuai Lu, Peter Mathé and Sergei V Pereverzev “Balancing principle in supervised learning for a general regularization scheme” In _Applied and Computational Harmonic Analysis_ 48.1 Elsevier, 2020, pp. 123–148
* [27] Attila Máté, Paul Nevai and Vilmos Totik “Szego’s extremum problem on the unit circle” In _Annals of Mathematics_ JSTOR, 1991, pp. 433–453
* [28] Attila Máté and Paul G Nevai “Bernstein’s Inequality in Lp for and (C, 1) Bounds for Orthogonal Polynomials” In _Annals of Mathematics_ JSTOR, 1980, pp. 145–154
* [29] Peter Mathé and Bernd Hofmann “Direct and inverse results in variable Hilbert scales” In _Journal of approximation theory_ 154.2 Elsevier, 2008, pp. 77–89
* [30] Peter Mathé and Sergei V Pereverzev “Geometry of linear ill-posed problems in variable Hilbert scales” In _Inverse problems_ 19.3 IOP Publishing, 2003, pp. 789
* [31] Shahar Mendelson and Joseph Neeman “Regularization in kernel learning” In _The Annals of Statistics_ 38.1 Institute of Mathematical Statistics, 2010, pp. 526–565
* [32] Stephen J Montgomery-Smith “The Hardy operator and Boyd indices” In _Lecture Notes Pure Appl. Math_ 175, 1995, pp. 359–364
* [33] Alain Pajor and Nicole Tomczak-Jaegermann “Subspaces of small codimension of finite-dimensional Banach spaces” In _Proceedings of the American Mathematical Society_ 97.4, 1986, pp. 637–642
* [34] Luboš Pick, Alois Kufner, Oldřich John and Svatopluk Fucı́k “Function spaces, 1” In _Function Spaces, 1_ de Gruyter, 2012
* [35] Evgeniy Pustylnik “Ultrasymmetric spaces” In _Journal of the London Mathematical Society_ 68.1 Cambridge University Press, 2003, pp. 165–182
* [36] Abhishake Rastogi “Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems” In _arXiv preprint arXiv:2002.01303_ , 2020
* [37] Abhishake Rastogi and Sivananthan Sampath “Optimal rates for the regularized learning algorithms under general source condition” In _Frontiers in Applied Mathematics and Statistics_ 3 Frontiers, 2017, pp. 3
* [38] Michael Ruzhansky and Niyaz Tokmagambetov “Nonharmonic analysis of boundary value problems” In _International Mathematics Research Notices_ 2016.12 Oxford University Press, 2016, pp. 3548–3615
* [39] Thomas Schuster, Barbara Kaltenbacher, Bernd Hofmann and Kamil S Kazimierski “Regularization methods in Banach spaces” In _Regularization Methods in Banach Spaces_ de Gruyter, 2012
* [40] Steve Smale and Ding-Xuan Zhou “Shannon sampling and function reconstruction from point values” In _Bulletin of the American Mathematical Society_ 41.3, 2004, pp. 279–305
* [41] Ingo Steinwart and Andreas Christmann “Support vector machines” Springer Science & Business Media, 2008
* [42] Ingo Steinwart and Clint Scovel “Mercer’s theorem on general domains: On the interaction between measures, kernels, and RKHSs” In _Constructive Approximation_ 35.3 Springer, 2012, pp. 363–417
* [43] Prem Talwai, Ali Shameli and David Simchi-Levi “Sobolev Norm Learning Rates for Conditional Mean Embeddings” In _International Conference on Artificial Intelligence and Statistics_ , 2022, pp. 10422–10447 PMLR
* [44] Vladimir Temlyakov “Greedy approximation” Cambridge University Press, 2011
* [45] Walter Van Assche “Asymptotics for orthogonal polynomials” Springer, 2006
* [46] Roman Vershynin and Convex Bodies “Lecture notes in geometric functional analysis” Citeseer, 2009
* [47] Holger Wendland “Scattered data approximation” Cambridge university press, 2004
* [48] Yuan Xu “Asymptotics for orthogonal polynomials and Christoffel functions on a ball” In _Methods and Applications of Analysis_ 3.2 International Press of Boston, 1996, pp. 257–272
* [49] Yiming Ying and Ding-Xuan Zhou “Learnability of Gaussians with flexible variances” Microtome Publishing, 2007
* [50] Ding-Xuan Zhou “The covering number in learning theory” In _Journal of Complexity_ 18.3 Elsevier, 2002, pp. 739–767
* [51] Ding-Xuan Zhou “Capacity of reproducing kernel spaces in learning theory” In _IEEE Transactions on Information Theory_ 49.7 IEEE, 2003, pp. 1743–1752
## Appendix A Proof of Lemma 1
We first demonstrate a two-sided Boas-type result (a two-sided Hardy-
Littlewood type inequality, see e.g. [23, 22]) for the radial Fourier
transform on rearrangement invariant spaces. We expect that this result is
well-known to experts, but could not find a definite source, and hence prove
it here:
###### Lemma 1.
Suppose $f=\tilde{f}(||\cdot||)\in\mathbb{R}^{d}$ is a radial function with
$\tilde{f}(r)r^{d-1}$ decreasing on $(0,\infty)$ and
$\tilde{f}(r)r^{\frac{1}{2}}\in L^{1}(0,\infty)$. Let $w$ be a positive weight
function on $\mathbb{R}^{+}$ and define $\tilde{w}(\cdot)=w(||\cdot||^{d})$ on
$\mathbb{R}^{d}$. For $1\leq p<\infty$, define:
$\displaystyle W^{1}_{p}(t)$
$\displaystyle=\int_{0}^{t}r^{p}w^{p}(r^{-1})\frac{dr}{r}$ (13) $\displaystyle
W^{2}_{p}(t)$ $\displaystyle=\int_{0}^{t}w^{p}(r)\frac{dr}{r}\ $ (14)
Suppose $\beta_{W^{1}_{p}},\beta_{W^{2}_{p}}<p$ (see section 2.2 for a
definition). Then:
$\int_{\mathbb{R}^{d}}\Big{(}|\mathcal{F}f(x)|\tilde{w}\Big{(}\frac{x^{\prime}}{||x||}\Big{)}||x||^{d}\Big{)}^{p}\frac{dx}{||x||^{d}}\asymp\int_{\mathbb{R}^{d}}(\tilde{w}(x)|f(x)|)^{p}\frac{dx}{||x||^{d}}$
(15)
where $x=||x||x^{\prime}$.
###### Proof.
We only show the backward direction as the forward direction follows
analogously (under the assumption $\beta_{W^{2}_{p}}>\frac{1}{d^{2}}$); the
latter may also be deduced from the proof of Theorem 2.3 in [11] through the
application of a specialized Hardy inequality (Theorem 2(i) in [32]; see
below) customized to our growth condition in (13) and (14). Hence, for the
reverse direction, we have, by the Fourier inversion formula (which holds
since $\tilde{f}(r)r^{\frac{1}{2}}\in L^{1}(0,\infty)$ by assumption; see [15,
16]):
$\displaystyle f(x)$
$\displaystyle=\int_{\mathbb{R}^{d}}\mathcal{F}f(||u||)e^{2\pi x\cdot u}du$
$\displaystyle=|S_{d-2}|\int_{0}^{\infty}s^{d-1}\mathcal{F}f(s)\int_{0}^{1}\cos(2\pi
s||x||t)(1-t^{2})^{\frac{d-3}{2}}dtds$
$\displaystyle=\int_{0}^{\frac{1}{||x||}}+\int_{\frac{1}{||x||}}^{\infty}s^{d-1}\mathcal{F}f(s)\int_{0}^{1}\cos(2\pi
s||x||t)(1-t^{2})^{\frac{d-3}{2}}dtds$
where $|S_{d-2}|$ is the surface area of the $(d-1)$-sphere. Then, analogously
to the proof of Theorem 2.3 in [11] (see (2.5)-(2.7) on p. 787-788), we can
demonstrate that:
$\displaystyle\Big{|}\int_{0}^{\frac{1}{||x||}}s^{d-1}\mathcal{F}f(s)\int_{0}^{1}\cos(2\pi
s||x||t)(1-t^{2})^{\frac{d}{3}}dtds\Big{|}$ $\displaystyle\leq
C_{1}\int_{0}^{\frac{1}{||x||}}|\mathcal{F}f(s)|s^{d-1}ds$ (16)
$\displaystyle\Big{|}\int_{\frac{1}{||x||}}^{\infty}s^{d-1}\mathcal{F}f(s)\int_{0}^{1}\cos(2\pi
s||x||t)(1-t^{2})^{\frac{d}{3}}dtds\Big{|}$ $\displaystyle\leq
C_{2}\int_{0}^{\frac{1}{||x||}}|\mathcal{F}f(s)|s^{d-1}ds$ (17)
for some constants $C_{1},C_{2}>0$. Therefore, we have that:
$\displaystyle\int_{\mathbb{R}^{d}}(w(||x||^{d})|f(x)|)^{p}\frac{dx}{||x||^{d}}$
$\displaystyle=\int_{0}^{\infty}(w(r^{d})|f(r)|)^{p}\frac{dr}{r}$
$\displaystyle\leq
C\int_{0}^{\infty}\Big{(}w(r^{d})\int_{0}^{\frac{1}{r}}|\mathcal{F}f(s)|s^{d-1}ds\Big{)}^{p}\frac{dr}{r}$
$\displaystyle=C\int_{0}^{\infty}r^{pd}w^{p}(r^{-d})\Big{(}\frac{1}{r^{d}}\int_{0}^{r}|\mathcal{F}f(s)|s^{d-1}ds\Big{)}^{p}\frac{dr}{r}$
$\displaystyle\leq
C\int_{0}^{\infty}r^{pd}w^{p}(r^{-d})(|\mathcal{F}f(r)|)^{p}\frac{dr}{r}$ (18)
$\displaystyle=C\int_{\mathbb{R}^{d}}\Big{(}|\mathcal{F}f(x)|\tilde{w}\Big{(}\frac{x^{\prime}}{||x||^{d}}\Big{)}||x||^{d}\Big{)}^{p}\frac{dx}{||x||^{d}}$
where in (18) we have applied Hardy’s inequality for rearrangement-invariant
spaces , i.e. Theorem 2(i) in [32] with $p=\frac{1}{d}$ and $r=1$ (using their
notation), and $p_{X}=\frac{p}{d\beta_{W^{1}_{p}}}$ (since here the r.i. space
$X=L^{p}(w^{p}(t^{-d})dt^{pd})$ in their notation). The requirement
$\frac{p}{d\beta_{W^{1}_{p}}}>\frac{1}{d}$ in this theorem is satisfied by
assumption. ∎
###### Remark.
We emphasize that the validity of Lemma 1 hinges on the condition that
$\tilde{f}(r)r^{d-1}$ is decreasing, which factors crucially in demonstrating
(16) and (17) (see the p. 787-788 in [11] for details). Simple counterexamples
to (15) may be found when this condition is violated.
###### Lemma 2.
Let $w(t)=t^{-p}$ for some $p\in(0,1)$. Suppose
$\rho=\alpha_{\Phi}=\beta_{\Phi}$. Then the fundamental function $\phi_{p}(t)$
of $L^{\Phi,w}$ is equivalent to $\frac{t^{\frac{2-p}{\rho}}}{\Phi^{-1}(t)}$.
###### Proof.
Note, that by a standard argument, we may find a function
$\bar{\Phi}\asymp\Phi$ such that
$\frac{t\bar{\Phi}^{\prime}(t)}{\bar{\Phi}(t)}=\rho$ and hence $\Phi(t)\sim
t^{\rho}$ (see e.g. [35]). Hence, we have
$\displaystyle\int_{0}^{\infty}\Phi\Big{(}\frac{\chi_{(0,s)}(t)}{\lambda}\Big{)}w(t)dt$
$\displaystyle=\int_{0}^{s}\Phi\Big{(}\frac{1}{\lambda}\Big{)}t^{-p}dt$
$\displaystyle=s^{1-p}\Phi\Big{(}\frac{1}{\lambda}\Big{)}$
$\displaystyle\asymp\frac{s^{2-p}}{s\lambda^{\rho}}$
$\displaystyle=\frac{1}{s}\cdot\Big{(}\frac{s^{\frac{2-p}{\rho}}}{\lambda}\Big{)}^{\rho}$
$\displaystyle\asymp\frac{1}{s}\cdot\Phi\Big{(}\frac{s^{\frac{2-p}{\rho}}}{\lambda}\Big{)}$
Setting the RHS of the last line to $1$ and solving for $\lambda$ we obtain
our result. ∎
###### Proof of Lemma 1.
Let $g=T\tilde{f}$, for some
$\tilde{f}\in\Lambda_{\text{rad}}(\mathbb{R}^{d})\cap
L^{p,q}(\mathbb{R}^{d})$. We may express $\tilde{f}=f(||\cdot||)$ with
$f(r)r^{d-1}$ decreasing. Then, we have that
$\mathcal{F}g(||\cdot||)=\mathcal{F}_{d}\kappa(||\cdot||^{\frac{1}{\alpha}})\mathcal{F}_{d}f(||\cdot||)$.
This follows from the fact that the elements of $\Lambda_{\text{rad}}$ are
radial, and hence so are their Fourier transforms. Moreover, we have that the
decreasing (symmetric) rearrangement $(\mathcal{F}g)^{*}(t)$ (with respect to
the $d$-dimensional Lebesgue measure) is given by
$(\mathcal{F}g)^{*}(t)\asymp\mathcal{F}_{d}\kappa(t^{\frac{1}{d\alpha}})\mathcal{F}_{d}f(t^{\frac{1}{d}})$.
By Lemma 2, the fundamental function of $L^{\psi,t^{\frac{\rho(p+1)}{p}-2}}$
is equivalent to $\Psi(t)=\frac{t^{\frac{p+1}{p}}}{\psi^{-1}(t)}$. Observe
that $\alpha_{\Psi}=\beta_{\Psi}=1+\frac{1}{p}-\frac{1}{\rho}\in(0,1)$ by
assumption. Hence, applying Lemma 1 we have:
$\displaystyle||g||_{\Lambda^{q}_{\Psi}}$
$\displaystyle\asymp\Big{\|}\Psi(t)\cdot
g^{*}(t)\Big{\|}_{L^{q}\big{(}\frac{dt}{t}\big{)}}$
$\displaystyle\asymp\Big{\|}t\Psi\Big{(}\frac{1}{t}\Big{)}\cdot(\mathcal{F}g)^{*}(t)\Big{\|}_{L^{q}\big{(}\frac{dt}{t}\big{)}}$
(19)
$\displaystyle\asymp\Big{\|}\frac{t^{-\frac{1}{p}}}{\psi^{-1}(t^{-1})}\cdot\mathcal{F}_{d}\kappa(t^{\frac{1}{d\alpha}})\mathcal{F}_{d}f(t^{\frac{1}{d}})\Big{\|}_{L^{q}\big{(}\frac{dt}{t}\big{)}}$
$\displaystyle=\Big{\|}\frac{t^{-1}}{\psi^{-1}(t^{-1})}\cdot\mathcal{F}_{d}\kappa(t^{\frac{1}{d\alpha}})t^{1-\frac{1}{p}}\mathcal{F}_{d}f(t^{\frac{1}{d}})\Big{\|}_{L^{q}\Big{(}\frac{dt}{t}\Big{)}}$
$\displaystyle\asymp||t^{1-\frac{1}{p}}\mathcal{F}_{d}f(t^{\frac{1}{d}})||_{L^{q}\Big{(}\frac{dt}{t}\Big{)}}$
(20)
$\displaystyle\asymp||t^{\frac{1}{p}}f(t^{\frac{1}{d}})||_{L^{q}\Big{(}\frac{dt}{t}\Big{)}}$
(21) $\displaystyle\asymp||\tilde{f}||_{L^{p,q}(\mathbb{R}^{d})}$ (22)
where (19) and (21) follow from Lemma 1 after noting that
$\frac{1}{p},1-\frac{1}{p}<1\leq q$ and
$\beta_{\Psi^{1}_{q}},\beta_{\Psi^{2}_{q}}<q$ (since here
$\Psi^{1}_{q}(t)\asymp t\Psi(t^{-1})$ and $\Psi^{2}_{q}(t)\asymp\Psi(t)$ by
the assumption $\alpha_{\psi}=\beta_{\psi}$), (20) follows from observing
$\frac{t^{-1}\mathcal{F}_{d}k(t^{\frac{1}{d\alpha}})}{\psi^{-1}(t^{-1})}\stackrel{{\scriptstyle
t=s^{\alpha}}}{{\asymp}}\frac{\mathcal{F}_{d}k(s^{\frac{1}{d}})}{s^{\alpha}\psi^{-1}(s^{-\alpha})}\asymp
1$ by Assumption 8*, and (22) follows from recalling that
$\tilde{f}\in\Lambda_{\text{rad}}(\mathbb{R}^{d})\cap L^{p,q}(\mathbb{R}^{d})$
is radially decreasing. The optimality of the domain space
$L^{\psi,\frac{\rho(p+1)}{p}-2}$ follows from the asymptotic equivalence in
(20), and Theorem 3.1 in [10]. ∎
###### Remark.
In Assumption 8*, the expression $s(t)$ can be interpreted as a change-of-
basis operation, that converts between the Fourier basis and the Mercer
eigensystem (this interpretation is exact when the eigenspace consists of
complex polynomials, see e.g. Theorem 1 in [27]). It is important to note that
our framework implicitly requires the nonequivalence of $\mu$ and the Lebesgue
(uniform) measure. To see this, suppose for simplicity, $\mathcal{X}=[0,1]$,
and $f_{2n}(x)=\sqrt{2}\cos(2\pi nx),f_{2n+1}(x)=\sqrt{2}\sin(2\pi nx)$. Then,
we have that
$\sup\Big{\\{}\Big{\|}\sum_{i=1}^{n}c_{i}f_{i}(\cdot)\Big{\|}_{\infty}:\sum_{i=1}^{n}c^{2}_{i}=1\Big{\\}}^{2}\asymp
n$, i.e. $s$ is the identity function, which does not satisfy Assumption 4 as
the harmonic series is not summable; in other words, the classical Nikolskii
inequality between $L^{\infty}$ and $L^{2}$ does not suffice. Intuitively, the
exclusion of Lebesgue-equivalent probability measures here is not problematic
— our framework is designed for the setting in which the entropy numbers of
$\mathcal{H}_{K}$ in $L^{2}(\mu)$ are unknown, which is typically not the case
when $\mu$ is Lebesgue-equivalent. Change-of-basis conditions appear
prominently in the study of Fourier multiplier operators associated with
anharmonic oscillators (see [12, 38]). Moreover, the study of the asymptotic
properties of the Christoffel function (square of the left-hand side of
Assumption 4) in relation to the properties of $\mu$ has been an active area
of research [28, 48, 3] with diverse applications to quadrature/interpolation
and random matrices [45] and [3].
## Appendix B Proof of Theorem 2
The main effort in the proof of Theorem 2 involves proving Lemma 3, from which
the Theorem 2 easily follows; hence we will begin with the proof of Lemma 3.
We start with the following bound on the bias:
###### Lemma 1.
$||f^{*}-f_{\lambda}||^{2}_{L^{2}(\nu)}\leq\phi(\lambda)||f^{*}||^{2}_{\phi}$
###### Proof.
Since $f\in\mathcal{H}^{\phi}$, we have that there exists a
$\\{a_{i}\\}_{i=1}^{\infty}\in\ell^{2}$ such that:
$f^{*}=\sum_{i}a_{i}\phi^{\frac{1}{2}}(\mu_{i})e_{i}$
Hence,
$\displaystyle
f^{*}-f_{\lambda}=\sum_{i=1}^{\infty}\frac{a_{i}\lambda\phi^{\frac{1}{2}}(\mu_{i})}{\mu_{i}+\lambda}e_{i}$
(23)
Therefore, by the Cauchy-Schwartz inequality:
$\displaystyle||f^{*}-f_{\lambda}||^{2}_{L^{2}(\nu)}$
$\displaystyle\leq\sum_{i=1}^{\infty}\Big{(}\frac{a_{i}\lambda\phi^{\frac{1}{2}}(\mu_{i})}{\mu_{i}+\lambda}\Big{)}^{2}$
$\displaystyle\leq\Big{(}\sup_{i}\frac{\lambda^{2}\phi(\mu_{i})}{(\mu_{i}+\lambda)^{2}}\Big{)}\sum_{i=1}^{\infty}a^{2}_{i}$
$\displaystyle\leq\Big{(}\sup_{i}\frac{\lambda^{2}\phi(\mu_{i})}{(\mu_{i}+\lambda)^{2}}\Big{)}||f^{*}||^{2}_{\phi}$
Now, observe that for $\mu_{i}<\lambda$:
$\sup_{i:\mu_{i}<\lambda}\frac{\lambda^{2}\phi(\mu_{i})}{(\mu_{i}+\lambda)^{2}}\leq\phi(\lambda)$
as $\phi$ is nondecreasing. For $\mu_{i}\geq\lambda$, we have that:
$\sup_{i:\mu_{i}\geq\lambda}\frac{\lambda^{2}\phi(\mu_{i})}{(\mu_{i}+\lambda)^{2}}=\sup_{i:\mu_{i}\geq\lambda}\frac{\lambda^{2}\mu_{i}^{2}\phi(\mu_{i})}{\mu^{2}_{i}(\mu_{i}+\lambda)^{2}}\leq\sup_{i:\mu_{i}\geq\lambda}\frac{\lambda^{2}\mu_{i}^{2}\phi(\lambda)}{\lambda^{2}(\mu_{i}+\lambda)^{2}}\leq\phi(\lambda)$
where the second inequality follows from the fact that $\frac{\phi(t)}{t^{2}}$
is nonincreasing (from Assumption 2). Putting this together, we obtain our
result. ∎
###### Lemma 2.
$||f^{*}-f_{\lambda}||^{2}_{\infty}\leq\frac{\phi(\lambda)||k_{\psi}||^{2}_{\infty}||f^{*}||^{2}_{\phi}}{\psi(\lambda)}$
###### Proof.
We have from (23):
$\displaystyle||f^{*}-f_{\lambda}||^{2}_{\infty}$
$\displaystyle\leq\Big{|}\sum_{i=1}^{\infty}\frac{a_{i}\lambda\phi(\mu_{i})^{\frac{1}{2}}e_{i}(\cdot)}{\mu_{i}+\lambda}\Big{|}^{2}_{\infty}$
$\displaystyle\leq\Big{(}\sum_{i}\frac{a^{2}_{i}\lambda^{2}\phi(\mu_{i})}{(\mu_{i}+\lambda)^{2}\psi(\mu_{i})}\Big{)}\Big{|}\sum_{i=1}^{\infty}\psi(\mu)e^{2}_{i}(\cdot)\Big{|}_{\infty}$
$\displaystyle\leq\frac{\phi(\lambda)||k^{\psi}||^{2}_{\infty}||f^{*}||^{2}_{\phi}}{\psi(\lambda)}$
where the last line follows from the definition of $||k^{\psi}||$ and the same
logic as in the proof of Lemma 1, after noting that
$\frac{\phi(t)}{t^{2}\psi(t)}$ is decreasing for $t\geq\lambda$ and
$\frac{\phi(t)}{\psi(t)}$ is increasing for $t<\lambda$ by Assumption 2. ∎
We now derive a bound on the variance of our estimator
$f_{\lambda}-f_{D,\lambda}$:
###### Lemma 3.
$||f_{\lambda}-f_{D,\lambda}||_{L^{2}(\nu)}\preceq\log(\delta^{-1})\sqrt{\frac{1152\sigma^{2}||k_{\psi}||^{2}_{\infty}||f^{*}||^{2}_{\phi}}{n\psi(\lambda)}\Big{(}\phi(\lambda)+\frac{1}{n}\Big{)}+\frac{s^{-1}(\psi(\lambda)^{-1})}{n}}$
with probability $1-3\delta$
###### Proof.
We begin with a standard decomposition applied in the proof of Theorem 16 in
[17] and Theorem 7 in [43]. We first observe that:
$f_{\lambda}-f_{D,\lambda}=f_{\lambda}-(C_{D}+\lambda)^{-1}g_{D}=(C_{D}+\lambda)^{-1}((C_{D}+\lambda)f_{\lambda}-g_{D})$
(24)
where $g_{D}=\sum_{i=1}^{n}y_{i}k(x_{i},\cdot)\in\mathcal{H}_{K}$. Hence, we
have that:
$\displaystyle||f_{\lambda}-f_{D,\lambda}||_{L^{2}(\nu)}$
$\displaystyle=||C^{\frac{1}{2}}_{\nu}(f_{\lambda}-f_{D,\lambda})||_{K}$ (25)
$\displaystyle\leq||C^{\frac{1}{2}}_{\nu}(C_{\nu}+\lambda)^{-\frac{1}{2}}||||(C_{\nu}+\lambda)^{\frac{1}{2}}(C_{D}+\lambda)^{-1}(C_{\nu}+\lambda)^{\frac{1}{2}}||||(C_{\nu}+\lambda)^{-\frac{1}{2}}((C_{D}+\lambda)f_{\lambda}-g_{D})||$
(26)
where (25) follows from the fact that for any $f\in\mathcal{H}_{K}$, $\langle
I_{\nu}f,I_{\nu}f\rangle_{2}=\langle f,I^{*}_{\nu}I_{\nu}f\rangle_{K}=\langle
f,C_{\nu}f\rangle_{K}$ and (4), and (26) follows from (24) and algebraic
manipulation. It is easy to see that:
$||C^{\frac{1}{2}}_{\nu}(C_{\nu}+\lambda)^{-\frac{1}{2}}||\leq 1$
We now analyze second factor in (26). Indeed, we have that:
$(C_{\nu}+\lambda)^{\frac{1}{2}}(C_{D}+\lambda)^{-1}(C_{\nu}+\lambda)^{\frac{1}{2}}=\Big{(}I-(C_{\nu}+\lambda)^{-\frac{1}{2}}(C_{\nu}-C_{D})(C_{\nu}+\lambda)^{-\frac{1}{2}}\Big{)}^{-1}$
by the same algebraic manipulation as in the proof of Theorem 16 in [17] (for
the sake of space, we avoid repeating the argument verbatim here). Hence,
estimating the middle factor in (26) boils down to estimating the
concentration of
$(C_{\nu}+\lambda)^{-\frac{1}{2}}(C_{\nu}-C_{D})(C_{\nu}+\lambda)^{-\frac{1}{2}}$.
We first note that:
$(C_{\nu}+\lambda)^{-\frac{1}{2}}(C_{\nu}-C_{D})(C_{\nu}+\lambda)^{-\frac{1}{2}}=\mathbb{E}_{X}[h(X,\cdot)\otimes
h(X,\cdot)]-\mathbb{E}_{\mathcal{D}}[h(X,\cdot)\otimes h(X,\cdot)]$
where $h(x,\cdot)=(C_{\nu}+\lambda)^{-\frac{1}{2}}k(x,\cdot)$. Note, by Lemma
2, we have that
$||h(x,\cdot)||_{K}\leq\sqrt{\frac{||k^{\psi}||^{2}_{\infty}}{\psi(\lambda)}}$.
Then, analogously to Lemma 17 in [17], we may demonstrate that:
$(C_{\nu}+\lambda)^{-\frac{1}{2}}(C_{\nu}-C_{D})(C_{\nu}+\lambda)^{-\frac{1}{2}}\leq\frac{2}{3}$
with probability $1-2\delta$, when $n\geq
8\log(\delta^{-1})||k^{\psi}||^{2}_{\infty}g_{\lambda}\psi(\lambda)^{-1}$,
where
$g_{\lambda}=\log(2e\mathcal{N}(\lambda)\Big{(}1+\frac{\lambda}{||C_{\nu}||}\Big{)})$,
where $\mathcal{N}(\lambda)=\text{tr}(C_{\nu}(C_{\nu}+\lambda)^{-1})$ Applying
a Neumann series expansion, we obtain (like in [17]):
$\|(C_{\nu}+\lambda)^{\frac{1}{2}}(C_{D}+\lambda)^{-1}(C_{\nu}+\lambda)^{\frac{1}{2}}\|=||\Big{(}I-(C_{\nu}+\lambda)^{-\frac{1}{2}}(C_{\nu}-C_{D})(C_{\nu}+\lambda)^{-\frac{1}{2}}\Big{)}^{-1}||\leq
3$
for $n\geq
8\log(\delta^{-1})||k^{\psi}||^{2}_{\infty}g_{\lambda}\psi(\lambda)^{-1}$.
Now, it remains to bound the third factor in (26). We observe that we may
write:
$\displaystyle(C_{\nu}+\lambda)^{-\frac{1}{2}}((C_{D}+\lambda)f_{\lambda}-g_{D})$
$\displaystyle=(C_{\nu}+\lambda)^{-\frac{1}{2}}((C_{D}-C_{\nu})f_{\lambda}+S_{\nu}f^{*}-g_{D})$
$\displaystyle=\mathbb{E}_{\mathcal{D}}[h(X,\cdot)(f_{\lambda}(X)-Y)]-\mathbb{E}_{XY}[h(X,\cdot)(f_{\lambda}(X)-Y)]$
which follow by the definition of $f_{\lambda}$ and $h(x,\cdot)$. In order to
estimate this concentration, we compute absolute moments:
$\displaystyle\mathbb{E}_{XY}[||h(X,\cdot)(f_{\lambda}(X)-Y)||_{K}^{p}]$
$\displaystyle\leq\mathbb{E}_{XY}[||h(X,\cdot)(f_{\lambda}(X)-f^{*}(X)+f^{*}(X)-Y)||_{K}^{p}]$
$\displaystyle\leq
2^{p-1}\mathbb{E}_{XY}[||h(X,\cdot)||^{p}_{K}(|f_{\lambda}(X)-f^{*}(X)|^{p}+|f^{*}(X)-Y|^{p})]$
$\displaystyle\leq
2^{p-1}\mathbb{E}_{X}\Big{[}||h(X,\cdot)||^{p}_{K}\Big{(}\|f_{\lambda}-f^{*}\|_{\infty}^{p-2}|f_{\lambda}(X)-f^{*}(X)|^{2}+\frac{p!L^{p-2}\sigma^{2}}{2}\Big{)}\Big{]}$
(27) $\displaystyle\leq
2^{p-1}\Big{(}\phi^{\frac{p}{2}}(\lambda)\Big{(}\frac{||k_{\psi}||_{\infty}||f^{*}||_{\phi}}{\psi(\lambda)}\Big{)}^{p}+\frac{p!\sigma^{2}}{2}\Big{(}\frac{||k_{\psi}||^{2}_{\infty}L^{2}}{\psi(\lambda)}\Big{)}^{\frac{p-2}{2}}\text{tr}(C_{\nu}(C_{\nu}+\lambda)^{-1})\Big{)}$
(28) $\displaystyle\preceq
2^{p-1}\Big{(}\Big{(}\frac{||k_{\psi}||_{\infty}||f^{*}||_{\phi}L}{\sqrt{\psi(\lambda)}}\Big{)}^{p-2}\Big{[}\Big{(}\frac{||k_{\psi}||^{2}_{\infty}||f^{*}||^{2}_{\phi}\phi(\lambda)}{\psi(\lambda)}\Big{)}+\sigma^{2}p!\Gamma\Big{(}\frac{p}{2}\Big{)}s^{-1}(\psi(\lambda)^{-1})\Big{]}\Big{)}$
(29) $\displaystyle\leq\frac{p!L_{\lambda}^{p-2}R^{2}}{2}$
with
$L_{\lambda}=\frac{2\sigma||f^{*}||_{\phi}||k_{\psi}||_{\infty}}{\sqrt{\psi(\lambda)}}$
and
$R^{2}=\frac{4||f^{*}||^{2}_{\phi}||k_{\psi}||^{2}_{\infty}\phi(\lambda)}{\psi(\lambda)}+4\sigma^{2}s^{-1}(\psi(\lambda)^{-1})$.
Here, (27) follows from Assumption 5, (28) invokes Lemmas 2, 1 and 2, and
finally (29) follows from Lemma 3 and the fact that
$\frac{\phi(\lambda)}{\psi(\lambda)}\to 0$ as $\lambda\to 0$. Hence, applying
Theorem 26 in [17], we have that:
$||(C_{\nu}+\lambda)^{-\frac{1}{2}}((C_{D}+\lambda)f_{\lambda}-g_{D})||^{2}_{K}\preceq\frac{32\log^{2}(\delta^{-1})}{n}\Big{(}\frac{4||f^{*}||^{2}_{\phi}||k_{\psi}||^{2}_{\infty}\phi(\lambda)}{\psi(\lambda)}+4\sigma^{2}s^{-1}(\psi(\lambda)^{-1})+\frac{4\sigma^{2}||k_{\psi}||^{2}_{\infty}||f^{*}||^{2}_{\phi}}{n\psi(\lambda)}\Big{)}$
with probability $1-2\delta$. Putting this all together, we have that:
$||f_{\lambda}-f_{D,\lambda}||_{L^{2}(\nu)}\preceq\log(\delta^{-1})\sqrt{\frac{1152\sigma^{2}||k_{\psi}||^{2}_{\infty}||f^{*}||^{2}_{\phi}}{n\psi(\lambda)}\Big{(}\phi(\lambda)+\frac{1}{n}\Big{)}+\frac{s^{-1}(\psi(\lambda)^{-1})}{n}}$
(30)
with probability $1-3\delta$, when $n\geq
8\log(\delta^{-1})||k^{\psi}||^{2}_{\infty}g_{\lambda}\psi(\lambda)^{-1}$. We
demonstrate that this latter condition is eventually met, when the
regularization program is
$\lambda_{n}\asymp(\frac{\phi}{\tilde{s}\circ\psi})^{-1}(n^{-1})$. Indeed, we
have that:
$\displaystyle\frac{8\log(\delta^{-1})||k^{\psi}||^{2}_{\infty}g_{\lambda_{n}}}{n\psi(\lambda_{n})}$
$\displaystyle=\frac{8\log(\delta^{-1})||k^{\psi}||^{2}_{\infty}\log\Big{(}2e\mathcal{N}(\lambda_{n})\Big{(}1+\frac{\lambda_{n}}{||C_{\nu}||}\Big{)}\Big{)}}{n\psi(\lambda_{n})}$
$\displaystyle\leq\frac{16\log(\delta^{-1})||k^{\psi}||^{2}_{\infty}\log\Big{(}2e\mathcal{N}(\lambda_{n})\Big{)}}{n\psi(\lambda_{n})}$
$\displaystyle\leq\frac{16\log(\delta^{-1})||k^{\psi}||^{2}_{\infty}\log\Big{(}s^{-1}(\psi(\lambda_{n})^{-1})\Big{)}}{n\psi(\lambda_{n})}$
where the last line follows by Lemma 3. Hence, it is sufficient to demonstrate
that
$\frac{\log\Big{(}s^{-1}(\psi(\lambda_{n})^{-1})\Big{)}}{n\psi(\lambda_{n})}\to
0$ for the regularization program. Indeed, since
$\lambda_{n}\asymp(\frac{\phi}{\tilde{s}\circ\psi})^{-1}(n^{-1})$, we have
that:
$\displaystyle\frac{\log\Big{(}s^{-1}(\psi(\lambda_{n})^{-1})\Big{)}}{n\psi(\lambda_{n})}$
$\displaystyle\preceq\frac{\log\Big{(}s^{-1}(\psi(\lambda_{n})^{-1})\Big{)}}{n\phi(\lambda_{n})}$
(31)
$\displaystyle\asymp\frac{\log\Big{(}s^{-1}(\psi(\lambda_{n})^{-1})\Big{)}}{\tilde{s}(\psi(\lambda_{n}))}$
$\displaystyle=\frac{\log\Big{(}s^{-1}(\psi(\lambda_{n})^{-1})\Big{)}}{s^{-1}(\psi(\lambda_{n})^{-1})}\to
0$ (32)
where (31) follows from Assumption 2 and (32) follows from the definition of
$\tilde{s}$.
###### Proof of Lemma 3.
This follows immediately by combining Lemma 3 with Lemma 1. ∎
###### Proof of Theorem 2.
From Lemma 3, we have:
$\displaystyle||f^{*}-f_{D,\lambda_{n}}||_{L^{2}(\nu)}$
$\displaystyle\preceq||f^{*}||_{\phi}\sqrt{\phi(\lambda_{n})}+\log(\delta^{-1})\sqrt{\frac{\sigma^{2}||k_{\psi}||^{2}_{\infty}||f^{*}||^{2}_{\phi}}{n\psi(\lambda_{n})}\Big{(}\phi(\lambda_{n})+\frac{1}{n}\Big{)}+\frac{s^{-1}(\psi(\lambda_{n})^{-1})}{n}}$
$\displaystyle\preceq||f^{*}||_{\phi}\sqrt{\phi(\lambda_{n})}+\log(\delta^{-1})\sqrt{\frac{s^{-1}(\psi(\lambda_{n})^{-1})}{n}}$
(33)
where the last line follows from the fact that
$\frac{\phi(\lambda_{n})}{\psi(\lambda_{n})}\to 0$ by Assumption 2 and
$\phi(\lambda_{n})\asymp\frac{\tilde{s}(\psi(\lambda_{n}))}{n}=\frac{s^{-1}(\psi(\lambda_{n})^{-1})}{n}\succeq\frac{1}{n}$
as $s^{-1}(\psi(\lambda_{n})^{-1})\to\infty$ when
$\lambda_{n}\asymp(\frac{\phi}{\tilde{s}\circ\psi})^{-1}(n^{-1})$. The result
follows immediately after substituting the latter choice of regularizer into
(33) (which was indeed proposed via balancing (33)). ∎
∎
## Appendix C Proof of Theorem 4
###### Lemma 1.
$b^{2}_{n-1}(\mathcal{H}_{K},L^{\infty})\leq||k^{\psi}||^{2}_{\infty}\Big{(}\frac{t}{\psi}\Big{)}(\mu_{n})$
###### Proof.
We first note that since $\frac{t}{\psi(t)}$ is concave by Assumption 2, we
have by the Gagliardo-Nirenberg interpolation inequality (Lemma 1), that for
all $f\in\mathcal{H}_{K}$:
$\frac{||f||^{2}_{\psi}}{||f||^{2}_{K}}\leq\Big{(}\frac{t}{\psi}\Big{)}\Big{(}\frac{||f||^{2}_{L^{2}(\nu)}}{||f||^{2}_{K}}\Big{)}$
By the embedding condition $\mathcal{H}^{\psi}\hookrightarrow L^{\infty}$, we
have that:
$\frac{||f||^{2}_{\infty}}{||f||^{2}_{K}}\leq||k^{\psi}||^{2}_{\infty}\Big{(}\frac{t}{\psi}\Big{)}\Big{(}\frac{||f||^{2}_{L^{2}(\nu)}}{||f||^{2}_{K}}\Big{)}$
Hence, we have that:
$\displaystyle\sup_{dim(Z)\geq n}\inf_{f\in
Z\cap\mathcal{H}_{K}}\frac{||f||^{2}_{\infty}}{||f||^{2}_{K}}$
$\displaystyle\leq\sup_{dim(Z)\geq n}\inf_{f\in
Z\cap\mathcal{H}_{K}}||k^{\psi}||^{2}_{\infty}\Big{(}\frac{t}{\psi}\Big{)}\Big{(}\frac{||f||^{2}_{L^{2}(\nu)}}{||f||^{2}_{K}}\Big{)}$
$\displaystyle=||k^{\psi}||^{2}_{\infty}\Big{(}\frac{t}{\psi}\Big{)}\Big{(}\sup_{dim(Z)\geq
n}\inf_{f\in
Z\cap\mathcal{H}_{K}}\frac{||f||^{2}_{L^{2}(\nu)}}{||f||^{2}_{K}}\Big{)}$
where the last step follows from the fact that $\frac{t}{\psi(t)}$ is
nondecreasing and continuous. Hence, by the definition of the Bernstein width
(10), and Proposition 5 in [29], we obtain our result. ∎
###### Lemma 2.
$b_{n-1}(\mathcal{H}_{K},L^{\infty})\geq\sqrt{C_{d}\mathcal{F}_{d}\kappa(n^{\frac{1}{d}})}$
where $C_{d}>0$ is constant depending on the dimension $d$ and independent of
$n$.
###### Proof.
Let $\\{z_{i}\\}_{i=1}^{2n}\subset\mathcal{X}$ be a set of $2n$ distinct
points in $\mathcal{X}$. Consider the subspace
$Z=\text{span}\\{k(z_{i},\cdot)\\}_{i=1}^{2n}\subset\mathcal{H}_{K}$. Since,
$k$ is positive definite by assumption, we have that $\text{dim}(Z)=2n$. Since
$\mathcal{H}_{K}$ is a Hilbert space, $Z$ is isomorphic to $\mathbb{R}^{2n}$
(with the Euclidean norm). By Theorem IV.3.1 in [46] (see also Theorem 1 in
[33]), there exists a subspace $\tilde{Z}\subset Z$ of dimension $n$ and
constant $C>0$, such that:
$||z||_{K}\leq\sqrt{\frac{C}{2}}M^{*}(B(L^{\infty})\cap Z)$
for all $z\in\tilde{Z}\cap B(L^{\infty})$ (we will absorb additional constants
into $C$ throughout this proof). Here, $M^{*}(B(L^{\infty})\cap Z)$ denotes
the mean width of $B(L^{\infty})\cap Z$ given by:
$M^{*}(B(L^{\infty})\cap Z)=\int_{\partial B(Z)}\sup_{x\in B(L^{\infty})\cap
Z}\langle x,y\rangle_{K}dS_{2n}(y)$
where $S_{2n}(\cdot)$ is the canonical uniform measure over the unit sphere
$\partial B(Z)$ in $Z$. Let $\gamma_{2n}$ denote the standard Gaussian measure
on $\mathbb{R}^{2n}$, and $\mu_{2n}$ its pushforward onto $Z$ via the
canonical isomorphism. We write $g=\sum_{i=1}^{2n}a_{i}(g)k(z_{i},\cdot)$ for
$g\in Z$ (where $\\{a_{i}(g)\\}_{i=1}^{2n}\in\mathbb{R}^{2n}$). Then may
compute, via a standard change of variables, that for an absolute constant
$C_{1}>0$:
$\displaystyle M^{*}(B(L^{\infty})\cap Z)$ $\displaystyle=\int_{\partial
B(Z)}\sup_{f\in B(L^{\infty})\cap Z}\langle f,g\rangle_{K}dS_{2n}(g)$
$\displaystyle=\frac{C_{1}}{\sqrt{2n}}\int_{Z}\sup_{f\in B(L^{\infty})\cap
Z}\langle f,g\rangle_{K}d\mu_{2n}(g)$
$\displaystyle=\frac{C_{1}}{\sqrt{2n}}\int_{Z}\sup_{f\in B(L^{\infty})\cap
Z}\Big{\langle}f,\sum_{i=1}^{2n}a_{i}(g)k(z_{i},\cdot)\Big{\rangle}_{K}d\mu_{2n}(g)$
$\displaystyle=\frac{C_{1}}{\sqrt{2n}}\int_{Z}\sup_{f\in B(L^{\infty})\cap
Z}\sum_{i=1}^{2n}a_{i}(g)f(z_{i})d\mu_{2n}(g)$
$\displaystyle\leq\frac{C_{1}}{\sqrt{2n}}\int_{Z}\sum_{i=1}^{2n}|a_{i}(g)|d\mu_{2n}(g)$
since $|f(z_{i})|\leq 1$ as $f\in B(L^{\infty})$. Now, observe that
$g\in\partial B(Z)$ iff $\mathbf{a}(g)\cdot(K_{2n}\mathbf{a}(g))=1$, where
$\mathbf{a}(g)=[a_{1}(g),\ldots a_{2n}(g)]^{T}$ and
$(K_{2n})_{i,j}=k(z_{i},z_{j})$ is the kernel Gram matrix on
$\\{z_{i}\\}_{i=1}^{2n}$. Hence, performing a change of variables (and
renaming), we obtain, for a constant $C_{1}>0$:
$\displaystyle M^{*}(B(L_{\infty})\cap Z)$
$\displaystyle\leq\frac{C_{1}}{\sqrt{2n}}\int_{\mathbb{R}^{2n}}|K^{-\frac{1}{2}}_{2n}\mathbf{a}|_{1}d\gamma_{2n}(\mathbf{a})$
$\displaystyle\leq
C_{1}\int_{\mathbb{R}^{2n}}\sqrt{||K^{-1}_{2n}||_{2}||\mathbf{a}||^{2}}d\gamma_{2n}(\mathbf{a})$
$\displaystyle\leq C_{1}\sqrt{2n||K^{-1}_{2n}||_{2}}$
since $\int_{\mathbb{R}^{2n}}||\mathbf{a}||^{2}d\gamma_{2n}(a)=2n$ as
$\gamma_{2n}$ is standard normal. Hence, putting this all together, we have:
$\displaystyle||z||_{K}\leq\sqrt{C_{3}n||K^{-1}_{2n}||_{2}}$
for all $z\in\tilde{Z}\cap B(L_{\infty})$. Hence, for all $z\in\tilde{Z}$, we
have that:
$||z||_{K}\leq\sqrt{C_{3}n||K^{-1}_{2n}||_{2}}||z||_{\infty}$
where $C_{3}=CC_{1}$. By the definition of the Bernstein width, we therefore
have:
$b_{n-1}(\mathcal{H}_{K},L^{\infty})\geq\sqrt{\frac{C}{n||K^{-1}_{2n}||_{2}}}$
(34)
We now wish to upper bound $||K^{-1}_{2n}||_{2}$ — we do this by using the
following lower bound on the minimum eigenvalue of $K_{2n}$, given in Theorem
12.3 of [47]:
$\lambda_{\min}(K_{2n})\geq
C_{d}\Big{(}\frac{12.76d}{q_{z}}\Big{)}^{d}\mathcal{F}_{d}\kappa\Big{(}\frac{12.76d}{q_{z}}\Big{)}$
where $q_{z}\equiv\min_{i\neq j}||z_{i}-z_{j}||$ and $C_{d}>0$ is an absolute
constant depending only the dimension $d$. We choose a maximal $2n$-set
$\\{z_{i}\\}_{i=1}^{2n}$, so that $q_{z}$ is the packing number
$p_{n}(\mathcal{X})$ of the domain $\mathcal{X}$. From the standard
equivalence $p_{n}(\mathcal{X})\asymp\epsilon_{n}(\mathcal{X})$ and Assumption
7*, we have that:
$\lambda_{\min}(K_{2n})\geq
C_{d}n\mathcal{F}_{d}\kappa\Big{(}12.76d(2n)^{\frac{1}{d}}\Big{)}$
From the integral expression for the $d$-dimensional Fourier transform (see
e.g. Theorem 5.26 in [47]), and the dilation condition in Assumption 6*, we
therefore have that:
$\lambda_{\min}(K_{2n})\geq C_{d}n\mathcal{F}_{d}\kappa(n^{\frac{1}{d}})$
where again we recall that the overhead constant $C_{d}$ depends only on the
ambient dimension in Assumption 7*. Noting that
$\|K^{-1}_{2n}\|_{2}\leq\lambda^{-1}_{\min}(K_{2n})$ and plugging back into
(34), we have that:
$b_{n-1}(\mathcal{H}_{K},L^{\infty})\geq\sqrt{C_{d}\mathcal{F}_{d}\kappa(n^{\frac{1}{d}})}$
∎
###### Corollary 2.1.
$\mu_{n}\asymp\psi^{-1}(s(n)^{-1})$
###### Proof.
From Assumption 4, for any $\alpha\in(0,1)$, we can choose a $c\in l^{n}_{2}$
with $\sum_{i=1}^{n}c^{2}_{i}=1$ and $x^{*}\in\mathcal{X}$, such that:
$\sum_{i=1}^{n}c_{i}e_{i}(x^{*})\succeq\alpha\sqrt{s(n)}$
Hence, noting that $\\{c_{i}\\}_{i=1}^{n}$ can always be chosen such that
$c_{i}e_{i}(x^{*})\geq 0$, for all $i\in[n]$, we have that that:
$\displaystyle\alpha\sqrt{\psi(\mu_{n})s(n)}$
$\displaystyle\preceq\sum_{i=1}^{n}\sqrt{\psi(\mu_{i})}c_{i}e_{i}(x^{*})$
$\displaystyle\leq||k^{\psi}||_{\infty}$
Observing that $||k^{\psi}||_{\infty}<\infty$ since $H^{\psi}\hookrightarrow
L^{\infty}$ in Assumption 1 and that $\alpha$ was chosen arbitrarily, we
obtain the upper estimate:
$\mu_{n}\preceq\psi^{-1}(s(n)^{-1})$
For the lower estimate, we apply Lemma 1 and Lemma 2 to obtain:
$\frac{t}{\psi}\Big{(}\mu_{n}\Big{)}\succeq
b^{2}_{n-1}(\mathcal{H}_{K},L^{\infty})\succeq\mathcal{F}_{d}k(n^{\frac{1}{d}})$
(35)
Now, from Assumption 8*, we have that:
$\displaystyle\psi^{-1}(s(t)^{-1})$
$\displaystyle=\Big{(}\frac{t}{\psi}\Big{)}^{-1}\Big{(}\Big{(}\frac{t}{\psi}\Big{)}(\psi^{-1}(s(t)^{-1}))\Big{)}$
$\displaystyle\Big{(}\frac{t}{\psi}\Big{)}^{-1}\Big{(}\frac{\psi^{-1}(s(t)^{-1})}{s(t)^{-1}}\Big{)}$
$\displaystyle\asymp\Big{(}\frac{t}{\psi}\Big{)}^{-1}\Big{(}\frac{s(t)\mathcal{F}_{d}k(t^{\frac{1}{d}})}{s(t)}\Big{)}$
$\displaystyle=\Big{(}\frac{t}{\psi}\Big{)}^{-1}(\mathcal{F}_{d}k(t^{\frac{1}{d}}))$
Hence, combining this with (35), we have obtain the lower estimate:
$\mu_{n}\succeq\psi^{-1}(s(n)^{-1})$
∎
###### Remark.
Observe, that in light of the proof of Lemma 2, Corollary 2.1 would still hold
if in Assumption 8*, $\mathcal{F}_{d}\kappa(t^{\frac{1}{d}})$ was replaced by
$\mathcal{F}_{d}\kappa\Big{(}12.76d(2t)^{\frac{1}{d}}\Big{)}$. However, in
order to streamline the analysis, we have chosen the simpler assumption, as
also discussed in section 2.4.
###### Proof of Theorem 4.
We follow the general outline of the proof of Theorem 2 in [17] that was
earlier utilized in [8] and [4]. Our goal is to construct a sequence of
probability measures $\\{P_{j}\\}_{j=1}^{M_{\epsilon}}$ on
$\mathcal{X}\times\mathbb{R}$ with $P_{j}|_{\mathcal{X}}=\nu$ (i.e. the
measures share the same marginal on $\mathcal{X}$) that are hard to learn,
i.e. their regression functions $f_{j}$ satisfy the source conditions but are
sufficiently “well spread out” in $L^{2}(\nu)$ so that it is sufficiently hard
to distinguish between these functions given only the sample data. We make
these ideas more precise in the subsequent analysis. Our candidate regression
functions take the following form — for some fixed $\epsilon\in(0,1)$ and
$m\in\mathbb{N}$, we consider the element:
$f_{\omega}=2\sqrt{\frac{8\epsilon}{m}}\sum_{i=1}^{m}\omega_{i}e_{i+m}$
where $\omega\in\\{0,1\\}^{m}$ is some binary string. Since the sum is finite,
we have that $f\in\mathcal{H}_{K}\subset L^{\infty}\cap\mathcal{H}^{\phi}$. We
will demonstrate that $||f||_{\phi}\leq B_{\phi}$ and $||f||_{\infty}\leq
B_{\infty}$ for sufficiently large choices of $m$. Indeed, we have that:
$\displaystyle||f_{\omega}||^{2}_{\phi}$
$\displaystyle=\frac{32\epsilon}{m}\sum_{i=1}^{m}\frac{\omega^{2}_{i}}{\phi(\mu_{i+m})}\leq\frac{32\epsilon}{\phi(\mu_{2m})}$
$\displaystyle||f_{\omega}||^{2}_{\infty}$
$\displaystyle\leq||k^{\psi}||_{\infty}^{2}||f_{\omega}||^{2}_{\psi}=\frac{32||k^{\psi}||_{\infty}^{2}\epsilon}{m}\sum_{i=1}^{m}\frac{\omega^{2}_{i}}{\psi(\mu_{i+m})}\leq\frac{32||k^{\psi}||_{\infty}^{2}\epsilon}{\psi(\mu_{2m})}$
Now, by Corollary 2.1, we have that:
$\displaystyle||f_{\omega}||^{2}_{\phi}$
$\displaystyle\leq\frac{32\epsilon}{\phi(C_{d}\psi^{-1}(s(m)^{-1}))}$
$\displaystyle||f_{\omega}||^{2}_{\infty}$
$\displaystyle\leq\frac{64||k^{\psi}||_{\infty}^{2}\epsilon}{C_{d}s(m)^{-1}}$
where $C_{d}>0$ depends only $d$ (and the constants in the $\Delta_{2}$
condition on $\phi,\psi$, and $s$). Hence, by the $\Delta_{2}$ condition on
$\phi$ and $\psi$, we have that $||f||^{2}_{\phi}\leq B^{2}_{\phi}$ and
$||f||^{2}_{\infty}\leq B^{2}_{\infty}$, when:
$m\leq
s^{-1}\Big{(}\min\Big{\\{}\frac{C_{d}B^{2}_{\infty}}{64||k^{\psi}||_{\infty}^{2}\epsilon},\Big{(}\psi\Big{(}C_{d}\phi^{-1}\Big{(}\frac{32\epsilon}{B^{2}_{\phi}}\Big{)}\Big{)}\Big{)}^{-1}\Big{\\}}\Big{)}$
(where we have renamed the constant $C_{d}$, as we will throughout this
proof). By Assumption 2, we have that $\frac{\phi(t)}{\psi(t)}\to 0$ as $t\to
0$. Hence, for sufficiently small $t>0$, we have that
$\frac{\psi(\phi^{-1}(t))}{t}=\frac{\psi(\phi^{-1}(t))}{\phi(\phi^{-1}(t))}\geq
1$. Hence, we may choose $\epsilon_{1}\in(0,1)$, such that for
$\epsilon\in(0,\epsilon_{1}]$, we have $||f||^{2}_{\phi}\leq B^{2}_{\phi}$ and
$||f||^{2}_{\infty}\leq B^{2}_{\infty}$, when:
$m\leq
U_{d,\infty,\phi}s^{-1}\Big{(}\frac{1}{\psi(\phi^{-1}(\epsilon))}\Big{)}=U_{d,\infty,\phi}\tilde{s}(\psi(\phi^{-1}(\epsilon)))$
where we have recalled the definition of
$\tilde{s}(t)=s^{-1}\Big{(}\frac{1}{t}\Big{)}$ and $U_{d,\infty,\phi}>0$ is a
constant depending only on $d,B_{\infty},B_{\phi}$ that we have again
extracted via the $\Delta_{2}$ conditions on $s$, $\psi$ and $\phi$. We can
ensure that there is an $m\geq 1$ satisfying this bound by choosing
$\epsilon\leq\epsilon_{2}=\min\\{\epsilon_{1},\phi(\psi^{-1}(s(U_{d,\infty,\phi}^{-1})^{-1}))\\}$.
Moreover, we have for any two binary strings
$\omega^{(1)},\omega^{(2)}\in\\{0,1\\}^{m}$:
$||f_{\omega^{(1)}}-f_{\omega^{(2)}}||^{2}_{L^{2}(\nu)}=\frac{32\epsilon}{m}\sum_{i=1}^{m}(\omega^{(1)}_{i}-\omega^{(2)}_{i})^{2}\leq
32\epsilon$
Now, observe that if we choose
$\epsilon\leq\epsilon_{3}\equiv\min\\{\epsilon_{2},\phi(\psi^{-1}(s(9U_{d,\infty,\phi}^{-1})^{-1}))\\}$,
and choose $m_{\epsilon}=\lfloor
U_{d,\infty,\phi}\tilde{s}(\psi(\phi^{-1}(\epsilon)))\rfloor$ like before, we
have $m_{\epsilon}\geq 9$. Then, by the Gilbert-Varshamov bound (see e.g.
Lemma 24 in [17]), there exists a $M_{\epsilon}\geq
2^{\frac{m_{\epsilon}}{8}}$, and some binary strings
$\omega^{(1)},\omega^{(2)},\ldots,\omega^{(M_{\epsilon})}$, such that:
$\sum_{i=1}^{m_{\epsilon}}(\omega^{(k)}_{i}-\omega^{(l)}_{i})^{2}\geq\frac{m_{\epsilon}}{8}$
for all $k,l\in[M_{\epsilon}]$. Therefore, we have:
$||f_{\omega^{(k)}}-f_{\omega^{(l)}}||^{2}_{L^{2}(\nu)}\geq 4\epsilon$ (36)
for $k,l\in[M_{\epsilon}]$. We now construct measures
$\\{P_{j}\\}_{j=1}^{M_{\epsilon}}$ on $\mathcal{X}\times\mathcal{Y}$ such that
$P_{j}|_{\mathcal{X}}=\nu=P_{X}$, with
$\mathbb{E}[Y|X]=f_{\omega^{(j)}}(X)\equiv f_{j}(X)$ and
$Y-\mathbb{E}[Y|X]\sim\mathcal{N}(0,\bar{\sigma}^{2})$ for $(X,Y)\sim P_{j}$
and all $j\in[M_{\epsilon}]$ (where $\bar{\sigma}=\max\\{\sigma,L\\}$). The
measures $P_{j}$ satisfy Assumption 5 by Lemma 21 in [17]. Letting $P_{0}$ be
such that $f_{0}=0$. Then, we have that:
$\frac{1}{M_{\epsilon}}\sum_{i=1}^{M_{\epsilon}}\text{KL}(P^{n}_{i}||P^{n}_{0})=\frac{n}{2\sigma^{2}M_{\epsilon}}\sum_{i=1}^{M_{\epsilon}}||f_{i}-f_{0}||^{2}_{L^{2}(\nu)}\leq\frac{16n\epsilon}{\sigma^{2}}\equiv\alpha^{*}$
Thus, for any measurable function
$\Theta:(\mathcal{X}\times\mathbb{R})^{n}\to\\{0,1,\ldots,M_{\epsilon}\\}$, we
have by Theorem 20 in [17] that:
$\max_{j\in[M_{\epsilon}]}P_{j}(\Theta(D)\neq
j)\geq\frac{\sqrt{M_{\epsilon}}}{1+\sqrt{M_{\epsilon}}}\Big{(}1-\frac{48n\epsilon}{\sigma^{2}\log
M_{\epsilon}}-\frac{1}{2\log M_{\epsilon}}\Big{)}$ (37)
Note that $M_{\epsilon}\geq 2^{\frac{m_{\epsilon}}{8}}\geq
2^{\frac{U_{d,\infty,\phi}\tilde{s}(\psi(\phi^{-1}(\epsilon)))}{9}}$ using the
definition of $m_{\epsilon}$ and the fact that $m_{\epsilon}\geq 9$.
Substituting this into (37), we obtain:
$\max_{j\in[M_{\epsilon}]}P_{j}(\Theta(D)\neq
j)\geq\frac{\sqrt{M_{\epsilon}}}{1+\sqrt{M_{\epsilon}}}\Big{(}1-\frac{432n\epsilon}{\tilde{s}(\psi(\phi^{-1}(\epsilon)))\cdot
U_{d,\infty,\phi}\sigma^{2}\log 2}-\frac{1}{2\log M_{\epsilon}}\Big{)}$
Now, we choose
$\epsilon_{n}=\tau\phi\Big{(}(\frac{\phi}{\tilde{s}\circ\psi})^{-1}(n^{-1})\Big{)}$
where $\tau<1$ has been chosen so that $\epsilon_{n}\leq\epsilon_{3}$. Define:
$\Theta(D)\equiv\text{arg}\min_{j\in[M_{\epsilon_{n}}]}||f_{D}-f_{j}||_{L^{2}(\nu)}$
Then for any $j\neq\Theta(D)$, we have by (36):
$2\sqrt{\epsilon_{n}}\leq||f_{\Theta(D)}-f_{j}||_{L^{2}(\nu)}\leq||f_{D}-f_{\Theta(D)}||_{L^{2}(\nu)}+||f_{D}-f_{j}||_{L^{2}(\nu)}\leq
2||f_{D}-f_{j}||_{L^{2}(\nu)}$
Thus, we have:
$\displaystyle\max_{j\in[M_{\epsilon_{n}}]}P_{j}(||f_{D}-f_{j}||_{L^{2}(\nu)}\geq\sqrt{\epsilon_{n}})$
$\displaystyle\geq\max_{j\in[M_{\epsilon_{n}}]}P_{j}(\Theta(D)\neq j)$
$\displaystyle\geq\frac{\sqrt{M_{\epsilon_{n}}}}{1+\sqrt{M_{\epsilon_{n}}}}\Big{(}1-\frac{432n\epsilon_{n}}{\tilde{s}(\psi(\phi^{-1}(\epsilon_{n})))\cdot
U_{d,\infty,\phi}\sigma^{2}\log 2}-\frac{1}{2\log M_{\epsilon_{n}}}\Big{)}$
$\displaystyle\geq\frac{\sqrt{M_{\epsilon_{n}}}}{1+\sqrt{M_{\epsilon_{n}}}}\Big{(}1-C_{\tau}-\frac{1}{2\log
M_{\epsilon_{n}}}\Big{)}$
after substituting the choice of $\epsilon_{n}$ and letting
$C_{\tau}=\frac{432\tau K^{u}}{U_{d,\infty,\phi}\sigma^{2}\log 2}$ (where
$u=\log\tau$ and $K=K(\psi,s,\phi)>1$ is a constant depending only on
$D^{1}_{\psi},D^{2}_{\psi},D^{1}_{\phi},D^{2}_{\phi},D^{1}_{s},D^{2}_{s}>1$).
Observing that $M_{\epsilon_{n}}\to\infty$ as $\epsilon_{n}\to 0$ (as
$n\to\infty$), we obtain our result. ∎
## Appendix D Auxiliary Results
###### Lemma 1.
Suppose $\frac{t}{\psi(t)}$ is concave. Then, if Assumption 1 holds, we have
for all $f\in\mathcal{H}_{K}$:
$\frac{||f||^{2}_{\psi}}{||f||^{2}_{K}}\leq\frac{t}{\psi}\Big{(}\frac{||f||^{2}_{L^{2}(\nu)}}{||f||^{2}_{K}}\Big{)}$
###### Proof.
Follows immediately from Proposition 7 in [29] with $\kappa=\sqrt{t}$ and
$\phi=\sqrt{\psi(t)}$ (note here $\kappa$ and $\phi$ denote notation used in
[29] and do not coincide with the objects $\kappa$ and $\phi$ in this paper).
∎
###### Lemma 2.
$\sup_{x\in\mathcal{X}}||(C_{\nu}+\lambda)^{-\frac{1}{2}}k(x,\cdot)||_{K}\leq\sqrt{\frac{||k^{\psi}||_{\infty}^{2}}{\psi(\lambda)}}$
###### Proof.
We first note that, for any linear operator $T$ on $\mathcal{H}_{K}$, we have:
$\displaystyle\sup_{x\in\mathcal{X}}||Tk(x,\cdot)||_{K}$
$\displaystyle=\sup_{||f||_{K}=1,x\in\mathcal{X}}\langle
Tk(x,\cdot),f\rangle_{K}$
$\displaystyle=\sup_{||f||_{K}=1,x\in\mathcal{X}}\langle
k(x,\cdot),T^{*}f\rangle_{K}$
$\displaystyle=\sup_{||f||_{K}=1,x\in\mathcal{X}}(T^{*}f)(x)$
$\displaystyle=\sup_{||f||_{K}=1}||T^{*}f||_{\infty}$
$\displaystyle=||T^{*}||_{\mathcal{H}\to L^{\infty}}$
Hence, since $(C_{\nu}+\lambda)^{-\frac{1}{2}}$ is self-adjoint, we have by
the interpolation inequality (Lemma 1), that:
$||(C_{\nu}+\lambda)^{-\frac{1}{2}}f||^{2}_{\infty}\leq\frac{||k^{\psi}||^{2}_{\infty}||(C_{\nu}+\lambda)^{-\frac{1}{2}}f||^{2}_{2}}{\psi\Big{(}\frac{||(C_{\nu}+\lambda)^{-\frac{1}{2}}f||^{2}_{2}}{||(C_{\nu}+\lambda)^{-\frac{1}{2}}f||^{2}_{K}}\Big{)}}$
(38)
Taking the supremum over $f\in\mathcal{H}_{K}$, and noting that the RHS of
(38) is jointly increasing in $||(C_{\nu}+\lambda)^{-\frac{1}{2}}f||_{2}$ and
$||(C_{\nu}+\lambda)^{-\frac{1}{2}}f||_{K}$ by the fact that
$\frac{t}{\psi(t)}$ is concave and nondecreasing in Assumption 2, we obtain:
$||(C_{\nu}+\lambda)^{-\frac{1}{2}}||^{2}_{\mathcal{H}\to
L^{\infty}}\leq\frac{||k^{\psi}||^{2}_{\infty}}{\psi(\lambda)}$
where we have used the fact that
$\sup_{f\in\mathcal{H}_{K}}||(C_{\nu}+\lambda)^{-\frac{1}{2}}f||^{2}_{2}=\sup_{f\in\mathcal{H}_{K}}||C^{\frac{1}{2}}_{\nu}(C_{\nu}+\lambda)^{-\frac{1}{2}}f||^{2}_{K}$.
∎
###### Lemma 3.
$\mathcal{N}(\lambda)\equiv\text{tr}(C_{\nu}(C_{\nu}+\lambda)^{-1})\preceq
s^{-1}(\psi(\lambda)^{-1})$
###### Proof.
From Corollary 2.1, we obtain $\mu_{i}\asymp\psi^{-1}(s(i)^{-1})$. Therefore,
there we have:
$\displaystyle\mathcal{N}(\lambda)$
$\displaystyle\equiv\text{tr}(C_{\nu}(C_{\nu}+\lambda)^{-1})$
$\displaystyle=\sum_{i=1}^{\infty}\frac{\mu_{i}}{\mu_{i}+\lambda}$
$\displaystyle=\sum_{i=1}^{\infty}\frac{1}{1+\lambda\mu^{-1}_{i}}$
$\displaystyle\leq\sum_{i=1}^{\infty}\frac{1}{1+\frac{C_{1}\lambda}{\psi^{-1}(s(i)^{-1})}}$
$\displaystyle\leq\int_{0}^{\infty}\frac{1}{1+\frac{C_{1}\lambda}{\psi^{-1}(s(t)^{-1})}}dt$
$\displaystyle=\int_{0}^{s^{-1}(\psi(\lambda)^{-1})}\frac{1}{1+\frac{C_{1}\lambda}{\psi^{-1}(s(t)^{-1})}}dt+\int_{s^{-1}(\psi(\lambda)^{-1})}^{\infty}\frac{1}{1+\frac{C_{1}\lambda}{\psi^{-1}(s(t)^{-1})}}dt$
We observe that, by Assumption 2 we have that:
$2\psi^{-1}\Big{(}\frac{x}{D^{\psi}_{2}}\Big{)}\leq\psi^{-1}(x)\leq
2\psi^{-1}\Big{(}\frac{x}{D^{\psi}_{1}}\Big{)}$
and by Assumption 4:
$\Big{(}D^{s}_{2}s\Big{(}\frac{x}{2}\Big{)}\Big{)}^{-1}\leq
s(x)^{-1}\leq\Big{(}D^{s}_{1}s\Big{(}\frac{x}{2}\Big{)}\Big{)}^{-1}$
Putting these together, we have:
$2\psi^{-1}\Big{(}s(D^{\psi,s}_{1}x)^{-1}\Big{)}\leq\psi^{-1}(s(x)^{-1})\leq
2\psi^{-1}\Big{(}s(D^{\psi,s}_{2}x)^{-1}\Big{)}$
where $D^{\psi,s}_{1}=2^{\log_{D^{s}_{1}}D^{\psi}_{2}}$ and
$D^{\psi,s}_{2}=2^{\log_{D^{s}_{2}}D^{\psi}_{1}}$. Hence, for
$t>s^{-1}(\psi(\lambda)^{-1})$, we have:
$\frac{\lambda}{\psi^{-1}(s(t)^{-1})}=\frac{\psi^{-1}(s(s^{-1}(\psi(\lambda)^{-1}))^{-1})}{\psi^{-1}(s(t)^{-1})}=\frac{\psi^{-1}(s(t^{-1}s^{-1}(\psi(\lambda)^{-1})t)^{-1})}{\psi^{-1}(s(t)^{-1})}\geq
2^{-\log_{D^{\psi,s}_{2}}t^{-1}s^{-1}(\psi(\lambda)^{-1})}$
Observing that
$2^{-\log_{D^{\psi,s}_{2}}t^{-1}s^{-1}(\psi(\lambda)^{-1})}=\Big{(}\frac{t}{s^{-1}(\psi(\lambda)^{-1})}\Big{)}^{r}$,
where $r\equiv\log_{D^{\psi,s}_{2}}(2)$. Hence, we have:
$\displaystyle\mathcal{N}(\lambda)$
$\displaystyle\leq\int_{0}^{s^{-1}(\psi(\lambda)^{-1})}\frac{dt}{1+\frac{C_{1}\lambda}{\psi^{-1}(s(t)^{-1})}}+\int_{s^{-1}(\psi(\lambda)^{-1})}^{\infty}\frac{dt}{1+\frac{C_{1}\lambda}{\psi^{-1}(s(t)^{-1})}}$
$\displaystyle\leq
s^{-1}(\psi(\lambda)^{-1})+\int_{s^{-1}(\psi(\lambda)^{-1})}^{\infty}\frac{dt}{1+C_{1}\Big{(}\frac{t}{s^{-1}(\psi(\lambda)^{-1})}\Big{)}^{r}}$
$\displaystyle\leq
s^{-1}(\psi(\lambda)^{-1})+s^{-1}(\psi(\lambda)^{-1})\int_{1}^{\infty}\frac{ds}{1+C_{1}s^{r}}$
$\displaystyle\leq Cs^{-1}(\psi(\lambda)^{-1})$
for some $C>0$, since the second integral converges as
$r=\log_{D^{\psi,s}_{2}}(2)>1$ (as we can always choose
$D^{s}_{2}>D^{\psi}_{1}$). ∎
|
[table]capposition=top
# A Model for Attribute Based Role-Role Assignment (ARRA)
aJiwan L. Ninglekhu and bRam Krishnan Department of Electrical and Computer
Engineering
The University of Texas at San Antonio
San Antonio, Texas 78249
Email<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract
Administrative Role Based Access Control (ARBAC) models specify how to manage
user-role assignments (URA), permission-role assignments (PRA), and role-role
assignments (RRA). Many approaches have been proposed in the literature for
URA, PRA and RRA. In this paper, we propose a model for attribute-based role-
role assignment (ARRA), a novel way to unify prior RRA approaches. We leverage
the idea that attributes of various RBAC entities such as admin users and
regular roles can be used to administer RRA in a highly-flexible manner. We
demonstrate that ARRA can express and unify prior RRA models.
###### Index Terms:
Roles, Role Hierarchy, RBAC, ARBAC, Access Control, Administration.
## I Introduction
Role-based access control (RBAC) [1, 2] model is well studied in academia[3]
and well adopted in the industry[4]. In RBAC, assignment of users and
permissions to roles allows users to execute permissions by activating roles.
Administrative RBAC (ARBAC), which involves in user-role assignment and
revocation (URA), permission-role assignment and revocation (PRA), and role-
role assignment and revocation (RRA), is a challenging task [5]. ARBAC has
been well explored [5, 6, 7, 8, 9]. All of these works explore URA and PRA.
However, only few cover RRA [5, 8]. Each of these models use few fixed set of
properties of RBAC in making assignment decisions. For example, In RRA97 [5],
an admin role is given admin authority over set of roles, called role range,
where admin role can perform role assignment.
Recently, Attribute-Based Access Control (ABAC) has gained popularity because
of its flexibility [10, 11, 12, 13, 14]. ABAC has also proven its ability to
represent different access control models [12]. However, it has rarely been
used in administration of RBAC.
Among three main aspects of ARBAC, we have explored models for attribute-based
user-role assignment (AURA) and attribute-based permission-role assignment
(ARPA) models [15].
In this paper we present our model for attribute-based role-role assignment
(ARRA), a follow-up work on AARBAC [15]. ARRA is driven by our objective to
design attribute-based model that allows us to express features from previous
RRA models and more. For example, it allows us to express features as a
combination of two prior models and in addition, allows us to add new
features. Thus, this work is motivated largely by two critical factors: (a) an
objective to build a coherent model, which can be configured to express prior
models and more (b) build a unified model that can be analyzed for various
desirable security features, which can provide a single codebase to express
prior models and beyond. The contributions of this paper are as follows:
* •
We develop an attribute-based administrative model for role-role assignment
(ARRA).
* •
We demonstrate that ARRA is capable of expressing prior approaches to RRA such
as RRA97 [5] and UARBAC’s RRA [8].
The remainder of this paper is organized as follows. In Section II, we discuss
related work. In Section III, ARRA model is presented. Section IV presents
algorithms that translate prior RRA instances into equivalent ARRA instances.
Section V concludes the paper.
## II Related work
Among many prominent works done for ARBAC [5, 6, 7, 9, 8], ARBAC97 [5] and
UARBAC [8] present role-role assignment (RRA) as part of their model.
In RRA97, a set of roles in the hierarchy called authority range is given to
an admin role for decentralized administration. A user with admin role can
perform operations like inserting an edge or creating a role, on roles in her
authority range. To confirm autonomy RRA97 requires role range to be
encapsulated [5].
UARBAC [8] redefines RBAC model with class objects. It treats each entity in
the RBAC system such as files, roles and users as objects. UARBAC’s RRA deals
with assigning/revoking role-role on the basis of admin user’s authority over
individual roles specified by access modes. Class permissions in UARBAC allows
an admin user to conduct an operation over every object of a class.
In both of these models, the policy for assigning a role to a role is based on
a fixed set of properties of the entities such as admin role, admin user and
regular roles, involved in assigning/revoking roles.
Crampton et. al [16] present models for RRA with administrative scopes. Admin
scopes are plausible approach for role hierarchy operations in RBAC. However,
admin scopes may not be intuitive to express as a role attribute. Many
literature [10, 17, 18] present benefits of integrating attributes into an
RBAC operational model. An operational model deals with making decisions on
user’s access request for objects. There are many works on operational aspects
of ABAC [19, 12, 11, 20]. In contrast, ARRA is an administrative model that
uses attributes of RBAC entities for assigning/revoking roles.
Table I: ARRA Model
o 1 | X[l] | – AU, AOP, ROLES, AR are finite sets of administrative users, administrative operations such as assign and revoke, regular roles and administrative roles, respectively.
---
– AUA $\subseteq$ AU $\times$ AR, administrative user to administrative role
assignment relation.
– RH $\subseteq$ ROLES $\times$ ROLES, a partial ordering on the set ROLES.
– ARH $\subseteq$ AR $\times$ AR, a partial ordering on the set AR.
– AATT, ARATT, and RATT are finite sets of administrative user attribute
functions, administrative role attribute functions, and regular role attribute
functions, respectively.
– For each att in AATT $\cup$ ARATT $\cup$ RATT, Scope(att) is a finite set of
atomic values from which the range
of the attribute function att is derived. – attType : AATT $\cup$ ARATT $\cup$
RATT $\rightarrow$ {set, atomic}, which specifies whether the range of a given
attribute
is atomic or set valued.
– Each attribute function maps elements in AU, AR and ROLES to atomic or set
values. $\forall aatt\in\mbox{AATT}.\
aatt:\mbox{AU}\rightarrow\left\\{\begin{array}[]{ll}\mbox{Scope}(aatt)\>\mbox{if}\>\mbox{attType}(aatt)=\mbox{atomic}\\\
2\textsuperscript{Scope({aatt})}\>\mbox{if}\>\mbox{attType}(aatt)=\mbox{set}\\\
\end{array}\right.$ $\forall aratt\in\mbox{ARATT}.\
aratt:\mbox{AR}\rightarrow\left\\{\begin{array}[]{ll}\mbox{Scope}(aratt)\>\mbox{if}\>\mbox{attType}(aratt)=\mbox{atomic}\\\
2\textsuperscript{Scope({aratt})}\>\mbox{if}\>\mbox{attType}(aratt)=\mbox{set}\\\
\end{array}\right.$ $\forall ratt\in\mbox{RATT}.\
ratt:\mbox{ROLES}\rightarrow\left\\{\begin{array}[]{ll}\mbox{Scope}(ratt)\>\mbox{if}\>\mbox{attType}(ratt)=\mbox{atomic}\\\
2\textsuperscript{Scope({ratt})}\>\mbox{if}\>\mbox{attType}(ratt)=\mbox{set}\\\
\end{array}\right.$ – is_ordered : AATT $\cup$ ARATT $\cup$ RATT $\rightarrow$
{True, False}, specifies if the scope is ordered for each of the
attributes.
– For each att $\in$ AATT $\cup$ ARATT $\cup$ RATT, if is_ordered(att) = True,
Hatt $\subseteq$ Scope(att) $\times$ Scope(att), a partially ordered attribute
hierarchy, and Hatt $\neq$ $\phi$, else, if is_ordered(att) = False, Hatt =
$\phi$
(For some att $\in$ AATT $\cup$ ARATT $\cup$ RATT for which attType(att) = set
and is_ordered(att) = True, if $\\{a,b\\}$, $\\{c,d\\}$ $\in$ 2Scope(att)
(where $a,b,c,d\in$ Scope($att$)), we infer $\\{a,b\\}$ $\geq$ $\\{c,d\\}$ if
$(a,c)$, $(a,d)$, $(b,c)$, $(b,d)$ $\in$ H${}^{\mbox{*}}_{\mbox{{att}}}$.)
ARRA model allows an administrator to perform an operation on a single role or
a set of roles at a time. The authorization rule for performing an operation
on a single role is as follows:
For each op in AOP, is_authorizedRop(au: AU, r1 : ROLES, r2 : ROLES) specifies
if the admin user au is allowed to perform the operation op (e.g. assign,
revoke, etc.) between the regular roles r1 and the role r2. Assigning a role
r1 to r2 makes r1 junior to r2. This rule is written as a logical expression
using attributes of admin user au, admin role, ar, and regular role, r.
The authorization rule for performing an operation on a set of users is as
follows:
For each op in AOP, is_authorizedRop(au: AU, $\chi$ : 2ROLES, r : ROLES)
specifies if the admin user au is allowed to perform the operation op (e.g.
assign, revoke, etc.) between the roles in the set $\chi$ and the role r. Here
$\chi$ is a set of roles that can be specified using a set-builder notation,
whose rule is written using role attributes.
## III ARRA Model
In this section, we present our approach for attribute-based role-role
assignment (ARRA). Inspired by prior RRA models we have included attributes
for admin users, admin roles and regular roles. Based on these attributes role
assignment and revocation decisions on are made.
Table II presents formal ARRA model. The entities involved in ARRA comprise of
admin users (AU), regular roles (ROLES) and their hierarchy (RH), admin roles
(AR) and their hierarchy (ARH), admin user to admin role relation (AUA) and
admin operations (AOP).
In ARRA, admin user in AU wants to perform admin operation such as assign or
revoke from AOP using attributes of entities in the model. We have developed
admin users attribute functions (AATT) and admin roles attribute functions
(ARATT). We have also introduced regular roles attribute functions (RATT)
based on the need we have observed. Attributes from other RBAC entities can
also be developed if needed. We will later see that we need attributes from
different entities in representing properties of RRA97 and UARBAC’s RRA in
ARRA.
The attribute functions or _simply_ attributes are defined as mapping from its
domain such as AU or AR to its range. Range of an attribute att can be atomic
or set valued, which is specified by function attType. It is derived from set
of scope or atomic values denoted by Scope(att). Furthermore, the scope of an
attribute can be either ordered or unordered, which is given by a function
called is_ordered(att). If it is ordered, we must specify that attribute’s
hierarchy denoted by Hatt on its scope Scope(att). Hatt is a partial ordering
on Scope(att). Note that even in the case of a set valued attribute att, the
hierarchy Hatt is specified on Scope(att) instead of 2Scope(att). We infer the
ordering between two set values given an ordering on atomic values.
H${}^{\mbox{*}}_{\mbox{{att}}}$ in Table II denotes the reflexive transitive
closure of Hatt.
In ARRA, there are two ways by which an admin user can select a set of regular roles for assignment to a target regular role. The first way allows an admin user to select a single role and a target role, and perform an admin operation like assign or revoke. The second way allows an admin user to select a set of regular roles, the target role and perform similar operation on those roles. In the latter case, the selection criteria for the set of regular roles can be expressed using a set-builder notation whose rule is based on the regular role attributes. For example, is_authorizedRassign(au, {r1 | r1 $\in$ ROLES $\wedge$ Lead $\in$ roleTitle(r1)}, r2) would specify a policy for an admin user au that selects the set of all the roles with role title Lead in order to assign those roles to a role r2. Assigning a role r1 to role r2 makes role r1 junior to r2. In other words, it adds an entry <r2, r1> in RH. It is also referred to as edge insetion.
Authorization rule is specified as a logical expression on the attributes of
admin users, admin roles, and that of regular roles considered for assignment.
## IV Mapping Prior RRA Models in ARRA
In this section, we demonstrate that ARRA can intuitively simulate the
features of prior RRA models. In particular, we have developed concrete
algorithms that can convert any instance of RRA97 and UARBAC’s RRA into an
equivalent ARRA instance. For each of these models, we first present an
example instance followed by that instance’s corresponding ARRA equivalent.
Then finally, we present algorithms for each to show a programmable mapping
procedure.
Figure 1: Role and Administrative Role Hierarchies in RRA97
### IV-A Mapping RRA97 to AARA
In this section, we present RRA97 example instance and manually convert it
into its equivalent ARRA instance. We then present a mapping algorithm for the
same, which demostrates a programmable procedure for translation of any RRA97
instance to its equivalent ARRA instance.
#### IV-A1 An RRA97 Example Instance
RRA97 [5] assumes an existing regular role hierarchy and an administrative
role hierarchy as depicted in Figure 1. An example instance for RRA97 model is
presented as follows: Sets and Functions:
* •
USERS = {u1, u2, u3, u4}
* •
ROLES = {ED, E1, PE1, QE1, PL1, E2, PE2,
* QE2, PL2, DIR}
* •
AR = {DSO, PSO1}
* •
AUA = {(u3, DSO), (u4, PSO1)}
* •
RH = {<ED, E1>, <E1, QE1>, <E1, PE1>,
* <PE1, PL1>, <QE1, PL1>, <ED, E2>,
* <E2, PE2>, <E2, QE2>, <PE2, PL2>,
* <QE2, PL2>, <PL1, DIR>, <PL2, DIR>}
* •
ARH = {<SSO, DSO>, <DSO, PSO1>,
<DSO, PSO2>} • can-modify = {(DSO, (ED, DIR)), (PSO1, (E1, PL1)), (PSO1, (E2,
PL2))} Where, (ED, DIR) = {E1, E2, PE1, PE2, QE1, QE2, PL1, PL2}, (E1, PL1)
= {PE1, QE1} and, (E2, PL2) = {PE2, QE2} USERS, ROLES, and AR are sets of
users, regular roles and admin roles, respectively. User u3 is given admin
role DSO and PSO1 is given to role u4. RH and ARH represent regular role and
admin role hierarchies as depicted in Figure 1. We have taken original can-
modify example from RRA97 paper [5] where each admin role is mapped with
authority ranges. The can-modify set in our instance depicts the same scenario
with three elements: (DSO, (ED, DIR)), (PSO1, (E1, PL1)) and (PSO1, (E2,
PL2)). It shows that role DSO is given an authority range (ED, DIR) while role
PSO1 is given two authority ranges (E1, PL1) and (E2, PL2). In RRA97, an
authority range, (x, y) = {r : ROLES $|$ x < r < y}, where x and y are end
points of the range. For every (ar, (x, y)) in can-modify, the range denoted
by (x, y) must be an encapsulated authority range. A range (x, y) is said to
be encapsulated if, $\forall$r1 $\in$ (x, y) $\wedge$ $\forall$r2 $\notin$ (x,
y). [r2 > r1 $\leftrightarrow$ r2 > y $\wedge$ r2 < r1 $\leftrightarrow$ r2 <
x] A new edge AB can be inserted between two incomparable roles A and B under
following two conditions: • ARimmediate(A) = ARimmediate(B). An immediate
authority range of a role r, ARimmediate(r) is the authority range (x,y) such
that r $\in$ (x, y) and for all authority ranges (x', y') junior to (x, y) we
have r $\notin$ (x', y'). or, • if (x, y) is an authority range such that (A =
y $\wedge$ B > x) $\vee$ (B = x $\wedge$ A < y) then insertion of AB must
preserve encapsulation of (x, y).
#### IV-A2 ARRA Instance Equivalent to RRA97 Instance
In this section, we show a manual translation of RRA97 example instance
presented in previous segment into an equivalent ARRA instance. Equivalent
ARRA instance is as follows:
Map RRA97 sets to ARRA sets
* •
AU = {u1, u2, u3, u4}
* •
AOP = {insertEdge, deleteEdge}
* •
ROLES = {ED, DIR, E1, PE1, QE1, PL1, E2,
* PE2, QE2, PL2, DIR}
* •
AUA = {(u3, DSO), (u4, PSO1)}
* •
RH = {<ED, E1>, <E1, QE1>, <E1, PE1>,
* <PE1, PL1>, <QE1, PL1>, <ED, E2>,
* <E2, PE2>, <E2, QE2>, <PE2, PL2>,
* <QE2, PL2>, <PL1, DIR>, <PL2, DIR>}
Define attributes
* •
AATT = {}, RATT = {}
* •
ARATT= {authRange}
* •
Scope(authRange) = RH+,
* attType(authRange) = set,
* is_ordered(authRange) = False, HauthRange = $\phi$
* •
authRange(DSO) = {ED, DIR},
* authRange(PSO1) = {(E1, PL1), (E2, PL2)}
Construct authorization rule for insertEdge To assign role r1 to role r2 (or
to insert an edge r1r2 and hence add <r2, r1> to RH), –
is_authorizedRinsertEdge(au : AU, r1 : ROLES, r2 : ROLES) $\equiv$
($\exists$(au, ar1) $\in$ AUA, $\exists$(s, t) $\in$ authRange(ar1). r1, r2
$\in$ $\lceil$s, t$\rceil$) $\wedge$ (($\exists$(m, n), (m', n') $\in$
$\bigcup\limits_{ar\textsubscript{2}\in\ \textrm{AR}}$ authRange(ar2). r1, r2
$\lceil$m, n$\rceil$ $\wedge$ ($\lceil$m', n'$\rceil$ $\subset$ $\lceil$m,
n$\rceil$ $\rightarrow$ r1, r2 $\notin$ (m', n'))) $\vee$ ($\exists$(x,y)
$\in$ $\bigcup\limits_{ar\textsubscript{3}\in\ \textrm{AR}}$ authRange(ar3).
((r1 = y $\wedge$ r2 > x) $\vee$ (r2 = x $\wedge$ r1 < y)) $\wedge$
($\forall$p $\in$ $\lceil$x, y$\rceil$ $\wedge$ $\forall$q $\notin$ $\lceil$x,
y$\rceil$. (<q, p> $\in$ (RH $\cup$ <r2, r1>)* $\leftrightarrow$ <q, y> $\in$
(RH $\cup$ <r2, r1>)*) $\wedge$ (<p, q> $\in$ (RH $\cup$ <r2, r1>)*
$\leftrightarrow$ <x, q> $\in$ (RH $\cup$ <r2, r1>)*))))
Construct authorization rule for deleteEdge – is_authorizedRdeleteEdge(au :
AU, r1 : ROLES, r2 : ROLES) $\equiv$ $\exists$(au, ar) $\in$ AUA, $\exists$(x,
y) $\in$ authRange(ar). r1, r2 $\in$ $\lceil$x, y$\rceil$ $\wedge$
$\forall$(u, v) $\in$ $\bigcup\limits_{ar\textsubscript{4}\in\ \textrm{AR}}$
authRange(ar4), $\exists$<r2, r1> $\in$ RH. (r1 $\neq$ u $\wedge$ r2 $\neq$ v)
As we can observe, the process of translation involves four main steps. First,
we map given sets from RRA97 to equivalent sets in ARRA. Second, required
attribute functions are defined. We have designed admin role attribute
authRange that maps authority ranges given to admin roles. Attribute authRange
is set valued and unordered. Scope of authRange is a transitive clousure of
regular role hierarchy, RH+. That is, if <a, b>, <b, c>, <c, d> $\in$ RH+,
then <a, d> is also true.
We know that authority range represents a set of roles over which an admin
role have authority to conduct operations such as assign and revoke. In ARRA,
we use symbolic representation of authority range with end points a and b as
(a, b). However, whenever we need to work with roles present in the set
represented by authority range (a, b), we denote the set of roles with
$\lceil$a, b$\rceil$. That is, $\lceil$a, b$\rceil$ = {r $|$ r $\in$ ROLES
$\wedge$ a < r < b}.
Next, we construct authorization rule for the admin operation insertEdge.
There are fundamentally three requirements that an admin user must consider to
assign role r1 to role r2. Firstly, both the roles r1 and r2 must be within
the authority range of an admin role assigned to a given admin user. Secondly,
both the roles taken for edge insertion must have same immediate authority
range, or (thirdly) if the edge is to be inserted between a role on the top
end point of an authority range and a role senior to the lower end point of
authority range, or if the edge is to be inserted between a role on the bottom
end of an authority range and a role junior to the top end of the authority
range, then that edge insertion must preserve encapsulation of an authority
range with those (top and bottom) end points. Finally, we construct an
authorization formula for operation deleteEdge. An admin user with an admin
role can delete an edge between any two roles in an authority range except
that any edge between the end points of an authority range can not be deleted.
Algorithm 1 MapRRA97
0: Input: RRA97 instance
0: Output: ARRA instance
0: /* Map basic sets and functions in AURA */
0: a. AU $\leftarrow$ USERS97
0: b. AOPA $\leftarrow$ {insertEdge, deleteEdge}
0: c. ROLESA $\leftarrow$ ROLES97 ; AUAA $\leftarrow$ AUAU
0: b. RHA $\leftarrow$ RH97
0: /* Map attribute functions in AURA */
0: a. AATTA $\leftarrow$ {} ; RATTA $\leftarrow$ {}
0: b. ARATTA $\leftarrow$ {authRange}
0: c. ScopeA(authRange) = RH+A
0: d. attTypeA(authRange) = set
0: e. is_orderedA(authRange) = False ; H${}^{\mbox{{A}}}_{\mbox{{authRange}}}$
= $\phi$
0: f. For each ar $\in$ ARA, authRange(ar) = $\phi$
0: g. For each (ar, (ri, rj)) $\in$ can-modify97,
0: authRange(ar) = authRange(ar) $\cup$ (ri, rj)
0: /* Construct assign rule in AURA */
0: a. assign_formula = $\phi$
0: b. For each (ar, (ri, rj)) $\in$ can-modify97,
0: assign_formula' = assign_formula $\vee$ ($\exists$(au, ar1) $\in$ AUA,
$\exists$(s, t) $\in$ authRange(ar1). r1, r2 $\in$ $\lceil$s, t$\rceil$)
$\wedge$ (($\exists$(m, n), (m', n') $\in$
$\bigcup\limits_{ar\textsubscript{2}\in\ \textrm{AR}}$ authRange(ar2). r1, r2
$\lceil$m, n$\rceil$ $\wedge$ ($\lceil$m', n'$\rceil$ $\subset$ $\lceil$m,
n$\rceil$ $\rightarrow$ r1, r2 $\notin$ (m', n'))) $\vee$ ($\exists$(x,y)
$\in$ $\bigcup\limits_{ar\textsubscript{3}\in\ \textrm{AR}}$ authRange(ar3).
((r1 = y $\wedge$ r2 > x) $\vee$ (r2 = x $\wedge$ r1 < y)) $\wedge$
($\forall$p $\in$ $\lceil$x, y$\rceil$ $\wedge$ $\forall$q $\notin$ $\lceil$x,
y$\rceil$. (<q, p> $\in$ (RH $\cup$ <r2, r1>)* $\leftrightarrow$ <q, y> $\in$
(RH $\cup$ <r2, r1>)*) $\wedge$ (<p, q> $\in$ (RH $\cup$ <r2, r1>)*
$\leftrightarrow$ <x, q> $\in$ (RH $\cup$ <r2, r1>)*))))
0: c. auth_assign = is_authorizedRinsertEdge(au : AUA,
0: r1 : ROLESA, r2 : ROLESA) $\equiv$ assign_formula'
0: /* Construct revoke rule for AURA */
0: a. revoke_formula = $\phi$
0: b. For each (ar1, (r1, r2)) $\in$ can-modify97,
0: revoke_formula' = revoke_formula $\vee$
0: $\exists$(au, ar) $\in$ AUA, $\exists$(x, y) $\in$ authRange(ar). r1, r2
$\in$ $\lceil$x, y$\rceil$ $\wedge$ $\forall$(u, v) $\in$
$\bigcup\limits_{ar\textsubscript{4}\in\ \textrm{AR}}$ authRange(ar4),
$\exists$<r2, r1> $\in$ RH. (r1 $\neq$ u $\wedge$ r2 $\neq$ v)
0: c. auth_revoke = is_authorizedRdeleteEdge(au : AUA,
0: r1 : ROLESA, r2 : ROLESA) $\equiv$ revoke_formula'
### IV-B MapRRA97
Algorithm 1 presents MapRRA97, a translation algorithm for any RRA97 instance
into an equivalent ARRA instance. Superscript labels 97 and A in the algorithm
represent sets and functions from RRA97 and ARRA, respectively. MapRRA97 takes
an instance of RRA97 as input. In particular, input consists of USERS97,
ROLES97, AR97, AUA97, RH97, ARH97, and can-modify97. The can-modify
instruction covers operations for inserting an edge, deleting an edge,
creating a role and deleting a role. ARRA can simulate inserting edge and
deleting an edge. However, creating and deleting roles are beyond the scope of
current ARRA model.
Output from MapRRA97 algorithm is an equivalent ARRA instance, with following
sets and functions. AUA, AOPA, ROLESA, AUAA, RHA, ARHA, AATTA, ARATTA, For
each attribute att $\in$ ARATTA, ScopeA(att), attTypeA(att), is_orderedA(att)
and HAatt, For each admin role ar $\in$ ARA and for each att $\in$ ARATTA,
att(ar), Authorization rule for assign (auth_assign), and Authorization rule
for revoke (auth_revoke).
Steps indicated in MapRRA97 correspond to the steps described in equivalent
ARRA instance for RRA97 in section IV-A2. We explain it here briefly. Step 1
maps sets from RRA97 to ARRA sets and functions. In Step 2, admin user
attributes and admin role attributes are expressed. AATT and RATT are left
empty as there is no use case for these attributes in translating RRA97. Admin
role attribute authRange captures the mapping between an authority range as
defined in RRA97, and an admin role. For each (ar, (ri, rj)) in set can-
modifySS97, authRange(ar) populates set of authority ranges for ar.
In Step 3, we construct assign_formula rule for ARRA that is equivalent to
inserting an edge in can_modify97. For each (ar, (ri, rj)) in set can-
modifySS97, logical expression that checks the assignment conditions is
constructed. Equivalent translation in ARRA is given by
is_authorizedRinsertEdge(au : AUA, rß1 : ROLESA, rß2 : ROLESA). This formula
must checks if roles rß1 and rß2 belong in an authority range of an admin
role, which is mapped to admin user au. It further checks if rß1 and rß2 have
same immediate authority range, or checks if any other assignment action
doesn’t violate encapsulation of admin role’s authority range. Similarly, In
Step 4, revoke_formula equivalent to deleting an edge from can_modify97 is
expressed. It checks if admin role mapped to admin role au has authority range
where rß1 and rß2 belong and restricts from deleting an edge between end
points of any authority range.
### IV-C Mapping UARBAC’s RRA to ARRA
In this section, we present an example instance for UARBAC’s RRA and its
equivalent ARRA instance. We then present an algorithm that outlines the
translation procedure.
#### IV-C1 An Example UARBAC’s RRA Instance
RBAC schema
* •
C = {role}
* •
OBJS(user) = USERS, OBJS(role) = ROLES
* •
AM(role) = {grant, empower, admin}
RBAC state
* •
USERS = OBJ(user) = {u1, u2, u3, u4}
* •
ROLES = OBJ(role)= {r1, r2, r3}
* •
P = [role, r1, grant], [role, r1, empower], [role, r1, admin], [role, r2,
grant], [role, r2, empower], [role, r2, admin], [role, r3, grant], [role, r3,
empower], [role, r3, admin], [role, grant], [role, empower], [role, admin]}
* •
RH = {<r2, r3>}
Administrative permissions of UARBAC’s RRA Following is the list of
administrative permissions each user has for role-role assignment:
* •
authorized_perms[u1] = {[role, r1, grant],
* [role, r2, admin], [role, r2, empower],
* [role, r3, admin]}
* •
authorized_perms[u2] = {[role, r1, admin],
* [role, r1, empower], [role, r2, empower],
* [role, r3, grant]}
* •
authorized_perms[u3] = {[role, admin]}
* •
authorized_perms[u4] = {[role, grant],
* [role, empower], [role, admin]}
Role-role assignment condition One can perform following operation to assign a
role r1 to another role r2.
* •
grantRoleToRole(r1, r2)
To perform this operation one needs one of the following two permissions:
* •
[role, r1, grant] and [role, r2, empower] or,
* •
[role, grant] and [role, r2, empower] or,
* •
[role, r1, grant] and [role, empower] or,
* •
[role, grant] and [role, empower]
Condition for revoking a role from another role To revoke a role r2 from a
role r3, admin user performs following operation.
* •
revokeRoleFromUser(r2, r3)
To conduct this operation one needs one of the following options:
* •
[role, r2, grant] and [role, r3, empower]
* •
[role, r2, admin]
* •
[role, r3, admin]
* •
[role, admin]
### IV-D ARRA instance equivalent to UARBAC’s RRA instance
This segment presents equivalent instance for UARBAC’s RRA example instance
presented in previous segment. Map UARBAC’s RRA sets to ARRA sets
* •
AU = {u1, u2, u3, u4}
* •
AOP = {assign, revoke}
* •
ROLES = {r1, r2, r3},
* •
AUA = {}
* •
RH = {<r2, r3>}
Define Attributes
* •
AATT = {grantAuth, empowerAuth, adminAuth, roleClassAuth}, ARATT = {}, RATT =
{}
* •
Scope(grantAuth) = {r1, r2, r3}
* attType(grantAuth) = set,
* is_ordered(grantAuth) = ROLES, HgrantAuth = RH
* •
grantAuth(u1) = {r1}, grantAuth(u2) = {r3},
* grantAuth(u3) = {}, grantAuth(u4) = {}
* •
Scope(empowerAuth) = {r1, r2, r3}
* attType(empowerAuth) = set,
* is_ordered(empowerAuth) = ROLES
* HempowerAuth = RH
* •
empowerAuth(u1) = {r2},
* empowerAuth(u2) = {r1, r2},
* empowerAuth(u3) = {}, empowerAuth(u4) = {}
* •
Scope(adminAuth) = {r1, r2, r3}
* attType(adminAuth) = set,
* is_ordered(adminAuth) = ROLES,
* HadminAuth = RH
* •
adminAuth(u1) = {r2, r3},
* adminAuth(u2) = {r1},
* adminAuth(u3) = {}, adminAuth(u4) = {}
* •
Scope(roleClassAuth) = AM(role)
* attType(roleClassAuth) = set,
* is_ordered(roleClassAuth) = False,
* HroleClassAuth = $\phi$
* •
roleClassAuth(u1) = {}, roleClassAuth(u2) = {},
* roleClassAuth(u3) = {admin},
* roleClassAuth(u4) = {grant, empower, admin}
Construct authorization rule for role-role assignment
For each op in AOP, authorization rule to assign/revoke role-role can be
expressed as follows: To assign any regular role r1 $\in$ ROLES to regular
role r2 $\in$ ROLES, – is_authorizedUassign(au : AU, r1 : ROLES, r2 : ROLES)
$\equiv$ (r1 $\in$ grantAuth(au) $\wedge$ r2 $\in$ empowerAuth(au)) $\vee$ (r1
$\in$ grantAuth(au) $\wedge$ empower $\in$ roleClassAuth(au)) $\vee$ (grant
$\in$ roleClassAuth(au) $\wedge$ r2 $\in$ empowerAuth(au)) $\vee$ (grant $\in$
roleClassAuth(au) $\wedge$ empower $\in$ roleClassAuth(au))
Construct authorization rule for revoking role-role
To revoke any regular role r1 $\in$ ROLES from another regular role r2 $\in$
ROLES, – is_authorizedUrevoke(au1 : AU, r1 : ROLES, r2 : ROLES) $\equiv$ (r1
$\in$ grantAuth(au) $\wedge$ r2 $\in$ empowerAuth(au)) $\vee$ r1 $\in$
adminAuth(au) $\vee$ r2 $\in$ adminAuth(au) $\vee$ admin $\in$
roleClassAuth(au)
There are four main stages in this translation process. First, we map main
sets from UARBAC’s RRA to ARRA model. Set of admin users consists of set of
users from UARBAC’s RRA, namely u1, u2, u3, and u4. There are two admin
operations, assign and revoke. Sets ROLES and RH map set of roles and role
hierarchy from UARBAC’s RRA, respectively. AUA is left empty as there is no
notion of admin roles in UARBAC.
Next, we define required attribute functions. We used admin user attributes to
express ARRA for UARBAC’s RRA. Among four admin user attributes, grantAuth,
empowerAuth and adminAuth are object level attributes. Each of these
attributes express the nature of authority an admin user has over different
roles. For example, empowerAuth over a role rx, allows admin user to assign
any object such as a user and/or another role to role rx. Class level
attribtue roleClassAuth captures the nature of access mode an admin user has
towards role class. The scope of this attribute is AM(role), which has three
different types of access modes, namely grant, empower and admin. For example,
an admin user with roleClassAuth of grant allows that admin user to grant
(assign) any role to other object such as a user and/or role. Translation of
UARBAC’s RRA doesn’t require admin role and regular role attributes, hence
left empty.
After attributes have been defined we construct the assignment formula
(assign_formula), using usual logical expression. As mentioned in the UARBAC’s
RRA instance, there are four different combination of permissions an admin
user can use to make role-role assignment. An equivalent logical expression is
captured as is_authorizedUassign(au : AU, r1 : ROLES, r2 : ROLES). Finally, we
construct a logical expression which captures four different permission
options that an admin user can use for revoking a role from another role. The
logical expression is equivalent to is_authorizedUrevoke(au1 : AU, r1 : ROLES,
r2 : ROLES).
Algorithm 2 MapRRA-UARBAC
0: Input: Instance of RRA in UARBAC
0: Output: ARRA instance
0: /* Map basic sets and functions in AURA */
0: a. AUA $\leftarrow$ USERSU ; AOPA $\leftarrow$ {assign, revoke}
0: b. ROLESA $\leftarrow$ ROLESU ; AUAA = $\phi$
0: c. RHA $\leftarrow$ RHU
0: /* Map attribute functions in AURA */
0: a. AATTA $\leftarrow$ {grantAuth, empowerAuth, adminAuth,
0: roleClassAuth}, ARATT = {}, RATT = {}
0: b. ScopeA(grantAuth) = ROLESA,
0: c. attTypeA(grantAuth) = set
0: d. is_orderedA(grantAuth) = True, HAgrantAuth = RHA
0: e. For each u in AUU, grantAuth(u) = $\phi$
0: f. For each u in USERSU and
0: for each [role, r, grant] $\in$ authorized_permsU[u],
0: grantAuth(u)' = grantAuth(u) $\cup$ (r)
0: g. ScopeA(empowerAuth) = ROLESA,
0: h. attTypeA(empowerAuth) = set
0: i. is_orderedA(empowerAuth) = True,
0: j. HAempowerAuth = RHA
0: k. For each u in AUU, empowerAuth(u) = $\phi$
0: l. For each u in USERSU and
0: for each [role, r, empower]
0: $\in$ authorized_permsU[u],
0: empowerAuth(u)' = empowerAuth(u) $\cup$ r
0: m. ScopeA(adminAuth) = ROLESA
0: n. attTypeA(adminAuth) = set
0: o. is_orderedA(adminAuth) = True,
0: p. HAadminAuth = RHA
0: q. For each u in USERSA, adminAuth(u) = $\phi$
0: r. For each u in USERSU and
0: for each [role, r, admin] $\in$ authorized_permsU[u],
0: adminAuth(u)' = adminAuth(u) $\cup$ r
0: s. ScopeA(roleClassAuth) = AMU(role)
0: t. attTypeA(roleClassAuth) = set
0: u. is_orderedA(roleClassAuth) = False
0: v. HAroleClassAuth = $\phi$
0: w. For each u in USERSA, roleClassAuth(u) = $\phi$
0: x. For each u in USERSU
0: for each [c, am] $\in$ authorized_permsU[u],
0: roleClassAuth(u)'= roleClassAuth(u) $\cup$ am
0: /* Construct assign rule in AURA */
0: a. assign_formula = (r1 $\in$ grantAuth(au) $\wedge$ r2 $\in$
empowerAuth(au)) $\vee$ (r1 $\in$ grantAuth(au) $\wedge$ empower $\in$
roleClassAuth(au)) $\vee$ (grant $\in$ roleClassAuth(au) $\wedge$ r2 $\in$
empowerAuth(au)) $\vee$ (grant $\in$ roleClassAuth(au) $\wedge$ empower $\in$
roleClassAuth(au))
Continuation of Algorithm 1 MapRRA-UARBAC
0:
0: b. auth_assign =
0: is_authorizedUassign(au : AU, r1 : ROLES, r2 : ROLES) $\equiv$
assign_formula
0: /* Construct revoke rule for AURA */
0: a. revoke_formula = r1 $\in$ grantAuth(au) $\wedge$ r2 $\in$
empowerAuth(au)) $\vee$ r1 $\in$ adminAuth(au) $\vee$ r2 $\in$ adminAuth(au)
$\vee$ admin $\in$ roleClassAuth(au)
0: b. auth_revoke =
0: is_authorizedUrevoke(au1 : AU, r1 : ROLES, r2 : ROLES) $\equiv$
revoke_formula
### IV-E MapRRA-UARBAC
We presented example instances for UARBAC’s RRA and its equivalent AARA
instance in previous segments. Algorithm IV-D presents MapRRA-UARBAC, a
procedure to map any instance of UARBAC’s RRA [8] to its equivalent ARRA
instance. It take instance of UARBAC’s RRA as its input and yields an
equivalent AARA instance as its output. Sets and functions from UARBAC and
ARRA are labeled with superscripts U and A, respectively. Input consists of
CU, USERSU, ROLESU, PU, RHU, AMU(role), For each u $\in$ USERSU,
authorized_permsU[u], and For every r1, r2 $\in$ ROLESU. Grant operation
grantRoleToRole(r1, r2) will be true if the granter has either [role, r2,
empower] and [role, r1, grant] or, [role, r2, empower] and [role, grant] or,
[role, empower] and [role, r1, grant] or, [user, empower] and [role, grant]
permissions towards roles.
For each r1, r2 $\in$ ROLESU, revokeRoleFromUser(r1, r2) is true if the
granter has either [role, r2, empower] and [role, r1, grant] or, [role, r1,
admin] or, [role, r2, admin] or, [role, admin] permission on roles.
MapRRA-UARBAC yields an ARRA instance consisting of AUA, AOPA, ROLESA, ARA,
AUAA, RHA, ARHA, AATTA, For each attribute att $\in$ AATTA, ScopeA(att),
attTypeA(att), is_orderedA(att) and HAatt, For each user au $\in$ AUA, and for
each att $\in$ AATTA, att(au), Authorization rule for assign (auth_assign),
and Authorization rule for revoke (auth_revoke).
In Step 1 primary sets from UARBAC’s RRA are mapped to AURA equivalent sets.
In Step 2, admin user attributes are defined. In UARBAC model, role-role
assignment decisions are based on the admin user’s _access modes_ such as
grant, empower and admin towards regular roles. For example, for each
permission [role, r, grant] in authorized_permsU[u], grantAuth(au) attribute
extracts set of roles towards which an admin user au has grant authority. In
other words, admin user can grant those roles to any other object such as user
and role. Similarly, empowerAuth yields set of roles to which an admin user
has empower access mode and adminAuth yields sets of roles about which admin
user has admin access mode. The meaning of each access mode is preserved by
UARBAC model. For each class level attribute [c, am] in authorized_permsU[u],
where c is a type of class and am is type of access mode, roleClassAuth(au)
specifies the nature of class level access mode such as grant or admin admin
user au has on role class.
In Step 3, an authorization formula equivalent to grantRoleToRole(r1, r2) in
UARBAC is expressed. Equivalent expression is_authorizedRassign(au : AUA, r1 :
ROLESA, r2 : ROLESA) checks if an admin user au grant authority over role r1
and empower authority over role r2. Similarly, in Step 4 revoke_formula
equivalent to revokeRoleFromRole(r1, r2) is constructed using revoke
permissions given to admin user. It checks whether an admin user au has an
authority to revoke roles r1 and r2.
### IV-F An Example Instance for ARRA with role attributes
Previous ARRA model simulation examples did not include regular role
attribute. In this section, we present a simple yet plausible example that
demonstrates a use case for role attributes in assigning role to role.
There are two admin users, Sam and Tom, and regular roles IT Director,
Development Manager, Quality Manager, Marketing Mganager, Finance Manager,
Support Engineer and System Analyst. There can be many other roles in an
organization. We have considered few roles, enough to illustrate our case.
Admin user attribute dept maps each admin user to set of departments they have
admin authority over. Although, it is likely that departments have hierarchy
in practice, we have not included hierarchy for departments for simplicity.
There are three departments in the organization, namely Operations, Account
and IT departments. Sam has admin authority over all the departments while Tom
has authority over IT department only. There is a regular role attribute dept
for mapping regular roles to their departments.
Authorization condition is that if an admin user au with admin role ar was
given authority over a department d and if roles r1 and r2 belonged to the
same department d then r1 can be assigned to r2.
Sets
– AU = {Sam, Tom}, AOP = {assign, revoke}
– ROLES = {IT Director, Devlopment Mgr., Quality
Mgr., Marketing Mgr., Finance Mgr., Support Engineer, System Analyst}
– AR = {}
– AUA = {}
– RH = {}, ARH = {}
Attributes definition
– AATT = {dept},
– Scope(dept) = {Operations, Account, IT},
attType(dept) = set, is_ordered(dept) = False,
Hdept = $\phi$
– dept(Sam) = {Operations, Account, IT},
dept(Tom) = {IT}
– ARATT = {}, RATT = {dept, level}
– Scope(dept) = {Operations, Account, IT},
attType(dept) = atomic,
is_ordered(dept) = False, Hdept = $\phi$
– dept(IT Director) = IT, dept(Dev. Manager) = IT,
dept(Marketing Mgr.) = Operations,
dept(QA Mgr.) = IT,
Authorization functions
– is_authorizedRassign(au : AU, r1 : ROLES, r2 : ROLES) $\equiv$ $\exists$d
$\in$ Scope(dept). dept(au) = dept(r1) = dept(r2)
– is_authorizedRrevoke(au : AU, r1 : ROLES, r2 : ROLES) $\equiv$
is_authorizedRassign(au : AU, r1 : ROLES, r2 : ROLES)
## V Conclusion
In this paper, we presented ARRA, a model for attribute based role-role
assignment. A design motivation behind ARRA model was to make it enough to
express prior RRA models. In particular, we took RRA97 and UARBAC’s RRA model
as foundation. For these models, we have presented MapRRA97 and MapRRA-UARBAC
algorithms, which map any instances of these models to ARRA model,
respectively.
ARRA model can not only express existing RRA models but also has a capability
to do more. For instance, prior models included in our research did not fit
the notion of regular role attributes. However, we acknowledge that it is an
essential attribute in making role assignment decisions to say the least.
Motivated by this notion, we presented a simple example scenario where we made
use of regular role attributes for role-role assignment.
## Acknowledgment
This work is partially supported by NSF grants CNS-1423481 and CNS-1553696.
## References
* [1] D. F. Ferraiolo, R. Sandhu, S. Gavrila, D. R. Kuhn, and R. Chandramouli, “Proposed NIST standard for role-based access control,” _ACM Transactions Inf. Syst. Secur._ , vol. 4, pp. 224–274, Aug. 2001.
* [2] R. S. Sandhu, E. J. Coyne, H. L. Feinstein, and C. E. Youman, “Role-based access control models,” _Computer_ , vol. 29, pp. 38–47, Feb 1996.
* [3] L. Fuchs, G. Pernul, and R. Sandhu, “Roles in information security - A survey and classification of the research area,” _Computers & Security_, vol. 30, pp. 748 – 769, 2011.
* [4] A. C. OConnor and R. J. Loomis, “2010 economic analysis of role-based access control,” _NIST, Gaithersburg, MD_ , 2010.
* [5] R. Sandhu, V. Bhamidipati, and Q. Munawer, “The ARBAC97 model for role-based administration of roles,” _ACM Transactions on Information and System Security (TISSEC)_ , vol. 2, pp. 105–135, 1999.
* [6] R. Sandhu and Q. Munawer, “The ARBAC99 model for administration of roles,” in _Proc. of the 15th Annual Computer Security Applications Conference (ACSAC’99)_. IEEE, 1999, pp. 229–238.
* [7] S. Oh and R. Sandhu, “A model for role administration using organization structure,” in _Proc. of the seventh ACM symposium on Access control models and technologies_. ACM, 2002, pp. 155–162.
* [8] N. Li and Z. Mao, “Administration in role-based access control,” in _Proc. of the 2nd ACM symposium on Information, computer and communications security_ , 2007, pp. 127–138.
* [9] P. Biswas, R. Sandhu, and R. Krishnan, “Uni-ARBAC: A unified administrative model for role-based access control,” in _International Conference on Information Security_. Springer, 2016, pp. 218–230.
* [10] D. R. Kuhn, E. J. Coyne, and T. R. Weil, “Adding attributes to role-based access control,” _Computer_ , vol. 43, pp. 79–81, 2010.
* [11] V. C. Hu, D. Ferraiolo, R. Kuhn, A. Schnitzer, K. Sandlin, R. Miller, K. Scarfone _et al._ , “Guide to attribute based access control (ABAC) definition and considerations,” _NIST special publication_ , vol. 800, no. 162, 2014.
* [12] X. Jin, R. Krishnan, and R. Sandhu, “A unified attribute-based access control model covering DAC, MAC and RBAC,” in _IFIP Annual Conference on Data and Applications Security and Privacy_. Springer, 2012, pp. 41–55.
* [13] S. Bhatt, F. Patwa, and R. Sandhu, “ABAC with group attributes and attribute hierarchies utilizing the policy machine,” in _Proc. of the 2nd ACM Workshop on Attribute-Based Access Control_. ACM, 2017, pp. 17–28.
* [14] P. Biswas, R. Sandhu, and R. Krishnan, “Attribute transformation for attribute-based access control,” in _Proc. of the 2nd ACM Workshop on Attribute-Based Access Control_. ACM, 2017, pp. 1–8.
* [15] J. Ninglekhu and R. Krishnan, “Attribute based administration of role based access control: A detail description,” _arXiv preprint arXiv:1706.03171_ , 2017.
* [16] J. Crampton, “Administrative scope and role hierarchy operations,” in _Proceedings of the seventh ACM symposium on Access control models and technologies_. ACM, 2002, pp. 145–154.
* [17] X. Jin, R. Sandhu, and R. Krishnan, “RABAC: role-centric attribute-based access control,” in _International Conference on Mathematical Methods, Models, and Architectures for Computer Network Security_. Springer, 2012, pp. 84–96.
* [18] Q. M. Rajpoot, C. D. Jensen, and R. Krishnan, “Attributes enhanced role-based access control model,” in _International Conference on Trust and Privacy in Digital Business_. Springer, 2015, pp. 3–17.
* [19] E. Yuan and J. Tong, “Attributed based access control (ABAC) for web services,” in _Proc. of the IEEE International Conference on Web Services (ICWS’05)_. IEEE, 2005.
* [20] D. Servos and S. L. Osborn, “HGABAC: Towards a formal model of hierarchical attribute-based access control,” in _International Symposium on Foundations and Practice of Security (FPS 2014)_. Springer, 2014, pp. 187–204.
|
∎
11institutetext: Devansh Mehta 22institutetext: Voicedeck Technologies
22email<EMAIL_ADDRESS>33institutetext: Kalika Bali 44institutetext:
Microsoft Research India
44email<EMAIL_ADDRESS>
# Learnings from Technological Interventions in a Low Resource Language
Enhancing Information Access in Gondi
Devansh Mehta∗∗ ** Equal Contribution Harshita Diddee∗11footnotemark: 1 Ananya
Saxena Anurag Shukla Sebastin Santy Ramaravind Kommiya Mothilal Brij Mohan
Lal Srivastava Alok Sharma Vishnu Prasad Venkanna U Kalika Bali
(Received: date / Accepted: date)
###### Abstract
The primary obstacle to developing technologies for low-resource languages is
the lack of representative, usable data. In this paper, we report the
deployment of technology-driven data collection methods for creating a corpus
of more than 60,000 translations from Hindi to Gondi, a low-resource
vulnerable language spoken by around 2.3 million tribal people in south and
central India. During this process, we help expand information access in Gondi
across 2 different dimensions (a) The creation of linguistic resources that
can be used by the community, such as a dictionary, children’s stories, Gondi
translations from multiple sources and an Interactive Voice Response (IVR)
based mass awareness platform; (b) Enabling its use in the digital domain by
developing a Hindi-Gondi machine translation model, which is compressed by
nearly 4 times to enable it’s edge deployment on low-resource edge devices and
in areas of little to no internet connectivity. We also present preliminary
evaluations of utilizing the developed machine translation model to provide
assistance to volunteers who are involved in collecting more data for the
target language. Through these interventions, we not only created a refined
and evaluated corpus of 26,240 Hindi-Gondi translations that was used for
building the translation model but also engaged nearly 850 community members
who can help take Gondi onto the internet.
###### Keywords:
Low-Resource Languages, Deployment, Applications
††journal: Language Resources and Evaluation
## 1 Introduction
In the present era of globalization and integration of technology into almost
every aspect of life, native speakers are turning to dominant languages at a
faster rate than ever before. As seen in Figure1, around 40% of all languages
in the world face the danger of extinction in the near future 31 , while 95%
have lost the capacity to ascend to the digital realm kornai2013digital . When
a language spoken in a particular community dies out, we lose a vital part of
the culture that is necessary to completely understand it. At a bare minimum,
languages need to be integrated with the Internet to give them a fighting
chance at survival.
In this work, we design, deploy and critically assess technological
interventions that aim to create a digital ecosystem of Gondi content, a
South-Central Dravidian tribal language spoken by the Gond tribe in Central
India. The case of Gondi is representative of many languages across the world
and presents a unique case study of how a language can be in danger despite
having all the ingredients of sustainability such as (1) long historical
continuity (2) a population of 3 million people speaking it and (3) widely
spoken in around 6 states of India with various dialects and forms. The
complexities arise as Gondi is a predominantly spoken language with no single
standard variety but a number of dialects, some mutually unintelligible
beine1994sociolinguistic . Our objectives with these interventions is creating
a repository of linguistic resources in Gondi that can be used for: (1)
building language technologies like machine translation or speech to text
systems that are essential for taking Gondi onto the internet; and (2)
expanding the information available to the Gond community in their language.
There needs to be groundwork and identification of the real problems that can
be solved by deployment of language technologies in minority communities
dearden2015ethical , as there are legitimate technological and ethical
concerns surrounding the use of technology in low-resource languages
joshi2019unsung ; 10.1145/3530190.3534792 . Technical systems aren’t simply
ethically neutral exchanges of information and those that work in isolation
from social contexts can be detrimental to the ecosystem of minority language
communities. At the same time, inaction on the part of language technologists
is an ethically dubious proposition in its own right, as minority communities
are currently forced to learn mainstream languages for availing the benefits
of the Internet and to access power structures enabling socio-economic
mobility. We believe it to be fairly non-controversial to maintain that anyone
should be able to surf the internet in the language that they are comfortable
in. Ethical issues only arise on the methods one uses to reach this goal.
Figure 1: UNESCO 2017 World Atlas: Demonstrating the endangered state of the
languages of the world.
Our focus when working with the Gond community is centered around devising
novel approaches for creation, evaluation and dissemination of Gondi content,
unlike well-resourced languages where the focus is more on engineering
language technologies using abundantly available data. As far as possible, we
have ensured that the technology interventions described in this study were
led by the Gond community and non-profit organizations working with them.
The first intervention is creating a 3,500 word Gondi dictionary that is
accessible to the community as an Android app. The second is 230 children’s
books that were translated by the Gond community in a 10 day workshop. The
third is crowdsourcing translations from community volunteers via an Android
app, the WhatsApp Business API and an assistive translation tool. The fourth
intervention is a phone number that community members could call to gain
awareness about a local election in their area, upon completion of which they
earn mobile credits. These interventions respectively resulted in facilitating
80-100 community members to create a 3,500 word dictionary; 20 community
members that translated 230 children’s stories from Hindi to Gondi; 219
community members that translated nearly 55,000 sentences; and 557 native
speakers of Gondi that learned about elections in their own language and who
can be called for future workshops.
This data was used to develop a Hindi-Gondi Machine Translation
Model111https://github.com/microsoft/INMT-lite, compressed for streamlined
operation on low-resource edge devices for use by the community. The model was
evaluated by community members that engaged with us through these
interventions and got an average Direct Assessment score specia-
etal-2020-findings-wmt of 63, indicating the semantic accuracy of our model’s
output. We also run preliminary experiments for evaluating the model’s ability
in assisting community members with translations. Our results show promising
efficacy with more than 50% of the suggestions being accepted.
## 2 Context
According to the 2011 census chandramouli2011census , the total population of
the Gond tribe is approximately 11.3 million. However, the total Gondi
speaking population is only around 2.7 million. That is, only about 25 percent
of the entire tribe now speaks it as a first language. UNESCO’s Atlas of the
Worlds Languages in Danger moseley2010atlas lists Gondi as belonging to the
vulnerable category. There is an added difficulty of creating resources for
Gondi due to the linguistic heterogeneity within the Gond community. As seen
in Figure2, Gondi is spread over 6 states in India. It is heavily influenced
by the dominant language of each state to the point where a Gond Adivasi from
Telangana (a Telugu speaking Southern state), finds it difficult to understand
a Gond Adivasi from Madhya Pradesh (a Central state with Hindi as the dominant
language). Recent scholarship has found the influence of these dominant
languages to be a major factor in language loss in the Gond community
boruah2020language .
Figure 2: Gondi speaking areas in India
As a predominantly oral language, the proportion of Gondi speakers is expected
to go down further as opportunities are shrinking to hear the language spoken
outside of their everyday surroundings. All India Radio, the only radio
station in India allowed to broadcast news, does not have regular Gondi news
bulletins. There is no TV station or channel catering to Gondi speakers. There
is also a severe dearth of online content in Gondi, resulting in members of
the tribe having to learn a mainstream language to enjoy the benefits of
internet connectivity. “The Wire”222Gondi Bulletin:
https://youtu.be/M3q2ycJ_U7g is one of the few Indian news outlet that
publishes a news bulletin in Gondi. However, the Gondi spoken in their
broadcast caters to the Gond Adivasis from the Central states of India and it
is difficult for Gonds from the Southern states to understand the content.
Gondi is also not included in the 8th Schedule of the Indian Constitution,
with the result that education and exams for government jobs cannot be
administered in the language. The deleterious effects of the marginalization
of Gondi on their community are manifold. Gondi is considered the lingua
franca of the local insurgents, who use their knowledge of the language and
the perceived neglect by the government to recruit candidates from the tribe
to join their civil war against the state kumar2019gondi . Further, there are
high dropout rates among monolingual children that speak Gondi as a first
language. A UNESCO study found that children whose mother tongue is not the
medium of instruction at primary school are more likely to fail in early
grades or drop out buhmann2008mother , which in turn increases the chances of
them joining the insurgency. Working with the Gond tribes on reviving their
language is thus important not just for cultural reasons, but may also serve
as an instrument for bringing peace to their society. These factors can be
mitigated provided there is political will and a standardized version of the
language that is agreed upon. While the younger, bilingual generation in the
Gond tribe attach less importance to their mother tongue, it was not difficult
to find community members worried about the extinction of their language and
it proved easy to recruit participants for the workshops we held, even without
any financial incentive.
## 3 Related Work
The first section focuses on how we address the constraints of efficiently
collecting data through traditional channels for low-resource languages. The
second section situates our study within existing language preservation and
revitalization programs and lays out our unique contributions to this prior
literature.
##### Data Collection Channels
Traditional channels of data collection can vary across 3 dimensions (a)
Channel of Data Collection (b) Quality of Collected Data and (c) Quantity of
Collected Data. There are two common channels of data collections: automated
mining and human reliant crowdsourcing. Mining includes a wide spectrum of
methods varying from crawling data off the web to utilizing content after
digitization by systems such as Optical Character Recognition. Unfortunately,
these channels assume the existence of data resources in a specific form: for
example, Automatic or Semi Automatic methods like Web mining expects the data
to be digitally readable, while OCR expects data to be digitally consumable by
an existing service, which might not be the case for many languages,
especially low-resource ones rijhwani2020ocr .
In our study, we rely on crowdsourcing large quantities of data through human
reliant channels. This approach is fraught with logistical challenges around
participant recruitment nangia2021ingredients , the cost of response
validation, and mitigating subjectivity in responses goldman-
etal-2018-strategies , as well as ethical considerations pertaining to
devising fair pay kummerfeld-2021-quantifying shmueli2021beyond and
maintaining response diversity by controlling for human bias in the inputs
provided by a crowdsourced audience liu-etal-2022-toward . Our study adds
novel insights to this prior literature by building a dataset from community
volunteers and staff at partner organizations without the provision of any
fiscal incentive, which is often a given attribute of human reliant data
crowdsourcing channels. We find that that a volunteer led workforce for data
generation can yield higher quality datasets by self-selecting for those
motivated to preserve their language.
Prior work has identified quality constraints for corpora arising out of
human-collection channels 10.1162/tacl_a_00447 caswell-etal-2020-language .
Specifically, noise derivative of code-mixing, non-linguistic characters,
short/meta-data segments, untranslated/copied content has been shown to
considerably harm the output of neural machine translation (NMT) systems
khayrallah2018impact . For high resource languages, the sheer quantity of a
corpus is usually significant enough to allow the selection of a subset of
high-quality sentences from the larger set using intelligent validation
techniques, especially since some percentage of noisy data is shown to improve
generalization robustness of NMT models. Training and evaluation corpora in
low-resource languages may not be as effective due to the paucity of data
10.1162/tacl_a_00447 . We observe this explicitly with our interventions,
since a large proportion of our data (nearly 60K data samples that we
collected) were sifted to a set of 27K samples, greatly reducing the effective
yield of our collection channels. Our work thus presents a case study on
developing alternative methods of data collection and evaluation for low-
resource languages.
##### Language Technologies for Revitalization and Preservation
Preservation programs generally attend to the need to develop data corpora in
that language abraham2020crowdsourcing , article , mus2021toward . However,
the notion that language documentation or artifact creation can independently
bring about revitalization is largely dismissed by the community zariquiey-
etal-2022-cld2 . Community-centric technical interventions that can infuse the
language technology into the community for sustained use are increasingly
coming under focus. One dimension of this focus has been towards developing
fundamental NLP tools for low-resource languages such as Part-Of-Speech
taggers finn-etal-2022-developing , parsers bowers-etal-2017-morphological ,
etc. Another dimension is the development of more end-user oriented language
technologies such as edge-deployments bettinson-bird-2017-developing
hermes2013ojibwe bird2014aikuma and interactive interfaces that can be
consumed rapidly by the community little-2017-connecting adams-etal-2021-user
. A critical point to note here is that access bottlenecks can limit the
consumption of outcomes from revitalization and preservation endeavours. For
instance, Roche highlights the political nature of language endangerment,
pointing out that low-resource language communities are usually under-
resourced in other aspects of life roche2020abandoning . Leibling et al
10.1145/3313831.3376261 discuss these constraints across the adaptation of
translation systems to edge devices across three globally distributed and
divergent demographic groups. We contribute to this identified gap by creating
a software artifact that displays NMT translations on low cost devices like
smartphones, which can be used by other speech communities and researchers
working with marginalized populations.
Due to the well-studied constraints of data collection through traditional
channels elazar-etal-2020-extraordinary millour-fort-2020-text liu-
etal-2022-toward , our work explores newer mechanisms for collecting low-
resource language data that are of wider relevance. For example, our Adivasi
Radio translation app allowed users to bulk download a corpus of sentences for
translation, which they could then complete even from areas without Internet.
Another area our work sheds new light on is reducing the cognitive burden
faced by communities in understanding an entirely novel interface for data
provision, as discussed in prior work like 10.1145/2858036.2858448
10.1145/2556288.2557155 . Our study uses low friction approaches for
collecting translations such as through the WhatsApp Business API, which is
well integrated into many communities. Lastly, our work briefly touches upon
the attempt to integrate assistive translation interfaces to provide sub-
optimal help in data collection efforts 10.1145/3290605.3300461
10.1145/3530190.3534792 .
Overall, this paper is a case study in collaborating with target communities
for creating linguistic resources to enable the development of language
technologies that can then give rise to a digital ecosystem for such
languages. In this regard, we follow the lead set by Kornai and Bhattacharyya
in building online tools for low-resource languages like spellchecks that can
be incrementally refined over time kornai2014indian . Our work distinguishes
itself by empirically studying methods of community engagement that are
cognizant of the constraints faced in low-resource environments.
## 4 Technological Interventions
The larger framework of our interventions is that language resource creation
feeds into building language technologies and enhancing access to information
in that language. We rolled out a series of interventions that created
linguistic resources useful to the community, such as children’s stories,
which in turn fed into building language technologies such as a Hindi-Gondi
translation system. These initiatives helped increase information access in
Gondi while supporting the end goal of taking Gondi onto the Internet.
### 4.1 Gondi Dictionary Development
This section discusses the need and complexity of building a common dictionary
across the various Gondi dialects.
#### 4.1.1 Motivation
Several researchers have found that mutual intelligibility of Gondi decreases
with distance, in part due to the influence of dominant state languages
creeping into the various dialects beine1994sociolinguistic ;
shapiro1981language ; tyler1969koya . Beine (1994) conducted mutual
intelligibility tests across the Gondi speaking areas and found there to be 7
mutually unintelligible dialects of the Gondi language (pp 89). He thus
recommended the creation of dialect centres to cover each Gondi speaking
region, with literacy materials separately developed for each center. These
workshops helped determine whether the community wanted separate efforts for
each dialect or a common effort towards one language.
#### 4.1.2 Intervention
Prior to the involvement of our research team, a citizen journalism platform
called CGNet Swara held 7 workshops beginning from 2014 to develop a Gondi
thesaurus containing all the different words used by native speakers from 6
states. Some words, such as water, had as many as 8 different words for it. At
the 8th workshop in 2018, which saw more than 80 people in attendance, the
thesaurus was developed into a dictionary containing 3,500 words.
The dictionary was made into an Android app, Gondi Manak
Shabdkosh333aka.ms/gondi-dictionary 444aka.ms/indian-express-gondi-dictionary,
depicted in Figure3. It allows users to enter a Hindi or Gondi word and hear
or read its equivalent translation, similar to the Ma! Iwaidja dictionary
app555ma-iwaidja-dictionary.soft112.com/ without the wheel based interface for
conjugation and sentence formation.
#### 4.1.3 Takeaways
CGNet Swara reported an overwhelming consensus by the community to remain as
one language. Gondi is primarily used within its local speech community and
participants recognized the need to develop a common vocabulary and spelling
for its emergence as a standard language. The three techniques commonly used
in standardizing languages are comparative (linguistic reconstruction to build
a mother tongue for all), archaizing (deriving a variety from older, written
texts) and the statistical (combining the different dialects having the widest
usage) haugen1966dialect , with most participants favoring the latter
approach.
CGNet’s initial aim with the dictionary app was studying whether it can allow
some basic communication and learning to take place in primary schools where a
teacher does not know Gondi while their student is a monolingual speaker.
However, our approach of finding a central dialect mediating between the
extremes resulted in a variety of Gondi promoted as everyone’s language that
became nobody’s language. For example, monolingual Gonds may only be aware of
their local word for water, but if it hasn’t been selected in the dictionary
then its use in such scenarios would be limited. In retrospect, a thesaurus
might have been more applicable for this particular use case. The dictionary
provided greater utility in pretraining our translation model. Following Wang
et al wang2022expand we augmented digitally consumable monolingual data to
improve performance of the model, as demonstrated in Table 2.
Figure 3: Gondi Manak Shabdkosh, a dictionary of 3,500 words
### 4.2 Creating children’s books in Gondi through Pratham Books
We narrate our efforts at translating children’s stories into Gondi for use in
schools and as a data source for language technologies.
#### 4.2.1 Motivation
Imparting basic, primary education in a child’s mother tongue inculcates pride
in their language and cultural identity formation among younger generations.
It also targets the most important stakeholder for ensuring continued language
use. More practically, having Gondi as a medium of instruction or a subject at
schools can help increase enrollment among monolingual Gond tribes within the
education system and decrease drop-out rates among indigenous children not
conversant in Hindi. This intervention focused on developing the necessary
linguistic resources in Gondi that could serve as educational material.
#### 4.2.2 Intervention
In 2018, our research team conducted a 10 day translation workshop with CGNet
Swara and Pratham Books, a nonprofit publisher with the motto ”Book in every
child’s hands”. 20 Gond tribals from three states of India came together and
translated 230 children’s books from Hindi to Gondi, some of which introduced
Gondi children to climate change for the first time 666Article:
aka.ms/news18-climatechange. These stories were published on the Storyweaver
platform 777storyweaver.org.in/, an initiative of Pratham Books that hosts
more than 15,000 children’s stories in various languages and dialects. To the
best of our knowledge, this is the first online repository of children’s
stories in Gondi 999Article: aka.ms/hindu-gondi-nextgen888aka.ms/storyweaver-
gondi. They were also printed out and distributed in primary schools in the
Gondi speaking districts of Chhattisgarh, with efforts now ongoing to convince
the state government to include them as part of the school curriculum across
the tribal belts of the state.
#### 4.2.3 Takeaways
Although no payment was provided to workshop participants (besides covering
food, travel and lodging), a total of around 20 volunteers translated 8000
sentences over 10 days. Having a volunteer only workforce helped in self-
selecting individuals that cared about the languages survival among future
generations. We found it most effective to create teams of 4-5 people for
translating a story, with more educated participants taking on the role of
writing the translation on paper or typing the written translation into the
computer interface. Monolingual Gond speakers would help find the right
vocabulary for translations, while bilingual speakers would help them
understand the Hindi text that needed translating. Having groups complete
translations (rather than individuals) resulted in considerable debate around
which Gondi words to use for translating a Hindi sentence. This intervention
yielded the most high quality data compared to the other efforts, as is
evident from our manual evaluation of the data referenced in Table 1.
### 4.3 Crowdsourcing Gondi translations through Adivasi Radio
Unlike the Pratham Books workshop featuring in-person group translations,
Adivasi Radio was geared towards the post pandemic world and allowed
volunteers to provide translations from the comfort of their homes.
#### 4.3.1 Motivation
At the children’s book workshop, we found that many participants wanted to
continue the translation work from home but there was no avenue for them to do
so. Taking inspiration from Aikumabird2014aikuma , we developed Adivasi Radio,
an Android application that presents users with Hindi words or sentences for
which they need to provide the Gondi translation. In addition to the
translation role, Adivasi Radio was designed as the go-to place for native
speakers to find outlets and sites publishing Gondi content.
Figure 4: Adivasi Radio app to collect translations and access Gondi content
#### 4.3.2 Intervention
The app was first deployed between March and June 2020, during the initial
months of the coronavirus induced lockdown. The interface of the app can be
seen in Figure4 while translations over time are shown in Figure5. In 4
months, 17,756 sentences were translated through the app from 164 unique
users.
6 superusers, most of whom were staff at our partner organization CGNet Swara,
accounted for the majority of the translations. Many volunteers gave up after
completing a few translations, indicating that sustaining data collection
efforts is a challenge, especially without fiscal incentives.
We uploaded the children stories produced with Pratham Books, Gondi stories on
CGnet Swara’s citizen journalism platform and The Wire’s Gondi news bulletin
into the app. A Devanagari text-to-speech system read out the written Gondi
text so that monolingual speakers could enjoy the content.
Figure 5: Number of translations collected per day.
#### 4.3.3 Takeaways
After Adivasi Radio’s deployment, we held meetings with the translators to
learn about their experience of using the application. We found that several
users resided in areas having weak internet connectivity, thus prompting us to
update the application so that they could bulk download a large corpus of
sentences that they could translate even when they were offline. Similar to
the feature used in d2014mobile ; mehta2020facilitating for enabling citizen
journalism in areas without internet, translated sentences would get sent to
our server whenever the user managed to connect to the internet. The attention
to providing accurate translations was reflected in their request for both a
’skip’ feature to avoid providing translations they were unsure of and an
option to edit their earlier submissions, as they would sometimes realize
mistakes only after completing a translation. However, we found that the skip
feature was used to avoid translating long sentences or short paras of three
or more sentences. This concern could be alleviated by having them translate
sentences forming a cohesive narrative like a Wikipedia article rather than
disjointed sentences unrelated to one another. Finally, we did not see
significant engagement with the uploaded Gondi content available on the app as
most users were bilingual and accustomed to a richer online environment in
Hindi.
Our evaluation of the quality of submitted translations reiterated the
importance of incorporating rigorous validation checks on data collected
through human-channels. Out of the nearly 18000 sentences collected through
this channel, we were only able to utilize 10105 sentences after weeding out
transliteration, skipped sentences and gibberish inputs with a high density of
special and/or numeric characters.
### 4.4 Crowdsourcing Translations through WhatsApp Business API
A growing body of research advocates integration with existing platforms for
coordinating social mobilization instead of creating custom made ones such as
an app or website lambton2020unplatformed ; saldivar2019online ;
starbird2012crowdwork . The majority of Adivasi Radio users made extensive use
of WhatsApp in their daily life, prompting us to explore how we could use this
medium to solicit translations from the community.
Figure 6: Payments were discontinued for translations received via WhatsApp
after finding users were gaming the system
#### 4.4.1 Motivation
During field testing with Adivasi Radio users, we found that some users did
not have space on their phones to install our app. In such cases, we had to
delete media files and unused apps before we could install the Adivasi Radio
application on their phone. Moreover, training had to be provided on how to
navigate the interface of our app. These factors limited our ability to reach
more members in the community without our physical presence.
We thus applied for - and received - permission to use the WhatsApp Business
API, which would enable users to provide translations over WhatsApp directly
to our server without any human intervention. We accessed the API through the
IMI Connect platform, which has a monthly charge of about USD 350 per month in
addition to variable costs per message.
#### 4.4.2 Intervention
To seed usage of the system, we needed users that had a smartphone with
regular internet connectivity and were conversant in both Hindi and Gondi.
From May 2021 to September 2021, we collected 37,173 sentences from 55
volunteers and staff members at CGNet Swara. 6 superusers accounted for nearly
half of the submitted translations. Many translations were written in the
latin script (as opposed to Devanagari), resulting in a little less than
12,000 sentences that could be fed into the translation model.
#### 4.4.3 Takeaways
To address declining interest in completing translations from home via Adivasi
Radio, we released a probe in February 2021 that for the first time paid users
for completing translations via WhatsApp. We found that payment per sentence
translated led to deteriorating quality by rewarding speed over quality.
Indeed, we saw some of the most devoted community members spend more than 10
minutes on a single sentence, even asking their neighbors for the right word,
so that they could get it right. When these members saw their peers provide
inaccurate translations and earn more than them, they withdrew their support.
We ended up discarding nearly 12,000 sentences collected via this process and
switched back to a volunteer driven model. We also observed frustration when
the sentence that translators were given timed out and they were provided a
new sentence upon logging back in, as they had researched the earlier sentence
but there was no option for them to provide it anymore.
We relaunched the WhatsApp chatbot in June 2021 without the option of earning
any money for translations. The majority of the translations came from staff
members at CGNet Swara. In the absence of strict oversight and punishments for
incorrect translations, we recommend paying a fixed salary to translators
rather than one varying by number of sentences translated or other such
quantitative metrics. It also proved effective to integrate data collection
into the day to day responsibilities of employees at partner organizations.
### 4.5 Data Quality
We requested community members to manually evaluate the data collected and
then compare the quality of data collected via each channel.
#### 4.5.1 Annotation Setup
We recruited 3 Gondi speaking annotators from the Indian states of
Maharashtra, Madhya Pradesh and Chhattisgarh to assist us with the manual
evaluations of our data and the translation model developed from it. This
selection was made bearing in mind the dialectal coverage of our dataset, as
most translations were provided by volunteers from these three states. The
annotation instructions were made visible to all annotators via the interface
we provide to them as shown in Figure7. We also discussed a set of samples and
the mechanism to acquaint them with the scoring system before the evaluation
exercise. The participating annotators performed 3 tasks where they scored a
provided sample with the Direct Assessment score described in specia-
etal-2020-findings-wmt . They were compensated using the following
system101010 All compensations are specified in Indian National Rupee \- per
ranking task, each annotator was given Rs. 5 (Rs. 3 base pay, and Rs. 2 upon
completion). Per assistive translation task, they were provided Rs. 5 and per
translation task, they were provided Rs.10 ( Rs. 7 base pay, Rs. 3 upon
completion ).
Figure 7: The scoring instructions are specified in Hindi on the annotation
screen for quick reference. A score slider is provided for limiting scores to
a valid range.
To analyze the quality of the data collected through Pratham Books, Adivasi
Radio and WhatsApp API, we requested each annotator to score 60 sentences
sampled across any two data sources i.e. 2 annotators score 30 instances per
source. These samples were taken from our training corpus, and we apply a
sequence length filter in the range of [4, 15] tokens to avoid undue weightage
to sentences coming from any one source.
#### 4.5.2 Results
We report average direct assessment scores for each data source in Table 1.
Channel | Average DA Score | Standard Deviation
---|---|---
Pratham Books | 65.4 | 16.1
Adivasi Radio | 58.6 | 19.1
WhatsApp | 36.5 | 21.9
Model(Hindi-Gondi) | 63.2 | 23.2
Table 1: Average Direct Assessment Scores with Standard Deviations for data
collected via all channels and the Hindi-Gondi model trained on the data
The average DA scores on the sample set indicate that the data collected from
Pratham Books is of the best quality, followed by Adivasi Radio and WhatsApp
respectively. This could be due to the setup in which participants were
placed: synchronous and supervised for Pratham Books and asynchronous and
unsupervised for WhatsApp or Adivasi Radio.
As is observed with the standard deviation, the inter-annotator score
agreement was the poorest for WhatsApp. The annotators report that WhatsApp
data inputs had a very high degree of code-switching, sometimes not even with
Hindi but other indic languages. For example, all annotators reported
observing Marathi tokens as a part of the provided translations.
### 4.6 Machine Translation
Using the data collected, we train a machine-translation model between Hindi-
Gondi and evaluate the efficacy of its output.
#### 4.6.1 Motivation
The development of the machine translation model aimed at developing a core-
technology that serves the larger goal of improving information access for the
community by enabling automated translation of web content from Hindi to
Gondi. It also let us evaluate the utility of the data when it is plugged into
a real-world NLP usecase. We envisage the usage of this model in other low-
resource languages like that of Bribri, Wixarica and Mundari
https://doi.org/10.48550/arxiv.2210.15184 to demonstrate the applicability of
our artifact across other data-deficient setups.
#### 4.6.2 Intervention
Using the data we had available from all channels, we finetuned MT5 xue2021mt5
for training our translation model in both the Hindi-Gondi and Gondi-Hindi
direction. The model used in this work was adapted from
https://doi.org/10.48550/arxiv.2210.15184 and we replicate the training setup
specified there. The models were evaluated using sacrebleu (v2.2.0) with the
spm tokenizer. As explored by wang2022expand , we used the Hindi-Gondi
dictionary to generate 200K noisy monolingual data to train our models, which
we further used for continual pretraining. We also generated forward
translated data using the same model to fuse with the lexicon-adapted data.
Additionally, we utilize the quantized model developed by
https://doi.org/10.48550/arxiv.2210.15184 to evaluate the performance of the
same model’s compressed version (400 MB in disk size compared to 2.28 GB), for
the purpose of commenting on the edge deployment feasibility in low-resource
environments.
The annotation setup is same as the one described in section 4.5.1. All
annotators scored 40 inferences generated from the model to analyze its
quality.
Model | BLEU | Size
---|---|---
mt5-small (Hindi-Gondi) | 14.3 | 1.2GB
mt5-small (Gondi-Hindi) | 30.9 | 1.2GB
mt5-base (Hindi-Gondi) | 15.6 | 2.2GB
mt5-small with continued pretraining | 14.9 | 1.2GB
mt5-small with fused data (With 50K forward translated) | 12.6 | 1.2GB
Quantized MT5 | 13.8 | 400 MB
Table 2: Performance of the Hindi-Gondi translation model with model
adaptations like lexicon-adaptation, continual pretraining and quantization.
#### 4.6.3 Takeaways
As shown in Table 1, the annotators report an average DA Score of 63 for the
Hindi-Gondi model which connotes that the model’s output was semantically
appropriate but contained an identifiable degree of grammatical errors and
typos. The annotators mentioned that the model’s output was visibly poor in
terms of producing appropriate punctuation. While the structural integrity of
the sentence seems to have been preserved, spelling differences and code-
switched tokens ( i.e. Hindi tokens on the target side ) were common.
Annotators also observed the occurrence of ambiguous-language tokens that they
were unable to identify as being correct in the given context, which could be
attributed to the dialectal differences that existed within the language.
The BLEU scores of the models are reported in Table 2. The significant drop in
the BLEU score in the Hindi-Gondi direction, in comparison to Gondi-Hindi,
appears to be most significantly associated with the inclusion of Hindi in the
pretraining corpus of MT5. The mC4 training corpus contains 1.21% Hindi data,
which greatly appears to aid the target side reconstruction in the Gondi-Hindi
model xue2020mt5 . Additionally, we posit that the mt5 tokenizer, which has
never seen Gondi, expectedly falters in the reconstruction of Gondi on the
target side ( as is seen in the Hindi-Gondi ) model, so we expect a high
degree of tokenization ambiguities in the Gondi output which would adversely
impact the BLEU evaluation. This is further substantiated by observation of
annotators who claimed that in some cases constructions of a Gondi token
emulated the Hindi construction of the token. We intend to do a deeper
investigation of such spurious correlations that massively-multilingual models
might develop when a dominant attribute of language, its script in this case,
matches other well-represented languages in the pretraining corpus despite
being linguistically quite divergent.
One of the primary learnings from this task was the need to integrate
linguistically-motivated validation structures in a data collection channel to
optimize the yield of the data collection process. In our specific case, the
lack of any validation structures around the dialects against which we
collected our data for led to the collection of a dataset with samples from at
least 3 dialects. This especially harmed the impact of the machine translation
system as it’s responses catered to neither dialect completely.
### 4.7 Assistive Translation
We built an interface showing suggestions from our translation model to see if
it improves the quality or speed of translation efforts.
#### 4.7.1 Motivation
We use the same Hindi-Gondi model to generate candidate words that could be
used when translating Hindi sentences to Gondi. The motivation for this task
was to understand if poor-accuracy models could be used to accelerate the data
collection process by providing assistive recommendations to the participating
translators. This task holds unique practical value because it allows us to
comment if sub-optimal machine translations (which would be an obvious
intermediate artifact in such community interventions) can help enhance the
yield of data-generation pipelines in low-resource communities.
Figure 8: Gisting Interface for providing recommendations to data-providers.
Our motivation to develop this system did not come directly from the Gond
community but through partners at an educational institution and the citizen
journalism platform CGNet Swara. They believed it could enable speedy
translation of books for primary education and media stories into Gondi.
#### 4.7.2 Intervention
We realized that developing such a language technology had to take local
constraints into account, which in this specific case entailed the lack of a
stable internet connection. We thus developed an offline variant of the model,
which was compressed enough to operate on relatively low-resource devices
without requiring an internet connection.
The interface for this task, as shown in Figure8, is a screen that shows users
the model’s output as a set of candidate tokens that they can opt for while
submitting the translation. We sample 150 sentences from our development set
and provided it for translation through the Karya app. Inferences for these
sample sentences were provided to the annotators as a panel of assistive
candidate tokens. The annotators were requested to post-edit these candidate
gists until they were correct using the provided suggestions wherever
possible. During this exercise we monitor the number of suggestions accepted
by the annotators over the total number of suggestions presented.
#### 4.7.3 Takeaways
We find that annotators use 3.66 out of an average of 5.56 options shown to
them in each sentence translation iteration. This is further substantiated by
the observations of the annotators, who reported that nearly half of the
suggestions provided by the model were accurate and useful in generating the
final output. They also commented on the interface design by mentioning that
they preferred to type a token displayed in the panel of candidate tokens
instead of clicking on the token to avoid disrupting their flow while typing
out a translation. This indicates that tracking the number of accepted tokens
via clicks would have provided us a pessimistic view of the model’s usefulness
since annotators were visually aided by tokens (which might have been
completely or partially correct) and our experimental design would not have
logged that as a successful instance of aid. Annotators also pointed out that
some stop words were useful candidates in the Bag Of Words interface as they
preferred opting for them while typing a translation (rather than post-editing
them as a part of a gist). Hence, providing even noisy assistance with the
right interface appears to be a promising direction.
The annotators overarching evaluation of the translation system deemed it to
be semantically accurate. This gives hope that we can setup a virtuous cycle
where generation of linguistic resources enables the development of community-
oriented language technologies, which in turn could help in generating more
linguistic resources that can then further improve language technologies.
Additionally, their feedback on finding the candidate translations being
visually assistive encourages a deeper investigation into if and how
relatively low-accuracy models could be leveraged to improve the experience of
data providers in low-resource communities.
### 4.8 Disseminating Gondi content via Interactive Voice Response
Collecting data and building language technologies is only one part of the
equation; the other is creating avenues for communities to hear their language
broadcast over mass media and other communication channels.
#### 4.8.1 Motivation
There is an established body of work on the role of Interactive Voice Response
(IVR) forums for reaching communities that are too poor to afford smartphones,
too remote to access the Internet or too low-literate to navigate text-driven
environments swaminathan2019learn2earn ; dipanjan2019 ; revisitCG ; designLess
; ivrFarmers ; raza2018baang ; sawaal ; raza2013job ; sangeet . As an example,
Learn2Earn swaminathan2019learn2earn is an IVR based system that was used as
an awareness campaign tool to spread farmers’ land rights in rural India, HIV
literacy and voter awareness mehta2020using . Learn2Earn awards mobile
talktime to users who call a toll-free number and answer all multiple-choice
questions on the message correctly. Starting from an initial set of just 17
users, it was successful in spreading land rights awareness to 17,000 farmers
in 45 days via additional rewards for successful referrals
swaminathan2019learn2earn .
We adapted Learn2Earn for spreading voter awareness among Gondi speakers in
Dantewada (a rural district in the state of Chhattisgarh in India), during the
time that a bypoll election was being held. Prior work has shown that larger
the contexts, identities and communicative functions associated with a
language, the more likely it is to thrive walsh2005will . Voter rights content
in local languages during election season has the potential to encourage
conversations on topics of wider contexts and functions as well as contribute
to establishing representative and effective governance.
#### 4.8.2 Intervention
The pilot obtained users through seeding activities and referrals. Figure9
shows the number of unique calls that were made to our system and the number
of users who passed the quiz over time. Further, the Figureshows that a
majority of quiz passers came to know about our system via direct seeding,
though people knew that additional credits can be earned through referrals.
Usage dropped sharply after the elections in September concluded and seeding
activities were discontinued.
Figure 9: Number of calls to the system (blue line) and the number of users
who answered all questions correctly (bar plot). The blue and orange bars
represent users onboarded via seeding and through referrals
---
Metrics | Value
Unique callers (total, during and after seeding) | (557, 480, 77)
Unique callers per day - during seeding (min, mean, median, max) | (13, 48, 49, 80)
Unique callers per day - after seeding (min, mean, median, max) | (0, 4, 3, 20)
Callers answering all questions correctly | 313
Callers answering all questions correctly in their first call | 104
Calls made by callers (min, mean, median, max) | (1, 3, 2, 64)
Table 3: Summary statistics for our Learn2Earn deployment in Gondi
Table 3 highlights some important statistics in our Learn2Earn pilot. We
spread voter awareness in Gondi to 557 speakers, of which a majority (86%)
were reached during active seeding. Nearly 60% of the users correctly answered
all questions, either in their first attempt or in successive attempts,
indicating the effectiveness of the system for content dissemination. Further,
many users called the system more than once, with one user calling 64 times
(to refer someone, users had to place another call to the system). A follow-up
survey revealed that 22 of 113 users surveyed could not vote as they did not
have a voter ID card, potentially useful information for authorities to
increase voter turnout. One disappointing finding was that only about 8.3
percent of users were women.
#### 4.8.3 Takeaways
We see three clear benefits from conducting Learn2Earn pilots in endangered or
vulnerable languages. First, since it is entirely in the spoken form, only
native speakers of an endangered or vulnerable language can comprehend the
content and earn the reward. It is also easily accessible in low-resource
environments as it requires a missed call and nothing more. Second, the phone
numbers collected are an important dataset of speakers of that language and
can be used for future translation workshops and related programs that help
their economic and linguistic development. Finally, an oral language has a
tendency to die out unless there is an opportunity for that language to be
used outside of everyday surroundings, which periodic Learn2Earn campaigns on
important issues can help achieve.
## 5 Discussion
Steven Bird alerts us to the dangers inherent in a technology driven approach
to language revival, contending that commodifying indigenous languages as data
alienates native speakers. He argues for an approach rooted in self-
determination where outside experts are only engaged with to help implement a
program or strategy that the community has already settled on
bird2020decolonising . This sentiment is echoed by Stebbins et al (2017) who
maintain that ‘the community must be the driver and driving interest of the
research and language projects”stebbins2017living . Lewis and Simons carve out
an additional role for reflective practitioners of language development, that
of ”facilitating opportunities for local language users to interact with each
other” book .
There are also concerns that experts profit off their domain expertise in an
indigenous language with little benefit accruing to the communities
themselves. For example, Bird asks ”where are the full implementations,
deployed in robust software, in active use to capture primary data, leading to
curated language products that are being mobilized in speech communities?” By
his yardstick, even the interventions described in our paper fall short as
none have translated to scale or persistent use by the community. With the
benefit of hindsight, we can now reflect on whether our initiatives have been
exploitative.
Our position is that these interventions avoided an exploitative approach by
providing tangible outputs advancing one of the five FAMED conditions laid out
by Lewis and Simons as an ethical guide to language practitioners. This
acronym expands to having Functions associated with the language that are in
use and recognized by the community; a means for community members to Acquire
proficiency in the language; a Motivation for community members to use the
language, which is frequently but not exclusively economic; a policy
Environment that is conducive to the growth of the language; and a
Differentiated sphere for use of the language in adherence to established
societal norms.
Our machine translation artifact can be built upon by the Gondi community and
other low resource communities around the world to expand the functions
associated with their language, through automated translation of digital
content. The assistive translation app can help in language acquisition and
documentation by improving the speed of translating literary material and
providing vocabulary that translators may themselves be unaware of. Our
Learn2Earn pilot provided incentive payments to those speaking Gondi, thus
increasing (at a small scale) the motivation to use and recognize the
language. Translating children’s stories into Gondi can create a fertile
policy environment that makes possible the use of tribal languages in primary
education. The dictionary development process kickstarted a differentiated
sphere of Gondi usage between tribals of different states, whereas it has
historically been used only within their own region. Our thematic approach of
collecting language data to build technologies while simultaneously creating
resources for the community that advance one of the FAMED conditions can help
avoid the pitfall of alienating local speakers by commodifying their language
as data.
Overall, our work raises interesting questions on the role of the outside
interventionist in building langugage technologies and the relationship they
should have with the community. For example, the dictionary development
workshops primarily comprised bilingual and educated Gond speakers, which may
have contributed to their desire for standardizing the language. However, we
realized that this approach would be discriminatory to monolingual Gond
speakers who would have to accept the language thrown at them by the
intellectuals within their community since the standardization efforts would
not be universally agreed upon or even known to the majority of Gond speakers.
As Bird writes, ”No community speaks with a single voice, and in building
relationships we can unwittingly align ourselves with agendas, clans and gate-
keepers.” This prompts reflection on whether the outside interventionist
should simply fill in ”white man’s paperwork” and let the community lead, or
if there is a deeper role that we need to play, one where we use our perceived
neutrality to arbitrate between the different sections of a community. The
latter role finds expression in Lewis and Simons’ contention that ”it may be
an appropriate role for an outsider to act as an intermediary between the
different speech communities.” While we do not presume to have answers to
these questions, we hope that our work helps stimulate wider debate in the
language development community on defining our relationship with the speech
communities we work with and depend upon.
## 6 Conclusion
Engineering is the primary obstacle to designing technologies for well-
resourced languages. For low-resource ones, more focus needs to be given
towards designing methods for robust data collection and evaluation upon which
the language technology can be built.
To keep community members motivated through the data collection process, our
team strived to achieve 2 simultaneous goals: the collection of data upon
which language technologies such as speech to text or machine translation
could be built, and building a literary resource that community members could
point to as an immediate, demonstrable success. For example, the Learn2Earn
pilot in Gondi not only provided the community with an opportunity to earn
money for answering a quiz in their language and referring others to it, but
it also provided a dataset of native Gondi speakers. Similarly, translating
children’s stories and creating a standardized dictionary resulted in both
data upon which a machine translation tool was built and also tangible
language resources that can be used by the community.
We made use of the data collected to develop a machine translation model and
an assistive translation interface. Community evaluators reported that the
model fared poorly in punctuation and grammar, and roughly half the suggested
words were applicable to the sentence they were translating. Our future goals
are improving the models accuracy by feeding it more data and testing its
performance with data from other low-resource languages.
In 2020, the Chhattisgarh government began allowing primary education in 10
tribal languages, including Gondi
111111https://indianexpress.com/article/governance/chhattisgarh-education-
reforms-tribal-languages-to-be-a-medium-of-education-in-pre-school-6271547/.
At the community level, our focus is on working with the government to
integrate the linguistic resources and technological artifacts we’ve created
into their apparatus. We also plan on integrating the translation tool with
CGNet Swara’s citizen journalism newsroom so that they can regularly put out
Gondi content.
## References
* (1) Abraham, B., Goel, D., Siddarth, D., Bali, K., Chopra, M., Choudhury, M., Joshi, P., Jyoti, P., Sitaram, S., Seshadri, V.: Crowdsourcing speech data for low-resource languages from low-income workers. In: Proceedings of The 12th Language Resources and Evaluation Conference, pp. 2819–2826 (2020)
* (2) Adams, O., Galliot, B., Wisniewski, G., Lambourne, N., Foley, B., Sanders-Dwyer, R., Wiles, J., Michaud, A., Guillaume, S., Besacier, L., Cox, C., Aplonova, K., Jacques, G., Hill, N.: User-friendly automatic transcription of low-resource languages: Plugging ESPnet into elpis. In: Proceedings of the 4th Workshop on the Use of Computational Methods in the Study of Endangered Languages Volume 1 (Papers), pp. 51–62. Association for Computational Linguistics, Online (2021). URL https://aclanthology.org/2021.computel-1.7
* (3) Aiton, G.: Translating fieldwork into datasets: The development of a corpus for the quantitative investigation of grammatical phenomena in eibela. Proceedings of the Workshop on Computational Methods for Endangered Languages 2 (2021). DOI 10.33011/computel.v2i.973
* (4) Alagarai Sampath, H., Rajeshuni, R., Indurkhya, B.: Cognitively inspired task design to improve user performance on crowdsourcing platforms. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, CHI ’14, p. 3665–3674. Association for Computing Machinery, New York, NY, USA (2014). DOI 10.1145/2556288.2557155. URL https://doi.org/10.1145/2556288.2557155
* (5) Beine, D.K.: A sociolinguistic survey of the gondi-speaking communities of central india. Ph.D. thesis, Univ. San Diego (1994)
* (6) Bettinson, M., Bird, S.: Developing a suite of mobile applications for collaborative language documentation. In: Proceedings of the 2nd Workshop on the Use of Computational Methods in the Study of Endangered Languages, pp. 156–164. Association for Computational Linguistics, Honolulu (2017). DOI 10.18653/v1/W17-0121. URL https://aclanthology.org/W17-0121
* (7) Bird, S.: Decolonising speech and language technology. In: Proceedings of the 28th International Conference on Computational Linguistics, pp. 3504–3519 (2020)
* (8) Bird, S., Hanke, F.R., Adams, O., Lee, H.: Aikuma: A mobile app for collaborative language documentation. In: Proceedings of the 2014 Workshop on the Use of Computational Methods in the Study of Endangered Languages, pp. 1–5 (2014)
* (9) Boruah, D.M.: Language loss and revitalization of gondi language: An endangered language of central india. Language in India 20(9) (2020)
* (10) Bowers, D., Arppe, A., Lachler, J., Moshagen, S., Trosterud, T.: A morphological parser for odawa. In: Proceedings of the 2nd Workshop on the Use of Computational Methods in the Study of Endangered Languages, pp. 1–9. Association for Computational Linguistics, Honolulu (2017). DOI 10.18653/v1/W17-0101. URL https://aclanthology.org/W17-0101
* (11) Bühmann, D., Trudell, B.: Mother tongue matters: Local language as a key to effective learning. France: UNESCO (2008)
* (12) Caswell, I., Breiner, T., van Esch, D., Bapna, A.: Language ID in the wild: Unexpected challenges on the path to a thousand-language web text corpus. In: Proceedings of the 28th International Conference on Computational Linguistics, pp. 6588–6608. International Committee on Computational Linguistics, Barcelona, Spain (Online) (2020). DOI 10.18653/v1/2020.coling-main.579. URL https://aclanthology.org/2020.coling-main.579
* (13) Chakraborty, D., Gupta, A., Team, G.V., Seth, A.: Experiences from a Mobile-based Behaviour Change Campaign on Maternal and Child Nutrition in Rural India. In: ICTD (2019)
* (14) Chandramouli, C., General, R.: Census of india 2011. Provisional Population Totals. New Delhi: Government of India (2011)
* (15) Dearden, A., Tucker, W.D.: The ethical limits of bungee research in ictd. In: 2015 IEEE International Symposium on Technology and Society (ISTAS), pp. 1–6. IEEE (2015)
* (16) Diddee, H., Bali, K., Choudhury, M., Mukhija, N.: The six conundrums of building and deploying language technologies for social good. In: ACM SIGCAS/SIGCHI Conference on Computing and Sustainable Societies (COMPASS), COMPASS ’22, p. 12–19. Association for Computing Machinery, New York, NY, USA (2022). DOI 10.1145/3530190.3534792. URL https://doi.org/10.1145/3530190.3534792
* (17) Diddee, H., Dandapat, S., Choudhury, M., Ganu, T., Bali, K.: Too brittle to touch: Comparing the stability of quantization and distillation towards developing lightweight low-resource mt models (2022). DOI 10.48550/ARXIV.2210.15184. URL https://arxiv.org/abs/2210.15184
* (18) D’Silva, K., Marathe, M., Vashistha, A., Borriello, G., Thies, W.: A mobile application for interactive voice forums: Design and pilot deployment in rural india. In: Proceedings of the Fifth ACM Symposium on Computing for Development, pp. 121–122 (2014)
* (19) Elazar, Y., Basmov, V., Ravfogel, S., Goldberg, Y., Tsarfaty, R.: The extraordinary failure of complement coercion crowdsourcing. In: Proceedings of the First Workshop on Insights from Negative Results in NLP, pp. 106–116. Association for Computational Linguistics, Online (2020). DOI 10.18653/v1/2020.insights-1.17. URL https://aclanthology.org/2020.insights-1.17
* (20) Ethnologue: https://www.ethnologue.com/world
* (21) Finn, A., Jones, P.L., Mahelona, K., Duncan, S., Leoni, G.: Developing a part-of-speech tagger for te reo Māori. In: Proceedings of the Fifth Workshop on the Use of Computational Methods in the Study of Endangered Languages, pp. 93–98. Association for Computational Linguistics, Dublin, Ireland (2022). DOI 10.18653/v1/2022.computel-1.12. URL https://aclanthology.org/2022.computel-1.12
* (22) Goldman, J.P., Clematide, S., Avanzi, M., Tandler, R.: Strategies and challenges for crowdsourcing regional dialect perception data for Swiss German and Swiss French. In: Proceedings of the Eleventh International Conference on Language Resources and Evaluation (LREC 2018). European Language Resources Association (ELRA), Miyazaki, Japan (2018). URL https://aclanthology.org/L18-1234
* (23) Haugen, E.: Dialect, language, nation 1. American anthropologist 68(4), 922–935 (1966)
* (24) Herbig, N., Pal, S., van Genabith, J., Krüger, A.: Multi-modal approaches for post-editing machine translation. In: Proceedings of the 2019 CHI Conference on Human Factors in Computing Systems, CHI ’19, p. 1–11. Association for Computing Machinery, New York, NY, USA (2019). DOI 10.1145/3290605.3300461. URL https://doi.org/10.1145/3290605.3300461
* (25) Hermes, M., King, K.A.: Ojibwe language revitalization, multimedia technology, and family language learning. Language Learning & Technology 17(1), 125–144 (2013)
* (26) Joshi, P., Barnes, C., Santy, S., Khanuja, S., Shah, S., Srinivasan, A., Bhattamishra, S., Sitaram, S., Choudhury, M., Bali, K.: Unsung challenges of building and deploying language technologies for low resource language communities. arXiv preprint arXiv:1912.03457 (2019)
* (27) Khayrallah, H., Koehn, P.: On the impact of various types of noise on neural machine translation. arXiv preprint arXiv:1805.12282 (2018)
* (28) Kornai, A.: Digital language death. PloS one 8(10), e77,056 (2013)
* (29) Kornai, A., Bhattacharyya, P.: Indian subcontinent language vitalization (2014)
* (30) Kreutzer, J., Caswell, I., Wang, L., Wahab, A., van Esch, D., Ulzii-Orshikh, N., Tapo, A., Subramani, N., Sokolov, A., Sikasote, C., Setyawan, M., Sarin, S., Samb, S., Sagot, B., Rivera, C., Rios, A., Papadimitriou, I., Osei, S., Suarez, P.O., Orife, I., Ogueji, K., Rubungo, A.N., Nguyen, T.Q., Müller, M., Müller, A., Muhammad, S.H., Muhammad, N., Mnyakeni, A., Mirzakhalov, J., Matangira, T., Leong, C., Lawson, N., Kudugunta, S., Jernite, Y., Jenny, M., Firat, O., Dossou, B.F.P., Dlamini, S., de Silva, N., Çabuk Ballı, S., Biderman, S., Battisti, A., Baruwa, A., Bapna, A., Baljekar, P., Azime, I.A., Awokoya, A., Ataman, D., Ahia, O., Ahia, O., Agrawal, S., Adeyemi, M.: Quality at a Glance: An Audit of Web-Crawled Multilingual Datasets. Transactions of the Association for Computational Linguistics 10, 50–72 (2022). DOI 10.1162/tacl˙a˙00447. URL https://doi.org/10.1162/tacl_a_00447
* (31) Kumar, S.R.: Gondi language - identity, politics and struggle in india. IOSR Journal 24(2), 51–57 (2019)
* (32) Kummerfeld, J.K.: Quantifying and avoiding unfair qualification labour in crowdsourcing. In: Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 2: Short Papers), pp. 343–349. Association for Computational Linguistics, Online (2021). DOI 10.18653/v1/2021.acl-short.44. URL https://aclanthology.org/2021.acl-short.44
* (33) Lambton-Howard, D., Olivier, P., Vlachokyriakos, V., Celina, H., Kharrufa, A.: Unplatformed design: A model for appropriating social media technologies for coordinated participation. In: Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems, pp. 1–13 (2020)
* (34) Lewis, M., Simons, G.: Sustaining language use: Perspectives on community-based language development. (2016)
* (35) Liebling, D.J., Lahav, M., Evans, A., Donsbach, A., Holbrook, J., Smus, B., Boran, L.: Unmet needs and opportunities for mobile translation ai. In: Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems, CHI ’20, p. 1–13. Association for Computing Machinery, New York, NY, USA (2020). DOI 10.1145/3313831.3376261. URL https://doi.org/10.1145/3313831.3376261
* (36) Little, A.N.: Connecting documentation and revitalization: A new approach to language apps. In: Proceedings of the 2nd Workshop on the Use of Computational Methods in the Study of Endangered Languages, pp. 151–155. Association for Computational Linguistics, Honolulu (2017). DOI 10.18653/v1/W17-0120. URL https://aclanthology.org/W17-0120
* (37) Liu, H., Thekinen, J., Mollaoglu, S., Tang, D., Yang, J., Cheng, Y., Liu, H., Tang, J.: Toward annotator group bias in crowdsourcing. In: Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp. 1797–1806. Association for Computational Linguistics, Dublin, Ireland (2022). DOI 10.18653/v1/2022.acl-long.126. URL https://aclanthology.org/2022.acl-long.126
* (38) Marathe, M., O’Neill, J., Pain, P., Thies, W.: Revisiting CGNet Swara and its Impact in Rural India. In: ICTD (2015)
* (39) Mehta, D., Mothilal, R.K., Sharma, A., Thies, W., Sharma, A.: Using mobile airtime credits to incentivize learning, sharing and survey response: Experiences from the field. In: Proceedings of the 3rd ACM SIGCAS Conference on Computing and Sustainable Societies, pp. 254–264 (2020)
* (40) Mehta, D., Sharma, A., Kommiya Mothilal, R.K., Shukla, A., Prasad, V., Thies, W., U, V., Scott, C., Sharma, A.: Facilitating media distribution with monetary incentives. In: Extended Abstracts of the 2020 CHI Conference on Human Factors in Computing Systems, pp. 1–7 (2020)
* (41) Millour, A., Fort, K.: Text corpora and the challenge of newly written languages. In: Proceedings of the 1st Joint Workshop on Spoken Language Technologies for Under-resourced languages (SLTU) and Collaboration and Computing for Under-Resourced Languages (CCURL), pp. 111–120. European Language Resources association, Marseille, France (2020). URL https://aclanthology.org/2020.sltu-1.15
* (42) Moitra, A., Das, V., Kumar, A., Seth, A.: Design Lessons from Creating a Mobile-based Community Media Platform in Rural India. In: ICTD (2016)
* (43) Moseley, C.: Atlas of the World’s Languages in Danger. Unesco (2010)
* (44) Mus, N., Metzger, R.: Toward a corpus of tundra nenets: stages and challenges in building a corpus. In: Proceedings of the Workshop on Computational Methods for Endangered Languages, vol. 2, pp. 4–9 (2021)
* (45) Nangia, N., Sugawara, S., Trivedi, H., Warstadt, A., Vania, C., Bowman, S.R.: What ingredients make for an effective crowdsourcing protocol for difficult nlu data collection tasks? arXiv preprint arXiv:2106.00794 (2021)
* (46) Patel, N., Chittamuru, D., Jain, A., Dave, P., Parikh, T.S.: Avaaj otalo - a field study of an interactive voice forum for small farmers in rural india. In: CHI (2010)
* (47) Raza, A.A., Saleem, B., Randhawa, S., Tariq, Z., Athar, A., Saif, U., Rosenfeld, R.: Baang: a viral speech-based social platform for under-connected populations. In: Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems, p. 643. ACM (2018)
* (48) Raza, A.A., Tariq, Z., Randhawa, S., Saleem, B., Athar, A., Saif, U., Rosenfeld, R.: Voice-based quizzes for measuring knowledge retention in under-connected populations. In: CHI (2019)
* (49) Raza, A.A., Ul Haq, F., Tariq, Z., Pervaiz, M., Razaq, S., Saif, U., Rosenfeld, R.: Job opportunities through entertainment: Virally spread speech-based services for low-literate users. In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems, pp. 2803–2812. ACM (2013)
* (50) Rijhwani, S., Anastasopoulos, A., Neubig, G.: Ocr post correction for endangered language texts. arXiv preprint arXiv:2011.05402 (2020)
* (51) Roche, G.: Abandoning endangered languages: Ethical loneliness, language oppression, and social justice. American Anthropologist 122(1), 164–169 (2020)
* (52) Saldivar, J., Daniel, F., Cernuzzi, L., Casati, F.: Online idea management for civic engagement: A study on the benefits of integration with social networking. ACM Transactions on Social Computing 2(1), 1–29 (2019)
* (53) Shapiro, M.C., Schiffman, H.F.: Language and society in South Asia, vol. 1. Motilal Banarsidass Publishe (1981)
* (54) Shmueli, B., Fell, J., Ray, S., Ku, L.W.: Beyond fair pay: Ethical implications of nlp crowdsourcing. arXiv preprint arXiv:2104.10097 (2021)
* (55) Specia, L., Blain, F., Fomicheva, M., Fonseca, E., Chaudhary, V., Guzmán, F., Martins, A.F.T.: Findings of the WMT 2020 shared task on quality estimation. In: Proceedings of the Fifth Conference on Machine Translation, pp. 743–764. Association for Computational Linguistics, Online (2020). URL https://aclanthology.org/2020.wmt-1.79
* (56) Starbird, K.: Crowdwork, crisis and convergence: How the connected crowd organizes information during mass disruption events. Ph.D. thesis, University of Colorado at Boulder (2012)
* (57) Stebbins, T.N., Eira, K., Couzens, V.L.: Living languages and new approaches to language revitalisation research. Routledge (2017)
* (58) Suh, H., Shahriaree, N., Hekler, E.B., Kientz, J.A.: Developing and validating the user burden scale: A tool for assessing user burden in computing systems. In: Proceedings of the 2016 CHI Conference on Human Factors in Computing Systems, CHI ’16, p. 3988–3999. Association for Computing Machinery, New York, NY, USA (2016). DOI 10.1145/2858036.2858448. URL https://doi.org/10.1145/2858036.2858448
* (59) Swaminathan, S., Medhi Thies, I., Mehta, D., Cutrell, E., Sharma, A., Thies, W.: Learn2earn: Using mobile airtime incentives to bolster public awareness campaigns. Proceedings of the ACM on Human-Computer Interaction 3(CSCW), 49 (2019)
* (60) Tyler, S.A.: Koya: an outline grammar: Gommu dialect, vol. 54. Berkeley: University of California Press (1969)
* (61) Vashistha, A., Cutrell, E., Borriello, G., Thies, W.: Sangeet Swara: A Community-Moderated Voice Forum in Rural India. In: CHI (2015)
* (62) Walsh, M.: Will indigenous languages survive? Annu. Rev. Anthropol. 34, 293–315 (2005)
* (63) Wang, X., Ruder, S., Neubig, G.: Expanding pretrained models to thousands more languages via lexicon-based adaptation. In: ACL (2022)
* (64) Xue, L., Constant, N., Roberts, A., Kale, M., Al-Rfou, R., Siddhant, A., Barua, A., Raffel, C.: mt5: A massively multilingual pre-trained text-to-text transformer. arXiv preprint arXiv:2010.11934 (2020)
* (65) Xue, L., Constant, N., Roberts, A., Kale, M., Al-Rfou, R., Siddhant, A., Barua, A., Raffel, C.: mT5: A massively multilingual pre-trained text-to-text transformer. In: Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, pp. 483–498. Association for Computational Linguistics, Online (2021). DOI 10.18653/v1/2021.naacl-main.41. URL https://aclanthology.org/2021.naacl-main.41
* (66) Zariquiey, R., Oncevay, A., Vera, J.: CLD2 language documentation meets natural language processing for revitalising endangered languages. In: Proceedings of the Fifth Workshop on the Use of Computational Methods in the Study of Endangered Languages, pp. 20–30. Association for Computational Linguistics, Dublin, Ireland (2022). DOI 10.18653/v1/2022.computel-1.4. URL https://aclanthology.org/2022.computel-1.4
|
# Extremal numbers of disjoint triangles in $r$-partite graphs
Junxue Zhang111Corresponding author
Center for Combinatorics and LPMC, Nankai University, Tianjin 300071, China
Email<EMAIL_ADDRESS>
###### Abstract
For two graphs $G$ and $F$, the extremal number of $F$ in $G$, denoted by
ex$(G,F)$, is the maximum number of edges in a spanning subgraph of $G$ not
containing $F$ as a subgraph. Determining ex$(K_{n},F)$ for a given graph $F$
is a classical extremal problem in graph theory. In 1962, Erdős determined
ex$(K_{n},kK_{3})$, which generalized Mantel’s Theorem. On the other hand, in
1974, Bollobás, Erdős, and Straus determined
ex$(K_{n_{1},n_{2},\dots,n_{r}},K_{t})$, which extended Turán’s Theorem to
complete multipartite graphs. In this paper, we determine
ex$(K_{n_{1},n_{2},\dots,n_{r}},kK_{3})$ for $r\geq 4$ and $10k-4\leq
n_{1}+4k\leq n_{2}\leq n_{3}\leq\cdots\leq n_{r}$.
Keywords: extremal numbers; multipartite graphs; disjoint triangles
## 1 Introduction
All graphs considered in this paper are finite and simple. We follow [12] for
undefined notation and terminology. Let $G=G(V(G),E(G))$ be a graph, where
$V(G)$ is the vertex set and $E(G)$ is the edge set. Denote $e(G)=|E(G)|$. For
any subset $S\subseteq V(G)$, we use $G[S]$ to denote the subgraph of $G$
induced by $S$ and let $G-S=G[V(G)\setminus S]$. Given two disjoint vertex
sets $V_{1}$ and $V_{2}$, the join $V_{1}\vee V_{2}$ is the edge set obtained
by joining each vertex of $V_{1}$ to each vertex of $V_{2}$. Furthermore,
given two graphs $G$ and $H$, the join $G\vee H$ is the graph obtained from
the disjoint union of $G$ and $H$ by joining each vertex of $G$ to each vertex
of $H$. Let $r,t,k$ be three positive integers. For convenience, we write
$[r]=\\{1,2,3,...,r\\}$ in the context. Denote by $K_{t}$ the complete graph
on $t$ vertices and $kK_{t}$ the disjoint union of $k$ copies of $K_{t}$.
Moreover, for $r$ positive integers $n_{1},n_{2},\dots,n_{r}$, we use
$K_{n_{1},n_{2},\dots,n_{r}}$ to denote the complete $r$-partite graph with
parts of sizes $n_{1},n_{2},\dots,n_{r}$.
Let ex$(G,H)=\max\\{e(G^{\prime}):$ $|V(G^{\prime})|=|V(G)|$,
$G^{\prime}\subseteq G$, and $H\nsubseteq G^{\prime}\\}$ and call it the
extremal number of $H$ in $G$. Let $n$ and $t$ be two integers with $n\geq t$.
The classical Turán problem considers the case $G=K_{n}$, i.e., determining
the value of ex$(K_{n},H)$ for a given $H$. For instance, the well-known
Mantel’s Theorem [8] and Turán’s Theorem [11] determined ex$(K_{n},K_{3})$ and
ex$(K_{n},K_{t})$, respectively. Turán’s Theorem is widely considered to be
the first extremal theorem on graphs and initiated the study of extremal graph
theory. Let $T_{r}(n)$ be a balanced complete $r$-partite graph on $n$
vertices, that is, any two parts of sizes differ by at most one. In 1959,
Erdős and Gallai [6] determined ex$(K_{n},kK_{2})$ for any positive integers
$n$ and $k$. Later, Erdős [5] proved ex$(K_{n},kK_{3})=e(K_{k-1}\vee
T_{2}(n-k+1))$ for $n>400(k-1)^{2}$ and Moon [9] proved that for
$n>\frac{9k}{2}+4$. Moreover, Moon [9] (only when $n-k+1$ is divisible by $p$)
and Simonovits [10] showed that $K_{k-1}\vee T_{r-1}(n-k+1)$ is the unique
extremal graph containing no copy of $kK_{r}$ for sufficiently large $n$. The
extremal problem on multipartite graphs was first considered by Bollobás,
Erdős, and Straus [1]. They determined ex$(K_{n_{1},n_{2},\dots,n_{r}},K_{t})$
for $r\geq t\geq 2$. Later, Chen, Li, and Tu [2] determined
ex$(K_{n_{1},n_{2}},kK_{2})$. Recently, De Silva, Heysse and Young [4] proved
ex$(K_{n_{1},n_{2},\dots,n_{r}},kK_{2})=(k-1)(n_{2}+n_{3}+\cdots+n_{r})$ for
$n_{1}\leq n_{2}\leq\dots\leq n_{r}$.
To make it easier to state the results in this paper, we define a function
$f$. Let $x_{1},x_{2},\dots,x_{r}$ be positive integers and $t\geq 2$ be an
integer. Note that $\\{x_{1},x_{2},\dots,x_{r}\\}$ is a multiset. For any
subset $P\subseteq[r]$, let $x_{P}:=\sum_{i\in P}x_{i}$. Now, we define the
desired function $f$ as follows. For $t=2$, let
$f_{2}(\\{x_{1},x_{2},\dots,x_{r}\\})=0$. For $t\geq 3$, let
$f_{t}(\\{x_{1},x_{2},\dots,x_{r}\\})=\max\limits_{\mathcal{P}}~{}\\{\sum\limits_{1\leq
i<j\leq t-1}x_{P_{i}}\cdot x_{P_{j}}\\}$, where
$\mathcal{P}=(P_{1},P_{2},\dots,P_{t-1})$ is a partition of $[r]$ into $t-1$
nonempty parts. For convenience, we simply write
$f_{t}(x_{1},x_{2},\dots,x_{r})$ for $f_{t}(\\{x_{1},x_{2},\dots,x_{r}\\})$.
Notice that $f_{r}(x_{1},x_{2},\dots,x_{r})=(\sum_{1\leq i<j\leq
r}x_{i}x_{j})-x_{1}x_{2}$ if $x_{1}\leq x_{2}\leq\dots\leq x_{r}$.
###### Theorem 1.1
[1] Let $n_{1},n_{2},\dots,n_{r},t$ be positive integers with $r\geq t\geq 2$.
Then
ex$(K_{n_{1},n_{2},\dots,n_{r}},K_{t})=f_{t}(n_{1},n_{2},\dots,n_{r})$.
By replacing the forbidden graph $K_{t}$ with $kK_{t}$, De Silva et al. [3]
considered a special case $t=r$ and raised the problem about
ex$(K_{n_{1},n_{2},\dots,n_{r}},kK_{t})$ for $r>t$.
###### Theorem 1.2
[3] For any integers $r\geq 2$ and $1\leq k\leq n_{1}\leq n_{2}\leq\cdots\leq
n_{r}$, we have
$\displaystyle{ex}(K_{n_{1},n_{2},\dots,n_{r}},kK_{r})$
$\displaystyle=\sum_{1\leq i<j\leq r}n_{i}n_{j}-n_{1}n_{2}+(k-1)n_{2}$
$\displaystyle{=}(k-1)(n-n_{1})+{f_{r}(n_{1}-(k-1),n_{2},\dots,n_{r}),}$
where $n=\sum n_{i}$.
###### Problem 1.3
[3] Determine ex$(K_{n_{1},n_{2},\dots,n_{r}},kK_{t})$ for $r>t$.
Han and Zhao [7] determined ex$(K_{n_{1},n_{2},n_{3},n_{4}},kK_{3})$ for a
sufficiently large integer $n_{1}$ with $n_{1}\leq n_{2}\leq n_{3}\leq n_{4}$.
###### Theorem 1.4
[7] Let $n_{1},\dots,n_{4}$ be sufficiently large and $n_{1}\leq n_{2}\leq
n_{3}\leq n_{4}$. For any integer $k\geq 1$, we have
$\displaystyle ex(K_{n_{1},n_{2},n_{3},n_{4}},kK_{3})$
$\displaystyle=(k-1)(n-n_{1})+f_{3}(n_{1}-{(k-1)},n_{2},n_{3},n_{4})$
$\displaystyle=\left\\{\begin{array}[]{lcl}n_{4}(n_{1}+n_{2}+n_{3})+(k-1)(n_{2}+n_{3}),&&{\text{if}~{}n_{4}>n_{2}+n_{3}};\\\
(n_{1}+n_{4})(n_{2}+n_{3})+(k-1)n_{4},&&{\text{if}~{}n_{4}\leq
n_{2}+n_{3}}.\\\ \end{array}\right.$
In addition, Han and Zhao [7] proposed a conjecture on
ex$(K_{n_{1},n_{2},\dots,n_{r}},kK_{t})$.
###### Conjecture 1.5
[7] Given three integers $k$, $r$, and $t$ with $r>t\geq 3$ and $k\geq 2$, let
$n_{1},\dots,n_{r}$ be sufficiently large. For $I\subseteq[r]$, write
$m_{I}:={\min}_{i\in I}n_{i}$. Given a partition $\mathcal{P}$ of $[r]$, let
$n_{\mathcal{P}}:={\max}_{I\in\mathcal{P}}\\{n_{I}-m_{I}\\}$. Then
$\displaystyle
ex(K_{n_{1},n_{2},\dots,n_{r}},kK_{t})=\mathop{max}\limits_{\mathcal{P}}\left\\{(k-1)n_{\mathcal{P}}+\sum_{I\neq{I^{\prime}}\in{\mathcal{P}}}n_{I}\cdot
n_{I^{\prime}}\right\\},$
where the maximum is taken over all partitions $\mathcal{P}$ of $[r]$ into
$t-1$ parts.
Actually, we can obtain an equivalent statement of Conjecture 1.5 as follows,
equivalence will be proved in Section 2.
###### Conjecture 1.6
Given three integers $k$, $r$, and $t$ with $r>t\geq 3$ and $k\geq 2$, let
$n_{1}\leq n_{2}\dots\leq n_{r}$ be integers with $\sum_{i=1}^{r}n_{i}=n$. If
$n_{1}$ is sufficiently large, then
ex$(K_{n_{1},n_{2},\dots,n_{r}},kK_{t})=(k-1)(n-n_{1})+f_{t}(n_{1}-(k-1),n_{2},\dots,n_{r})$.
To support Conjecture 1.6, we provide a simple construction, which gives a
lower bound for ex$(K_{n_{1},n_{2},\dots,n_{r}},kK_{t}).$
###### Theorem 1.7
Let $k$ and $t$ be two integers with $k\geq 2$ and $t\geq 3$. Let
$n_{1},n_{2},\dots,n_{r}$ be $r$ integers with $n_{1}\leq\min_{i\geq
2}\\{n_{i}\\}$, $n_{1}\geq k$, $r\geq t$, and $\sum_{i=1}^{r}n_{i}=n$. Then
ex$(K_{n_{1},n_{2},\dots,n_{r}},kK_{t})\geq(k-1)(n-n_{1})+f_{t}(n_{1}-(k-1),n_{2},\dots,n_{r})$.
Furthermore, with Theorem 1.7, we confirm Conjecture 1.6 for $t=3$, $r\geq 4$,
and $n_{1}+4k\leq n_{2}$.
###### Theorem 1.8
Let $r,n_{1},n_{2},\dots,n_{r}$ be integers with $r\geq 4$, $10k-4\leq
n_{1}+4k\leq\min_{i\geq 2}\\{n_{i}\\}$ and $\sum_{i=1}^{r}n_{i}=n$. Then
ex$(K_{n_{1},n_{2},\dots,n_{r}},kK_{3})=(k-1)(n-n_{1})+f_{3}(n_{1}-(k-1),n_{2},\dots,n_{r})$.
In Conjecture 1.6, it is required that “$n_{1}$ is sufficiently large”. But
how large it can be? By aforementioned results and Theorem 1.8, we conjecture
that the lower bound for $n_{1}$ is $k$.
###### Conjecture 1.9
Given three integers $k$, $r$, and $t$ with $r\geq t\geq 2$ and $k\geq 1$, let
$n_{1},n_{2},\dots,n_{r}$ be integers with $n_{1}\leq\min_{i\geq
2}\\{n_{i}\\}$ and $\sum_{i=1}^{r}n_{i}=n$. If $n_{1}\geq k$, then
ex$(K_{n_{1},n_{2},\dots,n_{r}},kK_{t})=(k-1)(n-n_{1})+f_{t}(n_{1}-(k-1),n_{2},\dots,n_{r})$.
The rest of this paper is organized as follows. In Section 2, we first prove
the equivalence of Conjectures 1.5 and 1.6. Later, some basic properties of
$f_{3}(n_{1},n_{2},\dots,n_{r})$ are provided, which will be used frequently
in the proof of Theorem 1.8. The proofs of Theorems 1.7 and 1.8 are presented
in Sections 3 and 4, respectively.
## 2 Equivalence and Properties of $f_{3}(n_{1},n_{2},\dots,n_{r})$
###### Proposition 2.1
Conjecture 1.5 and Conjecture 1.6 are equivalent.
Proof. Let $r>t\geq 3$, $k\geq 2$, $n_{1}\leq n_{2}\dots\leq n_{r}$. Our goal
is to prove
$\displaystyle\mathop{max}\limits_{\mathcal{P}}\left\\{(k-1)n_{\mathcal{P}}+\sum_{I\neq{I^{\prime}}\in{\mathcal{P}}}n_{I}\cdot
n_{I^{\prime}}\right\\}=(k-1)(n-n_{1})+f_{t}(n_{1}-(k-1),n_{2},\dots,n_{r}){.}$
(1)
First we show the $\geq$ direction of (1). Let $x_{1}=n_{1}-(k-1)$,
$x_{i}=n_{i}$ for any $i\geq 2$, and
$\mathcal{P}_{0}=(P_{1},P_{2},\dots,P_{t-1})$ be a partition of $[r]$
maximizing $f_{t}(x_{1},x_{2},\dots,x_{r})$. Assume $1\in P_{1}$. Hence
$\displaystyle(k-1)(n-n_{1})+f_{t}(n_{1}-(k-1),n_{2},\dots,n_{r})$
$\displaystyle=(k-1)(n_{P_{1}}-n_{1})+(k-1)(n-n_{P_{1}})+\sum_{i\neq
j\in[t-1]}x_{P_{i}}\cdot x_{P_{j}}$
$\displaystyle\leq(k-1)n_{\mathcal{P}_{0}}+\sum_{i\neq
j\in[t-1]}n_{P_{i}}\cdot n_{P_{j}}$
$\displaystyle\leq\mathop{max}\limits_{\mathcal{P}}\left\\{(k-1)n_{\mathcal{P}}+\sum_{I\neq{I^{\prime}}\in{\mathcal{P}}}n_{I}\cdot
n_{I^{\prime}}\right\\}.$
Now we prove the $\leq$ direction of (1). For any given partition
$\mathcal{P}=(P_{1},P_{2},\dots,P_{t-1})$ of $[r]$, let $\ell\in[t-1]$ such
that $n_{\mathcal{P}}=n_{P_{\ell}}-m_{P_{\ell}}$ and $\alpha\in P_{\ell}$ such
that $n_{\alpha}=m_{P_{\ell}}$.
Notice that
$\displaystyle(k-1)n_{\mathcal{P}}+\sum_{I\neq{I^{\prime}}\in{\mathcal{P}}}n_{I}\cdot
n_{I^{\prime}}$
$\displaystyle=(k-1)(n_{P_{\ell}}-m_{P_{\ell}})+n_{P_{\ell}}\cdot(n-n_{P_{\ell}})+\sum_{I\neq{I^{\prime}}\in{\mathcal{P}\setminus\\{P_{\ell}\\}}}n_{I}\cdot
n_{I^{\prime}}$
$\displaystyle=(k-1)(n-m_{P_{\ell}})+(n_{P_{\ell}}-(k-1))\cdot(n-n_{P_{\ell}})+\sum_{I\neq{I^{\prime}}\in{\mathcal{P}\setminus\\{P_{\ell}\\}}}n_{I}\cdot
n_{I^{\prime}}$
$\displaystyle\leq(k-1)(n-n_{\alpha})+f_{t}(\\{n_{1},\cdots,n_{r}\\}\cup\\{n_{\alpha}-(k-1)\\}\setminus\\{n_{\alpha}\\}).$
Next we assume that $\mathcal{P}$ be the partition maximizing the value
$(k-1)n_{\mathcal{P}}+\sum_{I\neq{I^{\prime}}\in{\mathcal{P}}}n_{I}\cdot
n_{I^{\prime}}$. It remains to show
$\displaystyle(k-1)(n-n_{\alpha})+f_{t}(\\{n_{1},\cdots,n_{r}\\}\cup\\{n_{\alpha}-(k-1)\\}\setminus\\{n_{\alpha}\\})$
$\displaystyle\leq(k-1)(n-n_{1})+f_{t}(\\{n_{1},\cdots,n_{r}\\}\cup\\{n_{1}-(k-1)\\}\setminus\\{n_{1}\\}).$
(2)
If $n_{\alpha}=n_{1}$, then we are done. We may assume that
$n_{\alpha}>n_{1}$. We can see $1\notin P_{\ell}$. We may assume $1\in{P}_{1}$
and $\ell\neq 1$. Now note that $m_{P_{1}}=n_{1}$ and
$m_{P_{\ell}}=n_{\alpha}$. Then $n_{P_{\ell}}-n_{\alpha}>n_{P_{1}}-n_{1}$ by
$\ell\neq 1$. Since $n_{\alpha}>n_{1}$ and
$n_{P_{\ell}}-n_{\alpha}>n_{P_{1}}-n_{1}$, we can switch $1$ and $\alpha$ in
the partition $\mathcal{P}$ and thus increases
$(k-1)n_{\mathcal{P}}+\sum_{I\neq{I^{\prime}}\in{\mathcal{P}}}n_{I}\cdot
n_{I^{\prime}}$, a contradiction. Thus the Inequation (2) holds.
Next we provided some properties of the function
$f_{3}(n_{1},n_{2},\dots,n_{r})$.
###### Proposition 2.2
Let $n_{1},n_{2},\dots,n_{r},\mu$ be positive integers with $\mu\in[r]$ and
$n_{\mu}-1\geq n_{1}+1$. Then
$f_{3}(\\{n_{1},n_{2},\dots,n_{r}\\}\setminus\\{n_{1},n_{\mu}\\}\cup\\{n_{1}+1,n_{\mu}-1\\})\leq
f_{3}(n_{1},n_{2},\dots,n_{r})+n_{\mu}-(n_{1}+1)$.
Proof. Let $x_{1}=n_{1}+1$, $x_{\mu}=n_{\mu}-1$, and $x_{j}=n_{j}$ for any
$j\in[r]\setminus\\{1,\mu\\}$. Then
$f_{3}(\\{n_{1},n_{2},\dots,n_{r}\\}\setminus\\{n_{1},n_{\mu}\\}\cup\\{n_{1}+1,n_{\mu}-1\\})=f_{3}(x_{1},x_{2},\dots,x_{r})$.
Let $\mathcal{P}=(P_{1},P_{2})$ be the partition of $[r]$ maximizing
$f_{3}(x_{1},x_{2},...,x_{r})$. Note that
$\sum_{j=1}^{r}x_{j}=\sum_{j=1}^{r}n_{j}$. If $\\{1,\mu\\}\subseteq P_{1}$ or
$\\{1,\mu\\}\subseteq P_{2}$, then $x_{P_{1}}=n_{P_{1}}$ and
$x_{P_{2}}=n_{P_{2}}$. Thus, $f_{3}(x_{1},x_{2},\dots,x_{r})=x_{P_{1}}\cdot
x_{P_{2}}=n_{P_{1}}\cdot n_{P_{2}}\leq f_{3}(n_{1},n_{2},\dots,n_{r})$. Hence,
without loss of generality, we may assume that $1\in P_{1}$ and $\mu\in
P_{2}$. Observe that $x_{P_{1}}=n_{P_{1}}+1$, $x_{P_{2}}=n_{P_{2}}-1$,
$x_{P_{1}\setminus\\{1\\}}=n_{P_{1}\setminus\\{1\\}}$, and
$x_{P_{2}\setminus\\{\mu\\}}=n_{P_{2}\setminus\\{\mu\\}}$. If
$x_{P_{1}\setminus\\{1\\}}\geq x_{P_{2}\setminus\\{\mu\\}}$, then
$n_{P_{1}\setminus\\{1\\}}\geq n_{P_{2}\setminus\\{\mu\\}}$, and we have
$\displaystyle f_{3}(x_{1},x_{2},\dots,x_{r})=$ $\displaystyle x_{P_{1}}\cdot
x_{P_{2}}$ $\displaystyle=$ $\displaystyle(n_{P_{1}}+1)\cdot(n_{P_{2}}-1)$
$\displaystyle=$ $\displaystyle n_{P_{1}}\cdot
n_{P_{2}}-n_{P_{1}}+n_{P_{2}}-1$ $\displaystyle\leq$ $\displaystyle
f_{3}(n_{1},n_{2},\dots,n_{r})-(n_{P_{1}\setminus\\{1\\}}+n_{1})+(n_{P_{2}\setminus\\{\mu\\}}+n_{\mu})-1$
$\displaystyle\leq$ $\displaystyle
f_{3}(n_{1},n_{2},\dots,n_{r})+n_{\mu}-(n_{1}+1).$
Now $x_{P_{1}\setminus\\{1\\}}<x_{P_{2}\setminus\\{\mu\\}}$, equivalently,
$n_{P_{1}\setminus\\{1\\}}<n_{P_{2}\setminus\\{\mu\\}}$. Recall that
$x_{1}=n_{1}+1\leq n_{\mu}-1=x_{\mu}$. If $n_{1}+1<n_{\mu}-1$, then
$|(x_{P_{1}\setminus\\{1\\}}+x_{\mu})-(x_{P_{2}\setminus\\{\mu\\}}+x_{1})|<|(x_{P_{1}\setminus\\{1\\}}+x_{1})-(x_{P_{2}\setminus\\{\mu\\}}+x_{\mu})|=|x_{P_{1}}-x_{P_{2}}|$,
which contradicts the choice of $\mathcal{P}$ that minimizes
$|x_{P_{1}}-x_{P_{2}}|$, i.e., maximizes $x_{P_{1}}\cdot x_{P_{2}}$. Thus, we
obtain $n_{1}+1=n_{\mu}-1$. This implies $x_{P_{1}}\cdot
x_{P_{2}}=(x_{P_{1}\setminus\\{1\\}}+x_{\mu})\cdot(x_{P_{2}\setminus\\{\mu\\}}+x_{1})$.
Then,
$\displaystyle f_{3}(x_{1},x_{2},\dots,x_{r})=$ $\displaystyle x_{P_{1}}\cdot
x_{P_{2}}$ $\displaystyle=$
$\displaystyle(x_{P_{1}\setminus\\{1\\}}+x_{\mu})\cdot(x_{P_{2}\setminus\\{\mu\\}}+x_{1})$
$\displaystyle=$
$\displaystyle(n_{P_{1}\setminus\\{1\\}}+n_{\mu}-1)\cdot(n_{P_{2}\setminus\\{\mu\\}}+n_{1}+1)$
$\displaystyle=$
$\displaystyle(n_{P_{1}\setminus\\{1\\}}+n_{\mu})\cdot(n_{P_{2}\setminus\\{\mu\\}}+n_{1})+(n_{P_{1}\setminus\\{1\\}}+n_{\mu})-(n_{P_{2}\setminus\\{\mu\\}}+n_{1})-1$
$\displaystyle<$ $\displaystyle
f_{3}(n_{1},n_{2},\dots,n_{r})+n_{\mu}-(n_{1}+1).$
###### Proposition 2.3
Let $n_{1},n_{2},\dots,n_{r}$ be $r$ integers with $n_{1}+2\leq\min_{i\geq
2}\\{n_{i}\\}$. For two indices $i,j\in[r]$ with $i\neq j$, let
$\mathcal{P}=(P_{1},P_{2})$ be the partition attaining
$f_{3}(\\{n_{1},\dots,n_{r}\\}\setminus\\{n_{i},n_{j}\\}\cup\\{n_{i}-1,n_{j}-1\\})$.
Then
$\displaystyle
f_{3}(\\{n_{1},\dots,n_{r}\\}\setminus\\{n_{i},n_{j}\\}\cup\\{n_{i}-1,n_{j}-1\\})\leq$
$\displaystyle f_{3}(n_{1},n_{2},\dots,n_{r})-\sum_{m=1}^{r}n_{m}$
$\displaystyle+max\\{n_{1}+2,n_{i}-n_{1}+1\\}$
Proof. Let $x_{i}=n_{i}-1$, $x_{j}=n_{j}-1$, and $x_{\ell}=n_{\ell}$ for any
$\ell\in[r]\setminus\\{i,j\\}$. Then we know
$f_{3}(\\{n_{1},\dots,n_{r}\\}\setminus\\{n_{i},n_{j}\\}\cup\\{n_{i}-1,n_{j}-1\\})=f_{3}(x_{1},x_{2},\dots,x_{r})$
and the partition $\mathcal{P}$ attains $f_{3}(x_{1},x_{2},\dots,x_{r})$.
Moreover, for any $\ell\in[r]\setminus\\{1\\}$, we have $x_{\ell}\geq
n_{\ell}-1>n_{1}\geq x_{1}$ since $n_{\ell}\geq n_{1}+2$.
If $\\{i,j\\}\nsubseteq P_{\zeta}$ for any $\zeta\in[2]$, we have
$x_{P_{\zeta}}=n_{P_{\zeta}}-1$ and $x_{P_{3-\zeta}}=n_{P_{3-\zeta}}-1$. Thus
$\displaystyle f_{3}(x_{1},x_{2},\dots,x_{r})=$ $\displaystyle
x_{P_{\zeta}}\cdot x_{P_{3-\zeta}}=(n_{P_{\zeta}}-1)\cdot(n_{P_{3-\zeta}}-1)$
$\displaystyle=$ $\displaystyle n_{P_{\zeta}}\cdot
n_{P_{3-\zeta}}-n_{P_{\zeta}}-n_{P_{3-\zeta}}+1$ $\displaystyle\leq$
$\displaystyle f_{3}(n_{1},n_{2},\dots,n_{r})+1-\sum_{m=1}^{r}n_{m}.$
Assume $\\{i,j\\}\subseteq P_{\zeta}$ for some $\zeta\in[2]$. Notice that
$x_{P_{\zeta}}=n_{P_{\zeta}}-2$ and $x_{P_{3-\zeta}}=n_{P_{3-\zeta}}$. Thus,
$\displaystyle f_{3}(x_{1},x_{2},\dots,x_{r})=$ $\displaystyle
x_{P_{\zeta}}\cdot x_{P_{3-\zeta}}=(n_{P_{\zeta}}-2)\cdot n_{P_{3-\zeta}}$
$\displaystyle=$ $\displaystyle n_{P_{\zeta}}\cdot
n_{P_{3-\zeta}}-2n_{P_{3-\zeta}}$ $\displaystyle\leq$ $\displaystyle
f_{3}(n_{1},n_{2},\dots,n_{r})-2n_{P_{3-\zeta}}.$ (3)
If $1\in{P_{\zeta}}$, we have $x_{P_{3-\zeta}}\geq x_{P_{\zeta}}-x_{1}$.
Otherwise $x_{P_{\zeta}}\cdot x_{P_{3-\zeta}}<x_{P_{\zeta}}\cdot
x_{P_{3-\zeta}}+x_{1}\cdot(x_{P_{\zeta}}-x_{1}-x_{P_{3-\zeta}})=(x_{P_{\zeta}}-x_{1})\cdot(x_{P_{3-\zeta}}+x_{1})$,
which contradicts to the choice of $\mathcal{P}$. With
$x_{P_{\zeta}}+x_{P_{3-\zeta}}=x_{\mathcal{P}}$, we know
$n_{P_{3-\zeta}}=x_{P_{3-\zeta}}\geq\frac{x_{\mathcal{P}}-x_{1}}{2}\geq\frac{\sum_{m=1}^{r}n_{m}-2-n_{1}}{2}.$
Thus, by the inequality (2),
$\displaystyle f_{3}(x_{1},x_{2},\dots,x_{r})\leq$ $\displaystyle
f_{3}(n_{1},n_{2},\dots,n_{r})-2n_{P_{3-\zeta}}$ $\displaystyle\leq$
$\displaystyle f_{3}(n_{1},n_{2},\dots,n_{r})+n_{1}+2-\sum_{m=1}^{r}n_{m}.$
It remains to consider the case $1\in{P_{3-\zeta}}$. Recall $x_{i}>x_{1}$ and
$i\in{P_{\zeta}}$, we have $x_{P_{3-\zeta}}-x_{1}\geq x_{P_{\zeta}}-x_{i}$.
Otherwise $x_{P_{\zeta}}\cdot x_{P_{3-\zeta}}<x_{P_{\zeta}}\cdot
x_{P_{3-\zeta}}+(x_{i}-x_{1})\cdot(x_{P_{\zeta}}-x_{i}-x_{P_{3-\zeta}}+x_{1})=(x_{P_{\zeta}}-x_{i}+x_{1})\cdot(x_{P_{3-\zeta}}-x_{1}+x_{i})$,
a contradiction to the choice of $\mathcal{P}$ again. With
$x_{P_{\zeta}}+x_{P_{3-\zeta}}=x_{\mathcal{P}}$, we know
$n_{P_{3-\zeta}}=x_{P_{3-\zeta}}\geq\frac{x_{\mathcal{P}}+x_{1}-x_{i}}{2}=\frac{\sum_{m=1}^{r}n_{m}-1+n_{1}-n_{i}}{2}.$
Thus, by the inequality (2),
$\displaystyle f_{3}(x_{1},x_{2},\dots,x_{r})\leq$ $\displaystyle
f_{3}(n_{1},n_{2},\dots,n_{r})-2n_{P_{3-\zeta}}$ $\displaystyle\leq$
$\displaystyle
f_{3}(n_{1},n_{2},\dots,n_{r})-n_{1}+n_{i}+1-\sum_{m=1}^{r}n_{m}.$
## 3 Proof of Theorem 1.7
We shall provide a construction to complete the proof. Let $x_{1}=n_{1}-(k-1)$
and $x_{i}=n_{i}$ for any $i\neq 1$. Then
$f_{t}(n_{1}-(k-1),n_{2},\dots,n_{r})=f_{t}(x_{1},x_{2},\dots,x_{r})$. Let
$\mathcal{P}=(P_{1},P_{2},\ldots,P_{t-1})$ be the partition of $[r]$ that
attains $f_{t}(x_{1},x_{2},\dots,x_{r})$. Assume that $V_{0}$ is a set of
$(k-1)$ vertices and $V_{i}$ is a set of $x_{i}$ vertices for any $i\in[r]$
such that $V_{0}$, $V_{1}$, …, $V_{r}$ are pairwise disjoint. According to the
partition $\mathcal{P}$ of $[r]$, we set $V_{P_{\ell}}=\cup_{i\in
P_{\ell}}V_{i}$ for each $\ell\in[t-1]$. So
$V_{[r]}=\cup_{\ell\in[t-1]}V_{P_{\ell}}$ and $|V_{P_{\ell}}|=x_{P_{\ell}}$.
Now we construct the graph $G$ as follows. Let $V(G)=V_{0}\cup V_{[r]}$ and
$E(G)=\\{V_{0}\vee(\cup_{i=2}^{r}V_{i})\\}\cup\\{\cup_{1\leq\ell<j\leq
t-1}(V_{P_{\ell}}\vee V_{P_{j}})\\}.$
We see $|V(G)|=(k-1)+\sum_{i=1}^{r}x_{i}=\sum_{i=1}^{r}n_{i}=n$, and
$\displaystyle|E(G)|=$ $\displaystyle(k-1)(n-n_{1})+\sum_{1\leq\ell<j\leq
t-1}x_{P_{\ell}}\cdot x_{P_{j}}$ $\displaystyle=$
$\displaystyle(k-1)(n-n_{1})+f_{t}(n_{1}-(k-1),n_{2},\dots,n_{r}).$
We can see $G$ is a subgraph of $K_{n_{1},n_{2},\dots,n_{r}}$ and $G-V_{0}$ is
a complete $(t-1)$-partite graph. So any copy of $K_{t}$ in $G$ must contain
at least one vertex of $V_{0}$. Since $|V_{0}|=k-1$, $G$ contains no copy of
$kK_{t}$. Thus
ex$(K_{n_{1},n_{2},\dots,n_{r}},kK_{t})\geq(k-1)(n-n_{1})+f_{t}(n_{1}-(k-1),n_{2},\dots,n_{r})$.
This completes the proof of Theorem 1.7.
## 4 Proof of Theorem 1.8
Recall that, in this section, $r,n_{1},n_{2},\dots,n_{r}$ are integers with
$r\geq 4$, $10k-4\leq n_{1}+4k\leq\min_{i\geq 2}\\{n_{i}\\}$, and
$\sum_{i=1}^{r}n_{i}=n$. By Theorem 1.7, it suffices to prove
$\displaystyle
ex(K_{n_{1},n_{2},\dots,n_{r}},kK_{3})\leq(k-1)(n-n_{1})+f_{3}(n_{1}-(k-1),n_{2},\dots,n_{r}).$
(4)
By way of contradiction, we suppose that $k$ is the minimum positive integer
such that
ex$(K_{n_{1},n_{2},\dots,n_{r}},kK_{3})>(k-1)(n-n_{1})+f_{3}(n_{1}-(k-1),n_{2},\dots,n_{r})$.
Thus, there exists a graph $G$, as a spanning subgraph of
$K_{n_{1},n_{2},\dots,n_{r}}$, containing no copy of $kK_{3}$ and satisfying
$\displaystyle
e(G)=ex(K_{n_{1},n_{2},\dots,n_{r}},kK_{3})>(k-1)(n-n_{1})+f_{3}(n_{1}-(k-1),n_{2},\dots,n_{r}).$
(5)
Moreover, we use $V_{1},V_{2},\dots,V_{r}$ to denote the parts of $G$ , where
$|V_{i}|=n_{i}$ for each $i\in[r]$. It is worth noting that the inequality (4)
holds for $k=1$ by Theorem 1.1. Hence, we may assume $k\geq 2$.
###### Claim 1
$G-\\{v\\}$ contains at least one copy of $(k-1)K_{3}$ for each $v\in V(G)$.
Proof. By contradiction, suppose that there is a vertex $v$ in $V(G)$ such
that $G-\\{v\\}$ contains no copy of $(k-1)K_{3}$. Suppose that $v\in
V_{\ell}$ for some $\ell\in[r]$. Let $n_{\ell}^{\prime}=n_{\ell}-1$,
$n_{j}^{\prime}=n_{j}$ for any $j\neq\ell$, and $n^{\prime}=n-1$. Then
$10(k-1)-4\leq n^{\prime}_{1}+4(k-1)\leq\min_{i\geq 2}\\{n^{\prime}_{i}\\}$.
Moreover, $G-\\{v\\}\subseteq
K_{n^{\prime}_{1},n^{\prime}_{2},\dots,n^{\prime}_{r}}$ by the construction.
Because $G-\\{v\\}$ contains no copy of $(k-1)K_{3}$, we obtain
$e(G-\\{v\\})\leq
ex(K_{n^{\prime}_{1},n^{\prime}_{2},\dots,n^{\prime}_{r}},(k-1)K_{3})$. By the
minimality of $k$, we have
$\displaystyle e(G)=$ $\displaystyle e(G-\\{v\\})+d_{G}(v)$
$\displaystyle\leq$ $\displaystyle
ex(K_{n^{\prime}_{1},n^{\prime}_{2},\dots,n^{\prime}_{r}},(k-1)K_{3})+d_{G}(v)$
$\displaystyle\leq$
$\displaystyle(k-2)(n^{\prime}-n_{1}^{\prime})+f_{3}(n_{1}^{\prime}-(k-2),n_{2}^{\prime},n_{3}^{\prime},\dots,n_{r}^{\prime})+d_{G}(v).$
(6)
If $v\in V_{1}$, then $n_{1}^{\prime}=n_{1}-1$ and
$d_{G}(v)=|N_{G}(v)|\leq|V(G)\setminus V_{1}|=n-n_{1}$ since $G\subseteq
K_{n_{1},n_{2},\dots,n_{r}}$. By the inequality (4), we have
$\displaystyle e(G)\leq$
$\displaystyle(k-2)(n-1-n_{1}+1)+f_{3}(n_{1}-1-(k-2),n_{2},n_{3},\dots,n_{r})+n-n_{1}$
$\displaystyle=$
$\displaystyle(k-1)(n-n_{1})+f_{3}(n_{1}-(k-1),n_{2},n_{3},\dots,n_{r}),$
which contradicts the inequality (5). Hence, we may assume $v\notin V_{1}$.
Then $d_{G}(v)\leq n-n_{\ell}$, $n_{\ell}^{\prime}=n_{\ell}-1$, and
$n_{i}^{\prime}=n_{i}$ for any $i\neq\ell$. Note that $n_{\ell}-1\geq
n_{1}+4k-1\geq n_{1}-k+2$. By the inequality (4) and Proposition 2.2, we have
$\displaystyle e(G)\leq$
$\displaystyle(k-2)(n-1-n_{1})+f_{3}(\\{n_{1}-(k-2),n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\ell}\\}\cup\\{n_{\ell}-1\\})+n-n_{\ell}$
$\displaystyle\leq$
$\displaystyle(k-2)(n-1-n_{1})+(f_{3}(n_{1}-(k-1),n_{2},n_{3},\dots,n_{r})+n_{\ell}-(n_{1}-k+2))+n-n_{\ell}$
$\displaystyle{=}$
$\displaystyle(k-1)(n-n_{1})-k+2+n_{1}+f_{3}(n_{1}-(k-1),n_{2},n_{3},\dots,n_{r})-(n_{1}-k+2)$
$\displaystyle=$
$\displaystyle(k-1)(n-n_{1})+f_{3}(n_{1}-(k-1),n_{2},n_{3},\dots,n_{r}),$
which is a contradiction to the inequality (5). This completes the proof of
Claim 1.
Since $k\geq 2$, the graph $G$ contains at least one triangle by Claim 1.
Arbitrarily choose a triangle $K^{*}_{3}$ in $G$ with vertices
$u_{1},u_{2},u_{3}$. Assume that $u_{1}\in V_{\alpha}$, $u_{2}\in V_{\eta}$,
and $u_{3}\in V_{\xi}$, where $\alpha,\eta,\xi$ are three distinct integers in
$[r]$. Moreover, let $S=\\{\alpha,\eta,\xi\\}$, $n^{\prime}_{\ell}=n_{\ell}-1$
for any $\ell\in S$, $n_{j}^{\prime}=n_{j}$ for any $j\in[r]\setminus S$, and
$n^{\prime}=\sum_{i=1}^{r}n^{\prime}_{i}$. Notice that
$n^{\prime}=\sum_{i=1}^{r}n_{i}-3=n-3$ and $G-\\{u_{1},u_{2},u_{3}\\}\subseteq
K_{n_{1}^{\prime},n_{2}^{\prime},\dots,n_{r}^{\prime}}$.
###### Claim 2
$e(G)\leq(k-2)(n^{\prime}-n^{\prime}_{1})+f_{3}(n^{\prime}_{1}-(k-2),n_{2}^{\prime},\dots,n_{r}^{\prime})+\sum_{1\leq
i<j\leq 3}|N_{G}(u_{i})\cap N_{G}(u_{j})|+n-3$.
Proof. Note that $G-\\{u_{1},u_{2},u_{3}\\}$ contains no copy of $(k-1)K_{3}$
since $G$ contains no copy of $kK_{3}$. Moreover, we have $10(k-1)-4\leq
n^{\prime}_{1}+4(k-1)\leq\min_{i\geq 2}\\{n^{\prime}_{i}\\}$ . By the
minimality of $k$, we know $e(G-\\{u_{1},u_{2},u_{3}\\})\leq
ex(K_{n_{1}^{\prime},n_{2}^{\prime},\dots,n_{r}^{\prime}},(k-1)K_{3})\leq(k-2)(n^{\prime}-n^{\prime}_{1})+f_{3}(n^{\prime}_{1}-(k-2),n_{2}^{\prime},\dots,n_{r}^{\prime})$.
Thus,
$\displaystyle e(G)=$ $\displaystyle
e(G-\\{u_{1},u_{2},u_{3}\\})+\sum_{i=1}^{3}|N_{G}(u_{i})|-3$
$\displaystyle\leq$
$\displaystyle(k-2)(n^{\prime}-n^{\prime}_{1})+f_{3}(n^{\prime}_{1}-(k-2),n_{2}^{\prime},\dots,n_{r}^{\prime})+\sum_{i=1}^{3}|N_{G}(u_{i})|-3.$
(7)
By the Principle of Inclusion-Exclusion,
$\displaystyle\sum_{i=1}^{3}|N_{G}(u_{i})|=$
$\displaystyle|\bigcup_{i=1}^{3}N_{G}(u_{i})|+\sum_{1\leq i<j\leq
3}|N_{G}(u_{i}))\cap N_{G}(u_{j})|-|\bigcap_{i=1}^{3}N_{G}(u_{i})|$
$\displaystyle\leq$ $\displaystyle|V(G)|+\sum_{1\leq i<j\leq
3}|N_{G}(u_{i})\cap N_{G}(u_{j})|$ $\displaystyle=$ $\displaystyle
n+\sum_{1\leq i<j\leq 3}|N_{G}(u_{i})\cap N_{G}(u_{j})|.$ (8)
Combining inequalities (4) and (4), we see
$\displaystyle
e(G)\leq(k-2)(n^{\prime}-n^{\prime}_{1})+f_{3}(n^{\prime}_{1}-(k-2),n_{2}^{\prime},\dots,n_{r}^{\prime})+\sum_{1\leq
i<j\leq 3}|N_{G}(u_{i})\cap N_{G}(u_{j})|+n-3{.}$
We finish our proof in the following two cases.
Case 1. $|N_{G}(u_{i})\cap N_{G}(u_{j})|\leq 6(k-1)$ for every two distinct
vertices $u_{i}$ and $u_{j}$ in $\\{u_{1},u_{2},u_{3}\\}$.
Recall that $G-\\{u_{1},u_{2},u_{3}\\}\subseteq
K_{n_{1}^{\prime},n_{2}^{\prime},\dots,n_{r}^{\prime}}$. We shall consider the
two cases when $1\in\\{\alpha,\eta,\xi\\}$ and $1\notin\\{\alpha,\eta,\xi\\}$.
First, assume $1\in\\{\alpha,\eta,\xi\\}$. Without loss of generality, say
$1=\alpha$. Then $n_{1}^{\prime}=n_{1}-1$ and
$\\{n_{2}^{\prime},n_{3}^{\prime},\dots,n_{r}^{\prime}\\}=\\{n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\eta},n_{\xi}\\}\cup\\{n_{\eta}-1,n_{\xi}-1\\}$.
By Claim 2, we have
$\displaystyle e(G)\leq$ $\displaystyle(k-2)(n-3-n_{1}+1)+18(k-1)+n-3$
$\displaystyle+f_{3}(\\{n_{1}-1-k+2,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\eta},n_{\xi}\\}\cup\\{n_{\eta}-1,n_{\xi}-1\\})$
$\displaystyle=$ $\displaystyle(k-1)(n-n_{1})-2(k-2)-(n-n_{1})+18(k-1)+n-3$
$\displaystyle+f_{3}(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\eta},n_{\xi}\\}\cup\\{n_{\eta}-1,n_{\xi}-1\\})$
$\displaystyle=$ $\displaystyle(k-1)(n-n_{1})+n_{1}+16(k-1)-1$
$\displaystyle+f_{3}(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\eta},n_{\xi}\\}\cup\\{n_{\eta}-1,n_{\xi}-1\\}).$
Now, we assume $1\notin\\{\alpha,\eta,\xi\\}$. Then
$n_{i}^{\prime}=n_{i}-1>n_{1}$ for any $i\in\\{\alpha,\eta,\xi\\}$, and
$n_{j}^{\prime}=n_{j}$ for any $j\in[r]\setminus\\{\alpha,\eta,\xi\\}$. By
Claim 2,
$\displaystyle e(G)\leq$ $\displaystyle(k-2)(n-3-n_{1})+18(k-1)+n-3$
$\displaystyle+f_{3}(\\{n_{1}-k+2,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\alpha},n_{\eta},n_{\xi}\\}\cup\\{n_{\alpha}-1,n_{\eta}-1,n_{\xi}-1\\})$
$\displaystyle=$ $\displaystyle(k-1)(n-n_{1})+15(k-1)+n_{1}$
$\displaystyle+f_{3}(\\{n_{1}-k+2,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\alpha},n_{\eta},n_{\xi}\\}\cup\\{n_{\alpha}-1,n_{\eta}-1,n_{\xi}-1\\}).$
(9)
Note that $n_{\alpha}-1\geq n_{1}-k+2$. By Proposition 2.2,
$\displaystyle f_{3}$
$\displaystyle(\\{n_{1}-k+2,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\alpha},n_{\eta},n_{\xi}\\}\cup\\{n_{\alpha}-1,n_{\eta}-1,n_{\xi}-1\\})$
$\displaystyle\leq
f_{3}(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\eta},n_{\xi}\\}\cup\\{n_{\eta}-1,n_{\xi}-1\\})+n_{\alpha}-(n_{1}-k+2).$
(10)
Thus, combining inequalities (4) and (4),
$\displaystyle e(G)\leq$
$\displaystyle(k-1)(n-n_{1})+15(k-1)+n_{1}+n_{\alpha}-(n_{1}-k+2)$
$\displaystyle+f_{3}(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\eta},n_{\xi}\\}\cup\\{n_{\eta}-1,n_{\xi}-1\\})$
$\displaystyle=$ $\displaystyle(k-1)(n-n_{1})+16(k-1)+n_{\alpha}-1$
$\displaystyle+f_{3}(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\eta},n_{\xi}\\}\cup\\{n_{\eta}-1,n_{\xi}-1\\}).$
(11)
In both cases ($1\in\\{\alpha,\eta,\xi\\}$ and
$1\notin\\{\alpha,\eta,\xi\\}$), we have inequality (4) holds. Since
$n_{1}-k+1+2\leq\min_{i\geq 2}\\{n_{i}\\}$, by Proposition 2.3 and the
inequality (4),
$\displaystyle e(G)\leq$
$\displaystyle(k-1)(n-n_{1})+n_{\alpha}+16(k-1)-1+f_{3}(n_{1}-k+1,n_{2},n_{3},\dots,n_{r})-(n-k+1)$
$\displaystyle+{\max\\{n_{1}-(k-1)+2,n_{\eta}-n_{1}+(k-1)+1\\}}$
$\displaystyle=$
$\displaystyle(k-1)(n-n_{1})+f_{3}(n_{1}-k+1,n_{2},n_{3},\dots,n_{r})$
$\displaystyle+{17k-18-n+\max\\{n_{\alpha}+n_{1}-k+3,n_{\eta}+n_{\alpha}-n_{1}+k\\}}.$
Since $n-n_{\alpha}-n_{1}\geq(r-2)n_{2}\geq 2n_{2}\geq 20k-8$ and
$n+n_{1}-n_{\eta}-n_{\alpha}\geq(r-1)n_{1}\geq 3n_{1}\geq 18k-12$, we have
$e(G)\leq(k-1)(n-n_{1})+f_{3}(n_{1}-k+1,n_{2},n_{3},\dots,n_{r}),$
which contradicts the inequality (5).
Case 2. There exist two distinct vertices $u_{i}$ and $u_{j}$ in
$\\{u_{1},u_{2},u_{3}\\}$ such that $|N_{G}(u_{i})\cap N_{G}(u_{j})|>6(k-1)$.
Without loss of generality, we may assume $|N_{G}(u_{2})\cap N_{G}(u_{3})|\geq
6(k-1)+1$. Then we have the following claim.
###### Claim 3
For any vertex set $A\subseteq V(G)$, if $|N_{G}(u_{2})\cap N_{G}(u_{3})\cap
A|\geq 6(k-1)+1$, then there is a vertex $u_{0}$ with $u_{0}\in A\cap
N_{G}(u_{2})\cap N_{G}(u_{3})$ such that $|N_{G}(u_{0})\cap N_{G}(u_{i})|\leq
3(k-1)$ for any $i\in\\{2,3\\}$.
Proof. By Claim 1, denote by $\mathcal{T}_{i}$ a copy of $(k-1)K_{3}$ in
$G\setminus\\{u_{i}\\}$ for any $i\in\\{2,3\\}$. Since $|N_{G}(u_{2})\cap
N_{G}(u_{3})\cap A|\geq 6(k-1)+1$ and
$|\mathcal{T}_{1}|=|\mathcal{T}_{2}|=3(k-1)$, there is a vertex, say $u_{0}$,
such that $u_{0}\in(N_{G}(u_{2})\cap N_{G}(u_{3})\cap
A)\setminus(\mathcal{T}_{1}\cup\mathcal{T}_{2})$. If $|N_{G}(u_{0})\cap
N_{G}(u_{i})|\geq 3(k-1)+1$ for some $i\in\\{2,3\\}$, then there exists a
vertex $w_{i}\in(N_{G}(u_{0})\cap N_{G}(u_{i}))\setminus\mathcal{T}_{i}$.
Notice that $\\{w_{i},u_{0},u_{i}\\}\cap\mathcal{T}_{i}=\emptyset$ and
$u_{0}u_{i}w_{i}u_{0}$ is a triangle. So the disjoint union of
$u_{0}u_{i}w_{i}u_{0}$ and $\mathcal{T}_{i}$ is a copy of $kK_{3}$ in $G$, a
contradiction to the choice of $G$. Thus, $|N_{G}(u_{0})\cap N_{G}(u_{i})|\leq
3(k-1)$ for any $i\in\\{2,3\\}$.
By setting $A=V(G)$ in Claim 3, there exists a vertex $u_{0}$ with $u_{0}\in
N_{G}(u_{2})\cap N_{G}(u_{3})$ such that $|N_{G}(u_{0})\cap N_{G}(u_{i})|\leq
3(k-1)$ for any $i\in\\{2,3\\}$. Note that $u_{0}u_{2}u_{3}u_{0}$ is a
triangle in $G$. Recall $u_{2}\in V_{\eta}$ and $u_{3}\in V_{\xi}$. Assume
$u_{0}\in V_{\alpha^{\prime}}$.
According to the arbitrariness of the choice of $K^{*}_{3}$ in $G$, we know
that Claims 2 and 3 are applicable to the triangle $u_{0}u_{2}u_{3}u_{0}$ by
replacing $K^{*}_{3}$ with $u_{0}u_{2}u_{3}u_{0}$ and replacing $S$ with
$\\{\alpha^{\prime},\eta,\xi\\}$. The remaining proof is divided into two
subcases.
Subcase 1. $1\in\\{\alpha^{\prime},\eta,\xi\\}$.
Note that $n_{1}^{\prime}=n_{1}-1$ and $n^{\prime}=n-3$. Moreover, we have
$|N_{G}(u_{2})\cap N_{G}(u_{3})|\leq n-n_{\eta}-n_{\xi}$ by $u_{2}\in
V_{\eta}$ and $u_{3}\in V_{\xi}$. Let
$\\{1,s,\ell\\}=\\{\alpha^{\prime},\eta,\xi\\}$. Due to
$\\{s,\ell\\}\cap\\{\eta,\xi\\}\neq\emptyset$, without loss of generality, we
may assume $s\in\\{\eta,\xi\\}$. Since $|N_{G}(u_{0})\cap N_{G}(u_{i})|\leq
3(k-1)$ for any $i\in\\{2,3\\}$, by Claim 2, we know
$\displaystyle e(G)\leq$
$\displaystyle(k-2)(n-3-n_{1}+1)+(6(k-1)+|N_{G}(u_{2})\cap N_{G}(u_{3})|)+n-3$
$\displaystyle+f_{3}(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{s},n_{\ell}\\}\cup\\{n_{s}-1,n_{\ell}-1\\})$
$\displaystyle\leq$
$\displaystyle(k-1)(n-n_{1})+4k-5+n+n_{1}-n_{\eta}-n_{\xi}$
$\displaystyle+f_{3}(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{s},n_{\ell}\\}\cup\\{n_{s}-1,n_{\ell}-1\\})$
$\displaystyle\leq$
$\displaystyle{(k-1)(n-n_{1})+4k-5+n+n_{1}-n_{\eta}-n_{\xi}+f_{3}(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\})}$
$\displaystyle{-(n-(k-1))+\max\\{(n_{1}-k+1)+2,n_{s}-(n_{1}-k+1)+1\\}}$
$\displaystyle\leq$
$\displaystyle{(k-1)(n-n_{1})+f_{3}(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\})}$
$\displaystyle{+5k-6+n_{1}-n_{\eta}-n_{\xi}+\max\\{n_{1}-k+3,n_{s}-n_{1}+k+2\\}}$
$\displaystyle\leq$
$\displaystyle{(k-1)(n-n_{1})+f_{3}(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\}),}$
where the third inequality holds by Proposition 2.3 and the last inequality
holds by $4k-3+2n_{1}\leq n_{\eta}+n_{\xi}$ and $6k-4+n_{s}\leq
n_{\eta}+n_{\xi}$.
Subcase 2. $1\notin\\{\alpha^{\prime},\eta,\xi\\}$.
By setting $A=V_{1}$ in Claim 3, if $|N(u_{2})\cap N(u_{3})\cap V_{1}|\geq
6(k-1)+1$, then there exists a vertex $u^{\prime}_{0}$ with $u^{\prime}_{0}\in
N_{G}(u_{2})\cap N_{G}(u_{3})\cap V_{1}$ such that $|N_{G}(u^{\prime}_{0})\cap
N_{G}(u_{i})|\leq 3(k-1)$ for any $i\in\\{2,3\\}$. Note that
$u^{\prime}_{0}u_{2}u_{3}u^{\prime}_{0}$ is also a triangle in $G$. This cases
reduces to Subcase 1. Hence, we may assume $|N(u_{2})\cap N(u_{3})\cap
V_{1}|\leq 6(k-1)$. Then
$\displaystyle|N_{G}(u_{2})\cap N_{G}(u_{3})|$
$\displaystyle=|N_{G}(u_{2})\cap N_{G}(u_{3})\cap(V(G)\setminus
V_{1})|+|N_{G}(u_{2})\cap N_{G}(u_{3})\cap V_{1}|$ $\displaystyle\leq
n-n_{\eta}-n_{\xi}-n_{1}+6(k-1).$
Recall that $n^{\prime}=n-3$, $n^{\prime}_{1}=n_{1}$, and $|N_{G}(u_{0})\cap
N_{G}(u_{i})|\leq 3(k-1)$ for any $i\in\\{2,3\\}$. By Claim 2,
$\displaystyle e(G)\leq$
$\displaystyle(k-2)(n-3-n_{1})+(6(k-1)+n-n_{\eta}-n_{\xi}-n_{1}+6(k-1))+n-3$
$\displaystyle+f_{3}(\\{n_{1}-k+2,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\alpha^{\prime}},n_{\eta},n_{\xi}\\}\cup\\{n_{\alpha^{\prime}}-1,n_{\eta}-1,n_{\xi}-1\\})$
$\displaystyle=$ $\displaystyle(k-1)(n-n_{1})+9k-9+n-n_{\eta}-n_{\xi}$
$\displaystyle+f_{3}(\\{n_{1}-k+2,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\alpha^{\prime}},n_{\eta},n_{\xi}\\}\cup\\{n_{\alpha^{\prime}}-1,n_{\eta}-1,n_{\xi}-1\\}).$
(12)
Since $n_{1}+4k\leq\min_{i\geq 2}\\{n_{i}\\}$, we know
$n_{1}-k+3\leq\min_{i\geq 2}\\{n_{i}\\}$. By Proposition 2.2 and the
inequality (4),
$\displaystyle e(G)\leq$
$\displaystyle(k-1)(n-n_{1})+9k-9+n-n_{\eta}-n_{\xi}+n_{\xi}-(n_{1}-k+2)$
$\displaystyle+f_{3}(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\alpha^{\prime}},n_{\eta}\\}\cup\\{n_{\alpha^{\prime}}-1,n_{\eta}-1\\})$
$\displaystyle=$ $\displaystyle(k-1)(n-n_{1})+10k-11+n-n_{\eta}-n_{1}$
$\displaystyle+f_{3}(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\alpha^{\prime}},n_{\eta}\\}\cup\\{n_{\alpha^{\prime}}-1,n_{\eta}-1\\}).$
(13)
Proposition 2.3 implies that
$\displaystyle f_{3}$
$\displaystyle(\\{n_{1}-k+1,n_{2},n_{3},\dots,n_{r}\\}\setminus\\{n_{\alpha^{\prime}},n_{\eta}\\}\cup\\{n_{\alpha^{\prime}}-1,n_{\eta}-1\\})$
$\displaystyle\leq
f_{3}(n_{1}-k+1,n_{2},\dots,n_{r})+{\max\\{n_{1}-(k-1)+2,n_{\eta}-(n_{1}-k+1)+1\\}-n+k-1.}$
(14)
Since $n_{1}\geq 6k-4$ and $n_{\eta}\geq n_{1}+4k\geq 10k-4$, combining
inequalities (4) and (4), we have
$\displaystyle e(G)\leq$ $\displaystyle(k-1)(n-n_{1})+10k-11+n-n_{\eta}-n_{1}$
$\displaystyle+f_{3}(n_{1}-k+1,n_{2},\dots,n_{r})+{\max\\{n_{1}-k+3,n_{\eta}-n_{1}+k\\}-n+k-1.}$
$\displaystyle=$
$\displaystyle(k-1)(n-n_{1})+f_{3}(n_{1}-k+1,n_{2},\dots,n_{r})+11k-12$
$\displaystyle-\max\\{n_{\eta}+k-3,2n_{1}-k\\}$ $\displaystyle\leq$
$\displaystyle(k-1)(n-n_{1})+f_{3}(n_{1}-k+1,n_{2},\dots,n_{r}),$
which contradicts to the inequality (5). This completes the proof of the
inequality (4), and so Theorem 1.8 follows.
Acknowledgements. The author would like to thank the editor and the two
referees very much for their valuable suggestions and comments. Junxue Zhang
was partially supported by the National Natural Science Foundation of China
(Nos. 12161141006 and 12111540249), the Natural Science Foundation of Tianjin
(Nos. 20JCJQJC00090 and 20JCZDJC00840), the Fundamental Research Funds for the
Central Universities, Nankai University (No. 63223063).
## References
* [1] B. Bollobás, P. Erdős, E. G. Straus, Complete subgraphs of chromatic graphs and hypergraphs, Utilitas Math. 6 (1974), 343–347.
* [2] H. Chen, X. Li, J. Tu, Complete solution for the rainbow numbers of matchings, Discrete Math. 309 (2009), 3370–3380.
* [3] J. De Silva, K. Heysse, A. Kapilow, A. Schenfisch, M. Young, Turán numbers of vertex-disjoint cliques in $r$-partite graphs, Discrete Math. 341 (2018), 492–496.
* [4] J. De Silva, K. Heysse, M. Young, Rainbow number for matchings in $r$-partite graphs, preprint.
* [5] P. Erdős, über ein Extremalproblem in der Graphentheorie (German), Arch. Math. (Basel) 13 (1962), 122–127.
* [6] P. Erdős, T. Gallai, On maximal paths and circuits of graphs, Acta Math. Acad. Sci. Hungar. 10 (1959), 337–356.
* [7] J. Han, Y. Zhao, Turán number of disjoint triangles in 4-partite graphs, Electron. J. Combin. 29 (2022), #P2.35.
* [8] W. Mantel, Problem 28, Wiskundige Opgaven 10 (1907), 60–61.
* [9] J. W. Moon, On independent complete subgraphs in a graph, Canad. J. Math. 20 (1968), 95–102.
* [10] M. Simonovits, A method for solving extremal problems in graph theory, Theory of Graphs, Proc. Colloq. Tihany, (1966), (P. Erdős and G. Katona, Eds.), 279–319, Acad. Press, New York, 1968.
* [11] P. Turán, On an extremal problem in graph theory, (Hungarian), Mat. Fiz. Lapok 48 (1941), 436–452.
* [12] D. West, Introduction to Graph Theory, Prentice hall, Upper Saddle River, 2001.
|
# Generalizing Yee’s method: Scalable geometric higher-order
FEEC algorithms for Maxwell’s equations on an unstructured mesh
Alexander S. Glasser Hong Qin
###### Abstract
The Yee algorithm for electromagnetic simulations is widely known to have many
advantages, including the following crucial two: (i) Its calculations are
local and therefore efficiently parallelizable—enabling simulations that
capitalize on the speed and scalability of high-performance computing
architecture. (ii) Yee’s method faithfully preserves the symplectic geometry
of Maxwell’s equations, improving its accuracy in long-time numerical
simulations. Whereas previous geometric generalizations of Yee’s method have
sacrificed its scalability, in this article the Yee algorithm is generalized
to higher order and unstructured meshes in a manner that fully preserves both
its scalability and geometric naturalness. This generalization is achieved by
prioritizing the locality of the algorithm, reflecting the physical locality
of Maxwell’s equations. Specifically, we demonstrate that Yee’s method is but
a special case of a larger family of symplectic, finite element exterior
calculus (FEEC) methods that use scalable, local approximations of mass
matrices. We discuss the numerical advantages of this family of methods, which
we call _scalable FEEC_ (SFEEC) methods.
[1]Princeton Plasma Physics Laboratory, Princeton University, Princeton, New
Jersey 08543 [2]Department of Astrophysical Sciences, Princeton University,
Princeton, New Jersey 08544
## 1 Introduction
The Yee algorithm [1, 2]—alternatively, the finite difference time domain
(FDTD) method—defines electromagnetic fields on a cubic mesh. It associates to
each edge a component of the electric field $\mathbf{E}$, and to each face a
component of the magnetic field $\mathbf{B}$. This discretization reflects a
natural geometric description of Maxwell’s equations, in which one defines
${\mathbf{E}\in\Lambda^{1}(\mathbb{R}^{3})}$ as a differential 1-form on
$\mathbb{R}^{3}$ and ${\mathbf{B}\in\Lambda^{2}(\mathbb{R}^{3})}$ as a
differential 2-form on $\mathbb{R}^{3}$. In this way, Yee’s method employs a
technique adopted in many _structure-preserving algorithms_ [3], wherein
differential $k$-forms are discretized by associating them with
$k$-dimensional features of a mesh [4, 5, 6, 7].
Structure-preserving algorithms have been widely adopted in many subfields of
computational physics, including gravitational simulations [8, 9, 10, 11, 12],
geophysics [13, 14] and plasma physics [15, 16, 17, 18, 19, 20, 21, 22, 23].
Such algorithms generally derive from variational principles or Hamiltonian
systems. As a result, they preserve essential mathematical features of their
underlying physical systems, including symplectic structure, topology,
symmetries, and conservation laws. These properties contribute to the
numerical fidelity of structure-preserving algorithms, especially in long-time
numerical simulations. Despite Yee’s omission of any overt Lagrangian or
Hamiltonian formulations in his original work [1], Yee’s method (apparently
serendipitously) is one of the most historically successful examples of a
structure-preserving algorithm [24].
More recently, the methodology of structure-preserving algorithms has led to
the development of advanced algorithms for the simulation of plasmas, whose
electromagnetic fields are constructed using the formalism of finite element
exterior calculus (FEEC) [6, 7, 20, 23]. Although such methods are in
principle readily generalizable to unstructured meshes and high order finite
elements, they lack the computational efficiency and scalability of Yee’s
method. In particular, the time evolution of electromagnetic fields in these
FEEC methods requires communication between all nodes of a simulation, thereby
destroying their parallelism. As we shall discuss and address, this problem
arises because sparse finite element mass matrices generally have dense
inverses.
On the other hand, there have also been numerous efforts (e.g. [25, 26, 27,
28]) to generalize Yee’s method using scalable finite element methods, called
finite element time domain (FETD) methods. Such efforts leverage the crucial
technique of sparse approximate inverse (SPAI) mass matrices, and preserve the
scalability of Yee’s method. However, these methods’ preservation of
symplectic structure has not been established.
Motivated by the desire to overcome the scalability limitation of structure-
preserving FEEC plasma methods, and to guarantee the structure preservation of
FETD methods, in this article we develop _scalable FEEC_ (SFEEC) methods, a
family of symplectic finite element methods for electromagnetic fields that
includes Yee’s method as a special case. SFEEC methods enable the higher order
simulation of electromagnetic fields on structured and unstructured meshes in
a manner that preserves the two aforementioned crucial advantages of Yee’s
method, namely: (i) its scalability on modern architectures; and (ii) its
symplectic geometry (and the resulting conservation of electric charge and
Gauss’ law).
From a finite element point of view, the scalability of Yee’s method will be
reframed as a result of its simplified (or ‘pruned’) approximation of finite
element mass matrices. To retain their scalability, SFEEC methods employ a
comparable, if more flexible treatment of mass matrices. Relative to Yee’s
method, SFFEC methods afford a greater flexibility to improve algorithmic
accuracy without sacrificing scalability. Their use of finite elements and the
FEEC formalism further enables their viability on more general meshes.
While Yee’s algorithm has been generalized numerous times in the literature,
including via higher order and finite element schemes (e.g. [24, 25, 27, 28,
29, 30, 31, 32]), no such generalization is known to us that simultaneously
affords higher order accuracy and scalability while ensuring the geometric
structure-preservation of Yee’s method. In particular, the Yee method’s
preservation of symplectic structure is an important aspect of the method’s
stability and long-term accuracy [33]. In this work, we will review this
symplectic structure of Yee’s method and demonstrate its exact preservation in
the SFEEC family of algorithms we define. In so doing, we also offer a means
to overcome the scalability limitations of structure-preserving FEEC
algorithms (e.g. [20, 23]) in a massively parallel, high-performance computing
architecture.
The remainder of this article is organized as follows. In Section 2, the
formalism of finite element exterior calculus (FEEC) [6, 7] and its
discretization of electromagnetic fields will be reviewed. In Section 3, it
will be demonstrated that Yee’s algorithm can be interpreted as an FEEC method
with simplified mass matrices. In Section 4 we will define SFEEC methods,
which extend Yee’s method to higher order finite element schemes on a general
mesh. In Section 5, numerical results will be presented that demonstrate the
improved higher order accuracy of the resulting SFEEC methods relative to
Yee’s method, without sacrificing its scalability. Section 6 will then
summarize and conclude.
## 2 Finite Element Exterior Calculus (FEEC)
for Electromagnetic Simulations
In this section, we briefly review aspects of FEEC [6, 7] and its application
in electromagnetic simulations. We refer the reader to [20, 23] for additional
background. We let $\Lambda^{p}(\mathcal{T}_{h})$ denote a vector space of
finite element differential $p\text{-form}$s on a simplicial or cubical
complex ${\mathcal{T}_{h}\subset\mathbb{R}^{n}}$ (where the subscript $h$
denotes the maximal diameter, or edge length, of any simplex in
$\mathcal{T}_{h}$). For all ${0\leq p\leq n}$, $\Lambda^{p}(\mathcal{T}_{h})$
may be defined as the span of a finite ($N_{p}$-dimensional) basis
${\mbox{\boldmath$\Lambda$}^{p}}$, whose $i^{\text{th}}$ basis element
${\Lambda^{p}_{i}\in\Lambda^{p}(\mathcal{T}_{h})}$ is a piecewise polynomial
$p\text{-form}$ on $\mathcal{T}_{h}$. Such a basis element typically has
support localized to one or more adjacent cells in ${\mathcal{T}_{h}}$. An
arbitrary $p\text{-form}$ ${\mathbf{S}\in\Lambda^{p}(\mathcal{T}_{h})}$ can be
expressed in the $\mbox{\boldmath$\Lambda$}^{p}$ basis as
$\displaystyle\begin{split}\mathbf{S}(\mathbf{x})&=\mathbf{s}\cdot\mbox{\boldmath$\Lambda$}^{p}(\mathbf{x})=s_{i}\Lambda^{p}_{i}(\mathbf{x})\end{split}$
(1)
$\forall$ ${\mathbf{s}\in\mathbb{R}^{N_{p}}}$ and
${\mathbf{x}\in\left|\mathcal{T}_{h}\right|}$, the convex hull of
${\mathcal{T}_{h}}$. (Einstein summation convention is used for the repeated
index in Eq. (1) and hereafter.) Individual components of
${\mathbf{S}\in\Lambda^{p}(\mathcal{T}_{h})}$ will be denoted
$\displaystyle\begin{split}\mathbf{S}(\mathbf{x})_{\mu_{1}\cdots\mu_{p}}&=\mathbf{s}\cdot\mbox{\boldmath$\Lambda$}^{p}(\mathbf{x})_{\mu_{1}\cdots\mu_{p}}=s_{i}\Lambda^{p}_{i}(\mathbf{x})_{\mu_{1}\cdots\mu_{p}}\end{split}$
(2)
where Greek letters denote coordinate indices. For example, given
${\left|\mathcal{T}_{h}\right|\subset\mathbb{R}^{3}}$, the $\mu^{\text{th}}$
component of the 1-form basis element $\Lambda^{1}_{i}(\mathbf{x})$ may be
written as ${\Lambda^{1}_{i}(\mathbf{x})_{\mu}}$ $\forall$
${\mu\in\\{1,2,3\\}}$, such that
${\Lambda^{1}_{i}(\mathbf{x})=\Lambda^{1}_{i}(\mathbf{x})_{\mu}\mathrm{d}x^{\mu}}$.
Because each basis ${\mbox{\boldmath$\Lambda$}^{p}}$ is finite $\forall$
${0\leq p\leq n}$, the exterior derivative
${\mathrm{d}:\Lambda^{p}(\mathcal{T}_{h})\rightarrow\Lambda^{p+1}(\mathcal{T}_{h})}$
of a finite element $p\text{-form}$ on ${\mathcal{T}_{h}}$ can be computed in
the ${\mbox{\boldmath$\Lambda$}^{0},\dots,\mbox{\boldmath$\Lambda$}^{n}}$
bases by straightforward matrix multiplication. A choice of basis for each
${\Lambda^{p}(\mathcal{T}_{h})}$ in three dimensions
(${\mathcal{T}_{h}\subset\mathbb{R}^{3}}$) determines, for example, matrices
that represent the gradient ($\mathbb{G}$), curl ($\mathbb{C}$), and
divergence ($\mathbb{D}$)—as defined in Table 1.
$p$-Form | Abstract $\mathrm{d}$ | Matrix $\mathrm{d}$ | Defined by
---|---|---|---
$\mathbf{S}=\mathbf{s}\cdot\mbox{\boldmath$\Lambda$}^{0}$ | - | - | -
$\mathbf{A}=\mathbf{a}\cdot\mbox{\boldmath$\Lambda$}^{1}$ | $\mathbf{A}=\mathrm{d}\mathbf{S}$ | $\mathbf{a}=\mathbb{G}\mathbf{s}$ | $\mathbb{G}^{T}\mbox{\boldmath$\Lambda$}^{1}=\mathrm{d}\mbox{\boldmath$\Lambda$}^{0}$
$\mathbf{B}=\mathbf{b}\cdot\mbox{\boldmath$\Lambda$}^{2}$ | $\mathbf{B}=\mathrm{d}\mathbf{A}$ | $\mathbf{b}=\mathbb{C}\mathbf{a}$ | $\mathbb{C}^{T}\mbox{\boldmath$\Lambda$}^{2}=\mathrm{d}\mbox{\boldmath$\Lambda$}^{1}$
$\mathbf{C}=\mathbf{c}\cdot\mbox{\boldmath$\Lambda$}^{3}$ | $\mathbf{C}=\mathrm{d}\mathbf{B}$ | ${\mathbf{c}=\mathbb{D}\mathbf{b}}$ | ${\mathbb{D}^{T}\mbox{\boldmath$\Lambda$}^{3}=\mathrm{d}\mbox{\boldmath$\Lambda$}^{2}}$
Table 1: FEEC matrix implementation of $\mathrm{d}$ on
${\mathcal{T}_{h}\subset\mathbb{R}^{3}}$. The property
${\mathrm{d}\circ\mathrm{d}=0}$ implies that
${\mathbb{C}\mathbb{G}=\mathbb{0}}$ and ${\mathbb{D}\mathbb{C}=\mathbb{0}}$.
To apply FEEC in a physical setting, it will also be essential to compute
_mass matrices_ on $\mathcal{T}_{h}$ for each basis
$\mbox{\boldmath$\Lambda$}^{p}$ of finite element $p\text{-form}$s.
Specifically, the mass matrix ${\mathbb{M}_{p}\in\mathbb{R}^{N_{p}\times
N_{p}}}$ for $\mbox{\boldmath$\Lambda$}^{p}$ is defined by
$\displaystyle\begin{split}(\mathbb{M}_{p})_{ij}&=\int_{\left|\mathcal{T}_{h}\right|}\mathrm{d}\mathbf{x}\Big{(}\Lambda^{p}_{i},\Lambda^{p}_{j}\Big{)}_{p}\end{split}$
(3)
where ${(\cdot,\cdot)_{p}}$ denotes the pointwise inner product on
$p\text{-form}$s induced by the metric $g_{\mu\nu}$—specifically
$\displaystyle\begin{split}(\alpha,\beta)_{p}&=\frac{1}{p!}\alpha_{\mu_{1}\cdots\mu_{p}}\beta^{\mu_{1}\cdots\mu_{p}}\\\
&=\frac{1}{p!}\alpha_{\mu_{1}\cdots\mu_{p}}\beta_{\nu_{1}\cdots\nu_{p}}g^{\mu_{1}\nu_{1}}\cdots
g^{\mu_{p}\nu_{p}}.\end{split}$ (4)
For ${p=1,2}$ on $\mathbb{R}^{3}$, for example, Eq. (4) defines the inner
product ${(\alpha,\beta)_{p}}$ as the standard inner product
${\alpha\cdot\beta}$. After all, 1- and 2-forms each have three independent
components in $\mathbb{R}^{3}$, and the metric ${g_{\mu\nu}}$ (and its inverse
${g^{\mu\nu}}$) is given by the Kronecker
delta—${g_{\mu\nu}=\delta_{\mu\nu}}$. We note that ${(\alpha,\beta)_{p}}$ is
symmetric, such that Eq. (3) is symmetric and
${\mathbb{M}_{p}^{T}=\mathbb{M}_{p}}$. Eq. (3) also implies that
$\mathbb{M}_{p}$ is typically sparse (so long as each basis element
$\Lambda_{i}^{p}$ has only local support). Importantly, however, its inverse
matrix $\mathbb{M}_{p}^{-1}$ is typically dense.
To apply the FEEC formalism to electromagnetic simulations, we will first
discretize the magnetic vector potential as an FEEC 1-form, that is,
$\displaystyle\begin{split}\mathbf{A}=\mathbf{a}\cdot\mbox{\boldmath$\Lambda$}^{1}\end{split}$
(5)
with degrees of freedom given by the vector of coefficients
${\mathbf{a}\in\mathbb{R}^{N_{1}}}$. (We work in the temporal gauge, such that
the electric potential $\phi$ vanishes, ${\phi=0}$.) The magnetic field is
then given by a 2-form,
$\displaystyle\begin{split}\mathbf{B}=\mathbf{b}\cdot\mbox{\boldmath$\Lambda$}^{2}=\mathbb{C}\mathbf{a}\cdot\mbox{\boldmath$\Lambda$}^{2}=\mathbf{a}\cdot\mathbb{C}^{T}\mbox{\boldmath$\Lambda$}^{2}=\mathbf{a}\cdot\mathrm{d}\mbox{\boldmath$\Lambda$}^{1}=\mathrm{d}\mathbf{A}\end{split}$
(6)
with ${\mathbf{b}\in\mathbb{R}^{N_{2}}}$, in keeping with the geometry of
Maxwell’s equations and the FEEC implementation of the curl, as in Table 1.
Lastly, the electric field will be defined as a discrete 1-form
$\displaystyle\begin{split}\mathbf{E}=\mathbf{e}\cdot\mathbb{M}_{1}^{-1}\cdot\mbox{\boldmath$\Lambda$}^{1}\end{split}$
(7)
with ${\mathbf{e}\in\mathbb{R}^{N_{1}}}$. Following [23], we use a convention
wherein the coefficients $\mathbf{e}$ determine the electric field
$\mathbf{E}$ with an additional factor of $\mathbb{M}_{1}^{-1}$. We shall
regard the pair ${(\mathbf{a},\mathbf{e})\in\mathbb{R}^{2N_{1}}}$ as defining
the degrees of freedom of the electromagnetic finite element dynamical system.
To describe the dynamics of these discrete fields via Maxwell’s equations, we
first recall the canonical symplectic structure of electromagnetic fields in
the continuum, defined with Poisson bracket and Hamiltonian in SI units as
[34]
$\displaystyle\begin{split}\\{F,G\\}_{\text{EM}}&=\frac{1}{\epsilon_{0}}\int\mathrm{d}\mathbf{x}\left(\frac{\delta
F}{\delta\mathbf{E}}\cdot\frac{\delta G}{\delta\mathbf{A}}-\frac{\delta
G}{\delta\mathbf{E}}\cdot\frac{\delta F}{\delta\mathbf{A}}\right)\\\
H_{\text{EM}}&=\frac{1}{2}\int\mathrm{d}\mathbf{x}\left(\epsilon_{0}\left|\mathbf{E}\right|^{2}+\frac{1}{\mu_{0}}\left|\nabla\times\mathbf{A}\right|^{2}\right).\end{split}$
(8)
We now substitute the FEEC fields
${\mathbf{A}=\mathbf{a}\cdot\mbox{\boldmath$\Lambda$}^{1}}$ and
${\mathbf{E}=\mathbf{e}\cdot\mathbb{M}_{1}^{-1}\cdot\mbox{\boldmath$\Lambda$}^{1}}$
into Eq. (8) to derive (see [23]) the following discrete Poisson bracket
${\\{\cdot,\cdot\\}_{\Delta}}$ and discrete Hamiltonian $H_{\Delta}$
approximating this system,
$\displaystyle\begin{split}\\{F,G\\}_{\Delta}&=\frac{1}{\epsilon_{0}}\left(\frac{\partial
F}{\partial\mathbf{e}}\cdot\frac{\partial
G}{\partial\mathbf{a}}-\frac{\partial
G}{\partial\mathbf{e}}\cdot\frac{\partial F}{\partial\mathbf{a}}\right)\\\
H_{\Delta}&=\frac{1}{2}\left(\epsilon_{0}\mathbf{e}^{T}\mathbb{M}_{1}^{-1}\mathbf{e}+\frac{1}{\mu_{0}}\mathbf{a}^{T}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}\mathbf{a}\right),\end{split}$
(9)
where we have applied
${\mathrm{d}\mbox{\boldmath$\Lambda$}^{1}=\mathbb{C}^{T}\mbox{\boldmath$\Lambda$}^{2}}$
as in Table 1.
Equations of motion are now readily calculated as usual with a Poisson
bracket. We plug the FEEC coefficient vectors $\mathbf{a}$ and $\mathbf{e}$
into ${\\{\cdot,\cdot\\}_{\Delta}}$ with $H_{\Delta}$, to find
$\displaystyle\begin{split}\dot{\mathbf{a}}&=\\{\mathbf{a},H_{\Delta}\\}_{\Delta}=-\mathbb{M}_{1}^{-1}\mathbf{e}\\\
\dot{\mathbf{e}}&=\\{\mathbf{e},H_{\Delta}\\}_{\Delta}=c^{2}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}\mathbf{a}.\end{split}$
(10)
These equations are discrete forms of Maxwell’s equations,
${\dot{\mathbf{A}}=-\mathbf{E}}$ and
${\dot{\mathbf{E}}=c^{2}\nabla\times\nabla\times\mathbf{A}}$, respectively. In
Section 5, the discrete approximation of the curl-of-curl operator
$\displaystyle\begin{split}\nabla\times\nabla\times\approx\mathbb{M}_{1}^{-1}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C},\end{split}$
(11)
whose calculation (until our later simplification) incorporates the dense
matrix $\mathbb{M}_{1}^{-1}$, will be a central focus of our study.
Note, it will also be convenient to rewrite Eq. (10) in terms of the magnetic
field by substituting ${\mathbf{b}=\mathbb{C}\mathbf{a}}$ (the discrete
realization of ${\mathbf{B}=\nabla\times\mathbf{A}}$) to find
$\displaystyle\begin{split}\dot{\mathbf{b}}&=-\mathbb{C}\mathbb{M}_{1}^{-1}\mathbf{e}\\\
\dot{\mathbf{e}}&=c^{2}\mathbb{C}^{T}\mathbb{M}_{2}\mathbf{b}.\end{split}$
(12)
Because of the gauge symmetry of Maxwell’s equations, Eq. (12) is an
equivalent formulation of Eq. (10) and exactly preserves the physical degrees
of freedom of interest [22, 23].
To algorithmically evolve Eq. (10) in discrete time, we follow [17] (see also
[22]) and define a splitting algorithm that decomposes $H_{\Delta}$ of Eq. (9)
into two pieces:
$\displaystyle\begin{split}H_{\Delta}&=H_{\mathbf{a}}+H_{\mathbf{e}}\\\
\text{where}~{}~{}~{}H_{\mathbf{a}}&=\frac{1}{2\mu_{0}}\mathbf{a}^{T}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}\mathbf{a}\\\
H_{\mathbf{e}}&=\frac{\epsilon_{0}}{2}\mathbf{e}^{T}\mathbb{M}_{1}^{-1}\mathbf{e}.\end{split}$
(13)
These ‘sub-Hamiltonians’ define two different subsystems which may be evolved
separately, that is,
$\displaystyle\begin{split}H_{\mathbf{a}}&\begin{cases}\dot{\mathbf{a}}&=\\{\mathbf{a},H_{\mathbf{a}}\\}_{\Delta}=0\\\
\dot{\mathbf{e}}&=\\{\mathbf{e},H_{\mathbf{a}}\\}_{\Delta}=c^{2}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}\mathbf{a}.\end{cases}\\\
H_{\mathbf{e}}&\begin{cases}\dot{\mathbf{a}}&=\\{\mathbf{a},H_{\mathbf{e}}\\}_{\Delta}=-\mathbb{M}_{1}^{-1}\mathbf{e}\\\
\dot{\mathbf{e}}&=\\{\mathbf{e},H_{\mathbf{e}}\\}_{\Delta}=0.\end{cases}\end{split}$
(14)
We see that $\mathbf{a}$ is constant in the subsystem $H_{\mathbf{a}}$, while
$\mathbf{e}$ is constant in the subsystem $H_{\mathbf{e}}$. As a result, each
subsystem is exactly solvable over a finite time interval $\Delta t$, in
particular,
$\displaystyle\begin{split}H_{\mathbf{a}}:~{}~{}~{}~{}\mathbf{e}(t_{0}+\Delta
t)&=\mathbf{e}(t_{0})+\Delta t\cdot
c^{2}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}\mathbf{a}(t_{0})\\\
H_{\mathbf{e}}:~{}~{}~{}~{}\mathbf{a}(t_{0}+\Delta
t)&=\mathbf{a}(t_{0})-\Delta
t\cdot\mathbb{M}_{1}^{-1}\mathbf{e}(t_{0}).\end{split}$ (15)
By choosing an appropriate sequence of these subsystems’ discrete time
evolutions, a splitting method may be implemented that approximates a timestep
of the entire electromagnetic system $H_{\Delta}$.
A typical example of such a splitting scheme is called Strang splitting [35],
which evolves $\mathbf{a}$ and $\mathbf{e}$ according to the approximation
$\displaystyle\begin{split}\left[\begin{matrix}\mathbf{a}\\\
\mathbf{e}\end{matrix}\right]_{\mathrlap{t_{0}+\Delta
t}}&~{}~{}~{}=\exp\left(\Delta
tH_{\Delta}\right)\cdot\left[\begin{matrix}\mathbf{a}\\\
\mathbf{e}\end{matrix}\right]_{t_{0}}\\\ &\approx\exp\left(\frac{\Delta
t}{2}H_{\mathbf{e}}\right)\exp\left(\Delta
tH_{\mathbf{a}}\right)\exp\left(\frac{\Delta
t}{2}H_{\mathbf{e}}\right)\cdot\left[\begin{matrix}\mathbf{a}\\\
\mathbf{e}\end{matrix}\right]_{t_{0}}.\end{split}$ (16)
Here, we use the notation ${\exp(\Delta tH_{i})}$ to denote a discrete time
mapping—as appears in Eq. (15)—that evolves according to the Hamiltonian
$H_{i}$ for a time interval $\Delta t$. Since $\exp(\Delta tH_{\mathbf{a}})$
and $\exp(\Delta tH_{\mathbf{e}})$ do not commute, the second line of Eq. (16)
is only an approximation of the first. Nevertheless, such a splitting method
is an effective algorithmic implementation of our dynamical system. It is
exactly solvable, accurate to second order, and preserves the symplectic
structure of the system as defined by the Poisson bracket of Eq. (9).
Splitting methods of arbitrarily high accuracy in $\Delta t$ can also be
constructed (see, e.g. [3]).
## 3 Reinterpreting Yee’s Method via FEEC
We now rederive Yee’s method using the finite element formalism of the
previous section. We proceed in three steps: (i) we first choose generalized
Whitney forms as an FEEC basis on a cubical mesh; (ii) we then maximally
simplify the mass matrices of the corresponding electromagnetic algorithm; and
finally (iii) we choose Strang splitting as our method of time evolution.
These three steps will exactly recover Yee’s method, demonstrating its
interpretation as an implementation of the FEEC electromagnetic algorithm
defined in Eq. (15)—with simplified mass matrices.
We begin by considering a cubical lattice
${\mathcal{T}_{h}\subset\mathbb{R}^{3}}$ with lattice spacings
${\\{\Delta_{x},\Delta_{y},\Delta_{z}\\}}$. We make the simplest possible
choice for an FEEC basis on $\mathcal{T}_{h}$, namely, the _generalized
Whitney forms_ , a family of piecewise polynomial finite elements denoted
${Q_{1}^{-}\Lambda^{p}(\mathcal{T}_{h})}$ [36]). This finite element basis is
also usefully defined in [37] Example 5.2.
A generalized Whitney $p\text{-form}$ can be readily understood by its 1-to-1
correspondence with a $p$-dimensional feature of a mesh. In particular, we
define the generalized Whitney $p\text{-form}$ ${\mathcal{W}_{\sigma^{p}}\in
Q_{1}^{-}\Lambda^{p}(\mathcal{T}_{h})}$ associated to a given $p$-face
${\sigma^{p}\subset\mathcal{T}_{h}\subset\mathbb{R}^{n}}$ by requiring that
$\displaystyle\begin{split}\frac{1}{\left|\tau^{p}\right|}\int_{\tau^{p}}\mathcal{W}_{\sigma^{p}}=\begin{cases}1&\tau^{p}=\sigma^{p}\\\
0&\tau^{p}\neq\sigma^{p}.\end{cases}\end{split}$ (17)
In this way, generalized Whitney $p\text{-form}$s are _dual_ to $p$-faces of
$\mathcal{T}_{h}$ via integration. Here, ${\left|\tau^{p}\right|}$ denotes the
$p$-volume of $\tau^{p}$ (defined as 1, length, area, and volume respectively
for ${p=0,1,2,3}$). This normalization is chosen to mimic fields as they are
typically represented in Yee’s method.
A concrete example of this is represented on
${\mathcal{T}_{h}\subset\mathbb{R}^{3}}$ in Fig. 1. The 1-form
${\mathcal{W}_{x_{1}x_{2}}\in Q_{1}^{-}\Lambda^{1}(\mathcal{T}_{h})}$ and the
2-form ${\mathcal{W}_{x_{1}x_{2}x_{4}x_{3}}\in
Q_{1}^{-}\Lambda^{2}(\mathcal{T}_{h})}$ are depicted, defined respectively by
$\displaystyle\begin{split}\mathcal{W}_{x_{1}x_{2}}&=\left(1-\tfrac{y}{\Delta_{y}}\right)\left(1-\tfrac{z}{\Delta_{z}}\right)\mathrm{d}x\\\
\mathcal{W}_{x_{1}x_{2}x_{4}x_{3}}&=\left(1-\tfrac{z}{\Delta_{z}}\right)\mathrm{d}x\wedge\mathrm{d}y.\end{split}$
(18)
As in Eq. (17), $\mathcal{W}_{x_{1}x_{2}}$ is the least-order piecewise
polynomial 1-form satisfying
${\int_{x_{1}x_{2}}\mathcal{W}_{x_{1}x_{2}}=\left|x_{1}x_{2}\right|}$ and
${\int_{\sigma^{1}}\mathcal{W}_{x_{1}x_{2}}=0}$ $\forall$ ${\sigma^{1}\neq
x_{1}x_{2}}$. $\mathcal{W}_{x_{1}x_{2}x_{4}x_{3}}$ is analogously defined so
that its integral satisfies
${\int_{x_{1}x_{2}x_{4}x_{3}}\mathcal{W}_{x_{1}x_{2}x_{4}x_{3}}=\left|x_{1}x_{2}x_{4}x_{3}\right|}$
and vanishes on any other face. In later calculations, it will also be useful
to explicitly define the following Whitney 2-form associated to the face
${x_{1}x_{5}x_{6}x_{2}}$ in Fig. 1,
$\displaystyle\begin{split}\mathcal{W}_{x_{1}x_{5}x_{6}x_{2}}&=\left(1-\tfrac{y}{\Delta_{y}}\right)\mathrm{d}z\wedge\mathrm{d}x.\end{split}$
(19)
0ptOlFFL
Figure 1: The generalized Whitney 1-forms $\mathcal{W}_{x_{1}x_{2}}$ and
$\mathcal{W}_{x_{1}x_{2}x_{4}x_{3}}$ are schematically depicted, respectively,
on the left and right. $\mathcal{W}_{x_{1}x_{2}}$ evaluates to the length
$\left|x_{1}x_{2}\right|$ when integrated along the edge ${x_{1}x_{2}}$ (in
blue) and vanishes on all other edges (in orange).
$\mathcal{W}_{x_{1}x_{2}x_{4}x_{3}}$ likewise yields the area
$\left|x_{1}x_{2}x_{4}x_{3}\right|$ when integrated over the blue face
${x_{1}x_{2}x_{4}x_{3}}$, and vanishes on all other faces.
Now consider the exterior derivative $\mathrm{d}$ of these forms. The curl
matrix ${\mathbb{C}:\mathbb{R}^{N_{1}}\rightarrow\mathbb{R}^{N_{2}}}$ is a
linear operator that precisely computes $\mathrm{d}$ for 1-forms, mapping the
coefficients of a discrete 1-form—expanded in the
${Q_{1}^{-}\Lambda^{1}(\mathcal{T}_{h})}$ basis—to coefficients of a discrete
2-form—expanded in the ${Q_{1}^{-}\Lambda^{2}(\mathcal{T}_{h})}$ basis. The
1-to-1 correspondence between these generalized Whitney forms and elements of
the mesh leads to a convenient interpretation of $\mathbb{C}$ in that basis,
as follows.
We find by directly computing the exterior derivative of
$\mathcal{W}_{x_{1}x_{2}}$ using Eqs. (18-19), for example, that
$\displaystyle\begin{split}\mathrm{d}\mathcal{W}_{x_{1}x_{2}}&=\frac{1}{\Delta_{y}}\mathcal{W}_{x_{1}x_{2}x_{4}x_{3}}-\frac{1}{\Delta_{z}}\mathcal{W}_{x_{1}x_{5}x_{6}x_{2}}.\end{split}$
(20)
We observe that a Whitney 2-form $\mathcal{W}_{\sigma^{2}}$ appears on the
right hand side above if and only if its associated face $\sigma^{2}$ contains
the edge $x_{1}x_{2}$ on its boundary. Its coefficient ${\pm
1/\Delta_{x^{\mu}}}$ is determined by the $\mu^{\text{th}}$ dimension in which
the associated face extends $x_{1}x_{2}$, and its sign is set by the relative
orientation between the edge and face.
Note, this interpretation of $\mathrm{d}$—as a mapping with coefficients ${\pm
1/\Delta_{x^{\mu}}}$ from 1-forms to the 2-forms of faces which contain them
on the boundary—is applicable only to the particular (cubical) FEEC mesh and
(Whitney form) FEEC basis that we have chosen. In this context, however, our
interpretation leads us to note that the curl operator $\mathbb{C}$, (which
implements $\mathrm{d}$ as defined in Table 1), acts on on the generalized
Whitney forms of a cubical lattice as nothing more than a finite difference
operator. To see this more explicitly, let us denote by
${\mathbf{a}_{\sigma^{1}}}$ the entry of ${\mathbf{a}\in\mathbb{R}^{N_{1}}}$
corresponding to the Whitney 1-form basis element
${\mathcal{W}_{\sigma^{1}}}$. Likewise, we let ${\mathbf{b}_{\sigma^{2}}}$
denote the coefficient in
${\mathbf{b}=\mathbb{C}\mathbf{a}\in\mathbb{R}^{N_{2}}}$ corresponding to
${\mathcal{W}_{\sigma^{2}}}$. With reference again to Fig. 1, we find that
$\displaystyle\begin{split}\mathbf{b}_{x_{1}x_{2}x_{4}x_{3}}&=(\mathbb{C}\mathbf{a})_{x_{1}x_{2}x_{4}x_{3}}\\\
&=\frac{1}{\Delta_{x}}\Big{(}\mathbf{a}_{x_{2}x_{4}}-\mathbf{a}_{x_{1}x_{3}}\Big{)}-\frac{1}{\Delta_{y}}\Big{(}\mathbf{a}_{x_{3}x_{4}}-\mathbf{a}_{x_{1}x_{2}}\Big{)}.\end{split}$
(21)
Under the map $\mathbb{C}$, the finite differences of the edge coefficients
are therefore summed to derive the coefficient on a face.
The transpose of the curl operator $\mathbb{C}^{T}$, yields an analogous
result, with ${(\mathbb{C}^{T}\mathbf{b})_{\sigma^{1}}}$ computed via finite
differences between the faces adjoining edge $\sigma^{1}$. Using Fig. 2 as a
reference, for example, we find
$\displaystyle\begin{split}(\mathbb{C}^{T}\mathbf{b})_{x_{1}x_{5}}=\frac{1}{\Delta_{x}}\Big{(}\mathbf{b}&{}_{x_{1}x_{5}x_{6}x_{2}}-\mathbf{b}_{x_{10}x_{12}x_{5}x_{1}}\Big{)}\\\
&-\frac{1}{\Delta_{y}}\Big{(}\mathbf{b}_{x_{1}x_{3}x_{7}x_{5}}-\mathbf{b}_{x_{9}x_{1}x_{5}x_{11}}\Big{)}.\end{split}$
(22)
Figure 2: A depiction of the degrees of freedom involved in
$\mathbb{C}^{T}$—the transposed curl operator—for generalized Whitney forms on
a cubic mesh.
The observation that $\mathbb{C}$ acts as a finite difference operator on
cubical generalized Whitney forms marks a first step toward recovering Yee’s
method from Eq. (15).
As a second step, let us first consider what $\mathbb{M}_{1}$ and
$\mathbb{M}_{2}$ look like for our chosen Whitney form FEEC basis on a cubical
mesh. Ignoring boundary cells for simplicity, we use Eq. (3) to integrate
inner products of forms such as those appearing in Eq. (18) to find that
$\displaystyle\begin{split}(\mathbb{M}_{1})_{\sigma^{1},\tau^{1}}&=\Delta_{V}\cdot\begin{cases}4/9&\sigma^{1}=\tau^{1}\\\
1/9&\sigma^{1}\parallel\tau^{1}\text{ and
}\exists~{}\tau^{2}\supset\\{\sigma^{1},\tau^{1}\\}\\\
1/36&\sigma^{1}\parallel\tau^{1}\text{ and
}\exists~{}\tau^{3}\supset\\{\sigma^{1},\tau^{1}\\}\end{cases}\\\
(\mathbb{M}_{2})_{\sigma^{2},\tau^{2}}&=\Delta_{V}\cdot\begin{cases}2/3&\sigma^{2}=\tau^{2}\\\
1/6&\sigma^{2}\parallel\tau^{2}\text{ and
}\exists~{}\tau^{3}\supset\\{\sigma^{2},\tau^{2}\\}\end{cases}\end{split}$
(23)
where ${\Delta_{V}=\Delta_{x}\Delta_{y}\Delta_{z}}$ denotes a cell volume.
Given these matrix entries, each row (and column) of $\mathbb{M}_{1}$ and
$\mathbb{M}_{2}$ can be readily shown to sum to $\Delta_{V}$. In the interest
of recovering Yee’s method, therefore, we define the simplified mass matrices
$\displaystyle\begin{split}\mathbb{M}_{1}^{Y}&=\Delta_{V}\cdot\mathbb{1}_{N_{1}\times
N_{1}}\\\ \mathbb{M}_{2}^{Y}&=\Delta_{V}\cdot\mathbb{1}_{N_{2}\times
N_{2}}.\end{split}$ (24)
Here, $\mathbb{M}_{p}^{Y}$ signifies the ‘Yee-modified’ mass matrix for
$p\text{-form}$s, and $\mathbb{1}$ denotes the identity matrix. This process
is often referred to as ‘lumping’ the mass matrix (of Eq. (23), for example)
into diagonal form.
Importantly, using such simplified mass matrices leaves the symplectic
structure defined by ${\\{\cdot,\cdot\\}_{\Delta}}$ in Eq. (9) entirely
undisturbed. Rather, a substitution of the simplified mass matrix
${\mathbb{M}_{p}\rightarrow\mathbb{M}_{p}^{Y}}$ is properly viewed as taking
place within the discrete Hamiltonian $H_{\Delta}$ in Eq. (9). While this
coarser approximation of $H_{\text{EM}}$ reduces the accuracy of the resulting
(Yee’s) algorithm, it has no deleterious effect on its characterization as a
structure-preserving algorithm.
We now complete our program to recover Yee’s method. Let us consider again the
Strang splitting defined in Eq. (16). We can rewrite this system more
explicitly as
$\displaystyle\begin{split}\left[\begin{matrix}\mathbf{a}\\\
\mathbf{e}\end{matrix}\right]_{\mathrlap{t_{0}+\Delta
t}}~{}~{}&=~{}~{}\mathbb{H}_{\mathbf{e}}^{\Delta
t/2}\mathbb{H}_{\mathbf{a}}^{\Delta t}\mathbb{H}_{\mathbf{e}}^{\Delta
t/2}\left[\begin{matrix}\mathbf{a}\\\
\mathbf{e}\end{matrix}\right]_{t_{0}}~{}~{}~{}\text{where}\\\
\mathbb{H}_{\mathbf{e}}^{\Delta t}&=\left[\begin{matrix}\mathbb{1}&-\Delta
t\mathbb{M}_{1}^{-1}\\\
\mathbb{0}&\mathbb{1}\end{matrix}\right]~{}~{}\text{,}~{}~{}\mathbb{H}_{\mathbf{a}}^{\Delta
t}=\left[\begin{matrix}\mathbb{1}&\mathbb{0}\\\ \Delta
tc^{2}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}&\mathbb{1}\end{matrix}\right]\end{split}$
(25)
or equivalently, in terms of ${\mathbf{b}=\mathbb{C}\mathbf{a}}$:
$\displaystyle\begin{split}\left[\begin{matrix}\mathbf{b}\\\
\mathbf{e}\end{matrix}\right]_{\mathrlap{t_{0}+\Delta
t}}&~{}~{}=~{}~{}\mathcal{H}_{\mathbf{e}}^{\Delta
t/2}\mathcal{H}_{\mathbf{b}}^{\Delta t}\mathcal{H}_{\mathbf{e}}^{\Delta
t/2}\left[\begin{matrix}\mathbf{b}\\\
\mathbf{e}\end{matrix}\right]_{t_{0}}~{}~{}~{}\text{where}\\\
\mathcal{H}_{\mathbf{e}}^{\Delta t}&=\left[\begin{matrix}\mathbb{1}&-\Delta
t\mathbb{C}\mathbb{M}_{1}^{-1}\\\
\mathbb{0}&\mathbb{1}\end{matrix}\right]~{}~{}~{}\text{,}~{}~{}~{}\mathcal{H}_{\mathbf{b}}^{\Delta
t}=\left[\begin{matrix}\mathbb{1}&\mathbb{0}\\\ \Delta
tc^{2}\mathbb{C}^{T}\mathbb{M}_{2}&\mathbb{1}\end{matrix}\right].\end{split}$
(26)
Eq. (26) is iterated in a simulation, so that after ${n+1}$ timesteps its
evolution operator is ${\mathcal{H}_{\mathbf{e}}^{\Delta
t/2}\left[\mathcal{H}_{\mathbf{e}}^{\Delta t}\mathcal{H}_{\mathbf{b}}^{\Delta
t}\right]^{n}\mathcal{H}_{\mathbf{e}}^{\Delta t/2}}$. Therefore, applying the
‘finite-difference operators’ $\mathbb{C}$ and $\mathbb{C}^{T}$ from Eqs.
(21-22) and the mass matrix approximations $\mathbb{M}_{p}^{Y}$ from Eq. (24),
let us examine our splitting method at ‘mid-step’—after
${[\mathcal{H}_{\mathbf{e}}^{\Delta t}\mathcal{H}_{\mathbf{b}}^{\Delta
t}]\mathcal{H}_{\mathbf{e}}^{\Delta t/2}}$—and compare it with Yee’s method.
Since ${\mathcal{H}_{\mathbf{e}}^{\Delta t/2}}$ only evolves $\mathbf{b}$, we
regard this as an ‘initialization step’ that gives us
$\displaystyle\begin{split}\mathcal{H}_{\mathbf{e}}^{\Delta
t/2}:\left(\begin{matrix}\mathbf{b}[t_{0}]\\\
\mathbf{e}[t_{0}]\end{matrix}\right)\mapsto\left(\begin{matrix}\mathbf{b}\left[t_{1/2}\right]\\\
\mathbf{e}\left[t_{0}\right]\end{matrix}\right).\end{split}$ (27)
In particular, this half-step establishes initial field data as it appears in
Yee’s method. Next, ${\mathcal{H}_{\mathbf{b}}^{\Delta t}}$ only evolves
$\mathbf{e}$, such that
$\displaystyle\begin{split}\mathcal{H}_{\mathbf{b}}^{\Delta
t}:\left(\begin{matrix}\mathbf{b}\left[t_{1/2}\right]\\\
\mathbf{e}\left[t_{0}\right]\end{matrix}\right)\mapsto\left(\begin{matrix}\mathbf{b}\left[t_{1/2}\right]\\\
\mathbf{e}\left[t_{1}\right]\end{matrix}\right)\\\\[5.0pt] \text{ where
}\mathbf{e}\left[t_{1}\right]=\mathbf{e}\left[t_{0}\right]+\Delta
t\Delta_{V}c^{2}\mathbb{C}^{T}\mathbf{b}[t_{1/2}].\end{split}$ (28)
This timestep
${(\mathbf{e}\left[t_{1}\right]-\mathbf{e}\left[t_{0}\right])/\Delta
tc^{2}=\Delta_{V}\mathbb{C}^{T}\mathbf{b}[t_{1/2}]}$ replicates the Yee
method’s evolution of the electric field, e.g.
$\displaystyle\begin{split}&\frac{1}{c^{2}\Delta
t}\left(E_{x}^{n+1}\left[i+\tfrac{1}{2},j,k\right]-E_{x}^{n}\left[i+\tfrac{1}{2},j,k\right]\right)\\\
&=\frac{1}{\Delta
y}\left(B_{z}^{n+\frac{1}{2}}\left[i+\tfrac{1}{2},j+\tfrac{1}{2},k\right]-B_{z}^{n+\frac{1}{2}}\left[i+\tfrac{1}{2},j-\tfrac{1}{2},k\right]\right)\\\
&-\frac{1}{\Delta
z}\left(B_{y}^{n+\frac{1}{2}}\left[i+\tfrac{1}{2},j,k+\tfrac{1}{2}\right]-B_{y}^{n+\frac{1}{2}}\left[i+\tfrac{1}{2},j,k-\tfrac{1}{2}\right]\right).\end{split}$
(29)
As depicted in Fig. 2 and described in Eq. (22), the operator $\mathbb{C}^{T}$
implements precisely the finite differences of Eq. (29). The additional
constant factor $\Delta_{V}$ in Eq. (28) has no effect other than changing the
effective unit in which $\mathbf{b}$ is expressed. Finally,
${\mathcal{H}_{\mathbf{e}}^{\Delta t}}$ only evolves $\mathbf{b}$, such that
$\displaystyle\begin{split}\mathcal{H}_{\mathbf{e}}^{\Delta
t}:\left(\begin{matrix}\mathbf{b}\left[t_{1/2}\right]\\\
\mathbf{e}\left[t_{1}\right]\end{matrix}\right)\mapsto\left(\begin{matrix}\mathbf{b}\left[t_{3/2}\right]\\\
\mathbf{e}\left[t_{1}\right]\end{matrix}\right)\\\\[5.0pt] \text{ where
}\mathbf{b}\left[t_{3/2}\right]=\mathbf{b}\left[t_{1/2}\right]-(\Delta
t/\Delta_{V})\mathbb{C}\mathbf{e}[t_{1}].\end{split}$ (30)
This timestep
${(\mathbf{b}\left[t_{3/2}\right]-\mathbf{b}\left[t_{1/2}\right])\Delta_{V}/\Delta
t=\mathbb{C}\mathbf{e}[t_{1}]}$ replicates Yee’s evolution of the magnetic
field, e.g.
$\displaystyle\begin{split}\frac{1}{\Delta
t}&\left(B_{x}^{n+\frac{1}{2}}\left[i,j+\tfrac{1}{2},k+\tfrac{1}{2}\right]-B_{x}^{n-\frac{1}{2}}\left[i,j+\tfrac{1}{2},k+\tfrac{1}{2}\right]\right)\\\
&=\frac{1}{\Delta
z}\left(E_{y}^{n}\left[i,j+\tfrac{1}{2},k+1\right]-E_{y}^{n}\left[i,j+\tfrac{1}{2},k\right]\right)\\\
&-\frac{1}{\Delta
y}\left(E_{z}^{n}\left[i,j+1,k+\tfrac{1}{2}\right]-E_{z}^{n}\left[i,j,k+\tfrac{1}{2}\right]\right).\end{split}$
(31)
As described in Eq. (21), $\mathbb{C}$ implements precisely the finite
differences of Eq. (31). The factor of $\Delta_{V}$ again only impacts the
unit of $\mathbf{b}$.
Consequently, we have demonstrated that Yee’s algorithm is equivalent to the
structure-preserving FEEC method of Eq. (15), using Whitney forms on a cubical
mesh, maximally simplified (diagonal) mass matrices, and Strang splitting time
evolution.
## 4 Generalizing Yee’s Method
| Yee’s Method | SFEEC methods
---|---|---
Splitting method | Strang | any (e.g. Lie-Trotter [38], Strang)
Mesh | cubical | any (e.g. simplicial, cubical)
Finite elements | Whitney forms | any FEEC basis (e.g. ${\mathcal{P}_{r}^{-}\Lambda^{k},\mathcal{P}_{r}\Lambda^{k}}$)
$\mathbb{M}_{1}^{-1}$ and $\mathbb{M}_{2}$ | diagonal approximation | any sparse approximation
Table 2: The SFEEC generalization of Yee’s method affords considerable
flexibility in the choice of splitting scheme, mesh, and finite element basis
in its implementation of Eq. (15). SFEEC methods are symplectic and are also
required to be scalable. Therefore, they require sparse mass matrices and
their sparse approximate inversions.
The previous section demonstrates that Yee’s method is a special case of the
FEEC structure-preserving algorithm defined by Eq. (15). Specifically, Yee’s
method is seen to make several selections for the FEEC algorithm, including
1. 1.
timesteps given by a Strang splitting scheme;
2. 2.
a cubical grid;
3. 3.
the least-order FEEC finite element basis—cubical Whitney forms; and
4. 4.
mass matrices simplified into diagonal form.
The last of these—the mass matrix approximation—can be seen not so much as a
‘selection’ but as a ‘departure’ from the algorithm defined by Eq. (15). As
previously described, however, a simplified treatment of mass matrices occurs
at the level of the Hamiltonian $H_{\Delta}$ in Eq. (9), and does not affect
the Poisson bracket ${\\{\cdot,\cdot\\}_{\Delta}}$. As a result, Yee’s method
remains symplectic and—though it loses accuracy—retains all of the advantages
of a structure-preserving algorithm. Yee’s method thereby preserves the
geometric structure of Maxwell’s equations while employing a parsimonious
description of electromagnetic fields. Moreover, its sacrifice of accuracy in
mass matrices is arguably compensated by the scalability the Yee method
achieves.
Here, we propose to relax all four of the above selections made for the Yee
scheme, and in so doing we define a new class of algorithms, _scalable FEEC_
(SFEEC) methods. In particular, Table 2 enumerates the generalizations of
Yee’s method afforded by SFEEC methods.
As defined, SFEEC methods offer considerable flexibility in generalizing the
Yee scheme. They have the full flexibility of FEEC behind them, and can
therefore handle quite general meshes and high order finite elements.
Moreover, the splitting scheme used for SFEEC methods can be chosen to be of
arbitrarily high order accuracy in time.
As we shall see, however, this improvement in accuracy is constrained by the
requirement that SFEEC methods remain scalable. In particular, while mass
matrices will no longer be maximally simplified into diagonal form (as in
Yee’s approach), $\mathbb{M}_{1}^{-1}$ will require sparse approximation. Even
with this caveat, we shall demonstrate that SFEEC methods offer a significant
improvement upon the accuracy of Yee’s method, at limited additional cost in
computational effort.
Let us consider in greater detail the need for simplified mass matrices in
large scale implementations of Eq. (15). We note that essentially any choice
of FEEC finite element has only local support, so that the mass matrices
$\mathbb{M}_{1}$ and $\mathbb{M}_{2}$ are generally sparse by definition.
However, Eq. (15) requires the use of $\mathbb{M}_{1}^{-1}$, and the inverse
of a sparse matrix is generally dense. Given $\mathbb{M}_{1}^{-1}$ dense,
however, a computation that evolves subsystem ${H_{\mathbf{e}}}$ of Eq. (15)
would require every node of a simulation to pass its local $\mathbf{e}$ data
to every other node, so that $\mathbf{a}$ could be evolved. This communication
would clearly spoil the efficacy of parallelization.
On the other hand, we may leverage the spirit of Yee’s algorithm as seen
through an FEEC lens, and consider more parsimonious approximations of the
matrix $\mathbb{M}_{1}^{-1}$. We emphasize from our discussion above that we
can ‘prune’ $\mathbb{M}_{1}^{-1}$ as minimally or as maximally as desired: Any
approximation of $\mathbb{M}_{1}^{-1}$ will produce, as Yee’s method produces,
a symplectic algorithm. The choice of approximation should be guided,
therefore, strictly by the trade-off between its scalability and its accuracy
in approximating the electromagnetic Hamiltonian, $H_{\Delta}\approx
H_{\text{EM}}$.
Before proceeding to describe our numerical results in the next section, we
first describe our method to find sparse approximations of
$\mathbb{M}_{1}^{-1}$. Our general strategy (following [26],[39]) is to
identify a desirable sparsity pattern a priori, and find a matrix
${\mathbb{Q}\approx\mathbb{M}_{1}^{-1}}$ of the desired sparsity pattern that
best approximates ${\mathbb{M}_{1}^{-1}}$. We shall denote the best fit for
$\mathbb{M}_{1}^{-1}$ with sparsity pattern ${S(\mathbb{A})}$ as
$\mathbb{Q}_{S(\mathbb{A})}$, where ${S(\mathbb{A})}$ denotes the sparsity
pattern of an appropriate matrix $\mathbb{A}$. The goodness of fit may be
determined by the Froebenius norm $\left\lVert\cdot\right\rVert_{F}$, in which
case $\mathbb{Q}$ can be solved for column-by-column, since
$\displaystyle\begin{split}\left\lVert\mathbb{M}_{1}\mathbb{Q}-\mathbb{1}\right\rVert_{F}^{2}=\sum\limits_{\ell=1}^{N_{1}}\left\lVert\mathbb{M}_{1}\mathbf{q}_{\ell}-1_{\ell}\right\rVert^{2}.\end{split}$
(32)
Here, $\mathbf{q}_{\ell}$ is the $\ell^{\text{th}}$ column of ${\mathbb{Q}}$
and $1_{\ell}$ is the $\ell^{\text{th}}$ column of the identity matrix, (a
standard basis vector). Thus, $\mathbf{q}_{\ell}$ can be found $\forall$
$\ell$ independently—in parallel, if desired—as the solution to a least
squares problem, in which the only entries of $\mathbf{q}_{\ell}$ permitted to
vary are those that fit the desired sparsity pattern for the
$\ell^{\text{th}}$ column of $\mathbb{Q}$.
More explicitly, consider the problem of solving for the entries of one such
column $\mathbf{q}_{\ell}$. We may define an index
${\mathcal{I}_{\ell}\subset\\{1,\dots,N_{1}\\}}$ corresponding to the row
numbers of nonzero entries permitted in $\mathbf{q}_{\ell}$ according to the
desired sparsity pattern $S(\mathbb{Q})$. Denote the cardinality of such an
index ${\left|\mathcal{I}_{\ell}\right|\leq N_{1}}$. Then let
${\mathbb{M}_{1}(:,\mathcal{I}_{\ell})}$ denote the
${N_{1}\times\left|\mathcal{I}_{\ell}\right|}$ submatrix of $\mathbb{M}_{1}$
comprised of the subset of its columns indicated by $\mathcal{I}_{\ell}$. For
each ${1\leq\ell\leq N_{1}}$ we minimize
${\left\lVert\mathbb{M}_{1}(:,\mathcal{I}_{\ell})\mathbf{q}_{\ell}(\mathcal{I}_{\ell})-1_{\ell}\right\rVert}$,
that is,
$\displaystyle\begin{split}\operatorname*{arg\,min}\limits_{\mathbf{q}_{\ell}(\mathcal{I}_{\ell})}\sum\limits_{\begin{subarray}{c}1\leq
i\leq N_{1}\\\
j\in\mathcal{I}_{\ell}\end{subarray}}\Big{(}(\mathbb{M}_{1})_{ij}(\mathbf{q}_{\ell})_{j}-\delta_{i\ell}\Big{)}^{2}.\end{split}$
(33)
In the numerical examples we describe below, we solve Eq. (33) directly via
$\displaystyle\begin{split}\mathbf{q}_{\ell}(\mathcal{I}_{\ell})=[\mathbb{M}_{1}(:,\mathcal{I}_{\ell})^{T}\mathbb{M}_{1}(:,\mathcal{I}_{\ell})]^{-1}\mathbb{M}_{1}(:,\mathcal{I}_{\ell})^{T}1_{\ell}.\end{split}$
(34)
Most implementations of Eq. (33) will be sparse enough to be solved this way.
Denser choices of the sparsity pattern (${\left|\mathcal{I}_{\ell}\right|\gg
1}$), however, for which the inversion in Eq. (34) is more difficult, can also
be readily solved using, for example, the conjugate gradient method (see [40])
or, if desired, constrained versions thereof (see [41]).
## 5 Numerical Results
Figure 3: The figures above depict the ${L^{2}\Lambda^{1}}$ log relative error
in an FEEC approximation of
$\dot{\mathbf{E}}=c^{2}\nabla\times\nabla\times\mathbf{A}$ vs. log cell size
$h$. The $x$-axis measures the number of cells (of size $h$) per wavelength
${\lambda_{\mathbf{A}}=2\pi/k_{n}}$ of the vector potential
${\mathbf{A}=\sin(k_{n}y)\mathrm{d}x}$. The left plot uses the first order
(Whitney form) ${\mathcal{P}_{1}^{-}\Lambda^{p}(\mathcal{T}_{h})}$ FEEC basis,
while the right plot uses a more accurate second order basis,
${\mathcal{P}_{2}^{-}\Lambda^{p}(\mathcal{T}_{h})}$. For each basis, we plot
the relative errors of four approximations of the FEEC curl-of-curl operator,
${\nabla\times\nabla\times\approx\mathbb{Q}_{i}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}}$,
${i\in\\{1,\dots,4\\}}$, and fit their power scalings against $h$. Here,
${\mathbb{Q}_{i}}$ is an approximation of the inverse mass matrix
$\mathbb{M}_{1}^{-1}$ with $i$ indicating one of four sparsity patterns:
diagonal, $\mathbb{M}_{1}$, $(\mathbb{M}_{1})^{2}$, and dense (which recovers
the exact FEEC operator). The left plot demonstrates that the accuracy of
Yee’s method, (which effectively uses Whitney forms and diagonal mass
matrices), is meaningfully improved upon by an approximation of
$\mathbb{M}_{1}^{-1}$ that has the sparsity pattern of $\mathbb{M}_{1}$. The
right plot demonstrates that approximating $\mathbb{M}_{1}^{-1}$ by a matrix
with the sparsity pattern of ${(\mathbb{M}_{1})^{2}}$ achieves much of the
improved accuracy possible with second order finite elements. It is seen that
the power scalings extend to a finite resolution, at which point relative
error saturates.
To measure the accuracy of an approximation
${\mathbb{Q}\approx\mathbb{M}_{1}^{-1}}$ in a manner relevant to our
electromagnetic problem, we examine the curl-of-curl operator
${\nabla\times\nabla\times}$, discretized by
${\mathbb{M}_{1}^{-1}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}}$ in the FEEC
setting, as in Eq. (11). For simplicity, we work with simplicial finite
elements in 2-D, and generate Delaunay triangulations ${\mathcal{T}_{h}}$ of a
2-torus domain
${\left|\mathcal{T}_{h}\right|=[0,L_{x}]\times[0,L_{y}]\subset\mathbb{R}^{2}}$
with periodic boundary conditions. On $\left|\mathcal{T}_{h}\right|$, we
consider a sinusoidal vector potential ${\mathbf{A}=\sin(k_{n}y)\mathrm{d}x}$
for ${k_{n}=2\pi n/L_{y}}$ and ${n\in\mathbb{N}}$, and canonically project it
onto some choice of discrete FEEC basis [6],
${\mathbf{a}\cdot\mbox{\boldmath$\Lambda$}^{1}\in\Lambda^{1}(\mathcal{T}_{h})}$.
We measure the ${L^{2}\Lambda^{1}}$ error between
${\nabla\times\nabla\times\mathbf{A}=k_{n}^{2}\sin(k_{n}y)\mathrm{d}x}$
(exactly representing ${\dot{\mathbf{E}}/c^{2}}$ in the continuum) and its
FEEC discretization,
${\mathbb{M}_{1}^{-1}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}\mathbf{a}}$
(representing ${\mathbb{M}_{1}^{-1}\dot{\mathbf{e}}/c^{2}}$). Finally, we
investigate how the error of approximating ${\mathbb{M}_{1}^{-1}}$ with
$\mathbb{Q}$—i.e., of approximating ${\nabla\times\nabla\times\mathbf{A}}$
with ${\mathbb{Q}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}\mathbf{a}}$—varies
with the sparsity pattern ${S(\mathbb{Q})}$.
We repeat this procedure over a range of frequencies ${k_{n}=2\pi n/L_{y}}$
and mesh diameters $h$, (the latter achieved by varying the number of vertices
in $\mathbb{T}^{2}$ and generating a Delaunay triangulation $\mathcal{T}_{h}$
for each). We examine two different FEEC bases on this 2-D simplicial mesh:
${\mathcal{P}_{1}^{-}\Lambda^{p}(\mathcal{T}_{h})}$, the Whitney 1-forms
(described above for a cubical mesh), and their counterparts at the next order
of accuracy, ${\mathcal{P}_{2}^{-}\Lambda^{p}(\mathcal{T}_{h})}$—the second-
order trimmed polynomial family of finite elements. For each basis, we compare
four different sparsity patterns for our approximation of
${\mathbb{M}_{1}^{-1}}$, including: diagonal (Yee’s implicit pattern),
$\mathbb{M}_{1}$ sparsity,
${(\mathbb{M}_{1})^{2}=\mathbb{M}_{1}\cdot\mathbb{M}_{1}}$
sparsity,111${(\mathbb{M}_{1})_{ij}}$ is nonzero whenever basis elements
$\Lambda^{1}_{i}$ and $\Lambda^{1}_{j}$ of $\mbox{\boldmath$\Lambda$}^{1}$
have nontrivial ‘overlap,’ in the sense of Eq. (3).
${(\mathbb{M}_{1})^{2}_{ij}}$ is generally nonzero whenever $\Lambda^{1}_{i}$
and $\Lambda^{1}_{j}$ each has nontrivial overlap with a third basis finite
element $\Lambda^{1}_{k}$ (where ${k\in\\{i,j\\}}$ is also permitted). In our
context, therefore, ${(\mathbb{M}_{1})^{2}}$ is strictly denser than
$\mathbb{M}_{1}$. By extension, the Neumann representation of
$\mathbb{M}_{1}^{-1}$ demonstrates that ${S((\mathbb{M}_{1})^{n})\rightarrow
S(\mathbb{M}_{1}^{-1})}$ as $n$ approaches $N_{1}$ [39]. and dense (the exact
inverse). The results of this investigation are depicted in Fig. 3.
With the setup described above, the two subplots of Fig. 3 display the log
relative $L^{2}\Lambda^{1}$ errors in our approximations of
${\dot{\mathbf{E}}=c^{2}\nabla\times\nabla\times\mathbf{A}}$. (For brevity, we
use ${\dot{\mathbf{E}}}$ merely as a shorthand notation.) In particular, given
the $L^{2}\Lambda^{1}$ norm (see Eq. (4))
$\displaystyle\begin{split}||\dot{\mathbf{E}}||_{L^{2}\Lambda^{1}}=\int_{\left|\mathcal{T}_{h}\right|}(\dot{\mathbf{E}},\dot{\mathbf{E}})_{1}\mathrm{d}\mathbf{x},\end{split}$
(35)
we compute
$\displaystyle\begin{split}||\hat{\dot{\mathbf{E}}}-\dot{\mathbf{E}}||_{L^{2}\Lambda^{1}}/||\dot{\mathbf{E}}||_{L^{2}\Lambda^{1}}\end{split}$
(36)
where
${\hat{\dot{\mathbf{E}}}=c^{2}\mathbb{Q}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}}$
denotes the relevant finite element approximation of the continuum 1-form
${\dot{\mathbf{E}}=c^{2}k_{n}^{2}\sin(k_{n}y)\mathrm{d}x}$.
The left plot demonstrates that the FEEC curl-of-curl operator
${\mathbb{M}_{1}^{-1}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}}$ is well
approximated for a Whitney form basis using
${\mathbb{Q}_{S(\mathbb{M}_{1})}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}}$
(where $\mathbb{Q}_{S(\mathbb{M}_{1})}$ denotes the approximation of
$\mathbb{M}_{1}^{-1}$ with a sparsity pattern of $\mathbb{M}_{1}$). In
particular, the relative error of
${\mathbb{Q}_{S(\mathbb{M}_{1})}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}}$
scales with cell size as ${h^{0.73}}$, while the exact FEEC operator
${\mathbb{M}_{1}^{-1}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}}$ error scales
comparably as ${h^{0.82}}$.
Remarkably, the first plot demonstrates that a substantial improvement in
accuracy over a diagonal sparsity pattern analogous to Yee’s method is
achieved by using only a slightly less parsimonious estimate of
$\mathbb{M}_{1}^{-1}$ in the curl-of-curl FEEC operator. In particular, while
${\mathbb{Q}_{S(\mathbb{M}_{1})}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}}$
scales as ${h^{0.73}}$ (as noted just above), the accuracy of
${\mathbb{Q}_{S(\mathbb{1})}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}}$ (with
$\mathbb{Q}_{S(\mathbb{1})}$ diagonal-patterned) scales as ${h^{0.56}}$.
More important than this modest improvement in power scaling, however, is its
applicability to finer meshes. The first plot of Fig. 3 shows that the power
scaling of the diagonal mass matrix curl-of-curl approximation can only be
assumed for meshes of relatively low resolution. The relative error in the
${\mathbb{Q}_{S(\mathbb{1})}}$ discrete differential operator is seen to
saturate when the number of cells per signal wavelength climbs above 5—that is
${\lambda_{\mathbf{A}}/h\gtrsim 5}$. This is in contrast to the
${\mathbb{Q}_{S(\mathbb{M}_{1})}}$ power scaling, which continues to hold even
at the higher resolution of ${\lambda_{\mathbf{A}}/h\sim 20}$. This could be a
particular advantage in simulations of small-scale electromagnetic structure,
such as in turbulent plasmas or nanomaterials.
It must be further emphasized that the additional computational cost
introduced by this less parsimonious $\mathbb{Q}_{S(\mathbb{M}_{1})}$
approximation of $\mathbb{M}_{1}^{-1}$ will generally be fairly modest. The
pairs of nodes that communicate in a simulation with an
$\mathbb{M}_{1}$-sparsity approximation are typically the same as those that
communicate with a diagonal-sparsity approximation, such that no ‘new
channels’ of communication are introduced. Each communication between nodes
will pass more data, however—in our 2-D example, roughly 5 times as much.
Nevertheless, the amount of data passed is small to begin with, as it scales
only with the number of boundary cells of a node.
The right plot of Fig. 3 demonstrates that, to achieve the accuracy of higher
order finite elements, denser approximations of $\mathbb{M}_{1}^{-1}$ must be
used in the curl-of-curl operator. Nevertheless, greater accuracy is readily
attained. In our 2-D simplicial example, we see that much of the improved
accuracy possible with second order finite elements is achieved by using a
curl-of-curl operator
${\mathbb{Q}_{S(\mathbb{M}_{1}^{2})}\mathbb{C}^{T}\mathbb{M}_{2}\mathbb{C}}$,
which approximates $\mathbb{M}_{1}^{-1}$ with a sparsity pattern of
${(\mathbb{M}_{1})^{2}}$. Indeed, this approximate FEEC operator achieves an
error scaling with cell size of $h^{1.70}$, while the exact FEEC curl-of-curl
operator error scales in our results as $h^{1.98}$. It is worth noting,
however, that the approximation’s scaling holds only until a mesh resolution
of roughly ${\lambda_{\mathbf{A}}/h\gtrsim 5}$. Nevertheless, due to the
higher order finite element, this saturation occurs at a significantly higher
accuracy overall.
## 6 Discussion
We have demonstrated that Yee’s method can be interpreted as a structure-
preserving splitting method using an FEEC formalism on a cubical mesh with
simplified mass matrices. In so doing, we have identified Yee’s algorithm to
be a special case of a larger family of methods we call SFEEC (scalable finite
element exterior calculus) methods, summarized by Eq. (15) and Table 2. These
methods are structure-preserving, as Yee’s method is, and are therefore
accurate in long-time numerical simulations. They also respect the topological
and geometric properties of Maxwell’s equations. Crucially, SFEEC methods are
also scalable; they adopt the strategy implicit in Yee’s method of using
sparse approximations of finite element mass matrices to achieve computational
efficiency.
The classification of SFEEC methods enabled us to identify higher order
extensions of Yee’s method that preserve its scalability nevertheless (see
Fig. 3). Depending on the computing architecture employed and the desired
accuracy of the problem of interest, these alternative methods may enable
greater long-time accuracy than Yee’s method provides, with little additional
computational cost. The applicability of SFEEC methods on general meshes may
also be of particular benefit in problems with irregular geometries.
The saturation of the relative errors plotted in Fig. 3 suggest that it would
be fruitful to investigate in future work whether improved sparse
approximations of the discrete codifferential operator,
${\mathbb{M}_{1}^{-1}\mathbb{C}^{T}\mathbb{M}_{2}}$ can be found. Such efforts
might be geared to preserve higher order finite element error scaling at
arbitrarily high mesh resolution.
## 7 Acknowledgments
Thank you to Josh Burby and Tyrus Berry for helpful discussions, and to Phil
Morrison for his support. This research was further supported by the U.S.
Department of Energy (DE-AC02-09CH11466), as well as the U.S. Department of
Energy Fusion Energy Sciences Postdoctoral Research Program administered by
the Oak Ridge Institute for Science and Education (ORISE) for the DOE. ORISE
is managed by Oak Ridge Associated Universities (ORAU) under DOE contract
number DE-SC0014664. All opinions expressed in this paper are the authors’ and
do not necessarily reflect the policies and views of DOE, ORAU, or ORISE.
## References
* [1] K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” _IEEE Transactions on Antennas and Propagation_ , vol. 14, no. 3, pp. 302–307, 1966.
* [2] A. Taflove and S. C. Hagness, _Computational electrodynamics: the finite-difference time-domain method_ , 3rd ed. Boston: Artech House, 2005.
* [3] E. Hairer, C. Lubich, and G. Wanner, _Geometric Numerical Integration_ , 2nd ed. Berlin: Springer, 2006.
* [4] H. Whitney, _Geometric Integration Theory_. Princeton, NJ: Princeton University Press, 1957.
* [5] M. Desbrun, A. N. Hirani, M. Leok, and J. E. Marsden, “Discrete exterior calculus,” _arXiv preprint math/0508341_ , 2005. [Online]. Available: https://arxiv.org/abs/math/0508341
* [6] D. N. Arnold, R. S. Falk, and R. Winther, “Finite element exterior calculus, homological techniques, and applications,” _Acta Numerica_ , vol. 15, 2006\.
* [7] ——, “Finite element exterior calculus: from Hodge theory to numerical stability,” _Bulletin of the American Mathematical Society_ , vol. 47, no. 2, pp. 281–354, 2010.
* [8] H. Kinoshita, H. Yoshida, and H. Nakai, “Symplectic integrators and their application to dynamical astronomy,” _CELESTIAL MECHANICS AND DYNAMICAL ASTRONOMY_ , vol. 50, no. 1, pp. 59–71, 1991.
* [9] B. Gladman, M. Duncan, and J. Candy, “Symplectic integrators for long-term integrations in celestial mechanics,” _Celestial Mechanics and Dynamical Astronomy_ , vol. 52, no. 3, pp. 221–240, 1991.
* [10] J. E. Chambers, E. V. Quintana, M. J. Duncan, and J. J. Lissauer, “Symplectic Integrator Algorithms for Modeling Planetary Accretion in Binary Star Systems,” _The Astronomical Journal_ , vol. 123, no. 5, pp. 2884–2894, 2002.
* [11] A. Bravetti, M. Seri, M. Vermeeren, and F. Zadra, “Numerical integration in celestial mechanics: a case for contact geometry,” _Celestial Mechanics and Dynamical Astronomy_ , vol. 132, no. 1, 2020.
* [12] E. Kur and A. S. Glasser, “Discrete gravity with local Lorentz invariance,” _Physical Review D_ , vol. 106, no. 6, p. 064001, 2022.
* [13] X. Li, W. Wang, M. Lu, M. Zhang, and Y. Li, “Structure-preserving modelling of elastic waves,” _Geophysical Journal International_ , vol. 188, no. 3, pp. 1382–1392, 2012.
* [14] S. Liu, X. Li, W. Wang, L. Xu, and B. Li, “A modified symplectic scheme for seismic wave modeling,” _Journal of Applied Geophysics_ , vol. 116, pp. 110–120, 2015.
* [15] J. Squire, H. Qin, and W. M. Tang, “Geometric integration of the Vlasov-Maxwell system with a variational particle-in-cell scheme,” _Physics of Plasmas_ , vol. 19, no. 8, p. 084501, 2012.
* [16] J. Xiao, H. Qin, J. Liu, Y. He, R. Zhang, and Y. Sun, “Explicit high-order non-canonical symplectic particle-in-cell algorithms for Vlasov-Maxwell systems,” _Physics of Plasmas_ , vol. 22, no. 11, p. 112504, 2015.
* [17] Y. He, H. Qin, Y. Sun, J. Xiao, R. Zhang, and J. Liu, “Hamiltonian time integrators for Vlasov-Maxwell equations,” _Physics of Plasmas_ , vol. 22, no. 12, p. 124503, 2015.
* [18] N. Crouseilles, L. Einkemmer, and E. Faou, “Hamiltonian splitting for the Vlasov–Maxwell equations,” _Journal of Computational Physics_ , vol. 283, pp. 224–240, feb 2015.
* [19] H. Qin, J. Liu, J. Xiao, R. Zhang, Y. He, Y. Wang, Y. Sun, J. W. Burby, L. Ellison, and Y. Zhou, “Canonical symplectic particle-in-cell method for long-term large-scale simulations of the Vlasov–Maxwell equations,” _Nuclear Fusion_ , vol. 56, no. 1, p. 014001, 2016.
* [20] M. Kraus, K. Kormann, P. J. Morrison, and E. Sonnendrücker, “GEMPIC: Geometric ElectroMagnetic Particle-In-Cell Methods,” _Journal of Plasma Physics_ , vol. 83, no. 4, 2017.
* [21] P. J. Morrison, “Structure and structure-preserving algorithms for plasma physics,” _Physics of Plasmas_ , vol. 24, no. 5, p. 055502, 2017.
* [22] A. S. Glasser and H. Qin, “The geometric theory of charge conservation in particle-in-cell simulations,” _Journal of Plasma Physics_ , vol. 86, no. 3, p. 835860303, 2020.
* [23] ——, “A gauge-compatible Hamiltonian splitting algorithm for particle-in-cell simulations using finite element exterior calculus,” _Journal of Plasma Physics_ , vol. 88, no. 2, p. 835880202, 2022.
* [24] A. Stern, “Geometric Discretization of Lagrangian Mechanics and Field Theories,” Ph.D. dissertation, California Institute of Technology, Pasadena, California, 2009.
* [25] Bo He and F. Teixeira, “Sparse and explicit FETD via approximate inverse Hodge (mass) matrix,” _IEEE Microwave and Wireless Components Letters_ , vol. 16, no. 6, pp. 348–350, Jun. 2006.
* [26] B. He and F. L. Teixeira, “Differential Forms, Galerkin Duality, and Sparse Inverse Approximations in Finite Element Solutions of Maxwell Equations,” _IEEE Transactions on Antennas and Propagation_ , vol. 55, no. 5, pp. 1359–1368, May 2007.
* [27] J. Kim and F. L. Teixeira, “Parallel and Explicit Finite-Element Time-Domain Method for Maxwell’s Equations,” _IEEE Transactions on Antennas and Propagation_ , vol. 59, no. 6, pp. 2350–2356, Jun. 2011.
* [28] F. L. Teixeira, “Differential Forms in Lattice Field Theories: An Overview,” _ISRN Mathematical Physics_ , vol. 2013, pp. 1–16, Feb. 2013\.
* [29] M. Hano, “Generalized time-domain method for solution of maxwell’s integral equations,” _AIP Conference Proceedings_ , vol. 391, no. 1, pp. 197–202, 1997.
* [30] J. Cole, “A high-accuracy realization of the Yee algorithm using non-standard finite differences,” _IEEE Transactions on Microwave Theory and Techniques_ , vol. 45, no. 6, pp. 991–996, 1997.
* [31] Z. Chen and S. Luo, “Generalization of the Finite-Difference-Based Time-Domain Methods Using the Method of Moments,” _IEEE Transactions on Antennas and Propagation_ , vol. 54, no. 9, pp. 2515–2524, 2006\.
* [32] F. L. Teixeira and S. Member, “Time-Domain Finite-Difference and Finite-Element Methods for Maxwell Equations in Complex Media,” _IEEE Transactions on Antennas and Propagation_ , vol. 56, no. 8, pp. 2150–2166, 2008.
* [33] A. Stern, Y. Tong, M. Desbrun, and J. E. Marsden, “Geometric Computational Electrodynamics with Variational Integrators and Discrete Differential Forms,” in _Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden_ , ser. Fields Institute Communications, D. E. Chang, D. D. Holm, G. Patrick, and T. Ratiu, Eds. New York, NY: Springer New York, 2015, pp. 437–475.
* [34] J. E. Marsden and A. Weinstein, “The Hamiltonian Structure of the Maxwell-Vlasov Equations,” _Physica D_ , vol. 4, no. 3, p. 394, 1982\.
* [35] G. Strang, “On the Construction and Comparison of Difference Schemes,” _SIAM Journal on Numerical Analysis_ , vol. 5, no. 3, pp. 506–517, 1968.
* [36] D. N. Arnold, “Periodic Table of the Finite Elements.” [Online]. Available: http://www-users.math.umn.edu/~arnold/femtable/index.html
* [37] J. Lohi and L. Kettunen, “Whitney forms and their extensions,” _Journal of Computational and Applied Mathematics_ , vol. 393, p. 113520, 2021.
* [38] H. F. Trotter, “On the Product of Semi-Groups of Operators,” _Proceedings of the American Mathematical Society_ , vol. 10, no. 4, pp. 545–551, 1959.
* [39] T. Huckle, “Approximate Sparsity Patterns for the Inverse of a Matrix and Preconditioning,” Technische Universitat Munchen, Tech. Rep. TUM-I9829, 1998.
* [40] L. N. Trefethen and D. Bau, _Numerical linear algebra_. Philadelphia: Society for Industrial and Applied Mathematics, 1997.
* [41] M. Shariff, “A constrained conjugate gradient method and the solution of linear equations,” _Computers & Mathematics with Applications_, vol. 30, no. 11, pp. 25–37, Dec. 1995.
|
# A One-Shot Reparameterization Method for Reducing the Loss of Tile Pruning
on DNNs
Yanchen Li1, Qingzhong Ai2, and Fumihiko Ino1 1: Osaka University, 1-5
Yamadaoka, Suita, Osaka 565-0871, Japan, {yc-li<EMAIL_ADDRESS>2:
University of Electronic Science and Technology of China, Chengdu, China,
<EMAIL_ADDRESS>
###### Abstract
Recently, tile pruning has been widely studied to accelerate the inference of
deep neural networks (DNNs). However, we found that the loss due to tile
pruning, which can eliminate important elements together with unimportant
elements, is large on trained DNNs. In this study, we propose a one-shot
reparameterization method, called TileTrans, to reduce the loss of tile
pruning. Specifically, we repermute the rows or columns of the weight matrix
such that the model architecture can be kept unchanged after
reparameterization. This repermutation realizes the reparameterization of the
DNN model without any retraining. The proposed reparameterization method
combines important elements into the same tile; thus, preserving the important
elements after the tile pruning. Furthermore, TileTrans can be seamlessly
integrated into existing tile pruning methods because it is a pre-processing
method executed before pruning, which is orthogonal to most existing methods.
The experimental results demonstrate that our method is essential in reducing
the loss of tile pruning on DNNs. Specifically, the accuracy is improved by up
to 17% for AlexNet while 5% for ResNet-34, where both models are pre-trained
on ImageNet.
###### Index Terms:
Deep learning, tile pruning, reparameterization
## I Introduction
Deep neural networks (DNNs) have demonstrated excellent performance on a wide
variety of tasks, such as computer vision [1, 2], speech recognition [3, 4],
and robotics [5, 6]. One backbone behind the growth in the abilities of DNNs
is an increasing number of trainable parameters. For instance, a recent state-
of-the-art model, CogView [7], achieved superior performance on text-to-image
generation tasks, thanks to its almost four billion parameters. However,
training a model with many parameters is time- and resource-intensive.
Therefore, the need for accelerating the computation on DNN is also
increasing. Nowadays, several mainstream acceleration methods, such as pruning
[8, 9, 10], mixed precision [11], and parallelization [12] have recently
attracted significant attention. Among these methods, mixed precision and
parallelization focus on the acceleration in the training phase of DNNs,
whereas pruning concentrates on the inference time.
This study focuses on pruning for the increasing need to accelerate the
application of DNNs. Pruning accelerates DNNs by deleting the trivial elements
in the weight matrix. The importance of each element is evaluated by the
importance score [13] so that the elements with small importance scores will
be deleted. Furthermore, to quantify the result of pruning, we calculate the
loss of pruning based on the importance score of the deleted elements,
generally summation. As the goal of pruning, we hope to get a minor pruning
loss, which brings higher inference accuracy, with a certain sparsity of
pruning.
However, there is a tradeoff between pruning loss and acceleration. On the one
hand, pruning loss increases when pruning sparsity increases. On the other
hand, acceleration from pruning is proportional to the sparsity. Consequently,
various pruning methods [14, 15, 16, 17] managed to better balance the
tradeoff between the pruning loss and acceleration.
Among those pruning methods, tile pruning [18] achieves a better tradeoff.
Tile pruning deletes the elements in the weight matrix by tiles while other
structured pruning methods eliminate the elements by a larger shape, such as a
row. The advantages of deleting elements by tiles can be reflected in the
following two aspects. First, owing to the smaller pruning unit of tiles, the
pruning loss is relatively smaller than other structured pruning methods.
Second, the computation for pruned tiles can be skipped with a small overhead,
and thus the pruned DNNs are accelerated. However, we found that the trained
DNNs empirically do not always fit well with tile pruning. Those important
elements are usually eliminated with unimportant elements, leading to large
pruning loss.
To this end, we propose a reparameterization method, called TileTrans, based
on the DNNs model for tile pruning. Instead of changing the pruning strategy,
TileTrans reparameterizes the DNNs model before the pruning. Specifically, we
use permutation to separate elements by their importance so that there are
either important or unimportant elements in the same tile. In this way, we
preserve more important elements after tile pruning and achieve minor pruning
loss. TileTrans is user-friendly in the sense that we reparameterize the DNN
model by a trick of matrix multiplication instead of retraining. Besides, we
keep the model architecture unchanged and therefore introduce no extra
computation for inference. Additionally, our algorithm is a straightforward
heuristic algorithm, which means that the algorithm’s efficiency is
guaranteed.
We evaluated TileTrans on two classical DNN models: AlexNet [19] and ResNet-34
[20], where both models are first pretrained on the ImageNet dataset [21]. We
reparameterized the DNN models by TileTrans before tile pruning on the models.
The experimental results show that TileTrans reduces the pruning loss and
improves accuracy by 17% on AlexNet and 5% on ResNet-34, respectively.
The main contributions of our study are as follows:
* •
We proposed a reparameterization method, named TileTrans, to permute the
weight matrix such that important and unimportant elements are permuted in the
same tile before tile pruning, respectively. As a result, we preserve more
important elements after tile pruning, and thus we reduce the pruning loss.
* •
We reparameterized DNN models in a novel way based on a trick of matrix
multiplication. Compared with other reparameterization methods, our method is
user-friendly because TileTrans requires no extra retraining on DNN models.
Furthermore, We kept the architecture of the model unchanged so that no
additional computation occurs after TileTrans.
* •
We proposed a heuristic method to build an appropriate permutation for
TileTrans to avoid time-consuming searching for the optimal permutation.
Besides, we verified that this heuristic method does reduce the loss of tile
pruning experimentally.
Eventually, we deduce that TileTrans is capable of building DNN models that
are more adaptable to tile pruning.
## II Related work
Notably, TileTrans can be fused with any tile pruning method seamlessly.
Various pruning strategies [22, 18, 14] have been proposed to reduce the loss
of tile pruning. For example, Guo et al. [18] deleted the tiles by the number
of important elements in the tile. TileTrans reparameterizes DNN before the
procedure of pruning; thus, it can be used together with these pruning
strategies to improve the performance of tile pruning.
Compared with the major reparameterization methods on DNNs, TileTrans needs no
extra training. Reparameterization has been used to change the data structure
of DNN for many purposes [23, 24, 25, 26]. Specifically, reparameterization is
also used to build a model with structural sparsity. Most reparameterization
methods design a particular loss function and train the model for a large
number of iterations. For example, Wen et al. [27] introduced the Lasso
regression to the loss function and trained a model with structural sparsity;
Ding et al.[28] proposed the ResRep, which punished unimportant channels in
the weight matrices during training. These studies achieved outstanding
performance in building structural sparsity, but they required extra training
on the models. In contrast, our TileTrans only reparameterizes DNN by matrix
multiplication without any training.
Some previous works also reduced the pruning loss by transforming the weight
matrix. However, these methods changed the model architectures; thus,
increasing the cost of computation. Liu et al. [29] added a transformation to
each layer, by which they transformed the format of each weight matrix into
the frequency domain. Guo et al. [18] duplicated each weight matrix and saved
the important elements in one of them. These methods improved the accuracy of
the pruned models but called for extra computation compared with the original
architecture. In contrast, TileTrans transforms weight matrices without
modifying the architecture. In other words, we add no extra computation to the
model.
## III Preliminary
Before introducing TileTrans, we have to present the preliminary about tile
pruning because we specialize TileTrans for reducing the loss of tile pruning.
Therefore, we will introduce the optimization objective and algorithm of tile
pruning as follows.
### III-A Importance Score
The importance score is a criterion used to evaluate the importance of an
element in the weight matrix. Given an element with weight $w$, we calculate
the importance score as $\mathcal{E}(w)$, where
$\mathcal{E}:\mathbb{R}\to\mathbb{R}$ denotes the criterion of importance
score. There are many criteria in previous works, such as calculating the
Hessian matrix [30] and deriving from Taylor expansion [13].
We can quantitatively define the loss of pruning with a certain importance
score. Generally, the pruning loss is an important score summation of the
deleted elements. Given the weights of the model as $\mathbf{W}$ and targeted
sparsity $s$, weights are pruned by $\mathit{f}(\mathbf{W},s;\mathcal{E})$,
where $f$ deletes the elements in $\mathbf{W}$ to the sparsity of $s$
depending on $\mathcal{E}$. Furthermore, pruning can be described as an
optimization problem with the objective as follows:
$\displaystyle\min_{f}$
$\displaystyle\mathcal{L}(\mathbf{W},s;\mathcal{E})=\left|\sum_{w}^{w\in\mathbf{W}}\mathcal{E}(w)-\sum_{w}^{w\in\mathit{f}(\mathbf{W},s;\mathcal{E})}\mathcal{E}(w)\right|$
(1) s.t. $\displaystyle\mathbf{W}\in\mathbb{W},0\leq s\leq 1,$
where $\mathcal{L}$ denotes the loss function of pruning and $\mathbb{W}$ is
the set of all possible weights. In other words, the loss of pruning can be
minimized by deleting the elements with the smallest importance scores. In
tile pruning, there are certain restrictions on the shape of the pruning unit.
Therefore, we evaluate the average importance score of the elements in the
tiles for tile pruning and delete the most unimportant tiles.
### III-B Tile Shape
The shape of tile pruning depends on the devices or libraries. For example,
Chen et al. [17] pruned the weights in the shape of the loading unit for the
tensor core so that the computation for pruned elements can be skipped with a
small overhead. Guo et al. [18] used a shape of $128\times 1$, because CUTLASS
[31] loads $128\times 1$ elements at once from the global memory to the shared
memory on the GPU system. Given the tile shape of $a\times b$, tile pruning
sometimes is called the “block pruning” [14] when $a=b$. Note that we regard
the block pruning as tile pruning in this paper. Especially, tile pruning in
the shape of $1\times 1$ is called unstructured pruning [32]. The pruning loss
of unstructured pruning is usually lower than that of tile pruning, where
$a>1$ and $b>1$. Unstructured pruning precisely deletes the unimportant
elements while tile pruning eliminates the important elements together with
the unimportant elements in the same tile. Therefore, unstructured pruning is
usually used as the baseline for tile pruning when evaluating the accuracy of
pruned models.
(a)
(b)
(c)
Figure 1: Examples for pruning the $4\times 4$ weight matrix to the sparsity
of 0.5 with the unstructured pruning (a), the $2\times 2$ tile pruning before
reparameterization (b), and the $2\times 2$ tile pruning after
reparameterization (c), respectively. We fill the background with different
colors for the corresponding tiles in (b) and (c) for easy observation. The
numbers in cells are the importance scores for elements. The values of pruning
loss are shown at the bottom of the figures.
### III-C Algorithm of Tile Pruning
For tile pruning, we evaluate the importance of every tile and delete
unimportant tiles according to the importance score, which is calculated from
the elements in the tile. We show details in Algorithm 1. From lines 2 to 6,
we traverse all weight matrices in the model and calculate the importance
scores for every tile. The importance score of a tile is defined as the
average of the importance scores of the elements in that tile. After
evaluating for tiles, we set a threshold according to the sparsity at line 7.
Because the sparsity is the percentage of zero elements in all matrices, we
delete unimportant tiles whose importance scores are smaller than the
threshold. Finally, we retrain the model to recover the accuracy for inference
after tile pruning.
Algorithm 1 TilePruning($\mathbf{W}$, $\mathcal{E}$, $a$, $b$, $s$)
1:Weight matrices of the model $\mathbf{W}$, metrics for importance score
$\mathcal{E}$, height of tile $a$, width of tile $b$, and targeted sparsity
$s$.
2:Pruned weight matrices $\mathbf{W}$.
3:$\mathbf{V}\leftarrow\emptyset$
4:foreach $W\in\mathbf{W}$ do $\triangleright$ Traverse the weights in the
layers
5: foreach $T\in W$ do
6: $\mathbf{V}\leftarrow\mathbf{V}\cup\\{v\leavevmode\nobreak\
|\leavevmode\nobreak\ v=\sum_{i}^{a}\sum_{j}^{b}\mathcal{E}(T_{i,j})/ab\\}$
7: end for
8:end for
9:Set threshold $t$ such that $|\\{v\leavevmode\nobreak\ |\leavevmode\nobreak\
v\in\mathbf{V},v<t\\}|/|\mathbf{V}|=s$
10:Delete the tile with importance score smaller than $t$.
## IV Weights Transformation
Naturally, the distribution of the elements in the weight matrices is usually
unfriendly to tile pruning. The important and unimportant elements are mixed
in the same tile. We show the examples in Figs. 1(a) and 1(b). For
unstructured pruning in Fig. 1(a), unimportant elements are precisely deleted,
and thus the loss for unstructured pruning is the smallest. For tile pruning
in Fig. 1(b), the elements that are alive in unstructured pruning are deleted.
Consequently, the loss for tile pruning is more significant than for
unstructured pruning. To separate the important and unimportant elements, we
propose TileTrans that reparameterizes DNN before the pruning by matrix
transformation.
### IV-A Idea of Permutation
The loss for tile pruning can be reduced by matrix transformation. We
transform the weight matrices to a better format, where the important elements
are combined together in the same tiles as far as possible. For example, in
Fig. 1(c), we switch the first row of the matrix with the third row.
Accordingly, more important elements are permuted in the same tiles than in
Fig. 1(b). Finally, the loss for the $2\times 2$ tile pruning on the
reparameterized weight matrix becomes smaller than that on the original weight
matrix.
The critical point is that the transformation must keep the model’s output
unchanged. The output of DNN is changed if we transform the weight matrix. One
option to recover the output is adding extra calculation after the inference.
However, we should avoid increasing the computational amount for inference;
thus, we keep the model architecture unchanged. Without modifying the
architecture, we managed to recover the model’s output by a trick of matrix
multiplication.
### IV-B Algorithm of TileTrans
We designed a reparameterization method based on matrix multiplication. The
main components of the DNN models are the linear layer [33] and the
convolutional layer [34], which can be represented as matrix multiplication.
Furthermore, the DNN model consists of multiple layers, where the calculations
are transitive. For example, Fig. 2 shows a partial architecture of ResNet,
where the output of a layer is the input of another layer. Utilizing this
transitivity of the calculations in DNN, we can reparameterize DNN without any
training by the properties of matrix multiplication.
Figure 2: Partial architecture of ResNet. A block denotes a layer with
weights. An arrow represents a data dependence inherent in the data flow.
The reparameterization procedure is a series of operations that transform two
layers. For a single operation, we first build an invertible transformation
$R$ for the previous layer and build its inverse $R^{-1}$ for the next layer.
In this way, the output of the transformed layer can be recovered by
calculating the next layer. We take Fig. 2 as an example to show how this
trick works. Suppose that the weights of the layer $i$ is $W_{i}$, the input
is $I_{i}$, and the output is $O_{i}$. The calculations in layer $A$ and $B$
then can be given as
$\begin{split}O_{A}=I_{A}W_{A}^{T},\end{split}$ (2) $O_{B}=O_{A}W_{B}^{T}.$
(3)
After that, we transform these weights by a matrix transformation $R$ as
follows
$\begin{split}O_{A}R^{T}=I_{A}(RW_{A})^{T},\end{split}$ (4)
$O_{B}=O_{A}R^{T}(W_{B}R^{-1})^{T}.$ (5)
In other words, we transform the previous layer and use the next layer to
recover the output. Therefore, the output of the two layers is unchanged, even
though we transform their weights. Finally, we reparameterize the entire DNN
model based on this trick. However, the restraint for the reparameterization
becomes more strict when the model architecture is more complex.
Some layers in the model have to share the same matrix transformation when
there are residuals in the model. Residual [20] is a calculation that reuses
the output of the previous layer as the input of the deeper layer. For
example, in Fig. 2, the output of layer A is the input of layers B and D.
Before calculating the matrix multiplication of layer D, we first calculate
the sum of $O_{A}$ and $O_{C}$ as follows:
$O_{A}=I_{A}W_{A}^{T},$ (6) $O_{B}=O_{A}W_{B}^{T},$ (7)
$O_{C}=O_{B}W_{C}^{T},$ (8) $O_{D}=(O_{A}+O_{C})W_{D}^{T}.$ (9)
Supposing the transformation for layer $i$ as $R_{i}$, the calculations for
the transformed layers are given as
$O_{A}R_{A}^{T}=I_{A}(R_{A}W_{A})^{T},$ (10)
$O_{B}R_{B}^{T}=O_{A}R_{A}^{T}(R_{B}W_{A}R_{A}^{-1})^{T},$ (11)
$O_{C}R_{C}^{T}=O_{B}R_{B}^{T}(R_{C}W_{C}R_{B}^{-1})^{T},$ (12)
$O_{D}=(O_{A}R_{A}^{T}+O_{C}R_{C}^{T})(W_{D}R_{C}^{-1})^{T},$ (13)
where $O_{D}=(O_{A}+O_{C})W_{D}^{T}$ only if $R_{A}=R_{C}$. Thus, layers A and
C must share the same transformation. The example of Fig. 2 is simple, but the
actual model is usually more sophisticated.
There may be groups of layers that must share the same transformation. For
simplicity, we define the layers that have to share the same transformation as
the layer group. This layer group is built according to the relations of
parents and children, which can be defined as follows.
###### Definition 1 (Parents and Children).
Given a layer A, parents of A are the layers that send their output to A as
the inputs of A. Meanwhile, children of A are the layers that receive the
output of A as their inputs.
Specifically, we present a theorem as follows.
###### Theorem 1.
Given a layer A in the DNN model, the parents of A must share the same
transformation.
###### Proof.
According to Eq. (13), assuming the weight is $W$, the transformation is $R$,
and the output is $O$, if there are $n$ parents for a layer, the output of the
layer is given by
$O=(O_{1}R_{1}^{T}+O_{2}R_{2}^{T}+...+O_{n}R_{n}^{T})(WR^{-1})^{T},$ (14)
where $O_{i}$ denotes the output of the $i$-th parent, and $R_{i}$ denotes the
transformation of the $i$-th parent. Then
$O=(O_{1}+O_{2}+...+O_{n})W^{T},$ (15)
only if $R_{1}=R_{2}=...=R_{n}=R$. ∎
Algorithm 2 TileTrans($\mathbf{M}$, $\mathcal{E}$)
1:DNN model $\mathbf{M}$ and metrics for importance score $\mathcal{E}$.
2:Reparameterized DNN model $\mathbf{M}$.
3:$\mathbb{G}\leftarrow\emptyset$ $\triangleright$ Initialize the layer group
4:foreach $L\in\mathbf{M}$ do
5: $\mathbf{G}\leftarrow$ the parents of $L$
6: $\mathbb{G}\leftarrow\mathbb{G}\cup\mathbf{G}$
7:end for
8:$\hat{\mathbb{G}}\leftarrow\\{\mathbf{G}\leavevmode\nobreak\
|\leavevmode\nobreak\
\mathbf{G}\leftarrow\mathbf{G}_{1}\cup\mathbf{G}_{2}\text{ if
}\mathbf{G}_{1}\cap\mathbf{G}_{2}\neq\emptyset,\text{ and
}\mathbf{G}_{1},\mathbf{G}_{2}\in\mathbb{G}\\}$ $\triangleright$ Merge the
intersecting groups
9:foreach $\mathbf{G}\in\hat{\mathbb{G}}$ do
10: $R\leftarrow$ BuildTrans($\mathbf{G},\mathcal{E}$) $\triangleright$ Build
the transformation
11: foreach $L\in\mathbf{L}$ do
12: Transform the weights of $L$ by $R$
13: end for
14: Transform the weights in the children of the layers in $\mathbf{G}$ by
$R^{-1}$
15:end for
According to Theorem 1, we design an algorithm for TileTrans, which is shown
in Algorithm 2. From lines 2 to 4, we first find the sets of weights that must
share the same transformation. Notably, some subsets in $\mathbb{G}$ may
intersect each other. For example, given two sets of layers
$\mathbf{G}_{0}\text{ and }\mathbf{G}_{1}$, $\mathbf{G_{0}}=\\{L_{A},L_{B}\\}$
and $\mathbf{G_{1}}=\\{L_{B},L_{C}\\}$, all $L_{A}$, $L_{B}$, and $L_{C}$ must
share the same transformation for their weights. Therefore, we merge the
subsets intersecting each other at line 6. According to
$\mathbf{G}\in\mathbb{G}$, we build the appropriate transformation $R$, which
is introduced later, for the weights at line 8. To obtain the correct output,
we transform the weights in the children of the layers in $\mathbf{G}$ by
$R^{-1}$ at line 12.
The modern DNN models improve their performance by introducing nonlinear
layers, such as ReLU [35], pooling [34], and normalization [36]. For such
nonlinear layers, we restraint the transformation $R$ to only the permutation
of rows in Algorithm 2. This restraint guarantees the correctness of the model
because permutation is a linear transformation that only changes the location
of the elements. By contrast, a linear transformation that changes the scale
of elements does not have this guarantee because it fails to recover the
output of the model.
Additionally, the permutation of columns works in the TileTrans, but Algorithm
2 needs to be “reversed.” In Algorithm 2, we define the permutation of rows
$R$ for the parents at line 8 and permute the weight matrices at line 10.
After that, we use the permutation of columns $R^{-1}$ to recover the output
for the children at line 12. Therefore, TileTrans in rows is a procedure from
parents to children. Meanwhile, TileTrans in columns is a procedure from
children to parents. Thus, for the permutation of columns, we build the
transformation for children and then recover the output by the calculations in
parents.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3: Loss of tile pruning on AlexNet (a, b, and c) and ResNet-34 (d, e,
and f) that are pre-trained on the ImageNet for different sparsity. The tile
shape in Figs. (a) and (d) are $256\times 1$, that in Figs. (b) and (e) are
$16\times 16$, and that in Figs. (c) and (f) are $32\times 32$. Specially, we
define the result for unstructured tile pruning, i.e., the $1\times 1$ tile
pruning, as the baseline for tile pruning. Besides, the legend with “trans”
denotes the result for the model reparameterized by TileTrans. Meanwhile,
“diff” denotes the difference between tile pruning and unstructured pruning,
which is defined as Eq. (16). For simplicity, we normalize the pruning loss
and difference by 0-1 normalization.
### IV-C Heuristic Permutation
Ideally, if all-important elements were aggregated in the same tile, all
important elements then exist after tile pruning. That is, the result of tile
pruning becomes the same as that of unstructured pruning. TileTrans aims to
make the result of tile pruning as close as possible to that of unstructured
pruning. Because the loss for unstructured pruning is the smallest, we assume
the loss for unstructured pruning is the baseline. Based on this baseline, we
give the optimization problem for the TileTrans as follows:
$\displaystyle\min\left|\sum_{w}^{w\in\mathit{f}_{u}(\mathbf{W},s;\mathcal{E})}\mathcal{E}(w)-\sum_{w}^{w\in\mathit{f}_{t}(\mathcal{R}(\mathbf{W}),s;\mathcal{E})}\mathcal{E}(w)\right|,$
(16) $\displaystyle\textrm{s.t.}\mathbf{W}\in\mathbb{W},0\leq s\leq 1,$
where $\mathit{f}_{u}:\mathbb{W}\to\mathbb{W}$ denotes the unstructured
pruning method, $\mathit{f}_{t}:\mathbb{W}\to\mathbb{W}$ denotes the tile
pruning method, and $\mathcal{R}:\mathbb{W}\to\mathbb{W}$ denotes the
reparameterization method. According to Eq. (16), given specific sparsity $s$,
we need to find the permutation that minimizes the difference between tile
pruning and unstructured pruning methods.
However, searching for the optimal solution for Eq. (16) is time consuming.
The most intuitive method to solve Eq. (16) is to try all possible
permutations and verify the effect, which costs too much time. Given a model
with $k$ layers and each layer with weights of $n$ rows, the time complexity
to traverse all possible permutations is $O((n!)^{k})$. Therefore, we propose
a heuristic method to build a relatively good permutation without searching.
Algorithm 3 BuildTrans($\mathbf{G}$, $\mathcal{E}$)
1:Group of layers $\mathbf{G}$ and metrics for importance score $\mathcal{E}$.
2:Permutation $R$.
3:$\mathbf{W}\leftarrow$ all weights of the layers in $\mathbf{G}$
4:$\hat{W}\leftarrow$ concatenate $W\in\mathbf{W}$
5:$\hat{W}_{R}\leftarrow\hat{W}$
6:Sort the rows in $\hat{W}_{R}$ $\triangleright$ According to $\mathcal{E}$
7:Build $R$ such that $R\hat{W}=\hat{W}_{R}$
We sort the rows according to their average importance scores, which are shown
in Algorithm 3. In line 1, we concatenate the weights that share the same
transformation in the dimension of rows to evaluate them together. These
weights can be concatenated because they have the same number of rows. After
that, we sort the rows of the concatenated weights according to the average
importance score, which is calculated by the metrics $\mathcal{E}$. Finally,
we build the transformation $R$ such that $R\hat{W}=\hat{W}_{R}$, where
$\hat{W}$ denotes the concatenated weights, and $\hat{W}_{R}$ denotes the
sorted concatenated weights.
The sorting procedure is based on two assumptions:
###### Assumption 1.
The averages of the rows follow the normal distribution.
###### Assumption 2.
The variance of importance scores for elements in a row is small.
We make the assumptions depending on previous studies and our experience.
Assumption 1 is inspired by previous studies on channel pruning [37], which
prunes the elements in rows. In channel pruning, the importance score for a
row is evaluated by the average of the importance scores for the elements in
the row. After channel pruning, rows are deleted without a few important rows,
meaning that a few important rows are in the matrix, whereas the other rows
are unimportant. Similarly, there are few large values in the normal
distribution, while the others are smaller. We use the normal distribution to
approximate the distribution of the importance scores for the rows. For
Assumption 2, we make it empirically because we observed that importance
scores for the elements in a row are usually all large when the row is
important. The unimportant and important elements are naturally separated
after sorting rows depending on the assumptions. Finally, the important
elements are preserved after tile pruning because the important elements are
in the same tiles. In other words, we reduce the loss of tile pruning.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4: Accuracy of the pruned AlexNet (a, b, and c) and ResNet-34 (d, e,
and f) that are pre-trained on the ImageNet for different sparsity. The tile
shape in Figs. (a) and (d) are $256\times 1$, that in Figs. (b) and (e) are
$16\times 16$, and that in Figs. (c) and (f) are $32\times 32$. Specially, we
define the result for unstructured tile pruning, i.e., the $1\times 1$ tile
pruning, as the baseline for tile pruning. Besides, the legend with “trans”
denotes the result for the model reparameterized by TileTrans.
## V Experiment
In the experiments, we evaluated the proposed TileTrans on the following
metrics:
* •
Loss. Pruning loss is our direct optimization objective. Therefore, we first
show the pruning loss before and after the reparameterization of TileTrans. At
the same time, we calculate the difference between tile pruning and
unstructured pruning for the following two reasons. First, the difference in
the pruning loss before and after TileTrans is negligible; therefore, we
illustrate the normalized difference in figures for better observation.
Second, we designed an optimization objective for TileTrans as Eq. (16), which
is the difference between tile pruning and unstructured pruning; thus, we
calculate this difference to verify the heuristic permutation does reduce the
pruning loss.
* •
Accuracy. The accuracy of the model is the most intuitive metric that
evaluates the performance of the model. We are unaware of the performance of
the pruned models only by evaluating pruning loss. Besides, the difference
between the pruning loss with different tile shapes on the same model is
negligible, so we cannot assess how tile shape affects the performance of
TileTrans only by loss. Therefore, we calculate the accuracy of the pruned
models to evaluate TileTrans further.
We design sets of controlled experiments on AlexNet and ResNet-34 to evaluate
the performance of TileTrans. For one set, we first reparameterized the model
by TileTrans and then pruned the model; for another set, we pruned the model
directly. For simplicity, we choose the L1 norm as the criteria of importance
score. Finally, we calculated the pruning loss and model accuracy of the
pruned models.
Besides, we evaluate TileTrans on tile pruning of different tile shapes to
assess how tile shape affects the performance of TileTrans. Expressly, we set
the tile shape as $256\times 1$, $16\times 16$, or $32\times 32$. For
$256\times 1$ and $16\times 16$, the sizes are the same, while the aspect
ratios are different; for $16\times 16$ and $32\times 32$, the aspect ratios
are the same, while the sizes are different. Therefore, we know how the size
and aspect ratio of tile affect the performance of TileTrans by comparing the
result of these tile shapes.
We realized TileTran with PyTorch and retrained the models on eight GeForce
GTX 1080 GPUs. Our code is available on https://github.com/LoonLi/TileTrans.
### V-A Loss
The results of pruning loss are shown in Fig. 3. Specially, we also illustrate
the result of unstructured pruning, i.e., the $1\times 1$ pruning, in Fig. 3
as the baseline of tile pruning. Accordingly, with specific sparsity, the
difference between tile pruning and unstructured pruning is the distance
between the point of tile pruning and unstructured pruning.
TileTrans reduced most pruning loss at the sparsity of about 60; however, it
reduced little loss when the sparsity was sufficiently small or large. This is
due to the minority tiles with all unimportant or important elements, which
exist naturally in weight matrices. These tiles with all unimportant elements
are deleted when the sparsity is small. In contrast, those with all important
elements are still preserved when the sparsity is large. Consequently,
TileTrans avoids degrading the pruning loss when the sparsity is sufficiently
small or large. Meanwhile, those tiles with important and unimportant elements
are permuted by TileTrans; thus, the pruning loss is reduced when the sparsity
is moderate.
TileTrans reduced less pruning loss on ResNet-34 than on AlexNet. This is
attributed to the following reasons: more complex architecture and smaller
weight matrices. First, the architecture of ResNet-34 is more complex than
that of AlexNet because there are multiple connections of residual in
ResNet-34. According to Theorem 1, layers with residuals must share the
transformation with other layers that send the residual to them. In contrast,
AlexNet is a simple architecture in which no layer needs to share the same
transformation. Thus, searching for good transformations for ResNet-34 is more
difficult. Second, the weight matrices of ResNet-34 are small for the
permutation. Specifically, the heights of most weight matrices in ResNet-34
are less than 512. Meanwhile, there are large weight matrices with a height of
4096 in AlexNet. In other words, we have fewer choices to permute the weight
matrices in rows on ResNet-34 than on AlexNet.
Pruning loss is the most direct metric to evaluate TileTrans, but it is not
intuitive enough for understanding the performance of the models.
Specifically, we are unaware of the difference between different tile shapes
since their results are very similar in the same model. Therefore, we also
evaluate TileTrans on the metric of the model accuracy.
### V-B Accuracy
We define accuracy as the probability that models correctly infer the types of
images in the test data. The accuracy $P_{a}$ is calculated as
$P_{a}=N_{c}/N_{t}$, where $N_{t}$ is the amount of test data $N_{t}$ and
$N_{c}$ is the amount of correctly inferred images.
TileTrans improved the accuracies of both AlexNet and ResNet-34 after tile
pruning, which is a natural result because TileTrans reduced the pruning loss.
We show results of model accuracy in Fig. 4. Specifically, the accuracies of
both AlexNet and ResNet-34 were improved most under the tile shape of
$32\times 32$. In detail, the pruned AlexNet was improved by up to 17% at the
sparsity of 60%; the pruned ResNet-34 was improved by up to 5% at the sparsity
of 70%. Specially, the accuracy of the pruned AlexNet decreased after
TileTrans at the sparsity of 80% under the tile shape of $16\times 16$.
However, in this case, the pruning loss was reduced according to Fig. 3(b).
Consequently, this decrease is not caused by increased pruning loss and may be
improved by changing the criteria of importance score.
Between the tile shapes of $256\times 1$ and $16\times 16$, TileTrans improved
the accuracy more for $256\times 1$ on AlexNet while for $16\times 16$ on
ResNet-34. According to this result, we draw two conclusions about the
performance of TileTrans. On the one hand, TileTrans improve more accuracy for
strip-shaped tile when the weight matrix is large enough. On the other hand,
TileTrans improves more accuracy for square-shaped tiles when the weight
matrix is too small. ResNet-34 consists of multiple convolutional layers with
heights smaller than $512$. As a result, TileTrans reduced little pruning loss
on ResNet-34 for $256\times 1$ because the permutation cannot exchange the
elements among tiles. Meanwhile, AlexNet contains two large linear layers
whose weight matrices are $4096\times 4096$. Therefore, the important and
unimportant elements were combined in different tiles on AlexNet for
$256\times 1$. In this case, the unimportant elements are deleted more
precisely for $256\times 1$ than $16\times 16$.
Between the tile shapes of $16\times 16$ and $32\times 32$, TileTrans improved
the accuracy more for $32\times 32$ on both models. Consequently, we conclude
that TileTrans improves accuracy more for larger tile sizes. The probability
of a tile in the weight matrix filled with important elements is low when the
tile size is large. Meanwhile, TileTrans builds weight matrices with more
tiles that are filled with important elements. In other words, TileTrans
reduces pruning loss more when the tile size is large.
In summary, we conclude three rules for TileTrans as follows:
1. 1.
TileTrans reduces more loss for strip-shaped than square-shaped tiles when the
sizes of tiles are the same, and the weight matrix is large enough for
permutation.
2. 2.
TileTrans reduces more loss for square-shaped than strip-shaped tiles when the
sizes of tiles are the same, and the weight matrix is small.
3. 3.
TileTrans reduces more loss for the tiles with large size.
## VI Conclusion
In this study, we proposed a one-shot reparameterization method, called
TileTrans, to reduce the loss of tile pruning on DNNs by permuting the weight
elements. We heuristically build an appropriate permutation to avoid time-
consuming searching for the optimal permutation. Besides, TileTrans requires
no retraining and keeps the model architecture unchanged. The experiments
demonstrated that TileTrans reduced the loss of tile pruning on AlexNet and
ResNet-34.
## Acknowledgments
This research was in part supported by “Program for Leading Graduate Schools”
of the Ministry of Education, Culture, Sports, Science and Technology, Japan
and the Japan Society for the Promotion of Science KAKENHI grant number
20K21794.
## References
* [1] S. Xiong et al., “Explore Visual Concept Formation for Image Classification,” in Proc. 22nd International Conference on Machine Learning (ICML 2021), pp. 11470–11479, 2021.
* [2] Z. Liu et al., “Swin transformer: Hierarchical vision transformer using shifted windows,” in Proceedings of the 18th IEEE/CVF International Conference on Computer Vision (ICCV 2021), pp. 10012–10022, 2021.
* [3] V. Popov et al., “Grad-TTS: A Diffusion Probabilistic Model For Text-to-Speech.” ArXiv, 2021. abs/2105.06337.
* [4] D. Min et al., “Meta-StyleSpeech: Multi-Speaker Adaptive Text-to-Speech Generation.” ArXiv, 2021. abs/2106.03153.
* [5] Y. Chebotar et al., “Actionable Models: Unsupervised Offline Reinforcement Learning of Robotic Skills.” ArXiv, 2021. abs/2104.07749.
* [6] N. Sünderhauf et al., “The Limits and Potentials of Deep Learning for Robotics,” The International Journal of Robotics Research, vol. 37, no. 4-5, pp. 405–420, 2018.
* [7] M. Ding et al., “CogView: Mastering Text-to-Image Generation via Transformers.” ArXiv, 2021. abs/2105.13290.
* [8] S. Han et al., “Learning Both Weights and Connections for Efficient Neural Networks.” ArXiv, 2015. abs/1506.02626.
* [9] W. Wang et al., “Accelerate CNNs from Three Dimensions: A Comprehensive Pruning Framework,” in Proc. 22nd International Conference on Machine Learning (ICML 2021), pp. 10717–10726, 2021.
* [10] A. Jordao et al., “Discriminative Layer Pruning for Convolutional Neural Networks,” IEEE Journal of Selected Topics in Signal Processing, vol. 14, no. 4, pp. 828–837, 2020.
* [11] P. Micikevicius et al., “Mixed Precision Training.” ArXiv, 2017. abs/1710.03740.
* [12] J. Dean et al., “Large scale distributed deep networks,” Advances in Neural Information Processing Systems, vol. 25, 2012.
* [13] P. Molchanov et al., “Importance Estimation for Neural Network Pruning,” in Proc. IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR 2019), pp. 11264–11272, 2019.
* [14] H. Peng et al., “Accelerating Transformer-based Deep Learning Models on FPGAs Using Column Balanced Block Pruning,” in Proc. 22nd International Symposium on Quality Electronic Design (ISQED 2021), pp. 142–148, 2021.
* [15] A. Ren et al., “DARB: A Density-Adaptive Regular-Block Pruning for Deep Neural Networks.,” in Proc. 34th Association for the Advancement of Artificial Intelligence (AAAI 2020), pp. 5495–5502, 2020.
* [16] S. Narang et al., “Block-Sparse Recurrent Neural Networks.” ArXiv, 2017\. abs/1711.02782.
* [17] S. Chen et al., “ET: Re-thinking Self-Attention for Transformer Models on GPUs,” in Proc. 34th International Conference for High Performance Computing, Networking, Storage and Analysis (SC 2021), pp. 1–18, 2021.
* [18] C. Guo et al., “Accelerating Sparse DNN Models without Hardware-Support via Tile-Wise Sparsity,” in Proc. 33rd International Conference for High Performance Computing, Networking, Storage and Analysis (SC 2020), pp. 1–15, 2020.
* [19] A. Krizhevsky et al., “Imagenet Classification with Deep Convolutional Neural Networks,” Advances in Neural Information Processing Systems, vol. 25, pp. 1097–1105, 2012.
* [20] K. He et al., “Deep residual Learning for Image Recognition,” in Proceedings of the 30th IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2016), pp. 770–778, 2016.
* [21] J. Deng et al., “Imagenet: A Large-Scale Hierarchical Image Database,” in Proc. 24th IEEE Conference on Computer Vision and Pattern Recognition (CPVR 2009), pp. 248–255, 2009.
* [22] S. Lin et al., “Towards Optimal Structured CNN Pruning via Generative Adversarial Learning,” in Proc. 32nd IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR 2019), pp. 2790–2799, 2019.
* [23] X. Ding et al., “Acnet: Strengthening the Kernel Skeletons for Powerful CNN via Asymmetric Convolution Blocks,” in Proc. 17th IEEE/CVF International Conference on Computer Vision (ICCV 2019), pp. 1911–1920, 2019\.
* [24] X. Ding et al., “Diverse Branch Block: Building a Convolution as an Inception-like Unit,” in Proc. 34th IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR 2021), pp. 10886–10895, 2021.
* [25] X. Ding et al., “Repmlp: Re-parameterizing convolutions into fully-connected layers for image recognition.” ArXiv, 2021. abs/2105.01883.
* [26] X. Ding et al., “Repvgg: Making Vgg-Style Convnets Great Again,” in Proc. 34th IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR 2021), pp. 13733–13742, 2021.
* [27] W. Wen et al., “Learning Structured Sparsity in Deep Neural Networks,” Advances in Neural Information Processing Systems, vol. 29, pp. 2074–2082, 2016.
* [28] X. Ding et al., “Resrep: Lossless CNN Pruning via Decoupling Remembering and Forgetting,” in Proc. 18th IEEE/CVF International Conference on Computer Vision (ICCV 2021), pp. 4510–4520, 2021.
* [29] Z. Liu et al., “Frequency-Domain Dynamic Pruning for Convolutional Neural Networks,” in Proc. 32nd International Conference on Neural Information Processing Systems (NeurIPS 2018), pp. 1051–1061, 2018.
* [30] Y. LeCun et al., “Optimal Brain Damage,” in Proc. 3rd Advances in Neural Information Processing Systems (NeurIPS 1990), pp. 598–605, 1990.
* [31] K. Andrew et al., “CUTLASS: Fast Linear Algebra in CUDA C++.” https://developer.nvidia.com/blog/cutlass-linear-algebra-cuda/, 2017. accessed on February 1, 2022.
* [32] Z. Liu et al., “Rethinking the Value of Network Pruning.” ArXiv, 2018\. abs/1810.05270.
* [33] W. Liu et al., “A Survey of Deep Neural Network Architectures and Their Applications,” Neurocomputing, vol. 234, pp. 11–26, 2017.
* [34] S. Albawi et al., “Understanding of A Convolutional Neural Network,” in Proc. 24th International Conference on Engineering and Technology (ICET 2017), pp. 1–6, 2017.
* [35] A. F. Agarap, “Deep Learning Using Rectified Linear Units (ReLu).” ArXiv, 2018\. abs/1803.08375.
* [36] S. Ioffe et al., “Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift,” in Proceedings of the 16th International Conference on Machine Learning (ICML 2015), pp. 448–456, 2015\.
* [37] J.-H. Luo et al., “Thinet: A Filter Level Pruning Method for Deep Neural Network Compression,” in Proc. 16th IEEE International Conference on Computer Vision (ICCV 2017), pp. 5058–5066, 2017.
|
# An example for Kuznetsov-Shinder conjecture
Tanya Kaushal Srivastava Indian Institute of Technology Gandhinagar,
Gandhinagar-382355, India<EMAIL_ADDRESS>
###### Abstract.
We give an example for the Kuznetsov-Shinder conjecture, with infinitely many
non-isomorphic but L-equivalent varieties.
## 1\. The Grothendieck ring of varieties
Let $k$ be a field of characteristic zero. One define the Grothendieck ring of
varieties over $k$ as the ring generated by isomorphism classes $[X]$ of
algebraic varieties $X$ over $k$ with relations $[X]=[Z]+[U]$ for every closed
subvariety $Z\subset X$ having open complement $U\subset X$. The product
structure of this ring is induced by the products of varieties. The ring is
denoted by $K_{0}(Var/k)$. The ring has a unit element given by the
equivalence class of a point $[Spec(k)]$. This ring was introduced by
Grothendieck in his correspondence with Serre [3]. Results of Bittner [1] and
Looijenga [16] give an alternate description of this ring, showing that the
ring is generated by isomorphism classes of smooth projective varieties
subject to the relation: $[X]+[Z]=[Y]+[E]$, whenever $Y$ is smooth projective,
$Z\subset Y$ is a smooth subvariety, $X$ is a blow-up of $Y$ along $Z$, and
$E\subset X$ is the exceptional divisor. The following lemmas are very useful
and easy consequences of the definition of Grothendieck ring of varieties.
###### Lemma 1.1.
[12, Lemma 2.1] Assume $M\rightarrow S$ is a Zariski locally trivial fibration
with fiber $F$. Then $[M]=[S][F]$ in $K_{0}(Var/k)$.
###### Lemma 1.2.
In the Grothendieck ring of varieties $K_{0}(Var/k)$, we have:
1. (1)
$[\emptyset]=0$
2. (2)
$[Spec\ k]=1$
3. (3)
$[\mathbb{P}^{n}]=1+\mathbb{L}+\ldots+\mathbb{L}^{n}$, where
$\mathbb{L}=[\mathbb{A}^{1}]$.
The above lemma is very easy to prove, see for example [18].
Poonen has shown that the ring $K_{0}(Var/k)$ is not a domain [19]. Let us
denote by $\mathbb{L}$ the equivalence class of the affine line. It has been
further shown that $\mathbb{L}$ is a zero-divisor in the ring $K_{0}(Var/k)$
[2], [17], [8].
###### Definition 1.3.
Let $X$ and $Y$ be smooth projective varieties. They are called $D$-equivalent
if their derived categories $D^{b}(X)$ and $D^{b}(Y)$ of bounded complexes of
coherent sheaves are equivalent as triangulated categories, i.e., there exists
an equivalence of categories $\phi:D^{b}(X)\rightarrow D^{b}(Y)$ which
commutes with the translations and sends any distinguished triangle to a
distinguished triangle.
###### Definition 1.4.
Two algebraic varieties $X$ and $Y$ are called $L$-equivalent if the
differences $[X]-[Y]$ vanishes in the localization
$K_{0}(Var_{\mathbb{C}})[\mathbb{L}^{-1}]$. Equivalently, for some $n>0$ we
have $([X]-[Y]).\mathbb{L}^{n}=0$.
Question(Kuznetsov-Shinder): Assume X and Y are $D$-equivalent smooth
projective varieties. What is the relation between the classes $[X]$ and $[Y]$
in the Grothendieck ring $K_{0}(Var/k)$?
A partial answer to the above question of theirs was conjectured by them.
Kuznetsov-Shinder Conjecture: If $X$ and $Y$ are smooth projective simply
connected varieties such that $D(X)$ and $D(Y)$, then there is a non-negative
integer $r\geq 0$ such that $([X]-[Y])\mathbb{L}^{r}=0$. Equivalently,
$D$-equivalence of simply connected varieties implies their $L$-equivalence.
###### Remark 1.5.
The condition of being simply connected is necessary. Efimov [5] and
independently Ito et. al [9], showed that there exist derived equivalent
Abelian varieties over $\mathbb{C}$ which are not $L$-equivalent.
For a list of known positive cases of Kusnetsov-Shinder conjecture, we refer
the reader to the survey by Ito et. al [9].
### 1.1. An example in support: Rational 3-folds
Lesieutre [14] gave an example of a rational threefold to show that a smooth
projective variety can have infinitely many Fourier-Mukai partners, i.e.,
there are infinitely many non-isomorphic smooth projective varieties (all
rational threefolds) whose derived category of coherent sheaves are all
isomorphic.
###### Theorem 1.6.
[14] Let p denote an ordered $8$-tuple of distinct points in $\mathbb{P}^{3}$,
and let $X_{\textbf{p}}$ be the blow-up of $\mathbb{P}^{3}$ at the points of
p. There is an infinite set $W$ of configurations of $8$ points in
$\mathbb{P}^{3}$ such that if p and q are distinct elements of $W$, then
$D(X_{\textbf{p}})\cong D(X_{\textbf{q}})$ but are not isomorphic.
A rational variety is simply connected (with the etale fundamental group being
trivial or if defined over $\mathbb{C}$ then with the usual topological
fundamental group) [4, Corollary 4.18]. In particular this example satisfies
the assumptions of Kuznetsov-Shinder conjecture. We can easily show that this
example provide support for the conjecture. The conjecture holds for it.
###### Proposition 1.7.
Each pair of non-isomorphic derived equivalent $X_{\textbf{p}}$ and
$X_{\textbf{q}}$ have the same class in the Grothendieck ring of varieties. In
particular, they are $\mathbb{L}$-equivalent.
###### Proof.
From [14, Lemma 1], we know that $X_{\textbf{p}}$ and $X_{\textbf{q}}$ are
isomorphic iff p and q coincide upto an automorphism of $\mathbb{P}^{3}$ and
permutation of points. From the alternate description of $K_{0}(Var/k)$ using
blowups (see [1] and [16]), we can write
(1) $[X_{\textbf{p}}]=[\mathbb{P}^{3}]-8[k]+[E],$
where $E$ is the exceptional divisor and $[k]$ is the equivalence class of a
point, which is our unit element. The exceptional divisor of the blow up is
Zariski locally isomorphic to $\\{pt\\}\times\mathbb{P}^{2}$ ([7, Theorem II
8.24]). Thus, from the relation above and lemma 1.2, we get
$\displaystyle[X_{\textbf{p}}]$
$\displaystyle=\mathbb{L}^{3}+\mathbb{L}^{2}+\mathbb{L}+1-8+\mathbb{L}^{2}+\mathbb{L}+1$
$\displaystyle=\mathbb{L}^{3}+2\mathbb{L}^{2}+2\mathbb{L}-6.$
Since this formula, does not depend on the configuration of the 8 points in
$\mathbb{P}^{3}$, we see that all the varieties $X_{\textbf{p}}$ have the same
class in the Grothendieck ring of varieties and hence are
$\mathbb{L}$-equivalent. ∎
###### Remark 1.8.
Note that the above example shows that there can be infinitely many
isomorphism classes of varieties which have the same equivalence class in
Grothendieck ring of varieties.
### 1.2. Grothendieck Ring of Varieties and Birationality
One can study rationality problems in algebraic geometry using the
Grothendieck ring of varieties. Consider the following lemma:
###### Lemma 1.9.
[6, Lemma 2.1] Let $X,X^{\prime}$ be smooth birationally equivalent varieties
of dimension $d$. Then we have an equality in the Grothendieck ring
$K_{0}(Var/k)$:
$[X^{\prime}]-[X]=\mathbb{L}.\mathcal{M}$
where $\mathcal{M}$ is a linear combination of classes of smooth projective
varieties of dimension $d-2$.
This immediately gives the following rationality condition as a corollary:
###### Corollary 1.10.
If $X$ is a rational smooth $d$-dimensional variety, then
(2) $[X]=[\mathbb{P}^{d}]+\mathbb{L}.\mathcal{M}_{X}$
where $\mathcal{M}_{X}$ is a linear combination of classes of smooth
projective varieties of dimension $d-2$.
###### Remark 1.11.
Localizing with respect to a zero divisor:
Further, there is another open conjecture of Larsen and Lunts [13].
Conjecture: If $[X]=[Y]$, then $X$ is birational to $Y$.
The above conjecture is trivially true for the example of Lesieutre as we have
constructed varieties which are birational to each other. The conjecture has
been proven true for Calabi-Yau’s by Liu and Sebag in [15, Corollary 1]. A
weaker result in this direction was already proven by Larsen and Lunts in
[13]:
###### Theorem 1.12.
([13]) The quotient-ring $K_{0}(Var/k)/\mathbb{L}$ is naturally isomorphic to
the free abelian group generated by classes of stable birational equivalence
of smooth projective connected varieties together with its natural ring
structure. In particular, if $X$ and $Y_{1},\ldots,Y_{m}$ are smooth
projective connected varieties and
$[X]=\sum_{j=1}^{m}n_{j}[Y_{j}]\mod\mathbb{L}$
for some $n_{j}\in\mathbb{Z}$, then $X$ is stably birationally equivalent to
one of the $Y_{j}$.
Recall that two smooth projective varieties $X$ and $Y$ are called stably
birationally equivalent if for some $m,n\geq 1$, $X\times\mathbb{P}^{m}$ is
birationally equivalent to $Y\times\mathbb{P}^{n}$.
###### Remark 1.13.
Generally the notion of stable birational equivalence is weaker than
birational equivalence. However, for varieties with non-negative Kodaira
dimension these two notions are the same [6, Lemma 2.6].
### 1.3. Comparison with K-equivalence
Kawamata introduced K-equivalence in [10].
###### Definition 1.14.
Let $X$ and $Y$ be smooth projective varieties. They are called $K$-equivalent
if they are birationally equivalent and if there exists a smooth projective
variety $Z$ with birational morphisms $f:Z\rightarrow X$ and $g:Z\rightarrow
Y$ such that the pull-backs of the canonical divisors are linearly equivalent:
$f^{*}K_{X}\sim g^{*}K_{Y}$ .
He further conjectured the following
Kawamata’s DK-Conjecture: [10] Let X and Y be birationally equivalent smooth
projective varieties. Then the following are equivalent.
1. (1)
There exists an equivalence of triangulated categories $D^{b}(X)\cong
D^{b}(Y)$.
2. (2)
There exists a smooth projective variety Z and birational morphisms
$f:Z\rightarrow X$ and $g:Z\rightarrow Y$ such that $f^{*}K_{X}\sim
g^{*}K_{Y}$.
In other words,for birational varieties D-equivalence is the same as
K-equivalence. For a survey of the current known examples for this conjecture
along with its more general form, see [11].
The above example of Lesieture satisfies the DK-conjecture as well. This can
be seen easily by taking $Z$ to be the blowup of $\mathbb{P}^{3}$ at all the
distinct points in the sets p and q of $X_{\textbf{p}}$ and $X_{\textbf{q}}$.
The maps $f$ and $q$ are just the respective blow downs. Note that pullback of
the canonical divisors of $X_{\textbf{p}}$ and $X_{\textbf{q}}$ are the same.
## Acknowledgements
This work was prepared for talk at 38th Annual Conference of Ramanujan
Mathematical Society, 2023 and is supported by the SERB POWER grant No.
SPG/2021/002307.
## References
* [1] Bittner F., The universal Euler characteristic for varieties of characteristic zero, Compos Math. 140(4), 1011-1032, 2004.
* [2] Borisov L., Class of the affine line is a zero divisor in the Grothendieck ring, arXiv:1412.6194.
* [3] Colmez P., Serre J.P., Grothendieck-Serre Correspondence: Bilingual Edition, American Mathematical Society, 2022.
* [4] Debarre, O, Higher-dimensional algebraic geometry, 1st Edition, Universitext, Springer, 2001.
* [5] Efimov A.I., Some remarks on L-equivalence of algebraic varieties, Sel. Math. New Series, 24:3753-3762, 2018.
* [6] Galkin, S., Shinder, E., The Fano variety of lines and rationality problem for a cubic hypersurface, arXiv: 1405.5154, 2014.
* [7] Hartshorne R., Algebraic Geometry, Graduate Text in Mathematics 52, Springer, 1977.
* [8] Ito, A., Miura, M., Okawa, S., Ueda, K., The class of the affine line is a zero divisor in the Grothendieck ring: Via $G_{2}$-Grassmannians, J. Algebraic Geom. 28 (2019), 245-250. DOI: https://doi.org/10.1090/jag/731
* [9] Ito, A., Miura, M., Okawa, S., Ueda, K., Derived equivalence and Grothendieck ring of varieties: the case of K3 surfaces of degree 12 and abelian varieties. Sel. Math. New Ser. 26, 38 (2020). https://doi.org/10.1007/s00029-020-00561-x
* [10] Kawamata, Y. D-equivalence and K-equivalence, J. Differential Geom., 61, no. 1, 147–171, 2002.
* [11] Kawamata Y., Birational geometry and derived categories, Surveys in Differential Geometry, 22(1):291–317, 2017.
* [12] Kuzentsov A., Shinder E., Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics, Sel. Math. New Series, 2017.
* [13] M. Larsen, V. Lunts, Motivic measures and stable birational geometry, Moscow Mathematical Journal, Vol.3 (1), 2003, 85–95.
* [14] Lesieture J., Derived-Equivalent Rational Threefolds, International Mathematics Research Notices, Volume 2015, Issue 15, 2015, Pages 6011–6020, https://doi.org/10.1093/imrn/rnu117.
* [15] Liu Q., Sebag J., The Grothendieck ring of varieties and piecewise isomorphisms, Math Z., 265, 321-342, 2010.
* [16] Looijenga, E., Motivic measures, Seminaire Bourbaki 42, 267-297 (1999-2000).
* [17] Martin N., The class of the affine line is a zero divisor in the Grothendieck ring: an improvement, Comptes Rendus Mathematique, Volume 354, Issue 9, 936-939, 2016. https://doi.org/10.1016/j.crma.2016.05.016.
* [18] Mustaţă M., Lecture 8 in Lecture notes on Zeta-functions in Algebraic Geometry, https://dept.math.lsa.umich.edu/~mmustata/lecture8.pdf
* [19] Poonen B., The Grothendieck ring of varieties is not a domain, Math. Res. Letters, 9(4):493-498, 2002.
|
# Fluid-gravity correspondence and causal first-order relativistic viscous
hydrodynamics
Luca Ciambelli<EMAIL_ADDRESS>Luis Lehner
<EMAIL_ADDRESS>Perimeter Institute for Theoretical Physics,
31 Caroline St North, Waterloo ON N2L 2Y5, Canada
###### Abstract
The fluid-gravity correspondence is a duality between anti-de Sitter Einstein
gravity and a relativistic fluid living at the conformal boundary. We show
that one can accommodate the causal first-order viscous hydrodynamics recently
developed by Bemfica, Disconzi, Noronha, and Kovtun in this framework, by
requiring a set of natural conditions for the geometric data at the horizon.
The latter hosts an induced Carrollian fluid, whose equations of motion are
shown to be tightly tied to the ones describing the fluid at the boundary.
Functional expressions for the transport coefficients are found –with those
associated to viscosity and heat flux uniquely determined–, satisfying a set
of known causality requirements for the underlying equations of motion.
## I Introduction
In a series of works by Bemfica, Disconzi, Noronha, and Kovtun (BDNK), a
formulation for viscous, relativistic hydrodynamics has been introduced where
dissipative corrections are accounted for via first-order derivatives of the
energy density and flow velocity [1, 2, 3], see also [4, 5, 6], and where
causality of the resulting equations of motion is achieved when choosing
transport coefficients within particular bounds. Such a formulation, at first
sight, is in tension with standard results where at least second-order
corrections are required to account for viscous relativistic hydrodynamics [7,
8, 9, 10, 11]. A key observation is that such results require a strictly non-
negative entropy change, while the first-order formulation does so up to
higher order in gradients. Arguably, this is not necessarily a significant
shortcoming as such higher order terms should be subleading in the effective
field theory regime where such theory can be written. Further, another key
aspect of BDNK is formulating such theory in a more general frame than the
typical Landau or Eckart frames where causality is violated in the first order
formulation.
The Landau frame [12] was introduced requiring that the heat current vanishes,
such that the fluid velocity is an eigenvector of the energy-momentum tensor.
On the other hand, in the Eckart frame [13] the fluid velocity is aligned with
the particle number flux, such that the equations are similar to those of non-
relativistic hydrodynamics. As pointed out by BDNK, the frame discussed above
should not be chosen driven by aesthetics, but instead requiring that the
resulting hydrodynamic equations lead to well-posed problems, thus the
equations of motions should be hyperbolic and causal.
In a parallel development, the celebrated fluid-gravity correspondence [14,
15, 16, 17, 18, 19] has linked the behavior of perturbed black holes (with
asymptotically anti-de Sitter boundary conditions) to viscous relativistic
hydrodynamics in one lower dimension. This remarkable correspondence, was
fully developed to second order in gradients, but specialized in the Landau
frame by judicious choices made when solving Einstein equations for a
perturbed black brane. Under restricted assumptions on the bulk duals, the
Landau frame was abandoned in [20, 21, 22], where the heat current was
considered in the fluid-gravity correspondence.
In the current work, with the aim of shedding light on the connection between
the fluid-gravity correspondence and the BDNK first order, viscous
relativistic hydrodynamics, we first show that the fluid-gravity
correspondence is well-suited to accommodate the full first order hydrodynamic
frame spectrum. This freedom was already present in [14], but the
correspondence was fully developed in the Landau frame. Given this freedom,
are there reasonable choices –in particular, from a gravity perspective–
leading to BDNK? To answer this question, we study the properties of the bulk
projected on the horizon, which is a null hypersurface.
It is known since the original “membrane paradigm” [23, 24] that the Einstein
equations projected to the horizon are conservation equations of a fluid,
which has been recently understood to be a Carrollian fluid [25, 26, 27, 28,
29, 30, 31]. We show that the Carrollian equations of motion, at first order,
are equal to those of perfect fluid conservation for a conformal fluid111In
particular, in agreement with the boundary fluid equations of motion for the
perfect fluid.. We also observe that requiring the null vector generating the
horizon to be aligned with the fluid velocity at first order in the derivative
expansion selects exactly the BDNK form of the heat current. Similarly, the
energy density at first order is the one used by BDNK if it is proportional to
the horizon expansion. Under these assumptions we derive the values induced by
Einstein gravity of most transport coefficients and a condition on a remaining
one for a conformal equation of state. We find that the transport coefficients
than can be fully fixed this way are within the causality and stability
bounds. These observations may open the door to unraveling a deeper connection
between the horizon Carrollian fluid and relativistic conformal fluids.
The rest of the manuscript is organized as follows. In section II we review
the construction of the fluid-gravity correspondence of [14], and extrapolate
the boundary energy-momentum tensor using the holographic prescription [32,
33]. In section III we discuss the horizon geometry using the framework of
[34], and study it at zeroth and first order derivative expansion. We then
make our geometrical choices, that we show in section IV lead to BDNK for the
boundary fluid. We conclude with final remarks in section V.
## II Setup
Following the presentation in [14], we consider a boosted black brane in 4+1
dimensions with asymptotically anti-de Sitter boundary conditions. In the
stationary case –zeroth order in the gradient expansion–, the spacetime metric
is given by
$\displaystyle\mathrm{d}s^{2}=-2u_{a}\mathrm{d}x^{a}\mathrm{d}r+r^{2}(P_{ab}-f(br)u_{a}u_{b})\mathrm{d}x^{a}\mathrm{d}x^{b}\,;$
(1)
with $f(r)=1-r^{-4}$, $u^{a}u^{b}\eta_{ab}=-1$ and
$P_{ab}=\eta_{ab}+u_{a}u_{b}$ the projector orthogonal to $u^{a}$,
$u^{a}P_{ab}=0$. The vector $u^{a}$ is constant and defines the boost, while
the function $f(br)$ describes a black brane with radius $r_{H}=b^{-1}$.
Perturbed solutions, in terms of a gradient expansion (also known as
derivative expansion) can be obtained by considering $(b,u^{a})$ as slowly
varying functions of $x^{a}$222We use bulk coordinates $x^{\mu}=(r,x^{a})$,
such that $x^{a}$ are coordinates of fixed-$r$ hypersurfaces, and in
particular of the boundary., inserting into Einstein equations and solving for
non-trivial functions of $r$ (see [14]).
As opposed to the treatment in that work, we refrain from adopting the
specific choice of vanishing “zero-modes” (as also considered in [6]). In
other terms, we allow for small coordinate changes in the $x^{a}$ sector,
which we capture in terms of a scalar function $\chi(x^{a})$ and a vector
$q^{a}(x^{b})$ transverse to $u^{a}$, $u^{a}q_{a}=0$. The resulting solution,
at first order in the gradient expansion is333Using the convention
$A_{(ab)}=\frac{1}{2}(A_{ab}+A_{ba})$.
$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle-2u_{a}\mathrm{d}x^{a}\mathrm{d}r+r^{2}(P_{ab}-f(br)u_{a}u_{b})\mathrm{d}x^{a}\mathrm{d}x^{b}$
(2)
$\displaystyle-\frac{2}{b^{4}r^{2}}u_{({a}}q_{{b})}\mathrm{d}x^{a}\mathrm{d}x^{b}+\frac{\chi}{b^{4}r^{2}}u_{a}u_{b}\mathrm{d}x^{a}\mathrm{d}x^{b}$
$\displaystyle+2r^{2}bF(br)\sigma_{ab}\mathrm{d}x^{a}\mathrm{d}x^{b}+\frac{2}{3}r\theta
u_{a}u_{b}\mathrm{d}x^{a}\mathrm{d}x^{b}$
$\displaystyle-2ru_{({a}}a_{{b})}\mathrm{d}x^{a}\mathrm{d}x^{b},$
where $F(br)=\frac{1}{br}-\frac{1}{4b^{4}r^{4}}$, and we introduced the shear,
expansion, and acceleration of $u^{a}$,
$\displaystyle\sigma^{ab}=P^{c(a}\partial_{c}u^{b)}-\frac{1}{3}P^{ab}\partial_{c}u^{c}$
(3) $\displaystyle
a_{a}=u^{b}\partial_{b}u_{a}\,,\qquad\theta=\partial_{a}u^{a},$ (4)
satisfying $u^{a}\sigma_{ab}=0$, $\sigma_{a}{}^{a}=0$ ($\eta_{ab}P^{ab}=3$),
$u^{a}a_{a}=0$.
Armed with this solution, we can now obtain the stress tensor induced at the
timelike asymptotic boundary. To do so, we follow the holographic prescription
discussed in [32, 33], of which we now recap the salient ingredients. Given a
bulk metric $g_{\mu\nu}$, we introduce a hypersurface at fixed $r$ and its
projector $h_{\mu\nu}$. Using the normalized normal form
$N=N_{\mu}\mathrm{d}x^{\mu}=\frac{\mathrm{d}r}{\sqrt{g^{rr}}}$, the projector
reads
$\displaystyle h_{\mu\nu}=g_{\mu\nu}-N_{\mu}N_{\nu}\quad
h_{\mu}{}^{\nu}N_{\nu}=0.$ (5)
The extrinsic curvature (second fundamental form) is defined as
$\displaystyle K_{ab}$ $\displaystyle=$ $\displaystyle
h_{a}{}^{\mu}h_{b}{}^{\nu}\frac{1}{2}\pounds_{N}g_{\mu\nu}$ (6)
$\displaystyle=$ $\displaystyle
h_{a}{}^{\mu}h_{b}{}^{\nu}\frac{1}{2}(\nabla_{\mu}N_{\nu}+\nabla_{\nu}N_{\mu}),$
(7)
where $\nabla$ is the bulk Levi-Civita connection. The induced inverse metric
is
$\displaystyle\overline{g}^{ab}=h_{\mu}{}^{a}h_{\nu}{}^{b}g^{\mu\nu},$ (8)
which can be used to define the trace on the hypersurface
$\displaystyle K=\overline{g}^{ab}K_{ab}.$ (9)
The traceless part of the extrinsic curvature defines the boundary stress
tensor
$\displaystyle
T^{a}{}_{b}=-2\lim_{r\to\infty}r^{4}\left(K^{a}{}_{b}-\frac{K}{4}\delta^{a}_{b}\right),$
(10)
where the $r$ pre-factor comes from the holographic dictionary and ensures its
finiteness approaching the boundary.
Applying this procedure to our line element (2), the final result is
$\displaystyle
T_{ab}=3\frac{1+\chi}{b^{4}}u_{a}u_{b}+\frac{1+\chi}{b^{4}}P_{ab}-\frac{2}{b^{3}}\sigma_{ab}-\frac{8}{b^{4}}u_{(a}q_{b)}.$
(11)
This is the stress tensor of a conformal viscous fluid with fluid velocity
$u^{a}$, which accounts for heat-flux through $q^{a}$, and a correction to the
perfect fluid energy density through $\chi$. Since a generic stress tensor is
decomposed as
$\displaystyle T_{ab}={\cal E}u_{a}u_{b}+{\cal
P}P_{ab}-2\eta\sigma_{ab}+u_{a}Q_{b}+Q_{a}u_{b},$ (12)
one has,
$\displaystyle{\cal E}$ $\displaystyle=$ $\displaystyle
3\frac{1+\chi}{b^{4}},\quad{\cal P}=\frac{1+\chi}{b^{4}}\quad$
$\displaystyle\eta$ $\displaystyle=$ $\displaystyle\frac{1}{b^{3}}\,,\quad
Q_{a}=-\frac{4}{b^{4}}q_{a}.$ (13)
We note that ${\cal E}=3{\cal P}$ is the conformal equation of state implied
by the asymptotic conformal symmetry, streaming from the vanishing of the
stress tensor trace. Notice that at equilibrium the temperature is given by
$T=\frac{1}{b}$.
From here, one could straightforwardly identify conditions for $(\chi,q^{a})$
to recover BDNK. This would amount to using a particular formulation of
viscous-relativistic hydrodynamics to fix conditions on the gravitational
sector. However, our goal is to go in the opposite way, namely to consider
arguably natural choices on the gravitational sector –specifically at the
horizon– and explore what they correspond to in the hydrodynamic side.
## III Choices
To consistently deal with a degenerate metric at the null hypersurface
describing the horizon, we adopt the null Rigging formalism described in [34].
The horizon is generically located at $r=r_{H}(x)$, and thus the one-form
normal to the horizon is
$\displaystyle\underline{n}=\tilde{\alpha}\mathrm{d}(r-r_{H}(x)),$ (14)
and we adopt $\tilde{\alpha}=1$ in the following. Next, we introduce the
vector $k=\partial_{r}$ with the defining properties
$\displaystyle\underline{n}(k)=1\,,\qquad\underline{k}(k)=0.$ (15)
This vector is called the null Rigging vector. We can then define the Rigging
projector as
$\displaystyle\Pi_{\mu}{}^{\nu}=\delta_{\mu}^{\nu}-n_{\mu}k^{\nu},$ (16)
such that
$\displaystyle\Pi_{\mu}{}^{\nu}n_{\nu}=0\qquad k^{\nu}\Pi_{\nu}{}^{\mu}=0.$
(17)
The Rigging projector projects to the null hypersurface, since indeed the form
$\underline{n}$ and the vector $k$ are normal to it.
The bulk metric duals $n$ and $\underline{k}=-u_{a}\mathrm{d}x^{a}$ satisfy
$\displaystyle\Pi_{\mu}{}^{\nu}k_{\nu}=k_{\mu}\qquad\ell^{\mu}=n^{\nu}\Pi_{\nu}{}^{\mu}.$
(18)
Furthermore, the projected metric is given by
$\displaystyle q_{\mu\nu}$ $\displaystyle=$
$\displaystyle\Pi_{\mu}{}^{\rho}\Pi_{\nu}{}^{\sigma}g_{\rho\sigma}=g_{\mu\nu}-n_{\mu}k_{\nu}-k_{\mu}n_{\nu}.$
(19)
The components intrinsic to the hypersurface, $(k_{a},\ell^{a},q_{ab})$, form
the ruled Carrollian structure discussed in [31] (with the same conventions).
In particular, $\ell^{a}$ is the Carrollian vector field, $k_{a}$ is the
Ehresmann connection, and $q_{ab}$ is the degenerate Carrollian metric
satisfying
$\displaystyle\ell^{a}q_{ab}=0$ (20)
at the horizon.
The other relevant quantities for the horizon physics are the surface gravity,
expansion, Hájiček connection, and acceleration. They are defined in the bulk
by,
(1) Surface gravity:444This quantity should be called inaffinity, but for non-
expanding horizons these two concepts coincide. Here, by construction at
zeroth order and as a consequence of the equations of motion at first order,
the horizon expansion vanishes so we are in this framework.
$\displaystyle\ell^{\mu}\nabla_{\mu}\ell^{\nu}=\kappa\ell^{\nu},\qquad
k_{\nu}\ell^{\mu}\nabla_{\mu}\ell^{\nu}=\kappa\,;$ (21)
(2) Expansion:
$\displaystyle\Theta=q_{\nu}{}^{\mu}\nabla_{\mu}n^{\nu}\,;$ (22)
(3) Hájiček connection:
$\displaystyle\pi_{\mu}=q_{\mu}{}^{\nu}k_{\rho}\nabla_{\nu}n^{\rho}\,;$ (23)
(4) Acceleration
$\displaystyle\varphi_{\mu}=n^{\nu}\nabla_{[\mu}k_{\nu]}\,.$ (24)
We now proceed to compute these quantities, for clarity we do so first in the
stationary solution (zeroth order), and then in the first order perturbed
case.
### III.1 Zeroth Order
At this order, the location of the horizon, and associated normal form and
vector are:
$\displaystyle r_{H}=\frac{1}{b}\,,\qquad\underline{n}=\mathrm{d}r\,,\qquad
k=\partial_{r}\,.$ (25)
The bulk metric duals are
$\displaystyle
n=r^{2}f(br)\partial_{r}+u^{a}\partial_{a}\,,\qquad\underline{k}=-u_{a}\mathrm{d}x^{a}$
(26)
and thus the Carrollian vector is exactly given by the boundary fluid
congruence (which is constant at zeroth order)
$\displaystyle\ell^{\mu}=n^{\nu}\Pi_{\nu}{}^{\mu}=u^{a}\delta_{a}^{\mu}.$ (27)
This implies
$\displaystyle\ell^{\mu}k_{\mu}=-u^{a}u_{a}=1.$ (28)
The degenerate metric on the null surface is
$\displaystyle q_{\mu\nu}=\begin{pmatrix}0&0\\\
0&r^{2}(P_{ab}-f(br)u_{a}u_{b})\end{pmatrix}\xrightarrow{r=r_{H}}\begin{pmatrix}0&0\\\
0&\frac{P_{ab}}{b^{2}}\end{pmatrix}$ (29)
which indeed satisfies at the horizon
$\displaystyle\ell^{\mu}q_{\mu\nu}=u^{a}q_{ab}\delta^{b}_{\nu}\xrightarrow{r=r_{H}}u^{a}\frac{P_{ab}}{b^{2}}=0.$
(30)
With the above quantities, it is straightforward to obtain:
$\displaystyle\kappa=\frac{2}{b},\quad\Theta=0,\quad\pi_{\mu}=0,\quad\varphi_{\mu}=0.$
(31)
Exactly like the relativistic conformal fluid at the boundary, the Carrollian
fluid at the horizon is a perfect fluid at zeroth order. Delving into the
properties of the Carrollian fluid on the horizon and its connection to the
boundary fluid would bring us too afield from the subject of this manuscript.
We leave this exploration to a future work.
### III.2 First Order
We now perturb the stationary solution using the first order gradient
expansion. Details on how to establish the location of the perturbed horizon
are in [15] (in particular subsection 2.3), so we just report the result here.
At first order, the horizon and associated normal form are
$\displaystyle
r_{H}=\frac{1}{b}+\frac{\theta}{6}+\frac{\chi}{4b}-\frac{u^{a}\partial_{a}b}{2b},\qquad\underline{n}=\mathrm{d}r+\frac{\mathrm{d}b}{b^{2}},$
(32)
where $\theta$, $\chi$, and $\mathrm{d}b$ are first order quantities.
Following the steps described above, we gather
$\displaystyle\underline{k}=-u_{a}\mathrm{d}x^{a},$ (33)
$\displaystyle\ell^{a}=u^{a}-ba^{a}-q^{a}+P^{ab}\partial_{b}b\,,\quad\ell^{r}=-u^{a}\frac{\partial_{a}b}{b^{2}};$
(34)
where the indices of the various quantities ($a^{a}$, $q^{a}$, and $P_{ab}$)
are raised using $\eta_{ab}$, and we note that $\ell^{r}$ is non-vanishing due
to the fact that the horizon position is now a function of $x^{a}$.
With the above, through a direct, but naturally more involved calculation, one
obtains:
$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\frac{2}{b}-\frac{2\theta}{3}+\frac{\chi}{2b}+\frac{u^{a}\partial_{a}b}{b}$
(35) $\displaystyle\Theta$ $\displaystyle=$
$\displaystyle\theta-3\frac{u^{a}\partial_{a}b}{b}$ (36)
$\displaystyle\pi_{a}$ $\displaystyle=$ $\displaystyle
2\frac{q_{a}}{b}-\frac{P_{a}{}^{b}\partial_{b}b}{b}$ (37)
$\displaystyle\varphi_{a}$ $\displaystyle=$ $\displaystyle a_{a}.$ (38)
With these, we are now ready to argue for some particular choices. First, one
could demand that at first order, the component of the null vector
$\ell^{\mu}$ orthogonal to $r$ should be aligned with $u^{a}$ (just as in the
zeroth order case). This allows one to still identify the Carrollian vector
with the boundary fluid velocity, even at first order. Such a choice implies
$\displaystyle q^{a}=-ba^{a}+P^{ab}\partial_{b}b\,.$ (39)
This, as we shall discuss below, is precisely in line with the hydrodynamic
choice in BDNK.
Before such discussion, we must address the choice of $\chi$. First, note that
$r_{H}$ can be re-expressed as,
$\displaystyle r_{H}=\frac{1}{b}+\frac{1}{6}\Theta+\frac{\chi}{4b}.$ (40)
We shall now show that, to first order, $\Theta=0$ as a consequence of the
Einstein equations projected on the horizon, specifically the Raychaudhuri
equation. Thus, the choice $\chi\propto\Theta$, conveniently keeps $r_{H}=1/b$
on-shell. Note that, with this choice, $\kappa$ receives non-trivial first-
order corrections. We discuss the consequences of choosing it to remain
unchanged in appendix V.1. Since $\kappa$ depends on the generators’
parameterization, we regard keeping $r_{H}$ unchanged as the more natural
choice.
To see that $\Theta=0$ to first order, let us recall that Raychaudhuri’s and
Damour’s equations in vacuum are,
$\displaystyle(\pounds_{\ell}+\Theta)[\Theta]=\mu\Theta-\sigma_{a}{}^{b}\sigma_{b}{}^{a},$
(41) $\displaystyle
q_{a}{}^{b}\left(\pounds_{\ell}+\Theta\right)[\pi_{b}]+\Theta\varphi_{a}=(\overline{D}_{b}+\varphi_{b})(\mu
q_{a}{}^{b}-\sigma_{a}{}^{b})$ (42)
where $\pounds_{\ell}$ is the Lie-derivative along $\ell$, $\overline{D}_{a}$
the (Carrollian) covariant derivative associated to $q_{ab}$,
$\mu=\Theta/2+\kappa$ and we used the conventions of [31]. Since here we will
be interested only in the first order expression, where most terms in these
equations vanish, we refer to this reference for an explanation of all the
quantities involved in general. Notice $\kappa$ has an order 0 contribution,
so, eq. (41) implies that at first order $\Theta=0$. A similar analysis of eq.
(42) implies $q_{a}{}^{b}\partial_{b}\kappa+\varphi_{a}\kappa=0$, where
$q_{a}{}^{b}$ is the projector orthogonal to $\ell^{a}$ at the horizon, which
at zeroth order is simply $P_{a}{}^{b}$, and thus (using (38)) this equation
is equal to $a_{a}=P_{a}{}^{b}\frac{\partial_{b}b}{b}$.
These observations have several consequences. First, since to the order we
work, $\Theta=0$, the choice stated above for $\chi$ indeed implies
$r_{H}=1/b$ to this order. Further, and importantly, they indicate that at
first order Raychaudhuri’s and Damour’s equations are exactly equal to the
conservation of the boundary perfect fluid stress tensor. Indeed, one can
easily show that $\partial_{a}T^{a}{}_{b}=0$, using (11) and the relationships
(35), (36), and (38), gives exactly the Raychuadhuri’s and Damour’s equations,
once projected on $u^{a}$ and $P_{a}{}^{b}$, respectively. This is ultimately
tied to the fact that these equations all come from the bulk Einstein
equations and their particular hierarchical structure arising from the
characteristic treatment along a timelike-null foliation of the spacetime.
To summarize then, examining the resulting structure at the horizon, our
choices are:
$\displaystyle\ell^{a}=u^{a}\Leftrightarrow
q^{a}=-ba^{a}+P^{ab}\partial_{b}b\,$ (43) $\displaystyle
r_{H}=\frac{1}{b}+\left(2+3\alpha\right)\frac{\Theta}{12}\Leftrightarrow\chi=\alpha\,b\,\Theta\,,$
(44)
with $\alpha$ a proportionality function that remains to be specified. We
reported these results off-shell of the conservation laws discussed above. If
we now impose these conservation laws, we obtain $q^{a}=0$ and $\chi=0$. This
is precisely the outcome of the intrinsic hydrodynamic BDNK analysis for a
conformal relativistic fluid: the heat current and the first-order correction
to the energy that implement causality are zero on-shell of the first order
conservation law. In what follows, we discuss in detail the structure implied
by the geometrical identifications/choices on the resulting hydrodynamical
equations.
## IV Consequences
We can now examine the consequences of these choices on the thermodynamic
quantities obtained in (II). First, note that
$\displaystyle{\cal E}^{(0)}=3{\cal P}^{(0)}$ $\displaystyle\equiv$
$\displaystyle e=\frac{3}{b^{4}}$ (45) $\displaystyle{\cal E}^{(1)}=3{\cal
P}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{3\alpha}{b^{3}}\left(\partial_{a}u^{a}-\frac{3}{b}u^{a}\partial_{a}b\right)$
(46) $\displaystyle Q^{a}$ $\displaystyle=$
$\displaystyle\frac{4}{b^{3}}\left(a^{a}-\frac{P^{ab}\partial_{b}b}{b}\right),$
(47)
where we introduced $\\{e,p=\frac{e}{3}\\}$ to denote the zeroth order
expressions for energy and pressure respectively.
The first order expressions can be re-expressed in terms of $e$ and $p$ as,
$\displaystyle{\cal E}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{3\alpha}{b^{3}}\left(\partial_{a}u^{a}+\frac{u^{a}\partial_{a}e}{(e+p)}\right)$
(48) $\displaystyle Q^{a}$ $\displaystyle=$
$\displaystyle\frac{4}{b^{3}}\left(a^{a}+\frac{P^{ab}\partial_{b}e}{3(e+p)}\right)$
(49)
(the expressions for the pressure is trivially set by the conformal
condition). We can now compare with the expressions adopted by BDNK for the
conformal case, as this is the one that corresponds to our case [3]. Namely,
denoting with an overbar their choices,
$\displaystyle\bar{\cal E}^{(1)}$ $\displaystyle=$
$\displaystyle\left(\chi_{2}\,\partial_{a}u^{a}+\chi_{1}\,\frac{u^{a}\partial_{a}e}{(e+p)}\right)$
(50) $\displaystyle\bar{Q}^{a}$ $\displaystyle=$
$\displaystyle\lambda\left(a^{a}+\frac{P^{ab}\partial_{b}e}{3(e+p)}\right),$
(51)
with $\lambda,\chi_{i}$ transport coefficients555 Which include
$\\{\chi_{3},\chi_{4}\\}$ analogously introduced for the first-order pressure
${\cal P}^{(1)}$. that are chosen to ensure causality of the underlying
equations, together with $\eta$ defined in (12).
Remarkably, the functional form for the first order corrections are in
excellent agreement with the proposed terms in [3]. Moreover, our choices
motivated by considerations at the horizon also imply for the transport
coefficients (for $\eta$ we recall (II)),
$\displaystyle\eta=\frac{1}{b^{3}}\,,\,\lambda=\frac{4}{b^{3}}\,,$
$\displaystyle\chi_{1}=\chi_{2}=3\chi_{3}=3\chi_{4}=\frac{3\alpha}{b^{3}}\,,$
(52)
where $\\{\chi_{3},\chi_{4}\\}$ are linked to $\\{\chi_{1},\chi_{2}\\}$ by
conformality.
Not only do the transport coefficients have the temperature dependency of
$T^{3}$ as expected from kinetic theory [3], but the shear viscosity and heat
transport coefficients are uniquely determined666The value of the viscous
transport coefficient is tied to the lowest-lying quasinormal modes of the
perturbed black brane (see, e.g. [35]).. In particular, they satisfy the
criteria for causality $\lambda\geq\eta$ identified in [3]. Notice however our
expressions make the transport coefficients $\chi_{i}$ all proportional to
each other but do not completely fix them, nor provide bounds for them which
need not be surprising. Namely, conditions on $\chi_{i}$ determined by the
causality analysis of [3], effectively, come from the high frequency limit
(through the standard analysis within PDE theory). This can be seen by
examining the dispersion relations for the shear and sound modes and their
dependency on $\\{\eta,\lambda,\alpha\\}$. Their roles appear at order
$k^{2}$, $k^{4}$ and $k^{6}$ respectively. On the other hand, the fluid-
gravity correspondence is obtained in the long wavelength regime of perturbed
black holes in General Relativity –which is a causal theory–, thus it is
natural to expect that in the regime where the duality can be established,
conditions on relevant parameters on the hydrodynamic side can be obtained
implying such property.
For the unfixed parameter, we only demand $\alpha>0$, as this choice ensures
equilibrium can be reached, i.e. ${\cal E}^{(0)}+{\cal
E}^{(1)}\rightarrow{\cal E}^{(0)}$ within a timescale given by $\alpha$. Of
course, one can choose a suitable value for $\alpha$ such that the full set of
requirements for causality are satisfied (e.g. $\alpha=4/3$, so that
$\chi_{\\{1,2\\}}=3\chi_{\\{3,4\\}}=\lambda$) but there is no geometric reason
at this order we can demand to argue for a specific value.
## V Final words
In this work we examined from a gravitation angle how the BDNK first order
formulation of relativistic, viscous hydrodynamics is connected to the fluid-
gravity correspondence. Such a formulation, which in practice is simpler to
deal with than standard, second order viscous formulations [36, 37], has
received significant attention in recent years both at the theoretical level
[3, 2, 38, 4, 5] and also in incipient numerical investigations (e.g. [39, 40,
41]). The results obtained also revealed new connections between relativistic
and Carrollian hydrodynamics as well as with gravity.
Our analysis unearthed a natural way to motivate the BDNK formulation from a
gravitational perspective. Further, the expected functional dependence of
transport coefficients was obtained and, for the viscous and heat-flux
coefficients, a unique expression was found. As well, our analysis revealed a
connection between the effective Carrollian hydrodynamic description of null
surfaces and the asymptotic relativistic fluid that is identified at the
timelike infinity of perturbed black branes in AdS. Such connection implies
that, at leading order, Raychaudhuri’s and Damour’s equations encode the
conservation of a conformal perfect fluid. The analysis of higher orders and
the exploration of Carrollian hydrodynamics from this perspective is an
interesting task which we defer to future work.
In a similar vein, it would be interesting to explore the horizon physics
deeper, as results there would also hold for asymptotically flat spacetimes.
Importantly, in the latter case, there is also an interesting relation between
the structure of interior null surfaces (like the horizon), and future null
infinity. However, the relationship between the horizon membrane paradigm and
the asymptotic (e.g. ${\cal I}^{+}$) null boundary Carrollian fluid is still
largely unexplored. The latter fluid however enjoys Weyl symmetry, which makes
it special. This could also help motivate a fluid interpretation of
(particular) quasi-normal modes in asymptotically flat spacetimes. Another
avenue for exploration is to consider a potential entropy current, both for
the relativistic fluid at the boundary and the horizon Carrollian fluid. This
current could help us connect with its microscopic origin and inform standing
questions on Carrolllian hydrodynamics. Finally, a deeper understanding of
potential connections between phenomena in non-linear gravity and
hydrodynamics can motivate new avenues to identify and study non-linear
gravitational behavior (e.g. [16, 42, 43, 44, 45, 46, 47, 48, 49, 30]).
###### Acknowledgements.
We thank F. Bemfica, M. Disconzi, L. Freidel, S. Giri, R. E. Hoult, P. Kovtun,
R. Leigh, J. Noronha, M. Petropoulos, E. Poisson, F. Pretorius, M. Rangamani,
and J. Senovilla for discussions. This work was supported in part by Perimeter
Institute for Theoretical Physics. Research at Perimeter Institute is
supported by the Government of Canada through the Department of Innovation,
Science and Economic Development Canada and by the Province of Ontario through
the Ministry of Economic Development, Job Creation and Trade. L. L. thanks
financial support via the Carlo Fidani Rainer Weiss Chair at Perimeter
Institute. L. L. receives additional financial support from the Natural
Sciences and Engineering Research Council of Canada through a Discovery Grant
and CIFAR. L. L. thanks KITP - UC Santa Barbara for its hospitality during
“The Many Faces of Relativistic Fluid Dynamics” Program, where this work’s
initial stages were completed.
### V.1 Alternative choice for $\chi$
An alternative choice for $\chi$, which fixes it completely, would be to
demand that to first order $\kappa=2/b$. This would imply
$\displaystyle\chi$ $\displaystyle=$ $\displaystyle
2\left(\frac{2b}{3}\partial_{a}u^{a}-u^{a}\partial_{a}b\right)$ (53)
$\displaystyle=$
$\displaystyle\frac{2b}{3}\left(2\partial_{a}u^{a}+\frac{u^{a}\partial_{a}e}{(e+p)}\right),$
and as a consequence, $\chi_{1}=3\chi_{3}=2/b^{3}$,
$\chi_{2}=3\chi_{4}=4/b^{3}$. These values however, (complemented by
$\lambda=4/b^{3},\eta=1/b^{3}$) are not within the causality bounds of [3].
Further, on-shell dynamical solutions have an associated energy density at
first order which differs from that at zeroth order.
## References
* [1] F. S. Bemfica, M. M. Disconzi, and J. Noronha, “Causality and existence of solutions of relativistic viscous fluid dynamics with gravity,” Phys. Rev. D 98 (2018) no. 10, 104064, arXiv:1708.06255 [gr-qc].
* [2] P. Kovtun, “First-order relativistic hydrodynamics is stable,” JHEP 10 (2019) 034, arXiv:1907.08191 [hep-th].
* [3] F. S. Bemfica, M. M. Disconzi, and J. Noronha, “Nonlinear Causality of General First-Order Relativistic Viscous Hydrodynamics,” Phys. Rev. D 100 (2019) no. 10, 104020, arXiv:1907.12695 [gr-qc]. [Erratum: Phys.Rev.D 105, 069902 (2022)].
* [4] R. E. Hoult and P. Kovtun, “Stable and causal relativistic Navier-Stokes equations,” JHEP 06 (2020) 067, arXiv:2004.04102 [hep-th].
* [5] F. S. Bemfica, M. M. Disconzi, and J. Noronha, “First-Order General-Relativistic Viscous Fluid Dynamics,” Phys. Rev. X 12 (2022) no. 2, 021044, arXiv:2009.11388 [gr-qc].
* [6] R. E. Hoult and P. Kovtun, “Causal first-order hydrodynamics from kinetic theory and holography,” Phys. Rev. D 106 (2022) no. 6, 066023, arXiv:2112.14042 [hep-th].
* [7] R. Geroch and L. Lindblom, “Causal theories of dissipative relativistic fluids,” Annals of Physics 207 (1991) no. 2, 394–416.
* [8] G. S. Denicol, T. Kodama, T. Koide, and P. Mota, “Stability and Causality in relativistic dissipative hydrodynamics,” J. Phys. G 35 (2008) 115102, arXiv:0807.3120 [hep-ph].
* [9] S. Pu, T. Koide, and D. H. Rischke, “Does stability of relativistic dissipative fluid dynamics imply causality?,” Phys. Rev. D 81 (2010) 114039, arXiv:0907.3906 [hep-ph].
* [10] L. Lehner, O. A. Reula, and M. E. Rubio, “Hyperbolic theory of relativistic conformal dissipative fluids,” Phys. Rev. D 97 (2018) no. 2, 024013, arXiv:1710.08033 [gr-qc].
* [11] M. Aguilar and E. Calzetta, “Causal Relativistic Hydrodynamics of Conformal Fermi-Dirac Gases,” Phys. Rev. D 95 (2017) no. 7, 076022, arXiv:1701.01916 [hep-ph].
* [12] L. D. Landau and E. M. Lifshitz, Fluid mechanics. 1959\.
* [13] C. Eckart, “The Thermodynamics of irreversible processes. 3.. Relativistic theory of the simple fluid,” Phys. Rev. 58 (1940) 919–924.
* [14] S. Bhattacharyya, V. E. Hubeny, S. Minwalla, and M. Rangamani, “Nonlinear Fluid Dynamics from Gravity,” JHEP 02 (2008) 045, arXiv:0712.2456 [hep-th].
* [15] S. Bhattacharyya, V. E. Hubeny, R. Loganayagam, G. Mandal, S. Minwalla, T. Morita, M. Rangamani, and H. S. Reall, “Local Fluid Dynamical Entropy from Gravity,” JHEP 06 (2008) 055, arXiv:0803.2526 [hep-th].
* [16] M. Van Raamsdonk, “Black Hole Dynamics From Atmospheric Science,” JHEP 05 (2008) 106, arXiv:0802.3224 [hep-th].
* [17] M. Haack and A. Yarom, “Nonlinear viscous hydrodynamics in various dimensions using AdS/CFT,” JHEP 10 (2008) 063, arXiv:0806.4602 [hep-th].
* [18] N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Dutta, R. Loganayagam, and P. Surowka, “Hydrodynamics from charged black branes,” JHEP 01 (2011) 094, arXiv:0809.2596 [hep-th].
* [19] M. Rangamani, “Gravity and Hydrodynamics: Lectures on the fluid-gravity correspondence,” Class. Quant. Grav. 26 (2009) 224003, arXiv:0905.4352 [hep-th].
* [20] M. M. Caldarelli, R. G. Leigh, A. C. Petkou, P. M. Petropoulos, V. Pozzoli, and K. Siampos, “Vorticity in holographic fluids,” PoS CORFU2011 (2011) 076, arXiv:1206.4351 [hep-th].
* [21] A. Mukhopadhyay, A. C. Petkou, P. M. Petropoulos, V. Pozzoli, and K. Siampos, “Holographic perfect fluidity, Cotton energy-momentum duality and transport properties,” JHEP 04 (2014) 136, arXiv:1309.2310 [hep-th].
* [22] J. Gath, A. Mukhopadhyay, A. C. Petkou, P. M. Petropoulos, and K. Siampos, “Petrov Classification and holographic reconstruction of spacetime,” JHEP 09 (2015) 005, arXiv:1506.04813 [hep-th].
* [23] T. Damour, Quelques proprietes mecaniques, electromagnetiques, thermodynamiques et quantiques des trous noir. PhD thesis, Paris U., VI-VII, 1979.
* [24] K. S. Thorne, R. H. Price, and D. A. Macdonald, eds., Black holes: the membrane paradigm. 1986\.
* [25] R. F. Penna, “Near-horizon Carroll symmetry and black hole Love numbers,” arXiv:1812.05643 [hep-th].
* [26] L. Donnay and C. Marteau, “Carrollian Physics at the Black Hole Horizon,” Class. Quant. Grav. 36 (2019) no. 16, 165002, arXiv:1903.09654 [hep-th].
* [27] L. Ciambelli, R. G. Leigh, C. Marteau, and P. M. Petropoulos, “Carroll Structures, Null Geometry and Conformal Isometries,” Phys. Rev. D 100 (2019) no. 4, 046010, arXiv:1905.02221 [hep-th].
* [28] L. Freidel and P. Jai-akson, “Carrollian hydrodynamics from symmetries,” Class. Quant. Grav. 40 (2023) no. 5, 055009, arXiv:2209.03328 [hep-th].
* [29] L. Freidel and P. Jai-akson, “Carrollian hydrodynamics and symplectic structure on stretched horizons,” arXiv:2211.06415 [gr-qc].
* [30] J. Redondo-Yuste and L. Lehner, “Non-linear black hole dynamics and Carrollian fluids,” JHEP 02 (2023) 240, arXiv:2212.06175 [gr-qc].
* [31] L. Ciambelli, L. Freidel, and R. G. Leigh, “Null Raychaudhuri: Canonical Structure and the Dressing Time,” arXiv:2309.03932 [hep-th].
* [32] M. Henningson and K. Skenderis, “The Holographic Weyl anomaly,” JHEP 07 (1998) 023, arXiv:hep-th/9806087.
* [33] V. Balasubramanian and P. Kraus, “A Stress tensor for Anti-de Sitter gravity,” Commun. Math. Phys. 208 (1999) 413–428, arXiv:hep-th/9902121.
* [34] M. Mars and J. M. M. Senovilla, “Geometry of general hypersurfaces in space-time: Junction conditions,” Class. Quant. Grav. 10 (1993) 1865–1897, arXiv:gr-qc/0201054.
* [35] E. Berti, V. Cardoso, and A. O. Starinets, “Quasinormal modes of black holes and black branes,” Class. Quant. Grav. 26 (2009) 163001, arXiv:0905.2975 [gr-qc].
* [36] I. Muller, “Zum Paradoxon der Warmeleitungstheorie,” Z. Phys. 198 (1967) 329–344.
* [37] W. Israel and J. M. Stewart, “Transient relativistic thermodynamics and kinetic theory,” Annals Phys. 118 (1979) 341–372.
* [38] F. S. Bemfica, M. M. Disconzi, C. Rodriguez, and Y. Shao, “Local well-posedness in Sobolev spaces for first-order conformal causal relativistic viscous hydrodynamics,” arXiv:1911.02504 [math.AP].
* [39] A. Pandya and F. Pretorius, “Numerical exploration of first-order relativistic hydrodynamics,” Phys. Rev. D 104 (2021) no. 2, 023015, arXiv:2104.00804 [gr-qc].
* [40] U. Danielsson, L. Lehner, and F. Pretorius, “Dynamics and observational signatures of shell-like black hole mimickers,” Phys. Rev. D 104 (2021) no. 12, 124011, arXiv:2109.09814 [gr-qc].
* [41] H. Bantilan, Y. Bea, and P. Figueras, “Evolutions in first-order viscous hydrodynamics,” JHEP 08 (2022) 298, arXiv:2201.13359 [hep-th].
* [42] C. Eling, I. Fouxon, and Y. Oz, “Gravity and a Geometrization of Turbulence: An Intriguing Correspondence,” arXiv:1004.2632 [hep-th].
* [43] H. Yang, A. Zimmerman, and L. Lehner, “Turbulent Black Holes,” Phys. Rev. Lett. 114 (2015) 081101, arXiv:1402.4859 [gr-qc].
* [44] A. Adams, P. M. Chesler, and H. Liu, “Holographic turbulence,” Phys. Rev. Lett. 112 (2014) no. 15, 151602, arXiv:1307.7267 [hep-th].
* [45] F. Carrasco, L. Lehner, R. C. Myers, O. Reula, and A. Singh, “Turbulent flows for relativistic conformal fluids in 2+1 dimensions,” Phys. Rev. D 86 (2012) 126006, arXiv:1210.6702 [hep-th].
* [46] S. R. Green, F. Carrasco, and L. Lehner, “Holographic Path to the Turbulent Side of Gravity,” Phys. Rev. X 4 (2014) no. 1, 011001, arXiv:1309.7940 [hep-th].
* [47] J. R. Westernacher-Schneider, “Fractal dimension of turbulent black holes,” Phys. Rev. D 96 (2017) no. 10, 104054, arXiv:1710.04264 [gr-qc].
* [48] S. Aretakis, Dynamics of Extremal Black Holes, vol. 33 of SpringerBriefs in Mathematical Physics. Springer International Publishing, Cham, 2018.
* [49] G. Moschidis, Two instability results in general relativity. PhD thesis, Princeton U., Math. Dept., 2018.
|
# Emergent inflation of the Efimov spectrum under three-body spin-exchange
interactions
J. van de Kraats<EMAIL_ADDRESS>Eindhoven University of Technology, P.
O. Box 513, 5600 MB Eindhoven, The Netherlands D. J. M. Ahmed-Braun
Eindhoven University of Technology, P. O. Box 513, 5600 MB Eindhoven, The
Netherlands J.-L. Li Institut für Quantenmaterie and Center for Integrated
Quantum Science and Technology IQ ST, Universität Ulm, D-89069 Ulm, Germany
S. J. J. M. F. Kokkelmans Eindhoven University of Technology, P. O. Box 513,
5600 MB Eindhoven, The Netherlands
###### Abstract
One of the most fascinating predictions of few-body quantum physics is the
Efimov effect, a universal accumulation of an infinite geometric series of
three-body bound states at a two-body scattering resonance. Ever since the
first experimental observation of such an Efimov state, the precise
characterization of their physical properties has continued to challenge few-
body theory. This is demonstrated most strongly by the lithium few-body
puzzle, a remarkable theoretical discrepancy with the observed Efimov spectrum
in 7Li. Here, we resolve this long-standing puzzle, demonstrating that the
discrepancy arises out of the presence of strong non-universal three-body
spin-exchange interactions. This conclusion is obtained from a thorough
numerical solution of the quantum mechanical three-body problem, including
precise interatomic interactions and all spin degrees of freedom for three
alkali-metal atoms. Our results show excellent agreement with the experimental
data regarding both the Efimov spectrum and the absolute rate constants of
three-body recombination, and in addition reveal a general product propensity
for such triatomic reactions in the Paschen-Back regime, stemming from
Wigner’s spin conservation rule.
There exists a general desire in physics to formulate accurate descriptions of
nature from a minimal number of adjustable parameters, thus uncovering the
presence of universal behavior. A paradigmatic example of a system where this
ideal picture is realized is the scattering of two particles at low energy.
Here, the wave function delocalizes to the point that the observable
properties of the system become insensitive to the exact microscopic detail of
the interaction, allowing for a description purely in terms of the $s$-wave
scattering length $a$ [1]. This remarkable universality carries through to
systems of more than two particles, most strikingly exemplified at the three-
body level by virtue of the Efimov effect. At a two-body scattering resonance,
where $a\rightarrow\infty$, the Efimov effect induces a universal emergence of
an infinite tower of geometrically spaced three-body bound states [2, 3]. The
resulting spectrum is fully determined by a single length scale, the three-
body parameter, typically expressed as the negative scattering length $a_{-}$
where the ground state trimer dissociates into the three-body scattering
continuum [1, 4, 5, 6]. In turn, the Efimov effect and three-body parameter
induce universal properties in few-body clusters of four or more particles,
further extending the applicability of universal theory [5].
The vast majority of experimental studies of the Efimov effect utilize
ultracold atomic gases, where the scattering length can be directly controlled
by means of a magnetic Feshbach resonance [7, 8]. Near such a resonance, the
three-body parameter can be extracted from a characteristic log-periodic
modulation of the rate of three-body recombination [9, 1, 4]. Interestingly,
although the precise value of $a_{-}$ is typically sensitive to non-universal
short-range physics, the Efimov spectrum in atomic systems possesses an
additional van der Waals universality $a_{-}\approx-9.7\ r_{\mathrm{vdW}}$
[10, 11, 12], where $r_{\mathrm{vdW}}$ gives the characteristic length scale
associated with the two-body interaction. Theoretical analyses have shown that
this universality is robust for Efimov states near broad Feshbach resonances,
and originates from a universal suppression of three-body probability in the
short range [13, 14].
For Feshbach resonances of intermediate to narrow width, both theoretical and
experimental works have demonstrated an increase of $\absolutevalue{a_{-}}$,
arising from the associated growth of the two-body effective range scale [15,
16, 17, 18, 19, 20, 21, 22]. However, a series of experiments in this regime
with the lightest bosonic alkali, 7Li, have failed to observe this behavior,
and in fact measured values for $\absolutevalue*{a_{-}}$ that, remarkably,
recede slightly below the universal van der Waals value [23, 24, 25, 26, 27].
While similar behavior can be obtained in some theoretical scenarios [28, 20],
it is generally unclear how to connect these to 7Li, and sophisticated
numerical models have so far failed to reproduce the data. The long-standing
challenge to explain this unexpected mismatch between theory and experiment is
now referred to as the lithium few-body puzzle [29, 30, 31].
Figure 1: Schematic representation of three-body recombination through two
distinct spin-exchange pathways, where the color of particles represents their
spin state and two connected particles represent a molecule. The upper pathway
contains purely two-body spin-exchange, where one of the particles conserves
its spin throughout the recombination. In the bottom pathway, all three
particles partake in spin-exchange such that no single spin is conserved. We
note that there is an additional pathway, not pictured here, in which all
particles preserve their initial spin state. This pathway is included in all
our calculations.
In this work, we investigate the connection between the anomalous value of
$\absolutevalue*{a_{-}}$ in 7Li and the presence of three-body spin-exchange
interactions. Here, spin-exchange refers to a process in which the internal
spin state of the atoms is altered by coupling of the valence electrons, and
thus necessarily occurs at short length scales. Due to the aforementioned
suppression of three-body probability in this regime, a useful distinction can
be made between two-body spin exchange, where the third particle only
spectates and its state is conserved, and three-body spin-exchange, where all
particles partake. As we illustrate in Fig. 1, this distinction can also
naturally be applied to three-body recombination. For many applications, the
contributions of three-body spin-exchange are negligible, which significantly
simplifies the three-body problem [15, 16, 32, 17, 33, 18, 19, 34, 20].
Recently however, studies have found three-body spin-exchange to contribute
significantly to three-body observables in 39K [35] and 7Li [36], both at
relatively large magnetic fields. Motivated by these findings we study the
Efimov spectrum in 7Li, using a recently developed numerical approach to the
quantum mechanical three-body problem which, together with high-performance
computing facilities, allows us to include all coupled three-body channels in
the Hamiltonian [37, 35] (see Methods).
## Inflation of the Efimov spectrum
Following the experiment of Ref. [26], we analyze two high-field Feshbach
resonances in spin-polarized ultracold 7Li gases, in the hyperfine states
$\ket*{f,m_{f}}_{\mathrm{in}}=\ket*{1,1}$ and
$\ket*{f,m_{f}}_{\mathrm{in}}=\ket*{1,0}$ respectively. Both resonances have
similar negative background scattering lengths on the order of
$r_{\mathrm{vdW}}=32.4863\ a_{0}$, and are both of intermediate to narrow
resonance width [26, 38]. In line with experiment we study the rate of three-
body loss of the trapped gas density $n$, typically expressed in terms of a
recombination rate constant $L_{3}$ as $\mathrm{d}n/\mathrm{d}t=-L_{3}\
n^{3}$. We calculate $L_{3}$ for a range of scattering lengths on the
attractive ($a<0$) side of the Feshbach resonance, and subsequently extract
the values of the three-body parameter $a_{-}$ and trimer width $\eta_{-}$ by
fitting the data to universal predictions from effective field theory [1]. To
highlight the role of three-body spin-exchange, we compare a full multichannel
spin (FMS) model with a fixed spectating spin (FSS) model [35], where the
latter is constrained to purely two-body spin exchange (i.e. the upper pathway
in Fig. 1, see Methods for more detail). To model the pairwise interactions we
use realistic singlet and triplet Born-Oppenheimer interaction potentials
[38]. It follows that the FMS model is quantum mechanically rigorous, save for
the omission of nonadditive three-body forces which are generally assumed to
be negligible in ultracold atomic gases [4].
Figure 2: Results of our three-body calculations, compared directly with the
experimental data of Ref. [26]. Table (a) shows the three-body parameter
$a_{-}$ and Efimov trimer width $\eta_{-}$ obtained using FSS and FMS models.
The associated values of the three-body recombination rate coefficient $L_{3}$
as a function of the scattering length $a$ are shown in Figs. (b) and (c), for
the two distinct incoming hyperfine states. The fits of $L_{3}$ to universal
theory giving the results in table (a) are shown as dash-dotted lines in
matching color. Insets show the two-body scattering length as a function of
magnetic field surrounding the Feshbach resonance, where the green shaded
region of field values matches the range of scattering lengths in the
enclosing figure. We note that next to Ref. [26], an additional independent
measurement of the Efimov trimer in the $\ket*{1,1}$ state obtained
$a_{-}/r_{\mathrm{vdW}}=-7.76(31)$ and $\eta_{-}=0.17$ [27].
Our results are shown in Fig. 2. In an FSS calculation, the value of
$\absolutevalue*{a_{-}}$ is significantly larger than both the universal van
der Waals value and the experimental data, suggesting a significant squeezing
of the spectrum for this narrow resonance which is in line with the majority
of multichannel three-body models in the current literature [17, 18, 22]. Our
main result is that upon including three-body spin-exchange processes, the
additional accessible states induce non-trivial multichannel physics that acts
to cancel the increase of $\absolutevalue*{a_{-}}$, and can even decrease
$\absolutevalue*{a_{-}}$ to below the universal van der Waals value thus
affecting an inflation of the spectrum. Consequently, our FMS calculations
significantly improve on the FSS results with respect to the experimental
data, and for the $\ket*{f,m_{f}}_{\mathrm{in}}=\ket*{1,1}$ state specifically
our FMS results for both $a_{-}$ and $\eta_{-}$ fall within the experimental
uncertainty of Ref. [26]. Due to the increased number of coupled channels, the
FMS calculations in the $\ket*{f,m_{f}}_{\mathrm{in}}=\ket*{1,0}$ state are
too intensive numerically to get fully converged, but the similarity in
behavior to the $\ket*{f,m_{f}}_{\mathrm{in}}=\ket*{1,1}$ state strongly
indicates that a similar match with the experiment is achievable if numerical
resources allow (see Supplementary Material).
Previous three-body studies have shown that the universal increase in
$\absolutevalue{a_{-}}$ near narrow resonances arises from a repulsive barrier
in the three-body potential scaling with the effective range, which
progressively squeezes the Efimov spectrum [39]. Recently however, the
unexpected observation of a trimer state above the atom-dimer disassociation
threshold has prompted further theoretical analysis, which indicates that the
Efimov state of 7Li may actually exist behind the universal repulsive barrier
[40]. While the universal effects of the barrier are evident in our FSS
results, the observed sensitivity of the value of $\absolutevalue{a_{-}}$ to
short-range three-body spin-exchange processes is in fact consistent with the
presence of a non-universal trimer state in the inner potential well. In this
sense, our FMS results can serve as important numerical confirmation of this
novel trimer binding mechanism. Such an identification furthermore indicates
that three-body spin-exchange couplings induce an effective attractive
interaction that can tug the trimer state into the inner potential well, thus
causing the inflation of the Efimov spectrum. In the future, it may be
interesting to analyze the exact nature of this trimer and the corresponding
potential in more detail, for which an approach similar to Ref. [22] could
prove a useful starting point.
Figure 3: Analysis of the electronic spin-propensity rule for three-body
recombination. In Figs. (a) and (b) we show the components of the incoming
two-body radial wave functions $r\psi_{\mathrm{in}}(r)$, highlighting the
electronic spin components $\ket*{{\downarrow}{\downarrow}}$ in green and
$\ket*{{\downarrow}{\uparrow}}$ in dash-dotted orange. The singlet
$\ket*{{\downarrow}{\uparrow}}$ components are resonantly enhanced by the
Feshbach level and thus dominate the wave function at short distance. In Fig.
(c) and (d) we show the averaged partial recombination rate
$\mathcal{L}_{3}(M_{S})$ to channels with definite total electronic projection
$M_{S}$, which automatically fixes the total nuclear projection $M_{I}$. The
fraction in blue originates from atom-dimer states accessible by purely two-
body spin exchange, while the red portion comes from states only accessible by
three-body spin exchange.
## Propensity rule for three-body recombination
Next to the excellent match with the three-body parameter our calculations
also show good agreement with the individual measurements of $L_{3}$. As
three-body recombination is an important and ubiquitous chemical process,
relevant far beyond the specific context of Efimov physics, this agreement
motivates us to analyze the recombination rates more closely [41].
Specifically we aim to characterize the nature of the spin-exchange pathways,
for which we decompose $L_{3}$ into partial recombination rates to all
distinct atom-dimer channels, providing a measure of the population
distribution of product states following three-body recombination.
Interestingly our calculations show that, even though the FSS approximation
fails, the actual number of strongly coupled channels remains remarkably
small, with the vast majority of product channels having near negligible
relative recombination rates (see Supplementary Material for more detail).
We will now show that this behavior results from a manifestation of Wigner’s
electronic-spin conservation rule [42, 43] for three atoms, originating from
the relatively weak coupling between the electronic and nuclear spins of the
atoms at large magnetic fields. In this Paschen-Back regime [44], the spins
independently precess around the magnetic field direction, such that single-
particle states are best described by the individual projection quantum
numbers $m_{s}$ and $m_{i}$. For simplicity let us briefly neglect the
subdominant contribution from the nuclear spin, whose coupling to the magnetic
field is relatively weak. Then both incoming single-particle states studied in
this work may be written as,
$\ket*{f,m_{f}}_{\mathrm{in}}\sim\ket*{\downarrow}+\delta\ket*{\uparrow},$ (1)
where $\ket*{{\downarrow}/{\uparrow}}$ represent the down and up electronic
spin states $m_{s}=-1/2$ and $m_{s}=1/2$ respectively, and $\delta$ scales
with the ratio of the hyperfine and Zeeman energies [45].
In the Paschen-Back regime $\delta$ is a small number, which motivates us to
expand the incoming three-body state into four distinct components scaling as
$\delta^{n}$, where $n$ equals the number of electronic spins pointing up.
Each component can be uniquely identified with a definite value of the total
electronic spin projection $M_{S}=m_{s_{1}}+m_{s_{2}}+m_{s_{3}}=-3/2+n$, which
is rigorously conserved in this basis as it is fully uncoupled from the
nuclear spin. Hence, if it is possible to identify a dominant incoming
projection $M_{S}$, then the outgoing product state distribution will show a
propensity to states that conserve this projection. To determine the dominant
incoming component we have to consider both the scaling with the small
parameter $\delta$ and the amplitude of the associated wave functions at small
nuclear separations, where recombination processes typically take place. As we
illustrate in Figs. 3(a) and 3(b), the dominant short-range components of the
incoming two-body wave functions are in the singlet two-body state
$\ket*{{\downarrow}{\uparrow}}$, correspondent with the spin character of the
resonantly coupled Feshbach level. It follows that recombination preferably
occurs through three-body states with (partial) singlet character, of which
the state $\ket*{{\downarrow}{\downarrow}{\uparrow}}$ ($n=1$) has the dominant
scaling with $\delta$. Thus, we finally deduce that three-body recombination
will show a propensity to product channels with $M_{S}=-\frac{1}{2}$.
To confirm the presence of this propensity in our numerics, we introduce the
quantity $\mathcal{L}_{3}(M_{S})$ as the average of the partial recombination
rates with respect to the projection of the associated outgoing state on a
three-body spin state with definite total electronic spin projection (see
Methods section). In this way $\mathcal{L}_{3}(M_{S})$ provides a measure for
the relative prevalence of a given value of $M_{S}$ in the product state
distribution following recombination. As shown in Figs. 3(c) and 3(d), the
electronic spin-conservation propensity is clearly present for both Feshbach
resonances we study. Note that in the argument outlined above all three
particles are treated equally, and no reference is made to a special role for
the spectating particle. Indeed, we observe that the
$\ket*{{\downarrow}{\downarrow}{\uparrow}}$ component of $\mathcal{L}_{3}$ is
split almost equally between channels in which the spectating spin is
conserved or changed, with the latter being slightly larger.
## Outlook
Our findings suggest several new avenues for future research. First the
excellent match between our 7Li results and the experimental data now provides
a new benchmark for the theoretical description of strongly interacting few-
body systems. Our method thus shows great promise for studying other systems
where measurements deviate from the current theoretical predictions [21].
Aside from these experimental concerns, there is also a more conceptual
challenge to now further characterize the physical mechanism underpinning the
formation of the non-universal Efimov trimer observed in this work, which will
require untangling the exact reshaping of the three-body potential in the
presence of three-body spin-exchange [22, 39, 40].
Our results also have interesting implications beyond the realm of Efimov
physics. The uncovered spin-propensity rule in the rate of three-body
recombination provides a remarkably simple picture of triatomic chemical
reactions in large magnetic fields, which can now aid in the understanding and
possible experimental control of state-to-state quantum chemistry in these
regimes [33, 34]. Further studies in this direction may also seek to elucidate
the more subtle role of the individual nuclear spins, which should have a
similar propensity to be conserved in the Paschen-Back regime.
###### Acknowledgements.
We thank Jose D’Incao for fruitful discussions. J.v.d.K. and S.J.J.M.F.K.
acknowledge financial support from the Dutch Ministry of Economic Affairs and
Climate Policy (EZK), as part of the Quantum Delta NL program. D.J.M.A.B.
acknowledges financial support from the Netherlands Organisation for
Scientific Research (NWO) under Grant No. 680-47-623. The results presented in
this work were obtained on the Dutch national supercomputer Snellius, with
support from TU/e HPC Lab and Surf.
## References
* Braaten and Hammer [2006] E. Braaten and H.-W. Hammer, Physics Reports 428, 259 (2006).
* Efimov [1970] V. Efimov, Phys. Lett. B 33, 563 (1970).
* Efimov [1971] V. Efimov, Sov. J. Nucl. Phys. 12, 589 (1971).
* Naidon and Endo [2017] P. Naidon and S. Endo, Rep. Prog. Phys. 80, 056001 (2017).
* Greene _et al._ [2017] C. H. Greene, P. Giannakeas, and J. Pérez-Ríos, Rev. Mod. Phys. 89, 035006 (2017).
* D’Incao [2018] J. P. D’Incao, J. Phys. B: At. Mol. Opt. Phys. 51, 043001 (2018).
* Köhler _et al._ [2006] T. Köhler, K. Góral, and P. S. Julienne, Rev. Mod. Phys. 78, 1311 (2006).
* Chin _et al._ [2010] C. Chin, R. Grimm, P. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
* Esry _et al._ [1999] B. D. Esry, C. H. Greene, and J. P. Burke, Phys. Rev. Lett. 83, 1751 (1999).
* Kraemer _et al._ [2006] T. Kraemer, M. Mark, P. Waldburger, J. G. Danzl, C. Chin, B. Engeser, A. D. Lange, K. Pilch, A. Jaakkola, H.-C. Nägerl, and R. Grimm, Nature 440, 315 (2006).
* Berninger _et al._ [2011] M. Berninger, A. Zenesini, B. Huang, W. Harm, H.-C. Nägerl, F. Ferlaino, R. Grimm, P. S. Julienne, and J. M. Hutson, Phys. Rev. Lett. 107, 120401 (2011).
* Wild _et al._ [2012] R. J. Wild, P. Makotyn, J. M. Pino, E. A. Cornell, and D. S. Jin, Phys. Rev. Lett. 108, 145305 (2012).
* Wang _et al._ [2012] J. Wang, J. P. D’Incao, B. D. Esry, and C. H. Greene, Phys. Rev. Lett. 108, 263001 (2012).
* Naidon _et al._ [2014] P. Naidon, S. Endo, and M. Ueda, Phys. Rev. A 90, 022106 (2014).
* Petrov [2004] D. S. Petrov, Phys. Rev. Lett. 93, 143201 (2004).
* Gogolin _et al._ [2008] A. O. Gogolin, C. Mora, and R. Egger, Phys. Rev. Lett. 100, 140404 (2008).
* Schmidt _et al._ [2012] R. Schmidt, R. Rath, and W. Zwerger, Eur. Phys. J. B 85, 386 (2012).
* Langmack _et al._ [2018] C. Langmack, R. Schmidt, and W. Zwerger, Phys. Rev. A 97, 033623 (2018).
* Chapurin _et al._ [2019] R. Chapurin, X. Xie, M. J. Van de Graaff, J. S. Popowski, J. P. D’Incao, P. S. Julienne, J. Ye, and E. A. Cornell, Phys. Rev. Lett. 123, 233402 (2019).
* Secker _et al._ [2021a] T. Secker, D. J. M. Ahmed-Braun, P. M. A. Mestrom, and S. J. J. M. F. Kokkelmans, Phys. Rev. A 103, 052805 (2021a).
* Etrych _et al._ [2023] J. c. v. Etrych, G. Martirosyan, A. Cao, J. A. P. Glidden, L. H. Dogra, J. M. Hutson, Z. Hadzibabic, and C. Eigen, Phys. Rev. Res. 5, 013174 (2023).
* van de Kraats _et al._ [2023] J. van de Kraats, D. J. M. Ahmed-Braun, J.-L. Li, and S. J. J. M. F. Kokkelmans, Phys. Rev. A 107, 023301 (2023).
* Pollack _et al._ [2009] S. E. Pollack, D. Dries, and R. G. Hulet, Science 326, 1683 (2009).
* Gross _et al._ [2009] N. Gross, Z. Shotan, S. Kokkelmans, and L. Khaykovich, Phys. Rev. Lett. 103, 163202 (2009).
* Gross _et al._ [2010] N. Gross, Z. Shotan, S. Kokkelmans, and L. Khaykovich, Phys. Rev. Lett. 105, 103203 (2010).
* Gross _et al._ [2011] N. Gross, Z. Shotan, O. Machtey, S. Kokkelmans, and L. Khaykovich, C. R. Phys. 12, 4 (2011).
* Dyke _et al._ [2013] P. Dyke, S. E. Pollack, and R. G. Hulet, Phys. Rev. A 88, 023625 (2013).
* Sørensen _et al._ [2012] P. K. Sørensen, D. V. Fedorov, A. S. Jensen, and N. T. Zinner, Phys. Rev. A 86, 052516 (2012).
* Yudkin and Khaykovich [2021] Y. Yudkin and L. Khaykovich, Phys. Rev. A 103, 063303 (2021).
* D’Incao _et al._ [2022] J. D’Incao, Y. Yudkin, P. Julienne, and L. Khaykovich, Bulletin of the American Physical Society (2022).
* Yudkin _et al._ [2023a] Y. Yudkin, P. S. Julienne, and L. Khaykovich, Phys. Rev. A 107, 053303 (2023a).
* Massignan and Stoof [2008] P. Massignan and H. T. C. Stoof, Phys. Rev. A 78, 030701(R) (2008).
* Wolf _et al._ [2017] J. Wolf, M. Deiß, A. Krükow, E. Tiemann, B. P. Ruzic, Y. Wang, J. P. D’Incao, P. S. Julienne, and J. H. Denschlag, Science 358, 921 (2017).
* Haze _et al._ [2022] S. Haze, J. P. D’Incao, D. Dorer, M. Deiß, E. Tiemann, P. S. Julienne, and J. H. Denschlag, Phys. Rev. Lett. 128, 133401 (2022).
* Secker _et al._ [2021b] T. Secker, J.-L. Li, P. M. A. Mestrom, and S. J. J. M. F. Kokkelmans, Phys. Rev. A 103, 022825 (2021b).
* Li _et al._ [2022] J.-L. Li, T. Secker, P. M. A. Mestrom, and S. J. J. M. F. Kokkelmans, Phys. Rev. Research 4, 023103 (2022).
* Secker _et al._ [2021c] T. Secker, J.-L. Li, P. M. A. Mestrom, and S. J. J. M. F. Kokkelmans, Phys. Rev. A 103, 032817 (2021c).
* Julienne and Hutson [2014] P. S. Julienne and J. M. Hutson, Phys. Rev. A 89, 052715 (2014).
* Tempest and Jonsell [2023] K.-M. Tempest and S. Jonsell, Phys. Rev. A 107, 053319 (2023).
* Yudkin _et al._ [2023b] Y. Yudkin, R. Elbaz, J. P. D’Incao, P. S. Julienne, and L. Khaykovich, arXiv 2308, 06237 (2023b).
* Mirahmadi and Pérez-Ríos [2022] M. Mirahmadi and J. Pérez-Ríos, International Reviews in Physical Chemistry 41, 233 (2022).
* Wigner [1927] E. Wigner, Nachr. Akad. Wiss. Goettingen, Math.-Physik Kl. IIa, 375 (1927).
* Moore [1973] J. H. Moore, Phys. Rev. A 8, 2359 (1973).
* Paschen and Back [1921] F. Paschen and E. Back, Physica 1, 261 (1921).
* Stoof _et al._ [1988] H. T. C. Stoof, J. M. V. A. Koelman, and B. J. Verhaar, Phys. Rev. B 38, 4688 (1988).
* Lee _et al._ [2007] M. D. Lee, T. Köhler, and P. S. Julienne, Phys. Rev. A 76, 012720 (2007).
* Moerdijk _et al._ [1996] A. J. Moerdijk, H. M. J. M. Boesten, and B. J. Verhaar, Phys. Rev. A 53, 916 (1996).
* Smirne _et al._ [2007] G. Smirne, R. M. Godun, D. Cassettari, V. Boyer, C. J. Foot, T. Volz, N. Syassen, S. Dürr, G. Rempe, M. D. Lee, K. Góral, and T. Köhler, Phys. Rev. A 75, 020702 (2007).
* Alt _et al._ [1967] E. Alt, P. Grassberger, and W. Sandhas, Nuclear Physics B 2, 167 (1967).
## Methods
### Three-body recombination rate
The three-body recombination rate constant $L_{3}$ in the zero-energy limit
can be formally expressed as [46, 47, 48, 37],
$L_{3}=\sum_{{\nu},c_{3}}L_{3}^{\mathrm{part}}(\varphi_{\nu},c_{3}),$ (2)
where,
$\displaystyle L_{3}^{\mathrm{part}}(\varphi_{\nu},c_{3})=\frac{12\pi
m}{\hbar}(2\pi\hbar)^{6}$ (3) $\displaystyle\times\lim_{E\rightarrow 0}\int
d\hat{\mathbf{q}}_{3}\
q_{3}\absolutevalue*{\matrixelement*{(\mathbf{q}_{3},c_{3}),\varphi_{\nu}}{U_{\alpha
0}(E)}{\Psi_{\mathrm{in}}}}^{2},$
reminiscent of Fermi’s golden rule. Each element
$\matrixelement*{(\mathbf{q}_{3},c_{3}),\varphi_{\nu}}{U_{\alpha
0}(z)}{\Psi_{\mathrm{in}}}$ describes transition from an incoming three-body
state with energy $E$ into a molecular state $\ket*{\varphi_{\nu}}$ of a pair
$\alpha$ and a third free particle with relative momentum $\mathbf{q}_{3}$ in
spin state $\ket*{c_{3}}$.
To calculate the operator $U_{\alpha 0}(E)$ we solve the inelastic three-body
scattering problem in momentum space, using the Alt-Grassberger-Sandhas (AGS)
equations [49] (see Supplemental Material). As any two-body subsystem can
transfer energy to the third particle, the three-body problem embeds the off-
shell solution to the two-body problem, which we obtain by exact
diagonalization of the two-body Hamiltonian. For the two-body interaction $V$
we use realistic Born-Oppenheimer singlet and triplet potentials, which allow
us to accurately account for the molecular level structure underpinning the
three-body recombination [38].
### Three-body spin models
To separate the contributions from two-body and three-body spin-exchange
processes, we use two different models for the spin coupling [35]. In a full
multichannel spin (FMS) model, we make no approximations and include all
three-body channels that conserve the total spin projection
$M_{F}=m_{f_{1}}+m_{f_{2}}+m_{f_{3}}$. In the fixed spectating spin (FSS)
model, we turn off three-body spin exchange processes by replacing the two-
body interaction $V$ with the restricted form,
$V_{\mathrm{FSS}}=V\ket*{c_{3}^{\mathrm{in}}}\bra{c_{3}^{\mathrm{in}}}.$ (4)
In this way the two-body interaction is turned off if the third particle is
not in the incoming spin state. In the AGS equations this replacement is
equivalent to restricting $\ket*{c_{3}}$ to $\ket*{c_{3}^{\mathrm{in}}}$
everywhere, such that $m_{f_{12}}=m_{f_{1}}+m_{f_{2}}$ and $m_{f_{3}}$ are
separately conserved.
### Fitting of the recombination rate
For large negative scattering length, universal theory predicts a power-law
increase of the three-body recombination rate coefficient, given as
$L_{3}=3C_{-}(a)\hbar a^{4}/m$. The function $C_{-}(a)$ gives the log-periodic
oscillation of the recombination rate arising from the Efimov effect. A two-
parameter analytical prediction for $C_{-}(a)$ can be obtained in the
formalism of effective field theory, which gives [1],
$C_{-}(a)=4590\
\frac{\sinh(2\eta_{-})}{\sin^{2}\left[s_{0}\ln(a/a_{-})\right]+\sinh^{2}(\eta_{-})}.$
(5)
To perform the fit of $L_{3}$ we first compute the dimensionless quantity
$mL_{3}/(3\hbar a^{4})$, and subsequently fit $C_{-}(a)$ directly. Provided
that we compute $L_{3}$ close to an Efimov resonance, then 3 data points are
typically sufficient to obtain a converged fit.
### Partial recombination probability
To examine the presence of spin-propensity for three-body recombination we
define an augmented partial recombination rate
$\mathcal{L}_{3}(\left\\{m_{s_{j}},m_{i_{j}}\right\\})$ as,
$\displaystyle\mathcal{L}_{3}(\left\\{m_{s_{j}},m_{i_{j}}\right\\})$
$\displaystyle=$
$\displaystyle\sum_{\nu,c_{3}}\absolutevalue*{\innerproduct*{m_{s_{1}},m_{i_{1}},m_{s_{2}},m_{i_{2}}}{\varphi_{\nu}}}^{2}$
$\displaystyle\times\absolutevalue*{\innerproduct*{m_{s_{3}},m_{i_{3}}}{c_{3}}}^{2}\frac{L_{3}^{\mathrm{part}}(\varphi_{\nu},c_{3})}{L_{3}}.$
By this definition $\mathcal{L}_{3}(\left\\{m_{s_{j}},m_{i_{j}}\right\\})$
averages the spin-projection
$\absolutevalue*{\innerproduct*{m_{s_{1}},m_{i_{1}},m_{s_{2}},m_{i_{2}}}{\varphi_{\nu}}}^{2}\absolutevalue*{\innerproduct*{m_{s_{3}},m_{i_{3}}}{c_{3}}}^{2}$
with respect to the discrete probability distribution
$L_{3}^{\mathrm{part}}(\varphi_{\nu},c_{3})/L_{3}$, and hence forms a measure
of the relative importance of a state defined by a set of quantum numbers
$\left\\{m_{s_{j}},m_{i_{j}}\right\\}$ for three-body recombination,
normalised to unity if summed over all available states. To obtain
$\mathcal{L}_{3}(M_{S})$ we sum
$\mathcal{L}_{3}(\left\\{m_{s_{j}},m_{i_{j}}\right\\})$ over all spin states
with definite $M_{S}$, which then finally gives the results shown in Figs.
3(c) and 3(d).
|
# Development of a sub-milimeter position sensitive gas detector ††thanks:
Supported by NSFC (11075095) and Shandong Province Science Foundation
(ZR2010AM015)
DU Yanyan1 XU Tongye1 SHAO Ruobin1
WANG Xu1 ZHU Chengguang1;1<EMAIL_ADDRESS>1 MOE key lab on particle physics
and particle irradiation,
Shandong University, Ji’nan 250100, China
###### Abstract
A position sensitive thin gap chamber has been developed. The position
resolution was measured using the cosmic muons. This paper presents the
structure of this detector, position resolution measurement method and
results.
###### keywords:
Thin gap chamber, Position resolution
###### pacs:
0
7.77.Ka, 29.40.Gx
## 1 Introduction
TGC( Thin Gap Chamber) used in ATLAS experiment [1] shows good performance in
the fast response and time resolution, but with limited position resolution.
The improvement of the position resolution with the timing performance
retained is straightforward for its flexibility to be used in the future
experiments and radiation measurement, for example, the upgrade of the trigger
system of ATLAS experiment. The main goal of the study described in this paper
aimed to build a prototype detector based on TGC, which can have a position
resolution better than 300${\mu}m$, while keeping timing performance not
deteriorated
The TGC detector operates in saturated mode by using a highly quenching gas
mixture of carbon dioxide and n-pentane, $55\%$ :$45\%$, which has many
advantages, such as small sensitivity to mechanical deformations, small
parallax, small Landau tails and good time resolution, but a position
sensitivity of around 1$cm$, decided by the geometrical width of the readout
channel and the strength of the induced signal. To improve the position
resolution, we concentrated on the improving of the method of signal readout
by fine tuning of the structure of the detector.
The new detector, named as pTGC(precision Thin Gap Chamber) based on the ATLAS
TGC, is constructed and tested. We found the position resolution can be
improved to be less than $300{\mu}m$, which meets the requirements.
In section 2, the structure of pTGC detector is described. Section 3 is
devoted to pTGC’s position resolution measurement. Results of the measurement
is summarized in section 4.
Figure 1: The schematic structure of pTGC-I chamber. Anode wires are placed in
the middle, with copper strip etched on the inner surface of the PCB board,
perpendicular to the wire direction.
Figure 2: The schematic structure of pTGC-II chamber. Compared to pTGC-I,
additional isolation layer and graphite layer cover the etched copper strips.
## 2 Construction of pTGC
In the pTGC development, two versions of detectors are constructed and tested,
which are referred as $pTGC-I$ in the first stage and $pTGC-II$ in the second
stage, respectively.
The schematic structure of $pTGC-I$ is shown in Fig. 2, similar to the
structure of ATLAS TGC, except that the position of the strips for signal
collection are modified. 48 copper strips of $0.8mm$ wide and $0.2mm$ spaced
are etched on the inner surface of the 2 parallel PCB boards, which form a
thin spaced chamber. The wires, segmented at $1.8mm$ interval and
perpendicular to the strip direction, are sandwiched in between the two PCB
boards. The resulted size of the detector is defined by the number of wires
and strips, which are $290mm{\times}50mm$.
In the test of $pTGC-I$, the discharge happened between wires and strips
resulted in fatal damage on the frontend electronics, even though we have
designed a protection circuit to insert between detector and frontend
electronics board. This means an instability for big detector and for long
time running. Besides, the induced charge on strips spread roughly $5$ to
$6mm$, which leaves limited rooms for reducing the quantity of the channels by
enlarging the width of strips. Based on $pTGC-I$, the $pTGC-II$ is developed
to deal with these problems.
The schematic structure of $pTGC-II$ is shown in the Fig. 2. The strip width
is enlarged to $3.8mm$ ($0.2mm$ spaced), and a thin ($100{\mu}m$) insolation
layer is pasted on the strip layer. The isolation layer is then coated with a
thin ( $30{\mu}m$) graphite layer as the electric ground to form the electric
field with wires. This graphite layer acts as the protector of the frontend
electronics from discharge and can enlarge the spreading size of the induced
charge on the strip layer. We tune the resistivity of the graphite layer to be
around $100k\Omega$, considering the diffusion speed of the charge, as well.
the resulted size of $pTGC-II$ is $290mm*200mm$.
Both detector use gas mixture of carbon dioxide and n-pentane, $55\%$ :$45\%$,
as working gas, and the anode wire is set to high voltage of 2900v, which are
all the same configuration as the ATLAS TGC detector to maintain the its
features relative to the time measurement of the detector.
## 3 Position resolution measurement
With 3 layers of identical chambers placed in parallel, and 2 layers of
scintillator detectors to build a muon hodoscope(see Fig 3), the $pTGC-I$ and
$pTGC-II$ detectors are tested. The induced charge on each strip is integrated
for the the position calculation based on the charge center-of-gravity
algorithms. The measured hitting position on the 3 layers of chamber are
supposed to be aligned into a straight line concerning the penetration power
of muons. The residue of the position relative to the straight line is then
used to calculate the position resolution of the detectors.
### 3.1 Signal definition
Using oscilloscope, we first observed the induced signal in one wire group and
3 adjacent strips (limited by channels of oscilloscope), as shown in Fig. 4.
It’s apparent that the signals are great significant above the noise and the
signals on the strip are in different magnitudes as expected.
For position resolution measurement, we designed a much more complicated
DAQ(data acquisition) system based on gassiplex frontend electronics [2] to
readout and digitize the induced charge from a quantity of channels of the 3
chambers in a more complex hodoscope [3]. Once the two scintillator detector
of the hodoscope are both fired, the DAQ is triggered. The trigger signal is
sent to the detector front end electronics, which then close the gate for the
discharge of capacitance which has integrated the signal charge on. The charge
on the capacitance are then read out one by one controlled by the clock
distributed from the DAQ system. The charge are then digitized and saved into
computer.
The digitized charge, denoted by $Q_{i}$ where $i$ is the channel number,
consists of three parts: electronic pedestal, noise, and charge induced by
muon hit.
First of all, we need to figure out the pedestal and noise for each channel.
The method is to histogram the integrated charge for each channel using a soft
trigger where no real muon induced signal appear in the data. Fitting the
histogram with a gaussian function to get the pedestal and the noise, denoted
by $P_{i}$ and $\sigma_{i}$, as shown in Fig. 5, where the height of the
histogram represents the pedestal and the error bar represents the noise of
that channel.
In the analysis, if $Q_{i}>P_{i}+3\sigma_{i}$, the channel is considered to be
fired by real muon hit, and the signal charge is calculated as:
$S_{i}=Q_{i}-P_{i},$ (1)
Figure 3: The comic muon hodoscope used for the chamber testing. Plastic
scintillator detector are used for trigger. 3 identical pTGC chambers placed
in parallel in between the 2 scintillator detectors. Figure 4: The observed
signals on wires and several copper strips induced by the same incident cosmic
ray. The signal on wire are negative, and the signal on strips are positive.
Figure 5: The noise and pedestal distribution of 96 signal strips in one
chamber (The x-axis is the signal strip number, the vertical coordinate is the
pedestal value and the error bars presents the noise of that channel.)
The signal magnitude distribution of the largest signal in each cluster
(cluster definition is in next section), named as the peak signal, is shown in
Fig. 6. The distribution of the second largest signal in each cluster, named
as second peak signal, is shown in Fig. 7. The distribution of the sum of all
charge in one cluster is shown in Fig. 8. The correspondence between the
magnitude of the signal and the charge is $1fC/3.6bits$. We can then calculate
that the maximum probable charge of the largest signal in one cluster is
$69fC$, the maximum probable total charge of one cluster is $470fC$, which is
consistent with the measurement in [1]
### 3.2 Cluster definition
The induced charge by the incident muons are distributed on several adjacent
strips, which are grouped in ”cluster” in the analysis and used for the hit
position calculation. In one event, we search all the channels of one
detector, and define group of fired adjacent strips without space as a
cluster. To suppress the fake signals from noise, if the cluster contains only
one strip, the cluster is dropped. The cluster size and number of cluster per
detector per event are shown in Fig. 9 for $pTGC-I$ and Fig. 10 for $pTGC-II$.
It can be seen that in both cases one cluster contain average six strips and
almost every event contains one cluster, which is consistent with the
expected. The hit position is then calculated for each cluster by
$x=\sum_{i}(S_{i}*x_{i})/\sum_{i}(S_{i}),$ (2)
where $x_{i}$ is the center coordination of the $i-th$ strip.
Figure 6: The distribution of the largest signal in one cluster. The x-axis is
the digitized charge collected.
Figure 7: The distribution of the second largest signal in one cluster
Figure 8: The distribution of total charge induced in one cluster
Figure 9: For pTGC-I: (Left) The distribution of cluster size (quantity of
strips in one cluster). (Right) The quantity of cluster in one chamber per
triggered event.
Figure 10: For pTGC-II: (Left) The distribution of cluster size (quantity of
strips in one cluster). (Right) The quantity of cluster in one chamber per
triggered event.
### 3.3 Position resolution
As redundant design, the strips are etched on both inner surface of the PCB
boards. Signals will be induced by the same avalanche on the 2 face-to-face
strips, which corresponds to an double measurements of a single hit. To
compare the two measurements, denoted as $x_{1}$ and $x_{1}^{\prime}$, we fill
$x_{1}-x_{1}^{\prime}$ into histogram to see the broadness of the
distribution. From a simple gaussian function fit, we observed a narrow width
of around $36{\mu}m$, which means that the electronics noise effect on the
resolution is much small. This is consistent with the expectation when to
compare Fig. 5 and Fig. 6, where it shows the signal is great significant
compared to the noise.
After the three hit positions $x_{1}$, $x_{2}$, $x_{3}$ are calculated for the
3 parallel chambers, to simplify the calculation, we first use $x_{1}$ and
$x_{3}$ to calculate the expected hit position on the second layer $x_{2c}$:
$x_{2c}=x_{1}\frac{L_{23}}{L_{12}+L_{23}}+x_{3}\frac{L_{12}}{L_{12}+L_{23}},$
(3)
where $L_{12}$ and $L_{23}$ are the vertical distance between the detector 1,2
and 2,3. To assume the same position resolution $\sigma$ for the 3 identical
detectors, we know the resolution of $x_{2c}$, with the error propagation, is:
$\sigma_{2c}=\sqrt{\frac{L_{23}^{2}}{(L_{12}+L_{23})^{2}}+\frac{L_{12}^{2}}{(L_{12}+L_{23})^{2}}}\sigma{\equiv}k\sigma,$
(4)
Filling $x_{2}-x_{2c}$ into the histogram and then fit with gaussian function,
the width is $w=\sqrt{1+k^{2}}\sigma$. So we can directly calculate the
position resolution of the detector as
$\sigma=\frac{w}{\sqrt{1+k^{2}}}.$ (5)
From Fig. 11 and Fig. 12, we can obtain that the position resolution are
$359um$ for $pTGC-I$ and $233um$ for $pTGC-II$. In both of the cases, the
detector resolution has reach our design requirement. In test, we see that
$pTGC-II$ are more stable with the graphite layer protection and achieve a
better resolution even with less channels.
Figure 11: The distribution of $x_{2}-x_{2c}$ for pTGC-I. The corresponding
position resolution of the chamber is
$\sigma=\frac{w}{\sqrt{1+k^{2}}}=\frac{439{\mu}m}{1.22}=359{\mu}m$.
Figure 12: The distribution of $x_{2}-x_{2c}$ for pTGC-II. The corresponding
position resolution of the chamber is
$\sigma=\frac{w}{\sqrt{1+k^{2}}}=\frac{286{\mu}m}{1.22}=233{\mu}m$.
To look at the dependence of the position sensitivity of the detector to the
incident angle of the muon, we divide the data into groups. Each group of data
contains the events of muon with specific incident angle. To redo the analysis
above, the result is shown in Fig. 13, which shows that the position
resolution of $pTGC-II$ is insensitive to the incident angle of muons.
To check the effect of the electronic noise, we use part of the top highest
signals in one cluster to calculate the position resolution. The result is
shown in Fig. 14, which shows that the resolution are similar and the
electronic noise doesn’t affect much.
Figure 13: The position resolution variance relative to the incident angle of
the cosmic rays. The x-axis is the incident angle of cosmic rays.
Figure 14: The position resolution variance relative to the quantity of strips
in one cluster used for position calculation. The x-axis is the quantity of
strips in one cluster used for position calculation.
## 4 Summary
Two pTGC version $pTGC-I$ and $pTGC-II$, have been constructed and tested.
With the basic structure and working gas unchanged, the detector can attains
the exiting features like good time resolution and fast response, which are
essential for trigger. By revising the signal collecting structure and method,
the position resolution is improved from the level of centimeter to be less
than $300{\mu}m$, which meet the requirement of design. To be noticed that the
resolution measured is a global resolution of the detector, which include the
effect of the non-uniformity of the detector all over the sensitive area. The
3 detectors are placed in parallel with mechanical method, the relative
rotation of the 3 detectors will deteriorate the final measured resolution,
which means that the measured resolution is much conservative.
## References
* [1] Atlas Collaboration, ATLAS muon spectrometer: Technical design report, 1997. CEAN/LHCC/97-22.
* [2] Liu Minghui $etal.$ Nuclear electronics and detection technology, 2008(5)
* [3] Xu Tongye $etal.$ arXiv:1308.5751
|
# On Stability and Denominators of $F$-pure Thresholds in Families of Diagonal
Hypersurfaces
Gari Lincoln Chua Department of Mathematics and Statistics, University of New
Mexico, 1 University of New Mexico, MSC01 1115
Albuquerque, NM 87131, United States of America<EMAIL_ADDRESS>
###### Abstract.
Given a prime number $p$ and a positive integer $m$, we provide a family of
diagonal hypersurfaces $\\{f_{n}\\}_{n=1}^{\infty}$ in $m$ variables, for
which the denominator of $\operatorname{fpt}(f_{n})$ (in lowest terms) is
always $p$ and whose $F$-pure thresholds stabilize after a certain $n$. We
also provide another family of diagonal hypersurfaces
$\\{g_{n}\\}_{n=1}^{\infty}$ in $m$ variables, for which the power of $p$ in
the denominator of $\operatorname{fpt}(g_{n})$ (in lowest terms) diverges to
$\infty$ as $n\to\infty$. This behavior of the denominator of the $F$-pure
thresholds is dependent on the congruence class of $p$ modulo the smallest two
exponents of $\\{f_{n}\\}$ and $\\{g_{n}\\}$.
## 1\. Introduction
Let $R=k[x_{1},\dots,x_{n}]$ where $k$ is a perfect field of characteristic
$p>0$ and denote the maximal ideal $\mathfrak{m}=(x_{1},\dots,x_{n})$. The
$F$-pure threshold of a polynomial $f\in R$, introduced in [10], is defined to
be
$\operatorname{fpt}(f)=\sup_{e}\left\\{\frac{N}{p^{e}}:f^{N}\not\in\mathfrak{m}^{[p^{e}]}\right\\}$,
where $\mathfrak{m}^{[p^{e}]}=(x_{1}^{p^{e}},\dots,x_{n}^{p^{e}})$ is the
$e$th Frobenius power of $\mathfrak{m}$. The $F$-pure threshold
$\operatorname{fpt}(f)$ is a rational number in $(0,1]$ that is an invariant
of positive characteristic singularities, with smaller values meaning that the
hypersurface is “more singular."
The $F$-pure threshold (fpt) is closely related to the log canonical threshold
(lct), an invariant of characteristic $0$ singularities. If
$f\in\mathbb{Q}[x_{1},\dots,x_{n}]$ and $\overline{f}$ is the reduction of $f$
modulo $p$, both $\operatorname{lct}(f)$ and
$\operatorname{fpt}(\overline{f})$ are rational numbers between $0$ and $1$.
Moreover,
$\operatorname{fpt}(\overline{f})\leq\operatorname{lct}(f)\hskip 10.0pt\text{
and }\hskip
10.0pt\lim_{p\to\infty}\operatorname{fpt}(\overline{f})=\operatorname{lct}(f).$
This motivates the following question (see for example, [8]): When is
$\operatorname{fpt}(\overline{f})=\operatorname{lct}(f)$, and if
$\operatorname{fpt}(\overline{f})\neq\operatorname{lct}(f)$, does $p$ always
divide the denominator of $\operatorname{fpt}(\overline{f})$?
The answer to the previous question is complicated. Canton, Hernandez,
Schwede, and Witt in [2] have shown a family of polynomials $f$ for which
$\operatorname{fpt}(\overline{f})\neq\operatorname{lct}(f)$ but the
denominator of $\operatorname{fpt}(\overline{f})$ is not a multiple of $p$. On
the other hand, [3], [5], [7] have shown that in many classes of polynomials,
if $\operatorname{fpt}(\overline{f})\neq\operatorname{lct}(f)$, then the
denominator of $\operatorname{fpt}(\overline{f})$ must be a multiple of $p$.
In particular, in [5], the authors have shown that if $f$ is a quasi-
homogeneous polynomial with an isolated singularity at the origin, then
$\operatorname{fpt}(\overline{f})$ is either $\operatorname{lct}(f)$ or its
denominator must be a power of $p$. The same paper also gives a bound on the
power of $p$ (depending on $\overline{f}$) appearing in the denominator of
$\operatorname{fpt}(\overline{f})$ depending on the choice of grading on
$k[x_{1},\dots,x_{n}]$ that turns $\overline{f}$ homogeneous.
The aformentioned bound in the power of $p$ in the $F$-pure threshold
motivates the following question: Given a family of quasi-homogeneous
polynomials $\\{f_{n}\\}$ in characteristic $p$ for which
$\operatorname{fpt}(f_{n})\neq\operatorname{lct}(f_{n})$ for all $n$, is there
a positive integer $M$ such that the power of $p$ in the denominator of
$\operatorname{fpt}(f_{n})$ is less than or equal to $M$ for all $n$?
In this paper, we consider families of diagonal hypersurfaces; i.e.,
hypersurfaces cut out by polynomials of the form
$x_{1}^{a_{1}}+\dots+x_{m}^{a_{m}}$, where $m>2$, the exponents $a_{1}$ and
$a_{2}$ are always fixed, while the other exponents may vary across the
natural numbers. We will see that the power of $p$ in the $F$-pure thresholds
of these families is either bounded by 1, or diverges to $\infty$ depending on
whether $p\equiv-1$ or $1\mod\operatorname{lcm}(a_{1},a_{2})$. This provides
some interesting behavior of $F$-pure thresholds. Our main results are as
follows:
###### Theorem 1.1.
Let
$f_{n}(x_{1},\dots,x_{m})=x_{1}^{a_{1}}+x_{2}^{a_{2}}+\dots+x_{m-1}^{a_{m-1}}+x_{m}^{n}\in
k[x_{1},\dots,x_{m}]$, where $1<a_{1}\leq a_{2}\leq\dots\leq a_{m-1}$ are
fixed integers, $n\geq a_{m-1}$, and $(a_{1},a_{2})\neq(2,2)$. If
$p\geq\frac{a_{1}a_{2}\left(a_{1}+a_{2}+\frac{1}{a_{1}}+\frac{1}{a_{2}}-4\right)}{a_{1}a_{2}-a_{1}-a_{2}}$
and $p\equiv-1\mod\operatorname{lcm}(a_{1},a_{2})$, then either
$\operatorname{fpt}(f_{n})=1$ or the denominator of
$\operatorname{fpt}(f_{n})$ (in lowest terms) is $p$. Moreover,
$\operatorname{fpt}(f_{n})=\operatorname{fpt}(x_{1}^{a_{1}}+x_{2}^{a_{2}}+x_{3}^{a_{3}}+\dots+x_{m-1}^{a_{m-1}}+x_{m}^{p})$
for all $n\geq p$.
###### Theorem 1.2.
Let
$f_{n}(x_{1},\dots,x_{m})=x_{1}^{a}+x_{2}^{b}+x_{3}^{n}+x_{4}^{n}+\dots+x_{m}^{n}$,
where $1<a\leq b$ are fixed integers, integers, $n\geq b$, and
$(a,b)\neq(2,2)$. If $p\equiv 1\mod\operatorname{lcm}(a,b)$ and $p^{d}\leq
n<p^{d+1}$ for some positive integer $d$ and
$\operatorname{fpt}(f_{n})\neq\frac{1}{a}+\frac{1}{b}+\frac{m-2}{n}$, then the
power of $p$ in the denominator of $\operatorname{fpt}(f_{n})$ (in lowest
terms) is at least $d$.
###### Theorem 1.3.
Let $s$ be a positive integer between 2 and $p-1$ inclusive. For $n\geq 2$,
let
$f_{n}(x_{1},\dots,x_{m})=x_{1}^{a}+x_{2}^{b}+x_{3}^{p^{n}-s}+x_{4}^{p^{n}-s}+\dots+x_{m}^{p^{n}-s}$,
where $1<a\leq b$ are integers and $(a,b)\neq(2,2)$. Let
$r=\left\lceil\log_{s}\left(\left(1-\frac{1}{a}-\frac{1}{b}\right)(p-1)\right)\right\rceil$.
If $p\equiv 1\mod\operatorname{lcm}(a,b)$ and $s^{r}<p-1$, then the power of
$p$ in the denominator of $\operatorname{fpt}(f_{n})$ is exactly
$d\left(r+1\right)-1$.
###### Corollary 1.4.
Given a prime $p>5$ and positive integers $\ell,m$ with $m>2$, there exists a
family of diagonal hypersurfaces $\\{f_{n}\\}_{n\in\mathbb{N}}$ in $m$
variables such that the power of $p$ in the denominator of
$\operatorname{fpt}(f_{n})$ is less than or equal to $\ell$ for all
$n\in\mathbb{N}$, and
$\operatorname{fpt}(f_{n})=\operatorname{fpt}(f_{p^{\ell}})$ for all $n\geq
p^{\ell}$. There also exists a different family of diagonal hypersurfaces
$\\{g_{n}\\}_{n\in\mathbb{N}}$ such that the power of $p$ in the denominator
of $\operatorname{fpt}(g_{n})$ diverges to $\infty$ as $n\to\infty$.
Acknowledgements. The author thanks Dr. Janet Vassilev for the many insightful
discussions.
## 2\. Nonterminating Base $p$ Expansions
In this section, we review relevant results regarding nonterminating base $p$
expansions. The content of this section is lifted from [3] and [5].
###### Definition 2.1.
Let $\alpha\in\mathbb{Q}\cap[0,1]$ and $p$ be a prime number. The
nonterminating base $p$ expansion of $\alpha$ is the expression
$\alpha=\sum_{e=1}^{\infty}\alpha^{(e)}p^{-e}$, where for each $e$,
$0\leq\alpha^{(e)}\leq p-1$, and the sequence
$\\{\alpha^{(e)}\\}_{e\in\mathbb{N}}$ is not eventually zero. The
nonterminating base $p$ expansion of $\alpha$ is unique, and for each positive
integer $e$, we call $\alpha^{(e)}$ the $e$-th digit of $\alpha$ (base $p$).
###### Definition 2.2.
Let $\alpha\in\mathbb{Q}\cap[0,1]$ and $p$ be a prime number. For
$e\in\mathbb{N}$, define the $e$-th truncation of $\alpha$ (base $p$) as
$\left\langle\alpha\right\rangle_{e}=\alpha^{(1)}p^{-1}+\dots+\alpha^{(e)}p^{-e}$,
and for convenience, we may also write
$\left\langle\alpha\right\rangle_{e}=.\alpha^{(1)}:\alpha^{(2)}:\dots:\alpha^{(e)}(\text{base
}p).$
By convention, $\left\langle\alpha\right\rangle_{0}=0$ and
$\left\langle\alpha\right\rangle_{\infty}=\alpha$.
Some relevant properties of nonterminating base $p$ expansions are encoded in
the following lemma:
###### Lemma 2.3.
Fix $\alpha,\beta\in\mathbb{Q}\cap[0,1]$.
1. (1)
If $\left\langle\alpha\right\rangle_{e}\neq 0$, then the denominator of
$\left\langle\alpha\right\rangle_{e}$ is less than or equal to $p^{d}$ for
some $d\leq e$.
2. (2)
$\alpha\leq\beta$ if and only if
$\left\langle\alpha\right\rangle_{e}\leq\left\langle\beta\right\rangle_{e}$
for all $e\geq 1$; if $\alpha<\beta$, these inequalities are strict for $e\gg
0$.
3. (3)
If $(p^{d}-1)\alpha\in\mathbb{Z}$ for some positive integer $d$, then the base
$p$ expansion of $\alpha$ is periodic with period dividing $d$.
###### Proof.
(1) and (2) follow from the definition; see [5, Lemmas 2.5 and 2.6] for (3). ∎
## 3\. $F$-pure Thresholds of Diagonal and Quasi-homogeneous Polynomials
From this point onward, we set $k$ to be a perfect field of characteristic
$p>0$ (typically, $k=\mathbb{F}_{p}$), $R=k[x_{1},\dots,x_{n}]$ the polynomial
ring in $n$ variables over $k$, and $\mathfrak{m}=(x_{1},\dots,x_{n})\subseteq
R$ be the maximal ideal of $R$ generated by the variables.
###### Definition 3.1.
Let $f\in\mathfrak{m}$. For each positive integer $e$, define
$\nu_{f}(p^{e})=\sup\left\\{N\in\mathbb{Z}:f^{N}\notin\mathfrak{m}^{[p^{e}]}\right\\}$
where $\mathfrak{m}^{[p^{e}]}=(x^{p^{e}}_{1},\dots,x^{p^{e}}_{n})$ denotes the
$e$-th Frobenius power of $\mathfrak{m}$. Define the $F$-pure threshold of $f$
as
$\operatorname{fpt}(f)=\lim_{e\to\infty}\frac{\nu_{f}(p^{e})}{p^{e}}.$
###### Remark 3.2.
For any $f\in\mathfrak{m}$, the sequence
$\\{p^{-e}\nu_{f}(p^{e})\\}_{e=1}^{\infty}$ is a nondecreasing sequence of
rational numbers contained in $(0,1]$ (see [6], Lemma 1.1 and Remark 1.2 for
the proof). Therefore, the limit defining the $F$-pure threshold of $f$
exists, and the $F$-pure threshold is in $(0,1]$.
Developing formulas and algorithms for $F$-pure threshold for certain classes
of polynomials has been an active area of research. For example, see [3] for
$F$-pure thresholds of diagonal hypersurfaces, [4] for $F$-pure thresholds of
binomial hypersurfaces, [1] and [7] for $F$-pure thresholds of Calabi-Yau
hypersurfaces, and [9] for $F$-pure thresholds of generic homogeneous
polynomials. We record some relevant results.
###### Theorem 3.3 ([3] Theorem 3.4, Remark 3.5).
Let $f\in k[x_{1},\dots,x_{n}]$ be a $k^{*}$-linear combination of the
monomials $x_{1}^{d_{1}},\dots,x_{n}^{d_{n}}$. Let $L=\inf\\{e\geq
0:(d_{1}^{-1})^{(e+1)}+\dots+(d_{n}^{-1})^{(e+1)}\geq p\\}$.
* •
If $L<\infty$, then $\operatorname{fpt}(f)=\left\langle
d_{1}^{-1}\right\rangle_{L}+\dots+\left\langle
d_{n}^{-1}\right\rangle_{L}+p^{-L}$.
* •
If $L=\infty$, then $\operatorname{fpt}(f)=d_{1}^{-1}+\dots+d_{n}^{-1}$.
###### Theorem 3.4 ([5], Theorem 3.5, Corollary 3.9).
Fix an $\mathbb{N}$-grading on the polynomial ring $k[x_{1},\dots,x_{n}]$, and
let $f$ be a homogeneous polynomial under this grading. Write
$\lambda=\frac{\deg x_{1}+\dots+\deg x_{n}}{\deg f}=\frac{s}{t}$ in lowest
terms.
* •
If $\operatorname{fpt}(f)\neq\lambda$, then
$\operatorname{fpt}(f)=\left\langle\lambda\right\rangle_{L}-\frac{E}{p^{L}}$
for some pair $(L,E)\in\mathbb{N}^{2}$ satisfying $L\geq 1$ and $1\leq E\leq
n-1-\frac{\left\lceil[sp^{L}\%t]+s\right\rceil}{t}$, where $[m\%t]$ denotes
the smallest positive residue of $m$ modulo $t$.
* •
If $\operatorname{fpt}(f)\neq\lambda$ and $p\not|t$, then the power of $p$ in
the denominator of $\operatorname{fpt}(f)$ (in lowest terms) is less than or
equal to $M=2\phi(t)+\left\lceil\log_{2}(n-1)\right\rceil$, where $\phi$
denotes Euler’s totient function.
These two important theorems tell us that the $F$-pure thresholds of certain
classes of polynomials can be computed off nonterminating base $p$ expansions.
In particular, the proofs of the main results will be based on finding $L$ and
then using these two theorems.
The second part of Theorem 3.4 says that the power of $p$ appearing in the
denominator of the $F$-pure threshold of a quasi-homogeneous polynomial $f$ is
bounded by a number depending on the $\mathbb{N}$-grading that turns $f$
homogeneous. This along with [7], Theorem 4.3, motivates the premise of the
paper: if we consider a family of quasi-homogeneous polynomials where one or
more exponents vary through the family, what is the growth of the power of $p$
in the denominator of the $F$-pure thresholds?
## 4\. Main Results
We now prove the main results.
###### Theorem 4.1.
Let
$f_{n}(x_{1},\dots,x_{m})=x_{1}^{a_{1}}+x_{2}^{a_{2}}+\dots+x_{m-1}^{a_{m-1}}+x_{m}^{n}\in
k[x_{1},\dots,x_{m}]$, where $1<a_{1}\leq a_{2}\leq\dots\leq a_{m-1}$ are
fixed integers, $n\geq a_{m-1}$, and $(a_{1},a_{2})\neq(2,2)$. If
$p\geq\frac{a_{1}a_{2}\left(a_{1}+a_{2}+\frac{1}{a_{1}}+\frac{1}{a_{2}}-4\right)}{a_{1}a_{2}-a_{1}-a_{2}}$
and $p\equiv-1\mod\operatorname{lcm}(a_{1},a_{2})$, then either
$\operatorname{fpt}(f_{n})=1$ or the denominator of
$\operatorname{fpt}(f_{n})$ (in lowest terms) is $p$. Moreover,
$\operatorname{fpt}(f_{n})=\operatorname{fpt}(x_{1}^{a_{1}}+x_{2}^{a_{2}}+x_{3}^{a_{3}}+\dots+x_{m-1}^{a_{m-1}}+x_{m}^{p})$
for all $n\geq p$.
###### Proof.
Since $p\equiv-1\mod\operatorname{lcm}(a_{1},a_{2})$, we must have
$p\equiv-1\mod a_{1}$ and $p\equiv-1\mod a_{2}$. Therefore, the first $2$
digits of the nonterminating base $p$ expansion of $\frac{1}{a_{1}}$ satisfies
$\left(\frac{1}{a_{1}}\right)^{(1)}=\frac{p-(a_{1}-1)}{a_{1}},\hskip
10.0pt\left(\frac{1}{a_{1}}\right)^{(2)}=\left\lfloor
p\left(\frac{a_{1}-1}{a_{1}}\right)\right\rfloor\geq(a_{1}-1)\left(\frac{p-(a_{1}-1)}{a_{1}}\right),$
and similarly for $\frac{1}{a_{2}}$. Now,
$(a_{1}-1)\left(\frac{p-(a_{1}-1)}{a_{1}}\right)+(a_{2}-1)\left(\frac{p-(a_{2}-1)}{a_{2}}\right)\geq
p\iff
p\geq\frac{a_{1}a_{2}\left(a_{1}+a_{2}+\frac{1}{a_{1}}+\frac{1}{a_{2}}-4\right)}{a_{1}a_{2}-a_{1}-a_{2}}.$
Therefore, for these $p$, we have $L<2$. On the other hand,
$\left(\frac{1}{a_{1}}\right)^{(1)}+\left(\frac{1}{a_{2}}\right)^{(1)}+\left(\frac{1}{n}\right)^{(1)}\geq
p\iff
n\leq\frac{p}{(p+2)\left(1-\frac{1}{a_{1}}-\frac{1}{a_{2}}\right)}<1-\frac{a_{1}+a_{2}}{a_{1}a_{2}-a_{1}-a_{2}}.$
Since $(a_{1},a_{2})\neq(2,2)$, $a_{1}a_{2}>a_{1}+a_{2}$, and so the last
inequality implies that to “carry at the first place," we must have $n<1$.
This is impossible. Therefore, $L\geq 1$. This then proves that $L=1$, and so
$\operatorname{fpt}(f)=\left\langle\frac{1}{a_{1}}\right\rangle_{1}+\left\langle\frac{1}{a_{2}}\right\rangle_{1}+\dots+\left\langle\frac{1}{a_{m-1}}\right\rangle_{1}+\left\langle\frac{1}{n}\right\rangle_{1}+\frac{1}{p}$
by Theorem 3.3. Since
$\left(\frac{1}{a_{1}}\right)^{(1)}+\left(\frac{1}{a_{2}}\right)^{(1)}=(p+1)\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}\right)-2<p$
for all $(a_{1},a_{2})\neq(2,2)$, this proves the result.
Finally, if $n\geq p$, then $\left(\frac{1}{n}\right)^{(1)}=0$, and
$\operatorname{fpt}(f)=\left\langle\frac{1}{a_{1}}\right\rangle_{1}+\left\langle\frac{1}{a_{2}}\right\rangle_{1}+\dots+\left\langle\frac{1}{a_{m-1}}\right\rangle_{1}+\frac{1}{p}=\operatorname{fpt}(x_{1}^{a_{1}}+x_{2}^{a_{2}}+x_{3}^{a_{3}}+\dots+x_{m-1}^{a_{m-1}}+x_{m}^{p}).$
∎
Theorem 4.1 shows that for certain families of hypersurfaces
$\\{f_{n}\\}_{n\in\mathbb{N}}$, the $F$-pure thresholds stabilize as
$n\to\infty$ and their denominators are always either 1 or $p$. On the other
hand, by considering different primes and indexing, we can make the power of
$p$ in the $F$-pure thresholds of a family of diagonal hypersurfaces become
unbounded as $n\to\infty$. One such method to do this is in the next result.
###### Theorem 4.2.
Let
$f_{n}(x_{1},\dots,x_{m})=x_{1}^{a}+x_{2}^{b}+x_{3}^{n}+x_{4}^{n}+\dots+x_{m}^{n}$,
where $1<a\leq b$ are fixed integers, integers, $n\geq b$, and
$(a,b)\neq(2,2)$. If $p\equiv 1\mod\operatorname{lcm}(a,b)$ and $p^{d}\leq
n<p^{d+1}$ for some positive integer $d$ and
$\operatorname{fpt}(f_{n})\neq\frac{1}{a}+\frac{1}{b}+\frac{m-2}{n}$, then the
power of $p$ in the denominator of $\operatorname{fpt}(f_{n})$ (in lowest
terms) is at least $d$.
###### Proof.
For all $e\geq 1$,
$\left(\frac{1}{a}\right)^{(e)}+\left(\frac{1}{b}\right)^{(e)}=(p-1)\left(\frac{1}{a}+\frac{1}{b}\right)<p$
for all $(a,b)$. Moreover, $p^{d}\leq n<p^{d+1}$ implies that
$\left(\frac{1}{n}\right)^{(e)}=0$ for all $1\leq e\leq d$. Therefore, $L\geq
d$. If $L=\infty$, then
$\operatorname{fpt}(f)=\frac{1}{a}+\frac{1}{b}+\frac{m-2}{n}.$
Otherwise,
$\operatorname{fpt}(f)=\left\langle\frac{1}{a}\right\rangle_{L}+\left\langle\frac{1}{b}\right\rangle_{L}+(m-2)\left\langle\frac{1}{n}\right\rangle_{L}+\frac{1}{p^{L}},$
which has denominator $p^{L}$ in lowest terms. ∎
Theorem 4.2 shows a linear asymptotic growth in the power of $p$ in the
denominator of the $F$-pure thresholds. If we restrict to a subsequence, we
can make the growth of the power of $p$ faster.
###### Theorem 4.3.
Let $s$ be a positive integer between 2 and $p-1$ inclusive. For $n\geq 2$,
let
$f_{n}(x_{1},\dots,x_{m})=x_{1}^{a}+x_{2}^{b}+x_{3}^{p^{n}-s}+x_{4}^{p^{n}-s}+\dots+x_{m}^{p^{n}-s}$,
where $1<a\leq b$ are integers and $(a,b)\neq(2,2)$. Let
$r=\left\lceil\log_{s}\left(\left(1-\frac{1}{a}-\frac{1}{b}\right)(p-1)\right)\right\rceil$.
If $p\equiv 1\mod\operatorname{lcm}(a,b)$ and $s^{r}<p-1$, then the power of
$p$ in the denominator of $\operatorname{fpt}(f_{n})$ is exactly
$d\left(r+1\right)-1$.
###### Proof.
Let $t=p^{n}-s$. We compute the nonterminating base $p$ expansion of
$\frac{1}{t}=\frac{1}{p^{n}-s}$:
$\frac{1}{p^{n}-s}=\left(\frac{1}{p^{n}}\right)\left(\frac{1}{1-\frac{s}{p^{n}}}\right)=\sum_{q=0}^{\infty}\frac{s^{q}}{p^{n(q+1)}}.$
Note that if $s^{q}\leq p-1$, then
$\left(\frac{1}{p^{n}-s}\right)^{(n(q+1))}=s^{q}$ and
$\left(\frac{1}{p^{n}-s}\right)^{(e)}=0$ for $e<n(q+1)$ and $d$ not dividing
$e$. Now,
$\left(\frac{1}{a}\right)^{(d(q+1))}+\left(\frac{1}{b}\right)^{(n(q+1))}+\left(\frac{1}{t}\right)^{(n(q+1))}>p-1\iff
q>\log_{s}\left((p-1)\left(1-\frac{1}{a}-\frac{1}{b}\right)\right).$
Therefore, we may choose
$q=\left\lceil\log_{s}\left(\left(1-\frac{1}{a}-\frac{1}{b}\right)(p-1)\right)\right\rceil$
(assuming $s^{q}<p-1$), and so, $L=n\left(q+1\right)-1$. ∎
###### Remark 4.4.
We consider the special case $m=3$ and fix the family of diagonal
hypersurfaces $f_{n}(x,y,z)=x^{a}+y^{b}+z^{n}$. Then Theorems 4.1 and either
4.2 or 4.3 take the following form: For any
$p\equiv-1\mod\operatorname{lcm}(a,b)$, then the denominator of
$\operatorname{fpt}(f_{n})$ is either $1$ or $p$, and
$\operatorname{fpt}(f_{n})=\operatorname{fpt}(f_{p})$ for all $n\geq p$. For
any $p\equiv 1\mod\operatorname{lcm}(a,b)$ and $n=p^{d}-s$ for some integer
$1<s<p-1$, then for
$r=\left\lceil\log_{s}\left(\left(1-\frac{1}{a}-\frac{1}{b}\right)(p-1)\right)\right\rceil$,
if $s^{r}<p-1$, then the power of $p$ in the denominator of
$\operatorname{fpt}(f_{n})$ is exactly $d\left(r+1\right)-1$.
###### Corollary 4.5.
Given a prime $p>5$ and positive integers $\ell,m$ with $m>2$, there exists a
family of diagonal hypersurfaces $\\{f_{n}\\}_{n\in\mathbb{N}}$ in $m$
variables such that the power of $p$ in the denominator of
$\operatorname{fpt}(f_{n})$ is less than or equal to $\ell$ for all
$n\in\mathbb{N}$, and
$\operatorname{fpt}(f_{n})=\operatorname{fpt}(f_{p^{\ell}})$ for all $n\geq
p^{\ell}$. There also exists a different family of diagonal hypersurfaces
$\\{g_{n}\\}_{n\in\mathbb{N}}$ such that the power of $p$ in the denominator
of $\operatorname{fpt}(g_{n})$ diverges to $\infty$ as $n\to\infty$.
###### Proof.
The case where $\ell=1$ is done by Theorems 4.1 and either 4.2 or 4.3. For the
case $\ell>1$, if the family
$f_{n}(x_{1},\dots,x_{m})=x_{1}^{a_{1}}+x_{2}^{a_{2}}+\dots+x_{m-1}^{a_{m-1}}+x_{m}^{n}$
satisfies Theorem 4.1, then consider the family
$g_{n}(x_{1},\dots,x_{m})=x_{1}^{p^{\ell}a_{1}}+x_{2}^{p^{\ell}a_{2}}+\dots+x_{m-1}^{p^{\ell}a_{m-1}}+x_{m}^{p^{\ell}n}$.
Since the nonterminating base $p$ expansion of $\frac{1}{p^{\ell}a}$ for any
positive integer $a$ satisfies
$\left(\frac{1}{a}\right)^{(e)}=\left(\frac{1}{p^{\ell}a}\right)^{(e+\ell)}$
for all positive integers $e$, we may then proceed similarly to the proof of
Theorem 4.1. ∎
We end with some computational remarks. All computations are made using the
FrobeniusThresholds package of Macaulay2.
###### Remark 4.6.
It is possible for the power of $p$ in the denominator of
$\operatorname{fpt}(f_{n})$ for some $n$ between $p^{d}$ and $p^{d+1}$ to be
higher than the prescribed power in Theorem 4.3. For example, let
$a=2,b=3,p=97,d=2,n=9216$. Then the power of $p$ in the denominator of
$\operatorname{fpt}(x^{2}+y^{3}+z^{9216})$ is $16$, but the power of $p$ in
the denominator of $\operatorname{fpt}(x^{2}+y^{3}+z^{97^{2}-s})$ for $2\leq
s\leq 96$ is at most $11$.
###### Remark 4.7.
Theorem 4.1 may inspire the conjecture that the denominator of
$\operatorname{fpt}(x_{1}^{a_{1}}+x_{2}^{a_{2}}+\dots+x_{m-1}^{a_{m-1}}+x_{m}^{n})$
for $p\equiv-c\mod\text{lcm }(a,b)$ and $c\leq\frac{p-1}{2}$ is $p^{e}$ for
some positive integer $e\leq c$. Unfortunately, this is false. For example,
let $a=3,b=5,n=699,p=71$: note that $p\equiv-4\mod 15$, but
$\operatorname{fpt}(x^{a}+y^{b}+z^{n})=\frac{964836811}{71^{5}}$.
###### Remark 4.8.
The growth of the denominator in Theorems 4.2 and 4.3 may not hold if $p\equiv
c\mod\operatorname{lcm}(a,b)$ and $c\leq\frac{p-1}{2}$. For example, let
$a=3,b=5,n=500,p=17$: note that $p\equiv 2\mod 15$, and the nonterminating
base $p$ expansions of $\frac{1}{3}$ and $\frac{1}{5}$ are
$\frac{1}{3}=.5:11:\dots$ and $\frac{1}{5}=.3:6:\dots,$ giving
$\operatorname{fpt}(x^{3}+y^{5}+z^{500})=\left<\frac{1}{3}\right>_{1}+\left<\frac{1}{5}\right>_{1}+\left<\frac{1}{500}\right>_{1}+\frac{1}{17}=\frac{9}{17}.$
###### Remark 4.9.
(Necessity of $s^{r}<p-1$) For $a=3,b=4,n=165,p=13$: By Macaulay2’s
computations,
$\operatorname{fpt}(x^{a}+y^{b}+z^{n})=\frac{480786743}{13^{8}}$. Note that
$n=p^{2}-4$, and the nonterminating base $p$ expansions of $\frac{1}{3}$ and
$\frac{1}{4}$ are $\frac{1}{3}=.\overline{4}$ and $\frac{1}{4}=.\overline{3}$.
Now, for $r=1$, $4+3+s^{r}=4+3+4=11<p$. However, for $r=2$, $s^{r}=16>p-1$, so
the “predicted" power would be $5$, which is wrong.
## References
* [1] Bhargav Bhatt and Anurag Singh, The $F$-pure threshold of a Calabi-Yau hypersurface, Math. Ann. 362 (2015), no. 1-2, 551-567.
* [2] Eric Canton, Daniel Hernandez, Karl Schwede, and Emily Witt, On the behavior of singularities at the $F$-pure threshold, Illinois J. Math 60 (2016), no. 3-4, 669-685.
* [3] Daniel Hernandez, $F$-invariants of diagonal hypersurfaces, Proc. Am. Math. Soc. 143 (2014), no. 1, 87-104.
* [4] Daniel Hernandez, $F$-pure thresholds of binomial hypersurfaces, Proc. Am. Math. Soc. 142 (2014), no. 7, 2227-2242.
* [5] Daniel Hernandez, Luis Nuñez-Betancourt, Emily Witt, and Wen-Liang Zhang, $F$-pure thresholds of homogeneous polynomials, Michigan Math. J. 65 (2016), no. 1, 57-87.
* [6] Mircea Mustata, Shunsuke Takagi, and Kei-ichi Watanabe, $F$-thresholds and Bernstein-Sato polynomials, European Congress of Mathematics (Ari Laptev, ed.), European Mathematical Society, Stockholm, 2005, pp. 341-364.
* [7] Susanne Müller, The $F$-pure threshold of quasi-homogeneous polynomials, J. Pure Appl. Algebra 222 (2017), no. 1, 75-96.
* [8] Karl Schwede, Generalized test ideals, sharp $F$-purity, and sharp test elements, Math. Res. Lett. 15 (2008), no. 6, 1251-1261.
* [9] Karen Smith and Adela Vraciu, Values of the $F$-pure threshold for homogeneous polynomials, J. London Math. Soc. 103 (2023), no. 3, 1004-1035.
* [10] Shunsuke Takagi and Kei-ichi Watanabe, On $F$-pure thresholds, J. Algebra 282 (2004), no. 52, 278-297.
|
# A Puzzle of Proportions: Two Popular Bayesian Tests Can Yield Dramatically
Different Conclusions
Fabian Dablander Department of Psychological Methods, University of Amsterdam
Karoline Huth Department of Psychological Methods, University of Amsterdam
Quentin F. Gronau Department of Psychological Methods, University of
Amsterdam Alexander Etz University of California, Irvine Eric-Jan
Wagenmakers Department of Psychological Methods, University of Amsterdam
(blueThis manuscript has been accepted for publication in Statistics in
Medicine.)
###### Abstract
Testing the equality of two proportions is a common procedure in science,
especially in medicine and public health. In these domains it is crucial to be
able to quantify evidence for the absence of a treatment effect. Bayesian
hypothesis testing by means of the Bayes factor provides one avenue to do so,
requiring the specification of prior distributions for parameters. The most
popular analysis approach views the comparison of proportions from a
contingency table perspective, assigning prior distributions directly to the
two proportions. Another, less popular approach views the problem from a
logistic regression perspective, assigning prior distributions to logit-
transformed parameters. Reanalyzing 39 null results from the New England
Journal of Medicine with both approaches, we find that they can lead to
markedly different conclusions, especially when the observed proportions are
at the extremes (i.e., very low or very high). We explain these stark
differences and provide recommendations for researchers interested in testing
the equality of two proportions and users of Bayes factors more generally.
blackThe test that assigns prior distributions to logit-transformed parameters
creates prior dependence between the two proportions and yields weaker
evidence when the observations are at the extremes. When comparing two
proportions, we argue that this test should become the new default.
Keywords: Bayesian testing, Evidence of absence, Equality of proportions, Data
reanalysis
## 1 Introduction
Researchers frequently wish to test whether two populations differ. In
medicine and public health, for example, the resulting statistical analysis
frequently concerns testing whether or not two proportions differ. Examples
include testing whether a vaccine decreases the number of infections compared
to a control [[, e.g.,]]polack2020safety, whether sexual minorities are more
prone to suicide compared to their heterosexual counterparts [[,
e.g.,]]ploderl2013suicide, or whether tightly or less-tightly controlling
hypertension leads to fewer miscarriages in pregnant women [19].
In these applications it is crucial to be able to discriminate between
evidence of absence and absence of evidence. For example, [19] conducted a
trial to investigate the effect of a tight (target diastolic blood pressure,
85 mm Hg) or a less-tight (target diastolic blood pressure, 100 mm Hg) control
of hypertension in pregnant woman on, among other outcomes, pregnancy loss.
They found no significant difference between the two conditions, with 15 out
of 493 women in the less-tight control condition and 13 out of 488 in the
tight control condition having lost their child, yielding an estimated odds
ratio of 1.14 (95% CI: [0.53, 2.45]). How confident are we that there is
indeed no difference between the two conditions rather than the data being
inconclusive? Bayesian statistics provides a principled way of quantifying
evidence via the Bayes factor [11, 18, 20], thus providing one avenue to
discriminate between evidence of absence and absence of evidence [[,
e.g.,]]keysers2020using.
The Bayes factor quantifies how well one hypothesis predicts the data compared
to another. Using the test of equality between two proportions as an example,
let $\mathcal{D}=(y_{1},y_{2},n_{1},n_{2})$ denote the combined data from the
two groups. We have:
$\displaystyle Y_{1}$ $\displaystyle\sim\text{Binomial}(n_{1},\theta_{1})$
$\displaystyle Y_{2}$
$\displaystyle\sim\text{Binomial}(n_{2},\theta_{2})\enspace,$
where the sample sizes $(n_{1},n_{2})$ are assumed fixed and under
$\mathcal{H}_{0}$ we have that $\theta\equiv\theta_{1}=\theta_{2}$ while under
$\mathcal{H}_{1}$ we have that $\theta_{1}\neq\theta_{2}$. By quantifying
relative predictive performance, the Bayes factor tells us how we should
update our prior beliefs about $\mathcal{H}_{0}$ relative to $\mathcal{H}_{1}$
after observing the data [14]:
$\underbrace{\frac{p(\mathcal{H}_{0}\mid\mathcal{D})}{p(\mathcal{H}_{1}\mid\mathcal{D})}}_{\text{Posterior
odds}}=\underbrace{\frac{p(\mathcal{D}\mid\mathcal{H}_{0})}{p(\mathcal{D}\mid\mathcal{H}_{1})}}_{\text{Bayes
factor}}\,\times\underbrace{\frac{p(\mathcal{H}_{0})}{p(\mathcal{H}_{1})}}_{\text{Prior
odds}}\enspace.$
blackA Bayes factor of, say, 15 means that the data are 15 times more likely
under one hypothesis compared to the other. While there exist verbal
guidelines that may aid in the interpretation of the Bayes factor [[, for
example, Bayes factors in the range from 1$-$3 constitute _weak_ evidence,
those in the range from 3$-$10 constitute _moderate_ evidence, and values
larger than $10$ constitute _strong_ evidence;]]jeffreys1939theory,
wasserman2000bayesian, lee2013book, the Bayes factor should be understood as a
continuous measure of evidence [20].
While the Bayes factor does not depend on the prior probability of hypotheses,
it does depend crucially on the prior over parameters in the models
instantiating $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$, which becomes apparent
when expanding:
$\frac{p(\mathcal{D}\mid\mathcal{H}_{0})}{p(\mathcal{D}\mid\mathcal{H}_{1})}=\frac{\int_{\theta}p(\mathcal{D}\mid\theta,\mathcal{H}_{0})\pi_{0}(\theta\mid\mathcal{H}_{0})\,\,\mathrm{d}\theta}{\int_{\theta_{1}}\int_{\theta_{2}}p(\mathcal{D}\mid\theta_{1},\theta_{2},\mathcal{H}_{1})\pi_{1}(\theta_{1},\theta_{2}\mid\mathcal{H}_{1})\,\,\mathrm{d}\theta_{1}\,\mathrm{d}\theta_{2}}\enspace,$
where $\pi_{0}$ and $\pi_{1}$ indicate the respective prior distributions.
There exist two main Bayes factor approaches for testing the equality of two
proportions. The more popular one comes from the analysis of contingency
tables, and assigns independent beta distributions directly to
$(\theta_{1},\theta_{2})$ [[, e.g.,]]gunel1974bayes, jamil2017default. We call
this approach the “Independent Beta” (IB) approach. The second approach is
less widely used, and assigns a prior to the average log odds $\beta$ and the
log odds ratio $\psi$ [15, 3]. We call this approach the “Logit
Transformation” (LT) approach. In this paper, we show that these two
approaches can yield markedly different results. This is especially the case
when the observed proportions are at the extremes (i.e., very low or very
high), as is the case for a large number of applications including the three
examples mentioned above. Consider the study by [19] again. The IB approach
yields a Bayes factor of 12.30 in favour of $\mathcal{H}_{0}$, while the LT
approach yields a mere 1.17. In other words, under the IB approach the data
are about 12 times more likely under the hypothesis that tightly or less-
tightly controlling hypertension have the same effect on miscarriages compared
to a hypothesis assuming a difference. Under the LT approach, however, the
data are about equally likely under both hypotheses, which constitutes
equivocal evidence. The answer to the question “is observing two equally small
proportions strong or weak evidence for the null hypothesis?” depends,
therefore, crucially — and non-trivially — on the prior setup.
This paper is structured as follows. In Section 2, we outline these two ways
of testing the equality of two proportions in more detail. In Section 3, we
highlight the occasionally stark differences of the two approaches by
reanalyzing 39 statistical tests reported in the New England Journal of
Medicine and explain why these differences occur. In Section 4, we end by
reviewing the implications of the prior setup and what users of Bayes factors
should be mindful of when testing the equality of two population parameters.
blackWe argue that the LT approach should become the default when testing the
equality of two proportions because it (a) induces prior dependence between
proportions which almost always are, in fact, dependent, and (b) yields a
sensibly milder assessment of the evidence compared to the IB approach when
the observations are at the extremes.
## 2 Two Ways of Testing the Equality of Two Proportions
In this section, we outline two ways of testing the equality of two
proportions. In Section 2.1 we describe the Independent Beta approach, and in
Section 2.2 we describe the Logit Transformation approach.
### 2.1 The Independent Beta (IB) Approach
In order to nest the null hypothesis under the alternative hypothesis, we
introduce the difference parameter $\eta=\theta_{2}-\theta_{1}$ and the grand
mean $\zeta=\frac{1}{2}\left(\theta_{1}+\theta_{2}\right)$. Using this
parameterization, we have that:
$\displaystyle\theta_{1}$ $\displaystyle=\zeta-\frac{\eta}{2}$
$\displaystyle\theta_{2}$ $\displaystyle=\zeta+\frac{\eta}{2}\enspace.$
The hypotheses are then specified as:
$\displaystyle\mathcal{H}_{0}$ $\displaystyle:\eta=0$
$\displaystyle\mathcal{H}_{1}$ $\displaystyle:\eta\neq 0\enspace.$
Under this approach, we assign independent $\text{Beta}(a,a)$ priors to
$\theta_{1}$ and $\theta_{2}$. Figure 1 visualizes the joint prior
distribution (top) under $\mathcal{H}_{1}$ for $a=1$ (left) and $a=2$ (right).
Increasing values of $a$ implies that the joint prior mass is more
concentrated around
$(\theta_{1},\theta_{2})=(\nicefrac{{1}}{{2}},\nicefrac{{1}}{{2}})$. The
bottom panels visualize the conditional prior distribution of $\theta_{2}$
given that we know that $\theta_{1}=0.10$. Knowing the value of $\theta_{1}$
does not change our prior about $\theta_{2}$, which follows from the
assumption of prior independence.
Figure 1: Top: Joint prior distribution assigned to $(\theta_{1},\theta_{2})$
under $a=1$ (left) and $a=2$ (right). Bottom: Conditional prior distribution
of $\theta_{2}$ given that $\theta_{1}=0.10$.
The IB Bayes factor is available in analytic form [[, see
e.g.,]]dickey1970weighted, gunel1974bayes, jeffreys1939theory. In the
literature on contingency tables, our setup corresponds to an independent
multinomial sampling scheme where the row (or column) sums of the contingency
table (here $n_{1}$ and $n_{2}$) are fixed; for an extension to other sampling
schemes and more than two groups, see [4] and [10]. [4] suggest $a=1$ as a
default value. Note that values $a<1$ lead to an undefined prior density for
$\theta=0$ and $\theta=1$, which implies model selection inconsistency in case
$\theta$ is indeed 1 or 0. Hence these values should be avoided on principle
grounds. Because the beta prior is conjugate for the binomial likelihood, the
posterior distributions of $\theta_{1}$ and $\theta_{2}$ are again
(independent) beta distributions.
### 2.2 The Logit Transformation (LT) Approach
The test proposed by [15] and implemented by [3] does not assign a prior
directly to $(\theta_{1},\theta_{2})$, but applies a logit transformation and
assigns priors to the transformed parameters ($\beta$, $\psi$). Specifically,
we write:
$\displaystyle\text{log}\left(\frac{\theta_{1}}{1-\theta_{1}}\right)$
$\displaystyle=\beta-\frac{\psi}{2}$
$\displaystyle\text{log}\left(\frac{\theta_{2}}{1-\theta_{2}}\right)$
$\displaystyle=\beta+\frac{\psi}{2}\enspace,$
where $\beta$ is a grand mean and $\psi$ is the difference in log odds (i.e.,
the log odds ratio):
$\displaystyle\beta$
$\displaystyle=\frac{1}{2}\left(\text{log}\left(\frac{\theta_{1}}{1-\theta_{1}}\right)+\text{log}\left(\frac{\theta_{2}}{1-\theta_{2}}\right)\right)$
$\displaystyle\psi$
$\displaystyle=\text{log}\left(\frac{\theta_{2}}{1-\theta_{2}}\right)-\text{log}\left(\frac{\theta_{1}}{1-\theta_{1}}\right)\enspace.$
While this is a more involved reparameterization than in the IB approach
above, another way to formulate this setup is by writing:
$\displaystyle\theta_{1}$
$\displaystyle=\frac{e^{\beta-\frac{\psi}{2}}}{1+e^{\beta-\frac{\psi}{2}}}$
$\displaystyle\theta_{2}$
$\displaystyle=\frac{e^{\beta+\frac{\psi}{2}}}{1+e^{\beta+\frac{\psi}{2}}}\enspace,$
which readers familiar with logistic regression may recognize. Using this
setup, we test the hypotheses:
$\displaystyle\mathcal{H}_{0}$ $\displaystyle:\psi=0$
$\displaystyle\mathcal{H}_{1}$ $\displaystyle:\psi\neq 0\enspace.$
Figure 2: Top: Joint prior assigned to $(\theta_{1},\theta_{2})$ for
$\sigma_{\psi}=1$ (left) and $\sigma_{\psi}=2$ (right). Bottom: Conditional
prior distribution of $\theta_{2}$ given that $\theta_{1}=0.10$. In both cases
we assume $\sigma_{\beta}=1$.
In contrast to above, we now assign priors to $\beta$ and $\psi$ rather than
to $\theta_{1}$ and $\theta_{2}$ directly. In particular, under both
hypotheses we assume $\beta\sim\mathcal{N}(0,\sigma_{\beta})$ with
$\sigma_{\beta}=1$. Under $\mathcal{H}_{1}$ we assume
$\psi\sim\mathcal{N}(0,\sigma_{\psi})$.111While this prior specification
assigns the log odds ratio $\psi$ a Gaussian distribution, the IB prior
specification results in a non-standard distribution on $\psi$. For more
details, see Appendix A. The top left panel in Figure 2 visualizes the implied
joint prior distribution on $(\theta_{1},\theta_{2})$ under $\mathcal{H}_{1}$
for $\sigma_{\psi}=1$. The prior mass is concentrated along the diagonal,
which indicates that $\theta_{1}$ and $\theta_{2}$ are dependent. The bottom
left panel illustrates this fact: if we know that $\theta_{1}=0.10$, then the
prior on $\theta_{2}$ shifts toward this value. Setting $\sigma_{\psi}=2$
removes the prior dependency, as the right column in Figure 2 shows. For
values of $\sigma_{\psi}>2$, $\theta_{1}$ and $\theta_{2}$ become anti-
correlated and hence observing a small value of $\theta_{1}$ results in a
prior that puts more mass on large values for $\theta_{2}$. Such an inverse
relation is undesirable in almost all empirical applications, and so values
$\sigma_{\psi}>2$ are therefore to be avoided. [3] developed software to
compute the Bayes factor using this prior specification, first proposed by
[15], suggesting $\sigma_{\psi}=1$ as a default value.
### 2.3 Comparison of Priors
A direct comparison of the two prior specifications may be helpful to get
further intuition for their differences. While the IB approach does not assign
a prior distribution to $(\beta,\psi)$ or $(\eta,\zeta)$ explicitly, assigning
a prior to $\theta_{1}$ and $\theta_{2}$ induces a prior distribution on these
quantities. Conversely, the LT approach assigns a prior to $\psi$ and $\beta$
and this induces a prior on $(\theta_{1},\theta_{2})$ and $(\eta,\zeta)$. The
induced prior distributions under both approaches are non-standard (see
Appendix A), but their densities can be calculated numerically.
Figure 3: Top: Prior distributions assigned to $\eta$ (left) and $\psi$
(right) under the LT (blue, vertical lines) and IB approach (red, horizontal
lines), for two different parameter settings, respectively. Bottom: Marginal
prior distribution of $\theta_{1}$ and $\theta_{2}$ under the two approaches.
blackThe density that is filled and has the highest peak corresponds to
$\sigma_{\psi}=1$.
The top left panel in Figure 3 shows the prior distribution assigned to $\eta$
by the LT approach for $\sigma_{\psi}=1$ (shaded blue) and $\sigma_{\psi}=2$
(striped blue) and the IB approach for $a=1$ (shaded red) and $a=2$ (striped
red) under $\mathcal{H}_{1}$. Similarly, the top right panel shows the prior
distribution assigned to $\psi$ for the two approaches and prior parameter
values. The (default) IB approach assigns comparatively more mass to large
values of $\eta$ and $\psi$, which in practice means that it expects larger
differences between the sample proportions. The bottom panel shows the
marginal priors for $\theta_{1}$ and $\theta_{2}$, where we find that the LT
approach assigns comparatively less mass to extreme values. The LT approach
cannot result in a uniform distribution on the proportions under
$\mathcal{H}_{0}$ because of the Gaussian prior on $\beta$. If it instead
would assign a (standard) logistic prior to $\beta$ (which has fatter tails),
the prior on the proportions would be uniform; see a related discussion in
Appendix B. In the next section, we discuss a somewhat surprising difference
between these two tests.
## 3 Practical Implications of the Prior Setup
To see the implications of the two different prior specifications in practice,
in Section 3.1 we reanalyze 39 null results published in the New England
Journal of Medicine, previously analyzed by [6] using the IB approach. We then
explain why these difference occur in Section 3.2. The data and code to
reproduce all analyses and figures are available from
https://github.com/fdabl/Proportion-Puzzle.
### 3.1 Reanalysis of New England Journal of Medicine Studies
[6] considered all 207 articles published in the New England Journal of
Medicine in 2015. The abstract of 45 of these articles contained a claim about
the absence or non-significance of an effect for a primary outcome measure,
and 37 of those allowed for a comparison of proportions, reporting 43 null
results in total. We focus on those results that can be reanalyzed using a
test between two proportions, which results in a total of 39 tests from 32
articles. The top left panel in Figure 4 contrasts Bayes factors in favour of
$\mathcal{H}_{0}$ computed using the IB approach (rectangles) across
$a\in[1,5]$ with Bayes factors computed using the LT approach (circles) across
$\sigma_{\psi}\in[1,2]$. In virtually all cases and across specifications, the
Bayes factor in favour of $\mathcal{H}_{0}$ is higher under the IB approach,
and this difference is frequently substantial.222In contrast, the conclusions
one would draw based on posterior distributions are very similar, see Appendix
C. As the parameter $a$ is increased under the IB approach, the expected
difference between the two groups is smaller (see top left panel in Figure 3).
Therefore, the predictions under $\mathcal{H}_{1}$ become more similar to the
predictions under $\mathcal{H}_{0}$, and the Bayes factor decreases.
Conversely, as $\sigma_{\psi}$ is increased under the LT approach, the
expected difference between the two group increases, and the Bayes factor in
favour of $\mathcal{H}_{0}$ increases.
Figure 4: Top: Bayes factors using the IB (rectangles) and LT approach
(circles) in favour of $\mathcal{H}_{0}$ across studies reported in [6] (left)
or for simulated equal proportions with $n=100$ (right) for values $a\in[1,5]$
and $\sigma_{\psi}\in[1,2]$ with $\sigma_{\beta}=1$. Bottom: Joint prior
distribution of $(\theta_{1},\theta_{2})$ under the IB approach with $a=1$
(left) and under the LT approach with $\sigma_{\psi}=1$ and $\sigma_{\beta}=1$
(right). Black dots and blue rectangles indicate the maximum likelihood
estimates of the proportions in the studies analyzed by [6].
The top left panel in Figure 4 shows that, for some studies, the IB and LT
Bayes factors cannot be brought into the vicinity of each other by changing
the prior parameters in the way specified above, while for other studies the
Bayes factors do overlap substantially. The studies without overlap are those
indexed as 1-12. They are shown as blue rectangles in the bottom panels in
Figure 4, which shows the joint prior density for the IB (left) and the LT
approach (right) with the symbols indicating the maximum likelihood estimates
for the individual studies. Note that the larger the proportions in the bottom
panels, the more likely are the two Bayes factors to overlap in the top left
panel.
While [6] are reassured by the fact that their Bayesian reanalysis (using the
IB approach) yields strong evidence in favour of $\mathcal{H}_{0}$ on average
(given that all studies reported a non-significant $p$-value), using the LT
approach yields a more uncertain picture. While the median Bayes factor across
the studies under the IB approach is 12.30, the median Bayes factor under the
LT approach is only 4.79. This difference is driven by the extremes, where the
two approaches suggest substantially different conclusions. We have already
seen one example in the introduction concerning hypertension and pregnancy
loss (analysis identifier 3). Another example is given by [13], which compared
the efficacy of the 9vHPV against the qHPV vaccine for preventing, among
others, cervical, vulvar, and varginal disease and persistent human
papillomavirus related infections in women. Comparing different outcome
measures in women who were HPV-negative and HPV-positive at baseline, the IB
approach yields Bayes factors consistently around 100 while the LT approach
never shows Bayes factors larger than 10 (see analysis identifiers 1, 2, 5,
and 6). The IB approach thus strongly suggests that the two vaccines have a
similar efficacy, while the LT approach suggests that more data is needed to
reach a firm conclusion.
The initially diverging and then converging pattern of the Bayes factors can
be seen neatly in the top right panel in Figure 4, which shows the Bayes
factors under the null hypothesis for increasing values of
$\nicefrac{{y}}{{100}}\equiv\nicefrac{{y_{1}}}{{100}}=\nicefrac{{y_{2}}}{{100}}$.
Note that the pattern is symmetric, so that for
$\nicefrac{{y}}{{100}}\in[\nicefrac{{1}}{{2}},1]$ the same holds, just
mirrored. At the extremes $\nicefrac{{y}}{{n}}\in\\{0,1\\}$, the IB approach
yields a Bayes factor in favour of $\mathcal{H}_{0}$ of 50.75, while the LT
Bayes factor gives a mere 1.40. The Bayes factor decreases as
$\nicefrac{{y}}{{n}}\rightarrow\nicefrac{{1}}{{2}}$ under the IB approach, but
increases under the LT approach. Consequently, the difference between the
Bayes factors becomes less pronounced as we move to more central proportions,
which matches the observation in the empirical analysis shown in the top left
panel in Figure 4. For $\nicefrac{{y}}{{n}}=\nicefrac{{1}}{{2}}$, the IB Bayes
factor yields 5.70 and the LT Bayes factor yields 3.67.
In summary, we have found that the two approaches give a very different answer
to the question “is observing two equally small proportions strong or weak
evidence in favour of the null hypothesis?”. The IB Bayes factor suggests that
the evidence is often orders of magnitude larger than the LT Bayes factor.
### 3.2 Explaining the Difference
Why does the IB Bayes factor yield so much stronger evidence than the LT Bayes
factor for $\mathcal{H}_{0}$ when the proportions are small and of roughly
equal size? And why does the IB Bayes factor decrease as the proportions get
closer to $\nicefrac{{1}}{{2}}$ while the LT Bayes factor increases? To answer
these questions, we need to zoom in on the differences in the respective prior
specifications. The two approaches differ in two key ways. First, while the IB
approach assigns independent priors to $\theta_{1}$ and $\theta_{2}$, the LT
approach assigns dependent priors to them. Second, the LT approach employs a
logit transformation while the IB approach does not. Appendix D shows that it
is not the prior dependence that underlies the difference between the two
approaches. Instead, as we will see below, it is the logit transformation.
The fact that the IB Bayes factor is larger than the LT Bayes factor even when
the latter approach expects a larger difference in the proportions (e.g., a
larger $\eta$ or $\psi$, compare $a=2$ with $\sigma_{\psi}=2$ in Figure 3)
means that focusing only on the difference parameter is not sufficient to
explain the difference between the Bayes factors. Instead, we turn to a
sequential predictive perspective. From such a perspective, we may first use
the prior to predict the data from group one, update the prior to a posterior,
and then predict the data from group two. One can rewrite the marginal
likelihoods to make this sequential perspective apparent, see Appendix E. This
perspective shows that there is a crucial difference in the predictions that
the CT and LT approach make under $\mathcal{H}_{1}$. Under both approaches
there is a common parameter $\theta$ under $\mathcal{H}_{0}$, and the prior
assigned to $\theta$ gets updated by data from the first group, after which
predictions about the second group are made. Under the alternative hypothesis,
however, the IB approach implies that observing data from group one does not
update our beliefs about likely values of $\theta_{2}$, and so data from group
one cannot inform the subsequent prediction about group two. Under the LT
approach, such information sharing does take place. This difference is shown
in the top panels in Figure 5, which visualizes the joint prior distribution
for $\theta_{1}$ and $\eta\equiv\theta_{2}-\theta_{1}$ under the IB (left) and
LT (right) approaches. We see that under the IB approach, learning about
$\theta_{1}$ does not influence our predictions about likely differences
between the two groups. In contrast, under the LT approach we find that when
we learn that $\theta_{1}$ is either small or large, we expect small
differences between the proportions compared to when we learn that
$\theta_{1}$ is about half. This explains why the LT Bayes factor in favour of
$\mathcal{H}_{0}$ increases as we move from extreme values of
$\nicefrac{{y}}{{n}}$ towards values around $\nicefrac{{1}}{{2}}$, as shown in
the top right panel in Figure 4: at the extremes, $\mathcal{H}_{1}$ expects
smaller differences between $\theta_{2}$ and $\theta_{1}$, and hence it more
closely resembles $\mathcal{H}_{0}$, leading to a more equivocal Bayes factor.
Moving towards more central values, $\mathcal{H}_{1}$ expects larger
differences and hence $\mathcal{H}_{0}$ outpredicts it by a larger margin.
Figure 5: Top: Joint prior distribution for $(\theta_{1},\eta)$ under the IB
approach with $a=1$ (left) and under the LT approach with $\sigma_{\psi}=1$
and $\sigma_{\beta}=1$ (right). Bottom: Joint prior distribution for
$(\theta_{1},\psi)$ under the two approaches.
While prior dependence is necessary to change one’s beliefs about likely
differences between the proportions upon learning $\theta_{1}$, the fact that
the expected difference decreases in the LT approach is due to the logit
transformation. When the two rates $(\theta_{1},\theta_{2})$ are small, say
$\theta_{1}=0.05$ and $\theta_{2}=0.10$, the log odds difference $\psi$ is
large, in this case $\psi=0.75$. The prior on $\psi$ in the LT approach
renders such large log odds differences to be unlikely, which means that a
smaller rate difference is expected. Conversely, if the rates are somewhere in
the center, say $\theta_{1}=0.50$ and $\theta_{2}=0.55$, then the log odds
difference $\psi$ is small, in this case $\psi=0.20$. The prior on $\psi$ in
the LT approach views such small log odds differences as likely; in fact, much
larger ones are possible. Therefore, the LT approach expects larger rate
differences when the rates are in the center. On the log odds scale, the
expected difference does not change markedly as $\theta_{1}$ varies, as the
bottom right panel in Figure 5 shows.333The fact that there is barely any
prior mass at extreme values of $\theta_{1}$ under the IB approach is due to
the Gaussian prior on $\beta$, as discussed in Section 2.3.
In contrast to the LT approach, the IB approach does not automatically reduce
the expectations about likely differences as we learn that $\theta_{1}$ takes
on extreme values. Taking slices at particular values for $\theta$ in the top
left panel in Figure 5 would show uniform distributions ranging from
$-\theta_{1}$ to $1-\theta_{1}$. Instead of reducing the size of an expected
difference at the extremes, the IB approach actually amplifies it. This occurs
because of the boundedness of the rate parameters. In particular, if
$\theta_{1}=0$, then the only values for which, say, $|\eta|<0.10$ are
$\theta_{2}\in[0,0.10]$. Differences in absolute magnitude between the two
rates of up to $0.10$ if $\theta_{1}=0$ are therefore assigned a probability
of $0.10$. If, however, $\theta_{1}=0.50$, the values of $\theta_{2}$ for
which $|\eta|<0.10$ are $\theta_{2}\in[0.40,0.60]$. This means that
differences in absolute magnitude between the two rates of up to $0.10$ if
$\theta_{1}=0.50$ are assigned a probability of $0.20$, or twice as much as in
the case that $\theta_{1}=0$. Another way to see this is to note that the
expected difference in the case that $\theta_{1}=0$ is $\eta=0.50$, while the
expected difference in the case that $\theta_{1}=0.50$ is $\eta=0$; more
generally, the expected difference increases linearly as we move to the
extremes, as can be seen in the top left panel in Figure 5. The bottom left
panel in Figure 5 shows an associated and pronounced increase in the expected
log odds difference as $\theta_{1}$ moves towards extreme values.
## 4 Discussion
Being able to discriminate between evidence of absence and absence of evidence
is key in medicine and public health. The Bayes factor, which pits the
predictive performance of two hypotheses against each other, is a method for
doing so. Reanalyzing 39 null results published in the New England Journal of
Medicine, we have found that the strength of evidence depends crucially on the
prior specification. Comparing an approach that assigns independent beta
distributions directly to the rate parameters (IB) to one that employs a logit
transformation (LT), we have found that the former approach suggests evidence
that is orders of magnitudes larger than the latter approach when observing
small proportions of roughly equal size. Consider again the effect of tightly
or less-tightly controlling hypertension in pregnant women [19]. The IB
approach suggests a Bayes factor about ten times larger in favour of the null
hypothesis than the LT approach. Similarly, the Bayes factor in favour of the
hypothesis that two vaccines against the human papillomavirus are equally
effective [13] is up to 38 times larger under the IB approach, depending on
the outcome measure. While the original statistical analysis by the authors is
more involved (e.g., adjusting for various covariates), these differences are
striking. As we have seen, they occur because of the boundedness of the
parameter space, which leads to the expectations of large differences at the
extremes under the IB approach. In contrast, the LT approach expects smaller
differences at the extremes.
Although the two Bayes factors frequently differed markedly in size, they
always both provided evidence in favour of $\mathcal{H}_{0}$ in the cases we
studied in this paper. This need not be the case, however. For example, a
large-scale investigation into the effect of aspirin found that 26 out of
$n_{1}=11,034$ who received a placebo suffered a fatal heart attack, but only
10 out of $n_{2}=11,037$ who received aspirin did [23]. The IB Bayes factor
yields strong evidence in favour of $\mathcal{H}_{0}$ ($\text{BF}_{01}=19.78$)
while the LT Bayes factor finds moderate evidence in favour of
$\mathcal{H}_{1}$ ($\text{BF}_{10}=5.36$). This is again owed to the fact that
the IB approach expects larger differences than the LR approach; a small
effect size is more likely under $\mathcal{H}_{0}$ than under
$\mathcal{H}_{1}$ in the IB approach simply because under $\mathcal{H}_{1}$
unrealistically large effect sizes are assumed to be plausible.
Our results have a number of implications. First, researchers should think
carefully about their prior setup. This not only implies thinking about the
parameter values for the priors, but also the nature of the priors and what
they imply for the data. As we have seen, one may be fooled by assessing how
sensitive the Bayes factor is by varying only prior parameters conditional on
the model. For all studies we re-analyzed that had proportions at the
extremes, varying the prior parameters of the IB approach still resulted in
substantial evidence in favour of the null hypothesis. However, changing the
prior setup to the LT approach, we found markedly reduced evidence. If there
are sensible alternative parameterizations, it may be prudent to explore how
sensitive the results are to these. In particular, our exposition suggests
that assessing the predictions that follow from the model can help get a
better intuition of its assumptions. In our case, this made clear that the IB
approach expects larger rate differences at the boundaries of the parameter
space compared to the center, while the LT approach expects the reverse. We
were struck by the initially puzzling contrast between these two tests, given
that the task — comparing two proportions — seemed so simple. This suggests
that even more caution needs to be applied when using Bayes factors for
comparing models that are invariably more complex. In the same spirit of
sensitivity analysis, it is prudent to compare inferences based on the Bayes
factor with inferences based on the posterior distributions of the parameters.
While testing precise hypotheses based on credible intervals is arguably
inappropriate since it makes inference conditional on $\mathcal{H}_{1}$,
ignoring $\mathcal{H}_{0}$ [[, e.g.,]]berger2006bayes,
wagenmakers2020principle, alarm bells should go off when they show stark
differences to the Bayes factor results, as in our case (compare Figure 4 and
6). blackResearchers who rely only on the posterior distribution for inference
or for trial design are unaffected by our results.
The second implication of our work is that the LT approach appears better
suited for testing the equality of two proportions than the currently more
popular IB approach. First, assuming prior dependence strikes us as a more
sensible approach especially for medicine and public health. This is because
it is generally unlikely that gaining information about one group does not
influence our beliefs about the other group. For example, if a particular
treatment for cancer yields a $30\%$ success rate, surely this should inform
our expectations for the success rate of a comparison treatment. Incorporating
this dependency results in more cautious predictions, which reduces the
eagerness with which the standard IB approach suggests strong evidence in
favour of $\mathcal{H}_{0}$ (see also Appendix D). While we have focused on
testing the sharp null hypothesis $\theta_{1}=\theta_{2}$, the difference
between dependent and independent priors is important also when testing other
hypotheses. [9] discusses the difference between a dependent and independent
prior in the context of comparing $\theta_{1}<\theta_{2}$ against
$\theta_{1}>\theta_{2}$, and similarly recommends dependent priors, as
independent priors lead to a test that is not “sufficiently cautious”.
Most importantly, however, is the fact that the LT approach avoids the issues
associated with a bounded parameter space. In particular, the IB approach is
not sensitive to the fact that the implications of a rate difference
$\theta_{2}-\theta_{1}$ depends on the actual values of the rates. For
example, $\theta_{2}=0.10$ and $\theta_{1}=0.05$ and $\theta_{2}=0.55$ and
$\theta_{1}=0.50$ imply the same rate difference, but while the former
represents a 100% increase, the latter represents only a 10% increase. Yet the
IB approach assigns more prior mass to large rate differences at the extremes
of the parameter space compared to its center. In contrast, the reverse holds
for the LT approach because of the logit transformation. This strikes us as
more sensible. With the recent development of accessible software that makes
the LT approach easily available [3, 8], we hope that the use of the LT
approach for testing the equality of two proportions becomes more widespread.
To sum up, we have seen an initially puzzling divergence in evidence when
reanalyzing 39 null results from the New England Journal of New Medicine with
two different Bayes factor tests. We have explained this divergence and sided
with the approach that employs a logit transformation and assumes prior
dependence between the rates. We suggest that this approach, rather than the
one motivated from a contingency table perspective, should be routinely
applied when testing the equality of two proportions. Our journey should also
act as a cautionary tale for users of Bayes factors, stressing the importance
of assessing the predictions of one’s models, conducting thorough sensitivity
analyses, and never relying on just a single quantity. We have demonstrated
the usefulness of this in the simple case of testing the equality of two
proportions. Most applications will arguably be more complex, however,
increasing the potential for puzzles and stressing the need for a holistic
evidence assessment.
#### Author contributions.
EJW proposed the study. KH, QFG, AE, and EJW worked on the problem initially.
After an interim period, FD and EJW resumed the work and solved the puzzle. FD
and KH analyzed the data. FD wrote the manuscript. All authors read, provided
comments, and approved the submitted version of the paper. They also declare
that there were no conflicts of interest.
#### Acknowledgements.
We thank Rink Hoekstra, Rei Monden, and Don van Ravenzwaaij for sharing the
New England Journal of Medicine data with us.
## References
* [1] J.. Berger “Bayes Factors” In _Encyclopedia of Statistical Sciences, vol. 1_ Hoboken, NJ: Wiley, 2006, pp. 378–386
* [2] James M Dickey and BP Lientz “The weighted likelihood ratio, sharp hypotheses about chances, the order of a Markov chain” In _The Annals of Mathematical Statistics_ JSTOR, 1970, pp. 214–226
* [3] Quentin F Gronau, A Raj and Eric-Jan Wagenmakers “Informed Bayesian inference for the A/B test” In _Journal of Statistical Software_ , in press
* [4] Erdogan Gunel and James Dickey “Bayes factors for independence in contingency tables” In _Biometrika_ 61.3 Oxford University Press, 1974, pp. 545–557
* [5] Max Hinne, Quentin F Gronau, Don van den Bergh and Eric-Jan Wagenmakers “A conceptual introduction to Bayesian model averaging” In _Advances in Methods and Practices in Psychological Science_ 3.2 SAGE Publications Sage CA: Los Angeles, CA, 2020, pp. 200–215
* [6] Rink Hoekstra, Rei Monden, Don van Ravenzwaaij and Eric-Jan Wagenmakers “Bayesian reanalysis of null results reported in medicine: Strong yet variable evidence for the absence of treatment effects” In _PloS One_ 13.4 Public Library of Science San Francisco, CA USA, 2018, pp. e0195474
* [7] Jennifer A Hoeting, David Madigan, Adrian E Raftery and Chris T Volinsky “Bayesian model averaging: A tutorial” In _Statistical Science_ JSTOR, 1999, pp. 382–401
* [8] Tabea Hoffmann and Eric-Jan Wagenmakers “Bayesian Inference for the A/B Test: Example Applications with R and JASP” PsyArXiv, 2021 DOI: 10.31234/osf.io/z64th
* [9] JV Howard “The 2 $\times$ 2 table: A discussion from a Bayesian viewpoint” In _Statistical Science_ JSTOR, 1998, pp. 351–367
* [10] Tahira Jamil et al. “Default “Gunel and Dickey” Bayes factors for contingency tables” In _Behavior Research Methods_ 49.2 Springer, 2017, pp. 638–652
* [11] Harold Jeffreys “Some tests of significance, treated by the theory of probability” In _Proceedings of the Cambridge Philosophical Society_ 31.2, 1935, pp. 203–222
* [12] Harold Jeffreys “Theory of Probability” Oxford, UK: Oxford University Press, 1939
* [13] Elmar A Joura et al. “A 9-valent HPV vaccine against infection and intraepithelial neoplasia in women” In _New England Journal of Medicine_ 372.8 Mass Medical Soc, 2015, pp. 711–723
* [14] Robert E Kass and Adrian E Raftery “Bayes factors” In _Journal of the American Statistical Association_ 90.430 Taylor & Francis, 1995, pp. 773–795
* [15] Robert E Kass and Suresh K Vaidyanathan “Approximate Bayes factors and orthogonal parameters, with application to testing equality of two binomial proportions” In _Journal of the Royal Statistical Society: Series B (Methodological)_ 54.1 Wiley Online Library, 1992, pp. 129–144
* [16] Christian Keysers, Valeria Gazzola and Eric-Jan Wagenmakers “Using Bayes factor hypothesis testing in neuroscience to establish evidence of absence” In _Nature Neuroscience_ 23.7 Nature Publishing Group, 2020, pp. 788–799
* [17] M.. Lee and E.–J. Wagenmakers “Bayesian Cognitive Modeling: A Practical Course” Cambridge University Press, 2013
* [18] Alexander Ly, Josine Verhagen and Eric-Jan Wagenmakers “Harold Jeffreys’s default Bayes factor hypothesis tests: Explanation, extension, and application in psychology” In _Journal of Mathematical Psychology_ 72 Elsevier, 2016, pp. 19–32
* [19] Laura A Magee et al. “Less-tight versus tight control of hypertension in pregnancy” In _New England Journal of Medicine_ 372.5 Mass Medical Soc, 2015, pp. 407–417
* [20] Richard D Morey, Jan-Willem Romeijn and Jeffrey N Rouder “The philosophy of Bayes factors and the quantification of statistical evidence” In _Journal of Mathematical Psychology_ 72 Elsevier, 2016, pp. 6–18
* [21] Matthew Oladejo Ojo “A remark on the convolution of the generalized logistic random variables” In _ASSET serves A_ 1.2, 2003
* [22] Thu Pham-Gia, Noyan Turkkan and P Eng “Bayesian analysis of the difference of two proportions” In _Communications in Statistics-Theory and Methods_ 22.6 Taylor & Francis, 1993, pp. 1755–1771
* [23] Steering Committee of the Physicians’ Health Study Research Group “Final report on the aspirin component of the ongoing Physicians’ Health Study” In _New England Journal of Medicine_ 321.3 Mass Medical Soc, 1989, pp. 129–135
* [24] Martin Plöderl et al. “Suicide risk and sexual orientation: a critical review” In _Archives of Sexual Behavior_ 42.5 Springer, 2013, pp. 715–727
* [25] Fernando P Polack et al. “Safety and efficacy of the BNT162b2 mRNA Covid-19 vaccine” In _New England Journal of Medicine_ 383.27 Mass Medical Soc, 2020, pp. 2603–2615
* [26] E.–J. Wagenmakers, M.. Lee, J.. Rouder and R.. Morey “The Principle of Predictive Irrelevance or Why Intervals Should not be Used for Model Comparison Featuring a Point Null Hypothesis” In _The Theory of Statistics in Psychology – Applications, Use and Misunderstandings_ Cham: Springer, 2020, pp. 111–129
* [27] Larry Wasserman “Bayesian model selection and model averaging” In _Journal of Mathematical Psychology_ 44.1 Elsevier, 2000, pp. 92–107
## Appendix A Induced Priors
Under the LT approach, the marginal priors for $\theta_{1}$ and $\theta_{2}$
as well as $\eta$ are given in integral representation by [3]. Under the IB
approach, the density of $\eta$ is given by:
$f(\eta;a)=\begin{cases}\frac{1}{B(a,a)}\eta^{2a-1}(1-\eta)^{2a-1}\,F_{1}\left({a},{4a-2},{1-a};{2a};{1-\eta},{1-\eta^{2}}\right)&0<\eta\leq
1\\\\[4.5pt]
\frac{1}{B(a,a)}(-\eta)^{2a-1}(1+\eta)^{2a-1}\,F_{1}\left({a},{1-a},{4a-2};{2a};{1-\eta^{2}},{1+\eta}\right)&-1\leq\eta<0\end{cases}\enspace,$
see [22]. In our case, we always have that $2a>1$ and so
$f(0)=\frac{B(2a-1,2a-1)}{B(a,a)^{2}}$ [22]. We can derive the density for
$\psi$ under prior independence for the special case of $a=1$. Note that,
because $\theta\sim\text{Beta}(1,1)$, it follows that:
$\text{log}\left(\frac{\theta}{1-\theta}\right)\sim\text{Logistic}(0,1)\enspace.$
Recall that:
$\psi=\text{log}\left(\frac{\theta_{2}}{1-\theta_{2}}\right)-\text{log}\left(\frac{\theta_{1}}{1-\theta_{1}}\right)\enspace.$
[21] gives the distribution function of the sum of $n$ i.i.d. logistic random
variables. Since the logistic distribution is symmetric, we have that if
$X\sim\text{Logistic}$ then $-X\sim\text{Logistic}$. Hence we can use the
results by [21] and write the density function of $\psi$ as:
$f(\psi;a=1)=\frac{e^{\psi}(e^{\psi}(\psi-2)+\psi+2)}{(e^{\psi}-1)^{3}}\enspace.$
## Appendix B Combining Approaches through Model-averaging
If one is uncertain as to which test one should employ, an approach would be
to combine both through model-averaging [[, e.g.,]]hoeting1999bayesian,
hinne2020conceptual. Let $\mathcal{M}_{0}^{\text{IB}}$ and
$\mathcal{M}_{0}^{\text{LT}}$ denote the null models under the IB and the LT
approach, respectively, and let $\mathcal{M}_{1}^{\text{IB}}$ and
$\mathcal{M}_{1}^{\text{LT}}$ denote the respective alternative models. The
model-averaged Bayes factor is given by:
$\text{BF}^{\text{AVG}}_{01}=\frac{p(\mathcal{D}\mid\mathcal{M}_{0}^{\text{IB}})\pi(\mathcal{M}_{0}^{\text{IB}})+p(\mathcal{D}\mid\mathcal{M}_{0}^{\text{LT}})\pi(\mathcal{M}_{0}^{\text{LT}})}{p(\mathcal{D}\mid\mathcal{M}_{1}^{\text{IB}})\pi(\mathcal{M}_{1}^{\text{IB}})+p(\mathcal{D}\mid\mathcal{M}_{1}^{\text{LT}})\pi(\mathcal{M}_{1}^{\text{LT}})}\enspace,$
where $\pi()$ gives the prior probability of the particular model. One has to
be careful when model-averaging. While the nuisance parameters generally do
not matter when testing nested models [15, 18], they do here. In particular,
the IB approach assigns $\beta$ a logistic distribution, while the LT approach
assigns $\beta$ a Gaussian distribution. The logistic distribution has fatter
tails, and thus puts more mass on extreme values of $\theta$ than the Gaussian
distribution. In our case, this can result in somewhat surprising results. Let
$\mathcal{D}_{1}=(y_{1},n_{1})$ and $\mathcal{D}_{2}=(y_{2},n_{2})$ denote
data in the two groups, respectively. For data $\mathcal{D}_{1}=(0,50)$ and
$\mathcal{D}_{2}=(0,50)$, for example, the IB model that assumes a difference
in the population ($\mathcal{M}^{\text{IB}}_{1}$) outpredicts the LT null
model ($\mathcal{M}^{\text{LT}}_{0}$) by a factor of seven! Since we want to
focus on differences under $\mathcal{H}_{1}$ between the IB and LT approach,
we would need to make the models identical under $\mathcal{H}_{0}$, for
example by assigning $\beta$ a logistic prior not only under the IB approach,
but also under the LT approach.
## Appendix C Inference based on Posterior Distributions
The contrasting results discussed in Section 3 do not occur when inferences
are based on the posterior distribution of the log odds difference $\psi$ or
the rate difference $\eta$. Figure 6 shows the posterior mean and 95% credible
interval of $\psi$ (top) and $\eta$ (bottom) for the IB (rectangles) and the
LT approach (circles) with the respective default parameterizations $a=1$ and
$\sigma_{\psi}=1$ across all studies. The conclusions one would draw from
these posterior distributions are very similar (and changing the prior
parameters does not affect them). This is as expected, since the effect of the
prior generally washes out as more and more data are observed, which is not
the case for Bayes factors.
Figure 6: Shows the posterior mean and 95% credible interval of the log odds
difference $\psi$ (top) and the rate difference $\eta$ (bottom) under the IB
approach (rectangles, $a=1$) and the LT approach (circles, $\sigma_{\psi}=1$).
## Appendix D A Dependent IB Approach
If prior dependence were responsible for the stark differences between the
Bayes factors at the extremes, we would find that a IB approach which assumes
prior dependence shows a pattern similar to the LT approach. Here, we show
that this is not the case. Recall that $\eta=\theta_{2}-\theta_{1}$ and
$\zeta=\frac{1}{2}\left(\theta_{1}+\theta_{2}\right)$. We define:
$\displaystyle\theta_{1}$
$\displaystyle=\text{min}\left(\text{max}\left(\zeta-\frac{\eta}{2},0\right),1\right)$
$\displaystyle\theta_{2}$
$\displaystyle=\text{min}\left(\text{max}\left(\zeta+\frac{\eta}{2},0\right),1\right)\enspace,$
and assign truncated Gaussian priors to $(\eta,\zeta$), i.e.,
$\eta\sim\mathcal{N}(0,\sigma_{\eta})_{\mathbb{I}(-1,1)}$ and
$\zeta\sim\mathcal{N}(0,\sigma_{\zeta})_{\mathbb{I}(0,1)}$. Using
$\sigma_{\eta}=\nicefrac{{1}}{{5}}$ and $\sigma_{\zeta}=\nicefrac{{1}}{{2}}$,
the left panel in Figure 7 shows the joint prior distribution for
$(\theta_{1},\theta_{2})$ from which a strong prior dependence between
$\theta_{1}$ and $\theta_{2}$ is apparent. The right panel, however, shows the
characteristic pattern of the IB approach as discussed in the main text,
rather than the characteristic pattern of the LT approach. Since the rates a
dependent a priori, it is not the prior dependence that is responsible for the
pattern. However, employing a prior that has a stronger dependency between the
two rates naturally reduces the size of their expected difference. Increasing
$\sigma_{\eta}$ results in a lower correlation (going from about 0.77 at
$\sigma_{\eta}=\nicefrac{{1}}{{5}}$ to zero at $\sigma_{\eta}=1$) and a
reduced Bayes factor in favour of $\mathcal{H}_{0}$, as the right panel in
Figure 7 shows.
Figure 7: Left: Joint distribution on $(\theta_{1},\theta_{2})$ using
$\sigma_{\eta}=\nicefrac{{1}}{{5}}$ and $\sigma_{\zeta}=\nicefrac{{1}}{{2}}$.
Right: Bayes factors in favour of $\mathcal{H}_{0}$ for simulated proportions
with $n=100$ for values $\sigma_{\eta}=[\nicefrac{{1}}{{5}},1]$ and
$\sigma_{\zeta}=\nicefrac{{1}}{{2}}$.
## Appendix E Sequential Predictive Perspective
Under $\mathcal{H}_{0}$ we have for both the IB (i.e., independent) and the LT
(i.e., dependent) setup that (suppressing conditioning on $\mathcal{H}_{0}$):
$\displaystyle p(\mathcal{D}_{2},\mathcal{D}_{1}\mid\mathcal{H}_{0})$
$\displaystyle=\int_{\theta}p(\mathcal{D}_{2},\mathcal{D}_{1}\mid\theta)\pi_{0}(\theta)\,\,\mathrm{d}\theta$
(1)
$\displaystyle=\int_{\theta}p(\mathcal{D}_{2}\mid\mathcal{D}_{1},\theta)p(\mathcal{D}_{1}\mid\theta)\pi_{0}(\theta)\,\,\mathrm{d}\theta$
(2)
$\displaystyle=\mathcal{Z}\int_{\theta}p(\mathcal{D}_{2}\mid\theta)p(\theta\mid\mathcal{D}_{1})\,\,\mathrm{d}\theta\enspace,$
(3)
where
$\mathcal{Z}=\int_{\theta}p(\mathcal{D}_{1}\mid\theta)\pi_{0}(\theta)\,\,\mathrm{d}\theta$
is the marginal likelihood for $\mathcal{D}_{1}$,
$p(\theta\mid\mathcal{D}_{1})$ is the posterior of $\theta$ after observing
$\mathcal{D}_{1}$, and we can remove the conditioning on $\mathcal{D}_{1}$ in
(2) because all relevant information is in $\theta$. Similarly, under
$\mathcal{H}_{1}$ for LT setup we have that:
$\displaystyle p(\mathcal{D}_{2},\mathcal{D}_{1}\mid\mathcal{H}_{1})$
$\displaystyle=\int_{\theta_{2}}\int_{\theta_{1}}p(\mathcal{D}_{2},\mathcal{D}_{1}\mid\theta_{1},\theta_{2})\pi_{1}(\theta_{1},\theta_{2})\,\mathrm{d}\theta_{1}\,\mathrm{d}\theta_{2}$
(4)
$\displaystyle=\int_{\theta_{2}}\int_{\theta_{1}}p(\mathcal{D}_{2}\mid\mathcal{D}_{1},\theta_{1},\theta_{2})p(\mathcal{D}_{1}\mid\theta_{1},\theta_{2})\pi_{1}(\theta_{1},\theta_{2})\,\mathrm{d}\theta_{1}\,\mathrm{d}\theta_{2}$
(5)
$\displaystyle=\mathcal{Z}\int_{\theta_{2}}\int_{\theta_{1}}p(\mathcal{D}_{2}\mid\theta_{1},\theta_{2})p(\theta_{1},\theta_{2}\mid\mathcal{D}_{1})\,\mathrm{d}\theta_{1}\,\mathrm{d}\theta_{2}\enspace,$
(6)
where $\mathcal{Z}$ is the marginal likelihood for $\mathcal{D}_{1}$,
$p(\theta_{1},\theta_{2}\mid\mathcal{D}_{1})$ is the posterior after observing
$\mathcal{D}_{1}$, and we can remove the conditioning on $\mathcal{D}_{1}$ in
(2) because all relevant information is in $(\theta_{1},\theta_{2})$. This is
in contrast to the IB setup, where:
$\displaystyle p(\mathcal{D}_{2},\mathcal{D}_{1}\mid\mathcal{H}_{1})$
$\displaystyle=\int_{\theta_{2}}\int_{\theta_{1}}p(\mathcal{D}_{2},\mathcal{D}_{1}\mid\theta_{1},\theta_{2})\pi_{1}(\theta_{1},\theta_{2})\,\mathrm{d}\theta_{1}\,\mathrm{d}\theta_{2}$
(7)
$\displaystyle=\int_{\theta_{2}}p(\mathcal{D}_{2}\mid\theta_{2})\pi_{1}(\theta_{2})\,\mathrm{d}\theta_{2}\int_{\theta_{1}}p(\mathcal{D}_{2}\mid\theta_{1})\pi_{1}(\theta_{1})\,\mathrm{d}\theta_{1}\enspace,$
(8)
and thus no sharing of information across the two groups takes place.
|
# High-dimensional entanglement for quantum communication in the frequency
domain
Meritxell Cabrejo-Ponce<EMAIL_ADDRESS>Fraunhofer
Institute for Applied Optics and Precision Engineering, Albert-Einstein-
Strasse 7, 07745 Jena Abbe Center of PhotonicsFriedrich Schiller University
JenaAlbert-Einstein-Str. 6, 07745 Jena, Germany André Luiz Marques Muniz
Fraunhofer Institute for Applied Optics and Precision Engineering, Albert-
Einstein-Strasse 7, 07745 Jena Marcus Huber<EMAIL_ADDRESS>Atominstitut, Technische Universität Wien, Stadionallee 2, 1020 Vienna,
Austria Institute for Quantum Optics and Quantum Information (IQOQI),
Austrian Academy of Sciences, 1090 Vienna, Austria Fabian Steinlechner
<EMAIL_ADDRESS>Fraunhofer Institute for Applied Optics
and Precision Engineering, Albert-Einstein-Strasse 7, 07745 Jena Abbe Center
of Photonics, Friedrich Schiller University Jena, Albert-Einstein-Str. 6,
07745 Jena, Germany
###### Abstract
High-dimensional photonic entanglement is a promising candidate for error-
protected quantum information processing with improved capacity. Encoding
high-dimensional qudits in the carrier frequency of photons combines ease of
generation, universal single-photon gates, and compatibility with fiber
transmission for high-capacity quantum communication. Recent landmark
experiments have impressively demonstrated quantum interference of a few
frequency modes, yet the certification of massive-dimensional frequency
entanglement has remained an open challenge. This study shows how to harness
the large frequency-entanglement inherent in standard continuous-wave
spontaneous parametric down-conversion processes. It further reports a record
certification of discretized frequency entanglement, combined with a novel
approach for certification that is both highly efficient and nonlocally
implementable. This technique requires very few measurements and does not
require assumptions on the state. The work opens the possibility for utilizing
this encoding in quantum communications and in quantum information science in
general.
## I Introduction
Entanglement is a unique and powerful quantum feature with a multitude of
applications in quantum information processing. The nonlocal correlations of
the entangled states may be used in quantum communications, imaging,
metrology, and quantum processors. In the case of photons, polarization-
entangled states have been traditionally used to demonstrate a multitude of
quantum gates and quantum information protocols [1] and the principles of
rapidly developing quantum networks [2, 3]. These qubit states are easy to
manipulate with linear optics and can be distributed in fiber or free-space
links because of their low interaction with the environment. Nevertheless,
larger alphabets in quantum communication are highly pursued not only to
increase the capacity of the quantum channel. High-dimensional encoding also
provides a stronger tolerance to noise [4], an essential asset to overcome the
transmission limits of polarization qubits. On the other hand, an increase in
dimensionality can also boost the computational power of quantum computers
[5]. In this context, a recent landmark experiment [6] used high-dimensional
entanglement of propagation paths on a silicon photonic chip to realise error-
protected logical qubits and thus improve the performance of a quantum phase
estimation algorithm. Harnessing the full state space of photonic degrees of
freedom (DOF) such as transverse spatial mode, time, and frequency, will be
key to future generations of photonic quantum information processors.
Figure 1: Schematic of our frequency-entangled photon pair source and the
state analysis parts. We generate broadband frequency entanglement at telecom
wavelength via spontaneous parametric down-conversion (SPDC) in a periodically
poled lithium niobate waveguide (ppLN). Afterwards, the photon pairs are
coherently manipulated with a pulse shaper and an electro-optic modulator
(EOM) to analyze the correlations in a superposition basis. CW: continuous-
wave; DEMUX: demultiplexer.
Although spatial [7, 8, 9] and temporal modes [10, 11, 12] have been
straightforward to operate in very large dimensional spaces, the frequency DOF
has remained behind in these advances. The frequency domain at optical scales
is of particular interest because of its parallelization capabilities, in
particular its compatibility with telecom multiplexing or frequency modulation
techniques. Earlier experiments exploiting electro-optic modulation were able
to demonstrate two-dimensional frequency entanglement [13], while the use of
pulse shapers and up-conversion processes allowed the characterization of up
to four-dimensional states [14, 15]. The combination of both components was
used later to coherently control the quantum states emerging from integrated
resonators [16]. These important building blocks opened the possibility to
measure in the superposition basis at optical scales and were used to
demonstrate discretized frequency entanglement with few dimensions (up to 6
[17]). Since then, approaches have been developed to perform arbitrary
manipulations of single-frequency qubits [18] and the first steps have been
taken towards full control of single qudits, its high-dimensional version [19,
20, 21]. While Hong-Ou-Mandel interference has also been used to verify two-
dimensional frequency entanglement [22], the requirement of local measurements
limits the utility of these methods in quantum networks.
Furthermore, it was recently shown that the HOM based approach cannot be used
to verify higher dimensional entanglement [23] , at least not without imposing
additional assumptions such as energy conservation [24].
Entangled frequency states can also be generated in combination with
entanglement in other DOFs like polarization, path [25] or even temporal,
whenever the relevant time and frequency properties can be manipulated
independently, because they refer to vastly different time scales. Such
hyperentangled states can also be used to enlarge the dimensionality of the
system, to generate cluster states [26] or to perform more advanced quantum
gates [27]. Yet, all these approaches are only able to manipulate a small
frequency mode set of typically extensive underlying quantum states.
Certifying real high-dimensional entanglement is not trivial, particularly in
the frequency domain, and its immense potential is still unexploited.
In this Letter, we show that it is not always necessary to design quantum
sources with a discretized frequency space, such as those built in cavities,
and therefore continuous spectra can also provide access to massive-
dimensional and well-controlled Hilbert space. We show quantum interference
with up to 7 modes and $>98\%$ visibility and demonstrate complete state
control in a 7-mode state space. Subsequently, based on previous work to
characterize time-bin qudits [28, 29], we certify genuine high-dimensional
frequency entanglement without prior assumptions regarding, e.g., the purity
of the quantum state. We can further alleviate the harsh requirements on the
number of measurements necessary to characterize the relevant correlations,
which grows with dimensionality, by taking into account energy-conservation.
This step is not necessary for our certification method, but it allows us
exploit a novel bucket detection approach that requires very few measurement
settings, in a similar fashion to the compressed measurements used to
characterize spatial correlations [30, 31]. Finally, by recovering the
information from high-quality quantum interference of two-dimensional (2D)
subspaces, we are able to certify a minimum of 33 entangled frequency modes,
the highest dimensionality of entanglement reported in time and frequency
degrees of freedom.
## II Results and Discussion
### II.1 Frequency comb generation
In our work, we analyze the frequency content of two-photon states generated
in standard $\chi^{(2)}$ non-linear crystals: periodically poled lithium
niobate (ppLN) waveguides (see Experimental Section). They provide continuous
and broadband spontaneous parametric down-converted photons (SPDC) with high
efficiency and 60 nm of bandwidth. The SPDC process is temperature tuned to a
degeneracy wavelength of 1548 nm to cover the entire C band with high
uniformity (see inset in Fig. 1).
At this point, the frequency space of the generated photon pairs would
typically be discretized, e.g., with etalon cavities to carve the spectrum
[32, 33]. This approach is useful for the isolation of frequency modes and for
performing sideband modulation. However, a considerable contribution of the
photon spectrum is directly rejected, reducing the total throughput. For this
reason, we avoid this step and employ the maximum bandwidth per frequency
mode.
Due to the energy conservation of the SPDC process, the emitted photon pairs
are strongly anticorrelated in frequency, and the state can be described as:
$|j\rangle_{s}|j\rangle_{i}=\int\Pi(\Omega-j\Delta\omega,\Omega+j\Delta\omega)|\omega_{0}+\Omega\rangle_{s}|\omega_{0}-\Omega\rangle_{i}\,d\Omega$
(1)
where $|j\rangle_{s,i}$ is the label of the $j^{\mathrm{th}}$ frequency mode
corresponding to the signal and the idler photon. $\omega_{0}$ is the
degeneracy frequency of the SPDC process, $\Delta\omega$ is the FSR between
modes and $\Pi$ is the spectral shape of each mode. We discretize the system
in bins of 25 GHz bandwidth and the same FSR over the whole C-band, yielding
the state:
$|\psi_{d}\rangle=\sum_{j=1}^{d}\alpha_{j}|j\rangle_{s}|j\rangle_{i}$ (2)
The term $\alpha_{j}$ refers to the phase and amplitude of the mode and is
determined by the spectral characteristics of the source. For a very broad and
uniform spectrum as here, $\alpha_{j}\approx 1$. Although after propagation,
each photon pair corresponding to mode $j$ accumulate different phases due to
material dispersion.
In a quantum key distribution (QKD) scenario, the frequency-entangled photon
pairs emerging from a single optical fiber can be distributed into different
paths, e.g., via wavelength- or polarization demultiplexing. In our
experiment, we use only one device simultaneously for both photons, but in
principle, manipulation would be just as easily possible at two separate
locations. To reveal the entanglement content, we use a commercial pulse
shaper to control the phase and amplitude of each of the frequency modes [16],
and electro-optic modulation to achieve mode superpositions [13] (see
Experimental Section).
### II.2 The superposition basis
For either quantum state characterization via full state tomography (FST),
evaluation of Bell-type tests, or implementation of QKD protocols,
measurements in superposition bases are fundamental to uncover quantum
correlations and statistics of the state. The eigenvectors of these bases may
be the superposition of some or all elements of the computational basis, here
the frequency basis, with certain phases for each mode. Here, we show that
with standard levels of RF signal amplification ($P_{\mathrm{max}}=26$ dBm),
it is possible to perform a full superposition of up to 7 modes with a low
contribution of accidental coincidence detection events. To prove this, we
performed high-dimensional Bell-type tests, also known as CGLMP [34], based on
the CHSH inequality for two-photon qubits [35]. Instead of measuring quantum
correlations only for fixed phase settings, we performed a phase scan for all
contributing modes [36]. The measurement projector we use for each photon is:
$|\Psi_{proj}\rangle=\frac{1}{\sqrt{d}}\sum_{j=1}^{d}\left(e^{ij\theta_{s,i}}|j\rangle_{s,i}\right)$
(3)
where $\theta_{s,i}$ is the phase applied to the signal or the idler photon,
and we use $\theta_{s}=\theta_{i}$. The phase of interference depends on the
sum of the signal and idler phases. By scanning their phase, we obtain the
quantum interferences shown in Fig. 2 for dimensions $d=2,3,5,7$. The
visibilities are 96.7%, 97.7%, 98.1% and 98.2%, respectively, without fitting
or subtraction of accidental coincidences and much above the thresholds 70.7%,
77.5%, 84.6% and 88.3% to rule out hidden variable theories [36]. To perform
these measurements, we have selected states centered on the $6^{\mathrm{th}}$
mode. Electro-optic modulation shifted photons from the neighboring modes into
the $6^{\mathrm{th}}$ mode, and demultiplexing filters (DEMUX) postselected
the superposition state. Our constrained RF signal only allowed a limited
efficiency of the frequency modulation of photons; thus, added mode loss by
the pulse shaper provided an equal contribution. Even dimensionality can also
be evaluated by using the same parameters as here.
Figure 2: Bell-type tests in the frequency domain for dimensions $d=2,3,5,7$.
We chose the modes centered on $|6\rangle$ for the signal and the idler
photons.
### II.3 Higher dimensionality certification
While these CGLMP tests provide a (partially) device-independent certification
of entanglement and demonstrate the reliability of our devices, they do not
easily test for actual entanglement dimensionality, i.e. the dimension of
entanglement needed to reproduce the correlations. Generically, to demonstrate
even higher levels of entanglement, one would need to project distant spectral
modes into superposition states, which may be limited or physically impossible
to perform. In our large frequency space and current setup, this would imply
unreachable RF power levels for electro-optic modulation. Although similar
practical limitations are to be expected for any type of encoding, there exist
other modulation methods that can provide frequency shifts of several
nanometers [37, 38]. On the other hand, common entanglement certification
techniques, such as FST, are expensive procedures that require measurements in
at least $(d+1)^{2}$ bases for bipartite systems, an equivalent of
$d^{2}(d+1)^{2}$ single-outcome measurement settings. The obvious consequence
is that the number of total measurements increases rapidly with dimension.
Adapting the techniques developed for the time-bin domain, we show that it is
sufficient to characterize the quantum coherence of a few 2D subspaces to
certify a high dimensionality of entanglement in the frequency domain.
The certification process is structured into two main blocks: the measurement
of some elements of the density matrix $\rho$ and the posterior lower bound of
the remaining unknowns. Measurements on the computational basis, that is, the
frequency basis, can be performed with standard filters and are, in fact,
directly related to the diagonal elements of the density matrix $\langle
j,k|\rho|j,k\rangle$. Only this characterization step would usually require
$d^{2}=10404$ filter settings, which can take arbitrary long times. Here, we
propose making good use of the frequency parallelization and bucket detection
of all uncorrelated frequencies with a single and broadband filter setting
$\sum_{k\neq j}\langle j,k|\rho|j,k\rangle$ (i.e. 101 modes $\times$ 25 GHz),
while still measuring the correlated set $|j,j\rangle$ with narrowband filter
settings (25 GHz). This method reduces the high number of measurements
required to only $2d$. The results of the maximum frequency correlation are
shown in Fig. 3a. The background noise measured for uncorrelated frequencies
originates from accidental coincidence detection events due to the high number
of single counts and imperfect filters. The detected coincidence to accidental
ratio (CAR) averaged throughout the space amounts to $1.4\times 10^{3}$.
Figure 3: (a) Intensity probabilities and (b) visibilities of the two-photon
interferences for the 2D subspaces of modes $|j,j+1\rangle$, $|j,j+2\rangle$
and $|j,j+6\rangle$. (c) Comparison of measured data with the expected lower
bound with our method. Note that, from all possible values of elements that
produce a positive density matrix $\rho$, we take the worst scenario.
Outside of the diagonal of $\rho$ we find two types of elements: those close
to zero due to energy conservation, as observed from the computational basis
measurements and upper limited by the accidentals in our system, and those
non-zero elements $\langle j,j|\rho|k,k\rangle$ that indicate the coherence of
the entangled state. We now measure some of the coherence elements to which we
have access with our measurement system. They can be estimated with the mode
amplitude from the computational basis and the strength of neighboring
interference. That is, we only need to measure the quantum interference
between any two modes of the whole system. We measure the interferences for
the first, second and sixth neighboring modes corresponding to the terms
$\langle j,j|\rho|j+i,j+i\rangle$ for $i=1,2,6$. The dispersion shifts the
interference patterns proportional to the frequency distance with respect to
the center wavelength [33]. Knowledge of the exact dispersion values would
allow us to directly measure the maxima and minima of the interference
patterns. For an unknown exact value as in our case, we perform a dispersion
calibration, a full phase scan of the interference for different frequency
modes. Finally, to collect statistics, we record 60 samples of the maxima and
minima of the expected interference for each subspace and calculate the
visibility. The total number of filter settings (not considering the
dispersion calibration) are now $2(d-1)+2(d-2)+2(d-6)=6(d-3)$ and the results
of all recorded visibilities are presented in Fig. 3b. The average visibility
for the 102 modes with 1st, 2nd, and 6th neighbors are, respectively,
96.85(7)%, 97.94(5)% and 96.8(1)%.
The fact that these high-contrast interference is preserved over the whole
investigated space indicates that the quantum state at hand is very close to a
maximally entangled state. However, in potentially adversarial scenarios, such
as QKD, we do not want to make any assumption on the distributed state. Thus,
we proceed with a rigorous analysis to demonstrate entanglement. Indeed, these
strong coherences allow us to finally certify a large amount of entanglement
without further measurements. Similarly to the methods proposed for
entanglement certification in the time domain [29, 28], we lower bound the
remaining unknown elements $\langle j,j|\rho|j+i,j+i\rangle$ (see Fig. 4a) by
using the fact that the density matrix must be positive semidefinite to
represent a valid quantum state. Thus, every element, submatrix, and therefore
subdeterminant of $\rho$ must be positive or equal to zero at the very least.
The magnitude of unknown density matrix elements can thus be lower-bounded
iteratively by solving $3\times 3$ subdeterminants, composed of measured or
estimated parameters and one unknown, and we keep the largest bound extracted
from all combinations of submatrices. The resulting submatrix $\langle
j,j|\rho|k,k\rangle$ is shown in Fig. 4b. Notice that this method causes a
rather fast loss of information, many of the elements are still lower-bounded
to zero, yet it is sufficient to certify high-dimensional entanglement. To
visualize this loss, we display in Fig. 3c the measured quantities for
$\langle j,j|\rho|j+2,j+2\rangle$ and the calculated bound if we would not use
those measurements.
Figure 4: (a) Simplified representation of the density matrix, considering
only the elements $\langle j,j|\rho|k,k\rangle$. While some visibilities
$V_{j}$ can be measured, the remaining unknowns in red are lower-bounded with
our method. (b) Lower bound of the density submatrix $\langle
j,j|\rho|k,k\rangle$ and (c) minimum certified dimensionality of our system,
according to the measured data. Notice that for low space dimension up to
$d=11$, we can certify the maximum dimensionality with very few measurement
settings.
We can now compare the lower-bounded density matrix with a target state
$|\Phi\rangle$ by computing the fidelity
$F(\rho,\Phi)=\mathrm{Tr}\left(\sqrt{\sqrt{\Phi}\rho\sqrt{\Phi}}\right)^{2}$.
There exists an upper bound for the fidelity of any state of Schmidt rank
$k\leq d$ [39, 40]. Fidelities with the maximally entangled state above the
threshold $B_{k}(\Phi)=k/d$ indicate a dimensionality of at least $k+1$. We
thus choose a maximally entangled state as the target state
$|\Phi\rangle=1/\sqrt{d}\sum_{j=1}^{d}|j,j\rangle$, iteratively calculate the
fidelity for $d=2$ to $d=102$ and compare it with the threshold values for
varying Schmidt rank. The final certification is plotted in Fig. 4c, where we
also show the dependence on the amount of measurement data. Certainly,
collecting the coherence information from more distant modes helps to improve
the certification. By using all the measured data, we find at least 33
entangled modes in a space of 101 to 102. To estimate the error of the
certification, we perform Monte Carlo simulations with our data error. On
average, the statistical method yields an even better certification, with 34.8
+- 0.7 entangled modes in a space of 102. It is worth emphasizing that with
only these very few measurement settings, $2d$ on the computational basis and
$\sim 6d$ on 2D subspaces, we are able to certify 11 dimensions in a space of
11 modes. Higher visibility would directly increase the amount of certified
entanglement per number of modes, and further measurements that could fill
more elements of the density matrix would also improve the certification.
## III Conclusion
The time-frequency domain straightforwardly provides entanglement of very high
dimensionality simply owing to energy conservation in spontaneous parametric
nonlinear processes. However, the characterization of this entanglement is a
key challenge. To date, high-dimensional frequency entanglement has never been
completely certified beyond a few dimensions. In our work, we have shown
methods on how to characterize these quantum states that unquestionably
possess a huge dimensionality. Similarly as in the time domain, the limits on
the dimensionality for continuously pumped processes depend on the resolution
of our devices, here the spectral filters.
In this work, we have shown full superpositions up to $d=7$ and great
interference visibilities over the full scanned spectrum. The used subspaces
are easily exploitable with few fiber-integrated optical components and can be
implemented nonlocally in quantum communication tasks. Due to physical
measurement constrains, limited amount of information is accessible from these
large states. Nevertheless, we have finally certified the full 11-dimensional
entanglement in a subspace of 11 modes with only $\sim 8d$ measurement
settings, and at least 33-dimensional entanglement in our frequency space of
102 frequency modes.
These results hold great promise for further enhancing the information
capacity of quantum information processing. Entanglement in the frequency
domain can also be used in combination with entanglement in time, path and
polarization DOFs, leading to huge state spaces. The main building blocks and
techniques can be readily combined with established spatial and temporal mode
approaches and thus provide the near-term prospect of more efficient hybrid
time-space photonic information processors.
We hope that these results on the quantification of quantum coherence in a
massive-dimensional frequency space will motivate the community to use the
frequency DOF beyond wavelength multiplexing schemes. Moreover, the further
development of photonic technology, in particular low-loss electro-optic
modulation and wave-shaping technologies, could render frequency encoding a
viable contender for near-term photonic quantum information processing with
massive bandwidth.
## IV Experimental section
To generate the frequency-entangled state, a commercial second harmonic
generation (SHG) module was used, a 40 mm type-0 ppLN waveguide (Covesion). To
pump the nonlinear process, a standard continuous-wave telecom laser was
upconverted with a second SHG module to better align the SPDC wavelength to
ITU channels. To process the frequency entanglement, a telecom pulse shaper
(Waveshaper 16000A) was used to select specific frequency subspaces for the
signal and the idler photons and to tune their relative phases. This device
limits the operational frequency range of the source to 40 nm of the telecom
C-band only. This space was then divided into 102 frequency modes for the
signal and the idler photon, with a standard free spectral range (FSR) of 25
GHz and the same bandwidth. This leads to a state space of $102\times 102$
dimensions. Note that subspace selection was an actual frequency
discretization procedure, and it could be employed as a resource for flexible
bandwidth allocation in reconfigurable QKD networks. The photon frequencies
were then modulated with an electro-optic modulator, driven by a radio-
frequency sine with the same frequency as the FSR. This sideband modulation
technique allowed us to distribute photons into neighboring modes according to
Bessel function amplitudes [41]. Choosing a low FSR and bandwidth allowed
slower modulation signals than in earlier work, where FSR values of 200 GHz
[16] down to 50 GHz [42] were used. Lower FSR would be better suited to
increase dimensionality, but would eventually be limited by the resolution of
the available optical filters and the number of photons per fraction of
spectral bandwidth. Lastly, due to photon scattering into distant spectral
modes, postselection was necessary to measure the right superposition states.
To reduce the noise that imperfect filters may introduce, a tunable and narrow
DEMUX of 22 GHz bandwidth centered at the corresponding signal and idler
frequencies was used prior to coincidence measurement. The colorful
illustrations at the bottom of Figure 1 depict the sideband modulation and
photon scattering.
## Funding
This work was supported by the Fraunhofer Internal Programs under Grant no.
Attract 066-604178, by the joint program of the Natural Sciences and
Engineering Research Council of Canada (NSERC) and the European Comission
under Grant no. HyperSpace 101070168, and by the Federal Ministry of Education
and Research of Germany (BMBF) through the project Quantum Photonics Labs
(QPL). The presented results were partially acquired using facilities and
devices funded by the Free State of Thuringia within the application center
Quantum Engineering. M.C.P is part of the Max Planck School of Photonics
supported by BMBF, Max Planck Society, and Fraunhofer Society.
## References
* [1] P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Dowling, G. J. Milburn, _Reviews of Modern Physics_ 2007, _79_ , 1 135.
* [2] J.-W. Pan, Z.-B. Chen, C.-Y. Lu, H. Weinfurter, A. Zeilinger, M. Żukowski, _Reviews of Modern Physics_ 2012, _84_ , 2 777.
* [3] A. Anwar, C. Perumangatt, F. Steinlechner, T. Jennewein, A. Ling, _Review of Scientific Instruments_ 2021, _92_ , 4 041101\.
* [4] S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, M. Huber, _Physical Review X_ 2019, _9_ , 4 041042.
* [5] B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C. Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, A. G. White, _Nature Physics_ 2009, _5_ , 2 134.
* [6] C. Vigliar, S. Paesani, Y. Ding, J. C. Adcock, J. Wang, S. Morley-Short, D. Bacco, L. K. Oxenløwe, M. G. Thompson, J. G. Rarity, A. Laing, _Nature Physics_ 2021, _17_ , 10 1137.
* [7] A. Mair, A. Vaziri, G. Weihs, A. Zeilinger, _Nature_ 2001, _412_ , 6844 313.
* [8] M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, A. Zeilinger, _Proceedings of the National Academy of Sciences_ 2014, _111_ , 17 6243.
* [9] N. K. Fontaine, R. Ryf, H. Chen, D. T. Neilson, K. Kim, J. Carpenter, _Nature Communications_ 2019, _10_ , 1 1865.
* [10] J. D. Franson, _Physical Review Letters_ 1989, _62_ , 19 2205.
* [11] J. Brendel, N. Gisin, W. Tittel, H. Zbinden, _Physical Review Letters_ 1999, _82_ , 12 2594.
* [12] D. Richart, Y. Fischer, H. Weinfurter, _Applied Physics B_ 2012, _106_ , 3 543.
* [13] L. Olislager, I. Mbodji, E. Woodhead, J. Cussey, L. Furfaro, P. Emplit, S. Massar, K. P. Huy, J.-M. Merolla, _New Journal of Physics_ 2012, _14_ , 4 043015.
* [14] A. Pe’er, B. Dayan, A. A. Friesem, Y. Silberberg, _Physical Review Letters_ 2005, _94_ , 7 073601.
* [15] C. Bernhard, B. Bessire, T. Feurer, A. Stefanov, _Physical Review A_ 2013, _88_ , 3 032322.
* [16] M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, R. Morandotti, _Nature_ 2017, _546_ , 7660 622.
* [17] H.-H. Lu, K. V. Myilswamy, R. S. Bennink, S. Seshadri, M. S. Alshaykh, J. Liu, T. J. Kippenberg, D. E. Leaird, A. M. Weiner, J. M. Lukens, _Nature Communications_ 2022, _13_ , 1 4338.
* [18] H.-H. Lu, E. M. Simmerman, P. Lougovski, A. M. Weiner, J. M. Lukens, _Physical Review Letters_ 2020, _125_ , 12 120503.
* [19] H.-H. Lu, J. M. Lukens, N. A. Peters, B. P. Williams, A. M. Weiner, P. Lougovski, _Optica_ 2018, _5_ , 11 1455.
* [20] H.-H. Lu, J. M. Lukens, B. P. Williams, P. Imany, N. A. Peters, A. M. Weiner, P. Lougovski, _npj Quantum Information_ 2019, _5_ , 1 24.
* [21] C. Cui, K. P. Seshadreesan, S. Guha, L. Fan, _Physical Review Letters_ 2020, _124_ , 19 190502.
* [22] Y. Chen, S. Ecker, J. Bavaresco, T. Scheidl, L. Chen, F. Steinlechner, M. Huber, R. Ursin, _Physical Review A_ 2020, _101_ , 3 032302.
* [23] N. B. Lingaraju, H.-H. Lu, S. Seshadri, P. Imany, D. E. Leaird, J. M. Lukens, A. M. Weiner, _Optics Express_ 2019, _27_ , 26 38683.
* [24] Y. Chen, S. Ecker, L. Chen, F. Steinlechner, M. Huber, R. Ursin, _npj Quantum Information_ 2021, _7_ , 1 1.
* [25] L. Zhang, Cui, Chaohan, Yan, Jianchang, Guo, Yanan, Wang, Junxi, Fan, Linran, On-chip parallel processing of quantum frequency combs for high-dimensional hyper-entanglement generation, Technical Report arXiv:2111.12784, 2021.
* [26] C. Reimer, S. Sciara, P. Roztocki, M. Islam, L. Romero Cortés, Y. Zhang, B. Fischer, S. Loranger, R. Kashyap, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, W. J. Munro, J. Azaña, M. Kues, R. Morandotti, _Nature Physics_ 2019, _15_ , 2 148.
* [27] P. Imany, J. A. Jaramillo-Villegas, M. S. Alshaykh, J. M. Lukens, O. D. Odele, A. J. Moore, D. E. Leaird, M. Qi, A. M. Weiner, _npj Quantum Information_ 2019, _5_ , 1 59.
* [28] A. Martin, T. Guerreiro, A. Tiranov, S. Designolle, F. Fröwis, N. Brunner, M. Huber, N. Gisin, _PHYSICAL REVIEW LETTERS_ 2017, 5.
* [29] A. Tiranov, S. Designolle, E. Z. Cruzeiro, J. Lavoie, N. Brunner, M. Afzelius, M. Huber, N. Gisin, _PHYSICAL REVIEW A_ 2017, 6.
* [30] J. Schneeloch, G. A. Howland, _Physical Review A_ 2018, _97_ , 4 042338.
* [31] J. Schneeloch, C. C. Tison, M. L. Fanto, P. M. Alsing, G. A. Howland, _Nature Communications_ 2019, _10_ , 1 2785.
* [32] Z. Xie, T. Zhong, S. Shrestha, X. Xu, J. Liang, Y.-X. Gong, J. C. Bienfang, A. Restelli, J. H. Shapiro, F. N. C. Wong, C. Wei Wong, _Nature Photonics_ 2015, _9_ , 8 536.
* [33] P. Imany, O. D. Odele, J. A. Jaramillo-Villegas, D. E. Leaird, A. M. Weiner, _Physical Review A_ 2018, _97_ , 1 013813.
* [34] D. Collins, N. Gisin, N. Linden, S. Massar, S. Popescu, _Physical Review Letters_ 2002, _88_ , 4 040404.
* [35] J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt, _Physical Review Letters_ 1969, _23_ , 15 880.
* [36] R. T. Thew, A. Acín, H. Zbinden, N. Gisin, _Physical Review Letters_ 2004, _93_ , 1 010503.
* [37] C. Joshi, A. Farsi, S. Clemmen, S. Ramelow, A. L. Gaeta, _Nature Communications_ 2018, _9_ , 1 847.
* [38] D. Zhu, C. Chen, M. Yu, L. Shao, Y. Hu, C. J. Xin, M. Yeh, S. Ghosh, L. He, C. Reimer, N. Sinclair, F. N. C. Wong, M. Zhang, M. Lončar, _Light: Science & Applications_ 2022, _11_ , 1 327\.
* [39] R. Fickler, R. Lapkiewicz, M. Huber, M. P. Lavery, M. J. Padgett, A. Zeilinger, _Nature Communications_ 2014, _5_ , 1 4502.
* [40] J. Bavaresco, N. Herrera Valencia, C. Klöckl, M. Pivoluska, P. Erker, N. Friis, M. Malik, M. Huber, _Nature Physics_ 2018, _14_ , 10 1032.
* [41] J. Capmany, C. R. Fernández-Pousa, _JOSA B_ 2010, _27_ , 6 A119.
* [42] P. Imany, J. A. Jaramillo-Villegas, O. D. Odele, K. Han, D. E. Leaird, J. M. Lukens, P. Lougovski, M. Qi, A. M. Weiner, _Optics Express_ 2018, _26_ , 2 1825.
|
# FRETting about Requirements: Formalised Requirements for an Aircraft Engine
Controller††thanks: The authors thank Georgios Giantamidis, Stylianos
Basagiannis, and Vassilios A. Tsachouridis (UTRC, Ireland) for their help in
requirements elicitation; and Anastasia Mavridou (NASA Ames Research Center,
USA) for her help with FRET. This research was funded by the European Union’s
Horizon 2020 research and innovation programme under the VALU3S project (grant
No 876852), and by Enterprise Ireland (grant No IR20200054). The funders had
no role in study design, data collection and analysis, decision to publish, or
preparation of the manuscript.
Marie Farrell Matt Luckcuck Oisín Sheridan Rosemary Monahan
(Department of Computer Science, Maynooth University, Ireland
<EMAIL_ADDRESS>
###### Abstract
[Context & motivation] Eliciting requirements that are detailed and logical
enough to be amenable to formal verification is a difficult task. Multiple
tools exist for requirements elicitation and some of these also support
formalisation of requirements in a way that is useful for formal methods.
[Question/problem] This paper reports on our experience of using the Formal
Requirements Elicitation Tool (FRET) alongside our industrial partner. The use
case that we investigate is an aircraft engine controller. In this context, we
evaluate the use of FRET to bridge the communication gap between formal
methods experts and aerospace industry specialists. [Principal ideas/results]
We describe our journey from ambiguous, natural-language requirements to
concise, formalised FRET requirements. We include our analysis of the
formalised requirements from the perspective of patterns, translation into
other formal methods and the relationship between parent-child requirements in
this set. We also provide insight into lessons learned throughout this process
and identify future improvements to FRET. [Contribution] Previous experience
reports have been published by the FRET team, but this is the first such
report of an industrial use case that was written by researchers that have not
been involved FRET’s development.
## 1 Introduction
Formal verification uses mathematically-based techniques to guarantee that a
system obeys certain properties, which is particularly useful when developing
safety-critical systems like those used in the aerospace domain. Developing a
correct set of requirements necessitates discussion with people who have
expertise in the system under development, who may not have skills in formal
methods. In which case, it can be beneficial to the requirements elicitation
process to write the requirements in an intermediate language. Tools like
NASA’s Formal Requirements Elicitation Tool (FRET) provide a gateway for
developing formal requirements with developers who are not familiar with
formal languages [7].
In this paper, we examine how FRET can be used in an industrial case study of
an aircraft engine controller that has been supplied by our industrial
partner, United Technologies Research Center (Ireland). FRET has previously
been used to formalise the requirements for the 10 Lockheed Martin Cyber-
Physical Challenges [11]. However, to the best of our knowledge this paper
provides the first experience report on FRET’s use on an industrial case
study, by a team not involved in FRET’s development.
Our approach provides external and internal traceability. Using a tool like
FRET to develop the requirements provides a link between natural-language
requirements and formally verified artefacts. FRET also enables the user to
describe a link between requirements at different levels of abstraction, which
means that this traceability is maintained within the developing set of
requirements. We also use FRET to collect information about the rationale
behind a requirement, further improving the traceability; either back to a
natural-language requirement, or forward to a more concrete requirement. These
traceability features encourage better explainability of a requirement’s
source, and the intermediate language improves the explainability of the
requirements themselves.
The rest of the paper is laid out as follows. §2 outlines the relevant
background material pertaining to FRET and the aircraft engine controller use
case. Then, we describe our requirements elicitation process and present
detailed requirements in §3. These requirements are analysed in §4. We discuss
the lessons that were learned through this work in §5 and §6 concludes. We
also provide a detailed appendix showing the full set of requirements, test
cases, Simulink model and fretish requirements.
## 2 Background
This section provides an overview of FRET and the aircraft engine controller
use case for which we were developing requirements.
#### FRET:
is an open-source tool that enables developers to write and formalise system
requirements [7]. FRET accepts requirements written in a structured natural-
language called fretish, in which requirements take the form:
scope condition component shall timing response
The condition, component, and response fields are mandatory; scope and timing
are optional fields. This allows responses that are tied to a scope, are
triggered by conditions, relate to a system component, and may have timing
constraints.
The underlying semantics of a fretish requirement is determined by the scope,
condition, timing, and response fields. There is a template for each possible
combination of a requirement’s fields, currently FRET provides 160 such
templates [8]. The selected template is used to generate formalisations of the
associated requirement in both past- and future-time metric Linear-time
Temporal Logic (LTL). FRET displays a diagramatic semantics for each
requirement, which shows: the time interval where it should hold, and its
triggering and stopping conditions (if they exist). Both versions of the
requirements are helpful for sanity-checking what has been written in fretish.
The user must give each fretish requirement an ID, which can be used to create
a many-to-many, hierarchical link between requirements: a parent requirement
may have many child requirements, and one child may have many parents. While
this link facilitates traceability, FRET does not define this relationship
(formally or otherwise). For example, a child requirement does not inherit
definitions from its parent. We discuss possible improvements to this link in
§4.2. FRET also allows the user to enter ‘Rationale’ and ‘Comments’ for a
requirement, which further supports traceability and encourages explainability
of requirements.
FRET can automatically translate requirements into contracts for a Simulink
diagram, written in CoCoSpec, which are checked during Simulink simulations by
the CoCoSim tool, using the Kind2 model checker [4]. FRET can also generate
runtime monitors for the Copilot framework [6].
#### Aircraft Engine Controller:
Our use case is a software controller for a high-bypass civilian aircraft
turbofan engine, provided by our industrial partner on the VALU3S [2] project,
based on existing controller designs [13, 14]. It is an example of a Full
Authority Digital Engine Control (FADEC) system, which monitors and controls
everything about the engine,using input from a variety of sensors. The engine
itself contains two compressors (high-pressure and low-pressure) turning a
central spool, which drives the engine.
As described in our prior work [9], the controller’s high-level objectives are
to manage the engine thrust, regulate the compressor pressure and speeds, and
limit engine parameters to safe values. It should continue to operate, keeping
settling time, overshoot, and steady state errors within acceptable limits,
while respecting the engine’s operating limits (e.g. the spool’s speed limit),
in the presence of:
* •
sensor faults (a sensor value deviating too far from from its nominal value,
or being unavailable),
* •
perturbation of system parameters (a system parameter deviating too far from
from its nominal value), and
* •
other low-probability hazards (e.g. abrupt changes in outside air pressure).
The controller is also required to detect engine surge or stall and change
mode to prevent these hazardous situations.
Our industrial partner has supplied us with 14 English-language requirements
(Table 1) and 20 abstract test cases, which provide more detail about the
controller’s required behaviour. The naming convention for requirements is:
$<$use case id$>$_R_$<$parent requirement id$>$.$<$child requirement id$>$
For example, because this is Use Case 5 in the VALU3S project111The VALU3S
project: https://valu3s.eu/, requirement one is named UC5_R_1. Note, we use a
similar naming convention for test cases. Table 2 shows the abstract test
cases for UC5_R_1. Our industrial partner also designed the controller in
Simulink222Simulink: https://mathworks.com/products/simulink.html, shown in
the extended version of this paper333This Paper Extended Version:
https://arxiv.org/abs/2112.04251.
For our Use Case, we collaborated with scientists in the System Analysis and
Assurance division of an aerospace systems company. The hour-long requirements
elicitation meetings were held monthly, over a period of 10 months, with
additional meetings as needed. In these meetings, our collaborators reviewed
the fretish versions of their natural-language requirements, validating our
formalisation and clarifying ambiguities for us. Since our collaborators were
already familiar with other formal tools we were able to introduce them to
FRET quite quickly. However, we produced a training video for other members of
the project consortium444“Formalising Verifiable Requirements” Presentation:
https://www.youtube.com/watch?v=FQGKbYCbxPY&list=PLGtGM9euw6A66ceQbywXGjVoTKEhP-
Of7&index=9.
## 3 Our Requirements Elicitation Process Using FRET
In this section we describe our requirements elicitation process. We begin by
outlining how this fits into our larger approach to verification for the
aircraft engine controller use case. We then describe our journey from
natural-language requirements to formalised fretish requirements.
### 3.1 Requirements-Driven Methodology
As part of the three-phase verification methodology outlined in our prior work
[9], we used FRET to elicit and formalise requirements for the aircraft engine
controller. Focussing on Phase 1, this paper includes the full set of fretish
requirements and presents lessons learnt, which were not discussed in [9].
Fig. 1 shows a high-level flowchart of our methodology, with an exploded view
of the relationship between the artefacts involved in Phase 1. The methodology
takes the natural-language requirements (Table 1), test cases, and Simulink
diagram of the engine controller as input, and enables the formal verification
of the system’s design against the requirements.
Figure 1: Flowchart of our three-phase requirements-driven verification
methodology [9] (left) with an exploded view of Phase 1’s artefacts (right).
The solid lines and arrowheads show direct information flow between artefacts
(and back into Phase 2A and 2B), the dashed lines and open arrowheads show an
artefact being implemented by another.
Phase 1 of our methodology involves formalising natural-language requirements
using FRET, eliciting further detail as we progress. Phase 2 consists of two,
potentially parallel, phases. Phase 2A uses FRET’s built-in support for
generating CoCoSpec contracts that can be incorporated into a Simulink diagram
for verification with the Kind2 model-checker. In Phase 2B, the formalised
requirements drive a (manual) translation into other formal methods, as chosen
by the verifier. These tools typically require the construction of a formal
model of the system, which is verified against the translated requirements.
This step requires translation of the fretish requirements into the formalism
of the chosen verification tool. Finally, Phase 3 produces a report collecting
the verification results and other useful artefacts, such as formal models,
Simulink diagrams, various versions of the requirements, counter-examples,
proofs, etc. This supports tracing the requirements through the system’s
development lifecycle.
The following subsections describe the requirements elicitation process (Phase
1) in more detail. Figure 1’s exploded view, shows how natural-language
requirements are translated into fretish parent requirements (solid arrow),
and the test cases are translated into child requirements. Since we view the
test cases as implementations of the natural-language requirements (dashed
arrow), the child requirements are similarly viewed as implementations of
their corresponding parent requirements. The left-hand side of Fig. 1 shows
how the work in this paper fits within our development and verification
methodology; the solid arrows from the fretish parent and child requirements,
to Phases 2A and 2B, show how the output of this work is consumed by the next
phase.
ID | Description
---|---
UC5_R_1 | Under sensor faults, while tracking pilot commands, control objectives shall be satisfied (e.g., settling time, overshoot, and steady state error will be within predefined, acceptable limits)
UC5_R_2 | Under sensor faults, during regulation of nominal system operation (no change in pilot input), control objectives shall be satisfied (e.g., settling time, overshoot, and steady state error will be within predefined, acceptable limits)
UC5_R_3 | Under sensor faults, while tracking pilot commands, operating limit objectives shall be satisfied (e.g., respecting upper limit in shaft speed)
UC5_R_4 | Under sensor faults, during regulation of nominal system operation (no change in pilot input), operating limit objectives shall be satisfied (e.g., respecting upper limit in shaft speed)
UC5_R_5 | Under mechanical fatigue conditions, while tracking pilot commands, control objectives shall be satisfied (e.g., settling time, overshoot, and steady state error will be within predefined, acceptable limits)
UC5_R_6 | Under mechanical fatigue conditions, during regulation of nominal system operation (no change in pilot input), control objectives shall be satisfied (e.g., settling time, overshoot, and steady state error will be within predefined, acceptable limits)
UC5_R_7 | Under mechanical fatigue conditions, while tracking pilot commands, operating limit objectives shall be satisfied (e.g., respecting upper limit in shaft speed)
UC5_R_8 | Under mechanical fatigue conditions, during regulation of nominal system operation (no change in pilot input), operating limit objectives shall be satisfied (e.g., respecting upper limit in shaft speed)
UC5_R_9 | Under low probability hazardous events, while tracking pilot commands, control objectives shall be satisfied (e.g., settling time, overshoot, and steady state error will be within predefined, acceptable limits)
UC5_R_10 | Under low probability hazardous events, during regulation of nominal system operation (no change in pilot input), control objectives shall be satisfied (e.g., settling time, overshoot, and steady state error will be within predefined, acceptable limits)
UC5_R_11 | Under low probability hazardous events, while tracking pilot commands, operating limit objectives shall be satisfied (e.g., respecting upper limit in shaft speed)
UC5_R_12 | Under low probability hazardous events, during regulation of nominal system operation (no change in pilot input), operating limit objectives shall be satisfied (e.g., respecting upper limit in shaft speed)
UC5_R_13 | While tracking pilot commands, controller operating mode shall appropriately switch between nominal and surge / stall prevention operating state
UC5_R_14 | During regulation of nominal system operation (no change in pilot input), controller operating mode shall appropriately switch between nominal and surge / stall prevention operating state
Table 1: Natural-language requirements for the aircraft engine controller as produced by the aerospace use case in the VALU3S project. These 14 requirements are mainly concerned with continued operation of the controller in the presence of sensor faults (UC5_R_1–UC5_R_4), perturbation of system parameters (UC5_R_5–UC5_R_8) and low probability hazards (UC5_R_9–UC5_R_12). There are also requirements for switching between modes if engine surge/stall is detected (UC5_R_13–UC5_R_14). Test Case ID | Requirement ID | Description
---|---|---
UC5_TC_1 | UC5_R_1 | Preconditions: Aircraft is in operating mode M and sensor S value deviates at most +/- R % from nominal value
Input conditions/steps: Observed aircraft thrust is at value V1 and pilot
input changes from A1 to A2
Expected results: Observed aircraft thrust changes and settles to value V2,
respecting control objectives (settling time, overshoot, steady state error)
UC5_TC_2 | UC5_R_1 | Preconditions: Aircraft is in operating mode M and sensor S value is not available (sensor is out of order)
Input conditions/steps: Observed aircraft thrust is at value V1 and pilot
input changes from A1 to A2
Expected results: Observed aircraft thrust changes and settles to value V2,
respecting control objectives (settling time, overshoot, steady state error)
Table 2: Abstract test cases corresponding to requirement UC5_R_1. Each
specifies the preconditions for the test case, the input conditions/steps and
the expected results.
### 3.2 Speaking FRETISH: Parent Requirements
ID | FRETISH
---|---
UC5_R_1 | if ((sensorfaults) & (trackingPilotCommands)) Controller shall satisfy (controlObjectives)
UC5_R_2 | if ((sensorfaults) & (!trackingPilotCommands)) Controller shall satisfy (controlObjectives)
UC5_R_3 | if ((sensorfaults) & (trackingPilotCommands)) Controller shall satisfy (operatingLimitObjectives)
UC5_R_4 | if ((sensorfaults) & (!trackingPilotCommands)) Controller shall satisfy (operatingLimitObjectives)
UC5_R_5 | if ((mechanicalFatigue) & (trackingPilotCommands)) Controller shall satisfy (controlObjectives)
UC5_R_6 | if ((mechanicalFatigue) & (!trackingPilotCommands)) Controller shall satisfy (controlObjectives)
UC5_R_7 | if ((mechanicalFatigue) & (trackingPilotCommands)) Controller shall satisfy (operatingLimitObjectives)
UC5_R_8 | if ((mechanicalFatigue) & (!trackingPilotCommands)) Controller shall satisfy (operatingLimitObjectives)
UC5_R_9 | if ((lowProbabilityHazardousEvents) & (trackingPilotCommands)) Controller shall satisfy (controlObjectives)
UC5_R_10 | if ((lowProbabilityHazardousEvents) & (!trackingPilotCommands)) Controller shall satisfy (controlObjectives)
UC5_R_11 | if ((lowProbabilityHazardousEvents) & (trackingPilotCommands)) Controller shall satisfy (operatingLimitObjectives)
UC5_R_12 | if ((lowProbabilityHazardousEvents) & (!trackingPilotCommands)) Controller shall satisfy (operatingLimitObjectives)
UC5_R_13 | if (trackingPilotCommands) Controller shall satisfy (changeMode(nominal)) $|$ (changeMode(surgeStallPrevention))
UC5_R_14 | if (!trackingPilotCommands) Controller shall satisfy (changeMode(nominal)) $|$ (changeMode(surgeStallPrevention))
Table 3: fretish parent requirements corresponding to the natural-language
requirements outlined in Table 1. The correspondance is clear to see and we
have used booleans to represent the ambiguous terms from the natural-language
requirements.
The inputs to our requirements elicitation process were the Simulink diagram,
14 natural-language requirements (Table 1), and 20 abstract test cases that
were supplied by our industrial partner. We elicited further information about
the requirements through regular team discussions with our industrial partner.
We started by translating the natural-language requirements into fretish,
producing the set of 14 fretish requirements in Table 3. The correspondence
between the fretish requirements and their natural-language counterparts is
clear. For example, requirement UC5_R_1 states that:
* Under sensor faults, while tracking pilot commands, control objectives shall be satisfied (e.g., settling time, overshoot, and steady state error will be within predefined, acceptable limits).
This became the corresponding fretish requirement:
* if ((sensorfaults) & (trackingPilotCommands)) Controller shall
satisfy (controlObjectives)
Producing this initial set of requirements enabled us to identify the
ambiguous parts of the natural-language requirements. For example, the phrase
“sensor faults” simply becomes a boolean in our fretish requirements,
highlighting that we need to elicit more details. We captured these additional
details as child requirements, as described in §3.3
### 3.3 Adding Detail: Child Requirements
Once the fretish parent requirements (Table 3) were complete, we added more
detail to make the requirements set more concrete. We paid particular
attention to ambiguous phrases translated from the natural-language
requirements. These extra details were drawn from the abstract test cases and
from detailed discussions with our industrial collaborators, who clarified
specific ambiguities.
We captured the extra details in 28 child requirements. As mentioned in §2, a
child requirement does not inherit definitions from its parent(s). However, we
use this hierarchical link to group the detail in the child requirements under
a common parent, which enables the detailed child requirements to be traced
back to the more abstract parent requirements.
For example, UC5_R_1 was distilled into three requirements (UC5_R_1.1,
UC5_R_1.2 and UC5_R_1.3), shown in Table 4. These three child requirements
each have the same condition and component, but differ in their responses.
Each child requirement specifies one of the “control objectives” (settling
time, overshoot and steady state error) mentioned in the natural-language
version of UC5_R_1. During elicitation discussions, it was revealed that these
were actually the only control objectives that were of concern for this use
case. Here, using FRET encouraged us to question exactly what the phrase
“control objectives” meant.
Each of these requirements includes the condition when (diff(r(i),y(i)) $>$ E)
and the timing constraint until (diff(r(i),y(i)) $<$ e), which were initially
overlooked in the natural-language requirements but revealed during
elicitation discussions with our industrial partner. The response must hold
when the difference between the reference sensor value, $r(i)$, and the
observed sensor value, $y(i)$, falls between specific bounds ($E$ and $e$).
This important detail was missing from the parent requirement but was
uncovered during our requirements elicitation.
The “Preconditions” of test cases UC5_TC_1 and UC5_TC_2 (Table 2) showed us
that the phrase “Under sensor faults” meant a period where a sensor value
deviates by ±R% from its nominal value or returns a null value. To represent
this, the child requirements use the function sensorValue(S) where S is a
parameter representing each of the 4 sensors for the engine controller. These
requirements are thus applied to all of the sensors in the model.
In UC5_TC_1 and UC5_TC_2, the “Input conditions/steps” refer to the aircraft
thrust and a change in the pilot’s input. We encoded this as the condition and
response pair (pilotInput $=>$ setThrust = V2) & (observedThrust = V1) and
satisfy (observedThrust = V2), where V1 and V2 are thrust variables and => is
logical implication. During elicitation discussions we found that this pair
corresponds to the condition, trackingPilotCommands. This was a particularly
important clarification because trackingPilotCommands models the phrase “while
tracking pilot commands”, which the natural-language requirements use
extensively. This underlines that it is possible for an ambiguous statement to
have a very precise meaning that was lost while drafting the requirements.
The thrust variables V1 and V2 in our fretish requirements correspond to
variables V1, V2, A1, and A2 in the test cases. During elicitation
discussions, we found that V1 and V2 alone were sufficient to capture the
requirement. V1 and A1 are used interchangeably as the initial thrust value,
which we label V1. Similarly, V2 and A2 refer to the updated thrust value,
which we label V2 for consistency. This is another ambiguity that our
translation from natural-language to fretish helped to clarify.
Our industrial partner checked the child requirements to ensure that there
were no errors or omissions. The intuitive meaning of fretish constructs
simplified this check, and features like the requirements’ diagramatic
semantics provided quick feedback when we edited the requirements during
elicitation discussions. The act of formalising the requirements helped us to
identify ambiguities in the requirements, prompting elicitation of further
detail from our industrial partner.
ID | Parent | FRETISH
---|---|---
UC5_R_1.1 | UC5_R_1 | when (diff(r(i),y(i)) $>$ E) if((sensorValue(S) $>$ nominalValue + R) $|$ (sensorValue(S) $<$ nominalValue - R) $|$ (sensorValue(S) = null) & (pilotInput $=>$ setThrust = V2) & (observedThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (settlingTime $>=$ 0) & (settlingTime $<=$ settlingTimeMax) & (observedThrust = V2)
UC5_R_1.2 | UC5_R_1 | when (diff(r(i),y(i)) $>$ E) if((sensorValue(S) $>$ nominalValue + R) $|$ (sensorValue(S) $<$ nominalValue - R) $|$ (sensorValue(S) = null)& (pilotInput $=>$ setThrust = V2) & (observedThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (overshoot $>=$ 0) & (overshoot $<=$ overshootMax) & (observedThrust = V2)
UC5_R_1.3 | UC5_R_1 | when (diff(r(i),y(i)) $>$ E) if((sensorValue(S) $>$ nominalValue + R) $|$ (sensorValue(S) $<$ nominalValue - R) $|$ (sensorValue(S) = null)& (pilotInput $=>$ setThrust = V2)& (observedThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (steadyStateError $>=$ 0) & (steadyStateError $<=$ steadyStateErrorMax) & (observedThrust = V2)
Table 4: We have three distinct child requirements for UC5_R_1 that capture
the correct behaviour with respect to each of settling time, overshoot and
steady state error.
## 4 An Analysis of Elicited Requirements
This section provides an analysis of the fretish requirements that we produced
for the aircraft engine controller use case. We note that the requirements
only refer to one component, the Controller, but this could be decomposed to
refer to specific blocks in the use case Simulink design.
### 4.1 Requirement Templates
Each of the 14 fretish parent requirements (Table 3) uses the same pattern:
condition component shall response. As described in §2, FRET maps each
requirement into a semantic template so that it can generate the associated
LTL specification. Our parent requirements all correspond to the template
[null, regular, eventually], which specifies the scope-option, condition-
option and timing-option, respectively (if the timing-option is omitted, then
eventually is the default). Specific details about templates in FRET are given
in [8].
We introduced until clauses into all of the 28 child requirements, although
with different timing constraints. The introduction of the until clauses was
identified through a combination of the information in the test cases and from
extensive discussions with our industrial parter. However, the specific timing
constraints required in-depth discussion with our industrial partner to
identify. Most of the child requirements correspond to the template [null,
regular, until]. However, some child requirements differed slightly as
outlined below.
ID | Parent | FRETISH
---|---|---
UC5_R_13.1 | UC5_R_13 | in nominal mode when (diff(setNL, observedNL) $>$ NLmax) if (pilotInput $=>$ surgeStallAvoidance) Controller shall until (diff(setNL, observedNL) $<$ NLmin) satisfy (changeMode(surgeStallPrevention))
UC5_R_13.2 | UC5_R_13 | in surgeStallPrevention mode when (diff(setNL, observedNL) $<$ NLmax) if (pilotInput $=>$ !surgeStallAvoidance) Controller shall until (diff(setNL, observedNL) $>$ NLmin) satisfy (changeMode(nominal))
UC5_R_14.1 | UC5_R_14 | in nominal mode when (diff(setNL, observedNL) $>$ NLmax) if (!pilotInput $=>$ surgeStallAvoidance) Controller shall until (diff(setNL, observedNL) $<$ NLmin) satisfy (changeMode(surgeStallPrevention))
UC5_R_14.2 | UC5_R_14 | in surgeStallPrevention mode when (diff(setNL, observedNL) $<$ NLmax) if (!pilotInput $=>$ !surgeStallAvoidance) Controller shall until (diff(setNL, observedNL) $>$ NLmin) satisfy (changeMode(nominal))
Table 5: Child requirements corresponding to UC5_R_13 and UC5_R_14. These
differ from the previous requirements because we use the scope field to assert
which mode of operation the controller is in.
UC5_R_13 and UC5_R_14 generated a lot of discussion, because they differ so
much from the other requirements; here, the system changes between modes of
operation, so we use the scope clause. This produced the child requirements
shown in Table 5. The when and until clauses differ from the other
requirements, because here the mode change is triggered by comparing the set
value of the low-pressure compressor’s spool speed (setNL) to the value
produced by the sensor (observedNL). It is necessary to differentiate between
the cause of the difference, i.e. whether it was directly caused by pilotInput
or by external factors (!pilotInput). In either case the system must change
modes, but our industrial partner felt that it was important that the
requirements distinguish the difference.
Figure 2: fretish and semantics diagram generated for UC5_R_14.2. Here, ‘M’
indicates the mode, ‘TC’ the triggering condition, and ‘SC’ the stopping
condition.
Fig. 2 contains the semantics diagram produced by FRET for UC5_R_14.2. The
semantic template that was used is [in, regular, until]. In a recent study
using FRET to formalise the 10 Lockheed Martin Cyber Physical Challenge
problems, the most commonly used semantic template was [null, null, always]
[11]. Of these 10 problems, the autopilot system is the closest to our case
study, and it was the only requirement set to use the in scope-option. The
timing-option in their requirements was different to ours; but we use until,
which was introduced into fretish after that study.
We used all of the fields available in fretish in our use case, although a lot
of our individual requirements used a subset of them. We only used scope in
the four child requirements of UC5_R_13 and UC5_R_14. FRET provides many ways
of specifying modes, but we only used in mode for this; there are many ways to
specify a condition, but we only used when and if. There are also multiple
ways to specify timing, but in this case study we only used until clauses.
Despite until providing timing constraints, we did not use explicit times
(e.g. ticks/timesteps/seconds) in our requirements. This is because the
natural-language requirements (Table 1) do not mention timing, and our
introduction of timing constraints came from elicitation discussions. However,
time points are implicit in some of the child requirements, e.g. comparing
r(i) and y(i) in the child requirements of UC5_R_1(Table 4), or the T1 and T2
variables in UC5_R_11.1 (Table 6). The timing clause was not intentionally
avoided, but we felt that the implicit time constraints better suited the
requirements and was closer to the description agreed with our industrial
partner.
ID | Parent | FRETISH
---|---|---
UC5_R_11.1 | UC5_R_11 | when (diff(r(i),y(i)) $>$ E) if (outsideAirPressure(T1) != outsideAirPressure(T2) & (diff(t2,t1) $<$ threshold) &(abs(outsideAirPressure (T1) - outsideAirPressure(T2)) $>$ pressureThreshold) &(observedThrust = V1) &(pilotInput $=>$ setThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (shaftSpeed $>=$ operatingLowerBound) & (shaftSpeed $<=$ operatingUpperBound) & (observedThrust = V2)
Table 6: Child requirement of UC5_R_11 which has timing implicit through the
use of the timestamp variables T1 and T2.
### 4.2 Parent-Child Relationship in our Use Case
As previously mentioned, FRET allows a requirement to be related to another as
a ‘parent’ or a ‘child’, but this relationship is not well defined, formally
or otherwise. The parent-child relationship in FRET could be viewed as formal
refinement [1]: a concept supported by a variety of formal methods that enable
formal specifications to be gradually made more concrete, while proving that
they still obey a more abstract version of the specification. Similar
approaches exist in the literature on refactoring goal-driven requirements [5,
15].
If viewed through the lens of refinement, we would need to introduce
abstraction invariants to relate the abstract and concrete specifications.
These invariants facilitate the proof that the concrete specification does not
permit any behaviours that the abstract specification forbids.
Here, we investigate whether FRET’s parent-child relationship can be expressed
as formal refinement. In particular, it is possible to formalise the following
abstraction invariant in relation to sensorfaults:
sensorfaults $\iff$ (sensorValue(S) $>$ nominalValue + R) $|$ (sensorValue(S)
$<$ nominalValue - R) $|$ (sensorValue(S) = null)
Intuitively this means that the boolean sensorfaults (from the parent
requirement) corresponds to the condition on the right of the ‘$\iff$’ (from
the child requirement). This kind of refinement is referred to as data
refinement.
Similarly, the abstraction invariant between trackingPilotCommands and the
condition and response pair (pilotInput $=>$ setThrust = V2) & (observedThrust
= V1) and satisfy (observedThrust = V2) could be specified as:
trackingPilotCommands $\iff$ pilotInput
The remainder of the condition-response pair above is then treated as
superposition refinement, which adds detail during refinement. This approach
is used because of the update of the observedThrust variable which is
difficult to express in an abstraction invariant because it provides a
behavioural update rather than a simple match between booleans. The additional
when and until clauses in the child requirement are also superposition
refinements.
The parent-child relationship in FRET appears to us to be formal refinement,
at least for our set of requirements. In which case UC5_R_1 is refined by its
three child requirements (UC5_R_1.1, UC5_R_1.2, UC5_R_1.3). We will examine
this further in future work, where we will seek to translate these
requirements into a formalism that supports refinement, and then examine
whether the appropriate proof obligations can be discharged by theorem
provers.
### 4.3 Translatable Requirements
As mentioned in §1, our aim is to formally verify the aircraft engine
controller system described in §2. It is often difficult to identify what
properties a system should obey, for example what does it mean for the system
to operate ‘correctly’. Identifying the properties to verify often causes
difficulties for non-domain experts. FRET helped to guide conversations with
the domain experts to facilitate the formalisation of the requirements.
FRET currently supports translation to the CoCoSim [4] and Copilot [12]
verification tools. We are particularly interested in using CoCoSim since it
works directly on Simulink diagrams. Thus, we have started to generate CoCoSim
contracts for these requirements automatically using FRET [10]. This is
described in [9] and corresponds to Phase 2A of the methodology outlined in
Fig. 1.
As described in §4.1, we didn’t rely heavily on timing constraints that
specified specific time steps, rather we used until constraints that could
potentially be translated into boolean flags in other formalisms. As such, we
believe that the vast majority of the requirements that we formalised in FRET
could be used by other formal methods. For example, we may need to model the
aircraft engine controller in an alternative formalism if some of these
properties fail to verify using CoCoSim due to the state space explosion. This
approach has been taken, manually, in previous work [3].
## 5 Lessons Learnt and Future Improvements
This section summarises some of the lessons that we learnt from this case
study.
Communication Barrier: We found that FRET and fretish provided a useful
conduit for conversation with our industrial partner. Formalising natural-
language requirements is often time-consuming because of contradictions and
ambiguities. fretish provides a stepping-stone between readable natural-
language requirements and their fully-formal counterparts, and helped us to
step-wise reduce ambiguity. This process produced requirements that are easier
to read than if they had been fully-formal, but which FRET can still
automatically formalise.
We used FRET during elicitation discussions to explain and update our
requirements, alongside our industrial partner. The diagramatic semantics gave
a useful visualisation of the requirements, helping both us and our industrial
partner to sanity-check updates. FRET also enabled our documentation of
information for each natural-language requirement, recording the reasoning for
any changes, alongside each fretish requirement, thus facilitating
requirements explainability.
Parent-Child Relationship: While not a formal relationship, the link between
parent and child requirements enabled us to gradually make the requirements
more concrete, by adding details and removing ambiguities. For example, the
term sensorfaults in UC5_R_1 was replaced with (sensorValue(S) $>$
nominalValue + R) $|$ (sensorValue(S) $<$ nominalValue - R) $|$
(sensorValue(S) = null) in its child requirements (Table 4). Documenting these
links, via the ‘Parent ID’ and ‘Rationale’ fields in FRET, provides a
structuring mechanism that enables traceability within the requirement set.
However, a more concrete definition of this link would be beneficial. We have
suggested a definition using formal refinement, but an object-oriented
inheritance relationship could also provide structure here.
Limitations of FRETISH: While a useful language, we note some limitations of
fretish. Logical quantifiers ($\forall,\exists$) would be a welcome addition
to fretish. For example, in UC5_R_1.1, we used sensorValue(S), where the
parameter S indicates that this condition applies to all sensors. This is
slight abuse of notation, it would have been more accurate to use a $\forall$
quantifier.
We also suggest that highlighting assignments to variables (which hold after
the requirement is triggered) would be beneficial. For example, in UC5_R_1.1
we use the observedThrust variable in both the condition and the response. We
expect that observedThrust has been updated by the response but this is not
obvious, and may have implications when translating to other verification
tools.
An Industrial Perspective: Our industrial partner had not used FRET before, so
we asked them about their experience with it. They felt that the fretish
requirements were ‘much more clear’ than the natural-language requirements,
and that using a ‘controlled-natural language with precise semantics is always
better than natural-language’. When asked if FRET was difficult to use or
understand they said that FRET was ‘very easy to use; interface is intuitive;
formal language is adequately documented inside the tool (along with usage
examples)’. Overall, they found that FRET was useful ‘because it forces you to
think about the actual meaning behind the natural-language requirements’.
Having installed FRET, our industrial partner found some usability
improvements that could be made. Some were problems with the GUI, which have a
low impact but happen very frequently. Other issues related to fretish; for
example, they would like to be able to add user-defined templates and
patterns, such as specifying timing within the condition component. Finally,
to aid interoperability they ‘would like to be able to export to a format
where the formalised requirements are machine readable (e.g. parse tree)’.
Impact: Formalising the requirements made them more detailed and less
ambiguous; crucially much of the detail came from elicitation discussions, not
from existing documents. FRET captures the links between requirements, and
explanations of their intent (which was often more detailed than what already
existed). These two things mean that the FRET requirements are a valuable
development artefact. We are currently pursuing Phase 2 of our methodology
(Fig. 1), in which we will assess the impact of the fretish requirements on
verification.
We believe that FRET can scale to larger requirements sets, with the parent-
child relationship providing a grouping function. However, for large sets of
requirements it might be necessary to modularise or refactor the requirements
so that they are easier to maintain. We are currently examining how fretish
requirements can be refactored for our use case.
## 6 Conclusions and Future Work
This paper provides an experience report of requirements elicitation and
formalisation of an aircraft engine controller in FRET. Our industrial partner
provided a set of natural-language requirements, test cases, and a Simulink
diagram. In close collaboration with our industrial partner, we clarified
ambiguous text in the requirements and test cases. This was essential, as we
had originally misunderstood some of the text. This iterative process produced
a set of detailed fretish requirements that we, and our industrial partner,
are confident correspond to the intent of the natural-language requirements.
The fretish requirements are now ready for use in formal verification
activities.
During this work we identified improvements that could be made to FRET, which
we plan to investigate in future work. First, our fretish requirements contain
quite a lot of repetition, so if changes were needed we often had to make the
change manually in several places. This was very time-consuming, so we propose
adding automatic requirement refactoring. Second, we plan to investigate how
to introduce globally-declared variable types. This would improve the
readability of requirements; clarifying what operations are valid for a
particular variable, while encapsulating definitions that might change in the
future. This could be made optional, to retain the ability to write very
abstract initial requirements. Finally, we would like to improve the
interoperability of FRET with other formal verification tools. For example,
adding a translator to the input language of a theorem prover to avoid the
state-explosion faced by model checkers (like Kind2, which is used to verify
CoCoSpec contracts); or outputting the requirement to a parse tree, as
suggested by our industrial partner.
## References
* [1] R.-J. Back and J. von Wright. Refinement Calculus: A Systematic Introduction. Springer, 1998.
* [2] R. Barbosa, S. Basagiannis, G. Giantamidis, H. Becker, E. Ferrari, J. Jahic, A. Kanak, M. L. Esnaola, V. Orani, D. Pereira, et al. The VALU3S ECSEL Project: Verification and Validation of Automated Systems Safety and Security. In Euromicro Conference on Digital System Design, pages 352–359. IEEE, 2020.
* [3] H. Bourbouh, M. Farrell, A. Mavridou, I. Sljivo, G. Brat, L. A. Dennis, and M. Fisher. Integrating Formal Verification and Assurance: An Inspection Rover Case Study. In NASA Formal Methods Symposium, pages 53–71. Springer, 2021.
* [4] H. Bourbouh, P.-L. Garoche, T. Loquen, É. Noulard, and C. Pagetti. CoCoSim, a code generation framework for control/command applications An overview of CoCoSim for multi-periodic discrete Simulink models. In European Congress on Embedded Real Time Software and Systems, 2020.
* [5] R. Darimont and A. Van Lamsweerde. Formal Refinement Patterns for Goal-Driven Requirements Elaboration. ACM SIGSOFT Software Engineering Notes, 21(6):179–190, 1996.
* [6] A. Dutle, C. Muñoz, E. Conrad, A. Goodloe, I. Perez, S. Balachandran, D. Giannakopoulou, A. Mavridou, T. Pressburger, et al. From Requirements to Autonomous Flight: An Overview of the Monitoring ICAROUS Project. In Workshop on Formal Methods for Autonomous Systems, pages 23–30. EPTCS, 2020.
* [7] D. Giannakopoulou, A. Mavridou, J. Rhein, T. Pressburger, J. Schumann, and N. Shi. Formal Requirements Elicitation with FRET. In International Conference on Requirements Engineering: Foundation for Software Quality, 2020.
* [8] D. Giannakopoulou, T. Pressburger, A. Mavridou, and J. Schumann. Automated formalization of structured natural language requirements. Information and Software Technology, 137, 2021.
* [9] M. Luckcuck, M. Farrell, O. Sheridan, and R. Monahan. A Methodology for Developing a Verifiable Aircraft Engine Controller from Formal Requirements. In IEEE Aerospace Conference, 2022.
* [10] A. Mavridou, H. Bourbouh, P. L. Garoche, D. Giannakopoulou, T. Pessburger, and J. Schumann. Bridging the Gap Between Requirements and Simulink Model Analysis. In International Conference on Requirements Engineering: Foundation for Software Quality, 2020.
* [11] A. Mavridou, H. Bourbouh, D. Giannakopoulou, T. Pressburger, M. Hejase, P.-L. Garoche, and J. Schumann. The Ten Lockheed Martin Cyber-Physical Challenges: Formalized, Analyzed, and Explained. In International Requirements Engineering Conference, pages 300–310. IEEE, 2020.
* [12] I. Perez, F. Dedden, and A. Goodloe. Copilot 3. Technical report, NASA/TM-2020-220587, National Aeronautics and Space Administration, 2020.
* [13] I. Postlethwaite, R. Samar, B. Choi, and D. Gu. A Digital Multimode H$\infty$ Controller for the Spey Turbofan Engine. In European Control Conference, 1995.
* [14] R. Samar and I. Postlethwaite. Design and Implementation of a Digital Multimode H$\infty$ Controller for the Spey Turbofan Engine. Journal of Dynamic Systems, Measurement, and Control, 132(1):011010, 2010.
* [15] P. Zave and M. Jackson. Four Dark Corners of Requirements Engineering. ACM Transactions on Software Engineering and Methodology (TOSEM), 6(1):1–30, 1997.
## Appendix Appendix A Simulink Diagram
This appendix contains a high-level view of the Simulink diagram of the
aircraft engine controller that we described in Sect. 2.
Figure 3: High-level view of the Simulink diagram that our industrial partner
constructed to model the controller for this use case.
## Appendix Appendix B Test Cases
This appendix contains all of the test cases for the aircraft engine
controller, provided by our industry partners. There are 20 test cases in
total and these are shown in Tables 7 – 10.
Test Case ID | Req. ID | Description
---|---|---
UC5_TC_1 | UC5_R_1 | Preconditions: Aircraft is in operating mode M and sensor S value deviates at most +/ R % from nominal value
Input conditions / steps: Observed aircraft thrust is at value V1 and pilot
input changes from A1 to A2
Expected results: Observed aircraft thrust changes and settles to value V2,
respecting control objectives (settling time, overshoot, steady state error)
UC5_TC_2 | UC5_R_1 | Preconditions: Aircraft is in operating mode M and sensor S value is not available (sensor is out of order)
Input conditions / steps: Observed aircraft thrust is at value V1 and pilot
input changes from A1 to A2
Expected results: Observed aircraft thrust changes and settles to value V2,
respecting control objectives (settling time, overshoot, steady state error)
UC5_TC_3 | UC5_R_2 | Preconditions: Aircraft is in operating mode M and sensor S value deviates at most +/ R % from nominal value
Input conditions / steps: Observed aircraft thrust is at value V1 and
perturbations in nonpilot input cause it to change to V2
Expected results: Observed aircraft thrust returns to value V1, respecting
control objectives (settling time, overshoot, steady state error)
UC5_TC_4 | UC5_R_2 | Preconditions: Aircraft is in operating mode M and sensor S value is not available (sensor is out of order)
Input conditions / steps: Observed aircraft thrust is at value V1 and
perturbations in nonpilot input cause it to change to V2
Expected results: Observed aircraft thrust returns to value V1, respecting
control objectives (settling time, overshoot, steady state error)
UC5_TC_5 | UC5_R_3 | Preconditions: Aircraft is in operating mode M and sensor S value deviates at most +/ R % from nominal value
Input conditions / steps: Observed aircraft thrust is at value V1 and pilot
input changes from A1 to A2
Expected results: Observed aircraft thrust changes and settles to value V2,
respecting operating limit objectives (e.g. upper limit in shaft speed)
UC5_TC_6 | UC5_R_3 | Preconditions: Aircraft is in operating mode M and sensor S value is not available (sensor is out of order)
Input conditions / steps: Observed aircraft thrust is at value V1 and pilot
input changes from A1 to A2
Expected results: Observed aircraft thrust changes and settles to value V2,
respecting operating limit objectives (e.g. upper limit in shaft speed)
Table 7: The test cases for the aircraft engine controller, provided by our industrial partner corresponding to requirements UC5_R_1, UC5_R_2 and UC5_R_3. Test Case ID | Req. ID | Description
---|---|---
UC5_TC_7 | UC5_R_4 | Preconditions: Aircraft is in operating mode M and sensor S value deviates at most +/ R % from nominal value
Input conditions / steps: Observed aircraft thrust is at value V1 and
perturbations in nonpilot input cause it to change to V2
Expected results: Observed aircraft thrust returns to value V1, respecting
operating limit objectives (e.g. upper limit in shaft speed)
UC5_TC_8 | UC5_R_4 | Preconditions: Aircraft is in operating mode M and sensor S value is not available (sensor is out of order)
Input conditions / steps: Observed aircraft thrust is at value V1 and
perturbations in nonpilot input cause it to change to V2
Expected results: Observed aircraft thrust returns to value V1, respecting
operating limit objectives (e.g. upper limit in shaft speed)
UC5_TC_9 | UC5_R_5 | Preconditions: Aircraft is in operating mode M and system parameter P deviates at most +/ R % from nominal value
Input conditions / steps: Observed aircraft thrust is at value V1 and pilot
input changes from A1 to A2
Expected results: Observed aircraft thrust changes and settles to value V2,
respecting control objectives (settling time, overshoot, steady state error)
UC5_TC_10 | UC5_R_6 | Preconditions: Aircraft is in operating mode M and system parameter P deviates at most +/ R % from nominal value
Input conditions / steps: Observed aircraft thrust is at value V1 and
perturbations in nonpilot input cause it to change to V2
Expected results: Observed aircraft thrust returns to value V1, respecting
control objectives (settling time, overshoot, steady state error)
UC5_TC_11 | UC5_R_7 | Preconditions: Aircraft is in operating mode M and system parameter P deviates at most +/ R % from nominal value
Input conditions / steps: Observed aircraft thrust is at value V1 and pilot
input changes from A1 to A2
Expected results: Observed aircraft thrust changes and settles to value V2,
respecting operating limit objectives (e.g. upper limit in shaft speed)
Table 8: The test cases for the aircraft engine controller, provided by our industrial partner corresponding to requirements UC5_R_4, UC5_R_5, UC5_R_6 and UC5_R_7. Test Case ID | Req. ID | Description
---|---|---
UC5_TC_12 | UC5_R_8 | Preconditions: Aircraft is in operating mode M and system parameter P deviates at most +/ R % from nominal value
Input conditions / steps: Observed aircraft thrust is at value V1 and
perturbations in nonpilot input cause it to change to V2
Expected results: Observed aircraft thrust returns to value V1, respecting
operating limit objectives (e.g. upper limit in shaft speed)
UC5_TC_13 | UC5_R_9 | Preconditions: Aircraft is in operating mode M
Input conditions / steps: Observed aircraft thrust is at value V1, pilot input
changes from A1 to A2, and outside air pressure abruptly changes from P1 to P2
Expected results: Observed aircraft thrust changes and settles to value V2,
respecting control objectives (settling time, overshoot, steady state error)
UC5_TC_14 | UC5_R_10 | Preconditions: Aircraft is in operating mode M
Input conditions / steps: Observed aircraft thrust is at value V1, small
perturbations in nonpilot input cause it to change to V2, and outside air
pressure abruptly changes from P1 to P2
Expected results: Observed aircraft thrust returns to value V1, respecting
control objectives (settling time, overshoot, steady state error)
UC5_TC_15 | UC5_R_11 | Preconditions: Aircraft is in operating mode M
Input conditions / steps: Observed aircraft thrust is at value V1, pilot input
changes from A1 to A2, and outside air pressure abruptly changes from P1 to P2
Expected results: Observed aircraft thrust changes and settles to value V2,
respecting operating limit objectives (e.g. upper limit in shaft speed)
UC5_TC_16 | UC5_R_12 | Preconditions: Aircraft is in operating mode M
Input conditions / steps: Observed aircraft thrust is at value V1, small
perturbations in nonpilot input cause it to change to V2, and outside air
pressure abruptly changes from P1 to P2
Expected results: Observed aircraft thrust returns to value V1, respecting
operating limit objectives (e.g. upper limit in shaft speed)
Table 9: The test cases for the aircraft engine controller, provided by our industrial partner corresponding to requirements UC5_R_8, UC5_R_9, UC5_R_10, UC5_R_11 and UC5_R_12. Test Case ID | Req. ID | Description
---|---|---
UC5_TC_17 | UC5_R_13 | Preconditions: Aircraft is in nominal operating mode
Input conditions / steps: Pilot input changes from A1 to A2, causing surge /
stall avoidance indicator signal to be set
Expected results: Aircraft switches to surge / stall prevention operating mode
UC5_TC_18 | UC5_R_13 | Preconditions: Aircraft is in surge / stall prevention operating mode
Input conditions / steps: Pilot input changes from A1 to A2, causing surge /
stall avoidance indicator signal to be cleared
Expected results: Aircraft switches to nominal operating mode
UC5_TC_19 | UC5_R_14 | Preconditions: Aircraft is in nominal operating mode
Input conditions / steps: Perturbations in nonpilot input cause surge / stall
avoidance indicator signal to be set
Expected results: Aircraft switches to surge / stall prevention operating mode
UC5_TC_20 | UC5_R_14 | Preconditions: Aircraft is in surge / stall prevention operating mode
Input conditions / steps: Perturbations in nonpilot input cause surge / stall
avoidance indicator signal to be cleared
Expected results: Aircraft switches to nominal operating mode
Table 10: The test cases for the aircraft engine controller, provided by our
industrial partner corresponding to requirements UC5_R_13, and UC5_R_14.
## Appendix Appendix C Child Requirements
This appendix contains the complete set of fretish child requirements for the
aircraft engine controller use case. These requirements are contained in
Tables 11 –15.
ID | Parent | FRETISH
---|---|---
UC5_R_1.1 | UC5_R_1 | when (diff(r(i),y(i)) $>$ E) if((sensorValue(S) $>$ nominalValue + R) $|$ (sensorValue(S) $<$ nominalValue - R) $|$ (sensorValue(S) = null) & (pilotInput $=>$ setThrust = V2) & (observedThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (settlingTime $>=$ 0) & (settlingTime $<=$ settlingTimeMax) & (observedThrust = V2)
UC5_R_1.2 | UC5_R_1 | when (diff(r(i),y(i)) $>$ E) if((sensorValue(S) $>$ nominalValue + R) $|$ (sensorValue(S) $<$ nominalValue - R) $|$ (sensorValue(S) = null)& (pilotInput $=>$ setThrust = V2) & (observedThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (overshoot $>=$ 0) & (overshoot $<=$ overshootMax) & (observedThrust = V2)
UC5_R_1.3 | UC5_R_1 | when (diff(r(i),y(i)) $>$ E) if((sensorValue(S) $>$ nominalValue + R) $|$ (sensorValue(S) $<$ nominalValue - R) $|$ (sensorValue(S) = null)& (pilotInput $=>$ setThrust = V2)& (observedThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (steadyStateError $>=$ 0) & (steadyStateError $<=$ steadyStateErrorMax) & (observedThrust = V2)
UC5_R_2.1 | UC5_R_2 | when (diff(r(i),y(i)) $>$ E) if((sensorValue(S) $>$ nominalValue + R) $|$ (sensorValue(S) $<$ nominalValue - R) $|$ (sensorValue(S) = null)& (!pilotInput $=>$ setThrust = V1)& (observedThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (settlingTime $>=$ 0) & (settlingTime $<=$ settlingTimeMax) & (observedThrust = V1)
UC5_R_2.2 | UC5_R_2 | when (diff(r(i),y(i)) $>$ E) if((sensorValue(S) $>$ nominalValue + R) $|$ (sensorValue(S) $<$ nominalValue - R) $|$ (sensorValue(S) = null)& (!pilotInput $=>$ setThrust = V1)& (observedThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (overshoot $>=$ 0) & (overshoot $<=$ overshootMax) & (observedThrust = V1)
UC5_R_2.3 | UC5_R_2 | when (diff(r(i),y(i)) $>$ E) if((sensorValue(S) $>$ nominalValue + R) $|$ (sensorValue(S) $<$ nominalValue - R) $|$ (sensorValue(S) = null)& (!pilotInput $=>$ setThrust = V1)& (observedThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (steadyStateError $>=$ 0) & (steadyStateError $<=$ steadyStateError Max) & (observedThrust = V1)
Table 11: Child requirements for UC5_R_1 and UC5_R_2. ID | Parent | FRETISH
---|---|---
UC5_R_3.1 | UC5_R_3 | when (diff(r(i),y(i)) $>$ E) if((sensorValue(S) $>$ nominalValue + R) $|$ (sensorValue(S) $<$ nominalValue - R) $|$ (sensorValue(S) = null) & (observedThrust = V1))& (pilotInput $=>$ setThrust = V2) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (shaftSpeed $>=$ operatingLowerBound) & (shaftSpeed $<=$ operatingUpperBound) & (observedThrust = V2)
UC5_R_4.1 | UC5_R_4 | when (diff(r(i),y(i)) $>$ E) if((sensorValue(S) $>$ nominalValue + R) $|$ (sensorValue(S) $<$ nominalValue - R) $|$ (sensorValue(S) = null)& (!pilotInput $=>$ setThrust = V1) & (observedThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (shaftSpeed $>=$ operatingLowerBound) & (shaftSpeed $<=$ operatingUpperBound) & (observedThrust = V1)
UC5_R_5.1 | UC5_R_5 | when (diff(r(i),y(i)) $>$ E) if((systemParameter(P) $>$ nominalValue + R) $|$ (systemParameter(P) $<$ nominalValue - R) $|$ (systemParameter(P) = null)& (observedThrust = V1)& (pilotInput $=>$ setThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (settlingTime $>=$ 0) & (settlingTime $<=$ settlingTimeMax) & (observedThrust = V2)
UC5_R_5.2 | UC5_R_5 | when (diff(r(i),y(i)) $>$ E) if((systemParameter(P) $>$ nominalValue + R) $|$ (systemParameter(P) $<$ nominalValue - R) $|$ (systemParameter(P) = null)& (observedThrust = V1)& (pilotInput $=>$ setThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (overshoot $>=$ 0) & (overshoot $<=$ overshootMax) & (observedThrust = V2)
UC5_R_5.3 | UC5_R_5 | when (diff(r(i),y(i)) $>$ E) if((systemParameter(P) $>$ nominalValue + R) $|$ (systemParameter(P) $<$ nominalValue - R) $|$ (systemParameter(P) = null)& (observedThrust = V1)& (pilotInput $=>$ setThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (steadyStateError $>=$ 0) & (steadyStateError $<=$ steadyStateErrorMax) & (observedThrust = V2)
Table 12: Child requirements for UC5_R_3, UC5_R_4 and UC5_R_5. ID | Parent | FRETISH
---|---|---
UC5_R_6.1 | UC5_R_6 | when (diff(r(i),y(i)) $>$ E) if((systemParameter(P) $>$ nominalValue + R) $|$ (systemParameter(P) $<$ nominalValue - R) $|$ (systemParameter(P) = null)& (observedThrust = V2)& (!pilotInput $=>$ setThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (settlingTime $>=$ 0) & (settlingTime $<=$ settlingTimeMax) & (observedThrust = V1)
UC5_R_6.2 | UC5_R_6 | when (diff(r(i),y(i)) $>$ E) if((systemParameter(P) $>$ nominalValue + R) $|$ (systemParameter(P) $<$ nominalValue - R) $|$ (systemParameter(P) = null)& (observedThrust = V2)& (!pilotInput $=>$ setThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (overshoot $>=$ 0) & (overshoot $<=$ overshootMax) & (observedThrust = V1)
UC5_R_6.3 | UC5_R_6 | when (diff(r(i),y(i)) $>$ E) if((systemParameter(P) $>$ nominalValue + R) $|$ (systemParameter(P) $<$ nominalValue - R) $|$ (systemParameter(P) = null)& (observedThrust = V2)& (!pilotInput $=>$ setThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (steadyStateError $>=$ 0) & (steadyStateError $<=$ steadyStateErrorMax) & (observedThrust = V1)
UC5_R_7.1 | UC5_R_7 | when (diff(r(i),y(i)) $>$ E) if((systemParameter(P) $>$ nominalValue + R) $|$ (systemParameter(P) $<$ nominalValue - R) $|$ (systemParameter(P) = null)& (observedThrust = V1)& (pilotInput $=>$ setThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (shaftSpeed $>=$ operatingLowerBound) & (shaftSpeed $<=$ operatingUpperBound) & (observedThrust = V2)
UC5_R_8.1 | UC5_R_8 | when (diff(r(i),y(i)) $>$ E) if((systemParameter(P) $>$ nominalValue + R) $|$ (systemParameter(P) $<$ nominalValue - R) $|$ (systemParameter(P) = null)& (!pilotInput $=>$ setThrust = V1)& (observedThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (shaftSpeed $>=$ operatingLowerBound) & (shaftSpeed $<=$ operatingUpperBound) & (observedThrust = V1)
Table 13: Child requirements for UC5_R_6, UC5_R_7 and UC5_R_8. ID | Parent | FRETISH
---|---|---
UC5_R_9.1 | UC5_R_9 | when (diff(r(i),y(i)) $>$ E) if ((outsideAirPressure(T1) != outsideAirPressure(T2) & (diff(t2,t1) $<$ threshold) & (abs(outsideAirPressure (T1) - outsideAirPressure(T2)) $>$ pressureThreshold) & (observedThrust = V1)& (pilotInput $=>$ setThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (settlingTime $>=$ 0) & (settlingTime $<=$ settlingTimeMax) & (observedThrust = V2)
UC5_R_9.2 | UC5_R_9 | when (diff(r(i),y(i)) $>$ E) if ((outsideAirPressure(T1) != outsideAirPressure(T2) & (diff(t2,t1) $<$ threshold) & (abs(outsideAirPressure (T1) - outsideAirPressure(T2)) $>$ pressureThreshold) & (observedThrust = V1)& (pilotInput $=>$ setThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (overshoot $>=$ 0) & (overshoot $<=$ overshootMax) & (observedThrust = V2)
UC5_R_9.3 | UC5_R_9 | when (diff(r(i),y(i)) $>$ E) if ((outsideAirPressure(T1) != outsideAirPressure(T2) & (diff(t2,t1) $<$ threshold) & (abs(outsideAirPressure (T1) - outsideAirPressure(T2)) $>$ pressureThreshold) & (observedThrust = V1)& (pilotInput $=>$ setThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (steadyStateError $>=$ 0) & (steadyStateError $<=$ steadyStateErrorMax) & (observedThrust = V2)
UC5_R_10.1 | UC5_R_10 | when (diff(r(i),y(i)) $>$ E) if ((outsideAirPressure(T1) != outsideAirPressure(T2) & (diff(t2,t1) $<$ threshold) & (abs(outsideAirPressure (T1) - outsideAirPressure(T2)) $>$ pressureThreshold) & (observedThrust = V2)& (!pilotInput $=>$ setThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (settlingTime $>=$ 0) & (settlingTime $<=$ settlingTimeMax) & (observedThrust = V1)
UC5_R_10.2 | UC5_R_10 | when (diff(r(i),y(i)) $>$ E) if ((outsideAirPressure(T1) != outsideAirPressure(T2) & (diff(t2,t1) $<$ threshold) & (abs(outsideAirPressure (T1) - outsideAirPressure(T2)) $>$ pressureThreshold) & (observedThrust = V2)& (!pilotInput $=>$ setThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (overshoot $>=$ 0) & (overshoot $<=$ overshootMax) & (observedThrust = V1)
UC5_R_10.3 | UC5_R_10 | when (diff(r(i),y(i)) $>$ E) if ((outsideAirPressure(T1) != outsideAirPressure(T2) & (diff(t2,t1) $<$ threshold) & (abs(outsideAirPressure (T1) - outsideAirPressure(T2)) $>$ pressureThreshold) & (observedThrust = V2)& (!pilotInput $=>$ setThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (steadyStateError $>=$ 0) & (steadyStateError $<=$ steadyStateErrorMax) & (observedThrust = V1)
Table 14: Child requirements for UC5_R_9 and UC5_R_10. ID | Parent | FRETISH
---|---|---
UC5_R_11.1 | UC5_R_11 | when (diff(r(i),y(i)) $>$ E) if ((outsideAirPressure(T1) != outsideAirPressure(T2) & (diff(t2,t1) $<$ threshold) & (abs(outsideAirPressure (T1) - outsideAirPressure(T2)) $>$ pressureThreshold) & (observedThrust = V1)& (pilotInput $=>$ setThrust = V2)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (shaftSpeed $>=$ operatingLowerBound) & (shaftSpeed $<=$ operatingUpperBound) & (observedThrust = V2)
UC5_R_12.1 | UC5_R_12 | when (diff(r(i),y(i)) $>$ E) if ((outsideAirPressure(T1) != outsideAirPressure(T2) & (diff(t2,t1) $<$ threshold) & (abs(outsideAirPressure (T1) - outsideAirPressure(T2)) $>$ pressureThreshold) & (observedThrust = V2)& (!pilotInput $=>$ setThrust = V1)) Controller shall until (diff(r(i),y(i)) $<$ e) satisfy (shaftSpeed $>=$ operatingLowerBound) & (shaftSpeed $<=$ operatingUpperBound) & (observedThrust = V1)
UC5_R_13.1 | UC5_R_13 | in nominal mode when (diff(setNL, observedNL) $>$ NLmax) if (pilotInput $=>$ surgeStallAvoidance) Controller shall until (diff(setNL, observedNL) $<$ NLmin) satisfy (changeMode(surgeStallPrevention))
UC5_R_13.2 | UC5_R_13 | in surgeStallPrevention mode when (diff(setNL, observedNL) $<$ NLmax) if (pilotInput $=>$ !surgeStallAvoidance) Controller shall until (diff(setNL, observedNL) $>$ NLmin) satisfy (changeMode(nominal))
UC5_R_14.1 | UC5_R_14 | in nominal mode when (diff(setNL, observedNL) $>$ NLmax) if (!pilotInput $=>$ surgeStallAvoidance) Controller shall until (diff(setNL, observedNL) $<$ NLmin) satisfy (changeMode(surgeStallPrevention))
UC5_R_14.2 | UC5_R_14 | in surgeStallPrevention mode when (diff(setNL, observedNL) $<$ NLmax) if (!pilotInput $=>$ !surgeStallAvoidance) Controller shall until (diff(setNL, observedNL) $>$ NLmin) satisfy (changeMode(nominal))
Table 15: Child requirements for UC5_R_11, UC5_R_12, UC5_R_13 and UC5_R_14.
|
# A portrait of the rotation of Ultra-Cool Dwarfs revealed by TESS
D. O. Fontinele Departamento de Física Teórica e Experimental, Universidade
Federal do Rio Grande do Norte, Campus Universitário, Natal, RN, 59072-970,
Brazil P. D. S. de Lima Departamento de Física Teórica e Experimental,
Universidade Federal do Rio Grande do Norte, Campus Universitário, Natal, RN,
59072-970, Brazil School of Physics, Trinity College Dublin, Dublin 2,
Ireland Y. S. Messias Departamento de Física Teórica e Experimental,
Universidade Federal do Rio Grande do Norte, Campus Universitário, Natal, RN,
59072-970, Brazil Université de Montréal, Département de Physique, IREX,
Montréal, QC H3C 3J7, Canada R. L. Gomes Departamento de Física Teórica e
Experimental, Universidade Federal do Rio Grande do Norte, Campus
Universitário, Natal, RN, 59072-970, Brazil Université de Montréal,
Département de Physique, IREX, Montréal, QC H3C 3J7, Canada C. E. Ferreira
Lopes Instituto de Astronomía y Ciencias Planetarias, Universidad de Atacama,
Copayapu 485, Copiapó, Chile Millennium Institute of Astrophysics, Nuncio
Monseñor Sotero Sanz 100, Of. 104, 7500000 Providencia, Santiago, Chile B. L.
Canto Martins Departamento de Física Teórica e Experimental, Universidade
Federal do Rio Grande do Norte, Campus Universitário, Natal, RN, 59072-970,
Brazil INAF - Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5,
50125 Firenze, Italy I. C. Leão Departamento de Física Teórica e
Experimental, Universidade Federal do Rio Grande do Norte, Campus
Universitário, Natal, RN, 59072-970, Brazil J. M. DE Araújo Departamento de
Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte,
Campus Universitário, Natal, RN, 59072-970, Brazil E. Janot Pacheco Instituto
de Astronomia, Geofísica e Ciências Atmosféricas, Universidade de São Paulo,
05509-090, São Paulo, SP, Brazil Observatoire de la Côte d’Azur, Laboratoire
Lagrange, Bâtiment Fizeau, Parc Valrose, 06108 Nice, France J. R. De Medeiros
Departamento de Física Teórica e Experimental, Universidade Federal do Rio
Grande do Norte, Campus Universitário, Natal, RN, 59072-970, Brazil
(Received December, 2023; Revised March, 2024; Accepted May)
###### Abstract
This study presents the results of a search for rotation signature in 250 Gaia
DR3 Ultra-Cool Dwarfs (UCDs) with TESS light curves. We identified 71 targets
with unambiguous periodicities, of which 61 present rotation signatures and a
single source behavior, with periods between 0.133 and 5.81 days. Five UCDs
show double-dip features, namely variations with two periods, one
approximately double or half the other. The remaining ten UCDs with
unambiguous variability present a likely non-single behavior. We also found 20
UCDs showing complex behavior in their light curves, with noticeable
fluctuations and irregular structure, with a few exhibiting apparent changes
in their temporal structure. The remaining 159 targets show noisy light curves
corresponding to low-amplitude signals, whose temporal variation cannot be
easily identified. The distribution of the UCDs with rotation signature in the
CMD diagram points to a lack of rotating objects within about
$11.5<M_{G}<12.5$ and $G-G_{RP}<1.5$ separating them into two regimes, one
mainly composed of less massive late-M stars with $P_{rot}\geq 1.0$ d, and
another mainly composed of more massive early-M stars with $P_{rot}<1.0$,d. It
is important to emphasize that by separating stars into age intervals, one
observes that UCDs with $P_{rot}\geq 1.0$ d tend to be located in regions of
younger objects, and, in contrast, those with $P_{rot}<1.0$ d are mainly
concentrated in regions of older objects. Whether these trends of stars
contrasting the sample separation is physical or produced by observational
biases is a question to be verified in future studies.
Low mass stars; Stellar rotation; Brown dwarfs; Stellar photometry; Space
observatories; Stellar astronomy
††journal: ApJ
## 1 Introduction
Ultra-Cool Dwarfs (UCDs) are defined as objects with spectral types M7 or
later, including L, T, and Y types, of stellar and substellar nature. These
objects are characterized by intense and broad potassium lines in the optical,
and water, methane, and ammonia absorption bands in the near-IR (e.g.,
Kirkpatrick et al., 1999; Burgasser et al., 2002; Burningham et al., 2008).
Among their most interesting characteristics is the formation of atmospheric
cloud-like structures, composed of iron and silicates for L dwarfs, chlorides
and sulfides for T and early-type Y dwarfs, and water clouds in the coolest Y
dwarfs (e.g., Saumon et al., 2000; Zahnle & Marley, 2014; Suárez & Metchev,
2023). Another fundamental characteristic of the UCDs concerns their well-
established variability, characterized by brightness changes caused by large-
scale atmospheric structures, such as spots or longitudinal bands, which can
provide fundamental information on the physical processes operating at their
atmospheres (e.g., Artigau et al., 2009; Radigan et al., 2014; Buenzli et al.,
2014; Metchev et al., 2015).
To date, the literature reports a growing list of studies on the photometric
variability of UCDs, based on observations from ground-based telescopes (e.g.,
Tinney & Tolley, 1999; Koen et al., 2005; Harding et al., 2013; Radigan et
al., 2014; Miles-Páez et al., 2017), and from space (e.g., Buenzli et al.,
2014; Metchev et al., 2015; Cushing et al., 2016; Tannock et al., 2021; Miles-
Páez et al., 2023; Vos et al., 2022; Petrucci et al., 2024). These studies
revealed essential characteristics on the behavior of the periodicities,
including a clustering of shortest rotation periods near 1 hr (Tannock et al.,
2021) and a lower envelope of rotation periodicities that runs from 2 h at
spectral type M8 down to 1 h at spectral type T (Miles-Páez et al., 2023).
Nevertheless, the astrophysical root cause for the variability in UCDs still
needs to be better understood. On the stellar side of the hydrogen-fusing
limit, starspots are the usual explanation for the observed variability (e.g.,
Rockenfeller et al., 2006; Irwin et al., 2011; Donati et al., 2023). For the
UCDs, the variability can have multiple origins, such as magnetic processes
(e.g., Hooten & Hall, 1990), condensated particles that form atmospheric
cloud-like structures (Tsuji et al., 1996), or even brightness changes due to
an atmosphere out of chemical equilibrium (Tremblin et al., 2015). Pineda et
al. (2017) pointed to evidence for a complex interplay between the starspots
driving the variability of UCDs and robust large-scale magnetospheric current
systems.
The NASA TESS space mission (Ricker et al., 2015), launched in 2018, is
producing differential photometric light curves (LCs) for hundreds of
thousands of stars. Primarily dedicated to the search for terrestrial planets
transiting nearby bright stars, the large number of observed targets combined
with the high quality of the acquired data is opening new horizons for the
study of a variety of astrophysical phenomena, including stellar rotation and
activity (e.g., Canto Martins et al., 2020; Barraza et al., 2022; Doyle et
al., 2020). Thanks to its high-quality photometry data and continuous
monitoring, TESS could also represent an optimal laboratory in the search for
rotation periodicities in UCDs.
Aiming to enlarge the current sample of Ultra-Cool Dwarfs with measured
rotation period, this study is dedicated to the search for rotation signatures
in sources identified as Ultra-Cool Dwarfs in the Gaia DR3 (Sarro et al.,
2023), observed by the TESS mission. This paper is part of a collaborative
effort to determine rotation periodicities for different families of stars
located throughout the HR Diagram, with LCs collected by the referred space
mission. The paper is organized as follows. Section 2 presents the stellar
sample and observational data set used in this study and the procedure used to
analyze the LCs. Section 3 provides the main results. A summary is presented
in Section 4.
## 2 The Ultra-Cool Dwarf Sample and Observational Data Analysis
The TESS primary mission plan is to survey almost the entire sky by monitoring
26 segments (or sectors) of $90^{\circ}\times 24^{\circ}$, each one with a
duration of 27 days at a time, with four on-board 10.5 cm telescopes in a red
(600–1000 nm) band-pass. A given sector is, therefore, revisited about every
two years. However, overlaps between sectors can increase the coverage, and
targets in the Continuous Viewing Zone are observed with almost unbroken
coverage for a year. The mission provides photometric time series at different
cadences, namely 2 and 30 minutes in Cycles 1 and 2, and 20 s, 2 minutes, and
10 minutes in Cycle 3, with a time baseline from 27 to 351 days, depending on
sector overlaps. While 2-minute cadence data, also known as Target Pixel (TP)
files, are available for a subset of targets, all CCDs, called full-frame
images (FFIs), are read out every 30 minutes (10 minutes in the extended
mission). Ten subsets of TESS targets were observed for multiple sectors, with
approximately 1–2 percent of targets located in the continuous viewing zone
during the primary mission (Barclay et al., 2018). Such an aspect makes these
targets invaluable for extracting rotation periodicities and analyzing the
persistence of stellar cycles (Ferreira Lopes et al., 2015).
For this study, searching for rotation signatures in UCDs and determining
their periodicities, we have used the sample of 7630 Gaia DR3 UCDs, given by
Sarro et al. (2023), as a starting point. For the definition of this sample,
those authors used the Gaia DR3 set of UCD candidates and complemented the
Gaia spectrophotometry with additional photometry to characterize the
candidates’ global properties. This procedure includes the determination of
the distances and their locus in the Gaia color-absolute magnitude diagram
(CAMD). More specifically, they present the typical (median) RP spectra in
spectral-type bins and compare them to UCD standards, ground-based high-
resolution spectra, and CAMDs, including Gaia and external photometry. Then,
that sample was compared with the TESS Input Catalogue, from which 7360
objects were identified out of the total sample of 7630. Of this latter list,
we found 250 targets with TESS 2-minute cadence LCs available on October 30th,
2023. On this date, the TESS data basis also reported observations of 674
targets in a 30-min cadence, 124 in a 10-min cadence, and one target in a
20-seg cadence; all these additional data are not suitable for our purpose.
The 2-min cadence data were downloaded from the FFI-TP-LC-DV Bulk Downloads
Page of the Mikulski Archive for Space
Telescopes111https://archive.stsci.edu/tess/bulk/downloads.html (STScI, 2018)
using the cURL scripts available for retrieving Pre-search Data Conditioned
Simple Aperture Photometry (PDCSAP) reduced light curves. Jenkins et al.
(2016) described the TESS science processing operations center (SPOC) pipeline
that produces the 2-min cadence light curves. The distribution of the spectral
types for the list of 250 UCDs with TESS LCs is displayed in Figure 1, which
ranges from M5 to L3. The referred spectral types were computed following the
calibration by Pecaut & Mamajek (2013). These targets are located in the
following star-forming regions, open clusters, or moving groups, as listed in
Table 6: Taurus (TAU), Carina-Near (CARN), Tucana-Horologium Association
(THA), $\epsilon$ Chamaeleontis Association (EPSC), AB Doradus moving group
(ABDMG), $\beta$ Pic Moving Group (BPMG), Upper Centaurus Lupus (UCL), Hyades
(HYA), TW Hya Association (TWA), Lower Centaurus Crux (LCC), Pleiades cluster
(PLE), Columba (COL), Argus (ARG), Coma Berenices(CBER), and Carina
Association (Car). The determination of membership to the referred star
clusters and moving groups were derived by Sarro et al. (2023), using the
BANYAN $\Sigma$ software tool (Gagné et al., 2018). Those authors have used
the Gaia DR3 UCD candidates, namely their celestial positions and proper
motions and parallaxes, as input to BANYAN $\Sigma$.
When required, additional processing on these LCs was performed manually to
avoid possible distortions in the signature of periodicities. Such processing
was explicitly the removal of outliers and correction of instrumental trends,
following the recipe by De Medeiros et al. (2013), Paz-Chinchón et al. (2015),
and Canto Martins et al. (2020). In short, flux measurements exceeding
3$\times$ the standard deviation of the light curve compared to the average
neighbor data were considered outliers and then removed from the time series.
The correction of instrumental trends consisted of detrending the light curve
of each sector from a polynomial fit of third order. In the analysis of the
post-processed LCs for the 250 UCDs with TESS observations, we have used the
same procedure applied by Canto Martins et al. (2020), which is based on a
manifold interactive platform built in the Interactive Data Language
(IDL222https://www.l3harrisgeospatial.com/Software-Technology/IDL). Readers
are referred to Canto Martins et al. (2020) and Barraza et al. (2022) for a
complete description of the referred manifold package. In short, our analysis
consisted of computing Fast Fourier Transforms (FFT) (e.g., Zhan et al.,
2019), Lomb-Scargle periodograms (e.g., Scargle, 1982; Horne & Baliunas, 1986;
Press & Rybicki, 1989), as well as wavelet maps and global spectra (Grossmann
& Morlet, 1984), of each light curve. Based on the known range of the rotation
periods of UCDs (e.g., Bailer-Jones, 2004; Tannock et al., 2021), we searched
for the main periodicities, namely peaks, computed from those methods. The
errors on the corresponding peak periods were estimated using Eq. (2) of Lamm
et al. (2004). The Lomb-Scargle method was also used to consider the false
alarm probability (FAP) of the periodicities, based on Eq. (22) of Horne &
Baliunas (1986), given by
$F=1-\left\\{1-exp[-(N_{0}/2)(1+xi^{-1})-1]\right\\}^{N_{i}}$. The formula
gives the periodogram power threshold, $F$, corresponding to a given FAP,
$\xi$, where $N_{0}$ is the number of observations of a light curve, $\xi$ is
the FAP, and $N_{i}$ is the number of independent frequencies, namely
$N_{i}=-6.362+1.193N_{0}+0.00098N_{0}^{2}$, according to Eq. (13) of Horne &
Baliunas (1986). In the present work, only peaks with FAP less than 1%,
corresponding to significance levels greater than 99%, were taken into
consideration. Additionally, wavelet maps were used to examine the persistence
of the main signals within the time span of TESS observations of a given
target in different sectors, and phase diagrams of the possible periodicities
were computed to inspect the global morphology of each signal.
After obtaining the FFT, Lomb–Scargle, wavelet analyses, and phase diagrams
results, we visually inspected each light curve to identify the present
modulation features, adopting the procedure applied by Canto Martins et al.
(2020).
Specifically, we separated the targets into three groups: (i) those with a
typical rotational modulation in the LC, namely a signature characterized by
semi-regular flux variability that used to be multi-sinusoidal; (ii) those
with an ambiguous, dubious, or complex variability profile, namely those
targets with LCs displaying an asymmetric shape in the flux variation, and
(iii) targets with noisy LCs. From the sample of 250 UCDs with bonafide TESS
LCs, listed in Table A of the Online Material, we have identified 54 UCDs
presenting unambiguous variability, with variability amplitudes ranging from
0.6 to 40.9 percent, for which we have computed the associated periodicities.
This list of 54 UCDs is presented in Table 1, which also contains the referred
periodicities. Figure 2 displays three examples of LCs with typical rotation
signatures identified in our sample of UCDs, with the corresponding phase
folded LC, FFT, Lomb–Scargle periodograms, and wavelet maps. Figures following
the same design as Figure 2 for all the UCDs with unambiguous variability are
provided in the figure set given in the Online Material. Among the UCDs with
evident variability, we identified 20 targets presenting a complex temporal
structure in their LCs. The referred LCs are also provided in the figure set
given in the Online Material.
### 2.1 A search for periodicities in Gaia DR3 UCDs with supposedly noisy LCs
Our manifold analysis has initially identified 176 Gaia DR3 UCDs with LCs
presenting supposedly noisy signatures, an aspect assumed particularly for LCs
with a low-amplitude signal. Indeed, as underlined by different studies (e.g.,
Canto Martins et al., 2020; Gilliland et al., 2011), a noisy behavior is a
complex combination of different factors, including, for example, instrumental
noise contributions and data reduction processes. Some UCDs classified as
noisy LCs may have been classified this way because of data reduction issues.
Considering that different studies point to a clustering of shortest rotation
periods of UCDs near 1 to 2 hr (e.g., Miles-Páez et al., 2023; Tannock et al.,
2021), we revisited the sample of 176 Gaia DR3 targets with LCs previously
classified as noisy in the search for short periods, specifically for periods
shorter than 1 d. First, we reanalyzed each LC using the Wavelet method to
search for persistent temporal structures on time scales shorter than 1 d, all
along the considered LC time span. Such a procedure revealed 17 targets with
potential low amplitude variability ranging from 0.22 to 21 percent, a
signature also clearly visible in the related phase folded light curve. For
these targets, it was possible to compute Wavelet and LS periodicities, listed
in Table 2, with the respective variability amplitude. Table 2 lists Sectors
where it was possible to extract the period variability and Sectors with
negligible or absent modulation. To reinforce the presence of variability with
potential physical meaning, not reflecting instrumental systematics, we have
analyzed the LCs of these 17 UCDs using the W Transform (Wang, 2021, 2022),
which computes a time-frequency as the Wavelet Transform, but with a
significant difference. The W Transform uses a nonstationary window function
to concentrate the spectral energy in the region of a dominant frequency and
to remove the singularity problem related to the zero frequency in the time-
frequency spectrum. Indeed, whereas the standard Wavelet transform shifts the
energy of a signal toward a higher frequency, the W transform centralizes the
referred energy around the dominant frequency. The periodicity values computed
from the W Transfom are also listed in Table 2, which reinforces those
obtained from Wavelet and LS methods. Figures following the same design as
Figure 2 for these 17 UCDs with likely rotation signatures and low amplitude
variability are provided in the figure set given in the Online Material.
### 2.2 Cleaning the sample of UCDs with modulation
We have checked all the stars with unambiguous periodicities to verify if the
variability signals obtained from their LCs were associated with the
considered UCD. Indeed, given the TESS large plate scale of 21′′ pixel-1,
analysis of signals obtained from the LCs of this mission should be made with
caution due to potential contamination by blended LCs, which can induce an
interpretation of the observed variability to a wrong source (Mullally et al.,
2022; Higgins & Bell, 2023; Pedersen & Bell, 2023). To assess the potential
impact of contamination by nearby sources, we analyzed the crowding metric
(CROWD) calculated by the SPOC pipeline (Caldwell et al., 2020), which
indicates the fraction of the light in the TESS aperture that comes from the
considered target given the positions and amplitudes of stars in the TESS
Input Catalog. A CROWD value near 1.0 indicates a potentially isolated star,
while lower values indicate a potential crowding from neighbors. This analysis
found 42 UCDs with a CROWD parameter larger than 0.7 within the total sample
of 71 UCDs with modulation listed in Tables 1 and 2. These tables also give
the CROWD parameters, with the listed values corresponding to an average for
stars observed in more than one sector. Hence, the variability amplitude of
UCDs with a CROWD parameter smaller than 0.7 should be taken cautiously since
these objects could be significantly affected by the flux contamination of
nearby targets.
## 3 The Gaia DR3 UCDs with TESS rotation periods
Among 250 DR3 Ultra-Cool Dwarfs with TESS bonafide available 2-min cadence
LCs, we have identified 71 targets presenting unambiguous periodicities,
composing two sub-samples. The first one is composed of 54 UCDs with high
amplitude modulation, showing persistent temporal structures on time scales
ranging from 0.133 to 5.81 days, and the second one is composed of 17 UCDs
with modulation signatures despite the low amplitude of their LCs, also with
persistent temporal structures on time scales shorter than one day. However,
an important aspect should be underlined: a closer look into the whole of the
LCs of eight of these 17 UCDs with low amplitude, namely TIC 448590406, TIC
299007548, TIC 356632694, TIC 371080717, TIC 275133493, TIC 201688405, TIC
302396416, and TIC 342967337, reveals that, in some epochs (TESS Sectors),
they show a quasi-sinusoidal modulation, while in other epochs, there is a
negligible or undetectable sign of variability. Tables 1 and 2 give the
periodicities for the referred samples of 54 and 17 UCDs, respectively,
including associated error in the period ($eP_{rot}$), effective time span
($t_{SPAN}$) of each LC (the total time span subtracted by the duration of
eventual gaps), the effective number of cycles of the rotational modulation
(defined as $N_{Cycle}=t_{SPAN}/P_{rot}$), and the TESS observation sectors.
We have also identified 20 DR3 UCDs with dubious or ambiguous variability
signatures in their LCs. The Dubious variability corresponds to targets
showing potential rotation signature but whose period could not be
disentangled among two or more possibilities (from periodogram peaks and
wavelet maps) and stars with $N_{Cycle}<$ 3\. Ambiguous variability
corresponds to objects showing visually large-amplitude variations with a very
irregular or complex structure in the LCs, noticeable fluctuations, or an
insufficient time span to correctly identify the nature of their variability
signature (see Section 2.5 of Canto Martins et al. (2020)). The referred UCDs
are listed in Table 2. A few of these targets, TIC 125882881, TIC 367089503,
TIC 58285782, and TIC 58229208, present apparent changes in their temporal
structure in the same TESS sector.
Tables 1 and 2 show a relatively high number of UCDs with short-period
measurements, 40 percent out of the 71 objects with computed periodicities. We
carefully analyzed if some of those short periods could be associated with
binarity. Then, we visually inspected the short-period LCs, searching for
eclipsing binary signatures. However, no sign of binarity was found in the
LCs. In this sense, we have evaluated the Gaia RUWE parameter (Gaia
Renormalized Unit Weight Error) for the entire list of 71 UCDs, which may be
an indicator of multiplicity (Lindegren et al., 2021). A RUWE value of around
1.0 is expected for a single-star astrometric solution. In contrast, a value
significantly greater than 1.0, typically greater than 1.4, could indicate a
non-single source or problematic for the astrometric solution (e.g., Kervella
et al., 2022). Ten targets with RUWE greater than 1.4 were identified, six
from Table 1 (TIC 17518894, TIC 268325250, TIC 56002511, TIC 58229181, TIC
56624850, and TIC 902237947), and four from Table 2 (TIC 356632694, TIC
281668854, TIC 388682292, and TIC 448590406). The star TIC 58229181 is listed
in the literature as an EB, from TESS observations (Prša et al., 2022),
whereas TIC 56624850 is classified as a circumbinary protoplanetary disk
system (Czekala et al., 2019). In addition, among the UCDs with dubious or
ambiguous variability signatures in their LCs, 18 present a RUWE greater than
1.4.
Then, for the present study, namely, the search for rotation signatures in
sources identified as UCDs in the Gaia DR3, observed by the TESS mission, we
consider as potential bonafide rotation signatures only those UCDs with RUWE
less than 1.4. This criterion amounts to 48 and 13 objects from the lists
given in Tables 1 and 2, respectively. The distribution of the computed
rotation periods for the corresponding sample of 61 UCDs with RUWE less than
1.4, namely those expected to be likely single-rotating objects, is displayed
in Figure 3. As it arises, the periodicities for these UCDs range from 0.133 d
to 5.81 d for the sample listed in Table 1 and from 0.116 d to 0.71 d for the
UCDs listed in Table 2, the latter with low LC amplitudes.
Five of the UCDs listed in Table 1 have two period values (TIC 150094175, TIC
20305594, TIC 359892714, TIC 58638214, and TIC 96680763), four of which
present a second period at half the main period (TIC 20305594, TIC 359892714,
TIC 58638214, and TIC96680763). These cases can be interpreted as double-dip
signatures (Basri, 2018; Basri & Nguyen, 2018), typically produced by two
groups of heterogeneities on the stellar surface, such as spots, clouds, or
large wave bands (e.g., Apai et al., 2017), with phases shifted by about 90°.
Those phase shifts may evolve along time, producing beating effects in the
light curves or a variation of the peak amplitudes in the periodograms that
could be caused by differential rotation (e.g., Apai et al., 2017; Basri,
2018; Basri & Nguyen, 2018). In wavelet maps, a distinctive pattern often
includes two dominant features over time, where one feature’s period is either
double or half of the other. Frequently, the longer period corresponds to the
stellar rotation period, while the shorter one results from the combination of
two semi-sinusoids linked to the double-dip signature (e.g., Canto Martins et
al., 2020).
A no less relevant result concerns the high number of 159 targets showing
noisy behavior in their LCs, corresponding to 64 percent of the original
sample of 250 targets. Although those targets present typically low-amplitude
signals whose physical periodicities cannot be easily identified, they can
hide essential information. Typically, a noisy signature is a complex
combination of instrumental noise contributions (related, for instance, with
Poisson statistics and readout noise) plus a relevant contribution of
intrinsic stellar noise, the Galactic position, light from neighboring stars,
and sky background contamination (e.g., Gilliland et al., 2011). When the LCs
considered in this work present a low-amplitude signal, we assumed them to be,
in principle, a noisy signature. The noisy behavior could reflect low activity
or long periodicities for some stars, particularly for targets with short
observational time spans. Table B in the Online Material gives the list of
UCDs supposedly presenting noisy LCs.
Miles-Páez et al. (2023) have analyzed the TESS LCs of a sample of 23 known
rapidly rotating UCDs, typically objects with a $V\sin i>30$ km/s to explore
their photometric variability. These authors used a Lomb-Scargle periodogram
in their study, applying a first-order correction by removing a 24-hour median
filter from the data and then a Gaussian Process regression. From such an
analysis, they claimed that significant periodicities in 18 targets are
compatible with rotation signature, with amplitudes ranging from 0.17 to 0.77
percent. For comparative purposes, even though only two UCDs from Miles-Páez
et al. (2023) are present in our Gaia DR3 UCDs whole list, we have then
applied our manifold procedure to the 21 targets of their sample with 2-min
cadence LCs available. The result of this analysis is presented in Table 4,
from where one observes that, for most cases, our variability periods are in
good agreement with those computed by Miles-Páez et al. (2023), except for
three targets (TIC 288506050, TIC 441000085, TIC 311188315), for which our
analysis points instead for a noisy behavior in their LCs. Further, it is
worth underlining that among the 21 UCDs listed in Table 4, ten have a RUWE
greater than 1.4. Recently, Petrucci et al. (2024) have claimed for rotation
periodicities for a sample of 87 UCDs from TESS LCs, based on a Lomb-Scargle
periodogram. Among these objects, eighty percent show $P_{rot}<1$ d. Four
objects from the referred sample (TIC 298907057, TIC 902237947, TIC 201688405,
and TIC 371080717) are in common with our study, showing a good agreement in
the corresponding $P_{rot}$ values.
The location of the Gaia DR3 UCDs with rotation periodicities and RUWE shorter
than 1.4, listed in Tables 1, 2, and 4, is shown in the color-magnitude
diagram (CMD) displayed in Figure 4, from where one observes an apparent trend
for a transition in the distribution of UCD rotation periods. In the left
panel UCDs with $P_{rot}\geq 1.0$ d tend to be located mainly in the regions
above $11.5<M_{G}<12.5$, whereas those with $P_{rot}<1.0$ d tend to populate
the region below this range. In addition, the CMD diagram in the referred
panel also appears to point to a dependence on the distribution of $P_{rot}$
with stellar mass, separated into two different regimes. The upper region,
populated mainly by UCDs with $P_{rot}\geq 1.0$ d, is composed of lower mass M
stars, whereas the lower region, mainly populated by UCDs with $P_{rot}<1.0$
d, is composed of larger mass M stars (Reylé, 2018). Thanks to the nature of
the present UCDs sample, which consists of objects located in stellar
associations with measured ages (Kenyon & Hartmann, 1995; Zuckerman et al.,
2006, 2011; Murphy et al., 2013; Bell et al., 2015; Pecaut & Mamajek, 2016),
the right panel in Figure 4 displays the same CMD. In this panel, stars are
separated into age intervals based on the values given in Tables 1 and 2, to
tentatively define regions where younger and older stars are expected to
dominate. The scenario emerging from the right panel in Figure 4 indicates an
interesting trend: UCDs with $P_{rot}\geq 1.0$ d tend to be located mainly in
the regions of younger objects, whereas those with $P_{rot}<1.0$ d tend to
populate the regions occupied by older objects.
Nevertheless, we should be cautious with the referred trend observed in the
CMD diagram, which could be affected by observational biases. The
incompleteness of the analyzed UCD sample and the instrumental sensitivity in
the observed wavelength band may contribute to the portrait observed in the
rotation period distribution (e.g., Canto Martins et al., 2023; Leão et al.,
2015). For instance, the UCD sample may be incomplete to periods longer than
24 hours because TESS is mainly sensitive to L0 dwarfs and, on the other side,
the lower period limits are also uncertain because most UCD campaigns have
been limited to $\sim$8–24-h cadences (Miles-Páez et al., 2023). Besides, the
region with an apparent lack of stars with rotation period measurements could
be populated by low-amplitude rotators below the TESS sensitivity. Despite
these potential biases, it is worth underlining that, based on the analysis of
the projected rotation velocity $V\sin i$ of UCDs, Mohanty & Basri (2003)
showed that this parameter tends to increase from mid-M to L types and that L
UCDs are probably rapidly rotating in general, with a lack of slowly rotating
in this latter spectral region.
## 4 Summary and Conclusions
Based on a manifold procedure, which considers the analysis of TESS LCs taking
into account Lomb-Scargle periodograms, Fast Fourier Transform (FFT), and
Wavelet Transform, we have analyzed a total of 250 Gaia DR3 UCDs presenting
public LCs, with short-cadence TESS observations in sectors 5 to 63. We
identified 48 UCDs with rotational modulation, presenting an unambiguous
rotation period. Among these UCDs, five show two periods, with four presenting
a second period that is approximately double or half of the rotation period.
This aspect may reveal the presence of differential rotation in the referred
targets. We have also identified 13 UCDs with typical modulation signatures,
presenting periodicities shorter than 1 d despite a low-amplitude behavior in
their LCs. Ten UCDs with unambiguous periodicities show a potential non-single
behavior, pointing to a dubious origin of the referred periodicities. The
present study also revealed 20 UCDs with ambiguous or dubious variability
signatures in their LCs, a few of them with LCs presenting apparent changes in
the temporal structure. The TESS light curves for the remaining 159 objects
show a noisy pattern with very low-amplitude signals.
Considering the whole sample of UCDs with rotation periods, this study shows
that 47.5 percent have a period shorter than 1 d. Another relevant result
arises from the CMD diagram, which seems to show a lack of rotating UCDs
within about $11.5<M_{G}<12.5$ and $G-G_{RP}<1.5$ that separates them into two
different regimes. One is mainly composed of less massive late-M stars with
$P_{rot}\geq 1.0$ d, and another regime is mainly composed of more massive
early-M stars with $P_{rot}<1.0$ d. Also, by separating the stars into age
intervals, we found that UCDs with $P_{rot}\geq 1.0$ d tend to be in regions
of younger objects, while those with $P_{rot}<1.0$ d are concentrated in
regions of older objects. Whether the apparent lack of stars contrasting the
sample separation is physical or produced by observational biases is a
question to be verified in future studies based on a more extensive and
homogeneous sample, as well as better statistics. If, on one side, the
signatures of rotation in UCDs are associated with multiple origins, including
magnetic processes, condensate particles that form atmospheric cloud-like
structures, or even brightness changes due to an atmosphere out of chemical
equilibrium, the objects with ambiguous variability or a noisy pattern appear
to reflect also the presence of different factors, such as low activity phases
or polar phenomena. In this context, the large sample of bonafide rapidly
rotating UCDs revealed in the present work represents a solid laboratory to
identify the cause of photometric variability in ultra UCDs through
spectroscopic analysis, in particular, to identify traces that can
differentiate between magnetically-induced star spots and inhomogeneous
atmospheric dust clouds. A first step in this direction should be the study of
their spectroscopic line profiles. If these targets are rapidly rotating,
their line profiles will be significantly Doppler-broadened. The large sample
of noisy UCDs also represents a relevant material for spectroscopic studies.
If such behavior reflects a low activity level and a slow rotation, these
objects will not show much line broadening. Identifying the root cause of
photometric variability in UCDs is also mandatory to understand the structural
stability of rapidly rotating Ultra Cool Dwarfs.
## 5 ACKNOWLEDGMENTS
Research activities of the Observational Astronomy Board at the Federal
University of Rio Grande do Norte (NAOS) are supported by continuous grants
from the Brazilian funding agency CNPq. This study was financed in part by the
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior $-$ Brasil (CAPES)
$-$ Finance Code 001. YSM and DOF acknowledge CAPES graduate fellowships, and
RLG acknowledges a CNPq PDE fellowship. PDSL and YSM acknowledge
UFRN/CAPES/Print fellowship, respectively, process 88887.838338/2023-00 and
88887.717782/2022-00. JMA acknowledges the CNPq Brazilian research agency for
funding (grant 311589/2021-9). BLCM, ICL, and JRM acknowledge CNPq research
fellowships. This work includes data collected by the TESS mission. The NASA
Explorer Program provides funding for the TESS mission. All the TESS data used
in this paper can be found in MAST:http://dx.doi.org/10.17909/fwdt-2x66
(catalog 10.17909/fwdt-2x66). We made use of data from the European Space
Agency (ESA) mission Gaia (https: //www.cosmos.esa.int/gaia), processed by the
Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, particularly the institutions
participating in the Gaia Multilateral Agreement. We want to thank the Referee
for carefully reading the manuscript and a number of suggestions that largely
improved the paper.
## References
* Apai et al. (2017) Apai, D., Karalidi, T., Marley, M. S., et al. 2017, Science, 357, 683. doi:10.1126/science.aam9848
* Artigau et al. (2009) Artigau, É., Bouchard, S., Doyon, R., et al. 2009, ApJ, 701, 1534. doi:10.1088/0004-637X/701/2/1534
* Bailer-Jones (2004) Bailer-Jones, C. A. L. 2004, A&A, 419, 703. doi:10.1051/0004-6361:20040965
* Barclay et al. (2018) Barclay, T., Pepper, J., & Quintana, E. V. 2018, ApJS, 239, 2. doi:10.3847/1538-4365/aae3e9
* Barraza et al. (2022) Barraza, L. F., Gomes, R. L., Messias, Y. S., et al. 2022, ApJ, 924, 117. doi:10.3847/1538-4357/ac3335
* Basri (2018) Basri, G. 2018, ApJ, 865, 142. doi:10.3847/1538-4357/aade45
* Basri & Nguyen (2018) Basri, G. & Nguyen, H. T. 2018, ApJ, 863, 190. doi:10.3847/1538-4357/aad3b6
* Bell et al. (2015) Bell, C. P. M., Mamajek, E. E., & Naylor, T. 2015, MNRAS, 454, 593. doi:10.1093/mnras/stv1981
* Buenzli et al. (2014) Buenzli, E., Apai, D., Radigan, J., et al. 2014, ApJ, 782, 77. doi:10.1088/0004-637X/782/2/77
* Burgasser et al. (2002) Burgasser, A. J., Kirkpatrick, J. D., Brown, M. E., et al. 2002, ApJ, 564, 421. doi:10.1086/324033
* Burningham et al. (2008) Burningham, B., Pinfield, D. J., Leggett, S. K., et al. 2008, MNRAS, 391, 320. doi:10.1111/j.1365-2966.2008.13885.
* Caldwell et al. (2020) Caldwell, D. A., Tenenbaum, P., Twicken, J. D., et al. 2020, Research Notes of the American Astronomical Society, 4, 201. doi:10.3847/2515-5172/abc9b3
* Canto Martins et al. (2020) Canto Martins, B. L., Gomes, R. L., Messias, Y. S., et al. 2020, ApJS, 250, 20. doi:10.3847/1538-4365/aba73f
* Canto Martins et al. (2023) Canto Martins, B. L., Messias, Y. S., Arruda Gonçalves, M. I., et al. 2023, Nature Astronomy, 7, 900. doi:10.1038/s41550-023-01976-0
* Cushing et al. (2016) Cushing, M. C., Hardegree-Ullman, K. K., Trucks, J. L., et al. 2016, ApJ, 823, 152. doi:10.3847/0004-637X/823/2/152
* Czekala et al. (2019) Czekala, I., Chiang, E., Andrews, S. M., et al. 2019, ApJ, 883, 22. doi:10.3847/1538-4357/ab287b
* De Medeiros et al. (2013) De Medeiros, J. R., Ferreira Lopes, C. E., Leão, I. C., et al. 2013, A&A, 555, A63. doi:10.1051/0004-6361/201219415
* Donati et al. (2023) Donati, J.-F., Lehmann, L. T., Cristofari, P. I., et al. 2023, MNRAS, 525, 2015. doi:10.1093/mnras/stad2301
* Doyle et al. (2020) Doyle, L., Ramsay, G., & Doyle, J. G. 2020, MNRAS, 494, 3596. doi:10.1093/mnras/staa923
* Ferreira Lopes et al. (2015) Ferreira Lopes, C. E., Leão, I. C., de Freitas, D. B., et al. 2015, A&A, 583, A134. doi:10.1051/0004-6361/201424900
* Gagné et al. (2018) Gagné, J., Mamajek, E. E., Malo, L., et al. 2018, ApJ, 856, 23. doi:10.3847/1538-4357/aaae09
* Gilliland et al. (2011) Gilliland, R. L., Chaplin, W. J., Dunham, E. W., et al. 2011, ApJS, 197, 6. doi:10.1088/0067-0049/197/1/6
* Grossmann & Morlet (1984) Grossmann, A. & Morlet, J. 1984, SIAM Journal on Mathematical Analysis, 15, 723. doi:10.1137/0515056
* Harding et al. (2013) Harding, L. K., Hallinan, G., Boyle, R. P., et al. 2013, ApJ, 779, 101. doi:10.1088/0004-637X/779/2/101
* Higgins & Bell (2023) Higgins, M. E. & Bell, K. J. 2023, AJ, 165, 141. doi:10.3847/1538-3881/acb20c
* Hooten & Hall (1990) Hooten, J. T. & Hall, D. S. 1990, ApJS, 74, 225. doi:10.1086/191497
* Horne & Baliunas (1986) Horne, J. H. & Baliunas, S. L. 1986, ApJ, 302, 757. doi:10.1086/164037
* Irwin et al. (2011) Irwin, J., Berta, Z. K., Burke, C. J., et al. 2011, ApJ, 727, 56. doi:10.1088/0004-637X/727/1/56
* Jenkins et al. (2016) Jenkins, J. M., Twicken, J. D., McCauliff, S., et al. 2016, Proc. SPIE, 9913, 99133E. doi:10.1117/12.2233418
* Kenyon & Hartmann (1995) Kenyon, S. J. & Hartmann, L. 1995, ApJS, 101, 117. doi:10.1086/192235
* Kervella et al. (2022) Kervella, P., Arenou, F., & Thévenin, F. 2022, A&A, 657, A7. doi:10.1051/0004-6361/202142146
* Kirkpatrick et al. (1999) Kirkpatrick, J. D., Reid, I. N., Liebert, J., et al. 1999, ApJ, 519, 802. doi:10.1086/307414
* Koen et al. (2005) Koen, C., Tanabé, T., Tamura, M., et al. 2005, MNRAS, 362, 727. doi:10.1111/j.1365-2966.2005.09280.x
* Lamm et al. (2004) Lamm, M. H., Bailer-Jones, C. A. L., Mundt, R., et al. 2004, A&A, 417, 557. doi:10.1051/0004-6361:20035588
* Leão et al. (2015) Leão, I. C., Pasquini, L., Ferreira Lopes, C. E., et al. 2015, A&A, 582, A85. doi:10.1051/0004-6361/201526085
* Lindegren et al. (2021) Lindegren, L., Klioner, S. A., Hernández, J., et al. 2021, A&A, 649, A2. doi:10.1051/0004-6361/202039709
* Metchev et al. (2015) Metchev, S. A., Heinze, A., Apai, D., et al. 2015, ApJ, 799, 154. doi:10.1088/0004-637X/799/2/154
* Miles-Páez et al. (2017) Miles-Páez, P. A., Pallé, E., & Zapatero Osorio, M. R. 2017, MNRAS, 472, 2297. doi:10.1093/mnras/stx2191
* Miles-Páez et al. (2023) Miles-Páez, P. A., Metchev, S. A., & George, B. 2023, MNRAS, 521, 952. doi:10.1093/mnras/stad273
* Mohanty & Basri (2003) Mohanty, S. & Basri, G. 2003, ApJ, 583, 451. doi:10.1086/345097
* Mullally et al. (2022) Mullally, S. E., Sloan, G. C., Hermes, J. J., et al. 2022, AJ, 163, 136. doi:10.3847/1538-3881/ac4bce
* Murphy et al. (2013) Murphy, S. J., Lawson, W. A., & Bessell, M. S. 2013, MNRAS, 435, 1325. doi:10.1093/mnras/stt1375
* Paz-Chinchón et al. (2015) Paz-Chinchón, F., Leão, I. C., Bravo, J. P., et al. 2015, ApJ, 803, 69
* Pecaut & Mamajek (2013) Pecaut, M. J. & Mamajek, E. E. 2013, ApJS, 208, 9. doi:10.1088/0067-0049/208/1/9
* Pecaut & Mamajek (2016) Pecaut, M. J. & Mamajek, E. E. 2016, MNRAS, 461, 794. doi:10.1093/mnras/stw1300
* Pedersen & Bell (2023) Pedersen, M. G. & Bell, K. J. 2023, AJ, 165, 239. doi:10.3847/1538-3881/accc31
* Petrucci et al. (2024) Petrucci, R. P., Gómez Maqueo Chew, Y., Jofré, E., et al. 2024, MNRAS, 527, 8290. doi:10.1093/mnras/stad3720
* Pineda et al. (2017) Pineda, J. S., Hallinan, G., & Kao, M. M. 2017, ApJ, 846, 75. doi:10.3847/1538-4357/aa8596
* Press & Rybicki (1989) Press, W. H. & Rybicki, G. B. 1989, ApJ, 338, 277. doi:10.1086/167197
* Prša et al. (2022) Prša, A., Kochoska, A., Conroy, K. E., et al. 2022, ApJS, 258, 16. doi:10.3847/1538-4365/ac324a
* Radigan et al. (2014) Radigan, J., Lafrenière, D., Jayawardhana, R., et al. 2014, ApJ, 793, 75. doi:10.1088/0004-637X/793/2/75
* Reylé (2018) Reylé, C. 2018, A&A, 619, L8. doi:10.1051/0004-6361/201834082
* Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003. doi:10.1117/1.JATIS.1.1.014003
* Rockenfeller et al. (2006) Rockenfeller, B., Bailer-Jones, C. A. L., & Mundt, R. 2006, A&A, 448, 1111. doi:10.1051/0004-6361:20054150
* Sarro et al. (2023) Sarro, L. M., Berihuete, A., Smart, R. L., et al. 2023, A&A, 669, A139. doi:10.1051/0004-6361/202244507
* Saumon et al. (2000) Saumon, D., Geballe, T. R., Leggett, S. K., et al. 2000, ApJ, 541, 374. doi:10.1086/309410
* Scargle (1982) Scargle, J. D. 1982, ApJ, 263, 835. doi:10.1086/160554
* STScI (2018) STScI 2018, TESS Input Catalog and Candidate Target List, STScI/MAST, doi:10.17909/FWDT-2X66
* Suárez & Metchev (2023) Suárez, G. & Metchev, S. 2023, MNRAS, 523, 4739. doi:10.1093/mnras/stad1711
* Tannock et al. (2021) Tannock, M. E., Metchev, S., Heinze, A., et al. 2021, AJ, 161, 224. doi:10.3847/1538-3881/abeb67
* Tinney & Tolley (1999) Tinney, C. G. & Tolley, A. J. 1999, MNRAS, 304, 119. doi:10.1046/j.1365-8711.1999.02297.x
* Tremblin et al. (2015) Tremblin, P., Amundsen, D. S., Mourier, P., et al. 2015, ApJ, 804, L17. doi:10.1088/2041-8205/804/1/L17
* Tsuji et al. (1996) Tsuji, T., Ohnaka, K., Aoki, W., et al. 1996, A&A, 308, L29
* Vos et al. (2022) Vos, J. M., Faherty, J. K., Gagné, J., et al. 2022, ApJ, 924, 68. doi:10.3847/1538-4357/ac4502
* Wang (2021) Wang, Y. 2021, Geophysics, 86, V31. doi:10.1190/geo2020-0316.1
* Wang (2022) Wang, Y. 2022, Geophysics, Time-Frequency Analysis of Seismic Signals: , Wiley. ISBN 978-1-119-89234-2. John Wiley & Sons, Ltd, 2022. doi:10.1002/9781119892373
* Zahnle & Marley (2014) Zahnle, K. J. & Marley, M. S. 2014, ApJ, 797, 41. doi:10.1088/0004-637X/797/1/41
* Zhan et al. (2019) Zhan, Z., Günther, M. N., Rappaport, S., et al. 2019, ApJ, 876, 127. doi:10.3847/1538-4357/ab158c
* Zuckerman et al. (2006) Zuckerman, B., Bessell, M. S., Song, I., et al. 2006, ApJ, 649, L115. doi:10.1086/508060
* Zuckerman et al. (2011) Zuckerman, B., Rhee, J. H., Song, I., et al. 2011, ApJ, 732, 61. doi:10.1088/0004-637X/732/2/61
Figure 1: Distribution of spectral types for the present working sample of 250
UCs with bonafide TESS LCs. Red color stands for UCDs with unambiguous
rotation and blue represents the entire UCD sample.
Figure 2: Examples of diagnostic plots displaying FFT and Lomb–Scargle
periodograms, LCs, LCs phase folded, and wavelet maps for three Gaia DR3
Ultra-Cool Dwarfs with typical TESS rotation signatures. Persistent
variability of rotation period 0.329, 0.519 and 0.472 days, respectively, for
TIC 170849193 (top panels), TIC 360329359 (middle panels), and TIC 359892714
(bottom panels) are observed in their wavelet maps and confirmed by FFT and
Lomb–Scargle peaks labeled A. Figures for the entire sample, following the
same design, are provided in the figure set given in the Online Material
Table 1: Gaia DR3 Ultra-Cool Dwarfs with TESS unambiguous periodicities
TIC ID | GAIA ID | Prot | eProt | tSPAN | NCycle | Sectors | CROWD | RUWE | Age
---|---|---|---|---|---|---|---|---|---
| | (days) | (days) | (days) | | | | | (Myr)
5630425 | 3493736924979792768 | 1.499 | 0.018 | 21.000 | 14.000 | 10 | 0.494 | 1.104 | 24aafootnotemark:
17308621 | 48193316928788352 | 1.332 | 0.020 | 43.981 | 33.019 | 43,44 | 0.589 | 1.357 | 2bbfootnotemark:
17518894 | 3314586142482134144 | 2.065 | 0.049 | 43.916 | 21.267 | 43,44 | 0.988 | 1.491 | 2bbfootnotemark:
20305594 | 1388684770725399296 | 0.281 | 0.001 | 73.210 | 260.534 | 23,24,50,51 | 0.994 | 1.183 | 149aafootnotemark:
45193132 | 5314960632252986880 | 2.486 | 0.025 | 123.355 | 49.620 | 8-10,35-37 | 0.093 | 1.01 | 149aafootnotemark:
49178529 | 5401822669314874240 | 0.967 | 0.022 | 21.503 | 22.237 | 36 | 0.436 | 1.085 | 10aafootnotemark:
56002511 | 3185321611286286976 | 0.377 | 0.003 | 23.785 | 63.089 | 5 | 0.956 | 2.317 | 149aafootnotemark:
56624850 | 163178353176600448 | 3.155 | 0.113 | 44.177 | 14.002 | 43,44 | 0.782 | 1.551 | 149aafootnotemark:
56658216 | 164802984685384320 | 2.606 | 0.077 | 44.140 | 16.938 | 43,44 | 0.823 | 1.013 | 2bbfootnotemark:
56658273 | 164800235906367232 | 5.81 | 0.382 | 44.160 | 7.601 | 43,44 | 0.894 | 0.888 | 2bbfootnotemark:
58229181 | 164513022853468160 | 2.11 | 0.050 | 44.140 | 20.919 | 43,44 | 0.978 | 1.54 | 2bbfootnotemark:
58285779 | 164513400810646912 | 1.669 | 0.063 | 22.000 | 13.182 | 44 | 0.183 | 1.205 | 2bbfootnotemark:
58436495 | 150501362066641664 | 0.58 | 0.004 | 43.885 | 75.664 | 43,44 | 0.270 | 0.979 | 2bbfootnotemark:
58538719 | 152516079683687680 | 2.447 | 0.068 | 44.046 | 18.000 | 43,44 | 0.872 | 1.036 | 2bbfootnotemark:
58601455 | 152108882425024128 | 4.66 | 0.251 | 43.211 | 9.273 | 44 | 0.568 | 0.976 | 2bbfootnotemark:
58638214 | 152917298349085824 | 0.752 | 0.004 | 67.494 | 89.753 | 19,43,44 | 0.726 | 1.372 | 2bbfootnotemark:
78772203 | 5424690587034982144 | 0.133 | 0.000 | 88.644 | 666.499 | 35,36,62,63 | 0.173 | 1.08 | 40ccfootnotemark:
96680763 | 156915878541979008 | 0.302 | 0.001 | 44.205 | 146.423 | 43,44 | 0.107 | 0.928 | 2bbfootnotemark:
101256057 | 216677698470254976 | 1.691 | 0.023 | 61.735 | 36.493 | 42,43,44 | 0.017 | 0.87 | $-$
102076870 | 3459372646830687104 | 0.982 | 0.011 | 41.944 | 42.713 | 10,37 | 0.412 | 1.068 | 10aafootnotemark:
118520413 | 145238687096970496 | 1.498 | 0.026 | 43.212 | 28.846 | 43,44 | 0.894 | 0.981 | 2bbfootnotemark:
125882806 | 148420639387738112 | 1.027 | 0.012 | 43.514 | 42.342 | 43,44 | 0.980 | 0.944 | 2bbfootnotemark:
125977598 | 147441558642852736 | 4.418 | 0.223 | 43.794 | 9.913 | 43,44 | 0.970 | 1.001 | 2bbfootnotemark:
150005829 | 148354733113981696 | 3.138 | 0.113 | 43.534 | 13.873 | 43,44 | 0.895 | 1.285 | 2bbfootnotemark:
150058659 | 148400229703257856 | 3.438 | 0.136 | 43.566 | 12.672 | 43,44 | 0.980 | 1.021 | 2bbfootnotemark:
150094175 | 157644373715415424 | 0.726 | 0.006 | 43.727 | 60.230 | 43,44 | 0.674 | 1.109 | 2bbfootnotemark:
150122702 | 148196510814073728 | 3.782 | 0.164 | 43.681 | 11.550 | 43,44 | 0.894 | 0.969 | 2bbfootnotemark:
167808464 | 5286310069347838080 | 0.279 | 0.000 | 145.412 | 583.983 | 7-13 | 0.223 | 0.99 | 45aafootnotemark:
170849193 | 4877824564574114304 | 0.329 | 0.002 | 23.768 | 72.243 | 5 | 0.845 | 1.079 | 45aafootnotemark:
232064183 | 6406967509044866560 | 0.718 | 0.006 | 41.292 | 57.510 | 27,28 | 0.064 | 1.062 | 24aafootnotemark:
268017136 | 151262876946558976 | 2.78 | 0.088 | 43.860 | 15.777 | 43,44 | 0.767 | 0.995 | 2bbfootnotemark:
268018471 | 152466120624336896 | 2.037 | 0.047 | 43.914 | 21.558 | 43,44 | 0.879 | 1.08 | 2bbfootnotemark:
268144659 | 152643240779301632 | 1.926 | 0.042 | 43.84 | 22.762 | 43,44 | 0.287 | 1.033 | 2bbfootnotemark:
268148996 | 151327159721125888 | 1.389 | 0.022 | 43.38 | 31.231 | 43,44 | 0.885 | 1.133 | 2bbfootnotemark:
268217504 | 149369139966814976 | 3.918 | 0.176 | 43.553 | 11.116 | 43,44 | 0.966 | 1.095 | 2bbfootnotemark:
268218180 | 146366442430208640 | 2.382 | 0.065 | 43.528 | 18.274 | 43,44 | 0.976 | 1.128 | 2bbfootnotemark:
268324394 | 146277553787186048 | 3.379 | 0.132 | 43.310 | 12.817 | 43,44 | 0.555 | 1.049 | 2bbfootnotemark:
268324578 | 147799209159857280 | 2.248 | 0.058 | 43.346 | 19.282 | 43,44 | 0.142 | 1.213 | 2bbfootnotemark:
268325250 | 151028990206478080 | 2.386 | 0.066 | 43.273 | 18.136 | 43,44 | 0.723 | 1.5 | 2bbfootnotemark:
268397898 | 147801339463632000 | 1.004 | 0.012 | 43.282 | 43.110 | 43,44 | 0.987 | 1.353 | 2bbfootnotemark:
268399069 | 151130591952773632 | 1.866 | 0.040 | 43.456 | 23.288 | 43,44 | 0.952 | 1.249 | 2bbfootnotemark:
298907057 | 3200303384927512960 | 0.484 | 0.005 | 23.879 | 49.337 | 5,32 | 0.825 | 1.099 | 200ddfootnotemark:
359892714 | 5841324542416019712 | 0.228 | 0.000 | 97.523 | 427.733 | 11,12,38,39 | 0.669 | 1.069 | 3.7 eefootnotemark:
360329359 | 5919792529769594240 | 0.502 | 0.005 | 26.857 | 53.500 | 39 | 0.142 | 0.974 | 149aafootnotemark:
367079252 | 2278720295036404736 | 2.769 | 0.029 | 132.993 | 48.029 | 17-19,24-26 | 0.906 | 1.026 | $-$
379097589 | 3724596914697213056 | 0.237 | 0.001 | 33.159 | 139.910 | 23,50 | 0.606 | 1.146 | 200ddfootnotemark:
397287296 | 3314309890186259712 | 2.655 | 0.080 | 43.785 | 16.492 | 43,44 | 0.826 | 1.125 | 2bbfootnotemark:
401838575 | 4954453580066220800 | 0.315 | 0.001 | 43.106 | 136.844 | 2,3 | 0.978 | 1.178 | 45aafootnotemark:
427770178 | 147614422487144960 | 1.479 | 0.025 | 43.355 | 29.314 | 43,44 | 0.726 | 1.198 | 2bbfootnotemark:
440859813 | 6224387727748521344 | 2.04 | 0.047 | 44.361 | 21.746 | 11,38 | 0.480 | 1.256 | 149aafootnotemark:
454227159 | 5201185574182015744 | 2.042 | 0.020 | 51.000 | 25.07 | 38,39 | 0.395 | 1.145 | 2bbfootnotemark:
454291779 | 5201129563515179520 | 3.23 | 0.075 | 70.000 | 21.672 | 11,38,39 | 0.848 | 1.024 | $-$
456944264 | 3314299238667410176 | 2.22 | 0.056 | 43.834 | 19.745 | 43,44 | 0.898 | 1.048 | 2bbfootnotemark:
902237947 | 3562157781229213312 | 0.241 | 0.001 | 21.439 | 88.884 | 36 | 0.210 | 3.117 | 200ddfootnotemark:
Targets with 2nd period | |
20305594 | 1388684770725399296 | 0.563 | 0.002 | 73.210 | 130.035 | 23,24,50,51 | 0.994 | 1.183 | 149aafootnotemark:
58638214 | 152917298349085824 | 0.367 | 0.001 | 67.494 | 183.758 | 19,43,44 | 0.726 | 1.372 | 2bbfootnotemark:
96680763 | 156915878541979008 | 0.117 | 0.000 | 44.205 | 377.821 | 43,44 | 0.107 | 0.928 | 2bbfootnotemark:
150094175 | 157644373715415424 | 0.966 | 0.011 | 43.727 | 45.266 | 43,44 | 0.674 | 1.109 | 2bbfootnotemark:
359892714 | 5841324542416019712 | 0.458 | 0.001 | 97.523 | 212.932 | 11,12,38,39 | 0.669 | 1.069 | 3.7eefootnotemark:
Note. — Ages are from (a)(Bell et al., 2015), (b)(Kenyon & Hartmann, 1995),
(c)(Zuckerman et al., 2011), (d)(Zuckerman et al., 2006), (e)(Murphy et al.,
2013)
Table 2: Gaia DR3 Ultra-Cool Dwarfs with low-amplitude and TESS unambiguous
periodicities.
TIC ID | GAIA ID | Prot (LS) | Amplitude | Prot (WT) | TESS sectors | Noisy | CROWD | RUWE | Age
---|---|---|---|---|---|---|---|---|---
| | (days) | (%) | (days) | with modulation | TESS sectors | | | (Myr)
26126812 | 68012529415816832 | 0.229 | 0.50 | 0.260 | 42,43,44 | | 0.974 | 1.217 | 24aafootnotemark:
50085560 | 2613754712222934656 | 0.427 | 0.80 | 0.444 | 42 | | 0.978 | 0.958 | 149aafootnotemark:
52256020 | 4691426694779173376 | 0.265 | 0.50 | 0.254 | 1,2,27-29 | | 0.971 | 1.292 | 45aafootnotemark:
58432630 | 152284735566828032 | 0.642 | 0.85 | 0.690 | 43,44 | | 0.511 | 0.945 | 2bbfootnotemark:
201688405 | 6473160651658069120 | 0.692 | 0.39 | 0.731 | 27 | 13 | 0.850 | 0.983 | 149aafootnotemark:
266012525 | 2556491429388743296 | 0.313 | 1.76 | 0.313 | 43 | | 0.971 | 1.079 | 45aafootnotemark:
275133493 | 3852362605385919616 | 0.438 | 0.73 | 0.446 | 8,35 | 45 | 0.808 | 1.042 | 149aafootnotemark:
281668854 | 4920330653311504256 | 0.116 | 0.56 | 0.118 | 2,28,29 | | 0.877 | 9.984 | 45aafootnotemark:
299007548 | 3230008650057256960 | 0.529 | 0.40 | 0.568 | 32 | 5 | 0.809 | 1.140 | 24aafootnotemark:
302396416 | 1113083282052337792 | 0.458 | 0.42 | 0.464 | 19,20,26,40,47 | 60,53 | 0.832 | 1.020 | 149aafootnotemark:
342967337 | 5316349173696052736 | 0.705 | 2.62 | 0.697 | 8 | 9,10 | 0.126 | 1.056 | 45aafootnotemark:
356632694 | 1640572714166158976 | 0.240 | 0.27 | 0.260 | 14,16,21,49 | 15,22-24,41,48,50 | 0.705 | 1.426 | 149 aafootnotemark:
371080717 | 898275195732381440 | 0.714 | 0.35 | 0.714 | 20 | 60 | 0.311 | 1.057 | 40ccfootnotemark:
388682292 | 5514929155583865216 | 0.533 | 9.80 | 0.543 | 8 | | 0.099 | 2.109 | 149aafootnotemark:
405514195 | 6059992460039778560 | 0.237 | 2.90 | 0.240 | 11 | | 0.160 | 1.047 | 15ddfootnotemark:
448590406 | 3478519134297202560 | 0.404 | 0.16 | 0.431 | 36 | 10 | 0.769 | 2.201 | 10aafootnotemark:
959690391 | 5856405272135505024 | 0.221 | 21.69 | 0.223 | 38 | | 0.005 | 1.144 | 149aafootnotemark:
Note. — Ages are from (a)(Bell et al., 2015), (b)(Kenyon & Hartmann, 1995),
(c)(Zuckerman et al., 2011), (d)(Pecaut & Mamajek, 2016)
Table 3: Gaia DR3 Ultra-Cool Dwarfs with TESS ambiguous or dubious variability. TIC ID | GAIA ID | Ambiguous or Dubious | Noisy | RUWE
---|---|---|---|---
| | TESS sectors | TESS sectors |
26390890 | 216704503361774080 | 42,43,44 | | 1.057
58108762 | 164474986623118592 | 43,44 | | 0.880
58175278 | 164783811951433856 | 43,44 | | 3.177
58229208 | 164702070133970944 | 19,43,44 | | 1.216
58285782 | 164502062096975744 | 19,43,44 | | 1.408
58544613 | 149629483705467008 | 43,44 | | 1.152
58601266 | 152029992465874560 | 43,44 | | 1.024
118710667 | 146874275068113664 | 43,44 | | 0.958
125882881 | 148449845165337600 | 43,44 | | 1.080
167670065 | 6118581861234228352 | 11 | 38 | 0.974
260304085 | 5496160698257650048 | 4 | 5,7,8,10-13 | 0.991
346721695 | 216575791780899200 | 42,44 | 43 | 1.004
348844681 | 5479241065435684352 | 13 | 5,7,8,12 | 0.987
367089503 | 2279554205887675392 | 17,19,24,25 | | 1.149
385557215 | 71504509626338048 | 42 | 43,44 | 0.909
387813466 | 2270043769607851264 | 16, 17,18,24, 25 | | 1.135
389053345 | 3311785239689504512 | 44 | 43 | 1.020
397367343 | 3480771277705948928 | 10 | | 1.041
436574130 | 3406128761895775872 | 43,44 | | 0.964
467753980 | 169163647804297216 | 44 | 43 | 1.010
Table 4: Gaia DR3 Ultra-Cool Dwarfs with previous TESS periodicities in the literature. TIC ID | Prot(h) | Amplitude(%) | TESS sectors | TESS sectors | Prot(h) | Amplitude(%) | TESS sectors | CROWD | RUWE
---|---|---|---|---|---|---|---|---|---
| this work | with modulation-this work | Noisy-this work | Miles-Páez et al. (2022) | |
8259901 | 3.072 $\pm$ 0.007 | 0.93 | 21 | — | 2.5267 $\pm$ 0.0002 | 0.44$\pm$0.01 | 21 | 0.240 | 2.785
8259902 | 3.072 $\pm$ 0.005 | 0.26 | 21,48 | — | 3.0889 $\pm$ 0.0002 | 0.44$\pm$0.01 | 21 | 0.829 | 1.498
16227031 | — | — | — | 44,45,46 | — | — | 44 | 0.663 | 1.212
17308656 | — | — | — | — | 4.9199$\pm$0.0035 | 0.58$\pm$0.08 | 43,44 | — | 2.794
27858644 | 2.832 $\pm$ 0.002 | 0.17 | 26,40,54 | 53 | 2.84140 $\pm$ 0.00039 | 0.34$\pm$0.03 | 26,40 | 0.703 | 1.096
32422219 | 2.112 $\pm$ 0.002 | 0.19 | 3,30 | — | 2.15012$\pm$0.00024 | 0.53$\pm$0.06 | 3,30 | 0.902 | 2.698
34014829 | 3.859 $\pm$ 0.007 | 0.24 | 2,29 | — | 3.8632 $\pm$ 0.0016 | 0.44$\pm$0.03 | 2,29 | 0.892 | 5.417
143029977 | 3.840 $\pm$ 0.014 | 0.1 | 31 | 4 | 3.8356 $\pm$0.0010 | 0.17$\pm$0.04 | 4,31 | 0.888 | 0.999
193974787 | 3.072 $\pm$ 0.009 | 0.24 | 16 | 23,24,49,51 | 3.0942$\pm$0.0010 | 0.48$\pm$0.08 | 16,23,24 | 0.651 | 0.998
198107795 | 11.292 $\pm$ 0.039 | 0.52 | 16,23,49,50 | — | 11.4334 $\pm$ 0.0058 | 0.59$\pm$0.03 | 16,23 | 0.845 | 11.011
202408306 | 15.336 $\pm$ 0.153 | 0.76 | 23,50 | — | 15.34$\pm$0.03 | 0.66$\pm$0.06 | 23 | 0.993 | 0.946
219095664 | 3.888 $\pm$ 0.001 | 0.24 | 14-21,24 | 22,23,40,41, | 3.89442 $\pm$ 0.00041 | 0.18$\pm$0.04 | 14-25,40,41 | 0.852 | 4.316
| | | 25,55,57,58 | 47-52,54,56,59 | | | | |
229653723 | — | — | — | — | — | $<$0.34 | 22 | — | 1.231
239097694 | — | — | — | 19,59 | — | — | 19 | 0.267 | 0.993
286963145 | — | — | — | 23 | — | $<$0.28 | 23 | 0.962 | 1.095
288506050 | — | — | — | 16,22,23 | 2.4221 $\pm$ 0.0003 | 0.70$\pm$0.05 | 16,22,23 | 0.986 | 1.178
307956653 | 3.614 $\pm$ 0.001 | 0.4 | 16-19,21,22,25,26,40 | 14,15,24,51 | 3.62703$\pm$0.00031 | 0.49$\pm$0.1 | 14-26,40,41 | 0.853 | 1.150
| | | 41,47-50,53,55-58,60 | 52,54,59 | | | | |
311188315 | — | — | — | 24,51 | 1.95976$\pm$0.00043 | 0.43$\pm$0.08 | 24 | 0.372 | 1.661
438883548 | — | — | — | — | 3.992$\pm$0.017 | 0.42$\pm$0.07 | 11 | — | 1.445
441000085 | — | — | — | 11,38 | 3.9614 $\pm$ 0.0009 | 0.77$\pm$0.06 | 11,38 | 0.831 | 1.185
902237947 | 6.000 $\pm$ 0.036 | 3.93 | 36 | — | 5.984$\pm$0.001 | 0.6$\pm$0.02 | 9 | 0.210 | 3.117
Figure 3: The distribution of rotation periods from the present analysis for
the UCDs with rotation signature and single source behavior. The dashed black
histogram stands for the 48 UCDs with unambiguous rotation periods, and the
red color for the 13 UCDs withunambiguous rotation and low amplitude
variability in the LCs.
Figure 4: The Gaia CMD for the sample of UCDs composing the present study,
with black circles illustrating the whole sample of 7630 Gaia DR3 UCDs, given
by Sarro et al. (2023). In the left panel circles in blue are for the UCDs
with $P_{rot}\geq 1.0$ d, red circles are for the UCDs with P${}_{rot}<1$ d,
and red open diamonds stand for the UCDs listed by Miles-Páez et al. (2023).
In the right panel, colors follow the color bar age intervals, which gives the
age in Myr.
|
# Towards Dynamic Urban Bike Usage Prediction for Station Network
Reconfiguration
Xi Yang and Suining He The University of ConnecticutStorrsConnecticut
xi.yang<EMAIL_ADDRESS>
###### Abstract.
Bike sharing has become one of the major choices of transportation for
residents in metropolitan cities worldwide. A station-based bike sharing
system is usually operated in the way that a user picks up a bike from one
station, and drops it off at another. Bike stations are, however, not static,
as the bike stations are often reconfigured to accommodate changing demands or
city urbanization over time. One of the key operations is to evaluate
candidate locations and install new stations to expand the bike sharing
station network. Conventional practices have been studied to predict existing
station usage, while evaluating new stations is highly challenging due to the
lack of the historical bike usage.
To fill this gap, in this work we propose a novel and efficient bike station-
level prediction algorithm called AtCoR, which can predict the bike usage at
both existing and new stations (candidate locations during reconfiguration).
In order to address the lack of historical data issues, virtual historical
usage of new stations is generated according to their correlations with the
surrounding existing stations, for AtCoR model initialization. We have
designed novel station-centered heatmaps which characterize for each target
station centered at the heatmap the trend that riders travel between it and
the station’s neighboring regions, enabling the model to capture the learnable
features of the bike station network. The captured features are further
applied to the prediction of bike usage for new stations. Our extensive
experiment study on more than 23 million trips from three major bike sharing
systems in US, including New York City, Chicago and Los Angeles, shows that
AtCoR outperforms baselines and state-of-art models in prediction of both
existing and future stations.
bike sharing, usage prediction, station-level, new stations, pick-ups and
drop-offs, attention, spatio-temporal
††copyright: none††conference: UrbComp ’20: The 9th SIGKDD International
Workshop on Urban Computing; August 24, 2020; San Diego, CA
## 1\. Introduction
Thanks to mobile networking and location-based services, bike sharing has
become one of the major transportation modalities for urban residents
worldwide due to its convenience and efficiency. As a representative product
of the sharing economy, it is often hailed as the excellent helper to solve
the “last mile” problem in citizen transportation. Given the social and
business importance, the bike sharing market is estimated to hit US$5 billion
by 2025111https://www.globenewswire.com/news-
release/2019/12/03/1955257/0/en/Bike-Sharing-Market-is-Predicted-to-
Hit-5-Billion-by-2025-P-S-Intelligence.html.
A station-based bike sharing system (each station is equipped with multiple
docks for bike parking) is usually operated in the way that a user or rider
picks up a bike from one station (pick-ups) and drops it off at another (drop-
offs). All the resultant bike trips (usage that consists of pairs of pick-ups
and drop-offs) connect different parts of the city, forming the bike sharing
station network. The bike stations are, however, not static, as the bike
sharing operators often reconfigure the stations to accommodate changing
demands or city urbanization over time (He and Shin, 2018; Liu et al., 2015),
where a key operation is to evaluate candidate locations and then install new
stations to expand the bike sharing station network (Zhang et al., 2016; Liu
et al., 2017) as shown in Figure 1. It is essential for the operators to know
the potential bike usage of a future station at a certain candidate location
beforehand, which helps predicting the station profitability as well as its
positive/negative effect upon the local mobility and traffic networks,
enhancing the bike sharing service quality to the local community. This
further benefits the future bike sharing network operations, including station
re-balancing (O’Mahony and Shmoys, 2015; Liu et al., 2016; Hulot et al., 2018;
Wang et al., 2018; Pan et al., 2019a) and bike route pre-planning (Zhang and
Philip, 2016).
Despite the business and social importance, predicting bike usage at future
stations is extremely challenging due to the lack of historical bike usage
data of those stations. Conventional practices include user survey,
crowdsourced feedbacks and public hearing, investigating the local demands,
which is often costly and time-consuming. Operating regulations are also
provided for bike sharing operators as guidance on candidate station
locations, which is based primarily on geo-information such as distance to
road intersections/public transit stations and station network
density222https://www.transformative-mobility.org/assets/publications/The-
Bikeshare-Planning-Guide-ITDP-Datei.pdf. However, such guidance is often too
general to take into account uniqueness of different cities, causing
inefficiency in station reconfiguration as well as reduced service quality and
profit loss. While most of the existing studies focus on predicting the bike
usage at the existing stations, only a few of them have comprehensively
explored how to forecast new stations for the station reconfiguration
purposes.
Figure 1. Illustration of bike sharing system and new station usage
prediction.
To fill this gap, in this work we propose a novel bike station prediction
algorithm called AtCoR, based on Attention Convolutional Recurrent Neural
Network, for predicting station-based bike usage of future stations. To tackle
the lack of historical data issues for future/new stations to be deployed,
virtual historical usage of new stations is generated according to their
correlation with the surrounding existing stations. To predict the usage of
new bike stations for reconfiguration decisions, we have designed novel
station-centered heatmaps which characterize for each target station centered
at the heatmap the trend that riders travel between it and the neighboring
regions, so that the model is able to capture the learnable common patterns of
the bike station network through a convolutional neural network (CNN). A Long
Short-Term Memory (LSTM) neural network with temporal attention mask leverages
the common patterns with integration of historical data and external factors,
such as weekends, holidays, and weather, to predict the bike usage for
existing stations. The captured bike usage features are then used to predict
the pick-ups and drop-offs for new stations along with the virtual historical
usage. An overview of information flow of this work is illustrated in Figure
2, summarizing the above process.
Our main technical contributions are as follows:
* 1)
Comprehensive bike data analysis for station prediction designs: We have
conducted comprehensive and detailed real-world data analysis on how the
weather conditions, regional bike usage and surrounding POIs impact the bike
usage in the metropolitan area like New York City (NYC), Chicago and Los
Angeles (LA), and visualized them to validate our design insights for station
reconfiguration prediction. These features serve as the shared patterns
leveraged for the bike usage prediction, including predicting for the new
stations.
* 2)
Dynamic urban bike usage prediction for reconfigured station networks: We have
designed a novel scheme called AtCoR to predict the station-level bike usage
for new/future stations (candidate locations for reconfiguration) as well as
existing/fixed ones. We propose a novel design of station-centered feature
heatmap representation as an input of AtCoR, which calculates the differences
between the features at the location of each station and those at the
surrounding areas. The heatmaps account for the trend that riders travel from
the center station to the neighborhood. The inclusion of the heatmaps
significantly improves the accuracy of usage predictions. Heatmaps are then
fed to AtCoR which consists of a deep Convolutional Neural Network (CNN) and a
Long Short-Term Memory (LSTM) network with integration of temporal attention
mechanism.
* 3)
Extensive experimental studies with real-world datasets: We have conducted
extensive experimental studies upon 23,955,989 trips in total, across three
major bike sharing systems in the United States: 20,551,697 from Citi Bike in
New York City (NYC), 3,113,950 from Divvy in Chicago, and 290,342 from Metro
Bike in Los Angeles (LA). Our experimental studies, upon both the existing and
new stations in the bike station network reconfiguration, have validated that
AtCoR outperforms the other baseline models in multi-station predictions,
often by more than 20% error reduction.
Figure 2. System overview and information flow of AtCoR.
While our studies focus on station-based bike sharing, the model and algorithm
derived can be further extended to other transportation platform with
deployment and expansion operations, including ride/car sharing (Chen et al.,
2017; Jauhri et al., 2017; He and Shin, 2019a) and scooter sharing (He and
Shin, 2020a).
The rest of the paper is organized as follows. We first conduct a brief survey
on related studies in Section 2. Then we analyze the datasets, and find out
the features and their corresponding representations for model input Section
3. Then the model is further presented in Section 4. Afterwards, we present
our extensive experimental studies to evaluate the performance of AtCoR for
both existing and new stations in Section 5. We will discuss the deployment in
Section 6, and conclude in Section 7.
## 2\. Related Work
Traffic flow prediction has been studied recently due to the advances of
intelligent transportation and smart cities. Gong et al. proposed a potential
passenger flow predictor to help the decision on the places to construct new
metro station (Gong et al., 2019). Tang et al. tackled the dense and
incomplete trajectories for citywide traffic volume inference (Tang et al.,
2019). For dockless bike sharing system, the potential bike distribution and
the detection of parking hotspots in a new city has been made by Liu et al.
(Liu et al., 2018a, b). Different from above works on dockless bike sharing,
we focus on station-based deployment due to its wider social acceptance, and
our studies provide highly comprehensive studies of predicting usage for
new/future stations before the bike sharing station reconfiguration (He and
Shin, 2018).
Based on bike usage prediction granularity, there are three categories of
prediction models in current works in the bike sharing systems: city-level,
cluster-level, and station-level (Li et al., 2019). In city-level prediction,
the aim is to predict bike usage for a whole city, while at the cluster level
the goal is to predict bike usage for clusters of bike stations. The station
cluster is generated by clustering algorithms such as the Bipartite Station
Clustering (Li et al., 2015), the Community Detection and the Agglomerative
Hierarchical Clustering method (Zhou, 2015), the K-means clustering and the
Latent Dirichlet Allocation (LDA). Chen et al. further considered clusters as
dynamic rather than static (Chen et al., 2016). While city-level and cluster-
level predictions save the computational cost by simplifying the problems,
station-level (Hulot et al., 2018; Chai et al., 2018; Chen et al., 2020; He
and Shin, 2020b) prediction still benefits the bike sharing system management
the most, including fine-grained station rebalancing, but yet is challenging.
Recent research efforts have also been made upon traffic prediction beyond the
bike sharing systems. Deep learning approaches have been studied for traffic
flow prediction (Lv et al., 2014; Huang et al., 2014; Yu et al., 2017;
Fadlullah et al., 2017; He and Shin, 2020a, 2019b). Yao et al. proposed a CNN-
LSTM based transfer learning method to predict traffic flows across different
cities (Yao et al., 2019). Pan et al. used a deep meta learning model to
predict urban traffic (Pan et al., 2019b). Ma et al. constructed spatial-
temporal graphs of urban traffic which was learned by a deep convolutional
neural network (Ma et al., 2017). In this work, we study a novel approach
based on deep learning designs and data-driven studies to handle the new
stations usage prediction problems. Specifically, we propose a novel approach
AtCoR which consists of a CNN component for modeling spatial characteristics
and a LSTM with integration of a temporal attention mechanism for capturing
temporal characteristics .
## 3\. Data Analysis & Feature Designs
Because no historical usage data of new stations is available, future mobility
patterns might only be predicted through accessible spatial and temporal
characteristics of the stations. In this work, we propose using station
specific regional usage, points of interest (POIs), weather conditions and
holidays as input features as they are all available once the locations and
launching time of new stations are provided by system operators.
In this section, we first present the preliminary concepts defined for the
feature designs of bike sharing stations in Section 3.1. Then in Section 3.2
we discuss our representation design of the features we choose, namely
station-centered heatmaps, which significantly improve the model performance.
We further cluster station networks based on such heatmaps to save
computational cost of training.
Table 1 summarizes all the symbols as well as their definitions in this work.
Table 1. List of symbols and definitions Symbols | Definitions
---|---
$n$ | New stations
$f$ | Existing stations
$\mathbf{F}$ | Set of existing stations
$j,k$ | Index of the stations
$T,S$ | Ranges of timestamps
$t,\tau$ | Indices of timestamps
$\mathcal{T}$ | Number of timestamps
$g_{lat},g_{lon}$ | Height and width of each grid
$\mathcal{G}_{lat},\mathcal{G}_{lon},\mathcal{P}$ | Dimensions of station-centered heatmaps
$\mathbf{H}$ | Station-centered heatmaps
$e$ | Entries of station-centered heatmaps
$L$ | Station usage
$h,w,c$ | Parameters of the convolutional layers
$\mathbf{x}$ | Input of LSTM
$\mathbf{h}$ | LSTM hidden state
$\mathbf{c}$ | LSTM cell state
$d$ | Number of hidden units of LSTM
$\mathbf{v},\mathbf{W},\mathbf{U},\mathbf{b}$ | Trainable parameters
$\lambda,\gamma$ | Attention scores between a pair of states
$\mathbf{d}$ | Decoder’s input
$\mathbf{ex}$ | External features
$\omega$ | | Similarity scores between a new station and
---
its surrounding existing peers
### 3.1. Preliminary for Station Feature Studies
#### 3.1.1. Overview of Datasets and New/Existing Stations
We conduct our data analytics upon three datasets: Citi Bike of
NYC333https://www.citibikenyc.com/ of 2019 (20,551,697 trips), Divvy Bike of
Chicago444https://data.cityofchicago.org/Transportation/Divvy-Trips/fg6s-gzvg
of the first three quarters of 2019 (3,113,950 trips), and Metro Bike of Los
Angeles555https://bikeshare.metro.net/ of 2019 (3,113,950 trips).
In this study, the new stations established in a set of certain time intervals
$[T,T^{\prime}]$ are defined as the ones that have no historical bike usage
data in the past 30 days, i.e. from $(T/24-30)$ days to $T$ considering $T$ on
hourly basis, while the active existing stations (or existing stations in
short) in a certain time interval $[S,S^{\prime}]$ are defined as the ones
that have historical usage data every day in this time interval. Note that
active existing stations focus on the existing stations which are populous
with everyday bike usage.
For example, there are 454 active existing stations at the New York City (NYC)
in 2019 in total, and from April 11, 2019 to July 19, 2019 there are 631
existing stations, while in June 2019 there are 8 new stations established,
compared to totally 1,047 unique bike station coordinates in the whole year
including those stations that are newly installed, removed, and relocated. The
monthly number of active existing stations and new stations are shown in
Figure 3 for the three datasets as demonstration.
Figure 3. The number of active existing stations and new stations per month in
NYC, Chicago and LA.
We select the following spatial and temporal features in our modeling:
regional usage in areas where bike stations are located, points of interest,
weather conditions and holidays and weekends. Since there is no historical
bike usage at the future locations of the new bike stations, it is difficult
to directly characterize the mobility patterns for the target locations.
However, the mobility patterns are strongly related to the spatial and
temporal features of those locations. Thus, in this work AtCoR incorporates
the correlations with the above features for the new station prediction.
#### 3.1.2. Regional Usage
To characterize the urban bike usage in a computationally efficient manner, we
discretize the neighborhood city map around each station into grids, each of
which is an $g_{lat}$ m $\times$ $g_{lon}$ m rectangular region. The total
bike pick-ups/drop-offs within the grid can represent the bike usage
popularity of this specific region. Bikes rented at the popular regions tend
to be returned at surrounding regions. Therefore, the station-centered
regional usage distribution is chosen an essential input feature.
#### 3.1.3. Points of Interest (POIs)
Another key insight of feature selection is that the differences in the POI
distributions around each station can steer the bike riders’ travels with the
corresponding preferences or purposes. Therefore, the POI distributions are
used as another important features. Following the manners of defining regional
station usage, the POI distributions are defined as the total numbers of POIs
within a $g_{lat}$ m $\times$ $g_{lon}$ m grid for each POI category.
Figure 4 illustrates some examples of the different distributions of different
categories of POIs in a 500m $\times$ 500m grid region. POIs around active
existing stations from April 11, 2019 to July 19, 2019 are shown here with the
numbers of POIs normalized by the min-max method.
Figure 4. POIs distributions around bike stations in NYC, Chicago and LA.
Specifically, the POIs data of NYC are obtained through NYC Open
Data666https://data.cityofnewyork.us/City-Government/Points-Of-
Interest/rxuy-2muj which contain 13 major categories, such as residential,
education facility, cultural facility, recreational facility, social services,
transportation facility, commercial, government facility (non public safety),
religious institution, health services, public safety, water and others. The
POIs of Chicago are obtained through OpenStreetMap Overpass
API777https://wiki.openstreetmap.org/wiki/Overpass_API where POIs are
categorized by the OSM tag of amenity including sustenance, education,
transportation, financial, healthcare, entertainment, arts $\&$ culture, and
others. The POIs of LA are obtained through LAC Open
Data888https://data.lacounty.gov/ which include communications,
transportation, private industry, health and mental health, social services,
postal, arts and recreation, community groups, municipal services, public
safety, education, government, emergency response, physical features and
environment.
#### 3.1.4. Weather Conditions, Holidays and Weekends
The station usage is highly correlated with the weather conditions. We have
collected and analyzed the weather condition data from open source weather
data API999https://api.weather.com. We have analyzed the correlations between
weather conditions and the station usage. Analysis of one-year bike usage
reveals that both high and low temperature decreases the bike usage. The
effect of daily temperature, precipitation and wind speed on daily overall
usage of the city is illustrated in Figures 5 and 6. Clearly, precipitation,
including rain, snow and large wind speed, significantly reduces the bike
usage. Given above, the weather conditions including temperature,
precipitation and wind speed are chosen as the input features. Besides weather
conditions, the bike usage has different patterns on federal holidays and
weekends from that on workdays. Specifically, we set the indicator as 1 if a
time interval belongs to holiday/weekend periods, or 0 for weekdays otherwise.
As an example, for the time interval of [0:00 a.m., 1:00 a.m.], 2019-01-01,
the external vector including temperature, wind speed, precipitation and
holiday/weekends, is given by [47 °F, 1.5 mph, 0.08 in, 1].
Figure 5. Temperature effect on daily overall bike usage. Figure 6. The effect
of precipitation and wind speed on city’s daily overall bike usage.
### 3.2. Feature Representations & Designs
Given the spatial and temporal features presented above for each station, we
present a representation design to integrate them as the model inputs, which
will be discussed in details as followed.
#### 3.2.1. Station-centered Heatmaps
We construct a $\mathcal{G}_{lat}\times\mathcal{G}_{lon}\times\mathcal{P}$
heatmap centered at the stations studied where each grid is a $g_{lat}$ m
$\times$ $g_{lon}$ m area on city map with $\mathcal{P}$ channels including
the regional usage and POIs distribution of the grid area. The first two
channels of the heatmap are the regional pick-ups and drop-offs, respectively,
with each of the following channel as the POIs distributions of one POIs
category. Every entry of the heatmaps is the regional usage or the POIs amount
for the corresponding grid location. Figure 7 shows a $11\times 11$ regional
pick-ups as the first channel of the station-centered heatmap for a candidate
station in NYC.
After construction of the heatmaps centered at a particular station, the
heatmaps are normalized by subtracting each one of the
$\mathcal{G}_{lat}\times\mathcal{G}_{lon}$ grid features by the features of
the center grid. This way, the normalized heatmaps represent the riders’
motivations or mobility trends departing from this station to the
neighborhoods, characterizing the spatial-temporal features near a station.
(a)
(b)
Figure 7. Regional pick-ups for a candidate station (red cycle) in Manhattan,
NYC on (a) Sunday 01-20-2019 from 8:00 a.m. to 9:00 a.m.; (b) Monday
01-21-2019 from 8:00 a.m. to 9:00 a.m.
#### 3.2.2. Station Clustering
To enhance the learning efficiency upon large-scale bike station network, we
design a clustering scheme for the bike stations and train the AtCoR for
certain cluster. We cluster existing stations and newly established ones all
together. This way, we can save computational cost by reducing the amount of
stations being trained, while the predictions upon new stations can be enabled
by the patterns learned from other stations.
We adopt the K-means clustering algorithm to find out the clusters from the
normalized station-centered heatmaps constructed as in Section 3.2. We first
calculate the sum of all the $\mathcal{G}_{lat}\times\mathcal{G}_{lon}$
entries at each channel, and the mean of heatmaps over the timeframe
$[T,T^{\prime}]$, generating vectors of length $\mathcal{P}$. The POI metric
scores between the heatmaps of stations $j$ and $k$ for the clustering are
calculated from the Euclidean distance between the generated vector:
(1)
$Sim_{j,k}=\left\|\sum_{e=1}^{\mathcal{G}_{lat}\times\mathcal{G}_{lon}}\overline{\mathbf{H}}_{j}^{{\textit{e}}}-\sum_{e=1}^{\mathcal{G}_{lat}\times\mathcal{G}_{lon}}\overline{\mathbf{H}}_{k}^{{\textit{e}}}\right\|.$
The combination of both existing and new stations in each city are then
clustered based on the above metric scores. The results of clustering existing
and new stations for NYC and Chicago based on $11\times 11\times\mathcal{P}$
heatmaps are shown below on Figure 8.
Figure 8. The cluster results of existing and new stations. The clusters
represented by dots contain new stations, while the cross ones have only
existing stations.
## 4\. AtCoR: Usage Prediction for Station Network Reconfiguration
We further discuss the details of AtCoR model in this section. We first define
our usage problem, followed by a description of model structure in Section
4.1. Our model is trained on the datasets of active existing stations, and the
details will be discussed in Section 4.2.
### 4.1. Problem & Model Definitions
#### 4.1.1. Problem Definition
The problem in this study can be defined as: given the input station-centered
heatmaps, $\mathbf{H_{n,\tau^{\prime}}}$, and the external features
$\mathbf{ex_{\tau^{\prime}}}$, including the weather conditions and
weekend/holidays information, for each station $n$ at the period of time
$\tau^{\prime}\in[1,\mathcal{T}]$, either an existing or new one after
reconfiguration, predict future usage (pick-ups and drop-offs), $L_{n,\tau}$,
where $\tau=\mathcal{T}+1$ is the target time interval for an existing station
or the one when the new station is established after reconfiguration.
#### 4.1.2. Model Overview
Our model consists of three major modules.
1. 1)
Spatial feature learning: A deep-channel CNN learns the station specific
heatmaps which describe the driving force of usage for each particular station
based on its spatial characteristics.
2. 2)
Temporal feature learning: The output of CNN combined with historical usage
data is then fed as input of a LSTM which learns the temporal pattern of
station’s usage.
3. 3)
Feature attention characterization: A temporal attention mechanism is applied
to further capture and differentiate the correlations between features across
different timestamps.
An overview of AtCoR is further illustrated in Figure 9. The details of each
module will be presented as follows.
Figure 9. Illustration of model designs in AtCoR.
#### 4.1.3. Convolutional Neural Network (CNN)
As described before, the station-specific or station-centered heatmaps as the
representation of spatial features, i.e. the POIs distribution and regional
usage popularity, indicate the motivation of people heading to places around
and hence correlate with the bike usage. Differences in this driving force may
result in different directions of bike usage and are represented by the
designed heatmaps. However, the mechanism behind this driving force of bike
usage can be too complicated to formulate, and simple vector concatenation is
not enough. Therefore, we propose a deep-channel CNN model to learn the
correlation between station specific spatial features in an area and station
bike usage. Note that different from previous studies (Ma et al., 2017), we
focus on learning station-centered heatmaps to further identify the useful
correlations between the center station’s usage and the neighborhood.
The CNN has $C$ convolutional layers, each of which is followed by a max
pooling layers. The size of the convolutional layers are $h\times w\times c$,
all with relu activation function. A fully connected layer is implemented as
the last layer to generate the output of CNN of size 1.
#### 4.1.4. Long Short-Term Memory (LSTM)
The output of the CNN is concatenated with historical usage data as the input
of LSTM. The last hidden state of the LSTM cell is connected to a fully
connected layer to generate predictions. The equations of an LSTM cell at
timestamp $t$ are listed below:
(2) $\displaystyle\mathbf{i}_{t}$
$\displaystyle=\sigma\left(\mathbf{W}^{i}\left[\mathbf{h}_{t-1},\mathbf{x}_{t}\right]+\mathbf{b}^{i}\right),$
$\displaystyle\mathbf{f}_{t}$
$\displaystyle=\sigma\left(\mathbf{W}^{f}\left[\mathbf{h}_{t-1},\mathbf{x}_{t}\right]+\mathbf{b}^{f}\right),$
$\displaystyle\mathbf{o}_{t}$
$\displaystyle=\sigma\left(\mathbf{W}^{o}\left[\mathbf{h}_{t-1},\mathbf{x}_{t}\right]+\mathbf{b}^{o}\right),$
$\displaystyle\mathbf{\tilde{c}}_{t}$
$\displaystyle=\text{tanh}\left(\mathbf{W}^{i}\left[\mathbf{h}_{t-1},\mathbf{x}_{t}\right]+\mathbf{b}^{i}\right),$
$\displaystyle\mathbf{c}_{t}$
$\displaystyle=\left(\mathbf{f}_{t}*\mathbf{c}_{t-1}+\mathbf{i}_{t}*\mathbf{\tilde{c}}_{t}\right),$
$\displaystyle\mathbf{h}_{t}$
$\displaystyle=\text{tanh}\left(\mathbf{c}_{t}\right)*\mathbf{o}_{t},$
where $\mathbf{x}_{t}\in\mathbb{R}^{l}$ is the input of this timestamp of size
$l$, which is the concatenation of historical usage and the CNN output,
$\mathbf{c}_{t}$ and $\mathbf{h}_{t}$ are the cell state and hidden state of
LSTM cell at time $t$, $\mathbf{i}_{t}$, $\mathbf{f}_{t}$, $\mathbf{o}_{t}$
and $\mathbf{\tilde{c}}_{t}$ are intermediate variables of LSTM cell, and
$\mathbf{W}^{q}\in\mathbb{R}^{d\times(d+l)}$ and
$\mathbf{b}^{q}\in\mathbb{R}^{d}$ where $q=\mathbf{i},\mathbf{f},\mathbf{o}$
are trainable variables, and $d$ is the number of hidden units of LSTM cell.
#### 4.1.5. Temporal Attention
The temporal attention mechanism captures correlation between features across
different timestamps. The intuition behind this is that there is a strong
temporal dependency of the usage of a station at time $t$, for example, on the
usage at the same time one day ago. The temporal attention scores between the
current decoder LSTM timestep $t$ and one of the previous encoder LSTM hidden
states, $\lambda_{t,t^{\prime}}$, are calculated as Eq. (3) by a concatenation
manner:
(3)
$\lambda_{t,t^{\prime}}=\mathbf{v}_{a}^{\intercal}\texttt{tanh}\left(\mathbf{W}_{a}[\mathbf{h}_{t-1};\mathbf{c}_{t-1}]+\mathbf{U}_{a}\mathbf{h}_{t^{\prime}}+\mathbf{b}_{a}\right),$
where $\mathbf{v}_{a},\mathbf{b}_{a}\in\mathbb{R}^{d}$,
$\mathbf{W}_{a}\in\mathbb{R}^{d\times 2d}$ and
$\mathbf{U}_{a}\in\mathbb{R}^{d\times d}$ are learnable parameters, $d$ is the
number of hidden units of LSTM cell, $\mathbf{h}_{t}$ and $\mathbf{c}_{t}$ are
hidden state and cell state of decoder at timestamp $t$, and
$\mathbf{h}_{t^{\prime}}$ is hidden state of encoder at timestamp
$t^{\prime}$, which is in range of $[1,\mathcal{T}]$. The attention weight,
denoted as $\gamma_{t,t^{\prime}}$, is then a softmax function of
$\lambda_{t,t^{\prime}}$ as in Eq. (4):
(4)
$\gamma_{t,t^{\prime}}=\frac{\texttt{exp}\left(\lambda_{t,t^{\prime}}\right)}{\sum_{t=1}^{\mathcal{T}}\texttt{exp}\left(\lambda_{t,t^{\prime}}\right)}.$
The weighted sum of the encoder LSTM hidden states, $\mathbf{d}_{t}$, shown in
Eq. (5) concatenated with external features $\mathbf{ex_{t}}$,
$[\mathbf{d}_{t}:\mathbf{ex_{t}}]$, serves as the input of decoder:
(5)
$\mathbf{d}_{t}=\sum_{t^{\prime}=1}^{\mathcal{T}}\gamma_{t,t^{\prime}}\mathbf{h}_{t^{\prime}},$
where $\mathcal{T}$ is the number of encoder timestamps.
### 4.2. Model Training & Prediction for New Stations
As conventional practices, we handle the existing station prediction based on
the historical data available. Given absence of historical data, the
predictions of the usage for new stations are based on the model trained on
the active existing stations. To address the initialization problem of the
model, during the training process, we generate the virtual trip data based on
the weighted average of those from multiple existing stations. This is
achieved by randomly picking a batch from the samples of all the existing
stations at each training epoch. This way, the model is able to learn their
shared bike usage patterns across all the existing stations, and such
knowledge learned can be used to predict the usage of new stations,
considering that the new stations share correlated spatial and temporal usage
features of existing stations.
The model is trained upon the existing stations within each cluster. To
address the model initialization without historical data, for the new stations
we design an efficient mechanism to generate the virtual historical usage from
those of surrounding existing stations. We note that the two adjacent stations
have similar bike usage and mobility patterns because of their similar
spatial-temporal characteristics. Therefore, we generate the virtual bike
usage by the distances between stations. Specifically, we compute the
geographic similarity score between a new station, $n$, and an existing peer
in its neighborhood, $f$, based on their mutual geographic distance in km:
(6) $Sim_{n,f}=\frac{1}{\text{distance}\left(n,f\right)}.$
Then the similarity scores across all the existing stations is normalized to
find the weights for the known bike usage assigned upon each existing station:
(7) $\omega_{n,f}=\frac{Sim_{n,f}^{2}}{\sum_{f=1}^{\mathbf{F}}Sim_{n,f}^{2}}.$
The virtual bike usage for the new station at time $\tau$ based on the
historical usage of those existing stations, $L_{f,\tau}$ is finally
calculated by:
(8) $L_{n,\tau}=\sum_{f\in\mathbf{F}}^{\mathbf{F}}\omega_{n,f}L_{f,\tau},$
where $\mathbf{F}$ is the set of existing stations.
## 5\. Experimental Studies
In this section, we present the experimental results based on the datasets of
Citi (NYC), Divvy (Chicago) and Metro (LA). The experimental settings are
first introduced in Section 5.1, followed by experimental results on
existing/new stations in Section 5.2.
### 5.1. Experimental Settings
We compare AtCoR with the following baselines and state-of-art models on the
given datasets:
1. (1)
ARIMA: Auto Regressive Integrated Moving Average for time-series forecasting.
The size of sliding window is set to be 24.
2. (2)
RNN: Simple Recurrent Neural Network for time-series predictions (Pan et al.,
2019c). The length of input sequence is 24.
3. (3)
LSTM: Long Short-Term Memory neural network (Lin et al., 1996) predicts future
usage with historical data of last 24 hour.
4. (4)
GRU: Gated Recurrent Units as another recurrent neural network for time-series
predictions (Cho et al., 2014).
5. (5)
GCN: Graph Convolutional Neural Networks with Data-driven Graph Filter (Chai
et al., 2018) predicts station-level usage taking into account of station
correlations.
We are leveraging the features for previous 24 hours to predict the station-
level bike usage for the next timestamp. For the evaluation of the existing
stations for NYC and Chicago datasets, our model as well as comparison schemes
are trained based on the hourly usage data from April 11, 2019 to July 19,
2019 (2,400 hours duration), and are tested on the hourly usage data from July
20, 2019 to August 18, 2019 (720 hours duration). For LA datasets, the models
are trained based on the 4-hour bike usage data in June 2019 (720 hours
duration), and are tested on the data for the following 30 days (720 hours
duration).
For the evaluation of the new stations, we find out all the deployed stations
from May to August 2019 for NYC and Chicago, and those from June to December
2019 for LA. Our model as well as the baselines are tested on the usage data
within 4 weeks (672 hours duration) for NYC and Chicago and 2 weeks for LA
from the first appearance of the hourly usage of the stations. The stations
with sufficient bike usage are of our interest here, and we discard those with
little usage (say, less than 10 pick-ups/drop-offs per day). For new stations
in LA, due to the large sparsity in the bike usage data, we consider the
appearance of consecutive bike usage as the first usage and the start of our
evaluations. The numbers of existing/new stations studied in our experimental
evaluations are further identified and listed as follows. The numbers of
existing and new stations studied, (existing, new), for each city are as
follows: (631, 15) for NYC, (229, 13) for Chicago, (48, 7) for LA. The
existing and new stations are clustered together in the manner described in
Section 3.2.2. We train AtCoR and baselines models on all the clusters of the
three bike sharing systems. Then we evaluate models regarding their
predictions of new stations.
Figure 10. Distributions of bike trip distance (meters) for NYC, Chicago and
LA.
Figure 10 shows that a majority of bikes rented at one station in NYC, Chicago
and LA, are returned at stations beyond 500m away, demonstrating that for a
specific station there is a small number ($<20\%$) of bike transitions between
the stations within a grid of 500m $\times$ 500m and the bikes rented at this
region are mostly returned to regions somewhere else. In addition, a majority
of bike trips are within the range of 500m and 5,000m as shown in Figure 10.
Therefore, the size of station-centered heatmaps discussed in Section 3.2.1 is
set to be $11\times 11\times\mathcal{P}$, where $\mathcal{P}$ is 2 (regional
pick-ups and drop-offs) plus the total number of POIs categories (13 for NYC,
7 for Chicago and 15 for LA), and each grid of the heatmaps is a 500m $\times$
500m area on the city map. This way, the heatmaps cover most of the areas
where the riders can reach by the bikes rented from the center station.
Other model parameters of AtCoR are set as followed. The CNN component has 3
layers convolutional layers with sizes $h\times w\times c$; we set $3\times
3\times 256$, $3\times 3\times 128$ and $2\times 2\times 64$, respectively,
and relu activation function for each of them. Each of the first two
convolutional layers is followed by a maxpooling layer. The final
convolutional layer is connected with a fully connected layer which converts
the CNN’s output into the one of size 1. The number of layers of LSTM is
1,024. The dropout rate for the LSTM is 0.5. The learning rate is 0.001. The
batch size is set to be 128. Total number of training epochs is 5,000. We have
implemented AtCoR and other schemes in Python 3.7 and Tensorflow 2.1, and the
models are trained and evaluated upon a desktop server with Intel i5-8700K,
16GB RAM, Nvidia GTX 1060/1050Ti and Windows 10.
We use mean square error (MSE) as the training metric, and we evaluate model
performance as well as the results based on both MSE and mean abosolute error
(MAE):
(9) $\displaystyle
MSE=\frac{1}{M}\sum_{i}^{M}\left(y_{i}-\hat{y}_{i}\right)^{2},\quad
MAE=\frac{1}{M}\sum_{i}^{M}\left|y_{i}-\hat{y}_{i}\right|,$
where M, $y_{i}$ and $\hat{y}_{i}$ are the total number of predictions made by
the evaluated model, the ground-truth of the bike usage and the predicted bike
usage, respectively.
### 5.2. Experimental Results
#### 5.2.1. Usage Prediction for Existing Stations
First, we test our model on all active existing stations for all three
datasets. The accuracy of bike usage predictions across the existing stations
is shown in Table 2. It is shown that our model achieves overall better
accuracy compared with other baselines for the systems in NYC, Chicago and LA.
It is mainly because the station-centered heatmaps incorporate the trend that
the bike riders travel between the stations. Focusing upon each station’s
neighborhood, a heatmap characterizes the mobility patterns, which are learned
and captured by AtCoR, and thus the bike usage at the new stations with
similar neighborhood patterns can be further predicted. For Metro in LA, GCN
achieves comparable or slightly better performance like AtCoR, likely due to
the sparsity of bike station network in LA. Despite this, AtCoR has shown high
accuracy and robustness with complicated mobility patterns.
(a) A station at NYC (40.7659∘, -73.9763∘).
(b) A station at Chicago (41.8782∘, -87.6319∘).
(c) A station at LA (34.085∘,-118.259∘).
Figure 11. Hourly bike usage prediction for the existing stations in the three
datasets.
Figure 11 illustrates the bike usage (pick-ups and drop-offs) predictions for
the three existing stations in NYC, Chicago and LA. We can observe that the
predictions of AtCoR are highly accurate, and close to the ground-truth
measurements. Such accuracy can enable advanced bike station operational
applications like station demand and supply rebalancing.
Table 2. Comparison between AtCoR and other baseline and state-of-art models
for existing stations of the three datasets.
Schemes | ARIMA | RNN | GCN | LSTM | GRU | AtCoR
---|---|---|---|---|---|---
Citi | Pick-up | MAE | 2.682 | 2.345 | 2.240 | 2.164 | 2.166 | 1.771
MSE | 30.028 | 18.711 | 12.226 | 16.080 | 16.143 | 10.599
Drop-off | MAE | 2.564 | 2.265 | 2.084 | 2.100 | 2.112 | 1.730
MSE | 24.762 | 16.514 | 11.318 | 14.210 | 14.245 | 9.447
Divvy | Pick-up | MAE | 1.964 | 1.832 | 2.300 | 1.437 | 1.418 | 1.263
MSE | 23.693 | 14.866 | 6.382 | 11.293 | 10.732 | 7.687
Drop-off | MAE | 1.927 | 1.633 | 1.067 | 1.247 | 1.404 | 1.247
MSE | 21.075 | 13.584 | 6.382 | 10.937 | 10.508 | 7.980
Metro | Pick-up | MAE | 2.126 | 3.258 | 1.276 | 1.878 | 1.936 | 1.462
MSE | 8.578 | 16.071 | 4.895 | 7.471 | 7.26 | 4.879
Drop-off | MAE | 2.154 | 1.478 | 1.480 | 1.251 | 1.005 | 1.435
MSE | 8.752 | 4.535 | 3.965 | 3.393 | 2.245 | 4.610
Table 3. Performance comparison for predictions of new stations.
Schemes | RNN | LSTM | GRU | AtCoR
---|---|---|---|---
Citi | Pick-up | MAE | 2.922 | 2.760 | 2.736 | 2.495
MSE | 20.961 | 18.652 | 18.461 | 15.883
Drop-off | MAE | 2.774 | 2.728 | 2.738 | 2.568
MSE | 17.833 | 18.930 | 19.016 | 15.835
Divvy | Pick-up | MAE | 2.092 | 1.814 | 1.825 | 1.627
MSE | 23.475 | 21.089 | 21.025 | 13.282
Drop-off | MAE | 1.812 | 1.589 | 1.639 | 1.442
MSE | 14.126 | 11.491 | 12.317 | 8.223
Metro | Pick-up | MAE | 1.552 | 1.249 | 1.246 | 1.098
MSE | 3.286 | 3.425 | 3.135 | 2.439
Drop-off | MAE | 1.588 | 1.583 | 1.216 | 1.043
MSE | 3.683 | 3.662 | 2.463 | 2.795
Table 4. Comparison of performance between AtCoR w/ and w/o virtual generated
data.
Schemes | Citi | Divvy | Metro
---|---|---|---
MAE | MSE | MAE | MSE | MAE | MSE
Pick-up | AtCoR w/ virtual | 2.384 | 15.018 | 1.638 | 11.983 | 1.305 | 4.639
AtCoR w/o virtual | 2.387 | 16.661 | 1.685 | 14.528 | 1.395 | 5.014
Drop-off | AtCoR w/ virtual | 2.666 | 15.892 | 1.589 | 9.320 | 1.561 | 5.797
AtCoR w/o virtual | 2.547 | 16.510 | 1.522 | 10.300 | 1.727 | 7.929
#### 5.2.2. Bike Usage Prediction for New Stations
The predictions of bike usage for new stations can be done using the knowledge
learned from the existing stations. Given the trained models from all the
existing stations, RNN, LSTM and GRU can be directly applied to predict each
individual new stations, and therefore we focus on comparing AtCoR with the
three approaches here. We leverage the knowledge learned from the existing
stations for the predictions of new stations given the bike station network
reconfiguration. MAE and MSE of new stations’ usage predictions, pick-ups and
drop-offs, by AtCoR as well as three baselines, RNN, LSTM and GRU, are shown
in Table 3. As is shown, AtCoR outperforms those three baseline models in
predicting the usage of new stations for all three bike sharing systems.
We illustrate the performance with and without the virtual bike usage data
generated. Virtual bike usage one day ahead of the first bike usage of new
stations are generated as in Section 4.2 as the starting points. The virtual
usage provides an initial inference for the models regarding how the mobility
pattern of a new station possibly look like, enhancing the model accuracy as
is shown in Table 4. Here we only compare the MAE and MSE for predicting the
next 24 hours after the first use of the stations given the data availability.
## 6\. Discussion
Incorporating Other Information: Despite the features selected in this work as
described in Section 3.1, other features may also be correlated with bike
usage, such as events and demographic distributions (Chen et al., 2016).
However, such information is not considered due to the limit of our current
resources. Our prediction accuracy could be further increased with the
inclusion of those features in our model. Nevertheless, the generic design of
station-centered heatmaps as the feature representations in this study allows
the easy integration of other information.
Sparsity of Usage Data: Though we focus on active existing stations in this
study, a lot of stations are not so active that their bike usage is low,
especially in LA . For experimental study of LA bike usage, we consider 4-hour
time interval as one timestamp to lower the influence of data sparsity.
Predicting the bike usage with sparsity is challenging, yet important for
system management. Further study on how to deal with sparse historical data is
necessary.
Similarity Scores for Virtual Data Generation: As mentioned in Section 5.2.2,
virtual historical usage is essential for the first few predictions, and their
accuracy depends on the quality of the initial inference in addition to the
model performance. In this work, we compute the virtual historical usage based
on the mutual geometric distances between stations, which has been shown to
benefit predicting the new stations. However, further enhanced inference can
be achieved by inclusion of other information such as crowd’s awareness of new
stations, where a comprehensive mechanism is needed for future study.
## 7\. Conclusion
In this work, we propose a novel bike station prediction algorithm called
AtCoR for predicting station-based bike usage of future stations given bike
station network reconfiguration. We design novel station-centered heatmaps
which characterize for each target station centered at the heatmap the trend
that the riders travel between it and the neighboring regions, making the
common patterns of the bike station network learnable. AtCoR further leverages
such knowledge learned to predict the usage for new stations with the aid of
virtual historical usage of new stations generated according to their
correlation to the surrounding existing stations. Extensive experiment study
on the bike sharing systems of NYC, Chicago and LA shows AtCoR is capable in
predicting usage at new/future stations as well as existing stations, and
outperforms the baseline and state-of-art models in our experimental
evaluations.
## References
* (1)
* Chai et al. (2018) Di Chai, Leye Wang, and Qiang Yang. 2018. Bike flow prediction with multi-graph convolutional networks. In _Proc. ACM SIGSPATIAL_. 397–400.
* Chen et al. (2016) Longbiao Chen, Daqing Zhang, Leye Wang, Dingqi Yang, Xiaojuan Ma, Shijian Li, Zhaohui Wu, Gang Pan, Thi-Mai-Trang Nguyen, and Jérémie Jakubowicz. 2016\. Dynamic cluster-based over-demand prediction in bike sharing systems. In _Proc. ACM UbiComp_. 841–852.
* Chen et al. (2017) Min Hao Chen, Abhinav Jauhri, and John Paul Shen. 2017\. Data driven analysis of the potentials of dynamic ride pooling. In _Proc. ACM SIGSPATIAL_. 7–12.
* Chen et al. (2020) Po-Chuan Chen, He-Yen Hsieh, Kuan-Wu Su, Xanno Kharis Sigalingging, Yan-Ru Chen, and Jenq-Shiou Leu. 2020. Predicting station level demand in a bike-sharing system using recurrent neural networks. _IET Intelligent Transport Systems_ (2020).
* Cho et al. (2014) Kyunghyun Cho, Bart Van Merriënboer, Dzmitry Bahdanau, and Yoshua Bengio. 2014. On the properties of neural machine translation: Encoder-decoder approaches. _arXiv preprint arXiv:1409.1259_ (2014).
* Fadlullah et al. (2017) Zubair Md Fadlullah, Fengxiao Tang, Bomin Mao, Nei Kato, Osamu Akashi, Takeru Inoue, and Kimihiro Mizutani. 2017\. State-of-the-art deep learning: Evolving machine intelligence toward tomorrow’s intelligent network traffic control systems. _IEEE COMST_ 19, 4 (2017), 2432–2455.
* Gong et al. (2019) Yongshun Gong, Zhibin Li, Jian Zhang, Wei Liu, and Jinfeng Yi. 2019. Potential Passenger Flow Prediction: A Novel Study for Urban Transportation Development. _arXiv preprint arXiv:1912.03440_ (2019).
* He and Shin (2018) Suining He and Kang G Shin. 2018. (Re) Configuring Bike Station Network via Crowdsourced Information Fusion and Joint Optimization. In _Proc. ACM MobiHoc_. 1–10.
* He and Shin (2019a) Suining He and Kang G. Shin. 2019a. Spatio-Temporal Adaptive Pricing for Balancing Mobility-on-Demand Networks. _ACM Trans. Intell. Syst. Technol._ 10, 4, Article 39 (July 2019), 28 pages.
* He and Shin (2019b) Suining He and Kang G Shin. 2019b. Spatio-temporal capsule-based reinforcement learning for mobility-on-demand network coordination. In _Proc. WWW_. 2806–2813.
* He and Shin (2020a) Suining He and Kang G Shin. 2020a. Dynamic Flow Distribution Prediction for Urban Dockless E-Scooter Sharing Reconfiguration. In _Proc. WWW_. 133–143.
* He and Shin (2020b) Suining He and Kang G Shin. 2020b. Towards Fine-grained Flow Forecasting: A Graph Attention Approach for Bike Sharing Systems. In _Proc. WWW_. 88–98.
* Huang et al. (2014) Wenhao Huang, Guojie Song, Haikun Hong, and Kunqing Xie. 2014\. Deep architecture for traffic flow prediction: deep belief networks with multitask learning. _IEEE T-ITS_ 15, 5 (2014), 2191–2201.
* Hulot et al. (2018) Pierre Hulot, Daniel Aloise, and Sanjay Dominik Jena. 2018\. Towards station-level demand prediction for effective rebalancing in bike-sharing systems. In _Proc. ACM KDD_. 378–386.
* Jauhri et al. (2017) Abhinav Jauhri, Brian Foo, Jerome Berclaz, Chih Chi Hu, Radek Grzeszczuk, Vasu Parameswaran, and John Paul Shen. 2017. Space-time graph modeling of ride requests based on real-world data. In _Proc. AAAI_.
* Li et al. (2015) Yexin Li, Yu Zheng, Huichu Zhang, and Lei Chen. 2015\. Traffic prediction in a bike-sharing system. In _Proc. ACM SIGSPATIAL_. 33.
* Li et al. (2019) Youru Li, Zhenfeng Zhu, Deqiang Kong, Meixiang Xu, and Yao Zhao. 2019. Learning Heterogeneous Spatial-Temporal Representation for Bike-Sharing Demand Prediction. In _Proc. AAAI_ , Vol. 33. 1004–1011.
* Lin et al. (1996) Tsungnan Lin, Bill G Horne, Peter Tino, and C Lee Giles. 1996\. Learning long-term dependencies in NARX recurrent neural networks. _IEEE TNNLS_ 7, 6 (1996), 1329–1338.
* Liu et al. (2015) Junming Liu, Qiao Li, Meng Qu, Weiwei Chen, Jingyuan Yang, Hui Xiong, Hao Zhong, and Yanjie Fu. 2015\. Station site optimization in bike sharing systems. In _Proc. IEEE ICDM_. IEEE, 883–888.
* Liu et al. (2016) Junming Liu, Leilei Sun, Weiwei Chen, and Hui Xiong. 2016\. Rebalancing bike sharing systems: A multi-source data smart optimization. In _Proc. ACM KDD_. 1005–1014.
* Liu et al. (2017) Junming Liu, Leilei Sun, Qiao Li, Jingci Ming, Yanchi Liu, and Hui Xiong. 2017\. Functional zone based hierarchical demand prediction for bike system expansion. In _Proc. ACM SIGKDD_. 957–966.
* Liu et al. (2018a) Zhaoyang Liu, Yanyan Shen, and Yanmin Zhu. 2018a. Inferring dockless shared bike distribution in new cities. In _Proc. ACM WDSM_. 378–386.
* Liu et al. (2018b) Zhaoyang Liu, Yanyan Shen, and Yanmin Zhu. 2018b. Where Will Dockless Shared Bikes be Stacked? —Parking Hotspots Detection in a New City. In _Proc. ACM KDD_. 566–575.
* Lv et al. (2014) Yisheng Lv, Yanjie Duan, Wenwen Kang, Zhengxi Li, and Fei-Yue Wang. 2014. Traffic flow prediction with big data: a deep learning approach. _IEEE T-ITS_ 16, 2 (2014), 865–873.
* Ma et al. (2017) Xiaolei Ma, Zhuang Dai, Zhengbing He, Jihui Ma, Yong Wang, and Yunpeng Wang. 2017\. Learning traffic as images: a deep convolutional neural network for large-scale transportation network speed prediction. _Sensors_ 17, 4 (2017), 818.
* O’Mahony and Shmoys (2015) Eoin O’Mahony and David B Shmoys. 2015. Data analysis and optimization for (Citi)bike sharing. In _Proc. AAAI_.
* Pan et al. (2019a) Ling Pan, Qingpeng Cai, Zhixuan Fang, Pingzhong Tang, and Longbo Huang. 2019a. A deep reinforcement learning framework for rebalancing dockless bike sharing systems. In _Proc. AAAI_ , Vol. 33. 1393–1400.
* Pan et al. (2019c) Yan Pan, Ray Chen Zheng, Jiaxi Zhang, and Xin Yao. 2019c. Predicting bike sharing demand using recurrent neural networks. _Procedia Computer Science_ 147 (2019), 562–566.
* Pan et al. (2019b) Zheyi Pan, Yuxuan Liang, Weifeng Wang, Yong Yu, Yu Zheng, and Junbo Zhang. 2019b. Urban traffic prediction from spatio-temporal data using deep meta learning. In _Proc. ACM KDD_. 1720–1730.
* Tang et al. (2019) Xianfeng Tang, Boqing Gong, Yanwei Yu, Huaxiu Yao, Yandong Li, Haiyong Xie, and Xiaoyu Wang. 2019. Joint modeling of dense and incomplete trajectories for citywide traffic volume inference. In _Proc. WWW_. 1806–1817.
* Wang et al. (2018) Shuai Wang, Tian He, Desheng Zhang, Yuanchao Shu, Yunhuai Liu, Yu Gu, Cong Liu, Haengju Lee, and Sang H Son. 2018\. BRAVO: Improving the rebalancing operation in bike sharing with rebalancing range prediction. _Proc. ACM IMWUT_ 2, 1 (2018), 1–22.
* Yao et al. (2019) Huaxiu Yao, Yiding Liu, Ying Wei, Xianfeng Tang, and Zhenhui Li. 2019. Learning from multiple cities: A meta-learning approach for spatial-temporal prediction. In _Proc. WWW_. 2181–2191.
* Yu et al. (2017) Bing Yu, Haoteng Yin, and Zhanxing Zhu. 2017. Spatio-temporal graph convolutional networks: A deep learning framework for traffic forecasting. _arXiv preprint arXiv:1709.04875_ (2017).
* Zhang et al. (2016) Jiawei Zhang, Xiao Pan, Moyin Li, and Philip S Yu. 2016\. Bicycle-sharing systems expansion: station re-deployment through crowd planning. In _Proc. ACM SIGSPATIAL_. 1–10.
* Zhang and Philip (2016) Jiawei Zhang and S Yu Philip. 2016. Trip route planning for bicycle-sharing systems. In _Proc. IEEE CIC_. IEEE, 381–390.
* Zhou (2015) Xiaolu Zhou. 2015\. Understanding spatiotemporal patterns of biking behavior by analyzing massive bike sharing data in Chicago. _PloS One_ 10, 10 (2015), e0137922.
|
# Ion-driven destabilization of a toroidal electron plasma - A 3D3VPIC
Simulation
S. Khamaru Institute for Plasma Research, Bhat, Gandhinagar 382428, India
<EMAIL_ADDRESS>Homi Bhabha National Institute, Training School
Complex, Anushaktinagar, Mumbai 400094, India R. Ganesh Institute for Plasma
Research, Bhat, Gandhinagar 382428, India Homi Bhabha National Institute,
Training School Complex, Anushaktinagar, Mumbai 400094, India M. Sengupta
Lawrence Livermore National Laboratory, Livermore, California, 94551, USA
###### Abstract
Ion resonance instability of toroidal electron plasmas in a tight aspect ratio
axisymmetric toroidal device is reported for ${Ar}^{+}$ ions of different
initial density values using a high fidelity 3D3V PIC solver. Stability of a
recently discovered quiescent quasi-steady state (QQS) of a toroidal electron
plasma obtained from “seed” solution as a result of entropy extremization at
zero inertia, is addressed to the presence of a small ion population. An ion
fraction ($f$) and corresponding number of secondary electrons are preloaded
into the system after the electron plasma attains a QQS state. Driven by the
ions, the electron plasma exhibits destabilized “center of charge motion” ($m$
$=$ $1$) along with increased poloidal mode coupling ($m$ $=$ $1$ to $9$) with
dominant $m$ $=$ $2$ mode. The growth in wall probe current is algebraic in
nature and increases for $f$ $\geq$ $0.005$, showing saturation at later time.
Higher values of ion temperatures than the electron temperatures indicate a
resonant energy transfer from electron plasma to ions via ion-resonance and
concomitant ion heating. The volume averaged temperature value of the electron
plasma rises with simulation time, attaining a quasi-steady nature near the
end of the simulation time. As can be expected from conservation of adiabatic
invariants, the flux tube averaged electron temperatures along parallel and
perpendicular directions are found to scale as $1/R^{2}$ and $1/R$
respectively, where $R$ is the major radial variable, though the plasma is
nearly collision-less.
††preprint: AIP/123-QED
## I Introduction
Instability induced confinement issues in charged particle traps are
ubiquitous and effective mitigation of such issues requires knowledge of the
underlying fundamental processes, which in turn often result in further
technical improvement of the device. Besides providing useful insight into
several fundamental physical processes, these particle trapping devices become
very useful in various applications, as discussed next.
External electric field and homogeneous magnetic field aided linear traps such
as Penning-MalmbergMalmberg and deGrassie (1975) (PM) trap, Paul trapPaul
(1990) and some modified PM trapsHurst _et al._ (2014); Yamada and Himura
(2016); Lane and Orperformedz (2019) are used extensively to trap pure
electron plasmaDavidson and Felice (1998), pure ion plasmaO’Neil and Dubin
(1998); Landa (2019) and unconventional charge species such as positron
plasmaGreaves and Surko (2000), anti-protron plasma.Ahmadi and et al. (2016);
Ahmadi _et al._ (2018) In straight cylindrical PM trap, radial and axial
confinement are provided to the pure electron (or ion) plasma, with the aid of
an axial homogeneous magnetic field and electrostatic end plugs, respectively.
The result achieved in such an arrangement is so robust that the electron
plasma plasma can be confined for a long time (over several days) in near
absolute equilibrium stateO’Neil and Driscoll (1979); Malmberg and Driscoll
(1980); Driscoll, Malmberg, and Fine (1988); Dubin and O’Neil (1999), allowing
numerous experiments to be performed. For example, pure electron plasmas share
fundamental properties with inviscid 2D Euler fluids at low density limit
($m\rightarrow 0$) and hence are often investigated experimentallyFine _et
al._ (1995) to understand complex nonlinear dynamics of such fluids. These
findings often corroborate with the corresponding theoreticalBriggs,
Daugherty, and Levy (1970) and numericalGanesh and Pahari (2006); Rome _et
al._ (2000) studies. Extensive numerical studiesSengupta and Ganesh (2014)
using two dimensional Particle-in-Cell (PIC)Birdsall and Langdon (2004)
simulation also provide essential insight into two dimensional electron plasma
dynamics in the cylindrical cross-section of the PM trap. Experimental
applications of pure electron plasmas are found in high-precision measurement
techniquesNiemann _et al._ (2019); Schuh _et al._ (2019), mass
spectrometryGutiérrez _et al._ (2019); Peurrung, Kouzes, and Barlow (1996)
etc. Positron, antiprotons are confined in separate PM traps with modified
electric fields and axial magnetic field, to synthesize antihydrogen used in
anti-matter investigationsAhmadi and et al. (2016); Ahmadi _et al._ (2018);
Fajans and Surko (2020). Linear Paul trapsPaul (1990); Landa (2019) are mainly
motivated to trap low density ions using time varying RF fields and axial
magnetic fields, whereas stabilization of the ions is performed with laser
coolingMorigi _et al._ (1999). Such stable ions are used in quantum
information processingMorigi and Walther (2001), mass spectrometryDouglas,
Frank, and Mao (2005) etc. Stable pair plasmas such as positron-electron,
found in astrophysical environment, are studied experimentally using combined
Penning-Paul trapGreaves and Surko (2002), inhomogeneous magnetic field in
magnetic mirror configurationHigaki _et al._ (2017) and recently reported
numerical experimentHicks, Bowman, and Godden (2019) with RF multipole
electric field. In a recent experiment, stable electron-ion coexistence is
achieved in a PM likeNakajima, Himura, and Sanpei (2021) trap under certain
conditions.
In contrast to the above discussed devices, toroidal trapsJanes _et al._
(1966); Daugherty, Eninger, and Janes (1969); Clark _et al._ (1976); Zaveri
_et al._ (1992); Khirwadkar _et al._ (1993); Lachhvani _et al._ (2017);
Stoneking _et al._ (2004) of different aspect ratios ($a/b$, $a$ is the major
radius, $b$ is the minor radius) apply an inhomogeneous external magnetic
field with inherent curvature to confine an electron plasma. The electron
plasma goes through rotational cross-field $\bf{E}\times\bf{B}$ drift where
$\bf{E}$ field is the self electric field of the plasma and $\bf{B}$ is the
external toroidal field. Because of the strong rotational $\bf{E}\times\bf{B}$
drift, electron plasma avoids the vertical loss of particles due to
${\bf{\nabla}}B$ and curvature drift. Additionally, the electron plasma
undergoes toroidal $m=1$ DiocotronDavidson (1990) rotation or center of charge
dynamics, which is similar to the Diocotron motion of cylindrical electron
plasma but due to magnetic field curvature are strongly coupled to elliptical
(i.e. $m=2$) mode and often to other higher order poloidal modes (i.e.
$m=3,4,5$ etc)Khamaru, Sengupta, and Ganesh (2019) depending on the aspect
ratio of the device and the amplitude of the $m=1$ mode. Apart from toroidal
magnetic field, electron plasma is also confined using magnetic surfaces in
stellaratorMarksteiner _et al._ (2008) and dipole magnetic fieldSaitoh _et
al._ (2010) based devices.
Earliest experiments of trapped electron plasmas in toroidal geometry were
performed by Janes et al. ($a/b$ $\sim$ 5)Janes _et al._ (1966), Daugherty et
al. ($a/b$ $\sim$ 4.6)Daugherty, Eninger, and Janes (1969), where the
axisymmetric toroidal device was proposed to be used as heavy ion plasma
accelerator (HIPAC) producing highly stripped heavy ions. The device was
intended to utilize the potential well of the electron plasma to trap and
accelerate heavy ions. Subsequent experiments were performed to confine
electron plasma in axisymmetric toroidal geometry of different aspect ratios
($a/b$ $\sim$ 6.2Clark _et al._ (1976) and $\sim$ 1.5Zaveri _et al._ (1992);
Khirwadkar _et al._ (1993)), followed by a number of experiments in partial
toroidal geometry: small aspect ratio device SMARTEX-C ($a/b$ $\sim$
1.6)Lachhvani _et al._ (2017), large aspect ratio device LNT II ($a/b$ $\sim$
13.7)Stoneking _et al._ (2004).
Such an electron plasma in inhomogeneous magnetic field is useful to
understand nonlinear couplingKhamaru, Sengupta, and Ganesh (2019); Khamaru,
Ganesh, and Sengupta (2021a, b) between different toroidal Diocotron modes,
collisional/non-collisional transport mechanismsCrooks and O’Neil (1996) of
electron plasma energy and also crucial to initiate an ongoing construction of
electron-positron plasma experiment using dipole magnetic fieldStoneking _et
al._ (2020); Singer _et al._ (2021). These systems can often serve as test-
beds to understand plasma dynamics in curved magnetic field which are often
found in astrophysical phenomenaBrambilla _et al._ (2018).
As indicated before, near-absolute equilibria of the electron plasma in PM
traps with uniform magnetic fields allows various experiments to be performed
in the device. Such near-absolute equilibrium in toroidal trap was recently
addressed and a quiescent quasi-steady (QQS)Khamaru, Ganesh, and Sengupta
(2021a, b) toroidal electron plasma has been found in a numerical experiment
with similar parametric space of that of an existing small aspect ratio
torusLachhvani _et al._ (2017). This finding also suggests the possibility of
realizing such a quiescent pure ion plasma in toroidal equilibrium, which
might provide excellent testbed/new directions for quantum information
processingLanda (2019); Morigi and Walther (2001) in future.
Coming back to electron plasmas, trapped pure electron plasmas are prone to
loss of electrons due to various instabilities associated with the electron
plasma dynamics. These instabilities are mainly resistive wall
instabilityWhite, Malmberg, and Driscoll (1982), electrostatic instability
induced by electron-neutral collisionsDavidson and Chao (1996) and ion
resonance instabilityLevy, Daugherty, and Buneman (1969); Davidson and Uhm
(1977); Fajans (1993); Peurrung, Notte, and Fajans (1993). Most of these
instabilities are subdued in devices with confining walls made of low
resistivity materials and better vacuum conditions (even at low working
pressure $\sim$ $10^{-9}$ torrLachhvani _et al._ (2016) or better). Amongst
these instabilities, ion resonance instability in cylindrical traps as well as
in toroidal traps, is more frequent and predominant over the rest. Even in the
low background pressure $\sim$ $10^{-9}$ torr, residual gases (mainly $N_{2}$
and $H_{2}$) remain in the system. Through the process of electron-impact
ionization of these background natural gas, ions and secondary electrons are
generated. It is believed that the ions are trapped in the device by the
potential well of the electron plasma leading to ion resonance instability,
inciting the destabilization of the electron plasma by the onset of an $m=1$
like toroidal Diocotron instability. To illuminate the underlying processes,
various analytical models of this instability that were constructed in the
past should be addressed first.
Initial analytical model of the cylindrical electron plasma in the presence of
few “trapped” ions was proposed by Levy et al.Levy, Daugherty, and Buneman
(1969) suggesting an exponential growth (i.e. mode amplitude is $\propto$
$exp(\gamma t)$, with $\gamma>0$) of the $m=1$ Diocotron mode of the electron
plasma. It was shown that the exact resonance condition arises if the
oscillation frequency of the ion trapped in the potential well of the electron
plasma matches the Diocotron frequency of the off centered electron plasma. In
another simplified linear model developed by Davidson and UhmDavidson and Uhm
(1977) for infinitely long cylindrical electron plasma under weak
neutralization by “trapped” ions, collisionless rotating two stream
instability between electron and ion plasma approach was adopted. The results
of this model are found to be in good agreement with those proposed by Levy el
al. showing the $m=1$ Diocotron mode of the electron plasma as fastest growing
mode among other exponentially growing modes. Later, for a 3D straight
cylindrical trap (i.e. PM trap) with electrostatic end plugs, ion resonance
instability of the cylindrical electron plasma in the presence of axially
drifting ions or “transient” ions (due to rapid loss of ions in the axial
direction as compared to an infinitely long cylinder) was investigated
analytically by FajansFajans (1993). This theory suggested algebraic growth of
the Diocotron instability, which was also reported to be confirmed in the
experiment performed by Peurrung et al.Peurrung, Notte, and Fajans (1993)
under similar experimental conditions. Earlier, simulation based studies using
2D Particle-in-Cell(PIC) code PEC2PICSengupta and Ganesh (2015, 2016, 2017),
have been performed addressing the ion resonance instability of cylindrically
confined partially neutralized electron plasma, under different initial
conditions. Through collisional and collisionless processes, the growth rate
of different Diocotron modes were addressed where the results were found to be
in good agreement with the Davidson and Uhm model. It was also shown that the
non-ionizing collisions do not initiate instabilities in contrary to then
existing theorySengupta and Ganesh (2016).
As of now, numerous experiments of cylindrical electron plasmasPeurrung,
Notte, and Fajans (1993); Eckhouse, Fisher, and Rostoker (1981); Bettega _et
al._ (2006); Kabantsev and Driscoll (2007) have exhibited ion resonance
instability under different configurations. These experiments show
exponentialEckhouse, Fisher, and Rostoker (1981); Kabantsev and Driscoll
(2007) or algebraic growthPeurrung, Notte, and Fajans (1993); Bettega _et
al._ (2006) depending on the nature of the ion entrapment. A toroidal electron
plasma, is also greatly affected by the presence of ions in the
systemLachhvani _et al._ (2016); Stoneking _et al._ (2002); Marksteiner _et
al._ (2008). Extensive investigation of the dependency of the growth on ion
species, magnetic field strength and pressure has been performed in
stellaratorMarksteiner _et al._ (2008) trap under magnetic surface trapping.
In SMARTEX-CLachhvani _et al._ (2016) trap, it is reported that the improved
vacuum condition of the device resulted in the damping of the growth of the
instability.
Recently, collisionless dynamics of toroidal electron plasma has been
addressed in tight aspect ratio device (in a similar parametric space as
SMARTEX-C) using the 3D3V Particle-in-Cell code PEC3PICKhamaru, Sengupta, and
Ganesh (2019), for both axisymmetricKhamaru, Sengupta, and Ganesh (2019);
Khamaru, Ganesh, and Sengupta (2021a, b) and partialSengupta, Khamaru, and
Ganesh (2021) or nonaxisymmetric configurations with electrostatic end-plugs
along the toroidal direction. Of late, the existence of a quiescent quasi-
steady state (QQS)Khamaru, Ganesh, and Sengupta (2021a, b) has been
discovered, for the toroidal electron plasma in axisymmetric configuration
using a maximum-entropy based zero-inertia solution as an initial condition to
the PEC3PIC, which then solves the full inertia-based equation of motion of
electrons. This slowly evolving state is shown to exhibit nearly free of
center of charge motion of the naturally shaped toroidal electron plasma with
finite parallel and perpendicular temperatures. We believe that the discovery
of such a quiescent quasi-steady state of the toroidal electron plasma may
become crucial to the future of quantum information processingLanda (2019);
Morigi and Walther (2001) and quantum computing. As a subsequent step,
collisionless ion-driven destabilization of the toroidal electron in QQS state
has been investigated in this paper. In an ongoing study, impact ionization
and various collisional mechanisms between electron-neutral, ion-neutral
collisions have been considered and shall be reported in future.
In the present study, we have addressed the effect of ion resonance
instability on the toroidal electron plasma in QQS state, via preloaded
$Ar^{+}$ ion plasma for different fractional neutralizationDavidson (1990)
values $f$ $=$ $n_{i}/n_{e}$ i.e. the ratio of ion density to electron
density, in an axisymmetric tight aspect ratio device using PEC3PICKhamaru,
Sengupta, and Ganesh (2019); Khamaru, Ganesh, and Sengupta (2021a, b);
Sengupta, Khamaru, and Ganesh (2021). It is shown that the electron plasma is
destabilized and the wall probe current shows algebraic growth. It is found
that the growth rate increases with increasing values of $f$. Along with
“center of charge motion” ($m$ $=$ $1$) of the electron plasma, mode coupling
between toroidal Diocotron modes ($m$ $=$ $1$ to $9$) has been observed with
$m$ $=$ $2$ as the dominant mode. The growth of the wall probe current later
saturates as ion (and electron) losses increase with simulation time. It is
shown that the calculated value of ion plasma rotational frequency of the
$Ar^{+}$ ion plasma in cylindrical approximation is at $\sim 31\%$ deviation
from the toroidal $m=1$ DiocotronDavidson (1990) frequency of the electron
plasma. Destabilization of the electron plasma and algebraic growth of the
wall probe current is brought out unambiguously. It has been observed that the
ion resonance instability of the electron plasma is possible for a certain
range of the ion plasma rotational frequency (via adding $N^{+}$ and $H^{+}$
ions into the system and observing the electron plasma dynamics, though the
results are not shown here). Collision-less heating of the ions has been
observed resulting in high ion temperature values, which is interpreted as due
to a resonant transfer of energy of the electron plasma to ions via toroidal
ion-resonance instability.
In Sec. II, the toroidal device configuration is given and the initial
conditions including the initialization steps of electron/ion plasma loading
in the toroidal device are explained. Sec. III various diagnostics implemented
for this study are shown. In Sec. IV the conclusion of the present study is
described.
## II Initial condition and simulation parameters
As discussed previously, 3D3V particle-in-cellBirdsall and Langdon (2004) code
PEC3PICKhamaru, Sengupta, and Ganesh (2019); Khamaru, Ganesh, and Sengupta
(2021a) has been used in the present study, addressing the dynamics of
toroidal electron plasma in tight aspect ratio axisymmetric toroidal device
(in a similar parametric space as SMARTEX-CLachhvani _et al._ (2017)).
Correct loading of the electron and ion plasma in this device is crucial for
the desired experiments and needs careful study along with corresponding
simulation parameters used in the study. In the next Subsec. II.1, the device
parameters and initialization process of the plasmas are explained in details.
### II.1 Device parameters and initialization of the electron and ion plasma
Figure 1: A schematic diagram of the axisymmetric toroidal vessel with
conducting boundary walls. The radius of the inner wall boundary is $R_{1}$
$=$ $0.05~{}m$, outer wall is $R_{2}$ $=$ $0.22~{}m$ (aspect ratio $\sim$ 1.6)
and the height is $L_{z}$ $=$ $0.32~{}m$. The vessel is closed by upper and
lower boundary walls. Wall probes located near the inner boundary wall,
separated by $\theta$ $=$ $\pi/2$ from each other are shown at $L_{z}$ $=$
$0.16~{}m$ mid-plane. The device sizes described are identical to SMARTEX-
CLachhvani _et al._ (2017) device. However unlike the experimental device,
here electrostatic endplugs in the toroidal direction are absent, though the
particle dynamics is fully 3D3V.
The schematic diagram of the axisymmetric toroidal vessel used in this study
is presented in Fig. 1 in cylindrical geometry. Inner wall radius of the torus
is $R_{1}$ $=$ $0.05~{}m$ and the outer wall radius is $R_{2}$ $=$ $0.22~{}m$,
making the device aspect ratio $\sim$ $1.6$. Height of the torus is $L_{z}$
$=$ $0.32~{}m$. The vessel is closed by upper and lower boundary walls. The
boundary walls of the torus are conducting. Wall probes are located near the
inner boundary wall, separated by $\theta$ $=$ $\pi/2$ locations at $L_{z}$
$=$ $0.16~{}m$ mid-plane are shown. The device size is identical to that of
SMARTEX-CLachhvani _et al._ (2017).
At background pressure of $10^{-9}$ to $10^{-7}$ torrMarksteiner _et al._
(2008); Lachhvani _et al._ (2016), ions and secondary electrons are generated
by the energetic primary electrons due to ionization of the background neutral
atoms. In typical tight aspect ratio toroidal experimentsLachhvani _et al._
(2016) with pressure $10^{-8}$ torr, the ion build up is upto $\sim$ $2.5\%$
within $5\times 10^{-4}$ sec time span. To investigate the effects of low
density ion population on the primary electron plasma in our study, the
primary electron plasma, $Ar^{+}$ ion plasma along with secondary electron
plasma are loaded in the system in two consecutive steps, described next in
detail.
Figure 2: Initial density distribution of the primary electrons in the 3D
toroidal vessel is shown at $\tau$ $=$ $0.00$, where $\tau$ $=$
$t/\overline{t}_{D}$ and $\overline{t}_{D}$ is toroidal Diocotron
periodKhamaru, Sengupta, and Ganesh (2019) ($\sim$ $1.8\times 10^{-6}s$) for
QQS stateKhamaru, Ganesh, and Sengupta (2021a, b). Only $3/4th$ of the
toroidal direction is shown to present the poloidal density contours along
with the toroidal density contours. The initial density of the electron plasma
has the radial variation of two orders of magnitude, viz $\sim 7.1\times
10^{12}m^{-3}-2.485\times 10^{14}m^{-3}$ i.e. $f_{b}\sim 0.002-0.07$, where
$f_{b}\sim 0.07$ corresponds to the mode (the locally maximum value of the
density) of the total density distribution at $z$ $=$ $L_{z}/2$ plane
($0.16~{}m$) of the torus. Few electrons are present outside the contour
region making the density negligible in that area. In the absence of ions,
this QQS state is shown to suffer very little radial transportKhamaru, Ganesh,
and Sengupta (2021a, b). Figure 3: (a)The density distributions of primary
electron plasma at $\tau$ $=$ $11.11$, where $\tau$ $=$ $t/\overline{t}_{D}$
and $\overline{t}_{D}$ is toroidal Diocotron period for the QQS state. The
density variation of the electron plasma is in the range $\sim 6.83\times
10^{12}m^{-3}-2.39\times 10^{14}m^{-3}$ i.e. $f_{b}\sim 0.0019-0.067$. Here
$f_{b}\sim 0.067$ corresponds to the mode of the total electron density
distribution at $z$ $=$ $L_{z}/2$ plane of the torus. (b) The density
distribution of $Ar^{+}$ ion plasma at $\tau$ $=$ $11.11$ for fractional
neutralization factor $f$ $=$ $0.007$ which corresponds to the ratio of ion
density to the electron density i.e. $n_{i}/n_{e}$. The density of the ion
plasma is in the range $\sim 4.78\times 10^{10}m^{-3}-1.67\times
10^{12}m^{-3}$. Here $1.67\times 10^{12}m^{-3}$ corresponds to the mode of the
total ion density distribution at $z$ $=$ $L_{z}/2$ plane of the torus.
As described in Ref.45Khamaru, Ganesh, and Sengupta (2021a), in the first
step, zero-inertia maximum entropy density distribution functionKhamaru,
Ganesh, and Sengupta (2021a) is used as the “seed” particle density profile of
the primary electron plasma in the system at the beginning of the simulation
which has been previously shown to result in a novel quiescent quasi-steady
state (QQS)Khamaru, Ganesh, and Sengupta (2021a) of the electron plasma. Thus
the initial spatial density distribution of the electron plasma, following
this initial “seed” solution, is shown to evolve into an axisymmetric
($\partial/\partial\theta=0$) nonuniform toroidal plasma plasma which now is
accurate to all orders in ${\rho_{{Le}}}/L$, where ${\rho_{{Le}}}$ is the
average electron Larmor radius and $L$ is a typical magnetic field gradient
length scale. Corresponding density distribution of the electrons in the 3D
toroidal vessel is shown in the Fig. 2 at $\tau$ $=$ $0.00$, where $\tau$ $=$
$t/\overline{t}_{D}$ and $\overline{t}_{D}$ is toroidal Diocotron
periodKhamaru, Sengupta, and Ganesh (2019) $\sim$ $1.8\times 10^{-6}s$ for the
QQSKhamaru, Ganesh, and Sengupta (2021a, b) state of the electron plasma. The
method to obtain $\overline{t}_{D}$ was elaborated in a previous studyKhamaru,
Sengupta, and Ganesh (2019). The quantity toroidal Diocotron period is
explained later in Sec. III.
The electrons are cold loaded with zero velocities. This method of initial
loading is in general different than the conventional electron injection
technique used in typical toroidal pure electron plasma experimentsLachhvani
_et al._ (2017); Stoneking _et al._ (2004), where a negatively biased
injector grid filament emit the electrons in toroidal direction, filling the
toroidal vessel via another positive biased grid placed “nearby”, followed by
axial confinement of electrons by negative end plugs. This procedure results
in a toroidal or parallel velocity such that toroidal or parallel bounce time
is much smaller than the typical $\bf{E}\times\bf{B}$ time scale. The parallel
kinetic energy of electrons often ionize the weak background neutrals, giving
rise to ions. In the present study, zero-inertia maximum entropy distribution
results in nonuniform density distribution of the electrons featuring
contoured structures with peaked centers in toroidal and poloidal planes, as
shown in the Fig. 2. Though the electrons fill up the whole torus, only
$3/4th$ of the whole torus is shown in this figure (Fig. 2) to present the
poloidal density contours along with the toroidal density contours. This
configuration of the density distribution will be used throughout this paper.
The initial density of the electron plasma is in the range $\sim 7.1\times
10^{12}m^{-3}-2.485\times 10^{14}m^{-3}$ i.e. $f_{b}\sim 0.002-0.07$. Here
$f_{b}$ is the “cylindrical” Brillouin fractionDavidson (1990) ratio,
$f_{b}=2{\omega_{p}}^{2}/{\omega_{c}}^{2}$, ($\omega_{p}$ is the plasma
frequency and $\omega_{c}$ is the cyclotron frequency for magnetic field
$B_{0}$ ($0.03T$) at inner wall radius). (Inferred range of $f_{b}$ is in the
range $f_{b}\sim 10^{-3}-10^{-4}$ in the experimentsLachhvani _et al._
(2016), which is typically an order of magnitude lower than our simulation
parameters.) As there are very few electrons in the outer regions of the
plasma, the density contours are seen to sharply fall to zero in this region.
The electron plasma attains QQS state shortly after the initialization and is
left to evolve upto $\tau$ $=$ $11.11$.
In the second step, the $Ar^{+}$ ions are introduced in the system at $\tau$
$=$ $11.11$ (which corresponds to $t$ $=$ $2\times 10^{-5}$ s) according to
same spatial distribution of QQS as the primary electrons have evolved into.
Though at $\tau$ $=$ $11.11$ the electrons acquire certain finite width of
velocity distribution function, ions are loaded cold at this instant. In
cylindrical experimentsPeurrung, Notte, and Fajans (1993); Bettega _et al._
(2006); Kabantsev and Driscoll (2007) in PM traps with ion resonance
instability, the values of the fractional neutralization factorDavidson (1990)
$f$, which corresponds to the fraction of ion population to the primary
electron population i.e. $n_{i}/n_{e}$, is reported as typically $\sim$
$10^{-5}$ to $10^{-4}$. For partial toroidal experimentsMarksteiner _et al._
(2008); Lachhvani _et al._ (2016), $f$ $\sim$ $10^{-2}$ to $10^{-1}$. In our
axisymmetric toroidal simulation, $f$ values are chosen between $0.001$ and
$0.01$, which is in the range of experimental estimates. The 3D density
distributions of primary electron plasma and $Ar^{+}$ ion plasma at $\tau$ $=$
$11.11$, is shown in the Fig. 3 (a) and (b) respectively for $f$ $=$ $0.007$.
In Fig. 3 (a), the density of the electron plasma is in the range $\sim
6.83\times 10^{12}m^{-3}-2.39\times 10^{14}m^{-3}$ i.e. $f_{b}\sim
0.0019-0.067$. Fig. 3 (b), the density of the ion plasma is in the range $\sim
4.78\times 10^{10}m^{-3}-1.67\times 10^{12}m^{-3}$. The ion densities used are
two orders of magnitude smaller than the electron density at any given spatial
location.
Figure 4: Inner wall probe current for fractional neutralization factor $f$
(i.e. $n_{i}/n_{e}$) ranging from $0.001$ to $0.01$ for $Ar^{+}$ upto $\tau$
$=$ $34$, where $\tau$ $=$ $t/\overline{t}_{D}$ and $\overline{t}_{D}$ is
toroidal Diocotron period for QQS state. The time of introduction of ions is
at $\tau$ $=$ $11.11$, as indicated by black dotted line. The inset plot shows
onset of the growth of wall probe current amplitude.
Along with the ions, cold (zero velocity) secondary electrons are also loaded
at $\tau$ $=$ $11.11$ with density value $f$ times the density of the primary
electrons i.e. the density of secondary electron plasma is same as the ion
plasma loaded. This method mimics the ionization process in the experiments
and conserves the total charge of the system at $\tau$ $=$ $11.11$. Total
energy of the system is not effected by addition of the secondary electron
population/ion population because of the low density $\sim$ $10^{-3}$ to
$10^{-2}$. After the loading processes, this nonneutral plasma consisting of
the primary and secondary electrons along with ions is left to evolve and the
dynamics is monitored through the simulation time period upto $34$
$\overline{t}_{D}$. In the next Subsec. II.2, the simulation parameters are
described.
### II.2 Simulation Parameters
In this study, 3D3V OPEN-MP parallelized particle-in-cellBirdsall and Langdon
(2004) (PIC) code PEC3PICKhamaru, Sengupta, and Ganesh (2019); Khamaru,
Ganesh, and Sengupta (2021a) is used. More details on the code and the
algorithm can be found in the recent QQS studyKhamaru, Ganesh, and Sengupta
(2021a) of electron plasmas where same PIC code has been used. Here initial
super particle number for the primary electron plasma is 2560640 in a grid
size of $192\times 192\times 192$ in Cartesian coordinates. At $\tau$ $=$
$11.11$, super particle number for the primary electron plasma becomes 2464401
due to initial loss of the electrons. Also initial super particle numbers for
the ion plasma and secondary electron plasma are 2464401 each at $\tau$ $=$
$11.11$. The density values of the ion plasma and secondary electron plasma
are made $f$ times lesser than the primary electron plasma using the
fractional neutralization factor $f$ $=$ $0.007$ in the code. At $\tau$ $=$
$11.11$, most of the electron and ion particles are confined in toroidal tube
with high density value which has lesser volume than the whole torus, as shown
in the Fig. 3. Thus the effective number of particles (electrons and ions) per
3D cell of this toroidal tube is $\sim 10$ and the number of particles per 3D
cell of the whole toroidal simulation device is $\sim 3.4$. Numerical
conservation tests for the current PEC3PIC solver has been performed in a
previous studyKhamaru, Sengupta, and Ganesh (2019). $Ar^{+}$ ion used in the
study have mass $72820.77$ $m_{e}$ where $m_{e}$ is the electron mass.
The toroidal magnetic field used in this device is spatially inhomogeneous. At
inner wall radius $R_{1}$, the magnitude of the magnetic field is highest
($B_{0}=0.03T$) and gradually falls with cylindrical radial coordinate $R$ as
$B={B_{0}R_{1}}/R$. At $R=R_{1}$ the estimated electron cyclotron time period
$T_{ce}=1.19\times 10^{-9}s$ and the ion cyclotron time period
$T_{ci}=8.68\times 10^{-5}s$. As is obvious, the cyclotron periods are
smallest at the largest magnetic field in the system, which is at the inner
wall at $R=R_{1}$. The simulation time step chosen is $10^{-10}s$ which is
about 10 times smaller than the smallest cyclotron time in the system.
The parameters used here are typical of recent experiments Lachhvani _et al._
(2016), though any collisional processes are not included in the present
study. As neutrals are not present in our simulation, collisional processes
with neutrals are exempted as well. Ion-ion collision are also exempted
because of their mass and extremely low density. Other possible processes are
the electron-electron and electron-ion collisions. Here we have estimated the
respective collisional parameters for electron-electron collision and
electron-ion collision (for $Ar^{+}$), $f=0.007$ case as follows: Large angle
electron-electron Coulomb collisions time scaleHuba (2013) ${\tau}_{col}^{e}$
$\simeq$ $\frac{{{T_{e}}^{3/2}}}{2.91\times 10^{-6}~{}n_{e}ln\Gamma}$ sec
$\simeq$ $0.099$ sec i.e. $\overline{\tau}_{col}^{e}$ $\simeq$
${{\tau}_{col}^{e}}/{\overline{t}_{D}}$ $\simeq$ $5.5$ $\times$ $10^{4}$.
Electron-ion Coulomb collisions time scale ${\tau}_{col}^{i}$ $\simeq$
$\frac{{{T_{i}}^{3/2}}}{4.8\times 10^{-8}~{}n_{i}\mu Z_{a}^{4}ln\Gamma}$ sec
$\simeq$ $141.44$ sec i.e. $\overline{\tau}_{col}^{i}$ $\simeq$ $7.86$
$\times$ $10^{7}$. (In the formulae for collision times, temperature is in eV
units, whereas all the other quantities are in CGS units). Consequently,
${\overline{t}_{D}}\sim 1.8\times 10^{-6}s$ is the toroidal Diocotron time
period, $ln\Gamma$ $\sim$ 10, $T_{max}^{e}$ $\sim$ $80.0~{}eV$, $n_{e}$ $\sim$
$2.485\times 10^{8}{cm}^{-3}$, $n_{i}$ $\sim$ $1.74\times 10^{6}{cm}^{-3}$,
$T_{max}^{i}$ $\sim$ $280.0~{}eV$, atomic number $Z_{a}$ $=$ $1$, ratio of ion
mass to proton mass $\mu$ $=$ $\frac{m_{i}}{m_{p}}$ $=$ 39.66. In the present
study, total simulation time is $\tau$ $=$ $t/\overline{t}_{D}$ $=$ $34$,
where $t$ is time. Both electron-electron Coulomb collisions time scale and
electron-ion collisions time scale are larger than the present simulation time
scale. Thus the plasma is considered as collision-less plasma.
Figure 5: Time evolution of primary electron plasma and ion plasma with
respective density values for $f$ $=$ $0.007$ at different simulation time
periods $\tau$ $=$ $16.67$, $21.11$, $23.33$ and $25.56$, where $\tau$ $=$
$t/\overline{t}_{D}$ and $\overline{t}_{D}$ is toroidal Diocotron period for
QQS state. The density contours of electron plasma along with the spatial
position of the plasma is shown in (a)-(d). These figures show displaced
electron plasma with “center of charge motion” ($m$ $=$ $1$ mode) in the
vertical and horizontal direction, with distinguishable compressed and
expanded forms of the plasma along with periodic peak density value evolution.
The time evolution of the ion plasma and density is shown in (e)-(h). (e) at
$\tau$ $=$ $16.67$ shows that the initially cold loaded ion plasma starts to
gain energy from the system and initial shape of the plasma (Fig. 3 (b) and
(d)) at $\tau$ $=$ $11.11$) is distorted. In (f)-(h), at $\tau$ $=$ $21.11$,
$23.33$, $25.56$ respectively, the ion plasma spreads out into the torus
representing rapid loss form the torus boundaries. This loss of ions is also
reflected in decrement of the peak poloidal and toroidal density values of the
ion plasma through the simulation time. Figure 6: (a)-(d) shows electrostatic
potential surface plots of the system for $f$ $=$ $0.007$ at different
simulation time periods $\tau$ $=$ $16.67$, $21.11$, $23.33$ and $25.56$,
where $\tau$ $=$ $t/\overline{t}_{D}$ and $\overline{t}_{D}$ is toroidal
Diocotron period for QQS state. The potential surface plots display closed
axisymmetric toroidal potential well of the system, with spatial location of
the well minima around the same spatial location of the electron plasma
density maxima (Fig. 5 (a)-(d)).
Wall probes similar to experimental devices are placed at four different
azimuthal locations (Fig. 1) (separated by $\theta$ $=$ $\pi/2$) near the
inner wall at $z$ $=$ $L_{z}/2$ plane. Wall probe current obtained from the
inner wall probe is used as a diagnostic method to analyse the electron plasma
dynamics in the vessel. This “wall probe current” depends mainly on the charge
induced by center of charge motion of the electron plasma and expressed as
$I_{wall}=\frac{d(CV)}{dt}$. Here $V(t)$ and $C(t)$ are the instantaneous
probe voltage and capacitance respectively. Thus the charge induced on the
probe is $C(t)V(t)$ at that instance. In this work, a constant capacitance
value is chosen to be $C=0.08pF$ so that the obtained order of magnitude of
probe current is similar to typical experimental values Ganesh and Pahari
(2006).
In the next Sec. III, the effect of the ion resonance instability on electron
dynamics is investigated via different diagnostics methods: (i) Growth in wall
probe current and dependency on $f$, electron/ion plasma spatial evolution in
the torus and corresponding density evolution, spectrogram analysis of the
wall probe current and electrostatic potential of the system, cylindrically
approximated ion plasma rotational frequency and corresponding toroidal $m=1$
Diocotron frequency of the electron plasma (ii) electron/ion particle loss in
the system (iii) energy of the electron/ion plasma (iv) growth rate of the
wall probe current at different periods of the simulation, and (v) temperature
analysis of the electron and ion plasmas.
## III Diagnostics
As indicated earlier, to investigate the electron and ion plasma dynamics, we
have implemented several diagnostics suitable for this study. The diagnostics
results covers the simulation time period $\tau$ $=$ $0.00$ to $\tau$ $=$
$34$.
### III.1 Growth in wall probe current and dependency on $f$
To study the electron dynamics in the presence of ions in the system, wall
probe signal (wall probe voltage in Volts) is obtained as a function of time
from one of the inner wall probe and the corresponding wall probe current is
shown in Fig. 4. Wall probe current from three other probes at three remaining
toroidal locations provide identical information which suggests absence of any
toroidal mode in the system (not shown here). In Fig. 4, wall probe current
growth is observed for $f$ values ranging from $0.001$ to $0.01$ upto $\tau$
$=$ $34$. As discussed earlier, at $\tau$ $=$ $0.00$, a cold electron plasma
density distribution obtained by extremizing inertia-less entropy is
initializedKhamaru, Sengupta, and Ganesh (2019); Khamaru, Ganesh, and Sengupta
(2021a), which evolves self consistently into a quiescent quasi-steady (QQS)
state, as reflected in the wall probe data shown in Fig. 4. As discussed
earlier, ions are then introduced “by hand” or “pre-loaded” at $\tau$ $=$
$11.11$ (along with equal number of secondary electrons, as it would happen in
an ionizing process), soon after which, the amplitudes of the probe currents
start to grow with time and the growth is more for higher values of $f$ for
any instant of time. This is clearly demonstrated in the inset plot displayed
at the onset of the growth. Also an initial phase difference between the probe
currents for different values of $f$ can be observed. The growth of the wall
probe current is algebraic in nature for all values of $f$ and saturates at
later time of the simulation, mainly for higher $f$ values. More analysis on
the growth rate is performed in Subsec. III.5.
To aid the above analysis, spatial positions of the electron plasma inside the
torus along with density contours are shown in Fig. 5 (a)-(d), at simulation
times $\tau$ $=$ $16.67$, $21.11$, $23.33$, $25.56$ respectively with $f$ $=$
$0.007$ case. Before $\tau$ $=$ $11.11$, the electron plasma has attained a
quiescent state with negligible displacement, shown in previous study of QQS
stateKhamaru, Ganesh, and Sengupta (2021a, b). After ions were introduced at
$\tau$ $=$ $11.11$, the “center of charge motion” ($m$ $=$ $1$) of the
electron plasma in the poloidal plane gains amplitude as time progresses. The
electron plasma is displaced along the vertical and horizontal directions,
with a distinguishable compression-expansion cycle due to toroidicity. The
peak density values at the toroidal/poloidal plane evolve periodically along
with the “center of charge motion” of the plasma. In these figures, the
poloidal shape of the plasma shows elliptic nature representing $m$ $=$ $2$
mode. Referring to Fig. 4, the probe currents attain “double peak” nature
which actually corresponds to the “center of charge motion” ($m$ $=$ $1$) of
the electron plasma in the poloidal plane and onset of strong compression-
expansion cycle. Due to this center of charge motion, electron plasma makes a
finite amplitude elliptical trajectory in the poloidal plane, which we dub
here as nonlinear toroidal Diocotron motion and effectively similar to the
cylindrical $m$ $=$ $1$ Diocotron motion of the electron plasma with a
remarkable difference that at such small aspect ratios as studies here, the
modes are coupled in a poloidal plane due to strong toroidicity or $1/R$
dependence of the toroidal magnetic field. Among these double peaks, the
higher amplitude peak corresponds to the electron plasma position near the
inner wall of the torus with most compressed form of the plasma and the lower
amplitude peak appears while the plasma is away from the inner wall in most
expanded formKhamaru, Sengupta, and Ganesh (2019). This can be seen from the
Fig. 4, (i) the growth rate of the amplitudes of the peaks increases with time
for every $f$ values (for higher $f$ values growth with time is more) and (ii)
at any instant of time, the amplitude values of the peaks depend on $f$ values
i.e. with high value of $f$ the peak amplitude is higher. This brings out
clearly the fact that with higher ion density the “center of charge” motion
becomes stronger, which means an extended elliptical trajectory of the
electron plasma from the QQS state, in the poloidal plane of the torus. This
displacement of the electron plasma increases rapidly with time for higher
values of the ion density. These signatures point out clearly that the physics
of destabilization of QQS state is strongly connected to the presence of ions
and its amplitude to the initial fraction of ions $f$ present in the system.
Following the above said arguments, we believe that the wall probe growth or
amplitude increase is indeed due to a toroidal ion resonance instability.
Figure 7: Spectrogram of the wall probe current for $f$ $=$ $0.0$ (QQS state),
$0.001$, $0.005$, $0.01$ upto $\tau$ $=$ $34$, where $\tau$ $=$
$t/\overline{t}_{D}$ and $\overline{t}_{D}$ is toroidal Diocotron period for
QQS state. White dotted line represents the introduction time of the ion
plasma at $\tau$ $=$ $11.11$. Left y axis presents the frequency of the data
where as in the right y axis the frequency is normalized by the nonlinear
toroidal Diocotron frequency ($\sim$ $0.58$ MHz) of the QQS state. For all
values of $f$, without the ion plasma before $\tau$ $=$ $11.11$, $m$ $=$ $1$
and $m$ $=$ $2$ Diocotron modes are present in the system. For $f$ $=$
$0.001$, $0.005$ and $0.01$ cases higher modes (upto $m$ $=$ $9$ for $f$ $=$
$0.01$) start to appear with time and $m$ $=$ $2$ mode becomes dominant. The
temporal evolution of the spectrogram plots for $f$ $=$ $0.001$, $0.005$ and
$0.01$ cases reveal the ion density dependency of the dominant modes ($m$ $=$
$1$ and $m$ $=$ $2$) and dynamical chirping infused coupling of low poloidal
modes ($m$ $=$ $1$ to $m$ $=$ $9$). For $f$ $=$ $0.01$ case, a high chirping
region between $\tau$ $\sim$ $15$ and $16$ is noted.
The time evolution of the ion plasma and density is shown in Fig. 5. At $\tau$
$=$ $16.67$ (see Fig. 5 (e)) the initially cold loaded ion plasma starts to
gain energy from the system and initial shape of the plasma (Fig. 3 (b) at
$\tau$ $=$ $11.11$) is distorted. In Fig. (f)-(h), at $\tau$ $=$ $21.11$,
$23.33$, $25.56$ respectively, the plasma spreads out into the torus
representing rapid loss of ions from the torus boundaries. Initial loss of the
ions is higher at the top boundary wall and at later simulation times from
other boundary walls also. This loss of ions is also reflected in decrement of
the peak poloidal and toroidal density values of the ion plasma through the
simulation time. Also rapid ion loss is discussed in the next Subsec. III.3.
The cold loaded ion plasma gains enough kinetic energy from the electron
plasma as a result of the ion resonance instability. As our simulation is
collisionless, we believe that the underlying exchange of energy between ions
and electrons is a collisionless resonant process. The thermal energy gain of
the ions results in increase of ion Larmour radii (for example, at
perpendicular ion temperature $50~{}eV$, average Larmour radius of an ion is
${\rho_{{Li}}}$ $\sim$ $0.214~{}m$ near the inner wall location and
${\rho_{{Li}}}$ $\sim$ $1.073~{}m$ near outer wall location). Also various
inertia driven drifts due to the toroidal inhomogeneous magnetic fields
($\bm{\nabla}B$ and curvature drift) aid the loss of ions. Thus the ions are
rapidly lost from the system. The energy transfer process is investigated
further in the Subsec. III.4.
Corresponding to the presented spatial locations and density plots of the
electron and ion plasma, electrostatic potential surface plots of the system
are shown in Fig. 6 (a)-(d). The potential surface plots display closed
axisymmetric toroidal potential well of the system, with spatial location of
the well minima around the same spatial location of the electron plasma
density maxima (Fig. 5 (a)-(d)). This potential well has parabolic nature in
the poloidal plane and the ions are initially trapped in the potential well.
Though the location of the maxima of the ion plasma density is near to that of
the electron plasma/potential minima at $\tau$ $=$ $16.67$, after that the ion
plasma density maxima do not exhibit the same. This observation suggests the
decoupling of the ion plasma from the electron plasma at later simulation time
period after $\tau$ $=$ $16.67$. More analysis on this is presented in Subsec.
III.4.
To further understand the dynamics of the dynamic toroidal Diocotron modes of
the system and possible mode coupling associated with the electron and ion
plasma dynamics, spectrogram analysis of the wall probe current has been
performed for $f$ $=$ $0.0$ (QQS state), $0.001$, $0.005$, $0.01$ upto $\tau$
$=$ $34$ and shown in Fig. 7. Spectrogram frequency is normalized by the
nonlinear toroidal Diocotron frequency ($\sim$ $0.58$ MHz) of the QQS state
and shown in the right y axis. For all values of $f$ considered, without the
ion plasma before $\tau$ $=$ $11.11$, $m$ $=$ $1$ and $m$ $=$ $2$ Diocotron
modes are the only dominant ones that are present in the system. For $f$ $=$
$0.001$, $0.005$ and $0.01$ cases higher modes (upto $m$ $=$ $9$ for $f$ $=$
$0.01$) start to appear with time with introduction of ions at $\tau$ $=$
$11.11$ and $m$ $=$ $2$ mode becomes dominant. The temporal evolution of the
spectrogram plots for $f$ $=$ $0.001$, $0.005$ and $0.01$ cases reveal the ion
density dependency of the dominant modes ($m$ $=$ $1$ and $m$ $=$ $2$) and
dynamical chirping infused coupling of low poloidal modes ($m$ $=$ $1$ to $m$
$=$ $9$). Similar feature of the poloidal modes with $m$ $=$ $2$ as dominant
mode, was also seen in our previous studyKhamaru, Sengupta, and Ganesh (2019)
with an initial loading of the electron plasma while the electron plasma had
stronger “center of charge” motion than that of the QQS state. Also for $f$
$=$ $0.01$ case, a high chirping region at very low power has been observed
between $\tau$ $\sim$ $15$ and $16$.
### III.2 Ion resonance instability and ion plasma rotational frequency
In straight cylinder geometryDavidson (1990), ion plasma rotational frequency
in a partially neutralized electron plasma is given by
$\omega_{ri}^{-}=-\frac{\omega_{ci}}{2}[1-(1+\frac{2\omega_{pe}^{2}}{\omega_{ce}^{2}}(1-f)\frac{m_{i}}{m_{e}})^{1/2}]$
(1)
where $\omega_{ci}$ and $\omega_{ce}$ are the cyclotron frequency of ion and
electron respectively at uniform $B$ field, $\omega_{pe}$ is the electron
plasma frequency, $m_{i}$ and $m_{e}$ are the ion mass and electron mass
respectively. The condition for ion resonance instabilityDavidson (1990) of
the electron plasma in cylindrical geometry is that the ion plasma rotational
frequency should be nearly equal to the cylindrical $m=1$ Diocotron frequency
of the electron plasma. In the present study with radially varying toroidal
magnetic field, the an approximated ion plasma rotational frequency has been
considered to compare with the toroidal $m=1$ Diocotron rotation of the
electron plasma, as discussed in several ion resonance instability
studiesMarksteiner _et al._ (2008); Lachhvani _et al._ (2016). With the
present parameters for $Ar^{+}$ ions, the ion plasma rotational frequency
(magnetic field at the minor axis location of the torus $B=0.011T$) $\sim$ 0.4
MHz under cylindrical approximation where “measured” toroidal Diocotron
frequency of the electron plasma $\sim$ 0.58 MHz in our system. To the lowest
order in inverse aspect ratio, thus the ion rotational frequency is at $\sim
31\%$ deviation from toroidal Diocotron frequency of the electron plasma. As
the electron plasma and ion plasma dynamics are in toroidal geometry with
inhomogeneous magnetic field, the presented deviation should be acceptable, if
a permissible range of the deviation can be obtained. To estimate the
permissible range of the deviation, $N^{+}$ ($m_{i}=25532.65m_{e}$) and
$H^{+}$ ($m_{i}=1836m_{e}$) ions are also introduced in the device replacing
$Ar^{+}$ ions to observe the electron plasma dynamics. For $N^{+}$ and $H^{+}$
ions, corresponding ion plasma rotational frequencies are $\sim$ 0.67 MHz
($\sim 9\%$ deviation) and $\sim$ 2.4 MHz ($\sim 182\%$ deviation)
respectively. Algebraic growth of the wall probe current similar to ${Ar}^{+}$
ions has been observed for $N^{+}$ ions where as for $H^{+}$ ions, no growth
is observed (the results are not shown here). These findings indicate the
dependency of the ion resonance instability on the ion mass. Thus we can
conclude that the permissible value of the ion plasma rotational frequencies
to initiate an ion resonance instability of the electron plasma, for the
current simulation parameters, lies between a certain range. Further analysis
in the present study mainly focused on the ${Ar}^{+}$ ions and detailed
investigation of the ion plasma rotational frequency range can be performed in
future with different ion species.
To investigate the ion resonance induced instability in this system in more
detail, loss of electron and ion particles and temporal evolution of the
ion/electron plasma’s energy with simulation time, are described next.
### III.3 Loss of electron and ion
Figure 8: Ratio of the instantaneous electron particle number ($N_{E}(t)$) to
the initial value of the electron particle number ($N_{E0}$) i.e.
${N_{E}(t)}/N_{E0}$ is shown for different values of $f$ upto $\tau$ $=$ $34$,
where $\tau$ $=$ $t/\overline{t}_{D}$ and $\overline{t}_{D}$ is toroidal
Diocotron period for QQS state. At $\tau$ $=$ $11.11$ (indicated by black
dotted line) introduction of ions results into a dip in ${N_{E}(t)}/N_{E0}$
values which is more for higher values of $f$. The inset plot shows the
effective electron loss, around $\tau$ $=$ $11.11$. Figure 9: Ratio of the
instantaneous ion particle number (${N_{I}(t)}$) to the ion particle number
($N_{I}(\tau=11.11)$) at $\tau=11.11$, i.e. ${N_{I}(t)}/N_{I}(\tau=11.11)$ is
shown for different values of $f$ upto $\tau$ $=$ $34$, where $\tau$ $=$
$t/\overline{t}_{D}$ and $\overline{t}_{D}$ is toroidal Diocotron period for
QQS state. After $\tau$ $=$ $11.11$ (indicated by black dotted line),
${N_{I}(t)}/N_{I}(\tau=11.11)$ show finite value (ion particles are introduced
in the system at $\tau$ $=$ $11.11$) and its value decreases with simulation
time. The decrement is more for higher values of $f$. The inset plot shows the
starting of ion loss, around $\tau$ $=$ $12.5$.
Though electron plasma in a quiescent quasi-steady state has better stability
properties, inertia effects i.e. various drifts due to the toroidal
inhomogeneous magnetic fields: $\bm{\nabla}B$ and curvature drift (and
$\overline{\rho}_{Le}/{L}$ to all ordersKhamaru, Ganesh, and Sengupta (2021a,
b)) are present in the system leading to some electron losses ($\sim$
$2\%$)Khamaru, Ganesh, and Sengupta (2021a) in the system. As ions are
introduced in the system, destabilization of electron plasma due to toroidal
ion resonance instability, initiates more rapid loss of electron particles
from the system. Such loss results in confinement issues of electron plasma in
the experimental studies. For the present study, primary electron particle
loss is shown in Fig. 8 where ratio of the instantaneous electron particle
number $N_{E}(t)$ to the initial value of the electron particle number
$N_{E0}$ i.e. ${N_{E}(t)}/N_{E0}$ for different values of $f$ upto $\tau$ $=$
$34$ is shown. Here $\tau$ $=$ $t/\overline{t}_{D}$ and $\overline{t}_{D}$ is
toroidal Diocotron period for QQS state. The time of introduction of ions is
at $\tau$ $=$ $11.11$, as indicated by black dotted line. Before $\tau$ $=$
$11.11$, the electron plasma evolves into QQS state and the the plasma losses
$\sim$ $3.5\%$ of initial total number of electrons. At $\tau$ $=$ $11.11$
introduction of ions results into a dip in the quantity ${N_{E}(t)}/N_{E0}$
and the loss rate increases with increasing value of $f$ or ion density. At
$\tau$ $=$ $34$, the quantity ${N_{E}(t)}/N_{E0}$ losses $\sim$ $5\%$ of the
initial value for $f$ $=$ $0.001$ case, and $\sim$ $8\%$ for $f$ $=$ $0.01$
case. The inset plot around $\tau$ $=$ $11.11$ shows the the effective
electron loss, around the introduction period of ions in the system.
Figure 10: Energy variations of the system upto $\tau$ $=$ $34$, where $\tau$
$=$ $t/\overline{t}_{D}$ and $\overline{t}_{D}$ is toroidal Diocotron period
for QQS state. The initial value of the total energy ${TE}_{0}$
($3.503\times{10}^{-5}$ J) of the system at $\tau$ $=$ $0.00$ is subtracted
from the total energy ($TE$), kinetic energy ($KE$) and potential energy
($PE$) of the system and shown for the values of $f$ ranging from $0.001$ to
$0.01$ in (a), (b) and (c) respectively. The time of introduction of ions is
at $\tau$ $=$ $11.11$, as indicated by black dotted line in all three plots.
The inset plot in (a) around $\tau$ $=$ $11.11$ shows that total energy of the
system starts to decrease before $\sim$ one toroidal Diocotron time period
after $\tau$ $=$ $11.11$. The inset plots in (b) and (c) shows the initial
values of ${KE}$ $-$ ${TE}_{0}$ and ${PE}$ $-$ ${TE}_{0}$ respectively.
In Fig. 9, ratio of the instantaneous ion particle number (${N_{I}(t)}$) to
the ion particle number ($N_{I}(\tau=11.11)$) at $\tau=11.11$, i.e.
${N_{I}(t)}/N_{I}(\tau=11.11)$ is shown for different values of $f$ upto
$\tau$ $=$ $34$, where $\tau$ $=$ $t/\overline{t}_{D}$ and $\overline{t}_{D}$
is toroidal Diocotron period for QQS state. The time of introduction of ions
is at $\tau$ $=$ $11.11$, as indicated by black dotted line. Upto $\tau$ $=$
$11.11$, ${N_{I}(t)}/N_{I}(\tau=11.11)$ shows zero value as the system
contained no ions at all, whereas after $\tau$ $=$ $11.11$,
${N_{I}(t)}/N_{I}(\tau=11.11)$ has finite value (ion particles are introduced
in the system at $\tau$ $=$ $11.11$). At $\tau$ $=$ $34$, ion plasma losses
$\sim$ $32\%$ of its initial value for $f$ $=$ $0.001$ case and $\sim$ $80\%$
for $f$ $=$ $0.01$ case. The inset plot around $\tau$ $=$ $11.11$ shows the
introduction period of ions and the starting point of the loss of ions from
the system. After $\tau$ $=$ $11.11$, the loss of ion particles starts after
$\sim$ one toroidal Diocotron cycle ($\sim$ $\tau$ $=$ $12.5$) and the
starting time varies with different values of $f$. In the case of electron
loss, the effective electron loss (after $\tau$ $=$ $11.11$) starts just
before $\sim$ one toroidal Diocotron cycle $\sim$ $\tau$ $=$ $12.5$). This
observation indicates that though the loss of ion at later stage of the
simulation is rapid and significantly high when compared to the electron loss
within the same simulation time period, initially the ion particles were
trapped and gained energy while the system endured electron particle loss.
Also, in spite of the ion loss at later stage of the simulation, the electrons
plasma remains destabilized throughout that simulation time period showing
saturation of the amplitude of the wall probe current (Fig. 4). This suggests,
the initial presence of the ion plasma at $\tau$ $=$ $11.11$ started the
process of electron plasma destabilization which became independent of the
presence of ion plasma in the system at a later time. It is also to be noted
that, as the ion loss rate is more than the electron loss rate along with the
simulation time, the dynamic $f$ values become lower than the initial values.
To understand the plasma dynamics in detail, temporal evolution of the kinetic
and potential energy of the total system as well as for electron/ion plasma
are described.
### III.4 Energy variation with simulation time
In Fig. 10 energy evolution of the system are shown upto $\tau$ $=$ $34$. The
initial value of the total energy ${TE}_{0}$ ($3.503\times{10}^{-5}$ J) of the
system at $\tau$ $=$ $0.00$ is subtracted from the total energy ($TE$),
kinetic energy ($KE$) and potential energy ($PE$) of the system and shown for
the values of $f$ ranging from $0.001$ to $0.01$ in Fig. 10 (a), (b) and (c)
respectively. The time of introduction of ions is at $\tau$ $=$ $11.11$, as
indicated by black dotted line in all three plots. Fig. 10 (a) shows ${TE}$
$-$ ${TE}_{0}$ $\sim$ ${10}^{-7}$, indicating that the total energy of the
system is conserved for all $f$ values (for lowest $f$ $=$ $0.001$, the
increment is $<$ $1\%$). The inset plot in Fig. 10 (a) around $\tau$ $=$
$11.11$ shows that total energy (${TE}$ $-$ ${TE}_{0}$) of the system starts
to decrease before one toroidal Diocotron time period after $\tau$ $=$
$11.11$. The inset plots in Fig. 10 (b) and (c) shows the initial values of
${KE}$ $-$ ${TE}_{0}$ and ${PE}$ $-$ ${TE}_{0}$ respectively. From $\tau$ $=$
$0.00$ to $\tau$ $=$ $11.11$, while the system remains in QQS state without
ion plasma, kinetic energy of the system/the electron plasma increases, along
with an increase oscillation amplitude, at the cost of electrostatic potential
energy of the system (Fig. 10 (c): $\tau$ $=$ $0.00$ to $\tau$ $=$ $11.11$).
After the introduction of cold ion plasma in the system at $\tau$ $=$ $11.11$,
average kinetic energy of the system (electrons and ions) rises with time
while the oscillation amplitude starts to grow with time for all values of
$f$. For higher values of $f$, the growth in the oscillation amplitude are
higher, where as average kinetic energy of the system reaches to close values
for all $f$ at $\tau$ $=$ $34$. In the same time interval, average potential
energy of the system decreases for all values of $f$ (Fig. 10 (c)), along with
growth of the oscillation amplitudes similar as the kinetic energy. However,
the average potential energy values drops with higher $f$ values at $\tau$ $=$
$34$, mainly because of its high transfer to the kinetic energy of the system
(electrons and ions) with higher $f$ values. Also loss of particles from the
system with time (Fig. 8 and 9) aids to small order decrement of the potential
energy of the system. Clearly, rise in the oscillation amplitude of kinetic
energy as well as potential energy with higher values of $f$ needs to be
addressed separately. In the ion resonance instability process, energy is
transferred from electron plasma to ion plasma. Thus the energy profile
variations of the electron and ion plasma are investigated separately and
explained in the next paragraph.
Figure 11: (a) and (b) represent volume averaged parallel and perpendicular
velocity distribution functions of electron plasma, $\langle
f_{e}(v_{\parallel})\rangle$ and $\langle f_{e}(v_{\bot})\rangle$
respectively, at different simulation time periods $\tau$ $=$ $12.22$,
$15.56$, $18.89$, $20.00$ and $34.00$ along with corresponding parallel and
perpendicular temperatures $\langle T_{\parallel}^{e}\rangle$ and $\langle
T_{\bot}^{e}\rangle$, for the case of $f$ $=$ $0.007$. In (c) and (d), volume
averaged parallel and perpendicular velocity distribution functions of ion
plasma, $\langle f_{i}(v_{\parallel})\rangle$ and $\langle
f_{i}(v_{\bot})\rangle$, at same simulation time periods, are shown
respectively along with corresponding parallel and perpendicular temperatures,
$\langle T_{\parallel}^{i}\rangle$ and $\langle T_{\bot}^{i}\rangle$, for same
$f$ value. In (a) and (b), the temperatures increase from ${\langle
T_{\parallel}^{e}\rangle}_{12.22\tau}$ $=$ $10~{}eV$ to ${\langle
T_{\parallel}^{e}\rangle}_{34\tau}$ $=$ $16.5~{}eV$ and from ${\langle
T_{\bot}^{e}\rangle}_{12.22\tau}$ $=$ $15~{}eV$ to ${\langle
T_{\bot}^{e}\rangle}_{34\tau}$ $=$ $21.03~{}eV$ respectively. In (c), parallel
distribution function of the ion plasma at $\tau$ $=$ $12.22$, results into
${\langle T_{\bot}^{i}\rangle}_{12.22\tau}$ $=$ $0.0004~{}eV$. The parallel
temperature attains ${\langle T_{\bot}^{i}\rangle}_{34\tau}$ $=$
$0.0546~{}eV$. In (d), for $\tau$ $=$ $12.22$, $15.56$ and $18.89$, the
distribution functions do not form near-Maxwellian shape, though
$v_{\perp}^{i}$ $\sim$ $10^{5}$ m/s, $10^{2}$ order $>$ $v_{\parallel}^{i}$.
For $\tau$ $=$ $20.00$ and $34.00$, the distribution function starts to form
near-Maxwellian shape resulting ${\langle T_{\bot}^{i}\rangle}_{20\tau}$ $=$
$218.46~{}eV$ and ${\langle T_{\bot}^{i}\rangle}_{34\tau}$ $=$ $280.36~{}eV$.
Figure 12: Normalized kinetic energy and potential energy variations of the
electron plasma and ion plasma are shown upto $\tau$ $=$ $34$ for $f$ $=$
$0.001$ to $0.01$. The energy values are normalized by initial total energy
${TE}_{0}$ ($3.503\times{10}^{-5}$ J) of the system at $\tau$ $=$ $0.00$. The
time of introduction of ions is at $\tau$ $=$ $11.11$, as indicated by black
dotted line in all plots. (a) and (b) represents the variation of kinetic
energy ${KE}_{e}$/${TE}_{0}$ and potential energy ${PE}_{e}$/${TE}_{0}$ of the
electron plasma respectively. The inset plot in (a) around $\tau$ $=$ $11.11$
shows that oscillation amplitude in the kinetic energy of the electron plasma
starts to increase at $\sim$ $\tau$ $=$ $11.11$. The inset plots in (b),
around $\tau$ $=$ $11.11$ shows $\sim$ ${10}^{-3}$ to ${10}^{-2}$ order
increase in the amplitude of the ${PE}_{e}$/${TE}_{0}$ values at $\tau$ $=$
$11.11$ because of the addition of secondary electrons. (c) and (d) shows
kinetic energy ${KE}_{i}$/${TE}_{0}$ and potential energy
${PE}_{i}$/${TE}_{0}$ of the ion plasma respectively from $\tau$ $=$ $11.11$
to $34$. ${KE}_{i}$/${TE}_{0}$ linearly increases after $\tau$ $=$ $11.11$ for
all $f$ values. The linear fits yield: growth rates ${\alpha}_{KE}^{0.001}$
$=$ $0.00087$ and ${\alpha}_{KE}^{0.01}$ $=$ $0.01052$. After the time period
$\tau$ $=$ $11.11$ to $\tau$ $\sim$ $19.17$, the fall of the quantity
${KE}_{i}$/${TE}_{0}$ is not followed by rise of the order $>$ ${10}^{-3}$ for
any values of $f$. In the period form $\tau$ $=$ $11.11$ to $\tau$ $\sim$
$19.17$ the ion plasma gains energy from the potential of the electron plasma.
In (d), the potential energy of the ion plasma, increases after $\tau$ $=$
$11.11$ (at the cost of the potential energy of the electron plasma) with
similar fluctuating trend as the kinetic energy. Around $\tau$ $\sim$ $19.17$,
the fluctuation reduces and the potential energy of the plasma slowly
decreases with simulation time.
In the above discussion, it is found that the kinetic energy of the system
(i.e. the electrons and ions) increases with simulation time. This suggests
that the average temperature also increases. To verify, volume averaged
parallel and perpendicular velocity distribution functions of the ion plasma
as well as electron plasma are shown in Fig. 11 to check if the electron or
ion plasma reach a certain equilibrium with a Maxwellian-like distribution
function. Here parallel velocity, $v_{\parallel}$ is defined as the 1D
velocity vector with velocity component along toroidal direction ($\theta$).
Perpendicular velocity, $v_{\bot}$ is defined as a 2D vector having a minor-
radial component and a poloidal component, or equivalently through a basis
transformation, a major-radial component, $v_{R}$ together with a cylindrical-
axial component, $v_{Z}$. Choosing the later basis for convenience, the
perpendicular speed is $|v_{\perp}|=\sqrt{{v_{R}}^{2}+{v_{Z}}^{2}}$. To
calculate corresponding volume averaged parallel and perpendicular
temperatures, volume averaged distribution functions, $\langle
f(v_{\parallel})\rangle$ and $\langle f(v_{\bot})\rangle$, which are near-
Maxwellian, are shown and fitted with respective perfect Maxwellian
distribution function (1D/2D) for a certain temperature value. Then $\langle
T_{\parallel}\rangle$ and $\langle T_{\bot}\rangle$ are calculated from the
respective fitted temperature values. Using the expressions given, for case of
$f$ $=$ $0.007$, volume averaged parallel and perpendicular velocity
distribution functions of electron plasma, $\langle
f_{e}(v_{\parallel})\rangle$ and $\langle f_{e}(v_{\bot})\rangle$, at
different simulation time periods (at $\tau$ $=$ $12.22$, $15.56$, $18.89$,
$20.00$ and $34.00$) are shown in Fig. 11 (a) and (b) respectively along with
corresponding parallel and perpendicular temperatures, $\langle
T_{\parallel}^{e}\rangle$ and $\langle T_{\bot}^{e}\rangle$. In Fig. 11 (c)
and (d), volume averaged parallel and perpendicular velocity distribution
functions of ion plasma, $\langle f_{i}(v_{\parallel})\rangle$ and $\langle
f_{i}(v_{\bot})\rangle$, at same simulation time periods, are shown
respectively along with corresponding parallel and perpendicular temperatures,
$\langle T_{\parallel}^{i}\rangle$ and $\langle T_{\bot}^{i}\rangle$, for same
$f$ value. In Fig. 11 (a) and (b), the widths of the electron distribution
functions spread with simulation time and the temperatures also increase as
${\langle T_{\parallel}^{e}\rangle}_{12.22\tau}$ $=$ $10~{}eV$ to ${\langle
T_{\parallel}^{e}\rangle}_{34\tau}$ $=$ $16.5~{}eV$ (Fig. 11 (a)) and
${\langle T_{\bot}^{e}\rangle}_{12.22\tau}$ $=$ $15~{}eV$ to ${\langle
T_{\bot}^{e}\rangle}_{34\tau}$ $=$ $21.03~{}eV$ (Fig. 11 (b)). The rise in
temperatures are also reflecting the rise in the kinetic energy of the
electron plasma. However, the ion distribution functions are more informative
about the energy transfer process and need to be discussed. In Fig. 11 (c),
parallel distribution function of the ion plasma at $\tau$ $=$ $12.22$, with
narrow width, corresponds to ${\langle T_{\parallel}^{i}\rangle}_{12.22\tau}$
$=$ $0.0004~{}eV$. The parallel temperature does not increase much and attains
${\langle T_{\parallel}^{i}\rangle}_{34\tau}$ $=$ $0.0546~{}eV$. This result
suggests that cold loaded ion plasma at $\tau$ $=$ $11.11$ does not acquire
enough parallel energy ($v_{\perp}^{i}$ $\sim$ $10^{3}$ m/s) obstructing the
ion particle motion in the toroidal plane. In Fig. 11 (d), for $\tau$ $=$
$12.22$, $15.56$ and $18.89$, the distribution functions do not form near-
Maxwellian shape, thus it is inaccurate to attribute any temperature
information to the plasma in these time periods. However, the spread in these
distribution functions ($v_{\perp}^{i}$ $\sim$ $10^{5}$ m/s, $10^{2}$ order
higher than $v_{\parallel}^{i}$) suggest the ion particle motion mainly in the
poloidal plane. For $\tau$ $=$ $20.00$ and $34.00$, the distribution function
starts to form near-Maxwellian shape which suggests that the ion plasma
approaches equilibrium state. By fitting the curves we obtain ${\langle
T_{\bot}^{i}\rangle}_{20\tau}$ $=$ $218.46~{}eV$ and ${\langle
T_{\bot}^{i}\rangle}_{34\tau}$ $=$ $280.36~{}eV$. As seen in this analysis,
the ion plasma forms near-Maxwellian distribution function in the region
between $\tau$ $=$ $18.89$ to $\tau$ $=$ $20.00$, indicating that the ion
plasma attains equilibrium after $\tau$ $=$ $18.89$. The temperature values
${\langle T_{\bot}^{i}\rangle}_{20\tau}$ $=$ $218.46~{}eV$ and ${\langle
T_{\bot}^{i}\rangle}_{34\tau}$ $=$ $280.36~{}eV$ are higher than that of the
corresponding electron temperatures. Such high temperature of ion plasma
indicates anomalous heating of ions via toroidal ion resonance instability
process and to the best of our knowledge has never been reported prior to this
work.
To get a better idea of the energy transfer processes, kinetic and potential
energies of the ion and electron plasmas have been obtained. Normalized
kinetic energy and potential energy variations of the electron plasma and ion
plasma with simulation time are shown in Fig. 12 upto $\tau$ $=$ $34$ for $f$
$=$ $0.001$ to $0.01$. The energy values are normalized by initial total
energy ${TE}_{0}$ of the system at $\tau$ $=$ $0.00$. The time of introduction
of ions is at $\tau$ $=$ $11.11$, as indicated by black dotted line in all
plots. Fig. 12 (a) and (b) represents the variation of kinetic energy
${KE}_{e}$/${TE}_{0}$ and potential energy ${PE}_{e}$/${TE}_{0}$ of the
electron plasma respectively, upto the simulation time $\tau$ $=$ $34$. The
inset plot in Fig. 12 (a) around $\tau$ $=$ $11.11$ shows that oscillation
amplitude in the kinetic energy of the electron plasma starts to increase at
$\sim$ $\tau$ $=$ $11.11$. In Fig. 12 (b), the inset plots around $\tau$ $=$
$11.11$ shows $\sim$ ${10}^{-3}$ to ${10}^{-2}$ order increase in the
amplitude of the ${PE}_{e}$/${TE}_{0}$ values at $\tau$ $=$ $11.11$. This is
because of the addition of secondary electrons of density $\sim$ ${10}^{-3}$
to ${10}^{-2}$ times the density of the primary electrons. After $\tau$ $=$
$11.11$, the kinetic energy and potential energy of the electron plasma show
similar trend as seen in the total kinetic energy and potential energy
variation of the system (Fig. 10 (b) and (c)) through the simulation time with
the values of $f$. Following initial drop after $\tau$ $=$ $11.11$, potential
energy of the electron plasma is drained out as the kinetic energy of the
plasma for all values of $f$. The drop is more for higher $f$ values as seen
in Fig. 10 (c). Loss of electron particles from the system with time (Fig. 8)
also aids to small order decrement of the potential energy of the plasma. The
orders of the values of ${KE}_{e}$/${TE}_{0}$ are that of the total kinetic
energy of the system throughout the simulation, as the ions and secondary
electrons are loaded cold contributing zero kinetic energies to the system.
Similarly, the orders of the values of ${PE}_{e}$/${TE}_{0}$ are that of the
total potential energy of the system as the ions and secondary electrons are
loaded with lower density than that of the primary electron plasma ($f$ $=$
$0.001$ to $0.01$). In Fig. 12 (a) and (b), $\tau$ $=$ $19.17$ is marked by
vertical dotted line in magenta and explained in next paragraph.
To understand the energy variation of ion plasma through the simulation time,
normalized kinetic energy ${KE}_{i}$/${TE}_{0}$ and potential energy
${PE}_{i}$/${TE}_{0}$ of the ion plasma are shown in Fig. 12 (c) and (d)
respectively for the values of $f$ ranging from $0.001$ to $0.01$ from
simulation time $\tau$ $=$ $11.11$ to $\tau$ $=$ $34$. As the ion plasma was
cold loaded at $\tau$ $=$ $11.11$, kinetic energy is zero at $\tau$ $=$
$11.11$ in Fig. 12 (c). The parameter ${PE}_{i}$/${TE}_{0}$ $\sim$ ${10}^{-3}$
to ${10}^{-2}$ times the total energy of the system at $\tau$ $=$ $11.11$ in
Fig. 12 (d). Demonstrating typical outcome of the ion resonance instability
process, the ion particles gain energy (kinetic/potential) from the potential
energy of the electron plasma. As our simulation is collisionless, we believe
that there exists an underlying collisionless process, resulting in
collisionless exchange of energy between ions and electrons. In Fig. 12 (c),
the parameter ${KE}_{i}$/${TE}_{0}$ linearly increases after $\tau$ $=$
$11.11$ for all $f$ values. To check the growth rate with respect to $f$
values, the parameter ${KE}_{i}$/${TE}_{0}$ is linearly fitted upto $\tau$ $=$
$11.8$ for $f$ $=$ $0.001$ and $0.01$ cases and shown in the inset plot. The
linear fits yield slope of ${KE}_{i}$/${TE}_{0}$ growth as
${\alpha}_{KE}^{0.001}$ $=$ $0.00087$ for $f$ $=$ $0.001$ and
${\alpha}_{KE}^{0.01}$ $=$ $0.01052$ for $f$ $=$ $0.01$, suggesting $\sim$
${10}$ time higher growth rate for one order of magnitude increase in $f$
value ($\frac{{\alpha}_{KE}^{0.01}}{{\alpha}_{KE}^{0.001}}$ $\sim$ ${12.09}$).
After $\tau$ $=$ $11.11$, the quantity ${KE}_{i}$/${TE}_{0}$ displays
fluctuating rise/fall profile with occasional gain/loss of ion kinetic energy,
for all values of $f$. We have identified a region, $\tau$ $=$ $11.11$ to
$\tau$ $\sim$ $19.17$ ($\tau$ $=$ $19.17$ marked by vertical dotted line in
magenta), after when the fall of the quantity ${KE}_{i}$/${TE}_{0}$ is not
followed by rise of the order $>$ ${10}^{-3}$ for any values of $f$. After
$\tau$ $\sim$ $19.17$, ${KE}_{i}$/${TE}_{0}$ slowly decreases with simulation
time with small amount of fluctuations (though for $f$ $=$ $0.001$ to $0.005$,
the gradual fall of the kinetic energy starts at later simulation time). In
the period form $\tau$ $=$ $11.11$ to $\tau$ $\sim$ $19.17$, the ion plasma
gains energy from the potential of the electron plasma. After $\tau$ $\sim$
$19.17$ the ion plasma starts to equilibrate. This transition time is within
the region between $\tau$ $=$ $18.89$ to $\tau$ $=$ $20.00$, as seen in the
velocity distribution function analysis of Fig. 11 (d). Further loss of
kinetic energy after $\tau$ $\sim$ $19.17$ can also be related to the loss of
ion particles from the system though the particle loss starts sooner than
$\tau$ $\sim$ $19.17$ (Fig. 9). The potential energy of the ion plasma, shown
in Fig. 12 (d), also increases after $\tau$ $=$ $11.11$ (at the cost of the
potential energy of the electron plasma) with similar fluctuating trend as the
kinetic energy. Around $\tau$ $\sim$ $19.17$, the fluctuation reduces and the
potential energy of the plasma slowly decreases with simulation time, though
the transition of the fluctuation amount around $\tau$ $\sim$ $19.17$ is not
strikingly distinguishable as the kinetic energy case. The region around
$\tau$ $\sim$ $19.17$ is also identified in Fig. 12 (a) and (b) (vertical
dotted line in magenta) for the electron plasma, but there is no noticeable
transition of kinetic/potential energy of the electron plasma around that time
period. The reason might be the low order magnitude of the ion plasma energies
compared to the electron plasma energies.
### III.5 Growth rate of the wall probe current at different periods of the
simulation
Figure 13: (a) Growth rate analysis of the inner wall probe current for $f$
$=$ $0.001$ and $0.01$ upto $\tau$ $=$ $34$. $\tau$ $=$ $t/\overline{t}_{D}$
and $\overline{t}_{D}$ is toroidal Diocotron period for QQS state. The time of
introduction of ions is at $\tau$ $=$ $11.11$, as indicated by black dotted
line. $\tau$ $=$ $19.17$ is indicated by vertical dotted line in magenta.
Algebraic growth rate of the wall probe current is calculated by linear
fitting of the maximum peak among the double peaks of the probe current, in
two regions: (1) $\tau$ $=$ $11.11$ to $19.17$ and (2) $\tau$ $=$ $19.17$ to
$34$, for both of $f$ $=$ $0.001$ and $0.01$. The fits yield
${\gamma}_{1}^{0.001}$ $=$ $0.0004$ $mA/\overline{t}_{D}$,
${\gamma}_{1}^{0.01}$ $=$ $0.008$ $mA/\overline{t}_{D}$ and
${\gamma}_{1}^{0.001}$ $=$ $0.0008$ $mA/\overline{t}_{D}$,
${\gamma}_{1}^{0.01}$ $=$ $0.0037$ $mA/\overline{t}_{D}$. (b) Growth rates for
the two regions (1) $\tau$ $=$ $11.11$ to $19.17$ and (2) $\tau$ $=$ $19.17$
to $34$ are shown as a function of $f$, as ${\gamma}_{1}$ and ${\gamma}_{2}$
respectively. Linear fit of ${\gamma}_{1}$ and ${\gamma}_{2}$ reveals growth
rate as ${\beta}_{1}$ $=$ $0.821$ $mA/\overline{t}_{D}$ and ${\beta}_{1}$ $=$
$0.263$ $mA/\overline{t}_{D}$.
Fig. 13 (a) shows growth rate analysis of the inner wall probe current for $f$
$=$ $0.001$ and $0.01$ upto $\tau$ $=$ $34$. Wall probe currents for extrema
values of $f$ are chosen for comparison purpose i.e. to reveal the dependency
of growth on lower and higher $f$ values. The time of introduction of ions is
at $\tau$ $=$ $11.11$, as indicated by black dotted line. $\tau$ $=$ $19.17$,
simulation time close to the transition time of the ion plasma to equilibrium
state, is indicated by vertical dotted line in magenta. Algebraic growth rate
of the wall probe current is calculated by linear fitting of the maximum peak
among the double peaks of the probe current, in two regions: (1) $\tau$ $=$
$11.11$ to $19.17$ and (2) $\tau$ $=$ $19.17$ to $34$, for both of $f$ $=$
$0.001$ and $0.01$. In the initial ($\tau$ $=$ $11.11$ to $19.17$) time
period, the linear fit reveals growth rates ${\gamma}_{1}^{0.001}$ $=$
$0.0004$ $mA/\overline{t}_{D}$ and ${\gamma}_{1}^{0.01}$ $=$ $0.008$
$mA/\overline{t}_{D}$. For later time period ($\tau$ $=$ $19.17$ to $34$),
growth rates are ${\gamma}_{1}^{0.001}$ $=$ $0.0008$ $mA/\overline{t}_{D}$ and
${\gamma}_{1}^{0.01}$ $=$ $0.0037$ $mA/\overline{t}_{D}$. As expected, the
growth rate is high in the initial region and attains lower value in the later
region, indicating saturation of the wall probe current for $f$ $=$ $0.01$
case. This result echos the same explanation of energy transfer given in the
ion plasma energy analysis of Subsec. III.4. For $f$ $=$ $0.001$ case though,
the growth rate is higher at later part (lower values than $f$ $=$ $0.01$
case). The discrepancy suggests that for lower $f$ values, the ion plasma
gains energy for longer time (low energy gain than the high $f$ value) and
equilibrates at a later time than $\tau$ $=$ $19.17$. This feature is also
seen in Fig. 12 (a), where for $f$ $=$ $0.001$ to $0.005$, the fall of the
kinetic energy started at later simulation time ($\sim$ $\tau$ $=$ $25$). To
get an idea about the variation of the growth rate for every $f$ values in the
present study, growth rate for $f$ $=$ $0.001$ to $0.01$ are shown as a
function of $f$ in Fig. 13 (b), for the two regions (1) $\tau$ $=$ $11.11$ to
$19.17$ and (2) $\tau$ $=$ $19.17$ to $34$ as ${\gamma}_{1}$ and
${\gamma}_{2}$ respectively. Linear fit of ${\gamma}_{1}$ and ${\gamma}_{2}$
values with respect to $f$ values shows that, ${\gamma}_{1}$ (${\beta}_{1}$
$=$ $0.821$ $mA/\overline{t}_{D}$) has higher slope value than ${\gamma}_{2}$
(${\beta}_{1}$ $=$ $0.263$ $mA/\overline{t}_{D}$) i.e. the average growth rate
of the wall probe current is higher at the initial time period $\tau$ $=$
$11.11$ to $19.17$. Interestingly, ${\gamma}_{1}$ and ${\gamma}_{2}$ values
decreases with low values of $f$ and approaches to a common value ($\sim$ $f$
$=$ $0.0015$).
As discussed in the Sec. I (Introduction), algebraic growthFajans (1993);
Peurrung, Notte, and Fajans (1993); Bettega _et al._ (2006) is seen in the
ion resonance instability with transient nature of the ion in the cylindrical
trap. In tight aspect ratio toroidal trap experimentLachhvani _et al._ (2016)
with toroidal end plugs, the ions are produced and lost along toroidal
direction and thus are effectively transient. Though the growth rate is high
but slowerLachhvani _et al._ (2016) than theoretically proposed rates by Levy
et al.Levy, Daugherty, and Buneman (1969). Here growth of the wall probe
current has been found out to be algebraic in nature. In our case there is no
continues source of ions but ion loss occurs throughout the simulation, which
is a unique situation never studied before. But it is evident that the nature
of the instability is similar to that of the transient ions.
In the next Subsec. III.6, the global temperature profiles of electron and ion
plasma are described as function time. Also the radial temperature profiles of
the electron plasma and their evolution is addressed.
### III.6 Temperature of the electron and ion plasma
To understand the temperature evolution of electron and ion plasma, volume
averaged parallel and perpendicular temperatures are evaluated through out the
simulation time and explained next. Average temperatures of electron flux
tubes at different radial locations are also found out and corresponding
radial variations are also discussed.
#### III.6.1 Global temperature of the electron plasma
Figure 14: Volume averaged parallel and perpendicular temperatures of the
electron plasma, $\langle T_{\parallel}^{e}\rangle$ and $\langle
T_{\bot}^{e}\rangle$ are shown for case of $f$ $=$ $0.007$ upto $\tau$ $=$
$34$. $\langle T_{\parallel}^{e}\rangle$ $\sim$ $13.8$ eV and $\langle
T_{\bot}^{e}\rangle$ $\sim$ $8.8$ eV at $\tau$ $=$ $11.11$ (indicated by black
dotted line), when the ion plasma was introduced. At $\tau$ $=$ $34$, $\langle
T_{\parallel}^{e}\rangle$ $\sim$ $21$ eV and $\langle T_{\bot}^{e}\rangle$
$\sim$ $17$ eV. Though the rise in the temperature values was almost steady
initially, after $\tau$ $\sim$ $25$ the fluctuations in $\langle
T_{\parallel}^{e}\rangle$ values were $\sim$ $5$ev and for $\langle
T_{\bot}^{e}\rangle$ $\sim$ $2.5$ev. Volume averaged parallel and
perpendicular distribution functions, $\langle f_{e}(v_{\parallel})\rangle$
and $\langle f_{e}(v_{\bot})\rangle$, at $\tau$ $=$ $11.11$, are shown in the
inset plot.
The volume averaged parallel and perpendicular temperatures of the electron
plasma, $\langle T_{\parallel}^{e}\rangle$ and $\langle T_{\bot}^{e}\rangle$
are shown in Fig. 14 for case of $f$ $=$ $0.007$ upto $\tau$ $=$ $34$. As the
electrons were loaded cold, the temperature values start from zero $\tau$ $=$
$0.00$ and reaches to finite temperature values ($\langle
T_{\parallel}^{e}\rangle$ $\sim$ $13.8$ eV and $\langle T_{\bot}^{e}\rangle$
$\sim$ $8.8$ eV), at $\tau$ $=$ $11.11$, at the instant the ion plasma was
introduced. Volume averaged parallel and perpendicular distribution functions,
$\langle f_{e}(v_{\parallel})\rangle$ and $\langle f_{e}(v_{\bot})\rangle$, at
$\tau$ $=$ $11.11$, are shown in the inset plot. After $\tau$ $=$ $11.11$, the
temperature values rise further and attains values as $\langle
T_{\parallel}^{e}\rangle$ $\sim$ $21$ eV and $\langle T_{\bot}^{e}\rangle$
$\sim$ $17$ eV. Though the rise was almost steady initially, after $\tau$
$\sim$ $25$ the fluctuations in $\langle T_{\parallel}^{e}\rangle$ values were
$\sim$ $5$ev and for $\langle T_{\bot}^{e}\rangle$ $\sim$ $2.5$ev. The orders
of these global temperature values are in the same range ($\simeq 10-20~{}eV$)
of the values for pure electron plasma, if the ion plasma was absent from the
system, though the oscillating features at later simulation time represents
the high amplitude oscillation of the electron plasma after $\tau$ $\sim$ $25$
seen in Fig. 12 (a). Volume averaged parallel and perpendicular temperatures
of the ion plasma are also calculated and discussed next.
#### III.6.2 Global temperature of the ion plasma
Figure 15: Volume averaged parallel and perpendicular temperatures of the ion
plasma, $\langle T_{\parallel}^{i}\rangle$ and $\langle T_{\bot}^{i}\rangle$
are shown for case of $f$ $=$ $0.007$ upto $\tau$ $=$ $34$. As seen in Fig.
11, the perpendicular distribution functions of ion plasma form near-
Maxwellian shape around $\tau$ $=$ $19.17$ (indicated by vertical dotted line
in magenta) with $\langle T_{\bot}^{i}\rangle$ $\sim$ $211$ eV at $\tau$ $=$
$19.17$. $\langle T_{\bot}^{i}\rangle$ $\sim$ $280$ eV at $\tau$ $=$ $34$. As
the ions were loaded cold, the $\langle T_{\parallel}^{i}\rangle$ values start
from zero at $\tau$ $=$ $11.11$ and reaches to low order finite temperature
values ($\langle T_{\parallel}^{i}\rangle$ $\sim$ $0.055$ eV ) at $\tau$ $=$
$34$. Steep rise in $\langle T_{\parallel}^{i}\rangle$ values is seen after
$\tau$ $=$ $19.17$. Volume averaged parallel and perpendicular distribution
functions, $\langle f_{i}(v_{\parallel})\rangle$ and $\langle
f_{i}(v_{\bot})\rangle$, at $\tau$ $=$ $34$, are shown in the inset plot.
As seen in Fig. 11, volume averaged temperature evolution of the ion plasma
also helps to clarify the equilibration process of the ion plasma. Thus, the
volume averaged parallel and perpendicular temperatures of the ion plasma,
$\langle T_{\parallel}^{i}\rangle$ and $\langle T_{\bot}^{i}\rangle$ are shown
for case of $f$ $=$ $0.007$ upto $\tau$ $=$ $34$. $\tau$ $=$ $19.17$ is
indicated by vertical dotted line in magenta. As seen in Fig. 11, the
perpendicular distribution functions of ion plasma form near-Maxwellian shape
around $\tau$ $=$ $19.17$ with $\langle T_{\bot}^{i}\rangle$ $\sim$ $211$ eV
at $\tau$ $=$ $19.17$. After $\tau$ $=$ $19.17$, $\langle T_{\bot}^{i}\rangle$
values increases and attains $\sim$ $280$ eV at $\tau$ $=$ $34$. As the ions
were loaded cold, the $\langle T_{\parallel}^{i}\rangle$ values start from
zero at $\tau$ $=$ $11.11$ and reaches to low order finite temperature values
($\langle T_{\parallel}^{i}\rangle$ $\sim$ $0.055$ eV) at $\tau$ $=$ $34$.
However, steep rise in $\langle T_{\parallel}^{i}\rangle$ values is seen after
$\tau$ $=$ $19.17$ suggesting energy gain of the ion plasma, though very small
in the parallel direction. Volume averaged parallel and perpendicular
distribution functions, $\langle f_{i}(v_{\parallel})\rangle$ and $\langle
f_{i}(v_{\bot})\rangle$, at $\tau$ $=$ $34$, are shown in the inset plot.
#### III.6.3 Radial variation of the electron plasma temperature
Figure 16: Averaged temperatures ${\bar{T}}_{\parallel}^{e}(R,t)$ and
${\bar{T}}_{\bot}^{e}(R,t)$ are shown as functions of $R$, at $\tau$ $=$
$16.67$ and $\tau$ $=$ $25.56$ for $f$ $=$ $0.007$ case. Z-averaged quantities
are expressed as ${\bar{T}}_{\parallel}^{e}(R,t)(Z_{avg})$,
${\bar{T}}_{\bot}^{e}(R,t)(Z_{avg})$ and flux tube averaged quantities are
expressed as ${\bar{T}}_{\parallel}^{e}(R,t)(Fluxtube)$,
${\bar{T}}_{\bot}^{e}(R,t)(Fluxtube)$ in all of these Figures. The magenta
colored vertical lines represents $1/e$ times value of the peak electron
density. For high accuracy the analysis is performed over different sized bins
along $R$ and the respective range of the temperatures are shown by the error
bars in these Figures. In (a) and (b),
${\bar{T}}_{\parallel}^{e}(R,t)(Z_{avg})$ and
${\bar{T}}_{\parallel}^{e}(R,t)(Fluxtube)$ values are higher at the inboard
side and gradually fall with $R$. The temperature values are fitted with
$\alpha/R^{2}$ and $\alpha_{2}/R^{2}$ ($\alpha$ and $\alpha_{2}$ are an
arbitrary constants) showing $1/R^{2}$ scaling with $R$. In (c) and (d),
${\bar{T}}_{\bot}^{e}(R,t)(Z_{avg})$ and ${\bar{T}}_{\bot}^{e}(R,t)(Fluxtube)$
values are also higher at the inboard side but showing non-monotonic variation
with $R$. The average temperature values are fitted with $\beta/R$ and
$\beta_{2}/R$ ($\beta$ and $\beta_{2}$ are arbitrary constants) showing $1/R$
scaling with $R$.
The electron plasma performs a center of charge motion in the poloidal plane,
the toroidal electron flux tubes passes across the $1/R$ varying toroidal
magnetic field resulting in periodic compression–expansion cycle of the tubes
(which is also known as magnetic pumping). Under this cross-field dynamics,
mechanisms like electron-electron large angle collisions, results into
asymmetric exchange of energies between parallel and perpendicular directions.
As a result, the parallel and perpendicular temperatures of the electron flux
tubes undergo radial variations which, in general, scales inversely with $R$.
As our simulation is collisionless, we believe that there exists an underlying
collisionless process, which introduces asymmetry in the compression–expansion
cycle (i.e. MP), resulting in collisionless exchange of energy between
parallel and perpendicular directions. More details of this mechanism remain
to be explored in future. Extensive analysis of the radial variation of
parallel and perpendicular temperature of pure toroidal electron plasma had
been performed in our previous work with QQS state. It was found out that the
parallel temperature of toroidal electron plasma flux tubes scales as
$1/R{{}^{2}}$ and the perpendicular temperature scales as $1/R$, which is
consistent with the angular momentum conservation and mean magnetic moment
conservation principles. In the present study, we have observed similar
scaling laws despite the presence of the ion plasmas in the system. As
performed in the previous study, parallel and perpendicular temperatures are
averaged in two different ways: 1. In toroidal and vertical directions
(Z-averaged). 2. In toroidal direction only, around mid-Z plane (averaged over
toroidal flux tubes). In the first method, particles from different vertical
coordinates contribute in the calculations. In the second method, particles
within different toroidal flux tubes along with radial coordinates, around
mid-z plane have been considered. In the second method, the particles within a
flux tube at certain R may be expected to follow similar dynamics to each
other. In Fig. 16 (a)-(d), these averaged temperatures
${\bar{T}}_{\parallel}^{e}(R,t)$ (Fig. 16 (a),(b)) and
${\bar{T}}_{\bot}^{e}(R,t)$ (Fig. 16 (c),(d)) are shown as functions of $R$,
at two different time periods $\tau$ $=$ $16.67$ and $\tau$ $=$ $25.56$ for
$f$ $=$ $0.007$ case. Z-averaged quantities are expressed as
${\bar{T}}_{\parallel}^{e}(R,t)(Z_{avg})$,
${\bar{T}}_{\bot}^{e}(R,t)(Z_{avg})$ and flux tube averaged quantities are
expressed as ${\bar{T}}_{\parallel}^{e}(R,t)(Fluxtube)$,
${\bar{T}}_{\bot}^{e}(R,t)(Fluxtube)$ in all of these Figures. The magenta
colored vertical lines represents $1/e$ times value of the peak electron
density. For high accuracy the analysis is performed over different sized bins
along $R$ and the respective range of the temperatures are shown by the error
bars in these figures. In Fig. 16 (a) and (b),
${\bar{T}}_{\parallel}^{e}(R,t)(Z_{avg})$ and
${\bar{T}}_{\parallel}^{e}(R,t)(Fluxtube)$ values are higher at the inboard
side and gradually fall with $R$. The temperature values are fitted with
$\alpha/R^{2}$ and $\alpha_{2}/R^{2}$ ($\alpha$ and $\alpha_{2}$ are an
arbitrary constants) showing $1/R^{2}$ scaling with $R$. In Fig. 16 (c) and
(d), ${\bar{T}}_{\bot}^{e}(R,t)(Z_{avg})$ and
${\bar{T}}_{\bot}^{e}(R,t)(Fluxtube)$ values are also higher at the inboard
side but showing non-monotonic variation with $R$. The average temperature
values are fitted with $\beta/R$ and $\beta_{2}/R$ ($\beta$ and $\beta_{2}$
are arbitrary constants) showing $1/R$ scaling with $R$. In these plots, the
scaling laws are preserved at later simulation times representing the
conservation of angular momentum and mean magnetic moment, thus suggesting the
conserved quantities of the system are not affected by the introduction of ion
plasma in the system. The reason for this weak non-monotonicity in the
perpendicular temperature profile is unknown and also seen in the analysis of
pure electron plasma in QQSKhamaru, Ganesh, and Sengupta (2021b) state.
## IV Conclusions and Discussions
In this paper, toroidal electron plasmas in a tight aspect ratio axisymmetric
toroidal device to the presence of small fraction of ions, has been
investigated with a three dimensional collisionless Particle-in-Cell
simulation study with $Ar^{+}$ ions as the ion species. The electron plasma is
loaded with a “seed” solution obtained by zero-inertia entropy maximization
and initialized in the PIC code to evolve to attain a quiescent quasi-steady
state (QQS). After a QQS state is attained, $Ar^{+}$ ions and secondary
electrons are introduced cold into the system. $Ar^{+}$ ions of different
number density values i.e. $f$ values (0.001 to 0.01) are used and the
dependency of the instability on the ion density is addressed via different
diagnostics methods.
We have demonstrated that the electron plasma dynamics and the ion plasma
dynamics are affected by presence of small number of ions, for mainly $f$
values $>$ 0.001 i.e. for such ion population in the system results in growth
of the wall probe current after the ions are introduced. Thus the results are
mainly interpreted on the basis of high $f$ value datum ($f$ $\geq$ 0.005).
The growth of the wall probe current is algebraic in nature and saturates at
later simulation time, when a substantial amount of ions are lost. In our
preloaded ion case there is no continuous source of ions but ions are being
lost with simulation time. This is a unique situation not belonging to
strictly trapped ions or transient ions in the system. Spatial evolution of
the electron plasma and corresponding spectrogram analysis demonstrate the
destabilized “center of charge motion” ($m$ $=$ $1$) of the electron plasma in
poloidal plane along with chirped coupling between toroidal Diocotron modes
($m$ $=$ $1$ to $9$ for $f$ $=$ $0.01$) with $m$ $=$ $2$ as dominant mode.
Investigation of the evolution of the three dimensional potential well of the
system, reveals interesting features. Based on the above, we have argued that
the ion fraction destabilizes the electron plasma in QQS state via a toroidal
ion resonance instability mechanism.
The variations in the ion and electron plasma energies show collisionless
transfer of electron potential energy to the kinetic energy of the ion plasma.
Though ions start to exhibit a dynamic nature in the poloidal direction,
toroidal motion of the ions are negligible. This is understandable as the
$E_{\parallel}=-\nabla_{\parallel}\phi\sim 0$, there is no parallel
acceleration or energizing dynamics for ions in the parallel direction. After
initial energy gain, the ion plasmas start to equilibrate and attains near-
Maxwellian distribution in the poloidal plane. We have identified a certain
time period after which the ion energy gain is trivial. Correlating the wall
probe growth in the same time period, it has been found out that the wall
probe current also starts to saturate after that time period. Also a linear
scaling of the growth rate of the instability with respect to the ion density
($f$ values) has been obtained for the above mentioned two time windows.
In this study, volume averaged parallel and perpendicular temperatures are
defined and used to calculate the temperatures of the ion and electron
plasmas. The volume averaged perpendicular temperature of the ion plasma is
obtained when ions start to equilibrate, shows high temperature value,
indicating ion heating due to gain of energy via the toroidal ion-resonance
instability. However, the exact process of equilibration of ions resulting
Maxwellian distributions is not clear. The volume averaged temperature values
of the electron plasma rises with simulation time though attaining a
fluctuating nature near the end of the simulation time.
For two different simulation time periods, the averaged temperatures
${\bar{T}}_{\parallel}^{e}$ and ${\bar{T}}_{\bot}^{e}$ show $1/R^{2}$ and
$1/R$ scaling with $R$ respectively, as can be expected from conservation of
angular momentum and conservation of $\mu$.
We have demonstrated toroidal ion resonance instability of toroidal electron
plasma and the underlying processes through extensive analysis and showed that
under controlled ion population, the growth of the instability can be
addressed via a scaling law of the growth rate with the ion density value. In
experimental conditions, ions are produced continuously via electron-impact
ionization and other collisional processes play a crucial role in the energy
exchange processes. An investigation considering the impact ionization,
electron-neutral and ion-neutral collision operations is required. In an
ongoing study we have implemented these collisional processes and the results
will be reported soon.
###### Acknowledgements.
All numerical results reported in this study were obtained using the ANTYA
cluster at IPR. The authors would like to thank the staff of the Computer
Center of IPR for facilities of the ANTYA HPC. The authors would like to thank
Dr. Nirmal Kumar Bisai for careful reading of the manuscript and for his
comments.
## V Data availability
The data that support the findings of this study are available from the
corresponding author upon reasonable request.
## References
* Malmberg and deGrassie (1975) J. H. Malmberg and J. S. deGrassie, “Properties of nonneutral plasma,” Phys. Rev. Lett. 35, 577–580 (1975).
* Paul (1990) W. Paul, “Electromagnetic traps for charged and neutral particles (nobel lecture),” Angewandte Chemie International Edition in English 29, 739–748 (1990), https://onlinelibrary.wiley.com/doi/pdf/10.1002/anie.199007391 .
* Yamada and Himura (2016) S. Yamada and H. Himura, “Note: Consecutive capture of images of ions and electrons using high-voltage vacuum relay,” Review of Scientific Instruments 87, 036109 (2016), https://aip.scitation.org/doi/pdf/10.1063/1.4944861 .
* Hurst _et al._ (2014) N. C. Hurst, J. R. Danielson, C. J. Baker, and C. M. Surko, “Electron plasma orbits from competing Diocotron drifts,” Phys. Rev. Lett. 113, 025004 (2014).
* Lane and Orperformedz (2019) R. A. Lane and C. A. Orperformedz, “Electrostatic equilibria of non-neutral plasmas confined in a penning trap with axially varying magnetic field,” Physics of Plasmas 26, 052511 (2019), https://doi.org/10.1063/1.5092136 .
* Davidson and Felice (1998) R. C. Davidson and G. M. Felice, “Influence of profile shape on the Diocotron instability in a non-neutral plasma column,” Physics of Plasmas 5, 3497–3511 (1998), https://doi.org/10.1063/1.873067 .
* O’Neil and Dubin (1998) T. M. O’Neil and D. H. E. Dubin, “Thermal equilibria and thermodynamics of trapped plasmas with a single sign of charge,” Physics of Plasmas 5, 2163–2193 (1998), https://doi.org/10.1063/1.872925 .
* Landa (2019) H. Landa, “Tuning nonthermal distributions to thermal ones in time-dependent paul traps,” Phys. Rev. A 100, 013413 (2019).
* Greaves and Surko (2000) R. G. Greaves and C. M. Surko, “Inward transport and compression of a positron plasma by a rotating electric field,” Phys. Rev. Lett. 85, 1883–1886 (2000).
* Ahmadi and et al. (2016) M. Ahmadi, M. Baquero-Ruiz, W. Bertsche, E. Butler, A. Capra, C. Carruth, C. L. Cesar, M. Charlton, A. E. Charman, S. Eriksson et al., Nature 529, 373–376 (2016).
* Ahmadi _et al._ (2018) M. Ahmadi, B. X. R. Alves, C. J. Baker, W. Bertsche, A. Capra, C. Carruth, C. L. Cesar, M. Charlton, S. Cohen, R. Collister et al., (ALPHA Collaboration), “Enhanced control and reproducibility of non-neutral plasmas,” Phys. Rev. Lett. 120, 025001 (2018).
* O’Neil and Driscoll (1979) T. M. O’Neil and C. F. Driscoll, “Transport to thermal equilibrium of a pure electron plasma,” The Physics of Fluids 22, 266–277 (1979), https://aip.scitation.org/doi/pdf/10.1063/1.862577 .
* Malmberg and Driscoll (1980) J. H. Malmberg and C. F. Driscoll, “Long-time containment of a pure electron plasma,” Phys. Rev. Lett. 44, 654–657 (1980).
* Driscoll, Malmberg, and Fine (1988) C. F. Driscoll, J. H. Malmberg, and K. S. Fine, “Observation of transport to thermal equilibrium in pure electron plasmas,” Phys. Rev. Lett. 60, 1290–1293 (1988).
* Dubin and O’Neil (1999) D. H. E. Dubin and T. M. O’Neil, Rev. Mod. Phys. 71, 87–172 (1999).
* Fine _et al._ (1995) K. S. Fine, A. C. Cass, W. G. Flynn, and C. F. Driscoll, Phys. Rev. Lett. 75, 3277–3280 (1995).
* Briggs, Daugherty, and Levy (1970) R. J. Briggs, J. D. Daugherty, and R. H. Levy, “Role of landau damping in crossed‐field electron beams and inviscid shear flow,” The Physics of Fluids 13, 421–432 (1970), https://aip.scitation.org/doi/pdf/10.1063/1.1692936 .
* Ganesh and Pahari (2006) R. Ganesh and S. Pahari, in _Frontiers in Turbulence and Coherent Structures_, Vol. 6 (World Scientific, 2006) p. 471.
* Rome _et al._ (2000) M. Rome, M. Brunetti, F. Califano, F. Pegoraro, and R. Pozzoli, “Motion of extended vortices in an inhomogeneous pure electron plasma,” Physics of Plasmas 7, 2856–2865 (2000), https://doi.org/10.1063/1.874135 .
* Sengupta and Ganesh (2014) M. Sengupta and R. Ganesh, “Inertia driven radial breathing and nonlinear relaxation in cylindrically confined pure electron plasma,” Physics of Plasmas 21, 022116 (2014), https://doi.org/10.1063/1.4866022 .
* Birdsall and Langdon (2004) C. Birdsall and A. Langdon, _Plasma Physics via Computer Simulation_, Series in Plasma Physics and Fluid Dynamics (Taylor & Francis, 2004).
* Niemann _et al._ (2019) M. Niemann, T. Meiners, J. Mielke, M. J. Borchert, J. M. Cornejo, S. Ulmer, and C. Ospelkaus, Measurement Science and Technology 31, 035003 (2019).
* Schuh _et al._ (2019) M. Schuh, F. Heiße, T. Eronen, J. Ketter, F. Köhler-Langes, S. Rau, T. Segal, W. Quint, S. Sturm, and K. Blaum, “Image charge shift in high-precision penning traps,” Phys. Rev. A 100, 023411 (2019).
* Gutiérrez _et al._ (2019) M. J. Gutiérrez, J. Berrocal, F. Domínguez, I. Arrazola, M. Block, E. Solano, and D. Rodríguez, “Dynamics of an unbalanced two-ion crystal in a penning trap for application in optical mass spectrometry,” Phys. Rev. A 100, 063415 (2019).
* Peurrung, Kouzes, and Barlow (1996) A. Peurrung, R. Kouzes, and S. Barlow, “The non-neutral plasma: an introduction to physics with relevance to cyclotron resonance mass spectrometry,” International Journal of Mass Spectrometry and Ion Processes 157-158, 39 – 83 (1996).
* Fajans and Surko (2020) J. Fajans and C. M. Surko, “Plasma and trap-based techniques for science with antimatter,” Physics of Plasmas 27, 030601 (2020), https://doi.org/10.1063/1.5131273 .
* Morigi _et al._ (1999) G. Morigi, J. Eschner, J. I. Cirac, and P. Zoller, “Laser cooling of two trapped ions: Sideband cooling beyond the lamb-dicke limit,” Phys. Rev. A 59, 3797–3808 (1999).
* Morigi and Walther (2001) G. Morigi and H. Walther, “Two-species coulomb chains for quantum information,” The European Physical Journal D - Atomic, Molecular, Optical and Plasma Physics 13 (2001), 10.1007/s100530170275.
* Douglas, Frank, and Mao (2005) D. J. Douglas, A. J. Frank, and D. Mao, “Linear ion traps in mass spectrometry,” Mass Spectrometry Reviews 24, 1–29 (2005) .
* Greaves and Surko (2002) R. G. Greaves and C. M. Surko, “Practical limits on positron accumulation and the creation of electron-positron plasmas,” AIP Conference Proceedings 606, 10–23 (2002), https://aip.scitation.org/doi/pdf/10.1063/1.1454263 .
* Higaki _et al._ (2017) H. Higaki, C. Kaga, K. Fukushima, H. Okamoto, Y. Nagata, Y. Kanai, and Y. Yamazaki, “Simultaneous confinement of low-energy electrons and positrons in a compact magnetic mirror trap,” New Journal of Physics 19, 023016 (2017).
* Hicks, Bowman, and Godden (2019) N. K. Hicks, A. Bowman, and K. Godden, “Particle-in-cell simulation of quasi-neutral plasma trapping by rf multipole electric fields,” Physics 1, 392–401 (2019).
* Nakajima, Himura, and Sanpei (2021) Y. Nakajima, H. Himura, and A. Sanpei, “Counter differential rigid-rotation equilibrium of electrically non-neutral two-fluid plasma with finite pressure,” Journal of Plasma Physics 87, 905870415 (2021).
* Janes _et al._ (1966) G. S. Janes, R. H. Levy, H. A. Bethe, and B. T. Feld, Phys. Rev. 145, 925–952 (1966).
* Daugherty, Eninger, and Janes (1969) J. D. Daugherty, J. E. Eninger, and G. S. Janes, “Experiments on the injection and containment of electron plasmas in a toroidal apparatus,” The Physics of Fluids 12, 2677–2693 (1969), https://aip.scitation.org/doi/pdf/10.1063/1.1692411 .
* Clark _et al._ (1976) W. Clark, P. Korn, A. Mondelli, and N. Rostoker, Phys. Rev. Lett. 37, 592–595 (1976).
* Zaveri _et al._ (1992) P. Zaveri, P. I. John, K. Avinash, and P. K. Kaw, “Low-aspect-ratio toroidal equilibria of electron plasmas,” Phys. Rev. Lett. 68, 3295–3298 (1992).
* Khirwadkar _et al._ (1993) S. S. Khirwadkar, P. I. John, K. Avinash, A. K. Agarwal, and P. K. Kaw, “Steady state formation of a toroidal electron plasma,” Phys. Rev. Lett. 71, 4334–4337 (1993).
* Lachhvani _et al._ (2017) L. Lachhvani, S. Pahari, S. Sengupta, Y. G. Yeole, M. Bajpai, and P. K. Chattopadhyay, Physics of Plasmas 24, 102132 (2017), https://doi.org/10.1063/1.5009013 .
* Stoneking _et al._ (2004) M. R. Stoneking, M. A. Growdon, M. L. Milne, and R. T. Peterson, Phys. Rev. Lett. 92, 095003 (2004).
* Davidson (1990) R. C. Davidson, _An introduction to the physics of nonneutral plasmas._ (IEEE, 1990).
* Marksteiner _et al._ (2008) Q. R. Marksteiner, T. S. Pedersen, J. W. Berkery, M. S. Hahn, J. M. Mendez, B. Durand de Gevigney, and H. Himura, Phys. Rev. Lett. 100, 065002 (2008).
* Saitoh _et al._ (2010) H. Saitoh, Z. Yoshida, J. Morikawa, Y. Yano, H. Hayashi, T. Mizushima, Y. Kawai, M. Kobayashi, and H. Mikami, “Confinement of electron plasma by levitating dipole magnet,” Physics of Plasmas 17, 112111 (2010), https://doi.org/10.1063/1.3514207 .
* Khamaru, Sengupta, and Ganesh (2019) S. Khamaru, M. Sengupta, and R. Ganesh, “Dynamics of a toroidal pure electron plasma using 3d pic simulations,” Physics of Plasmas 26, 112106 (2019), https://doi.org/10.1063/1.5111747 .
* Khamaru, Ganesh, and Sengupta (2021a) S. Khamaru, R. Ganesh, and M. Sengupta, “A novel quiescent quasi-steady state of a toroidal electron plasma,” Physics of Plasmas 28, 042101 (2021a), https://doi.org/10.1063/5.0032880 .
* Khamaru, Ganesh, and Sengupta (2021b) S. Khamaru, R. Ganesh, and M. Sengupta, “Erratum: “a novel quiescent quasi-steady state of a toroidal electron plasma” [phys. plasmas 28, 042101 (2021)],” Physics of Plasmas 28, 119901 (2021b), https://doi.org/10.1063/5.0074384 .
* Crooks and O’Neil (1996) S. M. Crooks and T. M. O’Neil, “Transport in a toroidally confined pure electron plasma,” Physics of Plasmas 3, 2533–2537 (1996), https://doi.org/10.1063/1.871971 .
* Brambilla _et al._ (2018) G. Brambilla, C. Kalapotharakos, A. N. Timokhin, A. K. Harding, and D. Kazanas, “Electron–positron pair flow and current composition in the pulsar magnetosphere,” The Astrophysical Journal 858, 81 (2018).
* Stoneking _et al._ (2020) M. R. Stoneking, T. S. Pedersen, P. Helander, H. Chen, U. Hergenhahn, E. V. Stenson, G. Fiksel, J. von der Linden, H. Saitoh, C. M. Surko, and et al., “A new frontier in laboratory physics: magnetized electron–positron plasmas,” Journal of Plasma Physics 86, 155860601 (2020).
* Singer _et al._ (2021) M. Singer, M. R. Stoneking, E. V. Stenson, S. Nißl, A. Deller, A. Card, J. Horn-Stanja, T. Sunn Pedersen, H. Saitoh, and C. Hugenschmidt, “Injection of positrons into a dense electron plasma in a magnetic dipole trap,” Physics of Plasmas 28, 062506 (2021), https://doi.org/10.1063/5.0050881 .
* White, Malmberg, and Driscoll (1982) W. D. White, J. H. Malmberg, and C. F. Driscoll, “Resistive-wall destabilization of Diocotron waves,” Phys. Rev. Lett. 49, 1822–1826 (1982).
* Davidson and Chao (1996) R. C. Davidson and E. H. Chao, “l=1 electrostatic instability induced by electron‐neutral collisions in a nonneutral electron plasma interacting with background neutral gas,” Physics of Plasmas 3, 3279–3287 (1996), https://doi.org/10.1063/1.871610 .
* Levy, Daugherty, and Buneman (1969) R. H. Levy, J. D. Daugherty, and O. Buneman, “Ion resonance instability in grossly nonneutral plasmas,” The Physics of Fluids 12, 2616–2629 (1969), https://aip.scitation.org/doi/pdf/10.1063/1.1692404 .
* Davidson and Uhm (1977) R. C. Davidson and H. Uhm, “Influence of strong self‐electric fields on the ion resonance instability in a nonneutral plasma column,” The Physics of Fluids 20, 1938–1946 (1977), https://aip.scitation.org/doi/pdf/10.1063/1.861813 .
* Fajans (1993) J. Fajans, “Transient ion resonance instability,” Physics of Fluids B: Plasma Physics 5, 3127–3135 (1993), https://doi.org/10.1063/1.860649 .
* Peurrung, Notte, and Fajans (1993) A. J. Peurrung, J. Notte, and J. Fajans, “Observation of the ion resonance instability,” Phys. Rev. Lett. 70, 295–298 (1993).
* Lachhvani _et al._ (2016) L. Lachhvani, S. Pahari, R. Goswami, M. Bajpai, Y. Yeole, and P. K. Chattopadhyay, Physics of Plasmas 23, 062109 (2016), https://doi.org/10.1063/1.4953440 .
* Eckhouse, Fisher, and Rostoker (1981) S. Eckhouse, A. Fisher, and N. Rostoker, “Possible observation of the ion‐resonance instability in a non‐neutral electron plasma,” Applied Physics Letters 38, 318–320 (1981), https://doi.org/10.1063/1.92354 .
* Bettega _et al._ (2006) G. Bettega, F. Cavaliere, M. Cavenago, F. De Luca, A. Illiberi, R. Pozzoli, and M. Romé, “Active control of the ion resonance instability by ion removing fields,” Physics of Plasmas 13, 112102 (2006), https://doi.org/10.1063/1.2363175 .
* Kabantsev and Driscoll (2007) A. Kabantsev and C. Driscoll, “Ion-induced instability of Diocotron modes in electron plasmas modelling curvature-driven flute modes,” Fusion Science and Technology 51, 96–99 (2007), https://doi.org/10.13182/FST07-A1324 .
* Stoneking _et al._ (2002) M. R. Stoneking, P. W. Fontana, R. L. Sampson, and D. J. Thuecks, “Electron plasmas in a “partial” torus,” Physics of Plasmas 9, 766–771 (2002), https://doi.org/10.1063/1.1445425 .
* Marksteiner _et al._ (2008) Q. R. Marksteiner, T. S. Pedersen, J. W. Berkery, M. S. Hahn, J. M. Mendez, B. Durand de Gevigney, and H. Himura, “Observations of an ion-driven instability in non-neutral plasmas confined on magnetic surfaces,” Phys. Rev. Lett. 100, 065002 (2008).
* Sengupta and Ganesh (2015) M. Sengupta and R. Ganesh, “Linear and nonlinear evolution of the ion resonance instability in cylindrical traps: A numerical study,” Physics of Plasmas 22, 072112 (2015), https://doi.org/10.1063/1.4927126 .
* Sengupta and Ganesh (2016) M. Sengupta and R. Ganesh, “Influence of electron-neutral elastic collisions on the instability of an ion-contaminated cylindrical electron plasma: 2d3v pic-with-mcc simulations,” Physics of Plasmas 23, 102111 (2016), https://doi.org/10.1063/1.4964913 .
* Sengupta and Ganesh (2017) M. Sengupta and R. Ganesh, “Destabilization of a cylindrically confined electron plasma by impact ionization of background neutrals: 2d3v pic simulation with monte-carlo-collisions,” Physics of Plasmas 24, 032105 (2017), https://doi.org/10.1063/1.4978473 .
* Khamaru, Sengupta, and Ganesh (2019) S. Khamaru, M. Sengupta, and R. Ganesh, “Dynamics of a toroidal pure electron plasma using 3d pic simulations,” Physics of Plasmas 26, 112106 (2019), https://doi.org/10.1063/1.5111747 .
* Sengupta, Khamaru, and Ganesh (2021) M. Sengupta, S. Khamaru, and R. Ganesh, “Self-organization of pure electron plasma in a partially toroidal magnetic-electrostatic trap: A 3d particle-in-cell simulation,” Journal of Applied Physics 130, 133305 (2021), https://doi.org/10.1063/5.0055828 .
* Huba (2013) J. D. Huba, _Plasma Physics_ (Naval Research Laboratory, Washington, DC, 2013) p. 28.
|
# Radiative stability of tiny cosmological constant from the Swampland and
quantized compactification
Cao H. Nam<EMAIL_ADDRESS>Phenikaa Institute for Advanced
Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha
Dong, Hanoi 12116, Vietnam
###### Abstract
We address a quantization mechanism that can allow us to understand why the
cosmological constant is not large under the quantum corrections from studying
the circle compactification solution of the Standard Model coupled to Einstein
gravity which is subject to the constraint of the Swampland conjectures. A
novel result in the present work compared to the previous investigations in
the literature is that the radius of the compactified dimension and the 4D
cosmological constant $\Lambda_{4}$ must in fact be quantized. The
quantization rule of the cosmological constant is given by $\Lambda_{4}\propto
n^{2}$ with $n=1,2,3,4,\ldots$, which means that the values of $\Lambda_{4}$
are not arbitrary but only its specific values are allowed. In general, the
quantum corrections as well as other effects would break this quantization
rule. Hence, it could prevent the quantum fluctuations from generating zero-
point energy contributions to the cosmological constant.
The current cosmological and astrophysical observations Planck2018 have
implied a positive cosmological constant of the value around $\sim
10^{-120}M^{4}_{\text{Pl}}\simeq\left(2.4\times 10^{-3}\text{eV}\right)^{4}$
Tanabashi2018 (responsible for the late accelerating expansion of the
universe) which is a very small number compared to the electroweak scale
around $\sim 10^{2}$ GeV and the Planck energy scale $M_{\text{Pl}}\sim
10^{19}$ GeV. There is no problem at all with respect to having such a tiny
value for the cosmological constant at the classical level. However, the
problem arises when the quantum corrections (and other corrections such as
non-perturbative effects coming from the QCD instantons) for the cosmological
constant are included, which would make the value of the cosmological constant
much bigger, and hence its small value is not stable to the large quantum
corrections Zeldovich6768 ; Birrell1994 . For instance, with the cutoff at the
Planck energy scale, the quantum fluctuations would additionally contribute to
the cosmological constant with a quantity around $\sim M^{4}_{\text{Pl}}$. In
order to yield the correct physical result, a cancellation between the bare
cosmological constant and the quantum corrections must be fine-tuned to an
unimaginable precision which is around $\sim 120$ orders of magnitude in
order. Therefore, the cosmological constant problem has been regarded as one
of the most important problems in modern physics Weinberg1989 ; Bousso2008 .
The traditional approach to understanding why the cosmological constant is not
large under quantum corrections is based on a certain symmetry (either
continuous or discrete), which provides a mechanism to cancel the quantum
contributions precisely. The most promising symmetry is perhaps supersymmetry
which if unbroken would lead to a very precise cancellation between boson and
fermions contributing to the vacuum energy since their contributions are
opposite signs. Unfortunately, the superpartners of the Standard Model (SM)
particles have not been observed so far since supersymmetry must be
spontaneously broken down at least at the TeV scale. It was also suggested
that the one-loop contribution to the cosmological constant might vanish in
certain string models with broken supersymmetry Kachru1998 ; Kachru1999 .
However, it was argued that the higher-order quantum corrections would spoil
this vanishing Harvey1999 ; Iengo2000 . In addition, these string models need
to have the supersymmetric non-Abelian sector Blumenhagen ; Angelantonj1999 ;
Angelantonj2004 . Another promising symmetry to solve the cosmological
constant problem is scale invariance or conformal symmetry which forbids the
quantities or the terms in Lagrangian with the length scale dimension and
hence the cosmological constant has to be zero. However, the observed world
simply is not invariant under the scale transformation. For other ideas which
have been proposed in the literature, see Nobbenhuis2006 ; Burgess2013 for
reviews. Although there have been attempts which have been made to solve the
cosmological constant problem there is indeed no known solution to this
problem at present.
The Swampland program aims Vafa2005 ; Ooguri2007 ; Palti2019 to seek the
universal features of quantum gravity and the consistency criteria which
distinguish effective field theories UV-completed consistent with quantum
gravity from those which cannot or are in the Swampland. In particular, this
program leads to interesting implications for inflationary cosmology
(corresponding to the early accelerating expansion of the universe) which is
constrained by the Swampland criterion for de Sitter Obied2018 ; Garg2019 and
trans-Planckian censorship conjecture Bedroya2020 ; Brandenberger2021 . In
this paper, based on the Swampland program we will point to a novel and simple
mechanism which could forbid quantum fluctuations (as well as other effects)
to generate zero-point energy contributions to the cosmological constant.
Although this mechanism is far from a solution to the cosmological constant
problem, we hope that it could give a step toward a complete solution.
If a theory satisfies the Swampland constraints and is completed into quantum
gravity in UV, then a lower-dimensional theory that is obtained from its
compactification to lower dimensions must do as well. On the contrary, if the
theory upon compactification is inconsistent with quantum gravity then its
parent higher dimensional theory itself is pathological. Because the SM
coupled to General Relativity (GR) is a good effective low energy theory and
thus cannot be in the Swampland, its compactification solution down to lower
dimensions Arkani-Hamed2007 ; Arnold2010 ; Fornal2011 must be consistent with
quantum gravity. In this sense, the Swampland conjectures impose the
constraints not only on the SM coupled to GR itself but also on its
compactification solution. This has motivated the intensive investigations in
the literature with the significant implications between the cosmological
constant and the neutrino masses Martin-Lozano2017 ; Valenzuela2017 ;
Hamada2017 ; Gonzalo2018a ; Gonzalo2018b arising when applying the non-SUSY
AdS conjecture Ooguri2017 (motivated by the weak gravity conjecture Arkani-
Hamed2007b ) and the AdS distance conjecture Lust2019 (realized in string
theory and also supported by the evidence of bottom-up physics Rudelius2021 ;
Gonzalo2021 ; Nam2022 ).
In the present work, in this direction, we study the compactification of the
SM coupled to GR on a circle $S^{1}$. However, a new point here is that we
consider the compactified coordinate dependence of the 3D tensor component of
the 4D metric in a natural way which is usually ignored in the literature.
Interestingly, from investigating the wavefunction profile of the 3D tensor
component along the compactified dimension we find a novel aspect that the
radius of the compactified dimension and the 4D cosmological constant are in
fact quantized. Here, we obtain a quantization rule of the 4D cosmological
constant $\Lambda_{4}$ as $\Lambda_{4}\propto n^{2}$ with $n=1,2,3,4,\ldots$.
In this way, the values of the 4D cosmological constant are not arbitrary but
only its specific values are allowed. This quantization or this discrete
spectrum could prevent the quantum fluctuations from generating zero-point
energy contributions to the 4D cosmological constant because in general, the
additional contribution from the quantum fluctuations would not respect the
quantization rule. This is in analogy to that the hydrogen atom would be
unable to absorb the energy which is not equal to the energy difference
between two initial and final states due to the quantization of its energy
levels.
The action of the SM coupled to GR in the presence of the observed
cosmological constant $\Lambda_{4}$ is given as
$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}\left[\frac{M^{2}_{\text{Pl}}}{2}\left(\mathcal{R}^{(4)}-2\Lambda_{4}\right)+\mathcal{L}_{SM}\right],$
(1)
where $\mathcal{R}^{(4)}$ and $\mathcal{L}_{SM}$ refer to the scalar curvature
of 4D spacetime and Lagrangian of the SM, respectively. We consider the above
system compactified on a circle $S^{1}$ where one of three spatial dimensions
of 4D spacetime is periodic with the period of $2\pi$, which means that the
points $x^{3}$ and $x^{3}+2\pi$ are identified. The 4D metric is generally
parametrized in terms of the 3D fields as follows
$\displaystyle
ds^{2}_{4}=g_{ij}dx^{i}dx^{i}+R^{2}\left[dx^{3}+g_{{}_{A}}A_{i}dx^{i}\right]^{2},$
(2)
where $g_{ij}$ is the 3D metric component, $A_{i}$ the graviphoton, $R$ is the
radion field which determines the radius of $S^{1}$, $g_{{}_{A}}$ is a
coupling constant, and $i,j=0,1,2$.111We use the metric sign as $(-,+,+,+)$.
Using this ansatz of the 4D metric, we can expand explicitly the 4D scalar
curvature $\mathcal{R}^{(4)}$ given in the action (1) as
$\displaystyle\mathcal{R}^{(4)}$ $\displaystyle=$
$\displaystyle\mathcal{R}+\frac{1}{4R^{2}}\left(\partial_{3}g^{ij}\partial_{3}g_{ij}+g^{ij}g^{kl}\partial_{3}g_{ij}\partial_{3}g_{kl}\right)$
(3) $\displaystyle-\frac{g^{2}_{{}_{A}}R^{2}}{4}F_{ij}F^{ij},$
where $\mathcal{R}\equiv
g^{ij}\left(\hat{\partial}_{k}\hat{\Gamma}^{k}_{ji}-\hat{\partial}_{j}\hat{\Gamma}^{k}_{ki}+\hat{\Gamma}^{k}_{ji}\hat{\Gamma}^{l}_{lk}-\hat{\Gamma}^{k}_{li}\hat{\Gamma}^{l}_{jk}\right)$
with
$\hat{\Gamma}^{k}_{ij}\equiv\frac{g^{kl}}{2}(\hat{\partial}_{i}g_{jl}+\hat{\partial}_{j}g_{il}-\hat{\partial}_{l}g_{ij})$,
$\hat{\partial}_{i}\equiv\partial_{i}-g_{{}_{A}}A_{i}\partial_{3}$, and
$F_{ij}=\partial_{i}A_{j}-\partial_{j}A_{i}$. It is important to emphasize
here that the previous investigations about the compactification of the SM
coupled with GR on $S^{1}$ Arkani-Hamed2007 ; Arnold2010 ; Fornal2011 ;
Martin-Lozano2017 ; Valenzuela2017 ; Hamada2017 ; Gonzalo2018a ; Gonzalo2018b
have not considered the compactified dimension dependence of the 3D metric
$g_{ij}$ and hence the second term in the right-hand side of Eq. (3) was
absent. But, in the present work, we will point out that the presence of this
term would lead to the quantization of the 4D cosmological constant which
could provide a mechanism to prevent the vacuum fluctuations as well as other
effects contributing additionally to the cosmological constant.
Let us find the wavefunction profile of the 3D metric $g_{ij}$. In order to do
this, first we obtain the equation of motion for $g_{ij}$ in the vacuum
$R=\text{constant}$, $A_{i}=0$, and the vanishing matter fields by varying the
action (1) in the 3D tensor component of the 4D metric as follows
$\displaystyle\bar{\mathcal{R}}_{ij}-\frac{1}{2}g_{ij}\bar{\mathcal{R}}+\Lambda_{4}g_{ij}+\frac{1}{4R^{2}}\left[g_{ik}g_{jl}\partial^{2}_{3}g^{kl}\right.$
$\displaystyle\left.-\partial^{2}_{3}g_{ij}+g^{kl}\partial_{3}g_{kl}\partial_{3}g_{ij}+2\partial_{3}\left(g_{ij}g^{kl}\partial_{3}g_{kl}\right)\right.$
$\displaystyle\left.-\frac{1}{2}g_{ij}\left\\{\partial_{3}g^{kl}\partial_{3}g_{kl}-\left(g^{kl}\partial_{3}g_{kl}\right)^{2}\right\\}\right]=0,$
(4)
where
$\bar{\mathcal{R}}_{ij}\equiv(\partial_{k}\bar{\Gamma}^{k}_{ji}-\partial_{j}\bar{\Gamma}^{k}_{ki}+\bar{\Gamma}^{l}_{ji}\bar{\Gamma}^{k}_{kl}-\bar{\Gamma}^{l}_{ki}\bar{\Gamma}^{k}_{jl})$
with
$\bar{\Gamma}^{k}_{ij}\equiv\frac{g^{kl}}{2}(\partial_{i}g_{lj}+\partial_{j}g_{ki}-\partial_{l}g_{ij})$
and $\bar{\mathcal{R}}\equiv g^{ij}\bar{\mathcal{R}}_{ij}$. One can solve Eq.
(4) by separating the variables as
$g_{ij}(x^{i},x^{3})=\chi(x^{3})g^{(3)}_{ij}(x^{i})$ where
$g^{(3)}_{ij}(x^{i})$ is identified as the metric of the 3D effective theory
and $\chi(x^{3})$ is its wavefunction profile along the compactified
dimension. Then, substituting this variable separation into Eq. (4), we obtain
the 3D Einstein field equations determining the 3D effective geometry of
spacetime and the equation for the wavefunction profile of the 3D metric as
follows
$\displaystyle\mathcal{R}^{(3)}_{ij}-\frac{1}{2}g^{(3)}_{ij}\mathcal{R}^{(3)}+\Lambda_{3}g^{(3)}_{ij}$
$\displaystyle=$ $\displaystyle 0,$ (5)
$\displaystyle\chi^{\prime\prime}+\frac{11}{4}\frac{\chi^{\prime
2}}{\chi}+\kappa^{2}\chi$ $\displaystyle=$
$\displaystyle\frac{\Lambda_{3}}{R^{-2}},$ (6)
where $\mathcal{R}^{(3)}_{ij}$ ($\mathcal{R}^{(3)}$) is the Ricci (scalar)
curvature of 3D effective spacetime, $\kappa\equiv\sqrt{\Lambda_{4}}/R^{-1}$,
and $\Lambda_{3}$ is a constant which characterizes the dynamics of the 3D
metric $g_{ij}$ along the compactified dimension and from Eq. (5) we see that
it plays the role of a 3D cosmological constant.
It is difficult to find an analytical solution for $\chi(x^{3})$ in the
general case. However, in the situation of $\Lambda_{3}/R^{-2}\ll 1$ (which
can be seen from Table. 1 in Ref. Nam2023 ), we can perturbatively solve Eq.
(6) in the order of $\Lambda_{3}/R^{-2}$. The solution of $\chi(\theta)$ at
the leading order is easily found as follows
$\displaystyle\chi(x^{3})=\left[\frac{1}{2}\left\\{1+\cos\left(\sqrt{15}\kappa
x^{3}\right)\right\\}\right]^{2/15}.$ (7)
Due to the topology of $S^{1}$, the wavefunction profile $\chi(x^{3})$ must be
periodic with the period of $2\pi$, i.e. $\chi(x^{3})=\chi(x^{3}+2\pi)$, which
implies $\sqrt{15}\kappa=n$ with $n=1,2,3,\ldots.$ This thus leads to the
quantization of the radius $R$ of the compactified dimension and 4D
cosmological constant $\Lambda_{4}$ as
$\displaystyle\Lambda_{4}R^{2}=\frac{n^{2}}{15}.$ (8)
Equation (8) means that the radius of the compactified dimension and the 4D
cosmological constant are not arbitrary but must obtain the discrete values
according to the quantization relation (8). On the other hand, the radius of
the compactified dimension and the 4D cosmological constant have no continuous
spectrum but a discrete one. This is a novel point of the circle
compactification solution of the SM coupled to GR which is pointed to the
first time.
The quantization relation (8) results from the nontrivial dynamics of the 3D
metric along the compact direction in the presence of the cosmological
constant. From Eq. (6) that describes the dynamic propagation of the 3D metric
along the compact direction, we see that the 4D cosmological constant
$\Lambda_{4}$ plays the role of a force proportional linearly to the profile
$\chi$ because its presence curves the spacetime geometry, whereas the inverse
radius $R^{-1}$ of the compactified dimension plays the role of a mass for the
3D metric because its propagation is confined along the compact direction.
(Note that, the term $11\chi^{\prime 2}/(4\chi)$ in Eq. (6) comes from the
nonlinear property of the gravitational field which means that gravity is
itself a source creating the curvature of the spacetime geometry.) As a
result, it leads to the oscillation behavior of the 3D metric along the
compactified dimension with the corresponding period depending on the ratio of
$\sqrt{\Lambda_{4}}$ to $R^{-1}$. In addition, the dynamic propagation of the
3D metric is constrained by the periodic property of the compactified
dimension. Hence, both of these aspects result in the quantization of the
radius $R$ of the compactified dimension and 4D cosmological constant
$\Lambda_{4}$.
In fact, the size of the compactified dimension can be physically determined
by the 3D effective potential of the radion field which is generated by the
dynamics of $3$D tensor component along the compactified dimension and the
one-loop quantum corrections with the Casimir energy density calculated in
Ref. Arkani-Hamed2007 (see Appendix D for detailed computation). Taking into
account the stabilization mechanism and from Eq. (8), we can find a general
quantization rule of the 4D cosmological constant in terms of the mass of the
light particles in the spectrum and the parameter $\Lambda_{3}$ as follows
$\displaystyle\Lambda_{4}=f(m_{i},n_{i},\Lambda_{3})n^{2}\propto n^{2},$ (9)
where $f(m_{i},n_{i},\Lambda_{3})$ is a function of $m_{i}$ (the mass of the
$i$th light particle), $n_{i}$ (the number of degrees of freedom corresponding
to the $i$th light particle), and $\Lambda_{3}$. Note that, the function
$f(m_{i},n_{i},\Lambda_{3})$ can be completely determined from the minimum of
the radion potential. The quantization rule (9) implies that the ratio of the
4D cosmological constant to a specific combination of the 3D cosmological
constant and the masses of the particles contributing to the radion potential
must lead to 1, 4, 9, 16, 25, $\ldots$. In general, under the quantum
corrections this ratio would not lead to 1, 4, 9, 16, 25, $\ldots$ which are
only the allowed values. Hence, we expect that the quantization rule (9) can
provide a mechanism to prevent the quantum fluctuations from generating
additional contributions to the 4D cosmological constant.
It is important to emphasize that Eq. (8) can be realized as a quantum gravity
sign that arises at the low-energy regime. Hence, one expects that the
quantization rule for the radius of the compactified dimension and the
cosmological constant could provide a step toward solving not only the
cosmological constant problem but also other important problems (which require
a complete theory of quantum gravity) such as a microscopic description of the
black hole entropy. By considering the compactification of $\text{dS}_{5}$ on
a circle $S^{1}$ to obtain 4D effective field theory, we can find a
quantization relation for the size of the extra dimension as
$R^{-1}\propto\sqrt{\Lambda_{5}}/n$ where $\Lambda_{5}$ is the bulk
cosmological constant and $n=1,2,3,\ldots$, in analogy to what we have done
for the $S^{1}$ compactification of the SM. Each value of the positive integer
$n$ corresponding to a possible value of the size of the extra dimension
implies a possible configuration of 4D effective field theory. Hence, we can
calculate the black hole entropy by summing over all relevant configurations.
We leave an exploration of this proposal to future work.
## Appendix A: The expansion of $\mathcal{R}^{(4)}$ in terms of 3D components
Because Eq. (3) is the main technical starting point, in this appendix we
present detailed computations of how to obtain it. It is easy to do the
calculations in the frame
$\\{\hat{\partial}_{i},\hat{\partial}_{3}\\}\equiv\\{\hat{\partial}_{\mu}\\}$
where $\hat{\partial}_{i}\equiv\partial_{i}-g_{{}_{A}}A_{i}\partial_{3}$ and
$\hat{\partial}_{3}\equiv\partial_{3}$ which transform as a 3D vector and a 1D
vector, respectively. The corresponding coframe is given by
$\\{dx^{i},dx^{3}+g_{{}_{A}}A_{i}dx^{i}\\}$ which dual to
$\\{\hat{\partial}_{i},\hat{\partial}_{3}\\}$, respectively. The components of
4D metric and its dual are given in these bases as
$\displaystyle\hat{g}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\textrm{diag}\left(g_{ij},R^{2}\right),$
$\displaystyle\hat{g}^{\mu\nu}$ $\displaystyle=$
$\displaystyle\textrm{diag}\left(g^{ij},R^{-2}\right).$ (10)
Then, we can determine the Christoffel connection
$\hat{\Gamma}^{\rho}_{\mu\nu}$ and the Riemann curvature tensor
$\hat{\mathcal{R}}^{\lambda}_{\mu\rho\nu}$ in these bases as follows
$\displaystyle\hat{\Gamma}^{\rho}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\hat{g}^{\rho\lambda}}{2}\left(\hat{\partial}_{\mu}\hat{g}_{\nu\lambda}+\hat{\partial}_{\nu}\hat{g}_{\mu\lambda}-\hat{\partial}_{\lambda}\hat{g}_{\mu\nu}\right)$
$\displaystyle+\frac{\hat{g}^{\rho\lambda}}{2}\left(C^{\sigma}_{\lambda\mu}\hat{g}_{\nu\sigma}+C^{\sigma}_{\lambda\nu}\hat{g}_{\mu\sigma}\right)+\frac{C^{\rho}_{\mu\nu}}{2},$
$\displaystyle\hat{\mathcal{R}}^{\lambda}_{\mu\rho\nu}$ $\displaystyle=$
$\displaystyle\hat{\partial}_{\rho}\hat{\Gamma}^{\lambda}_{\nu\mu}-\hat{\partial}_{\nu}\hat{\Gamma}^{\lambda}_{\rho\mu}+\hat{\Gamma}^{\sigma}_{\nu\mu}\hat{\Gamma}^{\lambda}_{\rho\sigma}-\hat{\Gamma}^{\sigma}_{\rho\mu}\hat{\Gamma}^{\lambda}_{\nu\sigma}$
(11) $\displaystyle-C^{\sigma}_{\rho\nu}\hat{\Gamma}^{\lambda}_{\sigma\mu},$
where $C^{\rho}_{\mu\nu}$ defines the commutation of two frame fields as
$\left[\hat{\partial}_{\mu},\hat{\partial}_{\nu}\right]=C^{\rho}_{\mu\nu}\hat{\partial}_{\rho}.$
(12)
The 4D scalar curvature $\mathcal{R}^{(4)}$ reads
$\displaystyle\mathcal{R}^{(4)}=\hat{g}^{\mu\nu}\hat{\mathcal{R}}^{\rho}_{\mu\rho\nu}=g^{ij}\hat{\mathcal{R}}^{\rho}_{i\rho
j}+g^{33}\hat{\mathcal{R}}^{\rho}_{3\rho 3}.$ (13)
The first and second terms are explicitly expanded as follows
$\displaystyle g^{ij}\hat{\mathcal{R}}^{\rho}_{i\rho j}$ $\displaystyle=$
$\displaystyle
g^{ij}\left(\hat{\partial}_{\rho}\hat{\Gamma}^{\rho}_{ji}-\hat{\partial}_{j}\hat{\Gamma}^{\rho}_{\rho
i}+\hat{\Gamma}^{\rho}_{ji}\hat{\Gamma}^{\lambda}_{\lambda\rho}-\hat{\Gamma}^{\rho}_{\lambda
i}\hat{\Gamma}^{\lambda}_{j\rho}\right.$ (14)
$\displaystyle\left.-C^{\rho}_{\lambda j}\hat{\Gamma}^{\lambda}_{\rho
i}\right)$ $\displaystyle=$ $\displaystyle
g^{ij}\left[\left(\hat{\partial}_{k}\hat{\Gamma}^{k}_{ji}-\hat{\partial}_{j}\hat{\Gamma}^{k}_{ki}+\hat{\Gamma}^{k}_{ji}\hat{\Gamma}^{l}_{lk}-\hat{\Gamma}^{k}_{li}\hat{\Gamma}^{l}_{jk}\right)\right.$
$\displaystyle+\left.\left(\hat{\partial}_{3}\hat{\Gamma}^{3}_{ji}+\hat{\Gamma}^{3}_{ji}\hat{\Gamma}^{k}_{k3}+\hat{\Gamma}^{k}_{ji}\hat{\Gamma}^{3}_{3k}+\hat{\Gamma}^{3}_{ji}\hat{\Gamma}^{3}_{33}\right)\right.$
$\displaystyle-\left.\left(\hat{\partial}_{j}\hat{\Gamma}^{3}_{3i}+\hat{\Gamma}^{k}_{3i}\hat{\Gamma}^{3}_{jk}+\hat{\Gamma}^{3}_{ki}\hat{\Gamma}^{k}_{j3}+\hat{\Gamma}^{3}_{3i}\hat{\Gamma}^{3}_{j3}\right)\right.$
$\displaystyle-\left.\left(C^{3}_{kj}\hat{\Gamma}^{k}_{3i}+C^{3}_{3j}\hat{\Gamma}^{3}_{3i}\right)\right],$
$\displaystyle g^{33}\hat{\mathcal{R}}^{\rho}_{3\rho 3}$ $\displaystyle=$
$\displaystyle
g^{33}\left(\hat{\partial}_{\rho}\hat{\Gamma}^{\rho}_{33}-\hat{\partial}_{3}\hat{\Gamma}^{\rho}_{\rho
3}+\hat{\Gamma}^{\rho}_{33}\hat{\Gamma}^{\lambda}_{\lambda\rho}-\hat{\Gamma}^{\rho}_{\lambda
3}\hat{\Gamma}^{\lambda}_{3\rho}\right.$ (15)
$\displaystyle\left.-C^{\rho}_{\lambda 3}\hat{\Gamma}^{\lambda}_{\rho
3}\right)$ $\displaystyle=$ $\displaystyle
g^{33}\left(\hat{\partial}_{i}\hat{\Gamma}^{i}_{33}-\hat{\partial}_{3}\hat{\Gamma}^{i}_{i3}+\hat{\Gamma}^{i}_{33}\hat{\Gamma}^{j}_{ji}+\hat{\Gamma}^{3}_{33}\hat{\Gamma}^{i}_{i3}\right.$
$\displaystyle\left.-\hat{\Gamma}^{j}_{i3}\hat{\Gamma}^{i}_{3j}-\hat{\Gamma}^{3}_{i3}\hat{\Gamma}^{i}_{33}-C^{3}_{i3}\hat{\Gamma}^{i}_{33}\right).$
In (14) and (15), we find the following combinations
$\displaystyle
g^{ij}\left(\hat{\partial}_{3}\hat{\Gamma}^{3}_{ji}+\hat{\Gamma}^{3}_{ji}\hat{\Gamma}^{k}_{k3}+\hat{\Gamma}^{3}_{ji}\hat{\Gamma}^{3}_{33}\right)\equiv\nabla_{\mu}Y^{\mu}_{1}$
$\displaystyle+\frac{g^{33}}{2}\hat{\partial}_{3}g^{ij}\hat{\partial}_{3}g_{ij},$
(16) $\displaystyle
g^{ij}\left(\hat{\partial}_{j}\hat{\Gamma}^{3}_{3i}-\hat{\Gamma}^{k}_{ji}\hat{\Gamma}^{3}_{3k}\right)-g^{33}\hat{\Gamma}^{i}_{33}\hat{\Gamma}^{3}_{3i}$
$\displaystyle\equiv$ $\displaystyle\nabla_{\mu}Y^{\mu}_{2},$ (17)
$\displaystyle
g^{33}\left(\hat{\partial}_{i}\hat{\Gamma}^{i}_{33}+\hat{\Gamma}^{i}_{33}\hat{\Gamma}^{j}_{ji}+\hat{\Gamma}^{i}_{33}\hat{\Gamma}^{3}_{3i}\right)\equiv\nabla_{\mu}Y^{\mu}_{3}$
$\displaystyle+\frac{g^{ij}}{2}\hat{\partial}_{i}g^{33}\hat{\partial}_{j}g_{33}+g_{{}_{A}}g^{ij}\hat{\partial}_{3}A_{i}g^{33}\hat{\partial}_{j}g_{33},$
(18) $\displaystyle
g^{33}\left(\hat{\partial}_{3}\hat{\Gamma}^{i}_{i3}-\hat{\Gamma}^{3}_{33}\hat{\Gamma}^{i}_{i3}\right)-g^{ij}\hat{\Gamma}^{3}_{ij}\hat{\Gamma}^{k}_{k3}$
$\displaystyle\equiv$ $\displaystyle\nabla_{\mu}Y^{\mu}_{4},$ (19)
where
$\displaystyle Y^{\mu}_{1}$ $\displaystyle\equiv$
$\displaystyle\left(0,g^{ij}\hat{\Gamma}^{3}_{ji}\right),$ $\displaystyle
Y^{\mu}_{2}$ $\displaystyle\equiv$
$\displaystyle\left(g^{ij}\hat{\Gamma}^{3}_{3j},0\right),$ $\displaystyle
Y^{\mu}_{2}$ $\displaystyle\equiv$
$\displaystyle\left(g^{33}\hat{\Gamma}^{i}_{33},0\right),$ $\displaystyle
Y^{\mu}_{4}$ $\displaystyle\equiv$
$\displaystyle\left(0,g^{33}\hat{\Gamma}^{i}_{i3}\right).$ (20)
The terms $\nabla_{\mu}Y^{\mu}_{a}$ with $a=1,2,3,4$ are divergences
corresponding to the boundary terms and as a result, they vanish at infinity.
Then, $\mathcal{R}^{(4)}$ is a sum of the remaining terms as
$\displaystyle\mathcal{R}^{(4)}$ $\displaystyle=$
$\displaystyle\mathcal{R}-g^{ij}\left(\hat{\Gamma}^{k}_{3i}\hat{\Gamma}^{3}_{jk}+\hat{\Gamma}^{3}_{ki}\hat{\Gamma}^{k}_{j3}+\hat{\Gamma}^{3}_{3i}\hat{\Gamma}^{3}_{j3}+\hat{\Gamma}^{3}_{ij}\hat{\Gamma}^{k}_{k3}\right.$
(21)
$\displaystyle\left.+C^{3}_{kj}\hat{\Gamma}^{k}_{3i}+C^{3}_{3j}\hat{\Gamma}^{3}_{3i}\right)-g^{33}\left(\hat{\Gamma}^{j}_{i3}\hat{\Gamma}^{i}_{3j}+\hat{\Gamma}^{3}_{i3}\hat{\Gamma}^{i}_{33}\right.$
$\displaystyle\left.+2\hat{\Gamma}^{3}_{3i}\hat{\Gamma}^{i}_{33}+C^{3}_{i3}\hat{\Gamma}^{i}_{33}\right)+\frac{g^{33}}{2}\hat{\partial}_{3}g^{ij}\hat{\partial}_{3}g_{ij}$
$\displaystyle+\frac{g^{ij}}{2}\hat{\partial}_{i}g^{33}\hat{\partial}_{j}g_{33}+g_{{}_{A}}g^{ij}\hat{\partial}_{3}A_{i}g^{33}\hat{\partial}_{j}g_{33}$
$\displaystyle=$
$\displaystyle\mathcal{R}+\frac{1}{4R^{2}}\left(\partial_{3}g^{ij}\partial_{3}g_{ij}+g^{ij}g^{kl}\partial_{3}g_{ij}\partial_{3}g_{kl}\right)$
$\displaystyle-\frac{g^{2}_{{}_{A}}R^{2}}{4}F_{ij}F^{ij},$
where $\mathcal{R}\equiv
g^{ij}\left(\hat{\partial}_{k}\hat{\Gamma}^{k}_{ji}-\hat{\partial}_{j}\hat{\Gamma}^{k}_{ki}+\hat{\Gamma}^{k}_{ji}\hat{\Gamma}^{l}_{lk}-\hat{\Gamma}^{k}_{li}\hat{\Gamma}^{l}_{jk}\right)\equiv
g^{ij}R_{ij}$ and $F_{ij}=\partial_{i}A_{j}-\partial_{j}A_{i}$. Note that, in
the calculation of the scalar curvature $\mathcal{R}^{(4)}$, we have
considered the components $g_{ij}$, $A_{i}$, and $R$ to be the general
functions of $(x^{i},x^{3})$. However, we observe that first the terms
relating to $\hat{\partial}_{3}A_{i}$ automatically cancel together and hence
they do not contribute to $\mathcal{R}^{(4)}$. This is because the spacetime
curvature which is caused by the non-triviality of the $U(1)$ principal bundle
is measured by the curvature of the connection $A_{i}$, i.e. $F_{ij}$. Second,
the first-order derivative terms of $R$ associated with its kinetic terms do
not appear in $\mathcal{R}^{(4)}$ because the curvature of the $S^{1}$ fiber
is zero. But, the kinetic terms of $R$ would appear in Einstein frame derived
by rescaling the 3D metric as $g_{ij}\rightarrow\Omega^{-2}g_{ij}$ with
$\Omega=R/r$ where $r$ is introduced to keep the rescaled 3D metric
dimensionless and would be fixed equal to the vacuum expectation value of the
radion field.
## Appendix B: Equations of motion for the 3D components of the bulk metric
First, let us obtain equations of motion for the 3D tensor component $g_{ij}$
from the following variation
$\displaystyle\delta_{g_{3}}S_{\text{EH}}$ $\displaystyle=$
$\displaystyle\delta_{g_{3}}\int
d^{4}x\sqrt{-g}\frac{M^{2}_{\text{Pl}}}{2}\left(\mathcal{R}^{(4)}-2\Lambda_{4}\right)$
(22) $\displaystyle=$ $\displaystyle\frac{M^{2}_{\text{Pl}}}{2}\int
d^{4}x\left[\left(\mathcal{R}^{(4)}-2\Lambda_{4}\right)\delta_{g_{3}}\sqrt{-g}\right.$
$\displaystyle\left.+\sqrt{-g}\delta_{g_{3}}\mathcal{R}^{(4)}\right]=0,$
where $\delta_{g_{3}}$ refers to the variation in terms of $g_{ij}$ and the
expansion of $\mathcal{R}^{(4)}$ is given in Eq. (21). We find
$\delta_{g_{3}}\sqrt{-g}=-\frac{1}{2}\sqrt{-g}g_{ij}\delta g^{ij}$ and
$\displaystyle\delta_{g_{3}}\mathcal{R}^{(4)}$ $\displaystyle=$
$\displaystyle\delta g^{ij}R_{ij}+g^{ij}\delta_{g_{3}}R_{ij}$ (23)
$\displaystyle+\frac{1}{4}\delta_{g_{3}}\left(\partial_{3}g^{ij}\partial^{3}g_{ij}+g^{ij}g^{kl}\partial_{3}g_{ij}\partial^{3}g_{kl}\right)$
$\displaystyle-\frac{g^{2}_{{}_{A}}R^{2}}{4}\delta_{g_{3}}\left(F_{ij}F^{ij}\right).$
The second variation term in (23) is calculated as
$\displaystyle g^{ij}\delta_{g_{3}}R_{ij}$ $\displaystyle=$
$\displaystyle\nabla_{k}\left(g_{ij}\nabla^{k}\delta g^{ij}-\nabla_{l}\delta
g^{kl}\right)$ (24) $\displaystyle=$
$\displaystyle\nabla_{\mu}X^{\mu}_{1}-\hat{\Gamma}^{3}_{3k}\left(g_{ij}\nabla^{k}\delta
g^{ij}-\nabla_{l}\delta g^{kl}\right)$ $\displaystyle=$
$\displaystyle\nabla_{\mu}X^{\mu}_{1}+\nabla_{\mu}X^{\mu}_{2}+\nabla_{\mu}X^{\mu}_{3}+\hat{\Gamma}^{3}_{3k}\hat{\Gamma}^{3}_{3l}g^{kl}g_{ij}\delta
g^{ij}$
$\displaystyle+\nabla_{k}\left(g^{kl}\hat{\Gamma}^{3}_{3l}g_{ij}\right)\delta
g^{ij}-\hat{\Gamma}^{3}_{3i}\hat{\Gamma}^{3}_{3j}\delta g^{ij}$
$\displaystyle-\nabla_{i}\left(\hat{\Gamma}^{3}_{3j}\right)\delta g^{ij},$
where
$\displaystyle X^{\mu}_{1}$ $\displaystyle=$
$\displaystyle\left(g_{ij}\nabla^{k}\delta g^{ij}-\nabla_{l}\delta
g^{kl},0\right),$ $\displaystyle X^{\mu}_{2}$ $\displaystyle=$
$\displaystyle\left(-g^{kl}\hat{\Gamma}^{3}_{3l}g_{ij}\delta g^{ij},0\right),$
(25) $\displaystyle X^{\mu}_{3}$ $\displaystyle=$
$\displaystyle\left(\hat{\Gamma}^{3}_{3l}\delta g^{lk},0\right).$
The variation of the third term in (23) reads
$\displaystyle\delta_{g_{3}}\left(\partial_{3}g^{ij}\partial^{3}g_{ij}+g^{ij}g^{kl}\partial_{3}g_{ij}\partial^{3}g_{kl}\right)=\nabla_{\mu}X^{\mu}_{4}+\nabla_{\mu}X^{\mu}_{5}$
$\displaystyle+g_{ik}g_{jl}\partial_{3}\partial^{3}g^{kl}-\partial_{3}\partial^{3}g_{ij}+g^{kl}\partial_{3}g_{kl}\partial^{3}g_{ij}\delta
g^{ij}$
$\displaystyle+2\partial_{3}\left(g_{ij}g^{kl}\partial^{3}g_{kl}\right)+\left(g^{kl}\partial_{3}g_{kl}\right)^{2}g_{ij}\delta
g^{ij}$
$\displaystyle-2\hat{\Gamma}^{3}_{33}\left(\partial^{3}g_{ij}+g_{ij}g_{kl}\partial^{3}g^{kl}\right)\delta
g^{ij},$ (26)
where
$\displaystyle X^{\mu}_{4}$ $\displaystyle=$
$\displaystyle\left(0,2\partial^{3}g_{ij}\delta g^{ij}\right),$ $\displaystyle
X^{\mu}_{5}$ $\displaystyle=$
$\displaystyle\left(0,2g_{ij}g_{kl}\partial^{3}g^{kl}\delta g^{ij}\right).$
(27)
The last variation term is
$\displaystyle\delta_{g_{3}}\left(F_{ij}F^{ij}\right)=2{F_{i}}^{k}F_{jk}.$
(28)
Finally, we find the equations of motion for $g_{ij}$ as follows
$\displaystyle\mathcal{R}_{ij}-\frac{1}{2}g_{ij}\mathcal{R}+\Lambda_{4}g_{ij}+\frac{1}{4R^{2}}\left[g_{ik}g_{jl}\partial^{2}_{3}g^{kl}\right.$
$\displaystyle\left.-\partial^{2}_{3}g_{ij}+g^{kl}\partial_{3}g_{kl}\partial_{3}g_{ij}+2\partial_{3}\left(g_{ij}g^{kl}\partial_{3}g_{kl}\right)\right.$
$\displaystyle\left.-\frac{1}{2}g_{ij}\left\\{\partial_{3}g^{kl}\partial_{3}g_{kl}-\left(g^{kl}\partial_{3}g_{kl}\right)^{2}\right\\}\right]$
$\displaystyle-\frac{g^{2}_{{}_{A}}R^{2}}{2}\left({F_{i}}^{k}F_{jk}-\frac{1}{4}F_{kl}F^{kl}g_{ij}\right)+f_{1}\left(\hat{\Gamma}^{3}_{3k},\hat{\Gamma}^{3}_{33}\right)$
$\displaystyle=$ $\displaystyle 0,$
where $f_{1}\left(\hat{\Gamma}^{k}_{k3},\hat{\Gamma}^{3}_{33}\right)$ refers
to the terms which are proportional to $\hat{\Gamma}^{3}_{3k}$,
$(\hat{\Gamma}^{3}_{3k})^{2}$, $\nabla_{k}(\hat{\Gamma}^{3}_{3l})$, and
$\hat{\Gamma}^{3}_{33}$ which are zero with $\hat{\partial}_{i}R=0$,
$\hat{\partial}_{3}R=0$, and $\partial_{3}A_{i}=0$.
Equations of motion for the 3D vector component $A_{i}$ are derived from the
following variation
$\displaystyle\delta_{A}S_{\text{EH}}$ $\displaystyle=$
$\displaystyle\frac{M^{2}_{\text{Pl}}}{2}\int
d^{4}x\sqrt{-g}\left[\delta_{A}\mathcal{R}-\frac{g^{2}_{{}_{A}}R^{2}}{4}\delta_{A}\left(F_{ij}F^{ij}\right)\right]$
(30) $\displaystyle=$ $\displaystyle 0$
This variation leads to the equations of motion for $A_{i}$ as
$\nabla_{i}F^{ij}+R^{-2}f_{2}\left(A_{j},\hat{\partial}_{j}g_{kl},\hat{\partial}_{3}g_{kl}\right)=0.$
(31)
Note that, the second term in Eq. (31) is negligible because it is strongly
suppressed by $R^{-2}$ which is in the order of $\Lambda_{4}$ ($\approx
5.06\times 10^{-84}$ GeV2 Tanabashi2018 ) as indicated in Appendix D. Thus,
Eq. (31) can be approximated as $\nabla_{i}F^{ij}=0$.
Because the radion field $R$ would get an effective potential (generated from
the cosmological constant term and one-loop quantum corrections) by which the
size of the compactified dimension is physically fixed, we write the equation
of motion for the radion field $R$ including this radion potential,
corresponding to the variation of the 3D action (given in Appendix D) as
follows
$\displaystyle\square R$ $\displaystyle=$
$\displaystyle\frac{1}{R}(\partial_{i}R)^{2}+\frac{R^{2}}{2M_{3}}\frac{\partial
V}{\partial R}+f_{3}(A_{i},F_{ij}),$ (32)
where $f_{3}(A_{i},F_{ij})$ is a functional of $A_{i}$ and $F_{ij}$ which
vanishes for $A_{i}=0$.
## Appendix C: A derivation of Eqs. (5) and (6)
In order to solve Eq. (4), we use the method of the variable separation as
$g_{ij}(x^{i},x^{3})=\chi(x^{3})g^{(3)}_{ij}(x^{i})$ where
$g^{(3)}_{ij}(x^{i})$ is the metric of the 3D effective theory whose
wavefunction profile along the compactified dimension is $\chi(x^{3})$. First,
by substituting $g_{ij}(x^{i},x^{3})=\chi(x^{3})g^{(3)}_{ij}(x^{i})$ into Eq.
(4), we find
$\displaystyle\mathcal{R}^{(3)}_{ij}-\frac{1}{2}g^{(3)}_{ij}\mathcal{R}^{(3)}+\Lambda_{4}\chi
g^{(3)}_{ij}+\frac{1}{4R^{2}}\left[\left(\frac{1}{\chi}\right)^{{}^{\prime\prime}}\chi^{2}-\chi{{}^{\prime\prime}}\right.$
$\displaystyle\left.+\frac{3\chi^{\prime
2}}{\chi}+6\chi^{\prime\prime}-\frac{3\chi}{2}\left\\{\left(\frac{1}{\chi}\right)^{{}^{\prime}}\chi^{\prime}-3\left(\frac{\chi^{\prime}}{\chi}\right)^{2}\right\\}\right]g^{(3)}_{ij},$
(33)
where
$\mathcal{R}^{(3)}_{ij}\equiv(\partial_{k}\Gamma^{k}_{ji}-\partial_{j}\Gamma^{k}_{ki}+\Gamma^{l}_{ji}\Gamma^{k}_{kl}-\Gamma^{l}_{ki}\Gamma^{k}_{jl})$
with $\Gamma^{k}_{ij}\equiv
g^{(3)kl}(\partial_{i}g^{(3)}_{lj}+\partial_{j}g^{(3)}_{ki}-\partial_{l}g^{(3)}_{ij})/2$
and $\mathcal{R}^{(3)}\equiv g^{(3)ij}\mathcal{R}^{(3)}_{ij}$, and ′ denotes
the derivative with respect to the coordinate $x^{3}$. After simplification,
we obtain
$\displaystyle\mathcal{R}^{(3)}_{ij}-\frac{1}{2}g^{(3)}_{ij}\mathcal{R}^{(3)}+\left[\frac{1}{R^{2}}\left(\chi^{\prime\prime}+\frac{11}{4}\frac{\chi^{\prime
2}}{\chi}\right)+\Lambda_{4}\chi\right]g^{(3)}_{ij}=0.$
This equation can be written in the following way
$\displaystyle\frac{\mathcal{R}^{(3)}}{6}=\frac{1}{R^{2}}\left(\chi^{\prime\prime}+\frac{11}{4}\frac{\chi^{\prime
2}}{\chi}\right)+\Lambda_{4}\chi.$ (35)
It is clear that the left-handed and right-handed sides of Eq. (35) depend on
the variables $x^{i}$ and $x^{3}$, respectively. This implies that these sides
are equal to a constant denoted by $\Lambda_{3}$, which leads to
$\displaystyle\mathcal{R}^{(3)}=6\Lambda_{3},$ (36)
$\displaystyle\chi^{\prime\prime}+\frac{11}{4}\frac{\chi^{\prime
2}}{\chi}+\kappa^{2}\chi$ $\displaystyle=$
$\displaystyle\frac{\Lambda_{3}}{R^{-2}},$ (37)
$\kappa\equiv\sqrt{\Lambda_{4}}/R^{-1}$. Note that, Eq. (36) is equivalent to
Eq. (5) in the paper.
## Appendix D: The radion potential
The dimensional reduction of the 4D Einstein gravity on $S^{1}$ given in the
Einstein frame reads
$\displaystyle S_{3D}$ $\displaystyle\supset$ $\displaystyle\int
d^{3}x\sqrt{-g_{3}}\left[\frac{M_{3}}{2}\left\\{\mathcal{R}^{(3)}-2\left(\frac{\partial_{i}R}{R}\right)^{2}\right\\}\right.$
(38)
$\displaystyle\left.-M_{3}\left(\frac{r}{R}\right)^{2}\Lambda_{3}\right],$
where $g_{3}\equiv\text{det}\left[g^{(3)}_{ij}\right]$ and the 3D Planck
energy scale is identified as
$\displaystyle M_{3}$ $\displaystyle\equiv$ $\displaystyle
rM^{2}_{\text{Pl}}\int^{\pi}_{-\pi}dx^{3}\chi^{1/2}$ (39) $\displaystyle=$
$\displaystyle\frac{2\sqrt{\pi}\Gamma(17/30)}{\Gamma(16/15)}rM^{2}_{\text{Pl}}.$
The last term in the action (38) is the tree-level potential of the radion
field generated by the dynamics of $3$D tensor component along the
compactified dimension. In addition, the one-loop quantum corrections would
contribute to the radion potential as
$\displaystyle V_{\text{1L}}(R)$ $\displaystyle=$
$\displaystyle\sum_{i}(-1)^{s_{i}}n_{i}R\left(\frac{r}{R}\right)^{3}\rho_{i}(R)\int^{\pi}_{-\pi}dx^{3}\chi^{\frac{3}{2}}.\
\ $ (40)
Here $s_{i}$ is equal to $0(1)$ for the fermions(bosons), $n_{i}$ is the
number of degrees of freedom corresponding to the $i$th particle, and the
Casimir energy density with respect to the $i$th particle is given by Arkani-
Hamed2007
$\displaystyle\rho_{i}(R)=\sum^{\infty}_{n=1}\frac{2m^{4}_{i}}{(2\pi)^{2}}\frac{K_{2}(2\pi
nm_{i}R)}{(2\pi nm_{i}R)^{2}},$ (41)
where $m_{i}$ and $K_{2}(z)$ are the mass of the $i$th particle and the
modified Bessel function, respectively. It should be noted here that due to
the function $K_{2}(z)$ suppressed for $z\ll 1$ the particles with their mass
which is much larger than $R^{-1}$ do not contribute significantly to the one-
loop term of the radion potential and hence we can ignore their contribution.
On the other hand, only the light degrees of freedom contribute significantly
to $V_{\text{1L}}(R)$.
The radion potential $V(R)$ thus is a sum of the tree and loop level
contributions, which is expanded in terms of $m_{i}R$ for $m_{i}R\ll 1$ as
$\displaystyle\frac{V(R)}{2\sqrt{\pi}r^{3}}$ $\displaystyle\simeq$
$\displaystyle\left[\frac{\Gamma(17/30)}{\Gamma(16/15)}\frac{M^{2}_{P}\Lambda_{3}}{R^{2}}+\frac{1}{16\pi^{2}}\frac{\Gamma(7/10)}{\Gamma(6/5)}\frac{1}{R^{6}}\right.$
$\displaystyle\left.\times\sum_{i}(-1)^{s_{i}}n_{i}\left\\{\frac{1}{90}-\frac{(m_{i}R)^{2}}{6}+\frac{(m_{i}R)^{4}}{48}\right\\}\right].$
The behavior of the radion potential is depicted in Fig. 1. Here, the bosonic
contributions come from graviton and photon, whereas the fermionic
contributions come from the lightest neutrino and an additional massless Dirac
fermion.
---
Figure 1: The scaled radion potential $\bar{V}(R)\equiv
V(R)/(2\sqrt{\pi}r^{3}\Lambda^{3}_{4})$ versus the scaled $S^{1}$ radius
$R\sqrt{\Lambda_{4}}$ for various values of $m_{\nu}/\sqrt{\Lambda_{4}}$ with
$M^{2}_{\text{Pl}}\Lambda_{3}/\Lambda^{2}_{4}=0.1$. The red and orange curves
correspond to the AdS vacuum and no vacuum, respectively, whereas the blue and
purple curves lead to the dS vacuum.
## References
* (1) N. Aghanim _et al._ (Planck Collaboration), Astron. Astrophys 641, A6 (2020).
* (2) M. Tanabashi _et al._ (Particle Data Group), Phys. Rev. D 98, 030001 (2018).
* (3) Y. B. Zeldovich, Sov. Phys. Usp. 11, 381 (1968) [JETP Lett. 6, 316 (1967)].
* (4) N. D. Birrell and P. C. W. Davies, _Quantum Fields in Curved Space_ (Cambridge University Press, Cambridge, England, 1994).
* (5) S. Weinberg, Rev. Mod. Phys. 61, 1 (1989).
* (6) R. Bousso, Gen. Relativ. Gravit. 40, 607 (2008).
* (7) S. Kachru and E. Silverstein, Phys. Rev. Lett. 80, 4855 (1998).
* (8) S. Kachru, J. Kumar, and E. Silverstein, Phys. Rev. D 59, 106004 (1999).
* (9) J. A. Harvey, Phys. Rev. D 59, 026002 (1998).
* (10) R. Iengo and C. Zhu, JHEP 04 (2000) 028.
* (11) R. Blumenhagen and L. Gorlich, Nucl. Phys. B 551, 601 (1999).
* (12) C. Angelantonj, I. Antoniadis, and K. Forger, Nucl. Phys. B 555, 116 (1999).
* (13) C. Angelantonj and I. Antoniadis, Nucl. Phys. B 676, 129 (2004).
* (14) S. Nobbenhuis, arXiv:gr-qc/0609011.
* (15) C. P. Burgess, _Post-Planck Cosmology: Lecture Notes of the Les Houches Summer School_ (Oxford Academic, New York, 2015).
* (16) C. Vafa, arXiv: hep-th/0509212.
* (17) H. Ooguri and C. Vafa, Nucl. Phys. B 766, 21 (2007).
* (18) E. Palti, Fortschr. Phys. 67, 1900037 (2019).
* (19) G. Obied, H. Ooguri, L. Spodyneiko, and C. Vafa, arXiv:1806.08362.
* (20) S. K. Garg and C. Krishnan, JHEP 11 (2019) 075.
* (21) A. Bedroya, R. Brandenberger, M. Loverde, and C. Vafa, Phys. Rev. D 101, 103502 (2020).
* (22) R. Brandenberger, LHEP 2021 (2021) 198.
* (23) N. Arkani-Hamed, S. Dubovsky, A. Nicolis, and G. Villadoro, JHEP 0706 (2007) 078.
* (24) J. M. Arnold, B. Fornal, and M. B. Wise, JHEP 12 (2010) 083.
* (25) B. Fornal and M. B. Wise, JHEP 07 (2011) 086.
* (26) L. E. Ibáñez, V. Martin-Lozano, and I. Valenzuela, JHEP 11 (2017) 066.
* (27) L. E. Ibáñez, V. Martin-Lozano, and I. Valenzuela, arXiv:1707.05811.
* (28) Y. Hamada and G. Shiu, JHEP 11 (2017) 043.
* (29) E. Gonzalo, A. Herráez, and L. E. Ibáñez, JHEP 06 (2018) 051.
* (30) E. Gonzalo and L. E. Ibáñez, Phys. Lett. B 786, 272 (2018).
* (31) H. Ooguri and C. Vafa, Adv. Theor. Math. Phys. 21 (2017) 1787.
* (32) N. Arkani-Hamed, L. Motl, A. Nicolis, and C. Vafa, JHEP 06 060 (2007)
* (33) D. Lüst, E. Palti, and C. Vafa, Phys. Lett. B 797, 134867 (2019).
* (34) T. Rudelius, JHEP 08 (2021) 041.
* (35) E. Gonzalo, L. E. Ibáñez, and I. Valenzuela, Phys. Lett. B 822, 136691 (2021).
* (36) C. H. Nam, Phys. Rev. D 107, L041901 (2023).
* (37) C. H. Nam, _A prediction for the mass of the lightest neutrino from the Swampland and quantized compactification_ , in preparation.
|
# Constraints on Spin-Spin-Velocity-Dependent Interaction
Wei Ji Helmholtz-Institut, GSI Helmholtzzentrum für Schwerionenforschung,
Mainz 55128, Germany State Key Laboratory of Low-Dimensional Quantum Physics,
Department of Physics, Tsinghua University, Beijing 100084, China Johannes
Gutenberg-Universität Mainz, Mainz 55128, Germany Weipeng Li State Key
Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua
University, Beijing 100084, China Pavel Fadeev Helmholtz-Institut, GSI
Helmholtzzentrum für Schwerionenforschung, Mainz 55128, Germany Johannes
Gutenberg-Universität Mainz, Mainz 55128, Germany Filip Ficek Institute of
Theoretical Physics, Jagiellonian University, Łojasiewicza 11, 30-348 Kraków,
Poland Jianan Qin Helmholtz-Institut, GSI Helmholtzzentrum für
Schwerionenforschung, Mainz 55128, Germany Key Laboratory of Geophysical
Exploration Equipment, Ministry of Education of China, Jilin University,
Changchun 130012, China Kai Wei<EMAIL_ADDRESS>School of Instrumentation
Science and Opto-electronics Engineering, Beihang University, Beijing, 100191,
China Hangzhou Innovation Institute, Beihang University, Hangzhou, 310051,
China Hangzhou Extremely Weak Magnetic Field Major Science and Technology
Infrastructure Research Institute, Hangzhou, 310051, China Yong-Chun Liu
<EMAIL_ADDRESS>State Key Laboratory of Low-Dimensional Quantum Physics,
Department of Physics, Tsinghua University, Beijing 100084, China Frontier
Science Center for Quantum Information, Beijing, China Dmitry Budker
Helmholtz-Institut, GSI Helmholtzzentrum für Schwerionenforschung, Mainz
55128, Germany Johannes Gutenberg-Universität Mainz, Mainz 55128, Germany
Department of Physics, University of California, Berkeley, CA 94720-7300, USA
###### Abstract
The existence of exotic spin-dependent forces may shine light on new physics
beyond the Standard Model. We utilize two iron shielded SmCo5 electron-spin
sources and two optically pumped magnetometers to search for exotic long-range
spin-spin-velocity-dependent force. The orientations of spin sources and
magnetometers are optimized such that the exotic force is enhanced and common-
mode noise is effectively subtracted. We set direct limit on proton-electron
interaction in the force range from 1 cm to 1 km. Our experiment represents
more than ten orders of magnitude improvement than previous works.
The nature of dark matter is one of the most profound mysteries in modern
physics. Many new light bosons introduced by theories beyond the Standard
Model are proposed to be dark matter candidates, such as spin-0 bosons
including axions and axion like particles (ALPs) axion-PhysRevLett.38.1440 ;
preskill1983cosmology ; graham2015experimental , spin-1 bosons including dark
photons darkPhoton.2010 ; Dark-Photon-AN2015331 , and Z′ bosons
(okada2020dark, ; croon2021supernova, ). Furthermore, the new bosons may
mediate new types of long-range fundamental forces (moody1984new, ;
dobrescu2006spin, ; Fadeev2019, ; fayet1986fifth, ).
If we consider the spin, relative position and velocity of two fermions, the
exotic interaction between them can be classified to 16 terms dobrescu2006spin
; Fadeev2019 , and then generally classified into static terms and velocity-
dependent terms. A conventional velocity-dependent force in classical physics
is the Lorentz force of a moving charged particle.
Many experimental methods have been used to search for exotic forces,
including experiments with torsional resonators heckel2008preferred ;
hammond2007new ; Loong-Nature2003 ; terrano2015short , nuclear magnetic
resonance petukhov2010polarized ; yan2015searching ; chu2013laboratory ;
arvanitaki2014resonantly , magnetometers based on hot atoms and nitrogen-
vacancy center in diamond Romalis-PRL2009 ; Ji2018 ; jiao2021experimental ;
hunter2014using ; su2021search ; almasi2020new ; kim2018experimental , and
other high-sensitivity technologies tullney2013constraints ;
serebrov2010search ; ficek2017constraints ; ren2021search ;
stadnik2018improved . Most of these efforts focus on static interactions,
while the velocity-dependent interactions have also been gaining attention in
recent years jiao2021experimental ; su2021search ; hunter2014using ;
wei2022new ; kim2018experimental .
In this experiment, we focus on one term of Spin-Spin-Velocity-Dependent
Interaction (SSVDI) proposed by dobrescu2006spin :
$\displaystyle V=$ $\displaystyle\frac{f\hbar}{4\pi
c}\left[(\hat{\bm{\sigma}}_{1}\cdot\bm{v})(\hat{\bm{\sigma}}_{2}\cdot\bm{v})\right]\
\frac{e^{-r/\lambda}}{r},$ (1)
where $f$ is a dimensionless coupling coefficient, $\hat{\bm{\sigma}}_{1}$,
$\hat{\bm{\sigma}}_{2}$ are the respective Pauli spin-matrix vectors of the
two fermions, $r$ and $\mathbf{v}$ are the relative position and velocity
between two fermions. (This potential is called $V_{8}$ in Ref.
dobrescu2006spin .) To search for this force, a spin polarized test object is
required as the spin source, and an ultra-sensitive magnetometer is required
as the sensor.
In this experiment, the spin sources are two iron shielded SmCo5 magnets
(ISSCs) that have high net electron spin and small magnetic leakage
ji2017searching . The sensor is a pair of optically pumped magnetometers (OPM)
that operate in the spin-exchange-relaxation-free (SERF) mode allred2002high ;
shah2007subpicotesla , which use spin polarized Rb as the sensing atoms. By
designing the setup, the experiment is sensitive to the exotic force, while
common-mode noise is reduced. Our experiment sets new limits on exotic SSVDI
for electron-proton coupling.
Figure 1 shows a schematic of the experimental setup. Each of the two spin
sources ISSC1,2 contains a 40.00 mm diameter cylindrical SmCo5 magnet enclosed
in three layers of pure iron. The magnetization of the SmCo5 magnets is about
1 T. The magnetic field of the magnet is shielded by the iron layers, and the
magnetic leakage outside the iron layers is smaller than 10 $\mu$T. However,
the net spin of the ISSCs is not canceled, which is mostly due to the fact
that the orbital magnetic moment and spin magnetic moments of the 4$f$ rare
earth metal (Sm) and 3$d$ metals (Co and Fe) are differently oriented, and
thus the total magnetic moments are canceled but the net orbital magnetic
moment and spin magnetic moment are not ji2017searching ; heckel2008preferred
. The net electron spin for each ISSC is 1.75 $(21)\times 10^{24}$
ji2017searching . The ISSCs are connected with titanium-alloy supports and are
driven with a motor to rotate clockwise (CW) and counterclockwise (CCW). The
motor frequency is controlled with a direct current (DC) power supply.
Figure 1: The experimental setup (not to scale). Two QuSpin OPMs noted as
OPM1 and OPM2 are enclosed in a five-layer magnetic shield. Their sensitive
axis orientations are antiparallel along the $\hat{x}$-axis. Two spin sources
noted as ISSC1,2 are put in the other, four-layer shield. The spin source is
driven with a motor to rotate clockwise or counterclockwise. The blue arrows
show the direction of net spin in OPMs and ISSCs.
The OPMs are QuSpin Vector Zero-Field Magnetometers (QZFM Gen-2) Quspin that
work in the SERF regime. They are placed in the center of a five-layer
$\mu$-metal magnetic shield. The arrows along the $\hat{z}$-axis demonstrate
the direction of the circularly polarized laser beam that passes through the
87Rb vapor cell. A narrow linewidth Rb Hannle resonance in near-zero field is
used to detect the magnetic field dupont1969detection . Because the
orientation of two OPMs along $\hat{x}$ are anti-parallel, their responses to
the magnetic field have opposite signs. If there is a magnetic field $B_{0}$
applied, the responses of the OPMs are $S_{1}=B_{0}+N_{C}+N_{1}$ and
$S_{2}=-B_{0}+N_{C}+N_{2}$, respectively, where $N_{C}$ is the common-mode
noise and $N_{1}$ and $N_{2}$ are other noises. Subtracting the readings of
the two sensors can diminish the common noise and yields a signal of
$S_{\mathrm{sub}}=(S_{1}-S_{2})/2=B_{0}+(N_{1}-N_{2})/2$.
To test the validity of the subtraction procedure, an 8 Hz and 1.5 pT uniform
magnetic field is applied along $\hat{x}$ with a set of Helmholtz coils. The
spectrum of the OPM signals and the subtraction result are shown in Fig. 2
(a). By taking the difference, the uniform magnetic field is unaffected, the
common-mode (for example, electrical or gradient) noise is reduced by as high
as a factor of five, and the 8 Hz target signal is successfully extracted. The
noise level around 8 Hz is about 13 fT/$\sqrt{\textrm{Hz}}$.
Figure 2: Up: A typical spectrum of two OPMs and the subtraction result. A
uniform AC magnetic field of 8 Hz is applied along the $\hat{x}$-axis. The
dashed-blue line and the red-dot-dashed line are the spectrum of the OPMs on
the left side and right side, respectively. The yellow-solid line is their
difference. Down: The OPMs’ response to the pseudomagnetic field along the
$\hat{x}$-axis. The blue-dot and red-dot line are the pseudomagnetic field
sensed by OPM1 and OPM2 respectively, the black-dash line is the subtraction
result. The subtraction result agrees well with the result from OPM1.
The SSVDIs will manifest as a pseudomagnetic fields that could be sensed by
the Rb atoms like the Zeeman effect. The potential can be expressed as
$V^{n}\zeta^{n}+V^{p}\zeta^{p}+V^{e}\zeta^{e}=-\bm{\mu}\cdot\textbf{B}$, where
$\bm{\mu}$ is the magnetic moment of the Rb atom, B is the pseudomagnetic
field from the exotic interaction, $\zeta^{n,p,e}$ are the neutron, proton and
electron’s fraction of spin polarization in 87Rb atoms, which could be
obtained by the the Russel-Saunders LS-coupling and the Schmidt model of
nuclear physics.
In this experiment, we search for the coupling between the proton, neutron and
electron spins in the Rb atoms and the electron spins in ISSCs. The
pseudomagnetic field sensed by the OPM can be obtained by integrating the
exotic interaction from the electron spins over the ISSCs:
$\displaystyle\begin{split}\mathbf{B}^{p,e,n}=&\frac{f\zeta^{p,e,n}\hbar}{4\pi\mu
c}\iiint\rho(\mathbf{r})(\hat{\bm{\sigma}}_{2}\cdot\mathbf{v})\frac{\mathbf{v}}{r}e^{-r/\lambda}\rm{d}\mathbf{r},\end{split}$
(2)
where $\textbf{B}^{p,n,e}$ are the fractions of B that couple to proton,
neutron and electron respectively,
$\mathbf{v}(\mathbf{r})=\mathbf{\omega}\times\mathbf{r}$ and
$\rho(\mathbf{r})$ are the velocity and spin density at location $\mathbf{r}$,
$\mathbf{\omega}$ is the angular velocity of the ISSCs. The proton and
electron fractions of polarization in 87Rb are $\zeta^{p}=0.29$ and
$\zeta^{e}=0.13$ respectively, and neutron polarization $\zeta^{n}$ is assumed
to be zero under the basic nuclear shell model. The calculation of the
fraction of spin polarization is explained in the supplemental document.
The experimental parameters and a benchmark coupling coefficients $f^{0}=1$
are put in the simulation to obtain $\textbf{B}^{p}$. The benchmark parameter
$f^{0}$ is set to $1$ for convenience; a different $f^{0}$ does not affect the
final result. The orientation of the OPMs and ISSC sources are optimized by
simulating different configurations, such that the OPMs can sense the maximum
pseudomagnetic field. The best configuration is shown in Figure 1 and Table 1.
The distance between the spin sources and the OPMs is much larger than the
distance between two OPMs, such that two OPMs experience almost the same
pseudomagnetic field. Thus the signal subtraction procedure works well for
this pseudomagnetic field. The simulated responses of the two OPMs and their
subtraction result are shown in Fig. 2 (b).
Table 1: Experimental parameters and the error budget of $f^{\rm{ep}}$. The origin of coordinates is at the midpoint between the centers of the two OPMs. The contributions to the error budget are evaluated for $\lambda=20$ m. The final systematic error is derived from the uncertainties of the parameters listed. Parameter | Value | $\Delta f^{\rm{ep}}(\times 10^{-{22}})$
---|---|---
ISSC net spin ($\times 10^{24}$) | $1.75(21)$ | 0.084
Position of ISSCs x(m) | $0.000(2)$ | $0.001$
Position of ISSCs y(m) | $-0.477(2)$ | $0.001$
Position of ISSCs z(m) | $0.000(2)$ | $0.001$
Distance between ISSCs(m) | $0.251(1)$ | $0.044$
Distance between OPM cells(m) | $0.017(1)$ | $0.004$
Rotation frequency CW (Hz) | $4.11(1)$ |
Rotation frequency CCW (Hz) | $4.09(1)$ |
phase uncertainty ($\deg$) | $\pm 2.8$ | $\pm 1.190$
Final $f^{\rm{exp}}(\times 10^{-22}$) | $-0.7$ | $\pm 10.1\ (stat.)$
$(\lambda=20\,m)$ | | $\pm 1.2(syst.)$
The ISSC spin sources are driven with a DC motor. The positions of the spin
sources are monitored with a photoelectronic encoder placed on the rotation
axle. The signals of the encoder and the OPMs are taken simultaneously and
recorded with a data-acquisition (DAQ) device. The motor is tuned to rotate
CCW and CW alternatively for every two hours. The DC motor works in a good
stability with frequency of $4.09(1)$ Hz and $4.11(1)$ Hz for CW and CCW
rotations.
The two OPMs signals are subtracted and then transformed to frequency domain
by fast Fourier transformations (FFT). The 50 Hz power line interference and
its 100 Hz and 200 Hz harmonics are removed in the frequency domain. The data
were then transformed back to the time domain with inverse FFT.
The signals are then cut to one-period-long segments based on the encoder
signal of the spin source rotation. The DC components in each period are
removed. The data are noted as $\mathbf{S}^{\rm{exp}}_{i}(t_{j})$, where $i$
represents the $i$-th period, and $t_{j}$ is the time of the $j$-th point in
this period.
The coupling coefficient $f^{\rm{ep}}$ can be obtained by a similarity
comparison method between the experimental data and simulation results:
$f^{\rm{ep}}_{i}=k_{i}\,\sqrt{\frac{\sum_{j}\left[\mathbf{S}^{\rm{exp}}_{i}(t_{j})\right]^{2}}{\sum_{j}\left[\mathbf{S}^{\rm{sim}}(t_{j})\right]^{2}}}\,,$
(3)
where $k_{i}$ is the similarity score to weigh the similarity between
$\mathbf{S}^{\rm{exp}}_{i}$ and $\mathbf{S}^{\rm{sim}}(t)$ krasichkov2015shape
, which is defined as
$k_{i}\equiv\frac{\sum_{j}\mathbf{S}^{\rm{sim}}(t_{j})\cdot{\mathbf{S}}_{i}^{\rm{exp}}(t_{j})}{\sqrt{\sum_{j}\left[\mathbf{S}^{\rm{sim}}(t_{j})\right]^{2}}\sqrt{\sum_{j}\left[{\mathbf{S}_{i}}^{\rm{exp}}(t_{j})\right]^{2}}}.$
(4)
The expectation values and standard error for the CW and CCW rotation are
$\langle f^{\rm{ep}}\rangle^{+}$, $\langle f^{\rm{ep}}\rangle^{-}$, and
$\sigma_{f}^{+}$, $\sigma_{f}^{-}$ respectivly, the final coupling coefficient
can be obtained by
$\langle f^{\rm{ep}}\rangle=\frac{\langle
f^{\rm{ep}}\rangle^{+}/{\sigma^{+}}^{2}+\langle
f^{\rm{ep}}\rangle^{-}/{\sigma^{-}}^{2}}{1/{\sigma^{+}}^{2}+1/{\sigma^{-}}^{2}}.$
(5)
Some systematic bias could be removed by averaging over CW an CCW. The
distributions of the $f^{\rm{ep}+}$ and $f^{\rm{ep}-}$ are shown in Fig. 3.
Figure 3: Statistical results of the $f^{\rm{ep}}$. Each data point represents
an average of about one 2.7-hour long data set. The distribution of
$f^{\rm{ep}}$ for one data set is shown in the insert. The result is well
fitted with a Gaussian distribution (red line) with $\bar{\chi}^{2}=1.18$.
Figure 4: Limits on the SSVDI coupling coefficients between electron and
proton. The black solid line is our constraints. The “Hunter2014 e-p” is from
Ref. hunter2014using that uses geo-electrons and atomic magnetometer; the
“Fadeev 2022 e-p” is from Refs. Fadeev2022Pseudovector that compare the
experimental and theoretical results of hydrogen spectroscopy; the “Chu 2020
e-p” chu2020search propose to use 3He as sensor and dysprosium iron garnet as
spin source, the line is based on their sensitivity at $3\times 10^{-17}$ T
and is rescaled using the fraction of spin polarisation
$\zeta_{p}^{{}^{3}\text{He}}=-0.027$ vasilakis2009limits .
The parameters of the experiment and their corresponding uncertainties on
$\Delta f^{\rm{exp}}$ for range $\lambda=20$ m are shown in Table 1. The
$f^{\rm{ep}}$ is determined to be $f^{\rm{ep}}=0.7\pm 10.1_{stat.}\pm
1.2_{syst.}(\times 10^{-22})$. No evidence of the SSVDI is observed. New
constraints on the $f$ between electron-proton is set to be $|f^{\rm{ep}}|\leq
2.0\times 10^{-21}$ by the 95% confidence level. The values for other
$\lambda$s are obtained with the same procedure, and the final limits are
shown in Fig. 4.
If the mediator of the SSVDI is a spin-1 boson such as Z′, which is a dark
matter candidate and may resolve other discrepancies such as that in the
anomalous magnetic moment of the muon arushi2021solving ; okada2020dark , the
coupling coefficient can be rewritten as $f^{\rm{ep}}=-g_{A}^{e}g_{A}^{p}/2$
dobrescu2006spin ; Fadeev2019 . For $\lambda=20$ m, $|g_{A}^{e}g_{A}^{p}|\leq
4.0\times 10^{-21}$, where to set the limit on one of these coupling-constant
products, we assume that the other one is zero. Note that the velocity-
independent term provides significantly tighter limit on $g_{A}g_{A}$
coefficients Fadeev2019 ; dobrescu2006spin , however, the SSVDI provides a
unique way to explore the velocity-dependent interactions.
A comparison between our results (black and dashed-red lines) and the
literature is shown in Fig. 4. With the same hydrogen-spectrum analysis used
in Ref. Fadeev2022Pseudovector we obtained a bound on the SSVDF of
$f^{ep}<2.0\times 10^{-11}$ for the range larger than 1 cm (the green line
“Fadeev 2022 e-p” in Fig. 4). Results on the couplings between other fermions,
such as the coupling between electron-electron ficek2017constraints ;
hunter2014using ; Ji2018 neutron-proton kimball2010constraints and electron-
antiproton ficek2018constraints are not plotted on Fig. 4.
The major advance of our experiment is that the ISSC spin sources have much
larger numbers of spins compared to those in precision-spectroscopy
experiments yielding data for the analyses in ficek2017constraints ;
ficek2018constraints and spin-exchange approaches kimball2010constraints ,
which are most sensitive to forces with ranges on the atomic to microscopic
scale. The other advantage is that the OPMs typically have energy resolution
on the order of $10^{-18}$ eV kimball2015nuclear , significantly better than
for the spectroscopy used in Refs.ficek2017constraints ; ficek2018constraints
. On the other hand, spectroscopy experiments have an advantage over
macroscopic once in the short range, because of the exponential decay of the
exotic force. Our search covers the range of parameters inaccessible for the
geoelectron experiment hunter2014using . Using the same method and data, we
also set limits for the electron-proton coupling on the V6+7, V15 and V16
terms of SSVDF dobrescu2006spin ; Fadeev2019 . The results are shown in the
supplemental document.
A major concern in this experiment was magnetic leakage from the ISSCs. With
the iron shielding, at a distance of 10 cm away from the ISSC’s surface, its
residual magnetic field was measured to be less than 10 $\mu$T. The shielding
factors for the magnetic shielding of the ISSCs and OPMs were measured to be
both greater than $10^{6}$. Considering all the decay and shielding factors,
we conservatively expect the magnetic leakage from the ISSCs to the position
of the OPMs to be smaller than 0.1 aT, which was insignificant with regards to
the error budget.
The stability of the OPM is monitored throughout the experiment. The DC drift
of the OPM is less than 2 pT within two hours. A servo motor has a better
frequency precision, however, commercial servo motor’s control systems have
electromagnetic coupling with the magnetometer Ji2018 . A DC motor is chosen
to diminish this coupling. The experiment can further be improved if a larger
size ISSC could be used. The vapor cell can also be replaced with a
magnetometer that uses a levitated ferromagnetic sphere and has orders of
magnitude better potential magnetic sensitivity vinante2021surpassing .
In summary, we utilized a pair of OPMs that can reduce the common noise and
have ultrahigh sensitivity to search for exotic spin-dependent physics.
Together with the high electron spin density iron-shielded SmCo5 spin source,
the new experiment sets new limits on SSVDI, with more than 10 orders of
magnitude improvement for the electron-proton coupling.
###### Acknowledgements.
We thank Dr. Derek F. Jackson Kimball and Dr. Changbo Fu for valuable
discussions. This work is supported by Key-Area Research and Development
Program of Guangdong Province (Grant No. 2019B030330001), the National Natural
Science Foundation of China (NSFC) (Grant Nos. 12275145, 92050110, 91736106,
11674390, and 91836302), and the National Key R&D Program of China (Grants No.
2018YFA0306504), the DFG Project ID 390831469: EXC 2118 (PRISMA+ Cluster of
Excellence), by the German Federal Ministry of Education and Research (BMBF)
within the Quantumtechnologien program (Grant No. 13N15064). and by the
QuantERA project LEMAQUME (DFG Project Number 500314265).
## References
* (1) R.D.Peccei & Quinn, H. Cp conservation in the presence of pseudoparticles. _Phys. Rev. Lett._ 38, 1440 (1977).
* (2) Preskill, J., Wise, M. B. & Wilczek, F. Cosmology of the invisible axion. _Phys. Lett. B_ 120, 127–132 (1983).
* (3) Graham, P. W., Irastorza, I. G., Lamoreaux, S. K., Lindner, A. & van Bibber, K. A. Experimental searches for the axion and axion like particles. _Annual Review of Nuclear and Particle Science_ 65, 485–514 (2015).
* (4) Jaeckel, J. & Ringwald, A. _Phys. Rev. Lett._ 40, 223 (1978).
* (5) An, H., Pospelov, M., Pradler, J. & Ritz, A. Direct detection constraints on dark photon dark matter. _Phys. Lett. B_ 747, 331 – 338 (2015).
* (6) Okada, N., Okada, S., Raut, D. & Shafi, Q. Dark matter z’ and xenon1t excess from u (1) x extended standard model. _Phys. Lett. B_ 810, 135785 (2020).
* (7) Croon, D., Elor, G., Leane, R. K. & McDermott, S. D. Supernova muons: new constraints on z’ bosons, axions and alps. _J. High Energy Phys._ 2021, 1–28 (2021).
* (8) Moody, J. & Wilczek, F. New macroscopic forces? _Phys. Rev. D_ 30, 130 (1984).
* (9) Dobrescu, B. A. & Mocioiu, I. Spin-dependent macroscopic forces from new particle exchange. _J. High Energy Phys._ 2006, 005 (2006).
* (10) Fadeev, P. _et al._ Revisiting spin-dependent forces mediated by new bosons : Potentials in the coordinate-space representation for macroscopic- and atomic-scale experiments. _Phys. Rev. A_ 022113, 1–7 (2019).
* (11) Fayet, P. The fifth interaction in grand-unified theories: a new force acting mostly on neutrons and particle spins. _Phys. Lett. B_ 172, 363–368 (1986).
* (12) Heckel, B. R. _et al._ Preferred-frame and c p-violation tests with polarized electrons. _Phys. Rev. D_ 78, 092006 (2008).
* (13) Hammond, G. D., Speake, C. C., Trenkel, C. & Patón, A. P. New constraints on short-range forces coupling mass to intrinsic spin. _Phys. Rev. Lett._ 98, 081101 (2007).
* (14) Long, J. _et al._ Upper limits to submillimetre-range forces from extra space-time dimensions. _NATURE_ 421, 922–925 (2003).
* (15) Terrano, W., Adelberger, E., Lee, J. & Heckel, B. Short-range, spin-dependent interactions of electrons: A probe for exotic pseudo-goldstone bosons. _Phys. Rev. Lett._ 115, 201801 (2015).
* (16) Petukhov, A., Pignol, G., Jullien, D. & Andersen, K. Polarized he 3 as a probe for short-range spin-dependent interactions. _Phys. Rev. Lett._ 105, 170401 (2010).
* (17) Yan, H. _et al._ Searching for new spin-and velocity-dependent interactions by spin relaxation of polarized he 3 gas. _Phys. Rev. Lett._ 115, 182001 (2015).
* (18) Chu, P.-H. _et al._ Laboratory search for spin-dependent short-range force from axionlike particles using optically polarized he 3 gas. _Phys. Rev. D_ 87, 011105 (2013).
* (19) Arvanitaki, A. & Geraci, A. A. Resonantly detecting axion-mediated forces with nuclear magnetic resonance. _Phys. Rev. Lett._ 113, 161801 (2014).
* (20) Vasilakis, G., Brown, J. M., Kornack, T. W. & Romalis, M. V. Limits on new long range nuclear spin-dependent forces set with a $\mathbf{K}\mathrm{\text{-}}^{3}\mathrm{He}$ comagnetometer. _Phys. Rev. Lett._ 103, 261801 (2009).
* (21) Ji, W. _et al._ New Experimental Limits on Exotic Spin-Spin-Velocity-Dependent Interactions by Using SmCo 5 Spin Sources. _Phys. Rev. Lett._ 121, 261803 (2018).
* (22) Jiao, M., Guo, M., Rong, X., Cai, Y.-F. & Du, J. Experimental constraint on an exotic parity-odd spin-and velocity-dependent interaction with a single electron spin quantum sensor. _Phys. Rev. Lett._ 127, 010501 (2021).
* (23) Hunter, L. & Ang, D. Using geoelectrons to search for velocity-dependent spin-spin interactions. _Phys. Rev. Lett._ 112, 091803 (2014).
* (24) Su, H. _et al._ Search for exotic spin-dependent interactions with a spin-based amplifier. _Science advances_ 7, eabi9535 (2021).
* (25) Almasi, A., Lee, J., Winarto, H., Smiciklas, M. & Romalis, M. V. New limits on anomalous spin-spin interactions. _Phys. Rev. Lett._ 125, 201802 (2020).
* (26) Kim, Y. J., Chu, P.-H. & Savukov, I. Experimental constraint on an exotic spin-and velocity-dependent interaction in the sub-mev range of axion mass with a spin-exchange relaxation-free magnetometer. _Phys. Rev. Lett._ 121, 091802 (2018).
* (27) Tullney, K. _et al._ Constraints on spin-dependent short-range interaction between nucleons. _Phys. Rev. Lett._ 111, 100801 (2013).
* (28) Serebrov, A. P. _et al._ Search for macroscopic cp violating forces using a neutron edm spectrometer. _JETP letters_ 91, 6–10 (2010).
* (29) Ficek, F. _et al._ Constraints on exotic spin-dependent interactions between electrons from helium fine-structure spectroscopy. _Phys. Rev. A_ 95, 032505 (2017).
* (30) Ren, X. _et al._ Search for an exotic parity-odd spin-and velocity-dependent interaction using a magnetic force microscope. _Phys. Rev. D_ 104, 032008 (2021).
* (31) Stadnik, Y., Dzuba, V. & Flambaum, V. Improved limits on axionlike-particle-mediated p, t-violating interactions between electrons and nucleons from electric dipole moments of atoms and molecules. _Phys. Rev. Lett._ 120, 013202 (2018).
* (32) Wei, K. _et al._ New constraints on exotic spin-velocity-dependent interactions. _arXiv preprint arXiv:2203.07050_ (2022).
* (33) Ji, W., Fu, C. & Gao, H. Searching for new spin-dependent interactions with smco 5 spin sources and a spin-exchange-relaxation-free comagnetometer. _Phys. Rev. D_ 95, 075014 (2017).
* (34) Allred, J., Lyman, R., Kornack, T. & Romalis, M. V. High-sensitivity atomic magnetometer unaffected by spin-exchange relaxation. _Phys. Rev. Lett._ 89, 130801 (2002).
* (35) Shah, V., Knappe, S., Schwindt, P. D. & Kitching, J. Subpicotesla atomic magnetometry with a microfabricated vapour cell. _Nature Photonics_ 1, 649–652 (2007).
* (36) GZFM Gen-2, QuSpin. http://quspin.com/products-qzfm. Accessed: 2021-03-30.
* (37) Dupont-Roc, J., Haroche, S. & Cohen-Tannoudji, C. Detection of very weak magnetic fields (10- 9gauss) by 87rb zero-field level crossing resonances. _Phys. Lett. A_ 28, 638–639 (1969).
* (38) Krasichkov, A. S., Grigoriev, E. B., Bogachev, M. I. & Nifontov, E. M. Shape anomaly detection under strong measurement noise: An analytical approach to adaptive thresholding. _Phys. Rev. E_ 92, 042927 (2015).
* (39) Fadeev, P., Ficek, F., Kozlov, M. G., Budker, D. & Flambaum, V. V. Pseudovector and pseudoscalar spin-dependent interactions in atoms. _Phys. Rev. A_ 105, 022812 (2022).
* (40) Chu, P.-H., Kim, Y. & Savukov, I. Search for exotic spin-dependent interactions using polarized helium. _arXiv preprint arXiv:2002.02495_ (2020).
* (41) Vasilakis, G., Brown, J., Kornack, T. & Romalis, M. Limits on new long range nuclear spin-dependent forces set with a k- he 3 comagnetometer. _Phys. Rev. Lett._ 103, 261801 (2009).
* (42) Arushi, B. & Rupert, C. Solving the electron and muon g-2 anomalies in z’ models. _Eur. Phys. J. C, Particles and Fields._ 81 (2021).
* (43) Kimball, D. J., Boyd, A. & Budker, D. Constraints on anomalous spin-spin interactions from spin-exchange collisions. _Phys. Rev. A_ 82, 062714 (2010).
* (44) Ficek, F. _et al._ Constraints on exotic spin-dependent interactions between matter and antimatter from antiprotonic helium spectroscopy. _Phys. Rev. Lett._ 120, 183002 (2018).
* (45) Kimball, D. J. Nuclear spin content and constraints on exotic spin-dependent couplings. _New Journal of Physics_ 17, 073008 (2015).
* (46) Vinante, A. _et al._ Surpassing the energy resolution limit with ferromagnetic torque sensors. _Phys. Rev. Lett._ 127, 070801 (2021).
|
# Non-relativistic torque and Edelstein effect in noncollinear magnets
Rafael González-Hernández1, Philipp Ritzinger2, Karel Výborný2, Jakub
Železný2, Aurélien Manchon2 1Grupó de Investigación en Física Aplicada,
Departamento de Física, Universidad del Norte, Barranquilla, Colombia; 2
Institute of Physics, Czech Academy of Sciences, Cukrovarnická 10, 162 00
Praha 6 Czech Republic; 3Aix-Marseille Univ, CNRS, CINaM, Marseille, France
###### Abstract
The Edelstein effect is the origin of the spin-orbit torque: a current-induced
torque that is used for the electrical control of ferromagnetic and
antiferromagnetic materials. This effect originates from the relativistic
spin-orbit coupling, which necessitates utilizing materials with heavy
elements. Here we show that in magnetic materials with non-collinear magnetic
order, the Edelstein effect and consequently also a current-induced torque can
exist even in the absence of the spin-orbit coupling. Using group symmetry
analysis, model calculations, and realistic simulations on selected compounds,
we identify large classes of non-collinear magnet candidates and demonstrate
that the current-driven torque is of similar magnitude as the celebrated spin-
orbit torque in conventional transition metal structures. We also show that
this torque can exist in an insulating material, which could allow for highly
efficient electrical control of magnetic order.
## I Introduction
In materials and heterostructures with spin-orbit coupling, the
interconnection between the spin and momentum degrees of freedom of the
electronic Bloch states underscores a rich landscape of microscopic ”spin-
orbitronics” phenomena, such as anomalous Hall effect Nagaosa _et al._ (2010)
and anisotropic magnetoresistance Ritzinger and Vyborny (2022), spin Hall
effect Sinova _et al._ (2015), Dzyaloshinskii-Moriya interaction or spin-
orbit torques Soumyanarayanan _et al._ (2016); Manchon _et al._ (2019). To
maximize these effects, materials displaying reasonably large spin-orbit
coupling are necessary, which implies using metals with large atomic numbers
Z, such as Pt, W, Bi, etc. Some of these elements are however scarce,
expensive, and environmentally unfriendly. In addition, arbitrarily large
spin-orbit coupling does not necessarily lead to arbitrarily large spin-
orbitronics phenomena Sunko _et al._ (2017); Železný _et al._ (2021) because
of the competition with crystal field and exchange.
Contrary to a common conception though, spin-orbit coupling is not a mandatory
ingredient to obtain spin-momentum locking. In fact, as noticed by Pekar and
Rashba in the mid-sixties Pekar and Rashba (1965), electronics states in
materials with a spatially inhomogeneous magnetization display a spin texture
in momentum space that share similarities with the one obtained through spin-
orbit coupling. In other words, non-collinear magnetism mimics spin-orbit
coupling to some extent and can support a number of phenomena that are well
known in spin-orbit coupled materials such as electric-dipole spin resonance
Pekar and Rashba (1965); Ramazashvili (2009), topological Hall effect Shindou
and Nagaosa (2001); Martin and Batista (2008); Ndiaye _et al._ (2019), spin
Hall effect Zhang _et al._ (2018a), and magnetic spin Hall effect Zelezný
_et al._ (2017); Kimata _et al._ (2019), the latter being specific to
magnetic materials. It is therefore natural to wonder whether another hallmark
of spin-orbit coupled materials, the Edelstein effect F.T. Vasko (1979);
Edelstein (1990) (also called the Rashba-Edelstein effect, inverse spin-
galvanic effect, or the magneto-electric effect), and its associated spin-
orbit torque can also be achieved in spin-orbit free non-collinear magnets.
The Edelstein effect refers to the generation of non-equilibrium spin density
by an applied electric field in non-centrosymmetric semiconducting or metallic
materials and heterostructures with spin-orbit coupling. The magnitude of the
nonequilibrium spin density is governed by the competition between the spin-
orbit coupling energy and the crystal field energy associated with inversion
symmetry breaking. In magnetic materials, the spin-momentum locking is
governed by the magnetic exchange between local and itinerant electrons,
rather than by the atomic spin-orbit coupling, suggesting that a large
Edelstein effect can be obtained in non-centrosymmetric magnetic materials. A
possible advantage of such a mechanism is that it does not require the
presence of heavy elements and it could exist even in materials with
negligible spin-orbit coupling such as organic magnetic materials. In
addition, since the magnitude of the Edelstein effect is directly related to
the magnetic configuration of the material, it should be highly tunable using
an external magnetic field.
A first step towards the realization of this non-relativistic Edelstein effect
was established recently in a model non-collinear coplanar kagomé
antiferromagnet with in-plane broken inversion symmetry Hayami _et al._
(2020). In this particular case, the current-driven spin density is polarized
out of plane. What makes this Edelstein effect particularly appealing is that
the nonequilibrium spin density can exert a torque on the local magnetization
that is responsible for generating this spin density. This feature suggests
that current-driven dynamics, high-frequency excitation, and reversal, might
exhibit properties fundamentally different from those usually observed in
spin-transfer and spin-orbit torque devices. We have recently noticed such a
torque in a work studying spin-transfer torque in magnetic junctions composed
of centrosymmetric non-collinear antiferromagnets Ghosh _et al._ (2022).
Here, we study the non-relativistic Edelstein effect in non-collinear systems
in detail, focusing on systems where the spin-orbit torque is normally
studied, i.e. bulk non-centrosymmetric magnets and heterostructures with
current flowing parallel to the interface.
We demonstrate that wide classes of antiferromagnets lacking a center of
inversion can support the current-driven Edelstein effect in the absence of
spin-orbit coupling, and therefore also current-driven spin torques. We
establish general symmetry principles to discover new materials, propose
selected promising candidates, demonstrate and quantify the effect in specific
materials, and extend the idea to the case of magnetic multilayers. We also
show the existence of non-relativistic antisymmetric spin-textures in
reciprocal space, which are in some cases directly related to the Edelstein
effect, and discuss their general symmetry properties. We implemented an
algorithm for determining the symmetry of the non-relativistic Edelstein
effect as well as other non-relativistic phenomena and we released it within
an open source code. Remarkably, we show that the non-relativistic Edelstein
effect can also be present in insulating materials. This could allow for
controlling the magnetic order by a voltage in the absence of any Ohmic
conduction, resulting in a much higher efficiency than the conventional
current-induced torques.
## II General principles
### II.1 Conditions for an antisymmetric spin texture
In non-magnetic materials lacking inversion symmetry, the relativistic
Edelstein effect is associated with an antisymmetric spin texture in the
reciprocal space Bihlmayer _et al._ (2022). These spin textures arise from
the spin-momentum locking imposed by the spin-orbit coupling and are
characterized by a spin direction that varies in momentum space. In the
absence of spin-orbit coupling and in the presence of non-collinear magnetism,
one expects non-relativistic analogs of the antisymmetric spin textures.
Therefore, before addressing the non-relativistic Edelstein effect and its
associated torque, we first consider the conditions of the emergence of such
antisymmetric spin textures. Recently, spin textures in the absence of spin-
orbit coupling have been studied in non-collinear Zelezný _et al._ (2017);
Yuan _et al._ (2021); Hayami _et al._ (2020) as well as in collinear
magnetic materials Hayami _et al._ (2019); Yuan _et al._ (2020, 2021);
Šmejkal _et al._ (2022); Egorov _et al._ (2021). In collinear systems,
however, the direction of spin is fixed and only the magnitude and sign of the
spin-splitting varies in momentum space. In addition, most of the non-
relativistic spin-textures studied so far (with the exception of Ref. Hayami
_et al._ , 2020) are typically symmetric in momentum $\mathbf{k}$,
$\mathbf{S}_{n\mathbf{k}}=\mathbf{S}_{n-\mathbf{k}}$, $n$ being the band
index, which forbids the realization of the non-relativistic Edelstein effect
at the level of the magnetic unit cell.
In the absence of relativistic spin-orbit coupling, the spin and orbital
degrees of freedom are decoupled, which also means that the spin is not
coupled to the lattice. In such a case the symmetry of magnetic systems is
described by the so-called spin space groups Brinkman and Elliott (1966);
Litvin and Opechowski (1974). In addition to crystallographic symmetry
operations that form the magnetic space groups, which describe the
relativistic symmetry of magnetic systems, the spin space groups contain also
pure spin rotations. Elements of the spin space groups can be written in the
form $\\{R_{s}||R|{\bm{\tau}}\\}$, where $R_{s}$ denotes the spin-rotation,
$R$ is a crystallographic point group operation, i.e. a proper or improper
rotation, and ${\bm{\tau}}$ is a translation. We denote symmetry operations
that contain time-reversal as $\\{R_{s}||R|{\bm{\tau}}\\}^{\prime}$.
In a 3D periodic system, the rules for the existence of spin-splitting are
simple to determine. For an arbitrary $\mathbf{k}$-point (that is, a
$\mathbf{k}$-point lying away from any high-symmetry lines or planes), the
only symmetry operations that can keep the spin invariant are the combined
space-inversion and time-reversal (the so-called $P\cal{T}$ symmetry), a pure
spin rotation, translation, or any combination of these symmetry operations.
In a $P\cal{T}$ symmetric system, the bands with opposite spin must be
degenerate, which is known as Kramers degeneracy and holds even in the
presence of spin-orbit coupling. If a pure spin rotation is present, the spin
of all states must lie along the spin-rotation axis. If more than one spin
rotation with different spin axes is present, this cannot be satisfied without
a spin degeneracy. This can also be seen from the fact that spin rotations
along different axes do not commute. Since translation does not change spin,
the same conclusions apply to symmetry operations that contain translation.
Thus spin-splitting exists in all systems, except those that have a $P\cal{T}$
symmetry or two spin rotation axes. In ferromagnetic systems, spin splitting
can exist anywhere in the Brillouin zone since no symmetry operations
connecting states with opposite spin exist. In antiferromagnetic materials,
there are specific lines or planes where the bands with opposite spin must be
degenerate, see Ref. Krempaský _et al._ (2023). This has been studied
systematically for collinear antiferromagnets Hayami _et al._ (2019); Yuan
_et al._ (2020, 2021); Šmejkal _et al._ (2020); Egorov _et al._ (2021). Note
that the spin-split collinear antiferromagnets have sometimes been referred to
as “altermagnets” Šmejkal _et al._ (2022).
In a collinear magnetic system, any spin rotation along the magnetic axis is a
symmetry. Thus if there exists another spin-rotation around a perpendicular
axis, the bands must be degenerate. Such a spin rotation must contain
translation (otherwise the system could not be magnetic) and can only be a
$180^{\circ}$ rotation, which in a collinear system has the same effect on the
magnetic order as the time-reversal. The existence of such a symmetry thus
implies that the system is invariant under a $\cal{T}{\bm{\tau}}$ (a combined
time-reversal and translation) symmetry. Collinear antiferromagnetic systems
can thus be separated into three types. In systems with $P\cal{T}$ symmetry,
bands are spin degenerate even with spin-orbit coupling. In systems with
$\cal{T}{\bm{\tau}}$ but broken $P\cal{T}$ symmetry, spin-splitting occurs
only when the spin-orbit coupling is present. Finally, in systems with broken
$P\cal{T}$ and $\cal{T}{\bm{\tau}}$ symmetries, a non-relativistic spin-
splitting can be present. In non-collinear magnets, this does not hold since
time reversal does not have the same effect as a $180^{\circ}$ spin rotation,
and spin rotations with different angles can also occur.
The existence of antisymmetric spin textures is governed by symmetries that
transform $\mathbf{k}\rightarrow-\mathbf{k}$. This involves in particular the
inversion symmetry, which implies
$\mathbf{S}_{n\mathbf{k}}=\mathbf{S}_{n-\mathbf{k}}$. In systems with
inversion symmetry, any spin texture thus must be symmetric. In a coplanar
system a combined spin-rotation and time-reversal operation
$\\{R_{s}(\hat{\mathbf{n}}_{\perp},180^{\circ})||E||E\\}^{\prime}$ is a
symmetry. Here $R_{s}(\hat{\mathbf{n}}_{\perp},180^{\circ})$ denotes a spin
rotation by $180^{\circ}$ around the direction perpendicular to the magnetic
plane $\hat{\mathbf{n}}_{\perp}$. As a consequence, it must hold that
$\mathbf{S}^{||}_{n\mathbf{k}}=\mathbf{S}^{||}_{n-\mathbf{k}}$ and
$\mathbf{S}^{\perp}_{n\mathbf{k}}=-\mathbf{S}^{\perp}_{n-\mathbf{k}}$, where
$\mathbf{S}^{||}$ and $\mathbf{S}^{\perp}$ denote the components of spin
parallel and perpendicular to the plane, respectively. In a coplanar system,
the only antisymmetric component is thus perpendicular to the magnetic plane,
as in the case studied by Hayami et al. Hayami _et al._ (2020). We note that
even when all magnetic moments lie within a plane, the electron spins can
contain an out-of-plane component. In a collinear magnetic system, this is not
possible, however, since in this case spin is a good quantum number and all
spins must lie along a single axis. There, any non-relativistic spin splitting
must thus be symmetric in momentum.
### II.2 Conditions for nonequilibrium spin densities and torques
Let us now turn our attention towards the non-relativistic Edelstein effect.
The nonequilibrium properties of materials obtained via the Kubo formula are
often parsed into so-called Fermi surface and Fermi sea contributions Železný
_et al._ (2017); Bonbien and Manchon (2020), the former being even under
$\cal{T}$ and the latter being odd. In the context of the Edelstein effect,
the $\cal{T}$-even Fermi surface contribution is related to the antisymmetric
spin texture in momentum space F.T. Vasko (1979); Edelstein (1990), whereas
the $\cal{T}$-odd Fermi sea contribution is related to the Berry curvature in
mixed spin-momentum space in the weak scattering limit Kurebayashi _et al._
(2014); Li _et al._ (2015); Hanke _et al._ (2017). As a consequence, in
spin-orbit coupled non-centrosymmetric magnetic heterostructures, the Fermi
surface contribution produces the so-called field-like torque whereas the
Fermi sea contribution is responsible for the antidamping-like torque Manchon
_et al._ (2019). Notice that the $\cal{T}$-odd Fermi sea contribution can also
be nonzero in $P\cal{T}$ symmetric antiferromagnets with Kramers degeneracy.
Furthermore, for manipulating the magnetic order in more complex magnetic
systems, especially in antiferromagnets, the nonequilibrium spin density one
should be concerned with is the local one, projected on the magnetic
sublattices, rather than the global one, at the level of the magnetic unit
cell Železný _et al._ (2014, 2017). The $\cal{T}$-even component of the local
Edelstein effect can be understood as originating from the antisymmetric spin
texture obtained upon projecting on the local atom. Such a “hidden” texture
can again exist even in systems with Kramers degeneracy Zhang _et al._
(2014). Consequently, the symmetry conditions that allow for the existence of
an Edelstein effect and a torque on the magnetic order are distinct from those
for the existence of anti-symmetric spin textures.
The symmetry of the non-relativistic global and local Edelstein effects and
the resulting torque can be determined in a similar fashion as for the
relativistic one, just replacing the magnetic space groups with spin groups.
The key symmetry that needs to be broken for the existence of the Edelstein
effect is the inversion symmetry. This holds regardless of the presence of
spin-orbit coupling. For the global Edelstein effect, the global inversion
symmetry must be broken, whereas for the local Edelstein effect, it has to be
broken locally (see, e.g., Zhang _et al._ (2014)). This means that for the
presence of the Edelstein effect on a given magnetic site, there must be no
inversion symmetry operation that would leave this site invariant.
As already mentioned, in magnets the Edelstein effect can be nonzero even
without spin-orbit coupling, similar to the spin-Hall effect, for example,
Zhang _et al._ (2018b). This applies even to collinear magnets; however, in
such a case the induced spin density must be oriented along the magnetic order
and does not lead to a torque (although it could play a role for example in
the presence of magnons). Consequently, we focus here on non-collinear
magnetic systems. In presence of a pure spin-rotation
$\\{R_{s}(\hat{\mathbf{n}},\theta)||E||{\bm{\tau}}\\}\\}$, where ${\bm{\tau}}$
could also be zero, the global Edelstein effect must obey
$\mathbf{S}||\mathbf{n}$. In presence of spin-rotation coupled with time-
reversal $\\{R_{s}(\hat{\mathbf{n}},\theta)||E||{\bm{\tau}}\\}\\}^{\prime}$ it
obeys $\mathbf{S}^{\text{even}}||\mathbf{n}$ and
$\mathbf{S}^{\text{odd}}\perp\mathbf{n}$. The same holds for the local
Edelstein effect as long as the site is invariant under ${\bm{\tau}}$.
Consequently in coplanar systems, $\mathbf{S}^{\text{even}}$ must be oriented
perpendicular to the magnetic plane and $\mathbf{S}^{\text{odd}}$ must lie
within the plane for both the global and the local Edelstein effects.
To determine the full symmetry of the non-relativistic Edelstein effect, it is
necessary to consider all the symmetry operations of the spin group. We have
implemented an algorithm for determining all spin group symmetry operations of
a given magnetic system within the freely available open-source _Symmetr_ code
Zelezný (2017). The process of determining the non-relativistic symmetry is
described in detail in Supplementary Materials. We have utilized this program
to explore the symmetry of non-collinear materials from the MAGNDATA database
of magnetic materials. We have analyzed the symmetry of 484 non-collinear
magnetic materials and have found that the global Edelstein effect is allowed
in 160 of these materials, whereas the local Edelstein effect on a magnetic
sublattice is allowed in 355 compounds. The full list is given in the
Supplementary Materials Sup . As also described in the Supplementary
Materials, the Symmetr code allows one to directly obtain the non-relativistic
symmetry of the Edelstein effect (as well as other phenomena) for materials
from the MAGNDATA.
Among the noticeable materials whose crystal structure admits both a global
and local (sublattice) torque, we identified ferroelectric antiferromagnets
such as orthorhombic DyFeO3, hexagonal HoMnO3, YbMnO3, and LuFeO3, as well as
metallic antiferromagnets such as $\alpha$-Mn, Tb3Ge5 and Tb5Ge4.
Interestingly, the centro-symmetric metallic antiferromagnets Mn5Si3, Mn3(Sn,
Ge, As), and Mn3CuN do not display a global torque but do admit a local torque
on the individual magnetic sublattices. These torques are expected to induce
magnetic excitations and potentially magnetic order reversal. In the
following, we explicitly compute the global and local Edelstein effects in
both LuFeO3 and Mn3Sn as an illustration of both cases.
## III Non-relativistic Edelstein effect in non-collinear antiferromagnets
To calculate the Edelstein effect and torque we use the Kubo formula within
the constant relaxation time approximation. We only consider an Edelstein
effect linear in an electric field: $\delta S_{i}=\chi_{ij}E_{j}$, where
$\delta S_{i}$ is the induced spin, $E_{j}$ is the electric field, and
$\chi_{ij}$ is a response tensor. The $\cal{T}$-even and $\cal{T}$-odd
components are computed using the Kubo formula derived in Refs. Freimuth _et
al._ (2014); Li _et al._ (2015) and replacing the torque operator with the
spin operator,
$\displaystyle\chi^{\rm even}_{ij}$ $\displaystyle=$
$\displaystyle-\frac{e\hbar}{\pi}\sum_{\textbf{k},m,n}\frac{\text{Re}[\Bra{\psi_{\textbf{k}n}}\hat{S}_{i}\Ket{\psi_{\textbf{k}m}}\Bra{\psi_{\textbf{k}m}}\hat{v}_{j}\Ket{\psi_{\textbf{k}n}}]\Gamma^{2}}{(\left(\varepsilon_{F}-\varepsilon_{\mathbf{k}n})^{2}+\Gamma^{2})((\varepsilon_{F}-\varepsilon_{\mathbf{k}m}\right)^{2}+\Gamma^{2})},$
$\displaystyle\chi^{\rm odd}_{ij}$ $\displaystyle=$
$\displaystyle{2e\hbar}\sum_{\mathbf{k},n\neq m}^{\begin{subarray}{c}\\\ n\
\text{occ.}\\\ m\
\text{unocc.}\end{subarray}}\text{Im}[\Bra{\psi_{n\mathbf{k}}}\hat{S}_{i}\Ket{\psi_{m\mathbf{k}}}\Bra{\psi_{m\mathbf{k}}}\hat{v}_{j}\Ket{\psi_{n\mathbf{k}}}]$
(2)
$\displaystyle\times\frac{\Gamma^{2}-(\varepsilon_{\mathbf{k}n}-\varepsilon_{\mathbf{k}m})^{2}}{[(\varepsilon_{\mathbf{k}n}-\varepsilon_{\mathbf{k}m})^{2}+\Gamma^{2}]^{2}}.$
Here $\psi_{\textbf{k}n}$ is the Bloch function of band $n$, k is the Bloch
wave vector, $\varepsilon_{\mathbf{k}n}$ is the band energy, $\varepsilon_{F}$
is the Fermi energy, $\hat{v}_{j}$ is the velocity operator, $e>0$ is the
elementary charge, $\hat{S}_{i}$ is the spin operator, and $\Gamma$ is a
parameter that describes the strength of disorder, which is related to the
relaxation time $\tau=\hbar/2\Gamma$. To calculate the local Edelstein effect
on a given sublattice, a local spin operator is used instead.
In the limit $\Gamma\rightarrow 0$, Eq. (1) goes to the semiclassical
Boltzmann constant relaxation formula, which scales as $1/\Gamma$ whereas Eq.
(2) goes to the so-called intrinsic formula, which is $\Gamma$ independent and
can be understood in terms of Berry curvature in mixed spin-momentum space
Hanke _et al._ (2017).
Figure 1: The 3Q triangular antiferromagnet: (a) Sketch of the triangular
lattice with 3Q non-coplanar configuration of the magnetic moments. The green
atoms break the planar inversion symmetry. (b) Top view of the triangular
lattice. The hopping parameters of the black and red bounds are $t$ and
$t^{\prime}$, respectively. (c, f) Band structure for $t^{\prime}=t$ and
$t^{\prime}=t/2$, respectively. (e) In-plane spin texture in momentum space at
energy $\varepsilon=-5$ eV, corresponding to the red dashed line in panel (f).
In the absence of inversion symmetry breaking, $t^{\prime}=t$, the degenerate
bands display compensating spin texture in momentum space. When
$t^{\prime}\neq t$, the band degeneracy is lifted and the spin textures do not
longer compensate. Notice that a momentum-asymmetric out-of-plane spin texture
is also present (not shown). (d, g) $\cal{T}$-even (d) $\cal{T}$-odd (g)
contributions for $t^{\prime}=t/2$ and $\varepsilon=-4eV$ when rotating the
electric field direction in the (x,y) plane. We set $t=1$ eV, $\Gamma=0.1$ eV
and the exchange is $J=-2$ eV.
### III.1 A non-coplanar 3Q antiferromagnet
An example of a non-relativistic Edelstein effect in a non-collinear coplanar
antiferromagnet was recently given by Hayami et al. Hayami _et al._ (2020).
In this case, the coplanarity of the magnetic texture imposes the current-
driven spin density to be oriented perpendicular to the magnetic plane. Here,
we adopt a triangular antiferromagnet with a 3Q spin texture, as displayed in
Fig. 1(a). This magnetic texture can be stabilized in the presence of 4-spin
interaction Kurz _et al._ (2001); Kato _et al._ (2010) and hosts quantum
anomalous Hall effect Shindou and Nagaosa (2001); Martin and Batista (2008);
Ndiaye _et al._ (2019). The 3Q texture is also commonly observed in three-
dimensional materials such as $\gamma$-FeMnEndoh and Ishikawa (1971) and
pyrochlores Gardner _et al._ (2010). We use a simple tight-binding model with
a 3Q spin texture to illustrate the physical properties of such systems. The
model is given by
$\displaystyle
H=-\sum_{<ab>\alpha}t_{ab}c^{\dagger}_{a\alpha}c_{b\alpha}+J\sum_{a\alpha,\beta}({\bm{\sigma}}\cdot\mathbf{m}_{a})_{\alpha\beta}c^{\dagger}_{a\alpha}c_{a\beta}.$
(3)
Here, $c^{\dagger}$ and $c$ denote the creation and annihilation operators
respectively; $a,b$ denote the site index and $\alpha,\beta$ the spin index.
The first term is the nearest neighbor hopping term, with $t_{ab}$
representing the hopping magnitude. The second term represents the coupling of
the conduction electrons to the on-site magnetic moments. Here
$\mathbf{m}_{i}$ is the magnetic moment direction, $J$ is the exchange
parameter and ${\bm{\sigma}}$ is the vector of Pauli matrices. We only
consider nearest-neighbor hopping. To break the inversion symmetry we use two
different hopping magnitudes as shown in Fig. 1(b). This could be understood
as due to the presence of another atom illustrated in Fig. 1(a).
The band structures of the 3Q antiferromagnet are given in Figs. 1(c) and (f),
without and with inversion symmetry breaking. In the absence of inversion
symmetry breaking, the band structure is doubly degenerate. Breaking the
inversion symmetry lifts the band degeneracy [Fig. 1(f)] and results in a
spin-texture, shown in Fig. 1(e). This spin texture contains both symmetric
and anti-symmetric components, the latter giving rise to the current-driven
Edelstein effect and its torque. When the inversion symmetry is broken, we
observe a finite Edelstein effect as shown in Fig. 1(d) and (g). Several
features are worth noticing. First, because the magnetic texture of the 3Q
antiferromagnet spans the 3D space, the current-driven spin density possesses
all three components, $S_{x}$, $S_{y}$ and $S_{z}$, which strikingly contrasts
with the result of Ref. Hayami _et al._ (2020). Second, both $\cal{T}$-even
and $\cal{T}$-odd components contribute with similar magnitude. Finally, for
the set of parameters adopted in this calculation, i.e., the exchange and
hopping energies are of comparable magnitude ($\Delta$ =2t=4t’= 2 eV), we
obtain a nonequilibrium spin density of about 10-11 $\hbar$m/V. For the sake
of comparison, in a two-dimensional Rashba gas, the nonequilibrium spin
density is Edelstein (1990) $S^{R}_{\rm surf}/eE=(\alpha_{\rm
R}/\hbar^{2})(m_{0}/\pi\Gamma)$. Taking $\Gamma=0.1$ eV, $m_{0}$ being the
free electron mass, $\alpha_{\rm R}=10^{-9}$ eV$\cdot$m as the typical Rashba
strength expected in transition metal heterostructures Manchon _et al._
(2015), and $(3\mskip 3.0mu\textrm{\AA})^{2}$ as a unit cell area, the
Edelstein effect yields $\chi_{S}\sim 3.6\times 10^{-11}\mskip
3.0mu\hbar\text{m/V}$, which is in the same range as our calculations for the
two-dimensional 3Q system reported in Fig. 1.
### III.2 A centrosymmetric antiferromagnet: ${\rm Mn_{3}Sn}$
In antiferromagnets, and in general in more complex magnetic systems, the
magnetic dynamics is not determined by the global Edelstein effect, but rather
by the local Edelstein effect on each magnetic sublattice. Consequently, in
antiferromagnets, it is the local rather than the global inversion symmetry
breaking that is necessary for the existence of the Edelstein effect and the
current-induced torque Železný _et al._ (2017). An example of an
antiferromagnet with a global inversion symmetry and a local inversion
symmetry breaking is the well-known non-collinear antiferromagnet Mn3Sn
Tomiyoshi (1982); Nakatsuji _et al._ (2015). In this material, the global
Edelstein effect vanishes but the local Edelstein effect is allowed on each
sublattice, even in the absence of spin-orbit coupling.
The crystal and magnetic structure of Mn3Sn is given in Fig. 2(c). Mn3Sn has
six magnetic sublattices, which are composed of three pairs of sites with
equivalent moments connected by inversion symmetry. The inversion partners are
denoted by ′ in Fig. 2(b). Due to the inversion symmetry, the Edelstein effect
on the two inversion-connected sites must be opposite. The local Edelstein
effect tends to drive the system into a state where the magnetic moments of
the inversion-connected sites are not parallel and thus it acts against the
exchange. As such, it is unlikely to reverse the magnetic order but it can
excite different magnon modes. Leaving the rigorous analysis of the magnetic
dynamics to future studies, we emphasize that the global inversion symmetry
can be broken, for example, by an interface. Then the two sites with the same
moments are no longer connected by inversion symmetry and consequently can
experience the Edelstein effect of the same sign, enabling the electric
manipulation of the magnetic order.
We evaluate the Edelstein effect in Mn3Sn using ab-initio calculations with
and without spin-orbit coupling (see the Methods section for details of the
calculation setup). The result of the calculation for $\Gamma=0.01\mskip
3.0mu\text{eV}$ is shown in Fig. 2(a). We find substantial $\cal{T}$-even and
$\cal{T}$-odd Edelstein effect on all sublattices. Including the spin-orbit
coupling does not change the results substantially, similar to previous
calculations of the spin Hall effect in this material Zhang _et al._ (2018a).
Our calculations agree well with the symmetry analysis shown in the
Supplementary Materials. Notice that, again, the magnitude of the current-
driven spin density is rather large, corresponding to a Rashba strength of
$10^{-9}-10^{-10}$ eV$\cdot$m.
We note that current-induced switching of Mn3Sn has been experimentally
observed in Mn3Sn/non-magnetic metal heterostructures Tsai _et al._ (2020);
Krishnaswamy _et al._ (2022); Pal _et al._ (2022); Xie _et al._ (2022). The
switching has been attributed to the spin-Hall effect from the non-magnetic
metal layer and to spin-transfer torque and inter-grain spin-transfer torque,
however, it is possible that the non-relativistic Edelstein effect also
contributes.
Figure 2: (a),(b) Calculation of the $\cal{T}$-even (a) and $\cal{T}$-odd (b)
local Edelstein effect in Mn3Sn with (dashed lines) and without (solid lines)
spin-orbit coupling. The individual lines correspond to different tensor
components, see Eq. (LABEL:eq:Kubo_even)-(2). (c) The crystal structure and
magnetic structure of Mn3Sn.
### III.3 A non-centrosymmetric antiferromagnet: ${\rm LuFeO_{3}}$
Figure 3: (a) The crystal and magnetic structure of the hexagonal LuFeO3. (b)
LuFeO3 band structure without spin-orbit coupling. The color denotes the
$S_{x}$ projection. $X$ points represent opposite $k_{x}$ coordenates, $Z$
points represent opposite $k_{z}$ coordenates and $P$ points represent
opposite $k_{x},k_{y},k_{z}$ coordenates in the Brillouin zone. Asymmetric
-$odd$ in k- and symmetric -$even$\- spin splitting is labeled in the
correspondent $k$-path. (c) The spin texture of LuFeO3 at the Fermi surface
for Fermi level 0.45 eV below the top of the valence band. We plot the spin-
texture for two planes corresponding to $k_{z}$ = 0.25 Å-1 and $k_{z}$ = -0.25
Å-1. The center of the figure lies at the $\Gamma$ point. The arrows show the
spin and we use the color to highlight the $z$-component of the spin since it
would be hard to distinguish otherwise.
As an example of a real non-collinear antiferromagnet that can exhibit a
global non-relativistic Edelstein effect and torques, we consider the
hexagonal LuFeO3, a multiferroic with antiferromagnetic order. In bulk, LuFeO3
is typically orthorhombic. However, the hexagonal phase has been stabilized in
thin layers Disseler _et al._ (2015) and can also be stabilized in the bulk
Suresh _et al._ (2018). It has a non-collinear coplanar antiferromagnetic
structure with magnetic space group (MSG) #185.201 (P$6_{3}$c’m’) as presented
in Figure 3(a) Disseler _et al._ (2015); Suresh _et al._ (2018). The
inversion symmetry is broken in this material by the crystal structure, which
suggests the possibility of non-relativistic spin-torques. The system has a
small net moment $\sim 0.02\mskip 3.0mu\mu_{\textrm{B}}$ along the $z$
direction (weak ferromagnetism). This moment is of relativistic origin, and
thus in the absence of spin-orbit coupling the magnetic structure is perfectly
compensated. Apart from the magnetic order, hexagonal LuFeO3 also exhibits a
ferroelectric order that is present below $\sim 1000\mskip 3.0mu\textrm{K}$
and the material has attracted large attention for its multiferroic properties
and the possibility of magneto-electric coupling Das _et al._ (2014);
Disseler _et al._ (2015); Suresh _et al._ (2018).
The non-relativistic electronic structure is shown in Fig. 3(b). The material
is insulating; here we only show the valence bands, which are relevant to our
calculations. As can be also seen in Fig. 3(b) the bands are spin-split and
thus there is also a non-relativistic spin-texture, shown in Fig. 3(c) for two
cuts through the Brillouin zone. Due to the coplanarity of the magnetic order,
the spin-texture is symmetric for the $S_{x}$ and $S_{y}$ components and anti-
symmetric for the $S_{z}$ component. The $S_{z}$ component is non-zero but
very small.
For the calculation of the Edelstein effect, we move the Fermi level into the
valence band to simulate doping. Our symmetry analysis shown in the
Supplementary Materials shows that the Edelstein effect is allowed in LuFeO3
even with no spin-orbit coupling. Results of the calculation for
$\Gamma=0.01\mskip 3.0mu\text{eV}$ with and without spin-orbit coupling are
given in Fig. 4. We calculate both the global Edelstein effect and the local
one for all Fe sublattices. For brevity though, we only show here the result
for the global effect and for one sublattice. The full results are shown in
the Supplementary Materials.
Our calculations reveal a large non-relativistic global and local Edelstein
effect in good agreement with the symmetry analysis. Without spin-orbit
coupling for the global effect only the $\cal{T}$-odd component is allowed
[Fig. 4(b)]. With spin-orbit coupling even the global $\cal{T}$-even component
appears [Fig. 4(a)]. We find that the effect of spin-orbit coupling is quite
small for the $\cal{T}$-odd component, but fairly large for the $\cal{T}$-even
component. For the local effect [Fig. 4(c,d)], both $\cal{T}$-even and
$\cal{T}$-odd components are allowed.
An important remark is in order. The $\cal{T}$-even component has to vanish
within the gap since it is a Fermi surface property [see Eq.
(LABEL:eq:Kubo_even)]. The global $\cal{T}$-odd component vanishes within the
gap as well [Fig. 4(b)]. However, we find that the local $\cal{T}$-odd
components on the Fe atoms are non-zero within the gap, as shown in Fig. 4(d)
and in the Supplementary Materials. Only the $xz$ and $yz$ components are non-
zero within the gap, reaching a constant value. Such a result is intriguing as
within the gap there is no Ohmic conduction and thus heat dissipation is
absent. This could consequently allow for electric field control of magnetic
order in the absence of Ohmic dissipation. The existence of spin-orbit torque
in an insulator was previously studied in topological materials Hanke _et
al._ (2017); Tang and Cheng (2023). Our results are similar except that in the
case of LuFeO3 the origin of the torque is non-relativistic, due to the
coexistence of the non-collinear magnetic order with inversion symmetry
breaking. We point out that the torque is not quantized, contrary to the
quantized magnetoelectric effect in topological insulators Qi _et al._
(2008), and therefore unlikely to be of topological origin. We are also not
aware of any topological properties of LuFeO3. The $\cal{T}$-odd torque is
governed by the Berry curvature in mixed spin-momentum space and involves
electrically-driven interband transitions, resulting in a finite (but not
quantized) value in the gap.
We note that in metals the torques induced by an electric field are
accompanied by an electric current and thus often referred to as “current-
induced” torques. However, even in metals, the torques are in fact due to the
electric field, rather than to the current flow, although the torque cannot
exist without Ohmic conduction. In non-centrosymmetric insulating magnets
though, Ohmic conduction is suppressed while the electrically-driven torque
remains sizable as demonstrated in LuFeO3. This opens promising perspectives
for the dissipation-free electrical control of magnetization.
Figure 4: Edelstein effect calculation in LuFeO3 with (dashed lines) and
without spin-orbit coupling (solid lines). Here zero energy corresponds to the
top of the valence band. (a) $\cal{T}$-even component of the total Edelstein
effect. (b) $\cal{T}$-odd component of the total Edelstein effect. (c)
$\cal{T}$-even component of the local Edelstein effect. (d) $\cal{T}$-odd
component of the local Edelstein effect.
### III.4 Non-centrosymmetric heterostructures
The examples we have discussed so far all have inversion symmetry (globally or
locally) broken in the bulk of their crystal structure. Such a constraint
however severely restricts the Edelstein effect to the materials listed in the
Supplemental Materials. For this reason, we propose to exploit the broken
inversion symmetry taking place at the interface between the non-collinear
antiferromagnet and an adjacent metal. Such heterostructures are commonly
utilized for spin-orbit torque, where the ferro- or antiferromagnet is
typically interfaced with a heavy element metal such as platinum Manchon _et
al._ (2019). This simple but instrumental configuration allows for observing
the spin-orbit torque in a wide variety of systems and enables interfacial
engineering of the spin-orbit torque properties.
The same concept can be applied to the non-relativistic Edelstein effect. When
a non-collinear magnetic material with inversion symmetry is interfaced with a
different material, the broken inversion symmetry can result in a non-
relativistic Edelstein effect, which in turn generates a torque on the
magnetic order. We illustrate this concept using the example of the well-known
non-collinear antiferromagnet Mn3Ir whose crystal and magnetic structures are
displayed in Fig. 5(a). In this material, each magnetic site is an inversion
center and thus no Edelstein effect is allowed in the bulk. To break the
inversion symmetry we consider a thin layer of Mn3Ir interfaced with a thin
layer of a non-magnetic material. When Mn3Ir is grown along the [001]
direction, the non-relativistic Edelstein effect in such a heterostructure is
only allowed for an electric field along the [001] direction. In such a case
no electric current can flow, however. Thus we instead consider Mn3Ir grown
along the [111] direction. For this orientation, the symmetry is lowered and
the Edelstein effect is allowed for an electric field oriented along the
interface. We consider a structure composed of 12 atomic layers of the Mn3Ir,
as shown in Fig. 5(b). The individual atomic layers are shown in Fig. 5(c).
We utilize a simple tight-binding model that is not meant to give quantitative
predictions but rather to confirm that the effect can exist and illustrate its
basic properties. The model is analogous to the one we have used for the 3Q
antiferromagnet. It is composed of s-electrons on each site with nearest
neighbor hopping and exchange coupling to the atomic magnetic moments. Similar
models have been utilized to demonstrate other properties of the Mn3Ir and
similar antiferromagnets such as the non-relativistic spin currents Zhang _et
al._ (2018a) or the anomalous Hall effect Chen _et al._ (2014). We do not
include any spin-orbit coupling in the model. The Hamiltonian is in Eq. (3),
where we consider nearest neighbor hopping $t=1\ \text{eV}$ and magnetic
exchange $J=1.7\ \text{eV}$ on Mn atoms (gray and gold atoms representing non-
magnetic layer and Ir atoms, respectively, are not endowed with magnetic
moment).
Figure 5: (a) Mn3Ir bulk structure. (b) Side view of the Mn3Ir [111] bilayer.
Here we do not show the magnetic moments. (c) Top-down view of a single atomic
layer in the Mn3Ir [111] bilayer. Here $A$, $B$ and $C$ denote the three
different sublattices. Note that we have constructed the system in such a way
that there are 6 Mn atoms in each atomic layer, however, the two atoms for
each sublattice are connected by translation and thus there are only three
non-equivalent sublattices. (d) Edelstein effect calculation in the Mn3Ir
[111] for each sublattice as a function of the atomic layer. The top row plots
show the $\cal{T}$-even component and the bottom plots show the $\cal{T}$-odd
component.
The calculated Edelstein effect is shown in Fig. 5(c) for each sublattice and
atomic layer. For this calculation we have used $\Gamma=0.01\mskip
3.0mu\text{eV}$ and $\varepsilon_{F}=0\mskip 3.0mu\text{eV}$. Our calculations
are fully in agreement with the symmetry analysis, shown in the Supplementary
Materials. Both $\cal{T}$-even and $\cal{T}$-odd components are present. We
find the largest effect close to the interfaces, although the current-driven
spin density remains sizable in the center of the Mn3Ir layer. A large effect
is found both at the interface with the non-magnetic layer and at the top
surface, which illustrates that the presence of another layer is in principle
not necessary.
In Figure 5, we only give the result of the calculation for the tensor
components that correspond to an in-plane electric field. Interestingly, the
out-of-plane components do not vanish even though no current can flow in the
out-of-plane direction, similarly to the case of the LuFeO3. In this case,
however, since the system is metallic, the out-of-plane electric field is
screened and thus the effect is hard to observe in practice.
## IV Discussion and conclusion
The torque induced by the non-relativistic Edelstein effect shares important
similarities with the conventional spin-orbit torque. Both torques are
electrically driven self-induced torques that necessitate inversion symmetry
breaking. The key difference though is that the torque due to the non-
relativistic Edelstein effect does not originate from spin-orbit coupling, but
rather from the non-collinear magnetic order. As a consequence, the
microscopic origin of these two torques is quite distinct. In the non-
relativistic limit, spin is conserved and the torque is directly associated
with a spin current: the torque corresponds to spin sources Ghosh _et al._
(2022) and can be understood as a local transfer of spin angular momentum
within the magnetic unit cell.
In the present work, we have computed the non-relativistic Edelstein effect in
four different systems, all displaying inversion symmetry breaking, either in
the magnetic unit cell or locally. It is quite remarkable that all the
examples discussed here display a sizable electrically induced spin density,
in spite of the absence of spin-orbit coupling. For the sake of comparison, in
our previous calculations of the relativistic Edelstein effect in a collinear
antiferromagnet Mn2Au, we found a magnitude of $\chi_{S}\sim 4.3\times
10^{-11}\mskip 3.0mu\hbar\text{m/V}$ for $\Gamma=0.01\mskip 3.0mu\text{eV}$
Železný _et al._ (2017), corresponding to a Rashba strength of $10^{-9}$
eV$\cdot$m, similarly to the magnitude reported in our realistic simulations
on LuFeO3 and Mn3Sn. Further systematic studies as necessary to determine the
conditions for a maximal non-relativistic Edelstein effect.
A central feature of the non-relativistic nature of the torque is its
dependence on the magnetic order. In most magnetic systems, the magnetic
exchange is much larger than any other magnetic interactions or torques acting
on the system. Hence, during the dynamics of the magnetic order, the angles
between the individual magnetic moments stay approximately unchanged.
Therefore, the dynamics of the magnetic order are described by an overall
rotation of all magnetic moments and a small canting. In the non-relativistic
limit (ignoring the small canting) the rotated states are connected by a spin
rotation and, consequently, the corresponding torques must also be transformed
by this spin rotation. Specifically, any torque acting on magnetic moment
$\mathbf{M}_{i}$ can be written as
$\mathbf{T}_{i}=\mathbf{M}_{i}\times\mathbf{B}_{i}$, where $\mathbf{B}_{i}$ is
an effective magnetic field. When the magnetic moments are rotated by rotation
$R$ then the torque reads $\mathbf{T}_{i}=R\mathbf{M}_{i}\times
R\mathbf{B}_{i}$. This is quite distinct from the conventional spin-orbit
torque for which the two most important terms are the field-like torque, in
which $\mathbf{B}_{i}$ is independent of $\mathbf{M}$, and the anti-damping
torque, in which $\mathbf{B}_{i}\sim\mathbf{M}_{i}\times\mathbf{p}$,
$\mathbf{p}$ being some constant direction Manchon _et al._ (2019).
Because of the dependence of the non-relativistic torque on the magnetic
order, it may be difficult to realize reversible switching since there can be
no magnetic configuration for which the effective field $\mathbf{B}_{i}$
vanishes. This might not be such a limitation in practice, however, since some
spin-orbit coupling is always present, which may enable deterministic
switching even in cases where the non-relativistic torque is dominant.
Furthermore, deterministic switching could be achieved by using field-assisted
switching or precise pulse timing. In the presence of antiferromagnetic domain
walls, the non-relativistic torque could provide an additional source of spin
current and therefore enhance or quench the domain wall mobility, depending on
the wall configuration. In fact, the very dependence of the torque on the
magnetic ordering makes the interplay between the flowing electrons and the
magnetic order particularly rich and, as such, the non-relativistic torque is
well adapted to excite magnetic modes and self-oscillations which, in
antiferromagnets, are particularly appealing for THz applications.
## V acknowledgments
We acknowledge the Grant Agency of the Czech Republic Grant No. 22-21974S. A.
M. acknowledges support from the Excellence Initiative of Aix-Marseille
Université - A*Midex, a French ”Investissements d’Avenir” program.
## VI Methods
The DFT calculations use the VASP code Kresse and Furthmüller (1996) and we
use the Wannier90 code Pizzi _et al._ (2020) to construct the Wannier
Hamiltonian that serves as an input to the linear response calculations. For
LuFeO3 we use 9x9x3 k-point mesh and $520\mskip 3.0mu\text{eV}$ cutoff energy.
For the Wannierization we use the $s$ and $d$ orbitals for Fe, $s$ and $p$
orbitals for O and $\varepsilon_{F}\pm 3\mskip 3.0mu\text{eV}$ frozen energy
window. For Mn3Sn we use 11x11x11 k-point mesh and set the cutoff energy to
$520\mskip 3.0mu\text{eV}$. For the Wannierization we use the $s$,$p$, and $d$
orbitals for the Mn atoms, $s$ and $p$ orbitals for the Sn atoms and we set
the frozen energy window to $\varepsilon_{F}\pm 2\mskip 3.0mu\text{eV}$.
For the linear response calculations we use the Linres code Železný . This
code uses the Wannier or tight-binding Hamiltonian as an input. This
Hamiltonian is then Fourier transformed to a dense mesh in the reciprocal
space, which is used for evaluating the Kubo formulas as described in Ref.
Zelezný _et al._ (2017).
## References
* Nagaosa _et al._ (2010) N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Review of Modern Physics 82, 1539 (2010).
* Ritzinger and Vyborny (2022) P. Ritzinger and K. Vyborny, arXiv:2212.03700v2 (2022).
* Sinova _et al._ (2015) J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Review of Modern Physics 87, 1213 (2015).
* Soumyanarayanan _et al._ (2016) A. Soumyanarayanan, N. Reyren, A. Fert, and C. Panagopoulos, Nature 539, 509 (2016), arXiv:1611.09521 .
* Manchon _et al._ (2019) A. Manchon, J. Zelezný, M. Miron, T. Jungwirth, J. Sinova, A. Thiaville, K. Garello, and P. Gambardella, Review of Modern Physics 91, 035004 (2019).
* Sunko _et al._ (2017) V. Sunko, H. Rosner, P. Kushwaha, S. Khim, F. Mazzola, L. Bawden, O. J. Clark, J. M. Riley, D. Kasinathan, M. W. Haverkort, T. K. Kim, M. Hoesch, J. Fujii, I. Vobornik, A. P. Mackenzie, and P. D. C. King, Nature 549, 492 (2017), arXiv:1708.03887 .
* Železný _et al._ (2021) J. Železný, Z. Fang, K. Olejník, J. Patchett, F. Gerhard, C. Gould, L. W. Molenkamp, C. Gomez-Olivella, J. Zemen, T. Tichý, T. Jungwirth, and C. Ciccarelli, Physical Review B 104, 054429 (2021), arXiv:2102.12838 .
* Pekar and Rashba (1965) S. Pekar and E. I. Rashba, Soviet Physics JETP 20, 1295 (1965).
* Ramazashvili (2009) R. Ramazashvili, Physical Review B 79, 184432 (2009), arXiv:arXiv:0809.1893v3 .
* Shindou and Nagaosa (2001) R. Shindou and N. Nagaosa, Physical Review Letters 87, 116801 (2001).
* Martin and Batista (2008) I. Martin and C. D. Batista, Physical Review Letters 101, 156402 (2008), arXiv:0807.3552 .
* Ndiaye _et al._ (2019) P. B. Ndiaye, A. Abbout, V. M. L. D. P. Goli, and A. Manchon, Physical Review B 100, 144440 (2019).
* Zhang _et al._ (2018a) Y. Zhang, J. Zelezny, Y. Sun, J. V. D. Brink, and B. Yan, New Journal of Physics 20, 073028 (2018a).
* Zelezný _et al._ (2017) J. Zelezný, Y. Zhang, C. Felser, and B. Yan, Physical Review Letters 119, 187204 (2017).
* Kimata _et al._ (2019) M. Kimata, H. Chen, K. Kondou, S. Sugimoto, P. K. Muduli, M. Ikhlas, Y. Omori, T. Tomita, A. H. Macdonald, S. Nakatsuji, and Y. Otani, Nature 565, 627 (2019).
* F.T. Vasko (1979) F.T. Vasko, Pis’ma Zh. Eksp. Teor. Fiz 30, 574 (1979).
* Edelstein (1990) V. M. Edelstein, Solid State Communications 73, 233 (1990).
* Hayami _et al._ (2020) S. Hayami, Y. Yanagi, and H. Kusunose, Physical Review B 101, 220403 (2020), arXiv:2001.05630 .
* Ghosh _et al._ (2022) S. Ghosh, A. Manchon, and J. Železný, Physical Review Letters 128, 097702 (2022).
* Bihlmayer _et al._ (2022) G. Bihlmayer, P. Noël, D. V. Vyalikh, E. V. Chulkov, and A. Manchon, Nature Reviews Physics (2022), 10.1038/s42254-022-00490-y.
* Yuan _et al._ (2021) L. D. Yuan, Z. Wang, J. W. Luo, and A. Zunger, Physical Review Materials 5, 014409 (2021), arXiv:2008.08532 .
* Hayami _et al._ (2019) S. Hayami, Y. Yanagi, and H. Kusunose, Journal of the Physical Sociecty of Japan 88, 123702 (2019).
* Yuan _et al._ (2020) L.-d. Yuan, Z. Wang, J.-w. Luo, E. I. Rashba, and A. Zunger, Physical Review B 102, 014422 (2020).
* Šmejkal _et al._ (2022) L. Šmejkal, J. Sinova, and T. Jungwirth, Phys. Rev. X 12, 040501 (2022).
* Egorov _et al._ (2021) S. A. Egorov, D. B. Litvin, and R. A. Evarestov, The Journal of Physical Chemistry C 125, 16147 (2021).
* Brinkman and Elliott (1966) W. F. Brinkman and R. J. Elliott, Proceedings of the Royal Society of London Series A 294, 343 (1966).
* Litvin and Opechowski (1974) D. B. Litvin and W. Opechowski, Physica 76, 538 (1974).
* Krempaský _et al._ (2023) J. Krempaský, L. Šmejkal, S. W. D’Souza, M. Hajlaoui, G. Springholz, K. Uhlířová, F. Alarab, P. C. Constantinou, V. Strokov, D. Usanov, W. R. Pudelko, R. González-Hernández, A. B. Hellenes, Z. Jansa, H. Reichlová, Z. Šobáň, R. D. G. Betancourt, P. Wadley, J. Sinova, D. Kriegner, J. Minár, J. H. Dil, and T. Jungwirth, (2023).
* Šmejkal _et al._ (2020) L. Šmejkal, R. González-hernández, T. Jungwirth, and J. Sinova, Science Advances 6, eaaz8809 (2020).
* Železný _et al._ (2017) J. Železný, H. Gao, A. Manchon, F. Freimuth, Y. Mokrousov, J. Zemen, J. Mašek, J. Sinova, and T. Jungwirth, Physical Review B 95, 014403 (2017).
* Bonbien and Manchon (2020) V. Bonbien and A. Manchon, Physical Review B 102, 085113 (2020).
* Kurebayashi _et al._ (2014) H. Kurebayashi, J. Sinova, D. Fang, A. Irvine, T. D. Skinner, J. Wunderlich, V. Novák, R. P. Campion, B. L. Gallagher, E. K. Vehstedt, L. P. Zârbo, K. Výborný, A. J. Ferguson, and T. Jungwirth, Nature Nanotechnology 9, 211 (2014).
* Li _et al._ (2015) H. Li, H. Gao, L. P. Zârbo, K. Výborný, X. Wang, I. Garate, F. Dogan, A. Cejchan, J. Sinova, T. Jungwirth, and A. Manchon, Physical Review B 91, 134402 (2015).
* Hanke _et al._ (2017) J.-P. Hanke, F. Freimuth, C. Niu, S. Blügel, and Y. Mokrousov, Nat Commun 8, 1479 (2017), arXiv:1701.08050 .
* Železný _et al._ (2014) J. Železný, H. Gao, K. Výborný, J. Zemen, J. Mašek, A. Manchon, J. Wunderlich, J. Sinova, and T. Jungwirth, Physical Review Letters 113, 157201 (2014).
* Zhang _et al._ (2014) X. Zhang, Q. Liu, J.-w. Luo, A. J. Freeman, and A. Zunger, Nature Physics 10, 387 (2014).
* Zhang _et al._ (2018b) Y. Zhang, S. Okamoto, and D. Xiao, Physical Review B 98, 035424 (2018b).
* Zelezný (2017) J. Zelezný, “symcode,” (2017).
* (39) “Supplemental Material,” .
* Freimuth _et al._ (2014) F. Freimuth, S. Blügel, and Y. Mokrousov, Physical Review B 90, 174423 (2014).
* Kurz _et al._ (2001) P. Kurz, G. Bihlmayer, K. Hirai, and S. Blügel, Physical Review Letters 86, 1106 (2001).
* Kato _et al._ (2010) Y. Kato, I. Martin, and C. D. Batista, Physical Review Letters 105, 266405 (2010).
* Endoh and Ishikawa (1971) Y. Endoh and Y. Ishikawa, Journal of the Physical Society of Japan 30, 1614 (1971).
* Gardner _et al._ (2010) J. S. Gardner, M. J. Gingras, and J. E. Greedan, Reviews of Modern Physics 82, 53 (2010), arXiv:0906.3661 .
* Manchon _et al._ (2015) A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, Nature Materials 14, 871 (2015).
* Tomiyoshi (1982) S. Tomiyoshi, Journal of the Physical Society of Japan 51, 803 (1982).
* Nakatsuji _et al._ (2015) S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 527, 212 (2015).
* Tsai _et al._ (2020) H. Tsai, T. Higo, K. Kondou, T. Nomoto, A. Sakai, A. Kobayashi, T. Nakano, K. Yakushiji, R. Arita, S. Miwa, Y. C. Otani, and S. Nakatsuji, Nature 580, 608 (2020).
* Krishnaswamy _et al._ (2022) G. K. Krishnaswamy, G. Sala, B. Jacot, C.-H. Lambert, R. Schlitz, M. D. Rossell, P. Nöel, and P. Gambardella, Physical Review Applied 18, 024064 (2022).
* Pal _et al._ (2022) B. Pal, B. K. Hazra, B. Göbel, J.-C. Jeon, A. K. Pandeya, A. Chakraborty, O. Busch, A. K. Srivastava, H. Deniz, J. M. Taylor, H. Meyerheim, I. Mertig, S.-H. Yang, and S. S. P. Parkin, Science Advances 8, eabo5930 (2022).
* Xie _et al._ (2022) H. Xie, X. Chen, Q. Zhang, Z. Mu, X. Zhang, B. Yan, and Y. Wu, Nature Communications 13, 5744 (2022).
* Disseler _et al._ (2015) S. M. Disseler, J. A. Borchers, C. M. Brooks, J. A. Mundy, J. A. Moyer, D. A. Hillsberry, E. L. Thies, D. A. Tenne, J. Heron, M. E. Holtz, J. D. Clarkson, G. M. Stiehl, P. Schiffer, D. A. Muller, D. G. Schlom, and W. D. Ratcliff, Phys. Rev. Lett. 114, 217602 (2015).
* Suresh _et al._ (2018) P. Suresh, K. Vijaya Laxmi, A. K. Bera, S. M. Yusuf, B. L. Chittari, J. Jung, and P. S. Anil Kumar, Phys. Rev. B 97, 184419 (2018).
* Das _et al._ (2014) H. Das, A. L. Wysocki, Y. Geng, W. Wu, and C. J. Fennie, Nature Communications 5, 2998 (2014).
* Tang and Cheng (2023) J. Tang and R. Cheng, arXiv preprint arXiv:2303.06181 (2023).
* Qi _et al._ (2008) X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Physical Review B 78, 195424 (2008).
* Chen _et al._ (2014) H. Chen, Q. Niu, and A. H. Macdonald, Physical Review Letters 112, 017205 (2014).
* Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).
* Pizzi _et al._ (2020) G. Pizzi, V. Vitale, R. Arita, S. Blügel, F. Freimuth, G. Géranton, M. Gibertini, D. Gresch, C. Johnson, T. Koretsune, J. Ibanez-Azpiroz, H. Lee, J. M. Lihm, D. Marchand, A. Marrazzo, Y. Mokrousov, J. I. Mustafa, Y. Nohara, Y. Nomura, L. Paulatto, S. Poncé, T. Ponweiser, J. Qiao, F. Thöle, S. S. Tsirkin, M. Wierzbowska, N. Marzari, D. Vanderbilt, I. Souza, A. A. Mostofi, and J. R. Yates, Journal of Physics Condensed Matter 32, 165902 (2020), arXiv:1907.09788 .
* (60) J. Železný, https://bitbucket.org/zeleznyj/wannier-linear-response.
|
# On the Ehrhart Theory of Generalized Symmetric Edge Polytopes
Robert Davis Department of Mathematics, Colgate University, Hamilton, NY, USA
<EMAIL_ADDRESS>, Akihiro Higashitani Department of Pure and Applied
Mathematics, Graduate School of Information Science and Technology, Osaka
University, Osaka, Japan<EMAIL_ADDRESS>and Hidefumi Ohsugi
Department of Mathematical Sciences, School of Science, Kwansei Gakuin
University, Sanda, Hyogo 669-1337, Japan<EMAIL_ADDRESS>
###### Abstract.
The symmetric edge polytope (SEP) of a (finite, undirected) graph is a
centrally symmetric lattice polytope whose vertices are defined by the edges
of the graph. SEPs have been studied extensively in the past twenty years.
Recently, Tóthmérész and, independently, D’Alí, Juhnke-Kubitzke, and Koch
generalized the definition of an SEP to regular matroids, as these are the
matroids that can be characterized by totally unimodular matrices. Generalized
SEPs are known to have symmetric Ehrhart $h^{*}$-polynomials, and Ohsugi and
Tsuchiya conjectured that (ordinary) SEPs have nonnegative $\gamma$-vectors.
In this article, we use combinatorial and Gröbner basis techniques to extend
additional known properties of SEPs to generalized SEPs. Along the way, we
show that generalized SEPs are not necessarily $\gamma$-nonnegative by
providing explicit examples. We prove these polytopes to be “nearly”
$\gamma$-nonnegative in the sense that, by deleting exactly two elements from
the matroid, one obtains SEPs for graphs that are $\gamma$-nonnegative. This
provides further evidence that Ohsugi and Tsuchiya’s conjecture holds in the
ordinary case.
## 1\. Introduction
A _lattice polytope_ $P$ in $\mathbb{R}^{n}$ is the convex hull of finitely
many points in $\mathbb{Z}^{n}$. That is, $P$ is a lattice polytope if and
only if there are some $v_{1},\dots,v_{k}\in\mathbb{Z}^{n}$ such that
$P=\operatorname{conv}\\{v_{1},\dots,v_{k}\\}=\left\\{\sum_{i=1}^{k}\lambda_{i}v_{i}\mid\lambda_{1},\dots,\lambda_{k}\geq
0,\,\sum_{j=1}^{k}\lambda_{j}=1\right\\}.$
The main objects of study in this paper will be lattice polytopes arising from
(finite, undirected) graphs and matroids. First, given an undirected graph
$G=([n],E)$ where $[n]=\\{1,\dots,n\\}$, the _symmetric edge polytope_ (or
_SEP_) associated to $G$ is
$\Sigma(G)=\operatorname{conv}\\{\pm(e_{i}-e_{j})\in\mathbb{R}^{n}\mid ij\in
E\\}.$
This polytope was introduced in [11] and has been the object of much study for
its algebraic and combinatorial properties [6, 9, 12, 13], as well as its
applications to problems arising from engineering [3, 4, 5].
SEPs have recently been generalized to regular matroids in two ways. Suppose
$A$ is a totally unimodular matrix representing a matroid $\mathcal{M}$.
Tóthmérész [17], working within the context of oriented matroids, defined the
_root polytope_ of $\mathcal{M}$ to be the convex hull of the columns of $A$.
Independently, D’Alì, Juhnke-Kubitzke, and Koch [7] defined the _generalized
symmetric edge polytope_ (or _generalized SEP_) associated to $\mathcal{M}$ as
$\Sigma(\mathcal{M})=\operatorname{conv}\\{\pm u\mid u\text{ is a column of
}A\\}.$
Thus, $\Sigma(\mathcal{M})$ may be considered the root polytope of the matroid
represented by the matrix $[A\mid-A]$, since this is a totally unimodular
matrix. Because we will not be working with oriented matroids, and because are
focused on the polytopes $\Sigma(\mathcal{M})$ in this work, we will follow
the terminology and notation in [7].
Although the definition of $\Sigma(\mathcal{M})$ depends on choice of $A$, the
polytope is well-defined up to unimodular equivalence ([17, Proposition 3.2]
and [7, Theorem 3.2]). In [7], the authors proved that many properties held by
ordinary SEPs also hold for generalized SEPs. Further, this more general
setting allowed the authors to show that two SEPs are unimodularly equivalent
if and only if the underlying matroids are isomorphic [7, Theorem 4.6]. In
addition to these aspects of SEPs, much attention has been given to their
Ehrhart theory, which we briefly introduce here.
The function $m\mapsto|mP\cap\mathbb{Z}^{n}|$, defined on the positive
integers, agrees with a polynomial $L_{P}(m)$ of degree $\dim(P)$, called the
_Ehrhart polynomial_ of $P$. Consequently, the _Ehrhart series_ of $P$, which
is the (ordinary) generating function for the sequence $\\{L_{P}(m)\\}_{m\geq
0}$, may be written as the rational function
$E(P;t)=\sum_{m\geq
0}L_{P}(m)t^{m}=\frac{h_{0}^{*}+h_{1}^{*}t+\cdots+h_{d}^{*}t^{d}}{(1-t)^{\dim(P)+1}}$
for some $d\leq\dim(P)$. We denote the numerator by $h^{*}(P;t)$ and refer to
it as the _$h^{*}$ -polynomial_ of $P$. The _$h^{*}$ -vector_ of $P$,
$h^{*}(P)$, is the list of coefficients $h^{*}(P)=(h^{*}_{0},\dots,h^{*}_{d})$
of $h^{*}(P;t)$.
The $h^{*}$-vectors of polytopes have been the focus of much study in recent
decades, as it lies within the intersection of polyhedral geometry,
commutative algebra, algebraic geometry, combinatorics, and much more (see,
e.g., [15, 16]). For example, when the $h^{*}$-vector is _symmetric_ , meaning
$h^{*}_{i}=h^{*}_{d-i}$ for each $i=0,\dots,d$, then the associated semigroup
algebra
$K[P]=K[x^{v}z^{m}\mid v\in mP\cap\mathbb{Z}^{n}]\subseteq
K[x_{1}^{\pm},\dots,x_{d}^{\pm},z],$
over a field $K$, is Gorenstein. In fact, the converse holds as well.
Determining when $K[P]$ is Gorenstein can be detected entirely using
polyhedral geometry, so, from this perspective, there is a complete
characterization of when $h^{*}(P)$ is symmetric.
On the other hand, a more mysterious property held by some $h^{*}$-vectors is
unimodality. We call $h^{*}(P)$ _unimodal_ if there is some index $0\leq r\leq
d$ for which $h^{*}_{0}\leq\cdots\leq h^{*}_{r}$ and $h^{*}_{r}\geq\cdots\geq
h^{*}_{d}$. Unlike symmetry, there is no complete characterization of when an
$h^{*}$-vector is unimodal. There are various sets of sufficient conditions
that ensure a polytope has a unimodal $h^{*}$-vector and various sets of
necessary conditions [2], but no set of conditions capturing both
simultaneously.
When an $h^{*}$-vector is already known to be symmetric, new avenues for
proving $h^{*}$-unimodality appear. For instance, in this setting, the
$h^{*}$-polynomial may be expressed
$\sum_{i=0}^{d}h^{*}_{i}t^{i}=\sum_{i=0}^{\lfloor
d/2\rfloor}\gamma_{i}t^{i}(1+t)^{d-2i}$
for certain choices of $\gamma_{i}\in\mathbb{Q}$. The polynomial on the right
is the _$\gamma$ -polynomial_ of $P$ and is often denoted $\gamma(P;t)$, and
the vector $\gamma(P)=(\gamma_{0},\dots,\gamma_{\lfloor d/2\rfloor})$ is the
_$\gamma$ -vector_ of $P$. Note that if $\gamma_{i}\geq 0$ for each $i$, then
$h^{*}_{i}\geq 0$ for each $i$ as well. This is not necessary, though: the
$h^{*}$-vector of $(1,1,1)$, realized by the lattice triangle with vertices
$(1,0)$, $(0,1)$, and $(-1,-1)$, is unimodal but has a $\gamma$-vector of
$(1,-1)$. Nevertheless, $\gamma$-vectors remain the topic of intense study,
due in part to wide-open conjectures surrounding their nonnegativity. The most
famous of these is _Gal’s Conjecture_ : that the $h$-vector of a flag homology
sphere is $\gamma$-nonnegative. To an Ehrhart theorist, the conjecture states
that the $h^{*}$-polynomial of a lattice polytope with a regular, unimodular,
flag triangulation is $\gamma$-nonnegative.
In this article, we use combinatorial and Gröbner basis techniques to extend
additional known properties of SEPs to generalized SEPs. Along the way, we
show that generalized SEPs are not $\gamma$-nonnegative by providing an
infinite class of explicit examples (Theorem 4.2). Computational evidence
strongly suggests that these examples are “nearly” $\gamma$-nonnegative in the
sense that, by deleting exactly two elements from the matroid, we obtain SEPs
for graphs that appear to be $\gamma$-nonnegative. We present and prove a
formula for their $h^{*}$-polynomials (Theorem 4.3), deduce the normalized
volumes of these SEPs (Corollary 4.6), and compute their $\gamma$-polynomials
(Theorem 4.4). This provides further evidence that Ohsugi and Tsuchiya’s
conjecture holds in the ordinary case.
A brief structure of the article is as follows. In Section 2 we discuss the
necessary background for matroids and the relationships among Ehrhart theory,
triangulations of polytopes, and toric ideals. Section 3 discusses how various
important results for ordinary SEPs extend to generalized SEPs. In Section 4
we present explicit examples of generalized SEPs whose $h^{*}$-polynomials are
not $\gamma$-nonnegative (Theorem 4.2) and examine the consequences of making
minor variations to them. We also state two of the main results of this
article: Theorem 4.3 and Theorem 4.4. In Section 5 we both prepare for and
present the full proof of Theorem 4.3, which follows a combinatorial argument.
Finally, in Section 6, we use a generating function argument to prove Theorem
4.4.
## 2\. Background
To keep this article as self-contained as possible, this section provides the
necessary background on matroids, Ehrhart theory, and generalized SEPs. For a
comprehensive reference on the first two topics, we refer the reader to [14]
for matroids and [1] for Ehrhart theory. Throughout this article we use the
convention of writing $A\cup x$ and $A-x$ for the union or deletion,
respectively, of $A$ by a one-element set $\\{x\\}$.
### 2.1. Matroids
A _matroid_ $\mathcal{M}$ consists of a pair $(E,\mathcal{I})$ where $E$ is
finite and $\mathcal{I}\subseteq 2^{E}$ satisfies the following three
properties:
1. (i)
$\emptyset\in\mathcal{I}$;
2. (ii)
if $I\in\mathcal{I}$ and $J\subseteq I$, then $J\in\mathcal{I}$; and
3. (iii)
if $I,J\in\mathcal{I}$ and $|I|<|J|$, then there is some $x\in J-I$ such that
$I\cup x\in\mathcal{I}$.
The set $E$ is called the _ground set_ of $\mathcal{M}$ and the elements of
$\mathcal{I}$ are called the _independent sets_ of $\mathcal{M}$.
Matroids are useful for unifying various notions of “independence” in
mathematics. For example, if $G$ is a (finite) graph, then there is a
corresponding matroid $M(G)$ whose ground set is $E(G)$, the edge set of $G$,
and whose independent sets are the sets of edges that form acyclic subgraphs
of $G$. This matroid is called the _cycle matroid_ of $G$, and any matroid
that is isomorphic to $M(G)$ for some $G$ is a _graphic matroid_.
Another fundamental example of a matroid uses the columns of a $k\times n$
matrix $A$ as the ground set. Here, the independent sets are the columns of
$A$ that form linearly independent sets. This matroid, denoted $M(A)$, is the
_vector matroid_ of $A$, and any matroid isomorphic to $M(A)$ for some matrix
$A$ is called a _representable matroid_. More precisely, for a field $K$, the
matrix $A$ is called _$K$ -representable_ if $M(A)$ is isomorphic to a vector
matroid whose ground set consists of vectors in $K^{k}$. Hence, to say a
matroid is representable is to say it is $K$-representable for some field $K$.
On the other hand, a matroid is _regular_ if it is $K$-representable over
every field $K$.
The _rank_ of a matroid $\mathcal{M}=(E,\mathcal{I})$ is defined as
$\operatorname{rank}(\mathcal{M})=\max_{I\in\mathcal{I}}|I|.$
When $\mathcal{M}$ is graphic, $\operatorname{rank}(\mathcal{M})$ is the size
of a maximal spanning forest for any $G$ satisfying $\mathcal{M}=M(G)$. When
$\mathcal{M}$ is representable,
$\operatorname{rank}(\mathcal{M})=\operatorname{rank}(A)$ for any matrix $A$
satisfying $\mathcal{M}=M(A)$. With these definitions in hand, we can now
describe some of the many benefits of working with regular matroids.
###### Theorem 2.1 ([7, Definition/Theorem 2.3]).
A positive-rank matroid $\mathcal{M}$ is regular if and only if any of the
following hold:
1. (i)
$\mathcal{M}$ is representable over every field.
2. (ii)
$\mathcal{M}=M(A)$ for some matrix $A$ in which every minor is $\pm 1$ or $0$.
3. (iii)
$\mathcal{M}=M(A)$ for some full-rank matrix $A$ in which every maximal minor
is $\pm 1$ or $0$.
Note that a matrix satisfying the minor condition in (ii) is called _totally
unimodular_ and a matrix satisfying the minor condition in (iii) is called
_weakly unimodular_. Borrowing more terminology from linear algebra, an
element of $\mathcal{I}$ of cardinality $\operatorname{rank}(\mathcal{M})$ is
called a _basis_ of $\mathcal{M}$. The set
$\mathcal{B}=\mathcal{B}(\mathcal{M})$ of bases, together with the ground set
set $E$, can be used to define a matroid without explicit mention of
independent sets; when this is done, we usually write
$\mathcal{M}=(E,\mathcal{B})$.
Now, from a matroid $\mathcal{M}=(E,\mathcal{B})$ we may produce its _dual_
matroid $\mathcal{M}^{*}=(E,\mathcal{B}^{*})$ by setting
$\mathcal{B}^{*}=\\{B-E\subseteq E\mid B\in\mathcal{B}\\}.$
Many properties of $\mathcal{M}^{*}$ can be easily deduced from $\mathcal{M}$.
For example, $\mathcal{M}^{*}$ is representable if and only if $\mathcal{M}$
is representable. This does not hold for graphic matroids, though: if
$\mathcal{M}$ is graphic, then $\mathcal{M}^{*}$ is graphic if and only if
$\mathcal{M}=M(G)$ for some planar graph $G$.
There are additional notions in graph theory that have corresponding matroidal
analogues. For example, a _circuit_ in a matroid is a minimal dependent set;
this is the analogue of a cycle in a graph. Letting
$\mathcal{C}=\mathcal{C}(\mathcal{M})$ denote the set of circuits of a
matroid, we define the matroid to be _bipartite_ if each of the circuits has
even size. This further allows one to call a matroid _Eulerian_ if it is the
dual of a bipartite matroid. These definitions reflect the fact that a
connected, planar graph is Eulerian if and only if its dual is bipartite.
A _loop_ of $\mathcal{M}$ is a single element of $E$ that forms a circuit;
equivalently, a loop is an element that is in no basis of $\mathcal{M}$. On
the other hand, a _coloop_ of $\mathcal{M}$, called by some authors an
_isthmus_ , is a single element of $E$ that is in every basis of
$\mathcal{M}$. Now, given a matroid $\mathcal{M}=(E,\mathcal{I})$ and an
element $e\in E$ that is not a coloop, the _deletion_ of $e$ from
$\mathcal{M}$ is the matroid $\mathcal{M}-e$ having ground set $E-e$ and
independent sets $\\{I\in\mathcal{I}\mid e\notin I\\}$. The _contraction_ of
$\mathcal{M}$ by $e\in E$, where $e$ is not a loop, is denoted $\mathcal{M}/e$
and has ground set $E-e$ and independent sets $\\{I-e\mid e\in
I\in\mathcal{I}\\}$. Deletion and contraction are dual operations: if $e$ is
neither a loop nor a coloop of $\mathcal{M}$, then
$(\mathcal{M}-e)^{*}=\mathcal{M}^{*}/e$. Many binary operations on graphs
extend to matroids as well: the _direct sum_ , or _1-sum_ , of matroids
$\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ having disjoint ground sets $E_{1}$
and $E_{2}$ is $\mathcal{M}_{1}\oplus\mathcal{M}_{2}=(E_{1}\cup
E_{2},\mathcal{I}_{3})$, where $\mathcal{I}_{3}$ consists of independent sets
of the form $I=I_{1}\cup I_{2}$ where $I_{1}\in\mathcal{I}_{1}$ and
$I_{2}\in\mathcal{I}_{2}$.
To define the next operation we assume that $E_{1}\cap E_{2}=\\{p\\}$ and call
this element the _basepoint_. The _parallel connection_ , denoted
$P(\mathcal{M}_{1},\mathcal{M}_{2})$, has ground set $E_{1}\cup E_{2}$ and is
most efficiently defined through its circuits: if $p$ is neither a loop nor a
coloop of $\mathcal{M}_{1}$, then
$\mathcal{C}(P(\mathcal{M}_{1},\mathcal{M}_{2}))=\,\mathcal{C}(\mathcal{M}_{1})\cup\mathcal{C}(\mathcal{M}_{2})\cup\\{(C_{1}\cup
C_{2})-p\mid p\in C_{i}\in\mathcal{C}(\mathcal{M}_{i})\text{ for each }i\\}.$
If $p$ is a loop of $\mathcal{M}_{1}$, then we set
$P(\mathcal{M}_{1},\mathcal{M}_{2})=\mathcal{M}_{1}\oplus(\mathcal{M}_{2}/p).$
Similarly, if $p$ is a coloop of $\mathcal{M}_{1}$, then we set
$P(\mathcal{M}_{1},\mathcal{M}_{2})=(\mathcal{M}_{1}-p)\oplus\mathcal{M}_{2}.$
Of course, if $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ have disjoint ground
sets, then a point of each ground set may be selected and relabeled $p$ to
fall into these definition, although the resulting parallel connection may
then depend on which points were selected. An important property of this
operation is that it preserves regularity: if $\mathcal{M}_{1}$ and
$\mathcal{M}_{2}$ are both regular matroids, then so is
$P(\mathcal{M}_{1},\mathcal{M}_{2})$ [14, Corollary 7.1.25].
### 2.2. Dissecting Tree Sets and Toric Algebra
It is frequently more tractable to gather the desired information about a
polytope by decomposing it into subpolytopes. A _dissection_ , or
_subdivision_ , of a polytope $P$ is a collection $\mathcal{S}$ of full-
dimensional subpolytopes of $P$ whose union is $P$ and for any
$S_{1},S_{2}\in\mathcal{S}$, the intersection $S_{1}\cap S_{2}$ is a face of
both. A _triangulation_ of $P$ is a dissection in which every
$S\in\mathcal{S}$ is a simplex. There are many ways to create triangulations
of a polytope; we will be concerned with one that has been recently introduced
in the study of ordinary SEPs in particular.
Given an oriented subgraph $H$ of a graph $G$ on $[n]$, let
$Q_{H}=\operatorname{conv}\\{e_{i}-e_{j}\mid ij\in H\\}$. Thus, if $T$ is a
spanning tree of $G$, then $Q_{T}$ is a simplex contained in $\Sigma(G)$. In
fact, $Q_{T}$ is a full-dimensional simplex contained in the boundary of
$\Sigma(G)$. Further still, $Q_{T}$ is a _unimodular_ simplex: it has
normalized volume $1$. It follows that the simplex
$\operatorname{conv}\\{Q_{T}\cup\\{0\\}\\}$ is a full-dimensional, unimodular
simplex contained in $\Sigma(G)$. It is desirable, then, to find a collection
$\mathcal{T}$ of oriented spanning trees of $G$ so that the set $\\{Q_{T}\mid
T\in\mathcal{T}\\}$ forms a triangulation of the boundary of
$\operatorname{\Sigma}(G)$. Such a set $\mathcal{T}$ is called a _dissecting
tree set_ for $G$.
Of course, if $\mathcal{T}$ is a dissecting tree set for $G$, then the
normalized volume of $\Sigma(G)$ is exactly $|\mathcal{T}|$. However, there is
more information about $\Sigma(G)$ to be extracted from $\mathcal{T}$. To
describe this information, consider a fixed vertex $v$ in an oriented tree
$T$. For each edge $e\in T$ we say that _$e$ points away from $v$ in $T$_ if
$v$ is contained in the same component as the tail of $e$ in $T-e$. Otherwise,
we say _$e$ points towards $v$ in $T$_.
###### Theorem 2.2 ([10, Theorem 1.2]).
Let $G$ be a connected graph, let $\mathcal{T}$ be a dissecting tree set for
$G$, and let $v$ be any vertex of $G$. Then the $i^{th}$ entry in
$h^{*}(\Sigma(G))$ is
$|\\{T\in\mathcal{T}\mid T\text{ has exactly }i\text{ edges pointing away from
}v\\}|.$
In order to determine when we have a dissecting tree set, we turn to Gröbner
basis techniques. Here, we will introduce the necessary algebraic background
to prove our main results. We largely follow the framework given in [9]. For
more details about monomial orders and their relationships to triangulations
of polytopes, see [16, Chapter 8].
Recall that, given a lattice polytope $P\subseteq\mathbb{R}^{n}$ and a field
$K$, we may construct a semigroup algebra
$K[P]=K[x^{v}z^{m}\mid v\in mP\cap\mathbb{Z}^{n}]$
where $x^{v}=x_{1}^{v_{1}}\cdots x_{n}^{v_{n}}$. The _toric ideal_ of $P$ is
defined as the kernel of the map
$\pi_{P}:K[t_{v}\mid v\in P\cap\mathbb{Z}^{n}]\to K[P]$
defined by setting $\pi_{P}(t_{v})=x^{v}z$; this ideal is denoted $I_{P}$.
Consider a monomial order $\prec$ on $K[t_{v}\mid v\in P\cap\mathbb{Z}^{n}]$
and any ideal $I$ of this algebra. For each polynomial $f\in I$, its _initial_
term with respect to $\prec$, denoted $\operatorname{in}_{\prec}(f)$, is the
term of $f$ that is greatest with respect to $\prec$. The _initial ideal_ of
$I$ with respect to $\prec$ is
$\operatorname{in}_{\prec}(I)=(\operatorname{in}_{\prec}(f)\mid f\in I).$
A _Gröbner basis_ of $I$, which we denote by $\mathscr{G}$, is a finite
generating set of $I$ such that
$\operatorname{in}_{\prec}(I)=(\operatorname{in}_{\prec}(g)\ |\
g\in\mathscr{G})$. We call $\mathscr{G}$ _reduced_ if each element has a
leading coefficient of $1$ and if for any $g_{1},g_{2}\in\mathscr{G}$,
$\operatorname{in}_{\prec}(g_{1})$ does not divide any term of $g_{2}$. There
are many Gröbner bases of $I$ but there is exactly one reduced Gröbner basis
of $I$.
Now suppose $P\subseteq\mathbb{R}^{n}$ is an $n$-dimensional lattice polytope
and $P\cap\mathbb{Z}^{n}=\\{l_{1},\ldots,l_{s}\\}$. Choose a weight vector
$w=(w_{1},\ldots,w_{s})\in\mathbb{R}^{s}$ such that the polytope
$P_{w}:=\operatorname{conv}\\{(l_{1},w_{1}),\ldots,(l_{s},w_{s})\\}\subseteq\mathbb{R}^{n+1}$
is $(n+1)$-dimensional, i.e., $P_{w}$ does not lie in an affine hyperplane of
$\mathbb{R}^{n+1}$. Some facets of $P_{w}$ will have outward-pointing normal
vectors with a negative last coordinate. By projecting these facets back to
$\mathbb{R}^{n}$, we obtain the facets of a polyhedral subdivision
$\Delta_{w}(P)$ of $P$. Any triangulation that can be obtained in this way for
an appropriate choice of $w$ is called a _regular_ triangulation. Importantly,
regular unimodular triangulations are in one-to-one correspondence with
reduced Gröbner bases whose initial terms are all squarefree [16, Corollary
8.9]. It is well-known that any monomial order obtained as a graded reverse
lexicographic (grevlex) order can be represented by a weight vector.
In the case of $P=\Sigma(\mathcal{M})$ for a regular matroid
$\mathcal{M}=(E,\mathcal{I})$, we can identify $K[t_{v}\mid v\in
P\cap\mathbb{Z}^{n}]$ with
(2.1) $K\left[\\{z\\}\cup\bigcup_{e\in E}\\{x_{e},y_{e}\\}\right].$
Here, we let $x_{e}$ correspond to the column of a fixed matrix $M$ realizing
$\mathcal{M}$ that corresponds to $e$, and we will let $y_{e}$ correspond to
the negative of this column. The variable $z$, then, corresponds to the
origin. So, for each $e\in E$, we can treat $x_{e}$ and $y_{e}$ as encoding an
orientation of $e$. For ease of notation, if we are considering $e\in E$ with
a particular orientation, then we let $p_{e}$ denote the corresponding
variable and $q_{e}$ the variable corresponding to the opposite orientation.
###### Lemma 2.3 (cf. [9, Proposition 3.8]).
Let $\mathcal{M}$ be a regular matroid on the ground set $E$, let
$z<x_{e_{1}}<y_{e_{1}}<\cdots<x_{e_{n}}<y_{e_{n}}$ be an order on the lattice
points of $\Sigma(\mathcal{M})$, and let $\prec$ be the grevlex order on (2.1)
induced from this ordering on the variables. The collection of all binomials
of the following types forms a Gröbner basis of $I_{\Sigma(\mathcal{M})}$ with
respect to grevlex order:
1. (i)
For every $2k$-element circuit $C$ of $\mathcal{M}$ and choice of orientation
for each $e\in C$, and any $k$-element subset $I\subseteq C$ not containing
the weakest element among those of $C$,
$\prod_{e\in I}p_{e}-\prod_{e\in C-I}q_{e}.$
2. (ii)
For every $(2k+1)$-element circuit $C$ of $\mathcal{M}$ and choice of
orientation for each $e\in C$, and any $(k+1)$-element subset $I\subseteq C$,
$\prod_{e\in I}p_{e}-z\prod_{e\in C-I}q_{e}.$
3. (iii)
For any $e\in E$, $x_{e}y_{e}-z^{2}$.
In each case, the binomial is written so that the initial term has positive
sign.
###### Proof.
This proof follows the same approach as the proof to [9, Proposition 3.8].
First, recognize we need to show that for any binomial $m_{1}-m_{2}$ in
$I_{\Sigma(\mathcal{M})}$, one of $m_{1}$ or $m_{2}$ is divisible by the
leading monomial of one of the binomials described above.
Fix a totally unimodular matrix $A$ representing $\mathcal{M}$, and label its
columns $a_{1},\dots,a_{n}$. Both monomials $m_{1}$ and $m_{2}$ correspond to
a matrix so that $a_{i}$ is a column of the matrix if and only if $x_{a_{i}}$
is present in the monomial, and $-a_{i}$ is a column of the matrix if and only
if $y_{a_{i}}$ is present in the monomial. Let $A_{1}$ (resp. $A_{2}$) be the
matrix corresponding to $m_{1}$ (resp. $m_{2}$). Of course, we may assume that
neither matrix of $A_{1}$ and $A_{2}$ has both $a_{i}$ and $-a_{i}$ for any
$i$, since then the monomial would be divisible by $x_{a_{i}}y_{a_{i}}$.
Consider $-A_{2}$. Since $m_{1}-m_{2}$ is contained in the toric ideal of
$I_{\Sigma(\mathcal{M})}$, there is some minimal linearly dependent set $C$ of
columns in the matrix $[A_{1}\mid-A_{2}]$. Denote by $C_{1}$ the set of
vectors in $C$ that are columns in $A_{1}$, and let $c_{1}=|C_{1}|$.
Similarly, denote by $C_{2}$ the set of vectors in $C$ that are columns in
$-A_{2}$, and let $c_{2}=|C_{2}|$. We may assume $c_{1}\geq c_{2}$.
First, suppose that $c_{1}+c_{2}=2k$ for some $k>1$. If $c_{1}>c_{2}$,
consider the product of the $k$ largest variables, with respect to $\prec$,
among those corresponding to vectors in $C_{1}$. The leading term of the
corresponding binomial in (i) then divides $m_{1}$. If $c_{1}=c_{2}$, we may
assume without loss of generality that the smallest variable corresponding to
a vector in $C$ belongs to $A_{2}$ and proceed as before. If
$c_{1}+c_{2}=2k+1$, then since $a>b$, the leading term of the binomial in (ii)
corresponding to $(k+1)$ vectors in $A_{1}$ divides $m_{1}$. ∎
## 3\. Extensions of known properties of SEPs
Before providing the extensions of known properties of ordinary SEPs to
generalized SEPs, we recall the formula of the dimensions of them.
###### Proposition 3.1 (cf. [7, Theorem 4.3]).
Let $\mathcal{M}$ be a regular matroid on the ground set $E$. Then
$\dim\Sigma(\mathcal{M})=\operatorname{rank}(\mathcal{M})$. In particular,
given a connected graph $G$ with $n$ vertices, we have $\dim\Sigma(G)=n-1$.
Moreover, we have
$\dim\Sigma(\mathcal{M}^{*})=|E|-\operatorname{rank}(\mathcal{M})$.
First, we extend [13, Proposition 4.4], using the same overall strategy used
in its proof.
###### Proposition 3.2 (cf. [13, Proposition 4.4]).
Let $\mathcal{M}$ be a bipartite regular matroid on $E$. For any $e\in E$,
$h^{*}(\operatorname{\Sigma}(\mathcal{M});t)=(1+t)h^{*}(\operatorname{\Sigma}(\mathcal{M}/e);t).$
###### Proof.
Let $A$ be a full-rank, totally unimodular matrix of rank $r$ representing
$\mathcal{M}$, and denote its columns by $a_{1},\dots,a_{n}$ so that
$e=a_{1}$. Also, let $A_{e}$ be the matrix representing $\mathcal{M}/e$. Since
$\mathcal{M}$ is bipartite, the Gröbner basis $\mathscr{G}$ from Lemma 2.3
consists of the binomials of the form (i) and (iii). Let $A^{\prime}$ be the
rank-$(r+1)$ matrix
$A^{\prime}=\left[\begin{array}[]{@{}c|c@{}}A_{e}&\begin{matrix}0\\\ \vdots\\\
0\end{matrix}\\\
\hline\cr\begin{matrix}0&\cdots&0\end{matrix}&1\end{array}\right]$
and let $\mathcal{M}^{\prime}$ be the matroid represented by $A^{\prime}$. The
Gröbner basis of $\Sigma(\mathcal{M}^{\prime})$ consists of all binomials $b$
satisfying one of the following:
$b=\prod_{a_{i}\in I}x_{a_{i}}-\prod_{a_{i}\in C-I}y_{a_{i}},$
where $C$ is a set of columns of $A$ corresponding to a circuit in
$\mathcal{M}$ of size $2k$ and $a_{1}\notin C$, and $I$ is a $k$-subset of $C$
such that $a_{j}\notin I$ where $j=\min\\{i\mid a_{i}\in C\\}$;
$b=\prod_{a_{i}\in I}x_{a_{i}}-z\prod_{a_{i}\in C-I}y_{a_{i}},$
where $C\cup a_{1}$ corresponds to a circuit in $\mathcal{M}$ of size $2k+2$
and $I$ is a $(k+1)$-subset of $C$;
$b=x_{a_{i}}y_{a_{i}}-z^{2}$
for $1\leq i\leq k$. Thus, the initial terms in the Gröbner bases for
$I_{\Sigma(\mathcal{M})}$ and $I_{\Sigma(\mathcal{M}/e)}$ are the same.
Following the proof strategy of [8, Theorem 3.1], it follows that
$h^{*}(\Sigma(\mathcal{M});t)=h^{*}(\Sigma(\mathcal{M}^{\prime});t)=h^{*}(\Sigma(e_{r});t)h^{*}(\Sigma(\mathcal{M});t)=(1+t)h^{*}(\Sigma(\mathcal{M});t),$
where $e_{r}$ is a standard basis vector, as claimed. ∎
Next we present an extension of [6, Theorem 44]. Its proof is entirely
analogous to the proof of [6, Theorem 44], in the same way that our proof of
Lemma 2.3 adapts the proof [9, Proposition 3.8], and our proof of Proposition
3.2 adapts the proof of [13, Proposition 4.4]. So, due to the length of the
proof and the similarity in how we adapt it from [6, Theorem 44], we omit its
details.
###### Theorem 3.3 (cf. [6, Theorem 44]).
Let $\mathcal{M}$ be a regular matroid and let $\mathcal{M}^{\prime}$ be a
bipartite regular matroid. Then
$h^{*}(\operatorname{\Sigma}(P(\mathcal{M}_{1},\mathcal{M}_{2}));t)=\frac{h^{*}(\operatorname{\Sigma}(\mathcal{M}_{1});t)h^{*}(\operatorname{\Sigma}(\mathcal{M}_{2});t)}{1-t}.$
## 4\. A counterexample of $\gamma$-nonnegativity for generalized SEPs
In [13], the authors conjectured that SEPs are $\gamma$-nonnegative. It is
natural to want to extend this conjecture to generalized SEPs, but the
conjecture in this setting is false. Throughout the remainder of this section,
we will be considering the matroids $M^{*}(K_{3,n})$ for $n\geq 3$.
###### Proposition 4.1.
Let $\mathcal{M}=M^{*}(K_{3,n})$ with $n\geq 3$ and let $E$ be the ground set
of $\mathcal{M}$.
1. (i)
$\mathcal{M}$ can be represented by $[I_{2(n-1)}\mid B_{n}]$, where
$B_{n}=\begin{bmatrix}1&1&0&-1&0&0&\cdots&0\\\ 1&0&1&-1&0&0&\cdots&0\\\
1&1&0&0&-1&0&\cdots&0\\\ 1&0&1&0&-1&0&\cdots&0\\\ 1&1&0&0&0&-1&\cdots&0\\\
1&0&1&0&0&-1&\cdots&0\\\
\vdots&\vdots&\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\\
1&1&0&0&0&0&\cdots&-1\\\ 1&0&1&0&0&0&\cdots&-1\\\ \end{bmatrix}.$
2. (ii)
$\mathcal{M}-e$ is not graphic for any $e\in E$ if $n\geq 4$.
###### Proof.
1. (i)
It is known that if a matroid $\mathcal{M}$ is represented by a matrix
$A=[I\mid D]$, then $\mathcal{M}^{*}$ is represented by $[I\mid-D^{T}]$, where
$\cdot^{T}$ denotes the transpose ([14, Theorem 2.2.8]). Hence, we prove that
the matrix $[I\mid D]$ represents $M(K_{3,n})$ where $D=-B_{n}^{T}$.
Let $V(K_{3,n})=\\{u_{1},u_{2},u_{3}\\}\cup\\{v_{1},\ldots,v_{n}\\}$ and
$E(K_{3,n})=\\{u_{i}v_{j}\mid 1\leq i\leq 3,1\leq j\leq n\\}$. Fix a spanning
tree consisting of the following edges:
$e_{1}=u_{1}v_{1},\quad e_{2}=v_{1}u_{2},\quad e_{3}=v_{1}u_{3},\quad
e_{4}=u_{1}v_{2},\,\ldots\,,e_{n+2}=u_{1}v_{n}.$
This corresponds to the identity matrix in the representing matrix $[I\mid
D^{\prime}]$ of $M(K_{3,n})$. Our goal is to show that $D^{\prime}=-D^{T}$. By
assigning the orientation of each of these edges as described (i.e., from the
first vertex to the second vertex), for the remaining edges of $K_{3,n}$, we
see that $u_{2}v_{j}$ (resp. $u_{3}v_{j}$) with $2\leq j\leq n$ form minimal
dependent sets with the opposite of $e_{2}$, the opposite of $e_{1}$, and
$e_{j+2}$. This dependent set corresponds to the $(2j-1)$-th (resp. $2j$-th)
column of $-D^{T}$, as claimed.
2. (ii)
It is known that given a matroid $\mathcal{M}$ and an element $e$ of its
ground set, we have $\mathcal{M}^{*}-e=(\mathcal{M}/e)^{*}$ (see [14, 3.1.1]).
Hence, it is enough to show that $K_{3,n}/e$ never becomes planar for any edge
of $K_{3,n}$ This is easy to see: the edge set of $K_{3,n}/u_{1}v_{1}$ is
$\\{u_{i}v_{j}\mid i=2,3,\,1\leq j\leq n\\}\cup\\{v_{1}v_{k}\mid 2\leq k\leq
n\\}$, which contains $K_{3,n-1}$ as a subgraph.
∎
###### Theorem 4.2.
In every dimension at least $10$ there is a generalized SEP that is not
$\gamma$-nonnegative.
###### Proof.
Consider the matroid $\mathcal{M}=M^{*}(K_{3,6})$. Since $K_{3,6}$ has $18$
edges, we know that $\dim\Sigma(M^{*}(K_{3,6}))=18-(9-1)=10$. From its matrix
description, it is possible to show that
$h^{*}(\Sigma(\mathcal{M}))=(1,26,297,1908,6264,9108,6264,1908,297,26,1),$
hence $\gamma(\Sigma(\mathcal{M}))=(1,16,124,596,914,-148)$.
Now let $\mathcal{M}_{k}=\mathcal{M}\oplus x_{1}\oplus\cdots\oplus x_{k}$
where the $x_{i}$ are pairwise distinct elements, none of which is already in
$\mathcal{M}$. The generalized SEP $\Sigma(\mathcal{M}_{k})$ is the $k$-fold
direct sum of $\Sigma(\mathcal{M})$ with copies of the interval $[-1,1]$.
Consequently, $\dim(\mathcal{M}_{k})=10+k$,
$h^{*}(\Sigma(\mathcal{M}_{k});t)=h^{*}(\mathcal{M};t)(1+t)^{k},$
from which it follows that
$\gamma(\Sigma(\mathcal{M}_{k}))=\gamma(\Sigma(\mathcal{M}))$. ∎
Computational evidence suggests that the $\gamma$-vectors for
$\Sigma(M^{*}(K_{3,n}))$ fail to be nonnegative for all even $n\geq 6$,
although we do not aim to prove so here. Instead, we examine how these
matroids are situated between ordinary SEPs and generalized SEPs: by deleting
exactly two elements from $M^{*}(K_{3,n})$, the resulting matroid is now
graphic. In fact, by invoking Whitney’s $2$-isomorphism theorem [14, Theorem
5.3.1], it is a brief exercise to show that there is, up to isomorphism, a
unique graph $\Gamma(n)$ such that
$M^{*}(K_{3,n})-\\{e,e^{\prime}\\}=M(\Gamma(n))$. We will be more precise
about what this graph looks like in Section 5. For now, we simply state our
main results regarding $\Sigma(\Gamma(n))$.
###### Theorem 4.3.
For each $n\geq 1$,
(4.1)
$\begin{split}h^{*}(\Sigma(\Gamma(n+1));t)&=\sum_{k=0}^{n}\binom{n}{k}(2t)^{n-k}\sum_{(p,q)\in
S_{k}}\binom{2\lfloor\frac{k}{2}\rfloor+1}{p+q+1}f_{p,q,k}(t)\\\
&=\sum_{\ell=0}^{\lfloor
n/2\rfloor}(2t)^{n-2\ell-1}\left(2t\binom{n}{2\ell}+(1+t)^{2}\binom{n}{2\ell+1}\right)\sum_{(p,q)\in
S_{2\ell}}\binom{2\ell+1}{p+q+1}f_{p,q,2\ell}(t),\end{split}$
where
$\displaystyle S_{k}$ $\displaystyle=\\{(x,y)\in\mathbb{Z}:x\geq 0,y\geq
0,x+y\leq k\\},$ $\displaystyle f_{p,q,2\ell}(t)$ $\displaystyle=t^{2\ell-
p-q}\sum_{i=0}^{p}\sum_{j=0}^{q}\binom{p}{i}\binom{q}{j}\binom{2\ell-
p-q}{\ell-q-i+j}t^{2(i+j)},\text{ and }$ $\displaystyle f_{p,q,2\ell+1}(t)$
$\displaystyle=(1+t)^{2}f_{p,q,2\ell}(t).$
For example, the following are the computations of $h^{*}(\Gamma(n+1);t)$ for
some small values of $n$:
$\displaystyle h^{*}(\Sigma(\Gamma(3));t)$ $\displaystyle=$
$\displaystyle\binom{2}{0}(2t)^{2}\binom{1}{1}f_{0,0,0}(t)+\binom{2}{1}(2t)^{1}(1+t)^{2}\left(\binom{1}{1}f_{0,0,0}(t)+\binom{1}{2}(f_{1,0,0}(t)+f_{0,1,0}(t))\right)$
$\displaystyle+$
$\displaystyle\binom{2}{2}(2t)^{0}\left(\binom{3}{1}f_{0,0,2}(t)+\binom{3}{2}(f_{1,0,2}(t)+f_{0,1,2}(t))+\binom{3}{3}(f_{1,1,2}(t)+f_{2,0,2}(t)+f_{0,2,2}(t))\right),$
$\displaystyle h^{*}(\Sigma(\Gamma(4));t)$ $\displaystyle=$
$\displaystyle\binom{3}{0}(2t)^{3}\binom{1}{1}f_{0,0,0}(t)+\binom{3}{1}(2t)^{2}(1+t)^{2}\left(\binom{1}{1}f_{0,0,0}(t)+\binom{1}{2}(f_{1,0,0}(t)+f_{0,1,0}(t))\right)$
$\displaystyle+$
$\displaystyle\binom{3}{2}(2t)^{1}\left(\binom{3}{1}f_{0,0,2}(t)+\binom{3}{2}(f_{1,0,2}(t)+f_{0,1,2}(t))+\binom{3}{3}(f_{1,1,2}(t)+f_{2,0,2}(t)+f_{0,2,2}(t))\right)$
$\displaystyle+$
$\displaystyle\binom{3}{3}(2t)^{0}(1+t)^{2}\left(\binom{3}{1}f_{0,0,2}(t)+\binom{3}{2}(f_{1,0,2}(t)+f_{0,1,2}(t))+\binom{3}{3}(f_{1,1,2}(t)+f_{2,0,2}(t)+f_{0,2,2}(t))\right).$
Our second main result regarding $\Gamma(n)$ is the following. For
readability, we defer its proof to Section 6.
###### Theorem 4.4.
For each $n\geq 1$,
$\displaystyle\gamma(\Gamma(n+1);t)$ $\displaystyle=\sum_{\ell=0}^{\lfloor
n/2\rfloor}(2t)^{n-2\ell-1}\left(2\binom{n}{2\ell}t+\binom{n}{2\ell+1}\right)\sum_{a=0}^{\ell}\binom{2a}{a}t^{a}$
$\displaystyle=\sum_{m=0}^{n}\binom{n}{m}(2t)^{n-m}\sum_{a=0}^{\lfloor
m/2\rfloor}\binom{2a}{a}t^{a}.$
In particular, $\Sigma(\Gamma(n+1))$ is $\gamma$-nonnegative for all $n$.
As a corollary of Theorem 4.3, we can obtain the normalized volume of
$\Sigma(\Gamma(n+1))$. For its proof, as well as the proof of Theorem 4.3, we
collect some well-known formulas on the summation of binomial coefficients.
###### Lemma 4.5.
Let $a,b,c,d\in\mathbb{Z}_{\geq 0}$. Then the following hold:
$\displaystyle\text{(i)}\;\sum_{i=0}^{c}\binom{a}{i}\binom{b}{c-i}=\binom{a+b}{c},\;\;\text{(ii)}\;\sum_{i=0}^{c}\binom{a+i}{b}\binom{c-i}{d}=\binom{a+c+1}{b+d+1},\;\;\text{(iii)}\;\binom{a}{b}\binom{b}{c}=\binom{a-c}{b-c}\binom{a}{c}.$
###### Corollary 4.6.
The normalized volume of $\Sigma(\Gamma(n+1))$ is
$2^{n}\sum_{k=0}^{n}(k+1)\binom{n}{k}\binom{k}{\lfloor\frac{k}{2}\rfloor}.$
###### Proof.
We may set $t=1$ in (4.1). For $f_{p,q,2\ell}(1)$, we see that
$f_{p,q,2\ell}(1)=\sum_{i=0}^{p}\sum_{j=0}^{q}\binom{p}{i}\binom{q}{j}\binom{2\ell-
p-q}{\ell-
q-i+j}=\sum_{j=0}^{q}\binom{q}{j}\binom{2\ell-q}{\ell-q+j}=\sum_{j=0}^{q}\binom{q}{j}\binom{2\ell-q}{\ell-j}=\binom{2\ell}{\ell}$
by using Lemma 4.5 (i) twice. Hence,
$\sum_{(p,q)\in
S_{2\ell}}\binom{2\ell+1}{p+q+1}\binom{2\ell}{\ell}=\binom{2\ell}{\ell}\sum_{m=0}^{2\ell}(m+1)\binom{2\ell+1}{m+1}=\binom{2\ell}{\ell}\sum_{m=0}^{2\ell}(2\ell+1)\binom{2\ell}{m}=2^{2\ell}(2\ell+1)\binom{2\ell}{\ell}.$
Therefore,
$\displaystyle h^{*}(\Sigma(\Gamma(n+1));1)$
$\displaystyle=\sum_{\ell=0}^{\lfloor
n/2\rfloor}2^{n-2\ell}\left(\binom{n}{2\ell}+2\binom{n}{2\ell+1}\right)\sum_{(p,q)\in
S_{2\ell}}\binom{2\ell+1}{p+q+1}\binom{2\ell}{\ell}$
$\displaystyle=\sum_{\ell=0}^{\lfloor
n/2\rfloor}2^{n-2\ell}\left(\binom{n}{2\ell}+2\binom{n}{2\ell+1}\right)2^{2\ell}(2\ell+1)\binom{2\ell}{\ell}$
$\displaystyle=2^{n}\left(\sum_{\ell=0}^{\lfloor
n/2\rfloor}(2\ell+1)\binom{n}{2\ell}\binom{2\ell}{\ell}+2\sum_{\ell=0}^{\lfloor
n/2\rfloor}(2\ell+1)\binom{n}{2\ell+1}\binom{2\ell}{\ell}\right)$
$\displaystyle=2^{n}\left(\sum_{\ell=0}^{\lfloor
n/2\rfloor}(2\ell+1)\binom{n}{2\ell}\binom{2\ell}{\ell}+\sum_{\ell=0}^{\lfloor
n/2\rfloor}(2\ell+2)\binom{n}{2\ell+1}\binom{2\ell+1}{\ell}\right)$
$\displaystyle=2^{n}\sum_{k=0}^{n}(k+1)\binom{n}{k}\binom{k}{\lfloor\frac{k}{2}\rfloor}.$
∎
## 5\. Proof of Theorem 4.3
In this section, we put all of the necessary pieces together for the proof of
Theorem 4.3. We will produce a dissecting tree set by identifying which
oriented spanning trees correspond to monomials that are not divisible by an
initial monomial of one of the types in Lemma 2.3. Before this, we collect
several identities needed in the proof of Theorem 4.3.
###### Lemma 5.1.
1. (i)
For all $n,p,k,\ell\geq 0$ such that $p+q\leq 2\ell\leq n$,
$\binom{n-(p+q)}{2\ell-(p+q)}\sum_{i=0}^{n}\binom{n-i}{p}\binom{i-1}{q-1}+\binom{n-1-(p+q)}{2\ell-1-(p+q)}\sum_{i=0}^{n}\binom{n-i}{p}\binom{i-1}{q}=\binom{n}{2\ell}\binom{2\ell+1}{p+q+1}.$
2. (ii)
For all $n,p,k,\ell\geq 0$ such that $p+q\leq 2\ell\leq n-1$,
$\binom{n-1-(p+q)}{2\ell-(p+q)}\sum_{i=0}^{n}\binom{n-i}{p}\binom{i-1}{q}=\binom{n}{2\ell+1}\binom{2\ell+1}{p+q+1}.$
###### Proof.
These straightforwardly follow by applying Lemma 4.5 (i) and (ii). ∎
Now, note that $M^{*}(K_{3,n+1})-e$ is never graphic for $n\geq 3$ (see
Proposition 4.1). However, by an appropriate choice $e,e^{\prime}$ from
$M^{*}(K_{3,n+1})$, we see that the matroid
$M^{*}(K_{3,n+1})-\\{e,e^{\prime}\\}$ may be represented by the matrix
$[I_{2n}\mid B^{\prime}_{n+1}]$ where $B^{\prime}_{n}$ is the matrix $B_{n}$
with the second and third columns removed. This matrix also represents the
graph $\Gamma(n+1)$ on the ground set
$\\{u_{1},v_{1},\dots,u_{n},v_{n},u_{n+1}\\}$ with edges $u_{i}u_{i+1}$,
$u_{i}v_{i}$, $v_{i}u_{i+1}$ for each $i=1,\dots,n$, as well as
$u_{1}u_{n+1}$. See Figure 1 for examples.
Figure 1. The graphs $\Gamma(3)$, left, and $\Gamma(4)$, right.
###### Remark 5.2.
We can obtain $\Gamma(n)$ as the graph representing
$M^{*}(K_{3,n})-\\{e,e^{\prime}\\}$ in a theoretical way. Although $K_{3,n}/e$
is not planar for any edge $e$ of $K_{3,n}$ as described in the proof of
Proposition 4.1 (ii), we can get a planar graph $\tilde{G}$ by the contraction
with a new edge of $K_{3,n}/e$. By taking the dual graph of $\tilde{G}$ (in
the sense of planar graphs), a planar graph $\Gamma(n)$ appears.
In what follows, we concentrate on the graph $\Gamma(n+1)$. Let
$\widetilde{e}=u_{1}u_{n+1}$. To describe these spanning trees, it will be
helpful to refer to a specific planar embedding of $\Gamma(n+1)$. We may view
$\Gamma(n+1)$ as the $(2n+1)$-cycle $u_{1}v_{1}u_{2}v_{2}\dots u_{n+1}u_{1}$,
together with the chords $u_{i}u_{i+1}$ for each $i=1,\dots,n$. In particular,
throughout the remainder of the article, we will use “chord” to refer to any
of the $n$ edges of the form $u_{i}u_{i+1}$.
Referring to a specific planar embedding will allow us to construct the
$h^{*}$-polynomial by applying Theorem 2.2. To do this, we need to identify a
dissecting tree set for $\Gamma(n+1)$. Let $\mathcal{G}_{n+1}$ be the reduced
Gröbner basis of $I_{\Sigma(\Gamma(n+1))}$ with respect to any grevlex order
in which the variable $z$ corresponding to the origin is weakest and the
variables $p_{\widetilde{e}}$ and $q_{\widetilde{e}}$ corresponding to
$\widetilde{e}$ are the second- and third-weakest. Using the framework
developed in Section 2, we can identify which oriented spanning trees
contribute to our dissecting tree set. To help us describe them, we introduce
some new terminology.
Observe that if $T$ is a spanning tree of $\Gamma(n+1)$, then for each of the
triangles with vertices $u_{i},v_{i},u_{i+1}$, there are exactly three
possibilities of edges appearing in $T$: $T$ contains either
* •
both $u_{i}v_{i}$ and $v_{i}u_{i+1}$, which we call a _chordless pair_ ,
* •
both $u_{i}v_{i}$ and $u_{i}u_{i+1}$ or both $u_{i}u_{i+1}$ and
$u_{i+1}v_{i}$, each of which we call a _chorded pair_ , or
* •
only $u_{i}v_{i}$ or only $v_{i}u_{i+1}$, each of which we call an _unpaired
edge_.
For example, the spanning tree of $\Gamma(5)$ in Figure 2 contains one
chordless pair of edges, two chorded pairs of edges, and one unpaired edge.
Figure 2. A spanning tree of $\Gamma(5)$. The edges $u_{1}v_{1}$ and
$v_{1}u_{2}$ form a chordless pair. The edges $u_{2}v_{2}$ and $u_{2}u_{3}$
form a chorded pair of type $\alpha$, while the edges $u_{4}v_{4}$ and
$u_{4}u_{5}$ form a chorded pair of type $\beta$. The edge $v_{3}u_{4}$ is an
unpaired edge.
###### Proposition 5.3.
An oriented spanning tree $T$ of $\Gamma(n+1)$ is part of the dissecting tree
set if and only if, for every cycle $C$ in $\Gamma(n+1)$, either $T$ contains
no more than $\lfloor\frac{|C|-1}{2}\rfloor$ edges of $C$ that are oriented in
the same direction, or the following conditions holds:
* When $|C|$ is even, both $\widetilde{e}\in T$ and $T$ contains exactly $|C|/2$ edges of $C$ that are oriented in the same direction.
###### Proof.
This follows from Lemma 2.3. ∎
###### Corollary 5.4.
Suppose $T$ is an oriented spanning tree in the dissecting tree set. Then the
edges in the chordless pair share the same tail or share the same head.
Moreover, their orientations may be reversed to obtain another oriented
spanning tree in the dissecting tree set.
###### Proof.
If the edges in the chordless pair, say, $u_{i}v_{i}$ and $v_{i}u_{i+1}$,
share neither the same tail nor the same head, then the $3$-cycle
$u_{i},v_{i},u_{i+1}$ of $\Gamma(n+1)$ contains two edges that are oriented in
the same direction. ∎
For a spanning tree $T$ of $\Gamma(n+1)$, let $c(T)$ denote the number of
chords $u_{i}u_{i+1}$ of $T$ and let
$\epsilon(T)=\begin{cases}1&\text{ if }\widetilde{e}\in T\\\ 0&\text{
otherwise.}\end{cases}$
###### Lemma 5.5.
Let $T$ be a member of our dissecting tree set.
1. (i)
If $\epsilon(T)=1$, then $T$ contains a total of $n-1$ chordless pairs and
chorded pairs of edges as well as exactly one unpaired edge.
2. (ii)
If $\epsilon(T)=0$, then $T$ contains a total of $n$ chordless and chorded
pairs of edges. Moreover, $c(T)$ must be always even in this case.
###### Proof.
Note that each spanning tree of $\Gamma(n+1)$ has $2n$ edges. Hence, the
statements on the numbers of chordless/chorded pairs and unpaired edge follow.
Regarding the evenness of $c(T)$, on the contrary, suppose that $c(T)=2a-1$
for some $a\in\mathbb{Z}_{>0}$. Then there are at least $a$ edges that have
the same direction. On the other hand, since there are $(n-2a+1)$ chordless
pairs that consist of $2(n-2a+1)$ edges, Corollary 5.4 claims that the exactly
half of these edges should have the same oriantation. Hence, we can find a
cycle $C$ consisting of those chords of chorded pairs, the edges of chordless
pairs and $\widetilde{e}$, that have length $2(n-2a+1)+(2a-1)+1=2(n-a+1)$,
such that there are at least $(n-a+1)$ edges in $C$ in the same direction, a
contradiction to Proposition 5.3. ∎
At this point we will take time here to closely examine some particular cases.
We do this with the goal of helping motivate and illustrate the key aspects of
the full proof, which becomes somewhat technical.
$c(T)+\epsilon(T)=0$: Let $T$ be a member of our dissecting tree set with
$c(T)+\epsilon(T)=0$. Then the (undirected) edges of $T$ consist entirely of
chordless pairs. When orienting this tree, each pair of edges $u_{i}v_{i}$,
$v_{i}u_{i+1}$ has two possibilities: either both edges point toward $v_{i}$
or both edges point away from $v_{i}$ (see Corollary 5.4). In either case,
there are $n$ edges that will point towards $u_{1}$. So, by Theorem 2.2 with
$v=u_{1}$, each such oriented tree corresponds to a summand of $t^{n}$ in
$h^{*}(\Sigma(\Gamma(n+1));t)$. There are $2^{n}$ valid orientations of this
tree, resulting in a summand of $(2t)^{n}$ in the $h^{*}$-polynomial after
over all of these trees.
$c(T)+\epsilon(T)=1$: Let $T$ be a member of our dissecting tree with
$c(T)+\epsilon(T)=1$. Then $\epsilon(T)=1$ and $c(T)=0$ (see Lemma 5.5 (ii)).
We can establish orientations on the chordless pairs as in the previous case,
although there are now only $n-1$ of these pairs. Any cycle containing one
edge of a chordless pair must also contain the other. By Corollary 5.4, these
two edges must have opposite orientations. For any cycle containing
$\widetilde{e}$, the orientations of $\widetilde{e}$ and the unpaired edge
should have the opposite direction; otherwise we can find at least $(n+1)$
edges with the same direction in $(2n+1)$-cycle of $\Gamma(n+1)$ (see
Corollary 5.4 and Lemma 5.5).
For $T$ to contribute to $h^{*}(\Sigma(\Gamma(n+1)))$, the orientation of
$\widetilde{e}$ and the unpaired edge must be opposite. To help describe which
summand of the $h^{*}$-polynomial corresponds to $T$, we will refer to two
portions of it relative to $u_{1}$. Given a spanning tree $T$ of $\Gamma(n+1)$
that contains $\widetilde{e}$, the graph $T-\widetilde{e}$ consists of two
components. Call the component containing $u_{1}$ the _left side_ of $T$ and
denote it by $L(T)$. The _right side_ of the tree, which we denote $R(T)$, is
the subgraph of $T$ consisting of $\widetilde{e}$ and all edges not in $L(T)$.
This definition makes it clear that $T$ can correspond to any of the following
summands of the $h^{*}$-polynomial, depending on its orientation and the
location of the unpaired edge, $e$:
* •
If $\widetilde{e}$ and $e$ are pointing away each other and $e\in L(T)$, then
$T$ corresponds to the summand $(2t)^{n-1}$.
* •
If $\widetilde{e}$ and $e$ are pointing away each other and $e\in R(T)$, then
$T$ corresponds to the summand $t(2t)^{n-1}$.
* •
If $\widetilde{e}$ and $e$ are pointing towards each other and $e\in L(T)$,
then $T$ corresponds to the summand $t^{2}(2t)^{n-1}$.
* •
If $\widetilde{e}$ and $e$ are pointing towards each other and $e\in R(T)$,
then $T$ corresponds to the summand $t(2t)^{n-1}$.
Thus, by considering all orientations on the edges of a spanning tree
satisfying $c(T)+\epsilon(T)=1$, the tree will correspond to a summand of
$(1+t)^{2}(2t)^{n-1}$. Note that $(1+t)^{2}=f_{0,0,1}(t)$.
$c(T)+\epsilon(T)=2$: Because of one additional subtlety that arises when
$c(T)+\epsilon(T)=2$, we will discuss this case as well. When this happens,
there are two subcases: the first being where $c(T)=2$ and $\epsilon(T)=0$ and
the second being where $c(T)=1$ and $\epsilon(T)=1$. In the first case, we can
repeat the same argument as when $c(T)+\epsilon(T)=0$, but we now have only
$n-2$ possible chordless pairs. As before, the orientations on the chordless
pairs result in a factor of $(2t)^{n-2}$ in the contribution $T$ makes to the
$h^{*}$-polynomial. Such a factor occurs $\binom{n}{2}$ times, since this is
the number of chords $u_{i}u_{i+1}$ that can appear as parts of chorded pairs.
Now, for each chorded pair, the orientation on the edge $u_{i}u_{i+1}$
dictates the orientation of the other edge $u_{j}u_{j+1}$ of the pair: they
have the opposite direction. However, whether the chorded pair consists of
$u_{i}v_{i}$ or $v_{i}u_{i+1}$ will make a difference in the summand of the
$h^{*}$-polynomial corresponding to $T$. We say that a chorded pair has type
$\alpha$ if the non-chord edge is incident to the vertex that is closer to
$u_{1}$ within $T$; otherwise, we say that a chorded pair has type $\beta$.
See Figure 2 for an example of each.
When summing the monomials of the $h^{*}$-polynomial corresponding to $T$, if
a chorded pair has type $\alpha$, then one of the orientations of its edges
results in a factor of $t^{2}$, while the other results in a factor of $1$. On
the other hand, if a chorded pair has type $\beta$, then each of the
orientations of its edges results in a factor of $t$. So, if we consider all
spanning trees $T$ of $\Gamma(n+1)$ satisfying $c(T)=2$ and $\epsilon(T)=0$,
then we know that there are $p$ chorded pairs of type $\alpha$ in $L(T)$ and
$q$ chorded pairs of type $\alpha$ in $R(T)$ for some nonnegative $p,q$
satisfying $p+q\leq 2$. This leaves $2-(p+q)$ chords to be part of a chorded
pair of type $\beta$. This is where the set $S_{k}$ in (4.1) will come from:
the pair $(p,q)$ will indicate the presence of $p$ chorded pairs of type
$\alpha$ in $L(T^{\prime})$ and $q$ of them in $R(T^{\prime})$, where the
notation $T^{\prime}$ is explained below.
Recognize that in any oriented spanning tree $T$ of $\Gamma(n+1)$, there is at
most one unpaired edge. Consequently, if an unpaired edge exists, then
$\widetilde{e}\in T$ and the unpaired edge must have endpoints $u_{i}$ and
$v_{j}$ for some $i$ and $j=i\pm 1$. Thus we may, without affecting the
monomial to which $T$ corresponds, modify $T$ by deleting the vertex $v_{j}$,
adding a new vertex $v_{n+1}$, and adding a new edge as follows:
* •
If $\widetilde{e}$ and $u_{i}v_{j}$ are pointing away each other and
$u_{i}v_{j}\in L(T)$, then add the directed edge $u_{1}v_{n+1}$.
* •
If $\widetilde{e}$ and $u_{i}v_{j}$ are pointing away each other and
$u_{i}v_{j}\in R(T)$, then add the directed edge $u_{n+1}v_{n+1}$.
* •
If $\widetilde{e}$ and $u_{i}v_{j}$ are pointing towards each other and
$u_{i}v_{j}\in L(T)$, then add the directed edge $v_{n+1}u_{1}$.
* •
If $\widetilde{e}$ and $u_{i}v_{j}$ are pointing towards each other and
$u_{i}v_{j}\in R(T)$, then add the directed edge $v_{n+1}u_{n+1}$.
See Figure 3 for illustrations of each of these cases. We call $T^{\prime}$ a
_modified (oriented) spanning tree_ of $\Gamma(n+1)$. Modified spanning trees
allow us to consider $\widetilde{e}$ and $u_{1}v_{n+1}$ or $u_{n+1}v_{n+1}$ as
a chorded pair, so that all edges in $T^{\prime}$ are part of a chorded or
chordless pair. Note that, in each case, the number of edges pointing towards
or away from $u_{1}$ in each of $L(T)$ and $R(T)$ is preserved by this
operation. We are now prepared to present the proof of Theorem 4.3.
Figure 3. Examples of modified spanning trees in $\Gamma(4)$. The original
spanning trees and their corresponding modified trees are written above and
below each other.
###### Proof of Theorem 4.3.
To begin, we first show the second equation of (4.1) holds. Consider a summand
$\binom{n}{k}(2t)^{n-k}\sum_{(p,q)\in
S_{k}}\binom{2\lfloor\frac{k}{2}\rfloor+1}{p+q+1}f_{p,q,k}(t).$
If $k=2\ell$ for some $\ell$, then adding this summand and the summand for
$2\ell+1$ becomes
$\binom{n}{2\ell}(2t)^{n-2\ell}\sum_{(p,q)\in
S_{2\ell}}\binom{2\ell+1}{p+q+1}f_{p,q,2\ell}(t)+\binom{n}{2\ell+1}(2t)^{n-2\ell-1}\sum_{(p,q)\in
S_{2\ell+1}}\binom{2\ell+1}{p+q+1}f_{p,q,2\ell+1}(t).$
Note that if $p+q=2\ell+1$, then the binomial coefficient
$\binom{2\ell+1}{p+q+1}$ becomes $0$. So, there is no harm in replacing
$S_{2\ell+1}$ with $S_{2\ell}$. Further, since
$f_{p,q,2\ell+1}(t)=(1+t)^{2}f_{p,q,2\ell}(t)$, the above expression may be
written
$\binom{n}{2\ell}(2t)^{n-2\ell}\sum_{(p,q)\in
S_{2\ell}}\binom{2\ell+1}{p+q+1}f_{p,q,2\ell}(t)+\binom{n}{2\ell+1}(2t)^{n-2\ell-1}\sum_{(p,q)\in
S_{2\ell}}\binom{2\ell+1}{p+q+1}(1+t)^{2}f_{p,q,2\ell}(t).$
Factoring appropriately results in the expression
$(2t)^{n-2\ell-1}\left(2t\binom{n}{2\ell}+(1+t)^{2}\binom{n}{2\ell+1}\right)\sum_{(p,q)\in
S_{2\ell}}\binom{2\ell+1}{p+q+1}f_{p,q,2\ell}(t).$
Thus, ranging over all $\ell=0,1,2,\dots,\lfloor n/2\rfloor$ yields the second
equation of (4.1).
Now, fix an arbitrary $\ell$. We will treat the summand corresponding to
$\ell$ in two pieces:
(5.1) $\sum_{(p,q)\in
S_{2\ell}}(2t)^{n-2\ell}\binom{n}{2\ell}\binom{2\ell+1}{p+q+1}f_{p,q,2\ell}(t)$
and
(5.2) $\sum_{(p,q)\in
S_{2\ell}}(2t)^{n-2\ell-1}(1+t)^{2}\binom{n}{2\ell+1}\binom{2\ell+1}{p+q+1}f_{p,q,2\ell}(t)$
since these will more closely resemble how to identify the oriented spanning
trees in our dissecting tree set, and their contributions to the
$h^{*}$-polynomial.
For (5.1), we interpret $2\ell$ as the number of chords in the modified tree
$T^{\prime}$. Then there is exactly one $i$ such that all of
$u_{n-i}u_{n-i+1}$, $u_{n-i}v_{n-i}$ and $v_{n-i}u_{n-i+1}$ are missing in
$T^{\prime}$. Namely, we treat both cases, $c(T)=2\ell$ and $\epsilon(T)=0$,
and $c(T)=2\ell-1$ and $\epsilon(T)=1$, simultaneously. Passing to the
modified spanning trees, we can say that $\widetilde{e}$ is part of a chorded
pair, and is of type $\alpha$ or $\beta$. Suppose first that $\widetilde{e}$
is part of a pair of type $\alpha$ in $T^{\prime}$. Then $p$ of the chords
$u_{1}u_{2},\dots,u_{n-i-1}u_{n-i}$ may be chosen to be parts of chorded pairs
of type $\alpha$, and these will all be part of $L(T^{\prime})$. We may then
choose $q-1$ of the chords $u_{n-i+1}u_{n-i+2},\dots,u_{n}u_{n+1}$ to be part
of chorded pairs of type $\alpha$, to ensure that $R(T^{\prime})$ contains $q$
of them. Of the remaining chords of the form $u_{j}u_{j+1}$ (excluding
$u_{n-i}u_{n-i+1}$), there are $n-(p+q)$ of them, and we must choose
$2\ell-(p+q)$ of them to be the chorded pairs of type $\beta$. All together we
have constructed
$\sum_{i=0}^{n}\binom{n-i}{p}\binom{i-1}{q-1}\binom{n-(p+q)}{2\ell-(p+q)}$
trees. On the other hand, suppose $\widetilde{e}$ is part of a chorded pair of
type $\beta$. Following an analogous argument, we obtain
$\sum_{i=0}^{n}\binom{n-i}{p}\binom{i-1}{q}\binom{n-1-(p+q)}{2\ell-1-(p+q)}$
spanning trees.
Now we may apply Lemma 5.1 (i) to obtain
$\binom{n}{2\ell}\binom{2\ell+1}{p+q+1}$
spanning trees that contribute to the $h^{*}$-polynomial when there are
$2\ell$ chords in the modified tree $T^{\prime}$.
For each of the edges $u_{i}u_{i+1}$ that are part of a chorded pair, the
structure of $\Gamma(n+1)$ and Proposition 5.3 imply that exactly half of them
must be pointing towards $u_{1}$ and exactly half of them must be pointing
away from $u_{1}$. The contribution to the $h^{*}$-polynomial depends on
whether those chorded pairs are of type $\alpha$ or type $\beta$.
For each of these trees, we can determine a valid orientation by considering
which orientations satisfy Proposition 5.3. For each spanning tree $T$ with
$p$ chorded pairs of type $\alpha$ in $L(T)$, we choose $i$ of them to point
toward $u_{1}$. Similarly, we choose $j$ of them from $R(T)$ to point toward
$u_{1}$. There are $2\ell-p-q$ chorded pairs remaining, all of type $\beta$;
we must orient these so that exactly $\ell$ of the chords $u_{r}u_{r+1}$ are
pointing in the same direction. By selecting which pairs are oriented so that
the chord is pointing counterclockwise in our planar embedding of
$\Gamma(n+1)$, this must be done in $\binom{2\ell-p-q}{\ell-(q-j)-i}$ ways,
since $i+(q-j)$ of the edges have already been given counterclockwise
orientations.
For each of these trees, there will be $(n-2\ell)$ edges oriented in
$T^{\prime}$ toward $u_{1}$ that come from the chordless pairs, and this is
true for either possible orientation of the pairs. From the paragraph above,
there will be $2(i+j)$ edges pointing toward $u_{1}$ that come from chorded
pairs of type $\alpha$, and $2\ell-p-q$ edges pointing toward $u_{1}$ that
come from chorded pairs of type $\beta$. Summing over all $i$ and $j$ gives us
a summand of
$(2t)^{n-2\ell}f_{p,q,2\ell}(t)$
in the computation of the $h^{*}$-polynomial. Further, because we are
considering the case of modified spanning trees, we have shown that there are
$\binom{n}{2\ell}\binom{2\ell+1}{p+q+1}$
of them. By multiplying the above two expressions and summing over all
$(p,q)\in S_{2\ell}$, we obtain (5.1).
Now, suppose $c(T)+\epsilon(T)=2\ell+1$. By Lemma 5.5, $\epsilon(T)=1$. We can
apply an argument analogous to the one above, using Lemma 5.1 (ii), to show
that there are
$\binom{n}{2\ell+1}\binom{2\ell+1}{p+q+1}$
oriented spanning trees with $p$ and $q$ choreded pairs of type $\alpha$ in
$L(T)$ and $R(T)$ in $T$, respectively. Note that the argument holds for each
possibility of whether $\widetilde{e}$ is of type $\alpha$ or $\beta$. So, we
must multiply each of these by the sum of the $h^{*}$-polynomial contributions
from these chorded pairs, which is $1+2t+t^{2}=(1+t)^{2}$. This establishes
(5.2), which completes the proof.
∎
## 6\. Proof of Theorem 4.4
To prove Theorem 4.4, we will use a generating function argument. We will make
regular use of the following three well-known identities:
(6.1) $\sum_{n\geq 0}\binom{n+k}{n}x^{n}=\frac{1}{(1-x)^{k+1}},$ (6.2)
$\sum_{k\geq 0}\binom{2k}{k}x^{k}=\frac{1}{\sqrt{1-4x}},\\\ $ (6.3)
$\sum_{n\geq
0}\binom{2n+k}{n}x^{n}=\frac{1}{\sqrt{1-4x}}\left(\frac{1-\sqrt{1-4x}}{2x}\right)^{k}.$
The first lemma we will need is the following.
###### Lemma 6.1.
For all $0\leq k\leq\ell$ we have
$\sum_{a=0}^{\ell}(-1)^{k-a}4^{\ell-a}\binom{2a}{a}\binom{\ell-k}{\ell-a}=\frac{\binom{\ell}{k}\binom{2\ell}{\ell}}{\binom{2\ell}{2k}}.$
###### Proof.
Using (6.1), (6.2) we compute the generating function for the left side of the
identity:
$\displaystyle\sum_{k\geq 0}\sum_{\ell\geq
0}\sum_{a=0}^{\ell}(-1)^{k-a}4^{\ell-a}\binom{2a}{a}\binom{\ell-k}{\ell-a}x^{k}y^{\ell}$
$\displaystyle=\sum_{k\geq 0}x^{k}\sum_{a\geq 0}\sum_{\ell\geq
a}(-1)^{k-a}4^{\ell-a}\binom{2a}{a}\binom{\ell-k}{\ell-a}y^{\ell}$
$\displaystyle=\sum_{k\geq 0}x^{k}\sum_{a\geq
0}(-1)^{k-a}\binom{2a}{a}\sum_{\ell\geq
0}4^{\ell}\binom{\ell+a-k}{\ell}y^{\ell+a}$ $\displaystyle=\sum_{k\geq
0}x^{k}\sum_{a\geq 0}(-1)^{k-a}\binom{2a}{a}\frac{y^{a}}{(1-4y)^{a-k+1}}$
$\displaystyle=\frac{1}{1-4y}\sum_{k\geq
0}\left(-x(1-4y)\right)^{k}\sum_{a\geq
0}\binom{2a}{a}\left(\frac{-y}{1-4y}\right)^{a}$
$\displaystyle=\frac{1}{(1-4y)(1-(-x(1-4y)))\sqrt{1-4\frac{-y}{1-4y}}}$
$\displaystyle=\frac{1}{\sqrt{1-4y}\ (1+x-4xy)}.$
Applying the Taylor expansion to this rational function, we find
$\displaystyle\frac{1}{\sqrt{1-4y}\ (1+x-4xy)}$ $\displaystyle=$
$\displaystyle\sum_{k\geq 0}(-1)^{k}k!(1-4y)^{k-\frac{1}{2}}\
\frac{x^{k}}{k!}$ $\displaystyle=$ $\displaystyle\sum_{k\geq 0}\sum_{\ell\geq
0}(-1)^{k}(-4)^{\ell}\left(k-\frac{1}{2}\right)\cdots\left(k-\frac{1}{2}-\ell+1\right)x^{k}\frac{y^{\ell}}{\ell!}$
$\displaystyle=$ $\displaystyle\sum_{k\geq 0}\sum_{\ell\geq
0}(-1)^{k}2^{\ell}(2\ell-2k-1)\cdots(-2k+1)x^{k}\frac{y^{\ell}}{\ell!}$
$\displaystyle=$ $\displaystyle\sum_{k\geq 0}\sum_{\ell\geq
0}\frac{(2\ell-2k)!(2k)!}{(\ell-k)!k!}x^{k}\frac{y^{\ell}}{\ell!}$
$\displaystyle=$ $\displaystyle\sum_{k\geq 0}\sum_{\ell\geq
0}\frac{\binom{\ell}{k}\binom{2\ell}{\ell}}{\binom{2\ell}{2k}}x^{k}y^{\ell}.$
∎
###### Lemma 6.2.
For all $k\geq 0$,
$\sum_{\ell\geq
0}\frac{2\ell+1}{2k+1}\binom{\ell}{k}\binom{2\ell}{\ell}x^{\ell}=\binom{2k}{k}\frac{x^{k}}{(1-4x)^{k+\frac{3}{2}}}.$
###### Proof.
By Taylor expansion, we have
$\displaystyle\binom{2k}{k}\frac{x^{k}}{(1-4x)^{k+\frac{3}{2}}}$
$\displaystyle=$ $\displaystyle\binom{2k}{k}x^{k}\sum_{m\geq
0}4^{m}\left(k+\frac{3}{2}\right)\left(k+\frac{5}{2}\right)\cdots\left(k+\frac{2m+1}{2}\right)\frac{x^{m}}{m!}$
$\displaystyle=$ $\displaystyle\binom{2k}{k}x^{k}\sum_{m\geq
0}2^{m}\left(2k+3\right)\left(2k+5\right)\cdots(2k+2m+1)\frac{x^{m}}{m!}$
$\displaystyle=$ $\displaystyle\binom{2k}{k}x^{k}\sum_{m\geq
0}2^{m}\frac{(2k+2m+1)!!}{(2k+1)!!}\frac{x^{m}}{m!}$ $\displaystyle=$
$\displaystyle x^{k}\sum_{m\geq
0}2^{m}\frac{(2k)!}{k!k!}\frac{(2k+2m+2)!}{2^{k+m+1}(k+m+1)!}\frac{2^{k+1}(k+1)!}{(2k+2)!}\frac{x^{m}}{m!}$
$\displaystyle=$ $\displaystyle x^{k}\sum_{m\geq
0}\frac{(2k+2m+2)!}{2(2k+1)(k+m+1)!k!m!}\ \ x^{m}$ $\displaystyle=$
$\displaystyle\sum_{\ell\geq
0}\frac{(2\ell+2)!}{2(2k+1)(\ell+1)!k!(\ell-k)!}x^{\ell}$ $\displaystyle=$
$\displaystyle\sum_{\ell\geq
0}\frac{2\ell+1}{2k+1}\binom{\ell}{k}\binom{2\ell}{\ell}x^{\ell}.$
∎
The second lemma we will need is the following.
###### Lemma 6.3.
For $\ell\geq 0$,
$\sum_{p,q\geq
0}\binom{2\ell+1}{p+q+1}x^{p}y^{q}=\frac{(x+1)^{2\ell+1}-(y+1)^{2\ell+1}}{x-y}.$
###### Proof.
We again use a generating function argument, beginning with the left side of
the desired identity multiplied by $x-y$.
$\displaystyle(x-y)\sum_{p,q\geq 0}\binom{2\ell+1}{p+q+1}x^{p}y^{q}$
$\displaystyle=$ $\displaystyle\sum_{p,q\geq
0}\binom{2\ell+1}{p+q+1}x^{p+1}y^{q}-\sum_{p,q\geq
0}\binom{2\ell+1}{p+q+1}x^{p}y^{q+1}$ $\displaystyle=$
$\displaystyle\sum_{p\geq 1}\sum_{q\geq
0}\binom{2\ell+1}{p+q}x^{p}y^{q}-\sum_{p\geq 0}\sum_{q\geq
1}\binom{2\ell+1}{p+q}x^{p}y^{q}$ $\displaystyle=$ $\displaystyle\sum_{p\geq
1}\binom{2\ell+1}{p}x^{p}-\sum_{q\geq 1}\binom{2\ell+1}{q}y^{q}$
$\displaystyle=$ $\displaystyle\left(1+\sum_{p\geq
1}\binom{2\ell+1}{p}x^{p}\right)-\left(1+\sum_{q\geq
1}\binom{2\ell+1}{q}y^{q}\right)$ $\displaystyle=$
$\displaystyle(x+1)^{2\ell+1}-(y+1)^{2\ell+1}.$
∎
###### Theorem 6.4.
For all $\ell\geq 0$ we have
$\sum_{p+q\leq 2\ell}\binom{2\ell+1}{p+q+1}t^{2\ell-
p-q}\sum_{i=0}^{p}\sum_{j=0}^{q}\binom{p}{i}\binom{q}{j}\binom{2\ell-
p-q}{\ell-
q-i+j}t^{2i+2j}=\sum_{a=0}^{\ell}\binom{2a}{a}t^{a}(t+1)^{4\ell-2a}.$
###### Proof.
For notational convenience, set
$F_{\ell}=\sum_{a=0}^{\ell}\binom{2a}{a}t^{a}(t+1)^{4\ell-2a}$
and
$G_{\ell,m}=\sum_{p+q\leq 2\ell}\binom{2\ell+1}{p+q+1}t^{2\ell-
p-q}\sum_{i=0}^{p}\sum_{j=0}^{q}\binom{p}{i}\binom{q}{j}\binom{2m-p-q}{m-q-i+j}t^{2i+2j}.$
We compute the generating function for $F_{\ell}$ as follows:
$\displaystyle\sum_{\ell\geq 0}F_{\ell}\lambda^{\ell}$ $\displaystyle=$
$\displaystyle\sum_{\ell\geq
0}\sum_{a=0}^{\ell}\binom{2a}{a}t^{a}(t+1)^{4\ell-2a}\lambda^{\ell}$
$\displaystyle=$ $\displaystyle\sum_{a\geq 0}\binom{2a}{a}t^{a}\sum_{\ell\geq
a}(t+1)^{4\ell-2a}\lambda^{\ell}$ $\displaystyle=$ $\displaystyle\sum_{a\geq
0}\binom{2a}{a}t^{a}\frac{(t+1)^{2a}\lambda^{a}}{1-(t+1)^{4}\lambda}$
$\displaystyle=$ $\displaystyle\frac{1}{1-(t+1)^{4}\lambda}\sum_{a\geq
0}\binom{2a}{a}t^{a}(t+1)^{2a}\lambda^{a}$ $\displaystyle=$
$\displaystyle\frac{1}{(1-(t+1)^{4}\lambda)\sqrt{1-4t(t+1)^{2}\lambda}}.$
Hence it is enough to show that
$\sum_{\ell\geq
0}G_{\ell,\ell}\lambda^{\ell}=\frac{1}{(1-(t+1)^{4}\lambda)\sqrt{1-4t(t+1)^{2}\lambda}}.$
Using Lemma 6.3, we have
$\displaystyle\sum_{m\geq 0}G_{\ell,m}\lambda^{m}$ $\displaystyle=\sum_{m\geq
0}\sum_{p,q\geq 0}\binom{2\ell+1}{p+q+1}t^{2\ell-
p-q}\sum_{i=0}^{p}\sum_{j=0}^{q}\binom{p}{i}\binom{q}{j}\binom{2m-p-q}{m-q-i+j}t^{2i+2j}\lambda^{m}$
$\displaystyle=\sum_{p,q\geq
0}\sum_{i=0}^{p}\sum_{j=0}^{q}\binom{p}{i}\binom{q}{j}t^{2\ell-
p-q+2i+2j}\binom{2\ell+1}{p+q+1}\sum_{m\geq
0}\binom{2m-p-q}{m-q-i+j}\lambda^{m}$ $\displaystyle=\sum_{p,q\geq
0}\sum_{i=0}^{p}\sum_{j=0}^{q}\binom{p}{i}\binom{q}{j}t^{2\ell-
p-q+2i+2j}\binom{2\ell+1}{p+q+1}\lambda^{q+i-j}\sum_{m\geq
0}\binom{2m-p+q+2i-2j}{m}\lambda^{m}$ $\displaystyle=\sum_{p,q\geq
0}\sum_{i=0}^{p}\sum_{j=0}^{q}\binom{p}{i}\binom{q}{j}t^{2\ell-
p-q+2i+2j}\binom{2\ell+1}{p+q+1}\frac{\lambda^{q+i-j}}{\sqrt{1-4\lambda}}\ \
L^{-p+q+2i-2j}$
$\displaystyle=\frac{t^{2\ell}}{\sqrt{1-4\lambda}}\sum_{p,q\geq
0}\binom{2\ell+1}{p+q+1}\left(\lambda
Lt^{+}\frac{1}{tL}\right)^{p}\left(\frac{\lambda L}{t}+\frac{t}{L}\right)^{q}$
$\displaystyle=\frac{t^{2\ell}}{\sqrt{1-4\lambda}}\frac{(\lambda
Lt+\frac{1}{tL}+1)^{2\ell+1}-(\frac{\lambda
L}{t}+\frac{t}{L}+1)^{2\ell+1}}{(\lambda Lt+\frac{1}{tL})-(\frac{\lambda
L}{t}+\frac{t}{L})},$
where $L=\frac{1-\sqrt{1-4\lambda}}{2\lambda}$. It follows that
$\frac{1}{L}=\frac{1+\sqrt{1-4\lambda}}{2}$, that
$\displaystyle\lambda Lt+\frac{1}{tL}+1$
$\displaystyle=\frac{1-\sqrt{1-4\lambda}}{2}t+\frac{1+\sqrt{1-4\lambda}}{2t}+1$
$\displaystyle=\frac{1}{2t}\left(\sqrt{1-4\lambda}(1-t^{2})+(t+1)^{2}\right)$
$\displaystyle=\frac{t+1}{2t}\left(\sqrt{1-4\lambda}(1-t)+(t+1)\right),$
and that
$\displaystyle\frac{\lambda L}{t}+\frac{t}{L}+1$
$\displaystyle=\frac{1-\sqrt{1-4\lambda}}{2t}+\frac{1+\sqrt{1-4\lambda}}{2}t+1$
$\displaystyle=\frac{1}{2t}\left(\sqrt{1-4\lambda}(t^{2}-1)+(t+1)^{2}\right)$
$\displaystyle=\frac{t+1}{2t}\left(\sqrt{1-4\lambda}(t-1)+(t+1)\right).$
Hence,
$\left(\lambda Lt+\frac{1}{tL}\right)-\left(\frac{\lambda
L}{t}+\frac{t}{L}\right)=\left(\lambda
Lt+\frac{1}{tL}+1\right)-\left(\frac{\lambda
L}{t}+\frac{t}{L}+1\right)=\frac{1-t^{2}}{t}\sqrt{1-4\lambda}.$
Thus
$\displaystyle\sum_{m\geq 0}G_{\ell,m}\lambda^{m}$
$\displaystyle=\frac{(t+1)^{2\ell+1}}{2^{2\ell+1}(1-t^{2})(1-4\lambda)}\left(\left(\sqrt{1-4\lambda}(1-t)+(t+1)\right)^{2\ell+1}+\left(\sqrt{1-4\lambda}(1-t)-(t+1)\right)^{2\ell+1}\right)$
$\displaystyle=\frac{(t+1)^{2\ell+1}}{2^{2\ell}(1-t^{2})(1-4\lambda)}\sum_{k\geq
0}\binom{2\ell+1}{2k}(\sqrt{1-4\lambda}(1-t))^{2\ell+1-2k}(t+1)^{2k}$
$\displaystyle=\frac{(t+1)^{2\ell}(1-t)^{2\ell}}{2^{2\ell}\sqrt{1-4\lambda}}\sum_{k\geq
0}\binom{2\ell+1}{2k}(1-4\lambda)^{\ell-k}\left(\frac{t+1}{1-t}\right)^{2k}$
$\displaystyle=4^{-\ell}(t+1)^{2\ell}(1-t)^{2\ell}\sum_{a\geq
0}\binom{2a}{a}\lambda^{a}\sum_{k\geq
0}\binom{2\ell+1}{2k}(1-4\lambda)^{\ell-k}\left(\frac{t+1}{1-t}\right)^{2k}$
$\displaystyle=4^{-\ell}(t+1)^{2\ell}(1-t)^{2\ell}\sum_{a\geq
0}\binom{2a}{a}\lambda^{a}\sum_{k\geq
0}\binom{2\ell+1}{2k}\sum_{i=0}^{\ell-k}\binom{\ell-k}{i}(-4\lambda)^{i}\left(\frac{t+1}{1-t}\right)^{2k}.$
Since $G_{\ell,\ell}$ is the coefficient of $\lambda^{\ell}$ in $\sum_{m\geq
0}G_{\ell,m}\lambda^{m}$, we have
$G_{\ell,\ell}=4^{-\ell}(t+1)^{2\ell}(1-t)^{2\ell}\sum_{a\geq 0}\sum_{k\geq
0}\binom{2a}{a}\binom{2\ell+1}{2k}\binom{\ell-k}{\ell-a}(-4)^{\ell-a}\left(\frac{t+1}{1-t}\right)^{2k}.$
Additionally, from Lemma 6.1,
$\displaystyle G_{\ell,\ell}$
$\displaystyle=4^{-\ell}(t+1)^{2\ell}(1-t)^{2\ell}\sum_{k=0}^{\ell}(-1)^{\ell-k}\frac{2\ell+1}{2\ell-2k+1}\binom{\ell}{k}\binom{2\ell}{\ell}\left(\frac{t+1}{1-t}\right)^{2k}$
$\displaystyle=4^{-\ell}(t+1)^{2\ell}(1-t)^{2\ell}\sum_{k=0}^{\ell}(-1)^{k}\frac{2\ell+1}{2k+1}\binom{\ell}{k}\binom{2\ell}{\ell}\left(\frac{t+1}{1-t}\right)^{2\ell-2k}.$
Moreover, from Lemma 6.2,
$\displaystyle\sum_{\ell\geq 0}G_{\ell,\ell}\lambda^{\ell}$
$\displaystyle=\sum_{\ell\geq
0}4^{-\ell}(t+1)^{2\ell}(1-t)^{2\ell}\lambda^{\ell}\sum_{k=0}^{\ell}(-1)^{k}\frac{2\ell+1}{2k+1}\binom{\ell}{k}\binom{2\ell}{\ell}\left(\frac{t+1}{1-t}\right)^{2\ell-2k}$
$\displaystyle=\sum_{k\geq
0}(-1)^{k}\left(\frac{t+1}{1-t}\right)^{-2k}\sum_{\ell\geq
0}\frac{2\ell+1}{2k-1}\binom{\ell}{k}\binom{2\ell}{\ell}\left(\frac{\lambda}{4}(t+1)^{4}\right)^{\ell}$
$\displaystyle=\sum_{k\geq
0}(-1)^{k}\left(\frac{t+1}{1-t}\right)^{-2k}\binom{2k}{k}\frac{\left(\frac{\lambda}{4}(t+1)^{4}\right)^{k}}{\left(1-(t+1)^{4}\lambda\right)^{k+\frac{3}{2}}}.$
$\displaystyle=\frac{1}{\left(1-(t+1)^{4}\lambda\right)^{\frac{3}{2}}}\sum_{k\geq
0}\binom{2k}{k}\left(-\left(\frac{1-t}{t+1}\right)^{2}\frac{\frac{\lambda}{4}(t+1)^{4}}{1-(t+1)^{4}\lambda}\right)^{k}$
$\displaystyle=\frac{1}{\left(1-(t+1)^{4}\lambda\right)^{\frac{3}{2}}}\sum_{k\geq
0}\binom{2k}{k}\left(-\frac{\lambda(t+1)^{2}(1-t)^{2}}{4(1-(t+1)^{4}\lambda)}\right)^{k}$
$\displaystyle=\frac{1}{\left(1-(t+1)^{4}\lambda\right)^{\frac{3}{2}}\sqrt{1+\frac{\lambda(t+1)^{2}(1-t)^{2}}{1-(t+1)^{4}\lambda}}}$
$\displaystyle=\frac{1}{(1-(t+1)^{4}\lambda)\sqrt{1-4t(t+1)^{2}\lambda}},$
as desired. ∎
###### Proof of Theorem 4.4.
The equations
$\displaystyle\gamma(\Sigma(\Gamma(n+1);t))=$
$\displaystyle\sum_{\ell=0}^{\lfloor
n/2\rfloor}(2t)^{n-2\ell-1}\left(2\binom{n}{2\ell}t+\binom{n}{2\ell+1}\right)\sum_{a=0}^{\ell}\binom{2a}{a}t^{a}$
$\displaystyle=\sum_{m=0}^{n}\binom{n}{m}(2t)^{n-m}\sum_{a=0}^{\lfloor
m/2\rfloor}\binom{2a}{a}t^{a}$
follow from substituting the equation from Theorem 6.4 into equation (4.1). ∎
## References
* [1] Matthias Beck and Sinai Robins. Computing the continuous discretely. Undergraduate Texts in Mathematics. Springer, New York, 2007. Integer-point enumeration in polyhedra.
* [2] Benjamin Braun. Unimodality problems in Ehrhart theory. In Recent trends in combinatorics, volume 159 of IMA Vol. Math. Appl., pages 687–711. Springer, 2016.
* [3] Tianran Chen. Directed acyclic decomposition of Kuramoto equations. Chaos, 29(9):093101, 12, 2019.
* [4] Tianran Chen, Robert Davis, and Evgeniia Korchevskaia. Facets and facet subgraphs of symmetric edge polytopes. Discrete Appl. Math., 328:139–153, 2023.
* [5] Tianran Chen, Robert Davis, and Dhagash Mehta. Counting equilibria of the Kuramoto model using birationally invariant intersection index. SIAM J. Appl. Algebra Geom., 2(4):489–507, 2018.
* [6] Alessio D’Alì, Emanuele Delucchi, and Mateusz Michałek. Many faces of symmetric edge polytopes. Electron. J. Combin., 29(3):Paper No. 3.24, 42, 2022.
* [7] Alessio D’Alì, Martina Juhnke-Kubitzke, and Melissa Koch. On a generalization of symmetric edge polytopes to regular matroids, 2023\. arXiv:2307.04933.
* [8] Takayuki Hibi and Akiyoshi Tsuchiya. Reflexive polytopes arising from perfect graphs. Journal of Combinatorial Theory, Series A, 157:233–246, 2018.
* [9] Akihiro Higashitani, Katharina Jochemko, and Mateusz Michałek. Arithmetic aspects of symmetric edge polytopes. Mathematika, 65(3):763–784, 2019.
* [10] Tamás Kálmán and Lilla Tóthmérész. $h^{*}$-vectors of graph polytopes using activities of dissecting spanning trees, 2023. arXiv:2203.17127.
* [11] Tetsushi Matsui, Akihiro Higashitani, Yuuki Nagazawa, Hidefumi Ohsugi, and Takayuki Hibi. Roots of Ehrhart polynomials arising from graphs. J. Algebr. Comb., 34(4):721–749, 2011.
* [12] Hidefumi Ohsugi and Takayuki Hibi. Centrally symmetric configurations of integer matrices. Nagoya Math. J., 216:153–170, 2014.
* [13] Hidefumi Ohsugi and Akiyoshi Tsuchiya. The $h^{*}$-polynomials of locally anti-blocking lattice polytopes and their $\gamma$-positivity. Discrete Comput. Geom., 66(2):701–722, 2021.
* [14] James G. Oxley. Matroid theory. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1992.
* [15] Richard P. Stanley. Combinatorics and commutative algebra, volume 41 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, second edition, 1996.
* [16] Bernd Sturmfels. Gröbner bases and convex polytopes, volume 8 of University Lecture Series. American Mathematical Society, Providence, RI, 1996.
* [17] Lilla Tóthmérész. A geometric proof for the root-independence of the greedoid polynomial of eulerian branching greedoids, 2022. arXiv:2204.12419.
|
# Segmentation of Breast Microcalcifications: a Multi-Scale Approach
Chrysostomos Marasinou, Bo Li, Jeremy Paige, Akinyinka Omigbodun, Noor
Nakhaei,
Anne Hoyt, William Hsu, Senior Member, IEEE This work was supported in part
by the National Science Foundation under Grant No. 1722516 and the Department
of Radiological Sciences through the generous support of the Iris Cantor
Foundation.C. Marasinou, L. Bo, J. Paige, A. Omigbodun, A. Hoyt, and W. Hsu
are with the David Geffen School of Medicine at UCLA, Department of
Radiological Sciences, Los Angeles, CA 90024, United States of America (email:
[email protected])N. Nakhaei is with the University of California, Los
Angeles, Department of Computer Science, Los Angeles, CA 90024, United States
of America
###### Abstract
Accurate characterization of microcalcifications (MCs) in 2D full-field
digital screening mammography is a necessary step towards reducing diagnostic
uncertainty associated with the callback of women with suspicious MCs.
Quantitative analysis of MCs has the potential to better identify MCs that
have a higher likelihood of corresponding to invasive cancer. However,
automated identification and segmentation of MCs remains a challenging task
with high false positive rates. We present Hessian Difference of Gaussians
Regression (HDoGReg), a two stage multi-scale approach to MC segmentation.
Candidate high optical density objects are first delineated using blob
detection and Hessian analysis. A regression convolutional network, trained to
output a function with higher response near MCs, chooses the objects which
constitute actual MCs. The method is trained and validated on 435 mammograms
from two separate datasets. HDoGReg achieved a mean intersection over the
union of 0.670$\pm$0.121 per image, intersection over the union per MC object
of 0.607$\pm$0.250 and true positive rate of 0.744 at 0.4 false positive
detections per $cm^{2}$. The results of HDoGReg perform better when compared
to state-of-the-art MC segmentation and detection methods.
###### Index Terms:
breast cancer, full-field digital mammography, microcalcifications,
segmentation
## I Introduction
Breast cancer is the most prevalent cancer among women. More than two million
new cases and over 620,000 deaths were estimated for 2018 worldwide [1].
Studies have shown that early detection using mammography reduces breast
cancer mortality [2]. In many countries, screening programs have been
established with sensitivity levels for detecting cancers ranging between
80-95% [3, 4]. However, screening also results in false positive outcomes,
including the identification of clinically insignificant cancers, raising
concerns about overdetection.
Microcalcifications (MCs), which are small calcium deposits, are common
mammographic findings where they typically appear as bright structures. Nearly
50% of the biopsied MCs are associated with ductal carcinoma in situ (DCIS)
[5], a largely benign form of cancer but a nonobligate precursor to invasive
cancer [6, 7]. MCs are reported by radiologists using a set of qualitative
descriptors related to shape (morphology) and distribution, as defined by the
American College of Radiology Breast Imaging Reporting and Data System (ACR
BI-RADS). Descriptors correspond to varying levels of suspicion for cancer.
For example, amorphous MCs are assigned a moderate suspicion level (i.e., BI-
RADS 4B) with a positive predictive value (PPV) of 21% [8]. However,
interpretation of MCs varies by radiologist due to subtle differences in their
size, shape, texture, and inhomogeneity of the background tissue [9]. Hence,
determining whether a group of calcifications is malignant is challenging, and
the current PPV of assessing suspicious MCs on 2D mammography is in the range
of 20-41% [10].
Many computerized methods have been developed to aid radiologists in detecting
MCs [11, 12, 13, 14, 15, 9]. These methods, generally categorized as computer-
aided detection (CADe) systems, automatically mark groups of suspicious MCs in
mammograms. Current CADe systems achieve high sensitivity but at the cost of a
large number of false positive marks per mammogram, increasing the
interpretation time. Furthermore, most of the automated methods identify
regions with suspicious MCs but do not delineate the exact shape of individual
MCs. Studies have shown that using shape and intensity features from segmented
MCs can improve malignancy classification [16, 17, 18, 19, 20]. Precise
segmentation also allows for more accurate quantitative characterization of
shape and distribution of MCs and texture analysis of the surrounding breast
parenchyma that could be used to classify cancerous regions better.
In this paper, a quantitative approach for characterizing MCs based on their
shape is demonstrated. Given a 2D full-field digital mammogram, we initially
identify bright salient structures using the difference of Gaussians’ (DoG)
blob detection algorithm. Hessian analysis is then applied to segment these
structures. Next, dense regression is applied to segment regions containing
structures that are likely to be MCs. Dense regression has been used for
similar tasks such as cell & nuclei detection [21, 22], retinal optical disc
and fovea detection [23], and focal vascular lesion localization on brain MRI
[24]. The idea is that human experts’ reference annotations are mapped to a
smooth proximity function that reaches its maximum value when corresponding to
the annotated points. Dense regression models are then trained to map the
input mammogram to the proximity function. The proximity function method is
advantageous when the manual annotations are incomplete (e.g., impractical to
annotate every possible MC) or when objects are annotated by a single pixel
rather than their actual boundaries (e.g., many MCs are tiny and time-
consuming to delineate the boundary). To perform dense regression, a
pretrained, fully convolutional network with pretrained weights is utilized.
The outputs of the dense regression model and the blob segmentation algorithm
are combined to generate the final MC segmentation.
The novelty of our work is summarized as follows:
* •
Proximity functions represent individual MCs, providing a way to accommodate
uncertainty in boundaries as part of classifier training.
* •
A dense regression model and a novel blob segmentation algorithm are applied
to generate MCs’ accurate segmentation while achieving fewer false positives
than other state-of-the-art algorithms.
### I-A Prior Work
Detection of MCs generally occurs in two steps: (i) detection of individual
MCs, (ii) identification of clusters by grouping individual MCs following a
set of spatial clustering rules. For detecting individual MCs, various
approaches have been investigated, ranging from traditional image processing
techniques to deep learning methods. [11] used a DoG filter for spot detection
and shape reconstruction by morphological filtering. [12] used 2D wavelet
transform to suppress the low frequency background and enhance small signals.
[13] applied a multiscale spot detector based on Laplacian filters and size
and contrast estimation based on filter response. [14] used support vector
machines with the input being local image windows to classify locations as MCs
or non-MCs. [15] extracted local features using a predefined filter bank and
applied supervised learning with a boosting classifier. Most of these methods
focused on detecting the point locations of MCs, disregarding the task of
extracting their shape. More recently, [9] developed a context-sensitive deep
convolutional neural network to classify MCs’ presence at locations. In their
approach, candidate locations were generated using DoG blob detection,
filtering out non-MCs using a convolutional neural network. In contrast, our
study performs MC segmentation (i.e., each MC is characterized by its shape
and spatial location).
Several studies have examined MC segmentation. Techniques include wavelet
transform to isolate high frequency components [25, 26], gray-level
morphological operations [11, 27, 28, 29, 30], fuzzy logic [31], and machine
learning for binary pixel classification [17]. Although segmentation of MCs is
performed in these studies, all but one [30]) evaluated the performance of
their algorithm as a detection task, not a segmentation task. Reference [30]
reported an intersection over the union of 70.8% between the segmented MCs and
the radiologist annotations on a set of 200 regions. We have reimplemented
their approach, comparing its performance to our HDoGReg approach using the
same dataset.
### I-B Paper Organization
The rest of the paper is organized as follows. In section II, we present the
data utilized for training and validating our model. In section III, we
formulate the problem and describe the methodology of Hessian Difference of
Gaussians Regression (HDoGReg) and the corresponding evaluation metrics. The
experiments for validating our models are described in section IV, where the
comparison methods are given. In section V, we present and discuss the
results. Finally, conclusions are drawn in section VI.
## II Data
Two distinct full-field digital mammography datasets were used in this work:
The publicly available INbreast dataset and cases seen at our institution
(UCLA).
INbreast: For model training and internal validation, we utilized a public
dataset called INbreast [32], a collection of 2D mammograms generated using a
Siemens MammoNovation system. A total of 115 screening cases with 410 images
were collected at a 0.070 mm per pixel resolution and 14-bit greyscale. The
dataset included detailed annotations provided by two experts for several
types of lesions (i.e., masses, MCs, asymmetries, and distortions). Fifty-six
cases had pathology-confirmed diagnoses, out of which 45 were cancerous (DCIS
and invasive). We used 294 images with annotations of individual MCs. MCs were
annotated in two ways: (1) small MCs were annotated by a single pixel to
denote their location and (2) larger MCs were annotated using pixel-wise
contours. It should be noted that the guideline of what was considered small
versus larger MC was not reported [32].
UCLA: As an additional external test set, we utilized data collected
retrospectively from patients who had a mammogram performed by our
institution, following an institutional review board-approved protocol. The
dataset consisted of 79 diagnostic cases with 141 full-field digital images
where MCs were present. All images were acquired using Hologic Selenia full-
field digital mammography devices at a 0.070 mm per pixel resolution and
12-bit greyscale. After collecting the data, suspicious MCs were annotated by
a breast fellowship-trained, board-certified radiologist. An open-source
medical image viewer, Horos, was utilized to generate the annotations.
Individual MCs were annotated by single pixels indicating their locations. To
assess the annotation task’s interreader reliability, a sample of 5 cases was
annotated by a second board-certified radiologist. The index of specific
agreement and the kappa statistic were determined. The two radiologists’
agreement was moderate, with an index of specific agreement of 0.664
(0.606-0.729, 95% confidence intervals), see supplementary
material111Supplementary material will be made available when published.
## III Methods
The overall approach to MC segmentation, HDoGReg, is illustrated in Figure 1.
Figure 1: HDoGReg: Approach for segmenting individual MCs. While the
segmentation is performed on entire 2D full-field mammograms, for
visualization purposes, a small patch is shown. In the upper branch, blob
segmentation is performed to segment bright blob-like and tubular structures.
In the lower branch, a regression convolutional neural network gives a
continuous function with higher response close to MCs. A threshold is then
applied to segment regions where MCs are likely present. The two branches’
output is combined based on an overlap criterion, resulting in the final
segmentation mask.
### III-A Blob segmentation
The first stage in HDoGReg is the segmentation of granular structures that are
candidate MCs. To generate candidate MC segments, we developed Hessian DoG for
blob segmentation. This module’s objective is to obtain an accurate
segmentation of bright salient structures that are candidate MC objects, as
shown in Figure 1.
Scale-space theory is a framework formulated to represent signals at multiple
scales. The Gaussian scale-space representation of an image $I(x,y)$ is
defined as [33]:
$\displaystyle L(x,y;\sigma)=G(x,y;\sigma)\ast I(x,y),$ (1)
where $\ast$ is the convolution and $G(x,y,\sigma)$ the two-dimensional
Gaussian function
$\displaystyle G(x,y;\sigma)$
$\displaystyle=\frac{1}{2\pi\sigma^{2}}e^{-(x^{2}+y^{2})/(2\sigma^{2})}$ (2)
In the DoG method, blobs with associated scale levels are detected from scale-
space maxima of the scale-normalized DoG function. The normalized DoG function
is defined as:
$\displaystyle DoG(x,y;\sigma)$
$\displaystyle=\frac{\sigma}{\Delta\sigma}\left(L(x,y;\sigma+\Delta\sigma)-L(x,y;\sigma)\right)$
(3)
where $\Delta\sigma$ is the difference between two scales. To construct the
DoG scale-space representation, a sequence of scales is considered
$\sigma_{n}=k^{n}\sigma_{\textrm{min}}$ where $k$ is a constant multiplicative
factor and $n=[0,1,\cdots,n_{\textrm{max}}]$. The DoG representations (3) are
computed for all adjacent scales (i.e.,
$\Delta\sigma=\sigma_{n+1}-\sigma_{n}$) forming a 3-dimensional
representation:
$\displaystyle DoG(x,y,n)$
$\displaystyle=\frac{\sigma_{n}}{\sigma_{n+1}-\sigma_{n}}\left(L(x,y;\sigma_{n+1})-L(x,y;\sigma_{n})\right)$
(4)
with $x,y$ the two spatial dimensions and $n=[0,1,\cdots,n_{\max-1}]$ a scale
dimension. Local-maxima in the 3-dimensional representation are computed
giving a blob set $(x^{(i)},y^{(i)},\sigma^{(i)})$ where $i$ identifies each
blob. The number of blob detections is controlled by a threshold,
$T_{\textrm{DoG}}$, that is applied as a lower bound on the stacked
representation before obtaining the local maxima. Moreover, in the case of
overlapping blobs, the smaller blob is eliminated if the overlapping fraction
is greater than the threshold $O_{\textrm{DoG}}$.
We extended this method using Hessian analysis to achieve blob segmentation.
The geometrical structure of a blob-like object can be described by the
eigenvalues of the Hessian [34]. In particular, a bright blob-like structure
corresponds to two negative and large eigenvalues, whereas a bright tubular
structure corresponds to one large negative eigenvalue and a small eigenvalue
of an arbitrary sign. These structures correspond to the target MC candidates.
The Hessian of the DoG representation at scale $\sigma$ is given by
$\displaystyle\textrm{HDoG}(x,y;\sigma)=\begin{pmatrix}\frac{\partial^{2}DoG(x,y;\sigma)}{\partial
x^{2}}&\frac{\partial^{2}DoG(x,y;\sigma)}{\partial x\partial y}\\\
\frac{\partial^{2}DoG(x,y;\sigma)}{\partial x\partial
y}&\frac{\partial^{2}DoG(x,y;\sigma)}{\partial y^{2}}\end{pmatrix}$ (5)
HDoG is computed across all scales in the sequence $\sigma_{n}$. At each scale
the following constraints are imposed:
$\displaystyle\textrm{tr}(H)<0\quad\land\quad\left(\det(H)<0\quad\lor\quad\frac{|\det(H)|}{\textrm{tr}(H)^{2}}\leq
h_{\textrm{thr}}\right)$ (6)
where $h_{\textrm{thr}}$ is a tunable parameter. The constraints ensure that
the Hessian is either negative definite or has a small positive eigenvalue. In
this way, only bright salient blob-like and tubular structures are segmented.
The constraint generates a binary mask at each scale. Iterating over the blob
set found in the DoG algorithm $(x^{(i)},y^{(i)},\sigma^{(i)})$, the
corresponding objects are found in the Hessian masks. More specifically, for
the Hessian mask at scale $\sigma^{(i)}$, the object spanning the location
$(x^{(i)},y^{(i)})$ is found. The final blob segmentation result is comprised
of all detected objects merged into a single binary mask.
Our HDoG algorithm works on a predetermined scale range
$[\sigma_{\min},\sigma_{\max}]$, which was fine-tuned on the validation set to
achieve a high true positive rate (>90%). The segmented objects from this
phase were considered candidate MC objects and filtered in the subsequent step
to reduce the number of false positives.
### III-B Regression Convolutional Neural Network
Figure 2: FPN with Inception-v4 encoder used for regression. In the encoding
branch the image is processed with the Inception-v4 classification network.
Skip connections (dashed lines) are inserted after layers where the output was
reduced in spatial size by factors of 4, 8, 16 and 32 respectively. The skip
connections feed FPN blocks where they are processed by convolutions. The
outputs are up-sampled independently by factors of 1, 2, 4 and 8 respectively.
Their outputs are added and inserted in a spatial dropout layer activated only
during training for regularization purposes. After dropout, convolutions are
followed by an up-sampling by a factor of 4 to match the original image size
and the sigmoid activation function.
In the second stage, we identified regions where MCs are most likely to be
present in the mammogram. The task here was to segment the MCs’ area and use
the segmentation to choose relevant MC objects from the previous stage. We
performed this task using a fully convolutional neural network, previously
used in image segmentation, as a regression model. The model’s output is a
smooth proximity map reaching a maximum value at the predicted MC locations.
Segmentation approaches commonly perform binary classification on each pixel.
However, applying this method requires precise pixel-wise annotations for all
objects of interest appearing in our images. As previously described,
generating precise annotations of MCs is challenging, and the reference
annotations only provide the point locations of MCs, not their exact
boundaries.
The task of MC region segmentation is analogous to that of cell & nuclei
detection in microscopy images. The two tasks share the following
characteristics: (1) they are highly imbalanced ( i.e., the positive class
[MCs or cells/nuclei] captures a small region compared to the background
within an image, and it often consists of many small structures), (2) the
background is highly inhomogeneous, (3) individual objects (MCs or
cells/nuclei) exhibit large variation in sizes, shapes, and textures, (4)
boundaries of the structures are often blurry, (5) the resolution of both
types of images is large, and (6) the annotations are usually a mixture of
individual points or exact boundaries.
Inspired by this analogy, we adapted methods previously used in cell & nuclei
detection [21, 22]. In [21], the authors proposed regression as a method for
detecting cell centers. The human-annotated binary masks containing cell
centers’ locations were transformed into a continuous function flat on the
background with localized peaks at each cell’s center. These functions were
then used to train a Random Forest Regression algorithm on a set of image
patches. The cell centers were identified with a local maxima in the model’s
output. In [22], the authors showed that the same technique could be applied
using a deep learning model. Their regression model was a fully convolutional
neural network with a large receptive field capable of encoding high-
resolution information.
Our MC segmentation model is formulated as follows: Given a mask generated
from reference annotations $M(x,y)\in\\{0,1\\}$, the MC locations are given by
$\\{(x_{i},y_{i})\\}$ where $M(x_{i},y_{i})=1$. The proximity function is then
defined as:
$\displaystyle P(x,y)$ $\displaystyle=\max_{i}g(x,y,x_{i},y_{i})$ (7)
$\displaystyle g(x,y,x_{i},y_{i})$
$\displaystyle=\begin{cases}(e^{\alpha(1-r/\xi)}-1)/(e^{\alpha}-1),&r\leq\xi\\\
0,&r>\xi\end{cases}$ (8) $\displaystyle r$
$\displaystyle=\sqrt{(x-x_{i})^{2}+(y-y_{i})^{2}}$ (9)
where $\alpha,\xi$ are tunable parameters. The function maps MC locations on
an exponentially curved surface, expanding to a distance $\xi$ with decay rate
$\alpha$ before it vanishes. An example of the transformation is illustrated
in Figure 3. This transformed mask compensated for the fact that we had mixed
quality annotations (i.e., point-like and exact) and forced the model to learn
information from both the precise locations of MCs and the surrounding
background.
(a)
(b)
(c)
Figure 3: (a) A mammographic image patch which includes MCs; (b) The
corresponding annotation mask; (c) The corresponding proximity function map
with parameters $\xi=10$ and $\alpha=1$.
We constructed a model which predicts the proximity function $P(x,y)$ given
the image $I(x,y)$. A feature pyramid network (FPN) [35] was used with
Inception-v4 [36] as the backbone. The FPN architecture was introduced for
applications such as region proposal, object detection, and instance
segmentation. It adopts a pyramidal shape structure similar to many
segmentation networks, such as the U-net [37], with an encoder that produces
semantic features at different scales and a decoder that combines the encoder
features by upsampling them.
The FPN structure is a suitable architecture because it allows features from
all scales to contribute to the final prediction independently. An
illustration of the network is presented in Figure 2. The network consists of
an encoding and a decoding branch. In the encoding branch, the Inception-v4
architecture was adopted with weights pretrained on ImageNet [38]. Features
were extracted at four different scales (down-sampled compared to the original
image by factors of 4, 8, 16, and 32). The features were then transferred to
the decoding branch via skip connections. They were upsampled by factors of 1,
2, 4, and 8, respectively, to match their spatial sizes. The resulting
features were aggregated using addition and further upsampled to match the
image size. The number of output channels was set to 1 and passed through a
sigmoid function to generate a value between 0 and 1. This value was
thresholded to achieve the final segmentation.
The model was trained using a soft Dice loss function, which was introduced as
an optimization objective in biomedical segmentation applications [39, 40].
The formulation in [40] was used:
$\displaystyle
L_{\textrm{DICE}}(\hat{P},P)=1-\frac{2\sum\limits_{x,y}P(x,y)\hat{P}(x,y)+\epsilon}{\sum\limits_{x,y}(P(x,y)+\hat{P}(x,y))+\epsilon}$
(10)
with $\epsilon$ set to 1 where $\epsilon$ was introduced for numerical
stability and $P$ and $\hat{P}$ correspond to the target and predicted
proximity map, respectively. A segmentation binary mask was generated by
applying a cut-off on the resulting proximity mask, $\hat{P}(x,y)\geq p_{thr}$
where $p_{thr}\in[0,1]$.
### III-C Combining approach
HDoG resulted in the segmentation of bright objects, whereas the regression
CNN outputted a mask of MC regions. These outputs were combined to achieve the
segmentation of individual MCs. The objects identified with HDoG were
candidate MC objects, and their shape had to remain intact. Our approach chose
a subset of these candidates in their entirety. We retained the HDoG objects
that have an overlap $\leq o_{\textrm{thr}}$ with the CNN region mask, where
$o_{\textrm{thr}}$ is a tunable percentage parameter.
### III-D Evaluation metrics
The performance of HDoGReg was assessed using Intersection over the Union
(IoU). We defined IoU per object as the averaged IoU between each reference
annotation object and the object that has the most overlap with in the
prediction mask222MC objects annotated by a single pixel were disregarded in
computing IoU per MC object. IoU is not well-defined in such cases.. To
evaluate the image-wise segmentation, the mean IoU between the background and
the positive MC class per image was computed. The IoU per MC object was
measured to examine the performance of segmenting individual MCs.
HDoGReg was also evaluated as a MC detection task using Free-Response
Operating Characteristic (FROC) analysis, similar to prior work [14, 9]. In
FROC analysis, the true positive detection rate was contrasted with false
positive detection per image. The analysis required the definition of
localization rules to determine true positives. We defined a detected object
as a true positive if its distance from a ground truth object was at most 5
pixels (0.35 mm)333Centroids of individual objects were used in computing
their distance. or if it demonstrated an IoU value of at least 0.3 with a
ground truth object.
## IV Experiments
(a)
(b)
(c)
(d)
Figure 4: Five 256x256 patches extracted from different diagnostic mammograms
showing the results of HDoGReg and a comparison method. From left to right:
(a) unannotated images, (b) reference annotations, (c) results using HDoGReg,
and (d) results using [30]. The first three rows are from INbreast data and
the last two are from UCLA data. For better visualization the patches were
normalized. Note the inherent difference in the appearance of the mammograms
between INbreast and UCLA data.
### IV-A Training, Validation, and Test Sets
We partitioned the INbreast dataset into a training set with 51 cases (173
images), a validation set with 17 cases (56 images), and a test set with 18
cases (65 images). The UCLA external test dataset was partitioned into a
training set used for fine-tuning the model with 112 images and a held-out
test set with 29 images. We used the INbreast validation set to fine-tune
HDoGReg, and the INbreast test set to assess the performance. Cases were kept
independent (i.e., all images from an individual case were included within the
same subset) to avoid potential bias.
### IV-B Blob Segmentation Optimization
With HDoG blob segmentation, our goal was to achieve a maximized IoU per
object and a true positive rate (TPR) higher than 90%. We extracted
representative patches from the INbreast validation set and optimized HDoG
hyperparameters on them to satisfy the above criteria.
Through experimentation on the validation set, we set $\sigma_{\min}=1.18$,
$\sigma_{\max}=3.1$, overlapping fraction $O_{\textrm{DoG}}=1$, DoG threshold
$T_{\textrm{DoG}}=0.006$ and Hessian threshold $h_{\textrm{ thr}}=1.4$. On the
validation set, our algorithm achieved an IoU per object 0.648 and a TPR of
0.903 at 136.5 false positives per unit area ($cm^{2}$).
### IV-C Regression Model Training and Tuning
To train the regression model, we extracted patches from the images and
corresponding masks. We applied the sliding window approach with patch size
512 pixels and stride 480 to permit overlapping patches. Only patches with
annotated MCs present were considered. From INbreast cases, a total of 1045
patches were extracted from the training set and 329 from the validation set.
From UCLA cases, a total of 252 patches from the training set were extracted.
The mask patches were transformed using the proximity function map (7). We set
$\xi=\\{6,8,10,12\\}$ for the characteristic distance and
$\alpha=\\{-1,-2,10^{-4},1,2\\}$ for the decay rate. The proximity function
(7) is not well defined when $\alpha=0$.
A regression neural network was trained using the image patches of the
INbreast training set as input and the corresponding proximity function maps
as output. Data augmentation was performed to enrich the training set by
randomly applying horizontal flipping, magnification, spatial translations in
both directions, cropping, contrast enhancement, brightness adjustment, and
gamma correction (details are given in Supplementary Materials). The resulting
patches had size 320x320 pixels. The soft Dice loss was used to compute the
error between target and predicted proximity functions (10). The model was
trained for 40 epochs using the adaptive moment estimation (Adam) optimization
method [41] with batch size 8, learning rate $10^{-4}$, $\beta_{1}=0.9$,
$\beta_{2}=0.999$ and $\epsilon=10^{-8}$. At the end of each epoch, the model
was evaluated on the image patches of the INbreast validation set, with the
average IoU per patch as the metric. The model achieving the highest IoU over
all epochs was kept. The Inception-v4 weights were initialized with weights
pretrained on ImageNet. The rest of the model weights were initialized
randomly following He initialization [42]. The configuration $\xi=10$,
$\alpha=1$ achieved the highest performance for HDoGReg on the validation set.
We updated our HDoGReg model to work with data from our institution by
updating the model with an additional 40 epochs using a combination of patches
from the INbreast and UCLA datasets.
### IV-D Combining Approach Optimization
The combination approach keeps the HDoG candidate objects based on how much
they overlap with the regression model mask. Optimizing the value of
$o_{\textrm{thr}}=\\{0.2,0.3,0.4,0.5,0.6\\}$, $o_{\textrm{thr}}=0.3$ achieved
the highest performance on the validation set.
### IV-E Comparison
We compared HDoGReg against two state-of-the-art methods. For the MC detection
task, we compared our approach to Wang and Yang [9], which used two
subnetworks, focusing on the local features and the other on features
extracted from the background tissue around the location. They reported a
detection performance of 80% true positive fraction (TPF) at a false positive
rate of 1.03 FPs/cm2. We implemented their context-sensitive deep neural
network, which classifies a location as MC or non-MC, training our
implementation on the INbreast dataset. We implemented DoG based on their
reported parameters adjusting the scales to the resolution of our dataset.
We compared our MC segmentation results with the approach presented in [30].
In [30], MC segmentation was performed using morphological operations. In the
first step, morphological operators were applied to the original image to
detect the MCs’ locations. Specifically, a morphological pyramid was generated
using the closing-opening filter. Differences in the pyramid representations
of the original image were obtained and combined using the extended maximum of
the original image and morphological reconstruction. In the second step, the
MC shapes were extracted using watershed segmentation, where the output of the
first step was utilized as a marker. We report the mean IoU per image and the
IoU per object for this approach.
## V Results
### V-A Regression Model Selection
The entire segmentation pipeline was evaluated using the INbreast validation
set. The FROC analysis is presented in Table I, and the mean IoU per image and
IoU per object are summarized in Table II. For the FROC analysis, a total of
100 bootstrap samples were used to find the partial area under the curve
(pAUC) in each experiment. The pAUC was computed for the range between 0 and 1
FPs per unit area, and the 95% confidence interval is reported. For the
computation of the segmentation metrics, a threshold on the predicted
proximity function was applied. To determine the optimal threshold for each
experiment, we referred to the corresponding FROC curve and found the point
closest to TPR 1 and false positives per unit area 0. All configurations
performed similarly in terms of the FROC analysis and segmentation metrics. We
chose the model with the highest mean value of the FROC pAUC with $\xi=10$ and
$\alpha=1$.
TABLE I: Individual MC FROC pAUC values for validating different regression models. The highest pAUC is bolded. $\alpha/\xi$ | 6 | 8 | 10 | 12
---|---|---|---|---
-2 | 0.804$\pm$0.048 | 0.812$\pm$0.049 | 0.793$\pm$0.040 | 0.773$\pm$0.045
-1 | 0.783$\pm$0.058 | 0.808$\pm$0.047 | 0.790$\pm$0.047 | 0.794$\pm$0.045
10-4 | 0.813$\pm$0.054 | 0.783$\pm$0.056 | 0.790$\pm$0.044 | 0.784$\pm$0.048
1 | 0.799$\pm$0.054 | 0.776$\pm$0.056 | 0.819$\pm$0.046 | 0.790$\pm$0.053
2 | 0.775$\pm$0.056 | 0.791$\pm$0.055 | 0.802$\pm$0.049 | 0.789$\pm$0.053
TABLE II: Segmentation results of different regression models on validation
set. The highest IoUs are bolded. Mean IoU per image
---
$\alpha/\xi$ | 6 | 8 | 10 | 12
-2 | 0.593$\pm$0.109 | 0.590$\pm$0.102 | 0.576$\pm$0.088 | 0.560$\pm$0.066
-1 | 0.590$\pm$0.105 | 0.579$\pm$0.094 | 0.572$\pm$0.089 | 0.566$\pm$0.077
10-4 | 0.596$\pm$0.115 | 0.584$\pm$0.098 | 0.586$\pm$0.097 | 0.584$\pm$0.093
1 | 0.619$\pm$0.125 | 0.601$\pm$0.109 | 0.583$\pm$0.105 | 0.587$\pm$0.098
2 | 0.610$\pm$0.123 | 0.588$\pm$0.108 | 0.592$\pm$0.105 | 0.580$\pm$0.094
IoU per Object
$\alpha/\xi$ | 6 | 8 | 10 | 12
-2 | 0.645$\pm$0.207 | 0.645$\pm$0.206 | 0.648$\pm$0.200 | 0.626$\pm$0.228
-1 | 0.647$\pm$0.203 | 0.648$\pm$0.201 | 0.643$\pm$0.208 | 0.640$\pm$0.212
10-4 | 0.648$\pm$0.201 | 0.644$\pm$0.207 | 0.641$\pm$0.214 | 0.644$\pm$0.207
1 | 0.649$\pm$0.200 | 0.646$\pm$0.206 | 0.647$\pm$0.203 | 0.643$\pm$0.208
2 | 0.650$\pm$0.197 | 0.648$\pm$0.202 | 0.649$\pm$0.200 | 0.643$\pm$0.208
### V-B Individual Microcalcifications
Table III reports the segmentation results of HDoGReg on the INbreast
validation and test sets. For comparison, the segmentation results of the
morphological method of Ciecholewski et al. (see IV-E), are also presented.
Using the paired Wilcoxon signed-rank test, we showed that HDoGReg achieved
superior performance in both mIoU per image and IoU per object for both
subsets with $p<0.01$. Figure 4 presents a sampling of model outputs.
TABLE III: Segmentation Results of Final Model on Validation and Test Sets
HDoGReg
---
Metric/dataset | Validation | Test
mean IoU per image | 0.583$\pm$0.105 | 0.670$\pm$0.121
IoU per object | 0.647$\pm$0.203 | 0.607$\pm$0.250
Ciecholewski [30]
Metric/dataset | Validation | Test
mean IoU per image | 0.517$\pm$0.037 | 0.524$\pm$0.034
IoU per object | 0.408$\pm$0.286 | 0.363$\pm$0.278
Figure 5: Individual MC FROC analysis for final HDoGReg model and comparison
with baseline model. Figure 6: Individual MC FROC analysis on UCLA data.
Figure 5 presents the individual MC FROC analysis between HDoGReg and [9]. In
this plot, the true positive detection rate on the y-axis was plotted against
the false positive counts per unit area (1 $cm^{2}$). The FROC analysis was
performed on the INbreast validation and test sets. HDoGReg achieved FROC pAUC
0.819$\pm$0.046 with a TPR of 0.852 at 0.4 false positives per unit area on
the validation set. On the test set, the FROC pAUC was 0.697$\pm$0.078 with a
TPR of 0.744 at 0.4 false positives per unit area. In comparison, our
implementation of the approach in [9] achieved FROC pAUC 0.703$\pm$0.057 and
0.581$\pm$0.072 in the validation and test sets.
Figure 6 shows the detection performance of the final model on the UCLA data.
We compared the performance of the model trained solely on INbreast data and
the model fine-tuned on data from UCLA. The performance of the two models was
comparable since the original model achieved 0.313$\pm$0.109 FROC pAUC and the
fine-tuned model achieved 0.420$\pm$0.107. However, for the range 0.2 to 0.6
FPs per unit area, the fine-tuned model outperformed the original based on
TPR.
### V-C Microcalcification Clustering Analysis
MCs can be clinically significant when they appear in a group or a
distribution that would suggest deposits in a duct. In this section, we
analyzed HDoGReg segmentation masks to obtain MC clusters and assessed the
relationship between the extracted clusters and five predefined groups based
on radiologist-interpreted mammographic distributions. To cluster individual
MCs into groups, we applied a method called Ordering Points To Identify the
Clustering Structure (OPTICS) [43]. OPTICS operates on density-based
considerations to identify clusters among a given set of points in a multi-
dimensional space. OPTICS defines a notion of neighborhood density of points,
based on a threshold for the minimum number of adjacent points within a region
of a fixed radius. To perform the clustering, we mapped all MC objects in our
segmentation mask to their centroid locations and used them as input to
OPTICS. Upon clustering, a total of 24 expert-defined features were extracted
describing each cluster’s size, shape, and density of MCs (details are given
in the supplementary material). The features for all clusters were
standardized and fed into K-means clustering with k=5, corresponding to the
number of predefined groups.
The correspondence between the clustering approach versus radiologist
descriptors was evaluated as follows: (1) A board-certified fellowship-trained
breast radiologist provided labels for mammographic distribution at the image
level. Some (n=13, 10.7%) images were given multiple labels based on the
dominant and secondary distribution types that appeared in the image. (2) Each
detected cluster was assigned into one of five groups using the k-means
algorithm. (3) To associate each point (defined by the features of each
cluster) with a distribution label, we duplicated the clusters appearing in
images with multiple distribution labels and associated each copy with one
distribution label. Thus, we ended up with a list of distributions and their
associated K-means group. (4) We then computed the homogeneity score h [44],
which is a measure describing the purity of all clusters. If all clusters
contain points of a single class, h has the value of 1.0. OPTICS was fine-
tuned using a random grid search of 100 experiments. We hypothesized that the
MC clustering with the highest homogeneity score is the one closest to
capturing the actual mammographic distributions. Our clustering approach
achieved a homogeneity score $h=0.074$ on the INbreast dataset. Using the
extracted quantitative features, we achieved a moderate level of
correspondence to predefined radiologist-provided labels of distribution.
## VI Discussion
We present an approach that combines the difference of Gaussians with Hessian
analysis and dense regression to achieve precise MC segmentation in full-field
2D digital mammograms. To our knowledge, this is one of the first works
applying a fully convolutional architecture for MC segmentation, which permits
concurrent prediction on multiple adjacent locations. The method was trained
and validated on 435 mammograms from two separate datasets. The results showed
that HDoGReg outperformed comparable approaches that have been recently
published. In terms of the FROC analysis using the INbreast dataset, our
method achieved a TPR of 0.744 at 0.4 false positives per unit area in
comparison with 0.35 TPR at the same false positive counts as what was
presented in [9]. On the segmentation task, HDoGReg achieved 0.670 mean IoU
per image and 0.607 IoU per object compared to 0.524 mean IoU per image and
0.363 IoU per object for the morphological approach presented in [30]. The
addition of UCLA data, even when coarsely annotated by a human reader,
improved the performance of HDoGReg. The ability to utilize a mixture of
annotations, including exact segmentation of larger calcifications or
individual points representing the centroid of smaller calcifications, is a
strength of our approach.
While HDoGReg achieved a lower number of false positives compared to other
approaches, the overall number of false positives per image is still high. The
majority of false positives occur near larger calcifications and correspond to
more irregular shapes compared to actual MCs. The irregular detection can be
attributed to the regression model which was designed to segment regions
containing calcifications. In the case of larger calcifications, the segmented
regions span larger areas, increasing the likelihood of retaining false-
positive objects. Additional filtering based on size and shape criteria in
areas where large calcifications are identified could lead to a substantial
false positive reduction. Given that annotating every possible MC is
impractical, our algorithm likely identified MCs that were not annotated by
human readers, contributing to the false positive count. HDoGReg also
undersegments or oversegments in certain scenarios. Undersegmentation occurs
most often in large objects due to: (1) interior regions of objects having
lower intensities that are omitted and (2) incorrect delineation of boundaries
due to subtle contrast differences between the MC and surrounding tissue.
Nevertheless, large calcifications are typically considered benign and not
clinically significant. Their undersegmentation will have little effect on
quantitative features that may predict invasive cancers. Oversegmentation
occurs primarily when bright objects identified with HDoG are close together
and erroneously combined into a single object when only part of it corresponds
to an actual MC.
Several limitations of our approach exist. Labeling all MCs in full-field
mammograms is a time-consuming process and prone to human error and inter-
annotator variability. Hence, our work is limited by the dataset size and by
variations in how MCs are annotated, ranging from point-like annotations to
detailed contours. Our use of a proximity function to reflect the uncertainty
associated with MC annotations allows our approach to be robust to training
data variations. Moreover, the inherent differences in mammograms acquired
with equipment manufactured by different vendors present another challenge.
The UCLA dataset was obtained using equipment manufactured by Hologic while
the public dataset INbreast were obtained using Siemens equipment. The
brightness and contrast levels of the images varied substantially between
manufacturers. Given that the INbreast dataset had four times as many cases as
the UCLA dataset, our model was fine-tuned with a limited number of training
patches from UCLA. Ongoing work includes annotating additional UCLA cases that
would allow us to further fine-tune the model and experiment with different
training strategies to improve the generalizability of our approach.
Currently, a model with an encoder initialized with pretrained weights was
trained first on INbreast data and fine-tuned on mixed UCLA and INbreast data.
Provided additional UCLA annotated data, an approach where mixed data are
utilized from the beginning of training can be examined. In addition, the
significance of using pretrained weights could be explored. Additionally,
magnification views acquired as part of diagnostic mammograms typically
improve the conspicuity of confirmed suspicious or benign regions. One
direction of future research is to improve the conspicuity of findings in
magnification views, potentially yielding more accurate segmentations.
In summary, we presented HDoGReg, a new quantitative approach for MC
segmentation, based on blob segmentation and dense regression. We showed that
HDoGReg achieves better performance in comparison with state-of-the-art MC
segmentation and detection methods. Additionally, we explored how segmented
MCs can be clustered in an unsupervised manner, resulting in five groups that
are loosely related to qualitative descriptors of MC distributions provided by
radiologists. Our results suggest that HDoGReg could be used in large-scale,
multi-center studies to define reliable quantitative descriptions of MCs
associated with malignancy. Our work serves as the basis for improved
quantitative characterization of MCs. A number of shape, intensity, and
texture features can be extracted from individually segmented MCs and used to
yield quantitative descriptors of MC morphology and distribution. We expect
that quantitative descriptors of morphology and distribution may differ from
the subjective labels assigned by radiologists. Further studies are needed to
evaluate the PPV of our quantitative features and their ability to identify
MCs that are associated with invasive cancers.
## Acknowledgment
We gratefully acknowledge the support of the NVIDIA Corporation. Also, we
would like to thank the Breast Research Group at University of Porto for
developing and sharing the INbreast dataset.
## References
* [1] F. Bray, J. Ferlay, I. Soerjomataram, R. L. Siegel, L. A. Torre, and A. Jemal, “Global cancer statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries,” _CA: A Cancer Journal for Clinicians_ , vol. 68, no. 6, pp. 394–424, 2018.
* [2] M. G. Marmot, D. G. Altman, D. A. Cameron, J. A. Dewar, S. G. Thompson, M. Wilcox, and T. I. U. P. on Breast Cancer Screening, “The benefits and harms of breast cancer screening: an independent review,” _British Journal of Cancer_ , vol. 108, no. 11, pp. 2205–2240, 2013.
* [3] K. Kemp Jacobsen, E. S. O’Meara, D. Key, D. S.M. Buist, K. Kerlikowske, I. Vejborg, B. L. Sprague, E. Lynge, and M. von Euler-Chelpin, “Comparing sensitivity and specificity of screening mammography in the united states and denmark,” _International Journal of Cancer_ , vol. 137, no. 9, pp. 2198–2207, 2015.
* [4] M. v. Euler-Chelpin, M. Lillholm, G. Napolitano, I. Vejborg, M. Nielsen, and E. Lynge, “Screening mammography: benefit of double reading by breast density,” _Breast Cancer Research and Treatment_ , vol. 171, no. 3, pp. 767–776, Oct 2018.
* [5] G. Farshid, T. Sullivan, P. Downey, P. G. Gill, and S. Pieterse, “Independent predictors of breast malignancy in screen-detected microcalcifications: Biopsy results in 2545 cases,” _British Journal of Cancer_ , vol. 105, no. 11, pp. 1669–1675, 2011.
* [6] S. Hofvind, B. F. Iversen, L. Eriksen, B. M. Styr, K. Kjellevold, and K. D. Kurz, “Mammographic morphology and distribution of calcifications in ductal carcinoma in situ diagnosed in organized screening,” _Acta Radiologica_ , vol. 52, no. 5, pp. 481–487, jun 2011.
* [7] R. F. Cox, A. Hernandez-Santana, S. Ramdass, G. McMahon, J. H. Harmey, and M. P. Morgan, “Microcalcifications in breast cancer: Novel insights into the molecular mechanism and functional consequence of mammary mineralisation,” _British Journal of Cancer_ , vol. 106, no. 3, pp. 525–537, 2012.
* [8] E. A. Sickles, D. C. J., B. L. W. _et al._ , “ACR BI-RADS® Mammography. In: ACR BI-RADS® Atlas, Breast Imaging Reporting and Data System,” 2013.
* [9] J. Wang and Y. Yang, “A context-sensitive deep learning approach for microcalcification detection in mammograms,” _Pattern Recognition_ , vol. 78, pp. 12 – 22, 2018.
* [10] L. Wilkinson, V. Thomas, and N. Sharma, “Microcalcification on mammography: Approaches to interpretation and biopsy,” _British Journal of Radiology_ , vol. 90, no. 1069, 2017.
* [11] J. Dengler, S. Behrens, and J. F. Desaga, “Segmentation of Microcalcifications in Mammograms,” _IEEE Transactions on Medical Imaging_ , vol. 12, no. 4, pp. 634–642, 1993.
* [12] H. Yoshida, K. Doi, and R. M. Nishikawa, “Automated detection of clustered microcalcifications in digital mammograms using wavelet processing techniques,” _Medical Imaging 1994: Image Processing_ , vol. 2167, no. May 1994, pp. 868–886, 1994.
* [13] T. Netsch and H. O. Peitgen, “Scale-space signatures for the detection of clustered microcalcifications in digital mammograms,” _IEEE Transactions on Medical Imaging_ , vol. 18, no. 9, pp. 774–786, 1999.
* [14] I. El-Naqa, Y. Yang, M. N. Wernick, N. P. Galatsanos, and R. M. Nishikawa, “A support vector machine approach for detection of microcalcifications,” _IEEE Transactions on Medical Imaging_ , vol. 21, no. 12, pp. 1552–1563, 2002\.
* [15] A. Oliver, A. Torrent, X. Lladó, M. Tortajada, L. Tortajada, M. Sentís, J. Freixenet, and R. Zwiggelaar, “Automatic microcalcification and cluster detection for digital and digitised mammograms,” _Knowledge-Based Systems_ , vol. 28, pp. 68–75, 2012.
* [16] H. Strange, Z. Chen, E. R. Denton, and R. Zwiggelaar, “Modelling mammographic microcalcification clusters using persistent mereotopology,” _Pattern Recognition Letters_ , vol. 47, pp. 157–163, 2014.
* [17] A. Bria, N. Karssemeijer, and F. Tortorella, “Learning from unbalanced data: A cascade-based approach for detecting clustered microcalcifications,” _Medical Image Analysis_ , vol. 18, no. 2, pp. 241–252, 2014.
* [18] Z. Chen, H. Strange, A. Oliver, E. R. Denton, C. Boggis, and R. Zwiggelaar, “Topological Modeling and Classification of Mammographic Microcalcification Clusters,” _IEEE Transactions on Biomedical Engineering_ , vol. 62, no. 4, pp. 1203–1214, 2015.
* [19] N. Alam, E. R. Denton, and R. Zwiggelaar, “Classification of microcalcification clusters in digital mammograms using a stack generalization based classifier,” _Journal of Imaging_ , vol. 5, no. 9, 2019\.
* [20] H. Cai, Q. Huang, W. Rong, Y. Song, J. Li, J. Wang, J. Chen, and L. Li, “Breast Microcalcification Diagnosis Using Deep Convolutional Neural Network from Digital Mammograms,” _Computational and Mathematical Methods in Medicine_ , vol. 2019, pp. 1–10, 2019.
* [21] P. Kainz, M. Urschler, S. Schulter, P. Wohlhart, and V. Lepetit, “You should use regression to detect cells,” in _Medical Image Computing and Computer-Assisted Intervention – MICCAI 2015_ , N. Navab, J. Hornegger, W. M. Wells, and A. F. Frangi, Eds. Cham: Springer International Publishing, 2015, pp. 276–283.
* [22] Y. Xie, F. Xing, X. Shi, X. Kong, H. Su, and L. Yang, “Efficient and robust cell detection: A structured regression approach,” _Medical Image Analysis_ , vol. 44, pp. 245 – 254, 2018.
* [23] M. I. Meyer, A. Galdran, A. M. Mendonça, and A. Campilho, “A pixel-wise distance regression approach for joint retinal optical disc and fovea detection,” _Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)_ , vol. 11071 LNCS, pp. 39–47, 2018.
* [24] K. M. H. van Wijnen, F. Dubost, P. Yilmaz, M. A. Ikram, W. J. Niessen, H. Adams, M. W. Vernooij, and M. de Bruijne, “Automated lesion detection by regressing intensity-based distance with a neural network,” in _Medical Image Computing and Computer Assisted Intervention – MICCAI 2019_ , D. Shen, T. Liu, T. M. Peters, L. H. Staib, C. Essert, S. Zhou, P.-T. Yap, and A. Khan, Eds. Cham: Springer International Publishing, 2019, pp. 234–242.
* [25] R. Strickland and Hee Il Hahn, “Wavelet transforms for detecting microcalcifications in mammograms,” _IEEE Transactions on Medical Imaging_ , vol. 15, no. 2, pp. 218–229, 2002.
* [26] E. Regentova, L. Zhang, J. Zheng, and G. Veni, “Microcalcification detection based on wavelet domain hidden Markov tree model: Study for inclusion to computer aided diagnostic prompting system,” _Medical Physics_ , vol. 34, no. 6, pp. 2206–2219, 2007.
* [27] D. Betal, N. Roberts, and G. H. Whitehouse, “Segmentation and numerical analysis of microcalcifications on mammograms using mathematical morphology,” _British Journal of Radiology_ , vol. 70, no. SEPT., pp. 903–917, sep 1997.
* [28] S. Halkiotis and J. Mantas, “Automatic detection of clustered microcalcifications in digital mammograms,” _Studies in Health Technology and Informatics_ , vol. 90, pp. 24–29, 2002.
* [29] S. Xu, H. Liu, and E. Song, “Marker-controlled watershed for lesion segmentation in mammograms,” _Journal of Digital Imaging_ , vol. 24, no. 5, pp. 754–763, 2011.
* [30] M. Ciecholewski, “Microcalcification Segmentation from Mammograms: A Morphological Approach,” _Journal of Digital Imaging_ , vol. 30, no. 2, pp. 172–184, apr 2017.
* [31] Heng-Da Cheng, Yui Man Lui, and R. Freimanis, “A novel approach to microcalcification detection using fuzzy logic technique,” _IEEE Transactions on Medical Imaging_ , vol. 17, no. 3, pp. 442–450, 2002.
* [32] I. C. Moreira, I. Amaral, I. Domingues, A. Cardoso, M. J. Cardoso, and J. S. Cardoso, “Inbreast: Toward a full-field digital mammographic database,” _Academic Radiology_ , vol. 19, no. 2, pp. 236 – 248, 2012.
* [33] T. Lindeberg, “Feature Detection with Automatic Scale Selection,” _International Journal of Computer Vision_ , vol. 30, no. 2, pp. 79–116, 1998\.
* [34] A. F. Frangi, W. J. Niessen, K. L. Vincken, and M. A. Viergever, “Multiscale vessel enhancement filtering,” in _Medical Image Computing and Computer-Assisted Intervention — MICCAI’98_ , W. M. Wells, A. Colchester, and S. Delp, Eds. Berlin, Heidelberg: Springer Berlin Heidelberg, 1998, pp. 130–137.
* [35] T. Lin, P. Dollár, R. Girshick, K. He, B. Hariharan, and S. Belongie, “Feature pyramid networks for object detection,” in _2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2017, pp. 936–944.
* [36] C. Szegedy, S. Ioffe, V. Vanhoucke, and A. A. Alemi, “Inception-v4, inception-ResNet and the impact of residual connections on learning,” _31st AAAI Conference on Artificial Intelligence, AAAI 2017_ , pp. 4278–4284, 2017.
* [37] O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” _Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)_ , vol. 9351, pp. 234–241, 2015.
* [38] P. Yakubovskiy, “Segmentation models pytorch,” https://github.com/qubvel/segmentation_models.pytorch (accessed 1 November 2020), 2020.
* [39] F. Milletari, N. Navab, and S. A. Ahmadi, “V-Net: Fully convolutional neural networks for volumetric medical image segmentation,” _Proceedings - 2016 4th International Conference on 3D Vision, 3DV 2016_ , pp. 565–571, 2016\.
* [40] M. Drozdzal, E. Vorontsov, G. Chartrand, S. Kadoury, and C. Pal, “The importance of skip connections in biomedical image segmentation,” _Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)_ , vol. 10008 LNCS, pp. 179–187, 2016.
* [41] D. P. Kingma and J. L. Ba, “Adam: A method for stochastic optimization,” _3rd International Conference on Learning Representations, ICLR 2015 - Conference Track Proceedings_ , pp. 1–15, 2015.
* [42] K. He, X. Zhang, S. Ren, and J. Sun, “Delving deep into rectifiers: Surpassing human-level performance on imagenet classification,” in _Proceedings of the 2015 IEEE International Conference on Computer Vision (ICCV)_ , ser. ICCV ’15. USA: IEEE Computer Society, 2015, p. 1026–1034.
* [43] M. Ankerst, M. M. Breunig, H.-p. Kriegel, and J. Sander, “OPTICS: Ordering Points To Identify the Clustering Structure,” _ACM SIGMOD Record_ , vol. 28, no. 2, pp. 49–60, 1999.
* [44] A. Rosenberg and J. Hirschberg, “V-measure: A conditional entropy-based external cluster evaluation measure,” in _Proceedings of the 2007 Joint Conference on Empirical Methods in Natural Language Processing and Computational Natural Language Learning (EMNLP-CoNLL)_. Prague, Czech Republic: Association for Computational Linguistics, Jun. 2007, pp. 410–420.
|
11institutetext: Department of Physics, Chungbuk National University,
Cheongju 28644, Republic of Korea
11email<EMAIL_ADDRESS>22institutetext: Max Planck Institute
for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany 33institutetext:
Department of Astronomy, The Ohio State University, 140 W. 18th Ave.,
Columbus, OH 43210, USA 44institutetext: Astronomical Observatory, University
of Warsaw, Al. Ujazdowskie 4, 00-478 Warszawa, Poland 55institutetext:
Institute of Information and Mathematical Sciences, Massey University, Private
Bag 102-904, North Shore Mail Centre, Auckland, New Zealand 66institutetext:
Dipartimento di Fisica ‘E.R. Caianiello’, Universit a di Salerno, Via Giovanni
Paolo 132, Fisciano I-84084, Italy 77institutetext: Korea Astronomy and Space
Science Institute, Daejon 34055, Republic of Korea 88institutetext: University
of Canterbury, Department of Physics and Astronomy, Private Bag 4800,
Christchurch 8020, New Zealand 99institutetext: Department of Particle Physics
and Astrophysics, Weizmann Institute of Science, Rehovot 76100, Israel
1010institutetext: Center for Astrophysics $|$ Harvard & Smithsonian 60 Garden
St., Cambridge, MA 02138, USA 1111institutetext: Department of Astronomy and
Tsinghua Centre for Astrophysics, Tsinghua University, Beijing 100084, China
1212institutetext: School of Space Research, Kyung Hee University, Yongin,
Kyeonggi 17104, Republic of Korea 1313institutetext: Korea University of
Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon, 34113, Republic
of Korea 1414institutetext: Department of Physics, University of Warwick,
Gibbet Hill Road, Coventry, CV4 7AL, UK 1515institutetext: Institute for
Space-Earth Environmental Research, Nagoya University, Nagoya 464-8601, Japan
1616institutetext: Astrophysics Science Division, NASA/Goddard Space Flight
Center, Greenbelt, MD20771, USA 1717institutetext: Laboratory for Exoplanets
and Stellar Astrophysics, NASA / Goddard Space Flight Center, Greenbelt, MD
20771, USA 1818institutetext: Department of Astronomy, University of Maryland,
College Park, MD 20742, USA 1919institutetext: Department of Earth and
Planetary Science, Graduate School of Science, The University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo 2020institutetext: Instituto de Astrofísica de
Canarias, Vía Láctea s/n, E-38205 La Laguna, Tenerife, Spain
2121institutetext: Department of Earth and Space Science, Graduate School of
Science, Osaka University, 1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan
2222institutetext: Department of Astronomy, Graduate School of Science, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
2323institutetext: Code 667, NASA Goddard Space Flight Center, Greenbelt, MD
20771, USA 2424institutetext: Zentrum für Astronomie der Universität
Heidelberg, Astronomisches Rechen-Institut, Mönchhofstr. 12-14, 69120
Heidelberg, Germany 2525institutetext: Department of Physics, University of
Auckland, Private Bag 92019, Auckland, New Zealand 2626institutetext:
Department of Physics, The Catholic University of America, Washington, DC
20064, USA 2727institutetext: Institute of Space and Astronautical Science,
Japan Aerospace Exploration Agency, 3-1-1 Yoshinodai, Chuo, Sagamihara,
Kanagawa, 252-5210, Japan 2828institutetext: University of Canterbury Mt. John
Observatory, P.O. Box 56, Lake Tekapo 8770, New Zealand
# Four sub-Jovian-mass planets detected by high-cadence microlensing surveys
Cheongho Han 0101 Doeon Kim 0101 Andrew Gould 02020303 Andrzej Udalski 0404
Ian A. Bond 0505 Valerio Bozza 0606 Youn Kil Jung
(Leading authors)
Michael D. Albrow 07070808 Sun-Ju Chung 0707 Kyu-Ha Hwang 0707 Yoon-Hyun Ryu
0707 In-Gu Shin 0707 Yossi Shvartzvald 0909 Jennifer C. Yee 1010 Weicheng Zang
1111 Sang-Mok Cha 07071212 Dong-Jin Kim 0707 Seung-Lee Kim 0707 Chung-Uk Lee
0707 Dong-Joo Lee 0707 Yongseok Lee 07071313 Byeong-Gon Park 07071313 Richard
W. Pogge
(The KMTNet Collaboration) 0303
Przemek Mróz 0404 Michał K. Szymański 0404 Jan Skowron 0404 Radosław Poleski
0404 Igor Soszyński 0404 Paweł Pietrukowicz 0404 Szymon Kozłowski 0404
Krzysztof Ulaczyk 1414 Krzysztof A. Rybicki 0404 Patryk Iwanek
(The OGLE Collaboration) 0404
Fumio Abe 1515 Richard K. Barry 1515 David P. Bennett 17171818 Aparna
Bhattacharya 17171818 Hirosane Fujii 1515 Akihiko Fukui 19192020 Yuki Hirao
161617172121 Yoshitaka Itow 1515 Rintaro Kirikawa 2121 Naoki Koshimoto 2121
Iona Kondo 2121 Yutaka Matsubara 1515 Sho Matsumoto 2121 Shota Miyazaki 2121
Yasushi Muraki 2121 Greg Olmschenk 2323 Arisa Okamura 2121 Clément Ranc 2424
Nicholas J. Rattenbury 2525 Yuki Satoh 2121 Stela Ishitani Silva 17172626
Takahiro Sumi 2121 Daisuke Suzuki 2727 Taiga Toda 2121 Paul J. Tristram 2828
Aikaterini Vandorou 2323 Hibiki Yama
(The MOA Collaboration)
2121
(Received ; accepted)
###### Abstract
Aims. With the aim of finding short-term planetary signals, we investigated
the data collected from the high-cadence microlensing surveys.
Methods. From this investigation, we found four planetary systems with low
planet-to-host mass ratios, including OGLE-2017-BLG-1691L, KMT-2021-BLG-0320L,
KMT-2021-BLG-1303L, and KMT-2021-BLG-1554L. Despite the short durations,
ranging from a few hours to a couple of days, the planetary signals were
clearly detected by the combined data of the lensing surveys. It is found that
three of the planetary systems have mass ratios of the order of $10^{-4}$ and
the other has a mass ratio slightly greater than $10^{-3}$.
Results. The estimated masses indicate that all discovered planets have sub-
Jovian masses. The planet masses of KMT-2021-BLG-0320Lb, KMT-2021-BLG-1303Lb,
and KMT-2021-BLG-1554Lb correspond to $\sim 0.10$, $\sim 0.38$, and $\sim
0.12$ times of the mass of the Jupiter, and the mass of OGLE-2017-BLG-1691Lb
corresponds to that of the Uranus. The estimated mass of the planet host
KMT-2021-BLG-1554L, $M_{\rm host}\sim 0.08\leavevmode\nobreak\ M_{\odot}$,
corresponds to the boundary between a star and a brown dwarf. Besides this
system, the host stars of the other planetary systems are low-mass stars with
masses in the range of $\sim[0.3$–$0.6]\leavevmode\nobreak\ M_{\odot}$. The
discoveries of the planets well demonstrate the capability of the current
high-cadence microlensing surveys in detecting low-mass planets.
###### Key Words.:
gravitational microlensing – planets and satellites: detection
## 1 Introduction
During the 2010s, microlensing entered an era of high-cadence surveys with the
instrumental upgrade of previously established experiments and the
participation of a new experiment. The new era started when the Microlensing
Observations in Astrophysics survey (MOA: Bond et al., 2001), which had
previously carried out a lensing survey using a 0.61 m telescope and a camera
with a 1.3 deg2 field of view (FOV) during the early phase, launched its
second phase experiment with the employment of a 1.8 m telescope equipped with
a wide-field camera yielding a 2.2 deg2 FOV. Since its first operation in
1992, the Optical Gravitational Lensing Experiment (OGLE) has been upgraded
multiple times, and it is in its fourth phase (OGLE-IV: Udalski et al., 2015)
with the employment of a 1.3 m telescope and a camera with a 1.4 deg2 FOV. The
Korea Microlensing Telescope Network (KMTNet: Kim et al., 2016), which was
launched in 2016, is being carried out with the use of three globally
distributed 1.6 m telescopes, each of which is mounted with a wide-field
detector providing a 4 deg2 FOV. With the use of wide-field cameras mounted on
multiple telescopes, the current lensing surveys can monitor lensing events
with a dramatically enhanced observational cadence.
Table 1: Coordinates, fields, alert dates, and baseline magnitudes
Event | (RA, decl)J2000 | $(l,b)$ | Field | Alert date | $I_{\rm base}$ (mag)
---|---|---|---|---|---
OGLE-2017-BLG-1691 | (17:34:22.52, -29:17:05.39) | $(-1^{\circ}\hskip-2.0pt.601,1^{\circ}\hskip-2.0pt.895)$ | OGLE (BLG654.31) | 2017 Sep 6 | 19.90
(KMT-2017-BLG-0752) | | | KMT (BLG14) | postseason |
KMT-2021-BLG-0320 | (17:57:33.15, -30:30:14.18) | $(2^{\circ}\hskip-2.0pt.095,-4^{\circ}\hskip-2.0pt.299)$ | KMT (BLG01, BLG41) | 2021 Apr 9 | 20.08
KMT-2021-BLG-1303 | (18:07:27.33, -29:16:53.69) | $(-0^{\circ}\hskip-2.0pt.027,-3^{\circ}\hskip-2.0pt.028)$ | KMT (BLG04) | 2021 Jun 14 | 19.67
(MOA-2021-BLG-182) | | | MOA (gb13) | 2021 Jun 17 |
KMT-2021-BLG-1554 | (17:51:12.82, -31:51:51.70) | $(-1^{\circ}\hskip-2.0pt.888,-2^{\circ}\hskip-2.0pt.543)$ | KMT (BLG01, BLG41) | 2021 Jul 1 | 22.39
111 ${\rm HJD}^{\prime}={\rm HJD}-2450000$.
The greatly increased observational cadence of the lensing surveys has brought
out important changes not only in the observational strategy but also in the
outcome of planetary microlensing searches. First, being able to resolve
short-lasting planetary signals from survey observations, the high-cadence
surveys can function without the survey+followup experiment mode, in which
low-cadence surveys mainly detect and alert lensing events and followup groups
densely observe the alerted events. Second, the number of microlensing planets
has substantially increased with the operation of the high-cadence surveys.
Among the total 135 microlensing planets with measured mass ratios222The
sample includes 119 planets listed in the NASA Exoplanet Archive
(https://exoplanetarchive.ipac.caltech.edu/index.html) plus 16 planets that
are not included in the archive: KMT-2020-BLG-0414Lb (Zang et al., 2021a),
OGLE-2019-BLG-1053Lb (Zang et al., 2021b), KMT-2019-BLG-0253Lb,
KMT-2019-BLG-0953Lb, OGLE-2018-BLG-0977Lb, OGLE-2018-BLG-0506Lb,
OGLE-2018-BLG-0516Lb, OGLE-2019-BLG-1492Lb (Hwang et al., 2022),
KMT-2017-BLG-2509Lb, OGLE-2017-BLG-1099Lb, OGLE-2019-BLG-0299Lb (Han et al.,
2021), OGLE-2016-BLG-1093Lb (Shin et al., 2022), KMT-2018-BLG-1988Lb (Han et
al., 2022b), KMT-2021-BLG-0912Lb (Han et al., 2022a), OGLE-2019-BLG-0468Lb,
and OGLE-2019-BLG-0468Lc (Han et al., 2022c)., 80 (59%) planets were found
with the use of the data from the KMTNet survey. This can be seen in Figure 1,
in which we present the histogram of microlensing planets as a function of the
planet-to-host mass ratio $q$. The histogram also shows that the high-cadence
surveys contribute to the detections of planets with very low mass ratios,
especially those with $q<10^{-4}$.
Figure 1: Histogram of published microlensing planets as a function of the
planet-to-host mass ratio. The histogram of planets detected with the data
from KMTNet survey is separately marked in brown. The solid dots with error
bars indicate the four planets reported in this work.
In this paper, we report four planetary systems with low planet-to-host mass
ratios detected from the microlensing surveys. Despite the short durations,
ranging from a few hours to a couple of days, the planetary signals were
clearly detected by the combined data of the lensing surveys. It is found that
all of the discovered planets have sub-Jovian masses lying in the range of
[0.05–0.38] $M_{\rm J}$, illustrating the importance of the high-cadence
surveys in detecting low-mass planets.
We present the analysis of the planetary lensing events according to the
following organization. In Sect. 2, we describe the observations of the
individual lensing events, instrument used for observations, and the process
of the data reduction. In Sect. 3, we explain the procedure of analysis that
is applied to each of the individual events. In the subsequent subsections, we
describe the detailed features of the planetary signals and present the
results of the analyses conducted for the individual lensing events. In Sect.
4, we specify the types of the source stars and estimate the angular Einstein
radii of the lens systems. In Sect. 5, we estimate the physical parameters of
the planetary systems by conducting Bayesian analyses. We summarize the
results of the analyses and conclude in Sect. 6.
## 2 Observations and data
The four lensing events for which we present analyses are (1)
OGLE-2017-BLG-1691 (KMT-2017-BLG-0752), (2) KMT-2021-BLG-0320, (3)
KMT-2021-BLG-1303 (MOA-2021-BLG-182), and (4) KMT-2021-BLG-1554. The source
stars of all events lie toward the Galactic bulge field. In Table 1, we list
the equatorial and galactic coordinates, observation fields of the surveys,
alert dates, and baseline magnitudes of the individual events. The baseline
magnitude $I_{\rm base}$ is approximately scaled to the OGLE-III photometry
system.
The events were detected from the combined observations of the lensing surveys
conducted by the KMTNet, OGLE, and MOA groups. The KMTNet survey utilizes
three identical 1.6 m telescopes that are distributed in three countries of
the Southern Hemisphere for the continuous coverage of lensing events. The
sites of the individual telescopes are the Siding Spring Observatory in
Australia (KMTA), the Cerro Tololo InterAmerican Observatory in Chile (KMTC),
and the South African Astronomical Observatory in South Africa (KMTS).
Observations by the OGLE survey were conducted using the 1.3 m telescope
located at Las Campanas Observatory in Chile. The MOA survey uses the 1.8 m
telescope at the Mt. John Observatory in New Zealand. Observations were done
mainly in the $I$ band for the KMTNet and OGLE surveys and in the customized
MOA-$R$ band for the MOA survey. A subset of images were acquired in the $V$
band for the purpose of estimating source colors. The detailed procedure of
the source color estimation will be discussed in Sect. 4.
Table 2: Error bar readjustment factors Event | Data set | $k$ | $\sigma_{\rm min}$ (mag)
---|---|---|---
OGLE-2017-BLG-1691 | OGLE | 1.010 | 0.010
| KMTA | 0.866 | 0.040
| KMTC | 1.170 | 0.010
| KMTS | 1.131 | 0.020
KMT-2021-BLG-0320 | KMTA (BLG01) | 1.127 | 0.040
| KMTA (BLG41) | 1.389 | 0.025
| KMTC (BLG01) | 1.091 | 0.020
| KMTC (BLG41) | 1.175 | 0.020
| KMTS (BLG01) | 1.356 | 0.010
| KMTS (BLG41) | 1.319 | 0.010
KMT-2021-BLG-1303 | KMTA | 1.304 | 0.020
| KMTC | 1.237 | 0.010
| KMTS | 1.364 | 0.010
| MOA | 1.034 | 0.020
KMT-2021-BLG-1554 | KMTA (BLG01) | 1.371 | 0.020
| KMTA (BLG41) | 1.446 | 0.020
| KMTC (BLG01) | 1.418 | 0.020
| KMTC (BLG41) | 1.367 | 0.020
| KMTS (BLG41) | 1.364 | 0.020
Reduction and photometry of the data were done using the pipelines of the
individual survey groups: Albrow et al. (2009) for the KMTNet survey, Woźniak
(2000) for the OGLE survey, and Bond et al. (2001) for the MOA survey. All
these pipelines apply the difference image analysis algorithm (Tomaney &
Crotts, 1996; Alard & Lupton, 1998) developed for the optimal photometry of
stars lying in very dense star fields. For a subset of the KMTC data set, we
carried out additional photometry utilizing the pyDIA code (Albrow, 2017) for
the specification of the source colors. Following the routine described in Yee
et al. (2012), we rescale the error bars of data by $\sigma=k(\sigma_{\rm
min}^{2}+\sigma_{0}^{2})^{1/2}$, where $\sigma_{0}$ represents the error
estimated from the photometry pipeline, $\sigma_{\rm min}$ is a factor used to
make the data consistent with the scatter of data, and the factor $k$ is used
to make $\chi^{2}$ per degree of freedom for each data set become unity. In
Table 2, we list the factors $k$ and $\sigma_{\rm min}$ of the individual data
sets.
## 3 Analyses
The light curves of the analyzed events share a common characteristic that
short-lived anomalies appear on the otherwise smooth and symmetric form of a
single-lens single-source (1L1S) event. Such anomalies in lensing light curves
can be produced by two channels, in which the first channel is a perturbation
induced by a low-mass companion, such as a planet, to the lens (Gould & Loeb,
1992), and the second channel is an anomaly caused by a faint companion to the
source (Gaudi, 1998). Hereafter we denote events with a binary lens and a
binary source as 2L1S and 1L2S events, respectively. We examine the origins of
the anomalies by modeling the light curve of the individual events under these
2L1S and 1L2S interpretations.
Lensing light curves are described by the combination of various lensing
parameters. For a 1L1S event, the light curve is described by three parameters
of $(t_{0},u_{0},t_{\rm E})$, which denote the time of the closest approach of
the source to the lens, the separation (scaled to the angular Einstein radius
$\theta_{\rm E}$) between the source and lens at that time, and the time scale
of the event, respectively. The event time scale is defined as the time
required for a source to transit $\theta_{\rm E}$.
In addition to these basic parameters, describing the light curve of a 2L1S
event requires one to include extra parameters of $(s,q,\alpha)$, which
represent the separation (scaled to $\theta_{\rm E}$) and mass ratio between
the binary lens components, and the angle between the binary-lens axis and the
direction of the source motion (source trajectory angle), respectively.
Because planet-induced anomalies are usually generated by the crossings over
or close approach of the source to the planet-induced caustics, an additional
parameter of the normalized source radius $\rho$, which is defined as the
ratio of the angular radius of the source $\theta_{*}$ to $\theta_{\rm E}$, is
needed to describe the deformation of the anomaly by finite-source effects
(Bennett & Rhie, 1996). In computing finite-source magnifications, we consider
the limb-darkening variation of the source.
One also needs extra parameters to describe the lensing light curve of a 1L2S
event. These parameters are $(t_{0,2},u_{0,2},q_{F})$, which represent the
closest approach time and separation between the lens and the second source,
and the flux ratio between the binary source stars, respectively. In the 1L2S
model, we designate the lensing parameters related to the primary source,
$S_{1}$, as $(t_{0,1},u_{0,1})$ to distinguish them from those related to the
source companion, $S_{2}$. In order to consider finite-source effects
occurring when the lens passes over either of the source stars, we add two
additional parameters of the normalized source radii $\rho_{1}$ and $\rho_{2}$
for the lens transits over $S_{1}$ and $S_{2}$, respectively (Dominik, 2019).
In modeling 1L1S and 1L2S light curves, for which the lensing magnification is
smooth with the variation of the lensing parameters, we search for the
solution of the lensing parameters via a downhill approach by minimizing
$\chi^{2}$ using the Markow Chain Monte Carlo (MCMC) algorithm. In modeling
2L1S light curves, for which the lensing magnification variation is
discontinuous due to the formation of caustics and there may exist multiple
local solutions caused by various types of degeneracy, we model the light
curves in two steps. In the first step, we conduct grid searches for the
binary parameters $s$ and $q$, construct a $\chi^{2}$ map on the $s$–$q$
parameter plane, and then identify local solutions appearing on the $\chi^{2}$
map. In the second step, we polish the individual local solutions using the
downhill approach. Below we present the details of the modeling conducted for
the individual lensing events.
Figure 2: Lensing light curve of OGLE-2017-BLG-1691. The bottom panel shows
the whole view, and the top panel displays the enlargement of the region
around the anomaly (the region enclosed by a box in the bottom panel). In the
top panel, curves of the three tested models, 2L1S (outer), 1L2S, and 1L1S
models, are drawn over the data points, and the residuals from the individual
models are presented in the three middle panels. The curve drawn in the bottom
panel is the best-fit model (outer 2L1S model). Colors of data points are set
to match those of the telescopes used for observations marked in the legend.
The curves drawn in the 1L2S and 1L1S residual panels represent the
differences from the 2L1S model.
### 3.1 OGLE-2017-BLG-1691 (KMT-2017-BLG-0752)
The lensing event OGLE-2017-BLG-1691 occurred during the 2017 bulge season. It
was first found by the OGLE survey, and later confirmed by the KMTNet survey
from the post-season analyses of the data collected during the season. The
KMTNet group designated the event as KMT-2017-BLG-0752. Hereafter, we use the
nomenclatures of events by the ID references of the surveys who first found
the events in accordance with the convention of the microlensing community.
The baseline magnitude of the event was $I_{\rm base}=19.900\pm 0.004$.
The lensing light curve of OGLE-2017-BLG-1691 is shown in Figure 2, in which
the bottom panel shows the whole view and the top panel displays the
enlargement of the region around the anomaly. The event was alerted by the
OGLE survey on 2017 September 6, ${\rm HJD}^{\prime}\equiv{\rm
HJD}-2450000\sim 8002.5$, which approximately corresponds to the time near the
peak. An anomaly occurred about one day after the peak, but it was not noticed
during the progress of the event because it was covered by just a single OGLE
data point, and the KMTNet data were not released during the lensing
magnification. The existence of the anomaly was identified five years after
the event, shortly after the re-reduction of all 2017 KMT light curves, from a
project of reinvestigating the previous KMTNet data conducted to find
unnoticed planetary signals (Han et al., 2022b). The top panel of Figure 2
shows that the anomaly, which lasted for $\sim 5$ hours centered at ${\rm
HJD}^{\prime}\sim 8003.6$, was additionally covered by three KMTC data points,
and this confirmed that the single anomalous OGLE point was real.
Figure 3: Lens system configurations of OGLE-2017-BLG-1691: upper panel
(inner solution) and lower panel (outer solution). The inset in each panel
shows the whole view of the lens system, and the main panel presents the zoom-
in view of the caustics. The red cuspy figure is the caustic and the line with
an arrow is the source trajectory. The orange circle on the trajectory is
drawn to represent the source scale with respect to the caustic. The blue
solid dots marked by $M_{1}$ and $M_{2}$ indicate the positions of the host
and planet, respectively. The dashed circle with unit radius in the inset
represents the Einstein ring. Table 3: Lensing parameters of
OGLE-2017-BLG-1691
Parameter | 2L1S (Inner) | 2L1S (Outer) | 1L2S
---|---|---|---
$\chi^{2}$ | 866.9 | 866.5 | 880.4
$t_{0}$ (HJD′) | $8002.540\pm 0.010$ | $8002.531\pm 0.010$ | $8002.499\pm 0.012$
$u_{0}$ ($10^{-2}$) | $4.83\pm 0.27$ | $4.95\pm 0.26$ | $6.55\pm 0.49$
$t_{\rm E}$ (days) | $19.42\pm 0.83$ | $18.94\pm 0.82$ | $18.77\pm 0.83$
$s$ | $1.058\pm 0.011$ | $1.003\pm 0.014$ | –
$q$ ($10^{-4}$) | $0.708\pm 0.23$ | $0.97\pm 0.34$ | –
$\alpha$ (rad) | $3.873\pm 0.012$ | $3.865\pm 0.010$ | –
$\rho$ ($10^{-3}$) | $3.40\pm 0.47$ | $3.54\pm 0.52$ | –
$t_{0,2}$ (HJD′) | – | – | $8003.587\pm 0.015$
$u_{0,2}$ ($10^{-2}$) | – | – | $0.029\pm 0.124$
$\rho_{2}$ ($10^{-3}$) | – | – | $3.39\pm 0.93$
$q_{F}$ ($10^{-3}$) | – | – | $6.11\pm 1.85$
333 ${\rm HJD}^{\prime}={\rm HJD}-2450000$.
We model the light curve of the event under the 1L1S, 2L1S and 1L2S
interpretations. The residuals of the individual tested models are compared in
Figure 2. It is found that the 2L1S model best describes the anomaly, being
favored over the 1L1S and 1L2S models by $\Delta\chi^{2}=456.1$ and 13.9,
respectively. From the comparison of the residuals, it is found that the 2L1S
model is confirmed not only by the 4 data points (3 KMTC plus 1 OGLE points)
with positive deviations near the peak of the bump but also by the 3
additional points (2 KMTS and 1 KMTC points) before the major bump with
slightly negative deviations.
We find two degenerate 2L1S solutions with binary parameters of
$(s,q)\sim(1.058,0.71\times 10^{-4})$ and $\sim(1.003,0.97\times 10^{-4})$,
which we designate as “inner” and “outer” solutions, respectively, for the
reason to be mentioned below. We list the full lensing parameters of the two
2L1S solutions in Table 3, together with the parameters of the 1L2S model. The
degeneracy is very severe and the outer solution is favored only by
$\Delta\chi^{2}=0.4$. The lensing parameters of the solutions indicate that
the anomaly was produced by a very small mass-ratio planetary companion to the
lens lying close to the Einstein ring of the primary regardless of the
solutions. From the fact that both solutions have separations greater than
unity and do not follow the relation of $s_{\rm inner}\times s_{\rm
outer}\simeq 1$, the degeneracy is different from the “close-wide” degeneracy
(Griest & Safizadeh, 1998; Dominik, 1999; An, 2005), which arises due to the
similarity between the central caustics of planetary lens systems with planet
separations $s$ and $1/s$. Instead, the planet separations of the two
degenerate solutions follow the relation
$\sqrt{s_{\rm in}\times s_{\rm out}}=s^{\dagger};\qquad
s^{\dagger}={\sqrt{u_{\rm anom}^{2}+4}+u_{\rm anom}\over 2},$ (1)
where $u_{\rm anom}^{2}=\tau_{\rm anom}^{2}+u_{0}^{2}$, $\tau_{\rm
anom}=(t_{\rm anom}-t_{0})/t_{\rm E}$, and $t_{\rm anom}$ indicates the time
of the planetary anomaly (Hwang et al., 2022; Zhang et al., 2022; Ryu et al.,
2022). With the values $t_{0}\simeq 8002.5$, $u_{0}\simeq 4.85\times 10^{-2}$,
$t_{\rm E}\simeq 19$ day and $t_{\rm anom}\simeq 8003.6$, one finds that
$s^{\dagger}\simeq 1.04$, which matches well $\sqrt{s_{\rm in}\times s_{\rm
out}}\simeq 1.03$. This indicates that the similarity between the light curves
of the two solutions is caused by the degeneracy identified by Yee et al.
(2021), who first mentioned the continuous transition between the “close-wide”
and “inner-outer” (Gaudi & Gould, 1997) degeneracies. Hereafter, we refer to
this degeneracy as “offset degeneracy” following Zhang et al. (2022).
Figure 3 shows the lens systems configurations, in which the source trajectory
(line with an arrow) with respect to the caustic and positions of the lens
components (blue solid dots marked by $M_{1}$ and $M_{2}$) is presented: inner
solution in the upper panel and outer solution in the lower panel. The source
passed the inner side (with respect to $M_{1}$) of the caustic according to
the inner solution, while the source passes the outer side according to the
outer solution. For both solutions, the source crossed the cusp of the
caustic, and thus the anomaly is affected by finite-source effects during the
caustic crossing, allowing us to precisely measure the normalized source
radius. For each solution, the source size relative to the caustic is
represented by an orange circle marked on the source trajectory. As will be
discussed in Sect. 4, the measurement of $\rho$ is important to estimate the
lensing observable of the angular Einstein radius $\theta_{\rm E}$, which can
be used to constrain the physical lens parameters. However, the microlens
parallax vector, $\pi_{\rm E}=(\pi_{\rm rel}/\theta_{\rm
E})/(\mbox{\boldmath$\mu$}/\mu)$, which is another observable constraining the
physical lens parameters, cannot be securely measured because the event time
scale, $t_{\rm E}\sim 19$ days, is not long enough to produce detectable
deviations induced by the orbital motion of Earth around the Sun (Gould,
1992). Here $\mu$ represents the vector of the relative lens-source proper
motion.
### 3.2 KMT-2021-BLG-0320
The lensing event KMT-2021-BLG-0320 was found on 2021 April 9 (${\rm
HJD}^{\prime}\sim 9313$), when the event had not yet reached its peak, with
the employment of the KMTNet AlertFinder system (Kim et al., 2018), which
began full operation since the 2019 season. Two days after the detection, the
event reached its peak with a magnification of $A_{\rm max}\sim 170$, and then
gradually declined until it reached its baseline of $I_{\rm base}=20.08$. The
source of the event lies in the two KMTNet prime fields of BLG01 and BLG41,
toward which observations were conducted with a 30 min cadence for each field,
and thus with a 15 min combined cadence. The areas covered by the two fields
overlap except for about 15% of the area of each field filling the gaps
between the chips of the camera. Because the data were taken from two fields
of three telescopes, there are 6 data sets: KMTA (BLG01), KMTA (BLG41), KMTC
(BLG01), KMT (BLG41), KMTS (BLG01), and KMTS (BLG41).
Figure 4: Lensing light curve of KMT-2021-BLG-0320. Notations are same as those in Fig. 2. Table 4: Lensing parameters of KMT-2021-BLG-0320 Parameter | Close | Wide
---|---|---
$\chi^{2}$ | 4075.5 | 4075.5
$t_{0}$ (HJD′) | $9315.615\pm 0.001$ | $9315.615\pm 0.001$
$u_{0}$ ($10^{-3}$) | $5.86\pm 0.21$ | $5.93\pm 0.21$
$t_{\rm E}$ (days) | $13.12\pm 0.39$ | $12.97\pm 0.40$
$s$ | $0.771\pm 0.014$ | $1.274\pm 0.024$
$q$ ($10^{-4}$) | $3.02\pm 0.42$ | $2.89\pm 0.44$
$\alpha$ (rad) | $0.604\pm 0.014$ | $0.602\pm 0.014$
$\rho$ ($10^{-3}$) | – | –
In Figure 4, we present the light curve constructed from the combination of
the 6 KMTNet data sets. Although it would be difficult to notice the anomaly
from a glimpse, we inspected the light curve because the event reached a very
high magnification at the peak, near which the light curve is susceptible to
perturbations induced by a planet (Griest & Safizadeh, 1998). From this
inspection, it was found that the light curve exhibited an anomaly that lasted
for about 4 hours with a negative deviation with respect to a 1L1S model. See
the enlarged view around the peak region of the light curve presented in the
top panel of Figure 4. It is known that a planetary companion to a lens can
induce anomalies with both positive and negative deviations, while a faint
companion to a source can induce anomalies with only positive deviations
(Gaudi, 1998). We, therefore, modeled the light curve under the 2L1S
interpretation.
It is found that the anomaly is well described by a 2L1S model, in which the
mass ratio between the binary lens components is very low. We found two sets
of solutions with $(s,q)_{\rm close}\sim(0.77,3.0\times 10^{-4})$ and
$(s,q)_{\rm wide}\sim(1.27,2.9\times 10^{-4})$. We designate the individual
solutions as “close” and “wide” solutions, because the former solution has
$s<1.0$ and the latter has $s>1.0$. In Table 4, we list the full lensing
parameters for the two sets of solutions. The fits of the two solutions are
nearly identical with $\chi^{2}$ values that are equal to the first digit
after the decimal point, indicating that the degeneracy between the two
solutions is very severe. The fact that the binary separations of the two
solutions follow the relation of $s_{\rm close}\times s_{\rm wide}\simeq 1.0$
indicates that the similarity between the solution stems from the close-wide
degeneracy. It is known that the relation between the planet separations of
the solutions under the offset degeneracy in Equation (1) applies to more
general cases, including resonant case, than the $s_{\rm close}\times s_{\rm
wide}\simeq 1.0$ relation of the close-wide degeneracy, and we confirm this.
In addition to the relation between $s_{\rm in}$ and $s_{\rm out}$, Hwang et
al. (2022) provided analytic formulas for the heuristic estimation of the
source trajectory angle and the mass ratio;
$\alpha=\tan^{-1}\left({u_{0}\over\tau_{\rm anom}}\right);\qquad
q=\left({\Delta t_{\rm anom}\over 4t_{\rm
E}}\right)\left({s\over|u_{0}|}\right)\left|\sin^{3}\alpha\right|,$ (2)
where $\tau_{\rm anom}$ denotes the duration of the planet-induced anomaly. We
confirm that the heuristically estimated values of $\alpha$ and $q$ match well
those estimated from the modeling. It was found that the normalized source
radius could not be constrained, not even the upper limit, and thus the line
for $\rho$ in Table 4 is left blank.
Figure 5: Lens system configurations of KMT-2021-BLG-0320. The upper and
lower panels show the configurations of the close and wide solutions,
respectively.
Figure 5 shows the lensing configuration of KMT-2021-BLG-0320 for the close
(upper panel) and wide (lower panel) solutions. It shows that the source
passed the back-end region of the tiny central caustic induced by a planet,
and this generated the negative deviation of the observed anomaly. The central
caustics of the close and wide solutions are very similar, resulting in nearly
identical deviations. Because the source passed well outside the caustic,
finite-source effects could not be detected, and thus we do not mark an orange
circle representing the source size. Microlens-parallax effects could not be
securely detected because the event time scale, $t_{\rm E}\sim 13$ days, is
much shorter than the orbital period of Earth, that is, 1 year.
Figure 6: Lensing light curve of KMT-2021-BLG-1303. Notations are same as
those in Fig. 2.
### 3.3 KMT-2021-BLG-1303 (MOA-2021-BLG-182)
The event KMT-2021-BLG-1303 was first found on 2021 June 14 (${\rm
HJD}^{\prime}\sim 9379$) by the KMTNet survey, and later identified by the MOA
survey on 2021 June 17 (${\rm HJD}^{\prime}\sim 9382$). The MOA survey
designated the event as MOA-2021-BLG-182. A day after the first discovery, the
event displayed an anomaly that lasted for about 2 days. After the anomaly,
the event reached peak at ${\rm HJD}^{\prime}\sim 9385.1$ with a magnification
of $A_{\rm max}\sim 46$, and then gradually declined to the baseline of
$I_{\rm base}=19.67$.
The light curve of KMT-2021-BLG-1303 constructed with the combined KMTNet and
MOA data is shown in Figure 6. Compared to the previous two events, the light
curve of this event displays a very obvious anomaly with a maximum deviation
of $\Delta I\sim 1.4$ mag from a 1L1S model. The rapid brightening at ${\rm
HJD}^{\prime}\sim 9381.0$ indicates that the event experienced a caustic
crossing. Because caustics form due to the multiplicity of lens components, we
rule out the 1L2S origin of the anomaly and test only the 2L1S interpretation.
Table 5: Lensing parameters of KMT-2021-BLG-1303 Parameter | Value
---|---
$\chi^{2}$ | 1654.9
$t_{0}$ (HJD′) | $9385.091\pm 0.003$
$u_{0}$ ($10^{-2}$) | $2.17\pm 0.06$
$t_{\rm E}$ (days) | $25.21\pm 0.68$
$s$ | $1.029\pm 0.001$
$q$ ($10^{-4}$) | $6.42\pm 0.45$
$\alpha$ (rad) | $6.138\pm 0.001$
$\rho$ ($10^{-3}$) | $0.95\pm 0.08$
We find that the anomaly is well explained by a 2L1S model with a planetary
companion lying close to the Einstein ring of the host. We find a unique
solution without any degeneracy, and the binary parameters of the solution are
$(s,q)\sim(1.029,6.4\times 10^{-4})$. Here again the planet-to-host mass ratio
is of the order of $10^{-4}$ like the two previous events. We double checked
the uniqueness of the solution by thoroughly inspecting the region around the
$s$ and $q$ parameters predicted by the relations in Equations (1) and (2),
and confirmed that there is only a single solution. The full lensing
parameters of the event are listed in Table 5.
Figure 7: Lens system configuration of KMT-2021-BLG-1303. Notations are same as those in Fig. 3. Table 6: Lensing parameters of KMT-2021-BLG-1554 Parameter | 2L1S (Close) | 2L1S (Wide) | 1L2S
---|---|---|---
$\chi^{2}$ | 3486.0 | 3485.7 | 3496.1
$t_{0}$ (HJD′) | $9394.737\pm 0.010$ | $9394.742\pm 0.010$ | $9394.719\pm 0.019$
$u_{0}$ | $0.051\pm 0.009$ | $0.051\pm 0.011$ | $0.003\pm 0.054$
$t_{\rm E}$ (days) | $5.04\pm 0.78$ | $5.15\pm 0.72$ | $6.55\pm 1.11$
$s$ | $0.888\pm 0.039$ | $1.189\pm 0.051$ | –
$q$ ($10^{-4}$) | $14.14\pm 4.48$ | $14.94\pm 5.41$ | –
$\alpha$ (rad) | $4.705\pm 0.038$ | $4.727\pm 0.037$ | –
$\rho$ ($10^{-3}$) | $6.98\pm 1.18$ | $7.00\pm 1.28$ | $98.8\pm 24.2$
$t_{0,2}$ (HJD′) | – | – | $9394.740\pm 0.002$
$u_{0,2}$ | – | – | $0.002\pm 0.002$
$\rho_{2}$ ($10^{-3}$) | – | – | –
$q_{F}$ | – | – | $0.048\pm 0.008$
The lensing configuration of the event is presented in Figure 7, which shows
that the planet lies very close to the Einstein ring and induces a single six-
sided resonant caustic near the primary of the lens. The source entered the
caustic at ${\rm HJD}^{\prime}\sim 9381.0$, and this produced the sharp rise
of the light curve at the corresponding time. According to the model, the
source exited the caustic at ${\rm HJD}^{\prime}\sim 9384.2$, but the caustic-
crossing feature at this epoch is not obvious in the lensing light curve due
to the combination of the weak caustic and the poor coverage of the region.
The pattern of the anomaly between the caustic entrance and exit deviates from
a typical “U”-shape pattern because the source passed along the fold of the
caustic. The light curve during the caustic entrance of the source was well
resolved by the KMTA data, and thus the normalized source radius,
$\rho=(0.95\pm 0.08)\times 10^{-3}$, is tightly constrained. However, it was
difficult to constrain the microlens parallax because the time scale, $t_{\rm
E}\sim 25$ days, is not long enough and the photometric precision of the faint
source with $I\sim 21$ is not high enough to detect subtle deviations induced
by microlens-parallax effects.
Figure 8: Lensing light curve of KMT-2021-BLG-1554. Notations are same as
those in Fig. 2.
### 3.4 KMT-2021-BLG-1554
This short event with a time scale of $t_{\rm E}\sim 5$ days and a faint
baseline magnitude of $I_{\rm base}=22.39$ was detected on 2021 Jul 1 (${\rm
HJD}^{\prime}\sim 9396$), about two days after the event peaked at $t_{0}\sim
9394.7$, by the KMTNet survey. The source lies in the prime fields of BLG01
and BLG41, and thus there are 6 data sets from the three KMTNet telescopes.
Among these data sets, the data set of the BLG41 field acquired by the KMTS
telescope was not used in the analysis due to its poor photometric quality
caused by bad seeing during the lensing magnification. Although the other data
set from KMTS and those from KMTA were included in the analysis, the results
are mainly derived from the KMTC data sets, which cover the peak region of the
light curve.
The light curve constructed with the available KMTNet data sets is shown in
Figure 8. In the peak region, it exhibits an anomaly that lasted for about 3
hours from a 1L1S model with a positive deviation of $\Delta I\sim 0.5$ mag.
The anomaly appears both in the BLG01 (two data points) and BLG41 (3 points)
data sets obtained from KMTC observations, confirming that the signal is real.
Because a positive anomaly can be produced by a binary companion to either a
lens or a source, we test both 2L1S and 1L2S models.
Table 7: De-reddened colors, magnitudes, and spectral types of source stars Event | $(V-I)_{0}$ | $I_{0}$ | Spectral type
---|---|---|---
OGLE-2017-BLG-1691 | $0.714\pm 0.065$ | $17.397\pm 0.039$ | G1 turnoff or subgiant
KMT-2021-BLG-0320 | $0.700\pm 0.040$ | $18.843\pm 0.002$ | G1V
KMT-2021-BLG-1303 | $0.849\pm 0.015$ | $19.809\pm 0.003$ | G9V
KMT-2021-BLG-1554 | $0.873\pm 0.061$ | $18.649\pm 0.008$ | K1V
Figure 9: Lens system configuration of KMT-2021-BLG-1554. Notations are same as those in Fig. 5. Table 8: Angular source radii, Einstein radii, and relative lens-source proper motions Event | $\theta_{*}$ ($\mu$as) | $\theta_{\rm E}$ (mas) | $\mu$ (mas yr-1)
---|---|---|---
OGLE-2017-BLG-1691 | $1.043\pm 0.100$ | $0.30\pm 0.05$ | $5.68\pm 1.01$
KMT-2021-BLG-0320 | $0.528\pm 0.042$ | – | –
KMT-2021-BLG-1303 | $0.401\pm 0.029$ | $0.42\pm 0.05$ | $6.13\pm 0.70$
KMT-2021-BLG-1554 | $0.703\pm 0.065$ | $0.10\pm 0.02$ | $7.30\pm 1.50$
Detailed analysis of the light curve indicates that the anomaly was produced
by a planetary companion to the lens. We find two sets of solutions with
planet parameters of $(s,q)_{\rm close}\sim(0.89,1.4\times 10^{-3})$ and
$(s,q)_{\rm wide}\sim(1.12,1.5\times 10^{-3})$ resulting from the close-wide
degeneracy. The full lensing parameters of both solutions are provided in
Table 6. It is found that the degeneracy between the two 2L1S solution is
severe with $\Delta\chi^{2}=0.3$, and the 2L1S solutions are favored over the
model under the 1L2S interpretation by $\Delta\chi^{2}\sim 11$. The lensing
parameters of the 1L2S model are listed in Table 6. We compare the residuals
of the 2L1S and 1L2S models in the two middle panels of Figure 8. It turns out
that the source crossed a caustic, and thus we are able to measure the
normalized source radius, $\rho=(6.98\pm 1.18)\times 10^{-3}$.
The lensing configurations of the event according to the close and wide
solutions are shown in the upper and lower panels of Figure 9, respectively.
The figure shows that the anomaly was produced by the crossing of the source
over the sharp tip of the central caustic induced by a planet lying near the
Einstein ring of the planet host. The gap between the caustic entrance and
exit is much smaller than the source size, and thus the individual caustic-
crossing features do not show up in the anomaly and instead the anomaly
appears as a single bump. The incidence angle of the source on the binary axis
is nearly $90^{\circ}$, and thus the features on the rising and falling sides
of the anomaly are symmetric.
## 4 Source stars and angular Einstein radii
Among the four analyzed planetary events, the anomalies of the three events
(KMT-2017-BLG-0752, KMT-2021-BLG-1303, and KMT-2021-BLG-1554) were affected by
finite-source effects, and thus the normalized source radii can be measured.
In this section, we estimate the angular Einstein radii for these events.
Estimating $\theta_{\rm E}$ from a measured $\rho$ value requires one to
specify the source type, from which the angular source radius $\theta_{*}$ is
deduced and the Einstein radius is determined by
$\theta_{\rm E}={\theta_{*}\over\rho}.$ (3)
Although the normalized source radius and thus $\theta_{\rm E}$ cannot be
measured for the event KMT-2021-BLG-0320, we specify the source type for the
sake of completeness.
The specification of the source type is done by estimating the color and
brightness of the source. We measure the $I$ and $V$-band magnitudes of the
source from the regression of the data processed using the pyDIA photometry
code (Albrow, 2017) with the variation of the lensing magnification. We then
place the source on the instrumental color-magnitude diagram (CMD) of stars
lying around the source constructed using the same photometry code. Following
the method of Yoo et al. (2004), the instrumental source color and magnitude,
$(V-I,I)$, are then calibrated using the centroid of red giant clump (RGC),
for which its reddening and extinction-corrected (de-reddened) color and
magnitude, $(V-I,I)_{\rm RGC,0}$, are known, on the CMD as a reference.
Figure 10 shows the positions of the source stars (marked by blue dots) with
respect to those of the RGC centroids (red dots) on the CMDs of the individual
events. For OGLE-2017-BLG-1691, the source color could not be securely
measured not only because the $V$-band data sparsely covered the light curve
during the lensing magnification but also because the photometry quality of
these data is low, although the $I$-band magnitude was relatively well
measured. In this case, we estimated the source color as the median color of
stars with the same $I$-band magnitude offset from the RGC centroid in the CMD
constructed from the images of Baade’s window taken from the observations
using the Hubble Space Telescope (Holtzman et al., 1998). From the offsets in
color and magnitudes between the source and RGC centroid, $\Delta(V-I,I)$,
together with the known de-reddened color and magnitude of the RGC centroid,
$(V-I,I)_{\rm RGC,0}$ (Bensby et al., 2013; Nataf et al., 2013), we estimate
the calibrated source color and magnitude as $(V-I,I)_{0}=(V-I,I)_{\rm
RGC,0}+\Delta(V-I,I)$. In Table 7, we list the source colors and magnitudes of
the individual events estimated from this procedure along with the spectral
types of the source stars. It is found that the source of OGLE-2017-BLG-1691
is a turnoff star or a subgiant of a G spectral type, while the source stars
of the other events are main-sequence stars with spectral types ranging from G
to K.
Figure 10: Source locations with respect to the centroids of the red giant
clump (RGC) on the instrumental color-magnitude diagrams of stars lying around
the source stars of the events OGLE-2017-BLG-1691, KMT-2021-BLG-0320,
KMT-2021-BLG-1303, and KMT-2021-BLG-1554. For OGLE-2017-BLG-1691, the CMD is
constructed by combining the two CMDs from KMTC (grey dots) and HST (brown
dots) observations. For each event, the locations of the source and RGC
centroid are marked by blue and red solid dots, respectively.
For each event, the angular source radius and Einstein radius were estimated
from the measured color and magnitude of the source. For this, the measured
$V-I$ color was converted into $V-K$ color using the color-color relation of
Bessell & Brett (1988), $\theta_{*}$ value was interpolated from the
($V-K$)–$\theta_{*}$ relation of Kervella et al. (2004), and then the angular
Einstein radius was estimated using the relation in Equation (3). With the
measured Einstein radius together with the event time scale, the value of the
relative lens-source proper motion was assessed as
$\mu={\theta_{\rm E}\over t_{\rm E}}.$ (4)
In Table 8, we list the estimated values of $\theta_{*}$, $\theta_{\rm E}$,
and $\mu$ for the individual events. The $\theta_{\rm E}$ and $\mu$ values for
the event KMT-2021-BLG-0320 are left blank because finite-source effects in
the lensing light curve were not detected and subsequently the values of
$\rho$, $\theta_{\rm E}$, and $\mu$ could not be measured. We note that the
angular Einstein radius of KMT-2021-BLG-1554, $\theta_{\rm E}\sim 0.10$ mas,
is substantially smaller than those of the other events, and this together
with its short time scale, $t_{\rm E}\sim 5$ days, suggests that the mass of
the lens would be very low. KMT-2021-BLG-1554 is the seventh shortest
microlensing event with a bound planet. See Table 3 of Ryu et al. (2021).
Table 9: Physical lens parameters Event | $M_{\rm host}$ ($M_{\odot}$) | $M_{\rm planet}$ ($M_{\rm J}$) | $D_{\rm L}$ (kpc) | $a_{\perp}$ (AU)
---|---|---|---|---
OGLE-2017-BLG-1691 | $0.45^{+0.36}_{-0.25}$ | $0.046^{+0.037}_{-0.025}$ | $7.29^{+1.01}_{-1.33}$ | $2.41^{+0.34}_{-0.44}$ (inner)
| – | – | – | $2.54^{+0.36}_{-0.46}$ (outer)
KMT-2021-BLG-0320 | $0.32^{+0.39}_{-0.21}$ | $0.10^{+0.13}_{-0.07}$ | $6.95^{+1.07}_{-1.33}$ | $1.54^{+0.24}_{-0.30}$ (close)
| – | – | – | $2.55^{+0.39}_{-0.49}$ (wide)
KMT-2021-BLG-1303 | $0.57^{+0.32}_{-0.29}$ | $0.38^{+0.22}_{-0.20}$ | $6.28^{+0.95}_{-1.43}$ | $2.89^{+0.44}_{-0.66}$
KMT-2021-BLG-1554 | $0.08^{+0.13}_{-0.04}$ | $0.12^{+0.20}_{-0.07}$ | $7.68^{+1.04}_{-1.10}$ | $0.72^{+0.13}_{-0.14}$ (close)
| – | – | – | $0.96^{+0.15}_{-0.16}$ (wide)
## 5 Physical parameters
In this section, we estimate the physical parameters of the planetary systems.
For the unique constraint of the physical lens parameters, one must
simultaneously measure the extra observables of $\theta_{\rm E}$ and $\pi_{\rm
E}$, from which the mass $M$ and distance to the lens, $D_{\rm L}$, are
determined as
$M={\theta_{\rm E}\over\kappa\pi_{\rm E}};\qquad D_{\rm L}={{\rm
AU}\over\pi_{\rm E}\theta_{\rm E}+\pi_{\rm S}}.$ (5)
Here $\kappa=4G/(c^{2}{\rm AU})$, $\pi_{\rm S}={\rm AU}/D_{\rm S}$, and
$D_{\rm S}$ denotes the distance to the source (Gould, 1992, 2000). The values
of the microlens parallax could not be measured for any of the events, and
thus the physical parameters cannot be unambiguously determined from the
relation in Equation (5). However, one can still constrain $M$ and $D_{\rm L}$
using the other measured observables of $t_{\rm E}$ and $\theta_{\rm E}$,
which are related to the physical lens parameters as
$t_{\rm E}={\theta_{\rm E}\over\mu};\qquad\theta_{\rm E}=(\kappa M\pi_{\rm
rel})^{1/2},$ (6)
where $\pi_{\rm rel}={\rm AU}(D_{\rm L}^{-1}-D_{\rm S}^{-1})$ denotes the
relative parallax between the lens and source. The event time scales were well
measured for all events, and the Einstein radii were assessed for three of the
four events. In order to estimate the physical lens parameters with the
constraints provided by the measured observables of the individual events, we
conduct Bayesian analyses using a Galactic model.
Figure 11: Bayesian posteriors of the host masses of the planetary systems.
For each distribution, the solid vertical line indicates the median value, and
the dotted lines represent 1$\sigma$ range of the distribution. The blue and
red curves represent the distributions contributed by disk and bulge
populations of lenses, respectively, and the black curve is the sum from the
two lens populations.
In the Bayesian analysis, we started by producing many lensing events from a
Monte Carlo simulation conducted with the use of a Galactic model, which
defines the matter-density and kinetic distributions, and mass function of
Galactic objects. We adopted the Galactic model constructed by Jung et al.
(2021). In this model, the density distributions of disk and bulge objects are
described by the Robin et al. (2003) and Han & Gould (2003) models,
respectively. The kinematic distribution of disk objects is based on the
modified version of the Han & Gould (1995) model, in which the original
version is modified to reconcile the changed density model, that is, the Robin
et al. (2003) model. The bulge kinematic distribution is modeled based on
proper motions of stars in the Gaia catalog (Gaia Collaboration, 2016, 2018).
For the details of the density and kinematic distributions, see Jung et al.
(2021). In the Galactic model, the mass function is constructed with the
adoption of the initial mass function and the present-day mass function of
Chabrier (2003) for the bulge and disk lens populations, respectively.
Figure 12: Bayesian posteriors of the distances to the planetary systems.
Notations are same as those in Fig. 11.
In the second step, we computed the time scales and Einstein radii of the
simulated events using the relations in Equation (6), and then constructed
posterior distributions of $M$ and $D_{\rm L}$ for the simulated events with
the $t_{\rm E}$ and $\theta_{\rm E}$ values that are consistent with the
observables of the individual lensing events. For the three events
OGLE-2017-BLG-1691, KMT-2021-BLG-1303, and KMT-2021-BLG-1554, we use the
constraints of both $t_{\rm E}$ and $\theta_{\rm E}$, and for
KMT-2021-BLG-0320, we apply the constraint of only $t_{\rm E}$ because
$\theta_{\rm E}$ is not measured for the event.
Figures 11 and 12 show the Bayesian posteriors of $M$ and $D_{\rm L}$ of the
individual events, respectively. In Table 9, we summarize the estimated values
of the host and planet masses, $M_{\rm host}$ and $M_{\rm planet}$, distance,
and projected host-planet separation, $a_{\perp}=sD_{\rm L}\theta_{\rm E}$,
for all of which medians are presented as representative values and the
uncertainties are estimated as 16% and 84% of the Bayesian posterior
distributions. The median values and uncertainty ranges of the individual
events are marked by solid and dotted vertical lines in the Bayesian
posteriors presented in Figures 11 and 12, respectively. For the events with
degenerate solutions, OGLE-2017-BLG-1691, KMT-2021-BLG-0320 and
KMT-2021-BLG-1554, we present the planetary separations $a_{\perp}$
corresponding to both the close and wide (or inner and outer) solutions. We
note that the uncertainties of the estimated physical parameters are big
because of the intrinsically weak Bayesian constraint. For example, from the
comparison of the posterior distributions of OGLE-2017-BLG-1691 and
KMT-2021-BLG-0320, one finds that the difference between the two posterior
distributions is not significant, although the mean lens mass for the former
event is slightly bigger than that of the latter event due to the longer time
scale and the uncertainty is smaller due to the additional constrain of
$\theta_{\rm E}$.
The estimated masses indicate that all of the discovered planets have sub-
Jovian masses. The planet masses KMT-2021-BLG-0320L, KMT-2021-BLG-1303L, and
KMT-2021-BLG-1554L correspond to $\sim 0.10$, $\sim 0.38$, and $\sim 0.12$
times of the mass of the Jupiter, and the planet mass of OGLE-2017-BLG-1691L
corresponds to that of Uranus. To be noted among the planet hosts is that the
estimated mass of the planetary system KMT-2021-BLG-1554L, $M_{\rm host}\sim
0.08\leavevmode\nobreak\ M_{\odot}$, corresponds to the boundary between a
star and a brown dwarf (BD). Together with the previously detected planetary
systems with very low-mass hosts444MOA-2011-BLG-262 (Bennett et al., 2014),
OGLE-2012-BLG-0358 (Han et al., 2013), MOA-2015-BLG-337 (Miyazaki et al.,
2018), OGLE-2015-BLG-1771 (Zhang et al., 2020), KMT-2016-BLG-1820 (Jung et
al., 2018a), KMT-2016-BLG-2605 (Ryu et al., 2021), OGLE-2017-BLG-1522 (Jung et
al., 2018b), OGLE-2018-BLG-0677 (Herrera-Martín et al., 2020),
KMT-2018-BLG-0748 (Han et al., 2020b), and KMT-2019-BLG-1339L (Han et al.,
2020a), the discovery of the system demonstrates the microlensing capability
of detecting planetary systems with very faint or substellar hosts. Besides
KMT-2021-BLG-1554L, the host stars of the other planetary systems are low-mass
stars with masses lying in the range of $\sim[0.3$–$0.6]\leavevmode\nobreak\
M_{\odot}$. Under the approximation that the snow line distance scales with
the host mass as $a_{\rm sl}\sim 2.7(M/M_{\odot})$ AU, all discovered planets
lie beyond the snow lines of the hosts regardless of the solutions,
demonstrating the high microlensing sensitivity to cold planets. The planetary
systems lie in the distance range of $\sim[6.3$–$7.6]$ kpc, demonstrating the
usefulness of the microlensing method in detecting remote planets.
## 6 Summary and conclusion
We presented the analyses of four planetary microlensing events
OGLE-2017-BLG-1691, KMT-2021-BLG-0320, KMT-2021-BLG-1303, and
KMT-2021-BLG-1554. The events share a common characteristic that the planetary
signals appeared as anomalies with very short durations, ranging from a few
hours to a couple of days, and they were clearly detected solely by the
combined data of the high-cadence lensing surveys without additional data from
followup observations.
From the detailed analyses of the events, it was found that the signals were
generated by planets with low planet-to-host mass ratios: three of the
planetary systems with mass ratios of the order of $10^{-4}$ and the other
with a mass ratio slightly greater than $10^{-3}$. In the histogram of
microlensing planets presented in Figure 1, we mark the positions of the four
planets discovered in this work. The estimated masses indicated that all
discovered planets have sub-Jovian masses, in which the planet masses of
KMT-2021-BLG-0320Lb, KMT-2021-BLG-1303Lb, and KMT-2021-BLG-1554Lb correspond
to $\sim 0.10$, $\sim 0.38$, and $\sim 0.12$ times of the mass of the Jupiter,
and the mass of OGLE-2017-BLG-1691Lb corresponds to that of the Uranus. It was
found that the host of the planetary system KMT-2021-BLG-1554L has a mass at
around the boundary between a star and a brown dwarf. Besides this system, it
was found that the host stars of the other planetary systems are low-mass
stars with masses in the range of $\sim[0.3$–$0.6]\leavevmode\nobreak\
M_{\odot}$. The discoveries of the planets well demonstrate the capability of
the current high-cadence microlensing surveys in detecting low-mass planets.
###### Acknowledgements.
Work by C.H. was supported by the grants of National Research Foundation of
Korea (2020R1A4A2002885 and 2019R1A2C2085965). This research has made use of
the KMTNet system operated by the Korea Astronomy and Space Science Institute
(KASI) and the data were obtained at three host sites of CTIO in Chile, SAAO
in South Africa, and SSO in Australia. The OGLE project has received funding
from the National Science Centre, Poland, grant MAESTRO 2014/14/A/ST9/00121 to
AU. he MOA project is supported by JSPS KAKENHI grant Nos. JSPS24253004,
JSPS26247023, JSPS23340064, JSPS15H00781, JP16H06287, JP17H02871, and
JP19KK0082.
## References
* Alard & Lupton (1998) Alard, C., & Lupton, R. H. 1998, ApJ, 503, 325
* Albrow (2017) Albrow, M. 2017, MichaelDAlbrow/pyDIA: Initial Release on Github,Versionv1.0.0, Zenodo, doi:10.5281/zenodo.268049
* Albrow et al. (2009) Albrow, M., Horne, K., Bramich, D. M., et al. 2009, MNRAS, 397, 2099
* An (2005) An, J. H. 2005, MNRAS, 356, 1409
* Bennett et al. (2014) Bennett, D. P., Batista, V., Bond, I. A., et al. 2014, ApJ, 785, 155
* Bennett & Rhie (1996) Bennett, D. P., & Rhie, S. H. 1996, ApJ, 472, 660
* Bensby et al. (2013) Bensby, T., Yee, J. C., Feltzing, S., et al. 2013, A&A, 549, A147
* Bessell & Brett (1988) Bessell, M. S., & Brett, J. M. 1988, PASP, 100, 1134
* Bond et al. (2001) Bond, I. A., Abe, F., Dodd, R. J., et al. 2001, MNRAS, 327, 868
* Chabrier (2003) Chabrier, G. 2003, PASP, 115, 763
* Dominik (1999) Dominik, M. 1999, A&A, 349, 108
* Dominik (2019) Dominik, M., Bachelet, E., Bozza, V., et al. 2019, MNRAS, 484, 5608
* Gaia Collaboration (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1
* Gaia Collaboration (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
* Gaudi (1998) Gaudi, B. S. 1998, ApJ, 506, 533
* Gaudi & Gould (1997) Gaudi, B. S., & Gould, A. 1997, ApJ, 486, 85
* Gould (1992) Gould, A. 1992, ApJ, 392, 442
* Gould & Loeb (1992) Gould, A., & Loeb, A. 1992, ApJ, 396, 104
* Gould (2000) Gould, A. 2000, ApJ, 542, 785
* Griest & Safizadeh (1998) Griest, K., & Safizadeh, N. 1998, ApJ, 500, 37
* Han et al. (2022a) Han, C. Bond, I. A., Yee, J. C., et al. 2022a, A&A, 658, A62
* Han & Gould (1995) Han, C., & Gould, A. 1995, ApJ, 447, 53
* Han & Gould (2003) Han, C., & Gould, A. 2003, ApJ, 592, 172
* Han et al. (2022b) Han, C., Gould, A., Albrow, M. D., et al. 2022b, A&A, 658, A94
* Han et al. (2013) Han, C., Jung, Y. K., Udalski, A., et al. 2013, ApJ, 778, 38
* Han et al. (2020a) Han, C., Kim, D., Udalski, A., et al. 2020a, AJ, 160, 64
* Han et al. (2020b) Han, C., Shin, I.-G., Jung, Y. K., et al. 2020b, A&A, 641, A105
* Han et al. (2021) Han, C., Udalski, A., Kim, D., et al. 2021, A&A, 655, A21
* Han et al. (2022c) Han, C., Udalski, A., Lee, C.-U., et al. 2022c, A&A, 658, A93
* Herrera-Martín et al. (2020) Herrera-Martín, A., Albrow, A., Udalski, A., et al. 2020, AJ, 159, 134
* Holtzman et al. (1998) Holtzman, J. A., Watson, A. M., Baum, W. A., Grillmair, C. J., Groth, E. J., Light, R. M., Lynds, R., & O’Neil, E. J.,Jr. 1998, AJ, 115, 1946
* Hwang et al. (2022) Hwang, K.-H., Zang, W., Gould, A., et al. 2022, AJ, 163, 43
* Jung et al. (2021) Jung, Y. K., Han, C., Udalski, A., et al. 2021, AJ, 161, 293
* Jung et al. (2018a) Jung, Y. K., Hwang, K.-H., Ryu, Y.-H., et al. 2018a, AJ, 156, 208
* Jung et al. (2018b) Jung, Y. K., Udalski, A., Gould, A., et al. 2018b, AJ, 155, 219
* Kervella et al. (2004) Kervella, P., Thévenin, F., Di Folco, E., & Ségransan, D. 2004, A&A, 426, 29
* Kim et al. (2018) Kim, D.-J., Kim, H.-W., Hwang, K.-H., et al. 2018, AJ, 155, 76
* Kim et al. (2016) Kim, S.-L., Lee, C.-U., Park, B.-G., et al. 2016, JKAS, 49, 37
* Miyazaki et al. (2018) Miyazaki, S., Sumi, T., Bennett, D. P., et al. 2018, AJ, 156, 136
* Nataf et al. (2013) Nataf, D. M., Gould, A., Fouqué, P., et al. 2013, ApJ, 769, 88
* Robin et al. (2003) Robin, A. C., Reylé, C., Derriére, S., & Picaud, S. 2003, A&A, 409, 523
* Ryu et al. (2021) Ryu, Y. H., Hwang, K.-H., Gould, A., et al. 2021, AJ, 162, 96.
* Ryu et al. (2022) Ryu, Y. H., Jung Y. K., Yang, H., et al. 2022, in preparation
* Shin et al. (2022) Shin, I.-G., Yee, J. C., Hwang K.-H., et al. 2022, AJ, in press (arXiv:2201.04312)
* Tomaney & Crotts (1996) Tomaney, A. B., & Crotts, A. P. S. 1996, AJ, 112, 2872
* Udalski et al. (2015) Udalski, A., Szymański, M. K., & Szymański, G. 2015, Acta Astron., 65, 1
* Woźniak (2000) Woźniak, P. R. 2000, Acta Astron., 50, 42
* Yee et al. (2012) Yee, J. C., Shvartzvald, Y., Gal-Yam, A., et al. 2012, ApJ, 755, 102
* Yee et al. (2021) Yee, J. C., Zang, W, Udalski, A., et al. 2021, AJ, 162, 180
* Yoo et al. (2004) Yoo, J., DePoy, D. L., Gal-Yam, A., et al. 2004, ApJ, 603, 139
* Zang et al. (2021a) Zang, W., Han, C., Kondo, I., et al. 2021, Rearcxh in Astron and Astroph., 2021a, 21, 239
* Zang et al. (2021b) Zang, W., Hwang, K.-H., Udalski, A., et al. 2021b, AJ, 162, 163
* Zhang et al. (2022) Zhang, K., Gaudi, B. S., & Bloom, J. S. 2022, Nature Astronomy, submitted, arXiv:2111.13696
* Zhang et al. (2020) Zhang, X., Zang, W., Udalski, A., et al. 2020, AJ, 159, 116
|
# Appraisal Theories for Emotion Classification in Text
Jan Hofmann1, Enrica Troiano1, Kai Sassenberg2,3 Roman Klinger1
1Institut für Maschinelle Sprachverarbeitung, University of Stuttgart, Germany
2Leibniz-Institut für Wissensmedien, Tübingen, Germany
3University of Tübingen, Germany
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract
Automatic emotion categorization has been predominantly formulated as text
classification in which textual units are assigned to an emotion from a
predefined inventory, for instance following the fundamental emotion classes
proposed by Paul Ekman (fear, joy, anger, disgust, sadness, surprise) or
Robert Plutchik (adding trust, anticipation). This approach ignores existing
psychological theories to some degree, which provide explanations regarding
the perception of events. For instance, the description that somebody
discovers a snake is associated with fear, based on the appraisal as being an
unpleasant and non-controllable situation. This emotion reconstruction is even
possible without having access to explicit reports of a subjective feeling
(for instance expressing this with the words “I am afraid.”). Automatic
classification approaches therefore need to learn properties of events as
latent variables (for instance that the uncertainty and the mental or physical
effort associated with the encounter of a snake leads to fear). With this
paper, we propose to make such interpretations of events explicit, following
theories of cognitive appraisal of events, and show their potential for
emotion classification when being encoded in classification models. Our
results show that high quality appraisal dimension assignments in event
descriptions lead to an improvement in the classification of discrete emotion
categories. We make our corpus of appraisal-annotated emotion-associated event
descriptions publicly available.
00footnotetext: This work is licensed under a Creative Commons Attribution
4.0 International License. License details:
http://creativecommons.org/licenses/by/4.0/.
## 1 Introduction
The task of emotion analysis is commonly formulated as classification or
regression in which textual units (documents, paragraphs, sentences, words)
are mapped to a predefined reference system, for instance the sets of
fundamental emotions fear, anger, joy, surprise, disgust, and sadness proposed
by ?), or by ?), which includes also trust and anticipation. Machine learning-
based models need to figure out which words point to a particular emotion
experienced by a reader, by the author of a text, or a character in it.
Depending on the resource which has been annotated, the description of an
emotion experience can vary. On Twitter, for instance, other than direct
reports of an emotion state (“I feel depressed”), hashtags are used as emotion
labels to enrich the description of events and stances (“I just got my exam
result #sad”). In news articles, emotional events are sometimes explicitly
mentioned (“couple infuriate officials” [Bostan et al., 2020]) and other times
require world knowledge (“Tom Cruise and Katie Holmes set wedding date”,
labeled as surprise [Strapparava and Mihalcea, 2007]). In literature, a
sequence of events which forms the narrative leads to an emotion in the
reader. In this paper, we focus on those texts which communicate emotions
without an explicit emotion word, but rather describe events for which an
emotion association is evident.
Such textual examples became popular in natural language processing research
with the use of the data generated in the ISEAR project [Scherer and Wallbott,
1997]. The project led to a dataset of descriptions of events triggering
specific affective states, which was originally collected to study event
interpretations with a psychological focus. In text analysis, to infer the
emotion felt by the writers of those reports, an event interpretation needs to
be accomplished. For instance, in the text “When a car is overtaking another
and I am forced to drive off the road”, the model needs to associate the event
with fear.
To date, nearly all computational approaches that associate text with emotions
are agnostic to the way in which emotions are communicated, they do “not know”
how to interpret events, but, presumably, they purely learn word associations
instead of actual event interpretations. One might argue that approaches
predicting fine-grained dimensions of affect, namely arousal and valence,
actually tackle this problem [Buechel and Hahn, 2017, Preoţiuc-Pietro et al.,
2016]. However, these typically do not infer downstream emotion categories.
Further, particularly regarding events, psychological theories offer more
detailed information. As an example, the emotion component model [Scherer,
2005] advocates that cognitive appraisal dimensions underly discrete emotion
classes [Smith and Ellsworth, 1985]. These appraisal dimensions111The examples
follow the results by ?), an excerpt is shown in Table 1. evaluate (1) how
pleasant an event is (pleasantness, likely to be associated with joy, but
unlikely to appear with disgust), (2) how much effort an event can be expected
to cause (anticipated effort, likely to be high when anger or fear is
experienced), (3) how certain the experiencer is in a specific situation
(certainty, low, e.g., in the context of hope or surprise), (4) how much
attention is devoted to the event (attention, likely to be low, e.g., in the
case of boredom or disgust), (5) how much responsibility the experiencer of
the emotion holds for what has happened (self-other responsibility/control,
high for feeling challenged or pride), and (6) how much the experiencer has
control over the situation (situational control, low in the case of anger).
Despite their richness, cognitive theories of appraisal and their empirical
results have not been exploited for emotion prediction in text yet. We fill
this gap with this paper and analyze the relation between appraisal dimensions
and emotion categories in a text classification setting. We post-annotate an
English emotion corpus of self-reports of emotion events [Troiano et al.,
2019], which already contains annotations related to the emotions of anger,
disgust, fear, guilt, joy, sadness, and shame, and add the appraisal
dimensions by ?) mentioned above. Further, we analyze if an automatic
prediction of these dimensions from text is possible with standard neural
methods, and if these predictions contribute to emotion classifications. Our
main contributions are: (1) the first event-centered text corpus annotated
with appraisal dimensions; (2) the evaluation of how well text classification
models can recognize these appraisal dimensions; (3) we show emotion
classification benefits from the information of appraisal dimensions, when
high quality predictions of these are available. Further, (4), we replicate
the study by ?) from a CL/NLP perspective, based on textual event
descriptions.
## 2 Background on Emotion Psychology and Analysis
### 2.1 Emotion and Affect Theories
As a component of humans’ life, emotions have been thoroughly studied in the
field of psychology, where they are generally deemed responses to salient
events. The debates surrounding their definition, however, has never come to a
consensus, producing a varied literature on the topic. This has a clear
implication for computational emotion analyses, for they must choose and
follow one of the available psychological theories in order to motivate the
emotion phenomenon that they research in language.
Some of such theories focus on the evolutionary function of emotions, and
accordingly, on their link to actions [Izard, 1971, Tooby and Cosmides, 2008].
The core idea is that emotions help humans accomplish every-day life tasks and
communicate socially relevant information by triggering specific physiological
symptoms. In particular, there are patterns of behaviour (e.g., smiling) that
reflect discrete emotion terms (e.g., joy), which suggests that emotional
states can be grouped based on a few natural language categories. One of the
most popular sources for a set of fundamental emotions is the theory by ?).
Ekman studied the relation between emotions and both culture and facial
expressions: he claimed that the set of fundamental emotions, namely, anger,
disgust, fear, joy, sadness, and surprise can be distinguished by facial
muscular movements across cultures (which is partially doubted these days,
?)). As an addition to this model, ?) makes explicit the assumption that
different fundamental emotions can occur together, for instance trust and joy,
which is the case when love is experienced. Such emotion mixtures, as well as
an opposition between anger and fear, joy and sadness, surprise and
anticipation, trust and disgust, has been included in this model. In natural
language processing, mostly a set of four to eight fundamental emotions is
used, where anger, fear, joy, and sadness are shared by most approaches (an
exception with 24 emotion classes is ?)).
A diametrically opposite view is held by the constructivist tradition
[Averill, 1980, Oatley, 1993, Barrett and Russell, 2015], in which actions and
physiological changes are the building blocks that construct emotions, rather
than their direct effect [Feldman Barrett, 2006]. Feeling an emotion means
categorizing the fluctuations of an affect system along some components. For
instance, the affect components _valence_ (degree of polarity), _arousal_
(degree of excitement), and _dominance_ (degree of control over a situation)
[Posner et al., 2005] are used as dimensions to describe affect experiences in
a 3-dimensional space, which can then be mapped to discrete emotion
categories.
### 2.2 Theories of Cognitive Appraisal
Extensions to this model in terms of underlying components are the works of
?), ?) and ?), who qualified emotions as component processes that arise to
face salient circumstances: an emotion is an “episode of interrelated,
synchronized changes in the states of all or most of the five organismic
subsystems in response to the evaluation of an external or internal stimulus-
event as relevant to major concerns of the organism” [Scherer et al., 2001].
According to this view, there is an appraisal, that is, an information
processing component, which enables people to determine the significance of a
situation with respect to their needs and values. In the context of this
appraisal (e.g., judging a snake as dangerous), the resources of four other
components are mobilitated to deal with the situation. These are, next to the
_cognitive component (appraisal)_ , a _neurophysiological component (bodily
symptoms)_ , a _motivational component (action tendencies)_ , a _motor
expression (facial and vocal expression)_ , and a _subjective feeling
component (emotional experience)_ [Scherer, 2005, Table 1].
While the notions of subjective experience and bodily symptoms were common to
other emotion theories, appraisal represents a novelty that fills in some
shortcomings of basic models. First, it explains how emotions are elicited.
The origin of emotions is to be seen in the stimulus as appraised rather than
in the stimulus as such. Second, appraisals provide a structured account for
the differences among emotions. For instance, anger and fear are experienced
when the evaluation of a negative event attributes it to external factors,
whereas guilt and shame are felt if the causes of such event are identified in
the self, as stable and uncontrollable personality traits, like in the case of
shame (e.g., “I’m dumb”), or unstable and controllable behaviours for guilt
(e.g., “I did not observe the speed limit”) [Tracy and Robins, 2006].
We argue in this paper that this makes appraisals particularly useful for
natural language processing, because they both provide a framework for
research and represent a way of enriching existing data. As a matter of fact,
few dimensions are sufficient to explain emotions based on cognitive
appraisal. ?) explain 15 emotions by leveraging pleasantness (polarity), self-
other responsibility/control (for initiating the situation), certainty (about
what is going on), attention (whether the emotion stimulus is worth
attending), anticipated effort (the amount of physical or mental activation
before the stimulus), and situational control (the ability to cope with the
situation). Compared to the valence-arousal-dominance model, where it is left
unclear if the polarity dimension refers to a quality of the emotion stimulus
or a quality of the feeling [Scherer, 2005], all these dimensions are
unambiguously event-directed. In this paper, we focus on modelling the
cognitive components described by ?). We show their main findings in Table 1,
limited to the emotions that are available in the corpus we use.
Emotion | Unpleasant | Responsibility | Uncertainty | Attention | Effort | Control
---|---|---|---|---|---|---
Happiness | $-$1.46 | 0.09 | $-$0.46 | 0.15 | $-$0.33 | $-$0.21
Sadness | 0.87 | $-$0.36 | 0.00 | $-$0.21 | $-$0.14 | 1.15
Anger | 0.85 | $-$0.94 | $-$0.29 | 0.12 | 0.53 | $-$0.96
Fear | 0.44 | $-$0.17 | 0.73 | 0.03 | 0.63 | 0.59
Disgust | 0.38 | $-$0.50 | $-$0.39 | $-$0.96 | 0.06 | $-$0.19
Shame | 0.73 | 1.31 | 0.21 | $-$0.11 | 0.07 | $-$0.07
Guilt | 0.60 | 1.31 | $-$0.15 | $-$0.36 | 0.00 | $-$0.29
Table 1: The locations of emotions along appraisal dimensions (according to a
principle component analysis) as published by ?), Table 6, filtered to those
emotions which are available in the text corpus we use.
### 2.3 Automatic Emotion Classification
Previous work on emotion analysis in natural language processing focuses
either on resource creation or on emotion classification for a specific task
and domain. On the side of resource creation, the early and influential work
of ?) is a dictionary of words being associated with different psychologically
relevant categories, including a subset of emotions. Later, ?) made WordNet
Affect available to target word classes and differences regarding their
emotional connotation, ?) released the NRC dictionary with more than 14,000
words for a set of discrete emotion classes, and a valence-arousal-dominance
dictionary was provided by ?). ?) have developed a methodological framework to
adapt existing affect lexicons to specific use cases. Other than dictionaries,
emotion analysis relies on labeled corpora. Some of them include information
relative to valence and arousal [Buechel and Hahn, 2017, Preoţiuc-Pietro et
al., 2016], but the majority of resources use discrete emotion classes, for
instance to label fairy tales [Alm et al., 2005], blogs [Aman and Szpakowicz,
2007], tweets [Mohammad et al., 2017, Schuff et al., 2017, Mohammad, 2012,
Mohammad and Bravo-Marquez, 2017, Klinger et al., 2018], Facebook posts
[Preoţiuc-Pietro et al., 2016], news headlines [Strapparava and Mihalcea,
2007], dialogues [Li et al., 2017], literary texts [Kim et al., 2017], or self
reports on emotion events [Scherer and Wallbott, 1997, Troiano et al., 2019].
We point the reader to the survey by ?) for a more an overview.
Most automatic methods that assign labels to text rely on machine learning
[Alm et al., 2005, Aman and Szpakowicz, 2007, Schuff et al., 2017, i.a.].
Recent shared tasks showed an increase in transfer learning from generic
representations [Klinger et al., 2018, Mohammad et al., 2018, Mohammad and
Bravo-Marquez, 2017]. ?) proposed to use emoji representations for
pretraining, and ?) performed pretraining on existing emotion corpora followed
by fine-tuning for a specific domain for which only little training data was
available.
We are only aware of one preliminary study which considered appraisal
dimensions to improve text-based emotion prediction, namely ?). In their
study, subjects labeled 200 stories with 38 appraisal features (which remain
unmentioned), to evaluate if a text-based representation adds on top of an
fMRI-based classification. Apart from this study, all previous machine
learning-based approaches used models to predict emotions or affect values
directly from text, without any access to appraisal dimensions. Only a couple
of works incorporated cognitive components, for instance those coming from the
OCC model (named after the authors Ortony, Clore and Collins’ initials), which
sees every appraisal as an evaluation of the pleasantness of events, objects,
or actions with respect to one’s goals, tastes or behavioural and moral
standards [Clore and Ortony, 2013]. Based on the OCC model, ?) devised a rule-
based approach to interpret text. They did not explicitly formulate their
model following appraisal theories, but they moved towards a cognitively-
motivated interpretation of events and interpersonal descriptions. Others have
adopted patterns of appraisal to predict the emotions triggered by actions, as
described in a text. Specifically, ?) and ?) have created EmotiNet, a
knowledge base of action chains that includes information about the elements
on which the appraisal is performed within an affective situation, namely, the
agent, the action and the object involved in a chain. We share their
motivation to delve into event representations based on the descriptions of
their experiencers. Unlike their work, ours explicitly encodes appraisal
dimensions and uses the classification into these categories for emotion
prediction.
We are not aware of any textual corpus annotated with appraisals according to
emotion theories, or associated computational models. However, some research
exists which studies these theories with added value for computational models.
?), e.g., explicitly encode realizations of appraisals of events and compare
those between production and perception of facial expressions. ?) model
emotions in video sequences computationally, based on appraisals of events. ?)
inform intelligent agents about emotions based on formalizations of cognitive
appraisal. Further, in ?), the chapter authored by ?) argues for an integrated
computational approach of emotions which also embeds appraisal theories.
## 3 Corpus
The main objective of this study is to understand the relation between
appraisal dimensions and emotion categories. Therefore, we build appraisal
annotations on top of enISEAR, an existing corpus of 1001 English event
descriptions which are already labeled with the discrete categories of anger,
disgust, fear, guilt, joy, sadness, and shame [Troiano et al., 2019]. Each
instance has been generated by a crowdworker on the platform FigureEight by
completing the sentence “I feel [emotion name], when …”. This corpus has an
advantage over the original ISEAR resource because it has a German counterpart
which can be used in further studies; moreover, its emotion labels have been
intersubjectively validated. The corpus, annotations, and our implementations
are available at http://www.ims.uni-stuttgart.de/data/appraisalemotion.
### 3.1 Annotation
One presumable challenge in the post-annotation of events regarding the
appraisal dimensions is that our annotators do not have access to the private
state of the event experiencers. However, under the assumption that events are
perceived similarly in subjective feeling and evaluated comparably based on
cognitive appraisal, we assume that this is not a major flaw in the design of
the study. An alternative would have been to perform the text generation task
as ?) did, but asking the authors of event descriptions for their appraisal in
addition. We opted against such procedure as it would have meant to reproduce
an existing study in addition to our research goal.
For the post-labeling of enISEAR, we aimed at formulating unambiguous and
intuitive descriptions of appraisal dimensions, which would be faithful to
those in ?). As opposed to the subjects of their study, however, our
annotators had to judge events that they did not personally experience. For
this reason, we simplified our annotation guidelines in two respects. First,
we opted for a binary setting, while ?) used continuous scales to rate
discrete emotion categories on the appraisal dimensions. Second, we split
control into _Control_ and _Circumstance_ (i.e., _self_ and situational
control), in line with the discussion of this variable by ?, p. 824f.), while
retaining the category of responsibility. This was motivated by the
observation of a low inter-annotator agreement in preliminary annotation
rounds and a series of discussions that revealed the difficulty for annotators
to separate the concepts of responsibility and self control. Then, the
annotators were instructed to read an event description, without having access
to the emotion label, and to answer the following questions:
Most probably, at the time when the event happened, the writer…
* •
…wanted to devote further attention to the event. (Attention)
* •
…was certain about what was happening. (Certainty)
* •
…had to expend mental or physical effort to deal with the situation. (Effort)
* •
…found that the event was pleasant. (Pleasantness)
* •
…was responsible for the situation. (Responsibility)
* •
…found that he/she was in control of the situation. (Control)
* •
…found that the event could not have been changed or influenced by anyone.
(Circumstance)
Each event description from enISEAR was judged by three annotators between the
age of 26 and 29. One of them is a female Ph.D. student of computational
linguistics, the others are male graduate students of software engineering.
Two of the annotators are co-authors of this paper. The judges familiarised
themselves with their task through four training iterations. At every
iteration, we hand-picked 15–20 samples from the ISEAR dataset [Scherer and
Wallbott, 1997], such that instances used for training would not be seen
during the actual annotation, but had a comparable structure. Dissimilarities
in the annotation were discussed in face-to-face meetings and the annotation
guideline was refined.
The agreement improved from $\kappa$=$0.62$ to 0.67 in the four iterations. In
one of them, we experimented with giving access to the emotion label, which
lead to a large improvement in agreement ($\kappa$=$0.83$). Nevertheless, we
decided to continue without this information, in order to evaluate the
annotator’s performance in a similar setting as we evaluate the automatic
model – to predict appraisal for emotion classification. We show the pairwise
inter-annotator scores of the final set in Table 2. The agreement scores
between the different annotator pairs are comparable.
| Cohen’s $\kappa$
---|---
| between annotators | annotator–majority
Appraisal Dimension | A1/A2 | A1/A3 | A2/A3 | avg. | A1 | A2 | A3 | avg.
Attentional Activity | .28 | .24 | .41 | .31 | .50 | .76 | .66 | .64
Certainty | .41 | .23 | .29 | .31 | .62 | .77 | .46 | .62
Anticipated Effort | .38 | .33 | .26 | .32 | .69 | .67 | .62 | .66
Pleasantness | .89 | .88 | .90 | .89 | .93 | .96 | .94 | .94
Responsibility | .68 | .57 | .63 | .63 | .80 | .88 | .76 | .81
Control | .65 | .56 | .52 | .58 | .84 | .81 | .70 | .78
Circumstance | .52 | .32 | .28 | .37 | .80 | .69 | .49 | .66
Average | .59 | .48 | .52 | .53 | .77 | .82 | .70 | .76
Table 2: Cohen’s $\kappa$ between all annotator pairs and between each
annotator and the majority vote.
These scores tell us that rating appraisal dimensions for given events is
challenging, and its difficulty varies depending on the categories. Given the
comparably low agreement obtained for a subset of dimensions, we opt for a
“crowd-sourcing”-like aggregation by taking the majority vote to form the
final annotation, included in Table 2, on the right side of the table. We
observe that the agreement between majority vote and each annotator is
constantly above $\kappa$=.62, which is an acceptable agreement
(avg.$\kappa$=.76).
| Appraisal Dimension
---|---
Emotion | Attention | Certainty | Effort | Pleasant | Respons. | Control | Circum.
Anger | 129 | .90 | 119 | .83 | 60 | .42 | 0 | .00 | 9 | .06 | 1 | .01 | 5 | .03
Disgust | 67 | .47 | 134 | .94 | 40 | .28 | 2 | .01 | 14 | .10 | 11 | .08 | 24 | .17
Fear | 129 | .90 | 13 | .09 | 121 | .85 | 4 | .03 | 43 | .30 | 18 | .13 | 66 | .46
Guilt | 55 | .38 | 132 | .92 | 36 | .25 | 0 | .00 | 133 | .93 | 88 | .62 | 11 | .08
Joy | 139 | .97 | 140 | .98 | 4 | .03 | 141 | .99 | 65 | .45 | 41 | .29 | 25 | .17
Sadness | 122 | .85 | 112 | .78 | 88 | .62 | 1 | .01 | 7 | .05 | 2 | .01 | 97 | .68
Shame | 32 | .22 | 111 | .78 | 51 | .36 | 1 | .01 | 106 | .74 | 67 | .47 | 12 | .08
Total | 673 | | 761 | | 400 | | 149 | | 377 | | 228 | | 240 |
Table 3: Instance counts and ratios across emotions and appraisal annotations.
### 3.2 Analysis
In Table 3 are the cooccurrence counts across emotion and appraisal dimension
pairs, as well as the relative counts normalized by emotion (enISEAR provides
143 descriptions per emotion). The most frequently annotated class is
certainty, followed by attention. Appraisal dimensions are differently
distributed across emotions: anger and fear require attention, guilt and shame
do not; disgust and anger show the highest association with certainty, in
opposition to fear. _Responsibility_ and control play the biggest role in
guilt and shame, while joy, non-surprisingly, strongly relates to
pleasantness. _Fear_ has a clear link with anticipated effort and, together
with sadness, it is characterized by the inability to control the
circumstance.
These numbers are particularly interesting in comparison with the findings of
?), who report the average scores along the appraisal dimensions (based on a
principle component analysis) for each emotion222We report the subset of
emotions that overlap with ours. Also note that their “Control” corresponds to
our “Circum.”.. Results are consistent in most cases. For instance, joy (or
happiness in Table 1) stands out as highly pleasant and barely related to
anticipated effort. Self responsibility is lowest in anger, an emotion that
arises when blame is externalized, and mostly present in shame and guilt,
which derive from blaming the self [Tracy and Robins, 2006]. These two are
also the emotions that annotators associated with control more than others.
Attention is prominent for events that elicited anger and which were under the
control of others, as suggested by the low situational control. The highest
situational control, on the contrary, appears with the data points labeled as
fear, also characterized by a strong feeling of uncertainty and anticipated
effort. There are also dissimilarities between the two tables, like the level
of attention, reaching the lowest score for disgust in their study and not in
ours. They also find that situational control is a stronger indicator for
shame than for guilt, while anticipated effort is more marked in our sadness-
related events than theirs.
These differences may partly be data-specific, partly due to the type of
metrics shown in the tables. Most importantly, they can be traced to the
annotation setup: while their subjects recalled and appraised personal events,
our annotators evaluated the descriptions of events that are foreign to them.
It should be noted that for a reader/annotator it is challenging to
impersonate in the writer: although some events have a shared understanding
(e.g., “I passed the exam” is most likely appraised as pleasantness), others
are tied to one’s personal background, values and preferences. This may
represent a source of disaccord both between the tables, among the annotators,
and with the emotion labels themselves (e.g., “I felt … when my mom offered me
curry” has a pleasant gold label, while the original author meant it as a
negative emotion, namely disgust).
## 4 Experiments
We now move to our evaluation if automatic methods to recognize emotions can
benefit from being informed about appraisal dimensions explicitly. We first
describe our models and how we address our research questions and then turn to
the results.
### 4.1 Model Configuration
Figure 1 illustrates the four different tasks addressed by our models. Task
T$\rightarrow$E is the prediction of _emotions from text_ , namely the
standard setting in nearly all previous work of emotion analysis. We use a
convolutional neural network (CNN) inspired by ?), with pretrained GloVe
(Glove840B) as a 300-dimensional embedding layer [Pennington et al.,
2014]333https://nlp.stanford.edu/projects/glove/ with convolution filter sizes
of 2, 3, and 4 with a ReLu activation function [Nair and Hinton, 2010],
followed by a max pooling layer of length 2 and a dropout of 0.5 followed by
another dense layer.
As another model to predict emotions, we use a pipeline based on two steps,
one to detect the appraisal from text, and the second to assign the
appropriate emotion to the appraisal. We refer to Task T$\rightarrow$A as the
step of identifying _appraisal from text_. We use the model configuration of
Task T$\rightarrow$E, except for the sigmoid activation function and binary
cross entropy loss instead of softmax and cross-entropy loss. As a second
step, Task A$\rightarrow$E predicts _emotion from appraisal_. The features are
seven boolean variables corresponding to the appraisal dimensions. We use a
neural network with two hidden layers with ReLU activation, followed by a
dropout of 0.5.444We do not perform any further hyperparameter search. We also
compared all model configurations to MaxEnt models to ensure that we do not
suffer from overfitting due to a large number of parameters, in comparison to
the limited training data. In all settings, the neural models were superior.
We therefore limit our explanations to those.
A disadvantage of the pipeline setting could be that the emotion prediction
needs to handle propagated errors from the first step, and that the first step
cannot benefit from what the second model learns. Therefore, we compare the
pipeline setting (T$\rightarrow$A, A$\rightarrow$E) with a multi-task learning
setting (T$\rightarrow$A/E). The model is similar to Task T$\rightarrow$E. The
convolutional layer is shared by the tasks of predicting _emotions from text_
and predicting _appraisal from text_ and we use two output layers, one for
emotion predictions with softmax activation and one for appraisal predictions
with sigmoid activation.
Figure 1: Tasks investigated in experiments on appraisal-driven emotion
analysis.
| T$\rightarrow$A
---|---
Appraisal | P | R | $\textrm{F}_{1}$
Attention | 81 | 84 | 82
Certainty | 84 | 86 | 85
Effort | 68 | 68 | 68
Pleasantness | 79 | 63 | 70
Responsibility | 74 | 68 | 71
Control | 63 | 49 | 55
Circumstance | 65 | 58 | 61
Macro avg. | 73 | 68 | 70
Micro avg. | 77 | 74 | 75
Table 4: Classifier performance on predicting appraisal dimensions.
| | | | | | | | | | | | | Oracle Ensembles
---|---|---|---|---|---|---|---|---|---|---|---|---|---
| T$\rightarrow$E | T$\rightarrow$A,A$\rightarrow$E | A$\rightarrow$E (Gold) | T$\rightarrow$A/E | T5pt 0pt 0pt -1pt$\rightarrow$A5pt 0pt 0pt -1pt$\rightarrow$E + T5pt 0pt 0pt -1pt$\rightarrow$E | T5pt 0pt 0pt -1pt$\rightarrow$A/E + T5pt 0pt 0pt -1pt$\rightarrow$E
Emotion | P | R | $\textrm{F}_{1}$ | P | R | $\textrm{F}_{1}$ | P | R | $\textrm{F}_{1}$ | P | R | $\textrm{F}_{1}$ | P | R | $\textrm{F}_{1}$ | P | R | $\textrm{F}_{1}$
Anger | 51 | 52 | 52 | 34 | 62 | 44 | 55 | 71 | 62 | 51 | 52 | 52 | 66 | 81 | 73 | 59 | 59 | 59
Disgust | 65 | 63 | 64 | 59 | 34 | 43 | 53 | 48 | 51 | 64 | 64 | 64 | 78 | 68 | 73 | 69 | 66 | 67
Fear | 69 | 71 | 70 | 55 | 55 | 55 | 79 | 78 | 78 | 70 | 68 | 69 | 76 | 77 | 77 | 73 | 75 | 74
Guilt | 47 | 42 | 44 | 38 | 50 | 43 | 57 | 70 | 63 | 45 | 42 | 44 | 60 | 63 | 62 | 58 | 54 | 56
Joy | 74 | 80 | 77 | 77 | 69 | 72 | 94 | 98 | 96 | 77 | 77 | 77 | 79 | 80 | 80 | 79 | 85 | 82
Sadness | 69 | 67 | 68 | 58 | 40 | 47 | 69 | 63 | 66 | 68 | 68 | 68 | 74 | 70 | 72 | 73 | 71 | 72
Shame | 44 | 45 | 45 | 36 | 24 | 29 | 56 | 35 | 43 | 43 | 43 | 43 | 58 | 51 | 54 | 51 | 52 | 52
Macro avg. | 60 | 60 | 60 | 51 | 48 | 48 | 66 | 66 | 65 | 60 | 59 | 59 | 70 | 70 | 70 | 66 | 66 | 66
Micro avg. | | | 60 | | | 48 | | | 66 | | | 59 | | | 70 | | | 66
Table 5: Comparison of the Text-to-Emotion baseline (T$\rightarrow$E) with the
performance of first prediction appraisal followed by emotion analysis
(T$\rightarrow$A,A$\rightarrow$E) and the multi-task setting
(T$\rightarrow$A/E). The oracle consists of a combination of two models and is
informed which model is more likely to make the correct prediction.
### 4.2 Results
We perform each experiment in a repeated 10$\times$10-fold cross-validation
setting and report average results. All partitions of training and test sets
are the same across all experiments.
Experiment 1 (Appraisal Prediction, T$\rightarrow$A) aims at understanding how
well appraisal dimensions can be predicted from text. Emotion classification
is an established task, and one might have some intuition on the expected
performance with a given data set, but the prediction of appraisal dimensions
has never been performed before. Hence, we report precision, recall, and
$\textrm{F}_{1}$ for each appraisal component considered in Table 4. The
prediction of certainty works best (85%$\textrm{F}_{1}$) followed by attention
(82%$\textrm{F}_{1}$). The lowest performance is seen for control
(55%$\textrm{F}_{1}$) and circumstance (61%$\textrm{F}_{1}$). These results
are only partially in line with the inter-annotator agreement scores. We
obtain a .75 micro average $\textrm{F}_{1}$ score.
| Appraisal |
---|---|---
| Emotion (G/P) | A | Ce | Ex | Px | Rx | Co | Ci | Text
Appr+Emo corr. | Anger | 1 | 1 | 0 | 0 | 0 | 0 | 0 | when my neighbour started to throw rubbish in my garden for no reason.
Disgust | 0 | 1 | 0 | 0 | 0 | 0 | 0 | to watch someone eat insects on television.
Fear | 1 | 0 | 1 | 0 | 0 | 0 | 1 | when our kitten escaped in the late evening and we thought he was lost.
Guilt | 0 | 1 | 0 | 0 | 1 | 1 | 0 | when I took something without paying.
Joy | 1 | 1 | 0 | 1 | 1 | 0 | 0 | when I found a rare item I had wanted for a long time.
Sadness | 1 | 1 | 1 | 0 | 0 | 0 | 1 | when my dog died. He was ill for a while. Still miss him.
Shame | 0 | 1 | 0 | 0 | 1 | 0 | 0 | when I remember an embarrassing social faux pas from my teenage years.
Emo incorr. | Anger/Fear | 1 | 0 | 1 | 0 | 0 | 0 | 0 | when someone drove into my car causing damage and fear to myself – then drove off before exchanging insurance details.
Disgust/Anger | 1 | 1 | 0 | 0 | 0 | 0 | 0 | when I saw a bird being mistreated when on holiday.
Fear/Sadness | 1 | 1 | 1 | 0 | 0 | 0 | 1 | a huge spider just plopped on down on the sofa besides me, staring me out.
Guilt/Disgust | 0 | 1 | 0 | 0 | 0 | 0 | 0 | when I watched a documentary that showed footage of farms of pigs and chickens and as a meat eater I felt awful guilt at how they are treated.
Sadness/Anger | 1 | 1 | 0 | 0 | 0 | 0 | 0 | when I saw a group of homeless people and it was cold outside.
Shame/Guilt | 0 | 1 | 0 | 0 | 1 | 1 | 0 | because I did something silly.
Ap+Emo incorr. | Anger/Shame | 0 | 1 | 0 | 0 | 1 | 0 | 0 | I feel … because I can’t stand when people lie.
Disgust/Anger | 1 | 1 | 0 | 0 | 0 | 0 | 0 | when I saw a medical operation on a TV show.
Fear/Guilt | 1 | 0 | 0 | 0 | 1 | 0 | 0 | when I was on a flight as I am … of flying.
Guilt/Shame | 0 | 1 | 1 | 0 | 1 | 1 | 0 | when I lost my sister’s necklace that I had borrowed.
Joy/Anger | 1 | 1 | 0 | 0 | 0 | 0 | 0 | when I saw bees coming back to my garden after few years of absence.
Sadness/Guilt | 1 | 1 | 0 | 0 | 1 | 0 | 0 | when I watched some of the sad cases of children in need.
Shame/Guilt | 0 | 1 | 0 | 0 | 1 | 1 | 0 | when I forgot a hairdressers appointment.
Table 6: Examples for the prediction of the pipeline setting (T$\rightarrow$A,
A$\rightarrow$E) . A: Attention, Ce: Certainty, E: Effort, P: Pleasantness, R:
Responsibility, Co: Control, Ci: Circumstance. First emotion mention is gold,
second is prediction. Appraisal shown is prediction with errors shown in bold.
Experiment 2 (Appraisal Interpretation, A$\rightarrow$E) aims at understanding
how well emotions can be predicted from appraisals. We compare the baseline
text-to-emotion setting (T$\rightarrow$E) to the pipeline setting that first
predicts the appraisal and then, from those, the emotion. In the pipeline
setting we train the second step (A$\rightarrow$E) on the gold appraisal
annotations, not on the predictions.555We also tested if training on the
prediction leads to better results, but it constantly underperformed. We
compare this setting to the performance of the appraisal-to-emotion model
(A$\rightarrow$E), when applied on gold appraisal annotations. This serves as
an upper bound which can be reached with the best-performing appraisal
prediction model.
We first turn to the results of the model which predicts the emotion based on
annotated (gold) appraisal dimensions (A$\rightarrow$E (gold)). Here, we
observe a clear improvement in contrast to the emotion classification which
has access to the text (T$\rightarrow$E). _Anger_ increases from .52 to .62,
disgust decreases from .64 to .51, fear increases from .70 to .78, guilt from
.44 to .63, joy from .77 to .96, sadness decreases from .68 to .66 and shame
decreased from .45 to .43. On micro average, the performance increases from
.60 to .66% $\textrm{F}_{1}$. These results are an upper-bound for the
performance that can be achieved with the pipeline model, under the assumption
of having access to perfect appraisal predictions.
When moving to the real-world setting of first predicting the appraisal
dimensions and then, based on those, predicting the emotion, the performance
scores drop from 66 to 48% $\textrm{F}_{1}$. This is an indicator that the
performance of our current appraisal prediction, though comparably reasonable
with 75% $\textrm{F}_{1}$, is not yet sufficient to support emotion
predictions, at least partially. The clear improvement in emotion prediction
based on perfect appraisal annotations and the performance drop in the real-
world setting for emotion prediction suggest that annotating more data with
appraisal dimensions is necessary to further develop our approach.
Finally, Experiment 3 (Multi-Task Learning, T$\rightarrow$A/E and Oracle
Ensembles) evaluates if the model which learns appraisals and emotions jointly
performs better than the pipeline model (both being models applicable in a
real-world scenario). We show these results also in Table 5 and see that the
multi-task learning model cannot improve over the text-only setting.
A remaining question is on the complementarity of the pipeline and the
multitask model to the T$\rightarrow$E model. To look into this, we define two
oracle ensembles (T$\rightarrow$A, A$\rightarrow$E and T$\rightarrow$E as well
as T$\rightarrow$A/E and T$\rightarrow$E), in which an oracle predicts which
of the two approaches will obtain the correct result. In this experiment, we
therefore accept a prediction as true positive if one of the two parts of the
ensemble is correct. These results are shown in Table 5 in the columns “Oracle
Ensemble”. We see a clear improvement over the isolated text-based prediction
for both oracle ensembles, while the pipeline model shows a higher
contribution in addition to the T$\rightarrow$E model (70% $\textrm{F}_{1}$ in
contrast to 66% $\textrm{F}_{1}$).
### 4.3 Discussion and Analysis
We have seen in the experiments and results that the approach of predicting
emotions based on appraisal shows a clear potential for a better performance.
Though we have not been able to reach a substantial improvement in a real-
world setting, in which appraisal dimensions are first predicted as a basis
for the emotion prediction or in the multi-task setting, we observe that text-
and appraisal-based models behave differently. Table 6 shows examples for the
prediction in the real-world setting (T$\rightarrow$A, A$\rightarrow$E). In
the top block, one example is shown for each emotion in which both appraisal
and emotion are correctly predicted. This does not only include cases in which
clear emotion indicators exist. The second block reports instances in which
the appraisal is correct, but the emotion prediction is not. Here, the first
sentence (“when someone drove into my car…”) is an example in which a flip of
certainty would have changed the emotion. Similarly, the change of attention
in “when I saw a group…” would have lead to the correct emotion prediction.
These are therefore untypical cases of appraisal assignment. The last block
shows examples where the wrong appraisal prediction leads to wrong emotion
assignment.
It is further interesting to look into those cases which are wrongly predicted
from text, but correctly predicted based on the gold appraisal annotations. We
show examples for such cases in Table 7. Several of these cases are examples
in which a word seems to indicate a particular emotion, which is actually not
relevant to infer the emotion in the first place (e.g., “animal”, “vomiting”,
“kids”, “high school”). Often, shame is wrongly predicted when the event is
about the self. This is particularly problematic if the actual word pointing
to an emotion appears to be non-typical (e.g., “crossword”, “anaesthetic”).
Gold Emotion | A$\rightarrow$E | T$\rightarrow$E | Text
---|---|---|---
Anger | Anger | Fear | because I was overlooked at work.
Anger | Anger | Disgust | when I saw someone mistreating an animal.
Anger | Anger | Fear | when someone overtook my car on a blind bend and nearly caused an accident.
Disgust | Disgust | Shame | because I ate a sausage that was horrible.
Disgust | Disgust | Fear | when I was on a ferry in a storm and lots of people were vomiting.
Disgust | Disgust | Shame | because the milk I put in my coffee had lumps in it.
Fear | Fear | Shame | because I had to have a general anaesthetic for an operation.
Fear | Fear | Sadness | when my 2 year old broke her leg, and we felt helpless to assist her.
Fear | Fear | Anger | because we were driving fast in the rain in order to get somewhere before it shut, and the driver was going over the speed limit.
Guilt | Guilt | Shame | when I took something without paying.
Guilt | Guilt | Joy | for denying to offer my kids what they demanded of me.
Guilt | Guilt | Anger | when I had not done a job for a friend that I had promised to do.
Joy | Joy | Sadness | when witnessing the joy on my children’s face on Christmas morning.
Joy | Joy | Shame | when I managed to complete a cryptic crossword.
Joy | Joy | Disgust | when I found a twenty pound note on the ground outside.
Sadness | Sadness | Fear | when it was raining this morning as I been planning to go on a camping trip.
Sadness | Sadness | Joy | I feel … when I see the Christmas decorations come down, and know they won’t be up again for another year.
Sadness | Sadness | Shame | when my friend’s eye was watering after an injection into it and I could do nothing to help.
Shame | Shame | Joy | when I failed my ninth year at high school.
Shame | Shame | Guilt | when I had too much to drink in a pub, fell over and had to go to hospital.
Shame | Shame | Anger | when my mom caught me lying.
Table 7: Examples in which the appraisal model (on gold appraisal annotation)
predicts the correct emotion and the baseline system does not.
## 5 Conclusions and Future Work
We investigated the hypothesis that informing an emotion classification model
about the cognitive appraisal regarding a situation is beneficial for the
model performance. We were able to show that emotion classification performs
better than text-based classification, under the assumption that perfect
appraisal predictions are possible and shows complementary correct
predictions. Yet, neither in a multi-task learning nor a pipeline, in which
the appraisal was predicted as a basis, could we show an improvement in
emotion classification. This provides evidence that, though our appraisal
predictor is of reasonable performance, the model suffers from error
propagation. This is still an encouraging result, suggesting that future work
should further investigate the combination of appraisal information with
emotion prediction, particularly in the light of our oracle ensemble that
indicated a clear improvement.
This first study on the topic raises a couple of research questions: Would
there be other neural architectures which are better suited for including the
appraisal information? Will more annotated data improve the prediction quality
sufficiently? Finally, it should be analyzed if giving the annotators access
to the emotion label when making the appraisal annotation could have changed
the results.
## Acknowledgements
This work was supported by Leibniz WissenschaftsCampus Tübingen “Cognitive
Interfaces” and Deutsche Forschungsgemeinschaft (project SEAT, KL 2869/1). We
thank Laura Ana Maria Oberländer for inspiration and fruitful discussions and
Valentino Sabbatino for his annotation work. Further, we thank the three
anonymous reviewers for their constructive and helpful criticism.
## References
* [Abdul-Mageed and Ungar, 2017] Muhammad Abdul-Mageed and Lyle Ungar. 2017\. EmoNet: Fine-grained emotion detection with gated recurrent neural networks. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 718–728, Vancouver, Canada, July. Association for Computational Linguistics.
* [Alm et al., 2005] Cecilia Ovesdotter Alm, Dan Roth, and Richard Sproat. 2005\. Emotions from text: Machine learning for text-based emotion prediction. In Proceedings of Human Language Technology Conference and Conference on Empirical Methods in Natural Language Processing, pages 579–586, Vancouver, British Columbia, Canada. Association for Computational Linguistics.
* [Aman and Szpakowicz, 2007] Saima Aman and Stan Szpakowicz. 2007\. Identifying expressions of emotion in text. In International Conference on Text, Speech and Dialogue, pages 196–205. Springer.
* [Averill, 1980] James R Averill. 1980\. A constructivist view of emotion. In Theories of emotion, pages 305–339. Elsevier.
* [Balahur et al., 2011] Alexandra Balahur, Jesús M. Hermida, Andrés Montoyo, and Rafael Muñoz. 2011\. Emotinet: A knowledge base for emotion detection in text built on the appraisal theories. In Rafael Muñoz, Andrés Montoyo, and Elisabeth Métais, editors, Natural Language Processing and Information Systems, pages 27–39, Berlin, Heidelberg. Springer Berlin Heidelberg.
* [Balahur et al., 2012] Alexandra Balahur, Jesus M. Hermida, and Andrew Montoyo. 2012\. Building and exploiting emotinet, a knowledge base for emotion detection based on the appraisal theory model. IEEE Transactions on Affective Computing, 3(1):88–101.
* [Barrett and Russell, 2015] Lisa Feldman Barrett and James A Russell. 2015\. The psychological construction of emotion. Guilford Publications, New York.
* [Bostan and Klinger, 2018] Laura-Ana-Maria Bostan and Roman Klinger. 2018\. An analysis of annotated corpora for emotion classification in text. In Proceedings of the 27th International Conference on Computational Linguistics, pages 2104–2119, Santa Fe, New Mexico, USA. Association for Computational Linguistics.
* [Bostan et al., 2020] Laura Ana Maria Bostan, Evgeny Kim, and Roman Klinger. 2020\. GoodNewsEveryone: A corpus of news headlines annotated with emotions, semantic roles, and reader perception. In Nicoletta Calzolari, Khalid Choukri, Thierry Declerck, Hrafn Loftsson, Bente Maegaard, Joseph Mariani, Asuncion Moreno, Jan Odijk, and Stelios Piperidis, editors, Proceedings of the 12th International Conference on Language Resources and Evaluation (LREC’20), Marseille, France. European Language Resources Association (ELRA).
* [Broekens et al., 2008] Joost Broekens, Doug DeGroot, and Walter A. Kosters. 2008\. Formal models of appraisal: Theory, specification, and computational model. Cognitive Systems Research, 9(3):173 – 197.
* [Buechel and Hahn, 2017] Sven Buechel and Udo Hahn. 2017\. EmoBank: Studying the impact of annotation perspective and representation format on dimensional emotion analysis. In Proceedings of the 15th Conference of the European Chapter of the Association for Computational Linguistics: Volume 2, Short Papers, pages 578–585, Valencia, Spain. Association for Computational Linguistics.
* [Buechel et al., 2016] Sven Buechel, Johannes Hellrich, and Udo Hahn. 2016\. Feelings from the Past—Adapting affective lexicons for historical emotion analysis. In Proceedings of the Workshop on Language Technology Resources and Tools for Digital Humanities (LT4DH), pages 54–61, Osaka, Japan, December. The COLING 2016 Organizing Committee.
* [Campero et al., 2017] Andres Campero, Bjarke Felbo, Joshua B. Tenenbaum, and Rebecca Saxe. 2017\. A first step in combining cognitive event features and natural language representations to predict emotions. In Conference on Cognitive Computational Neuroscience.
* [Cevher et al., 2019] Deniz Cevher, Sebastian Zepf, and Roman Klinger. 2019\. Towards multimodal emotion recognition in german speech events in cars using transfer learning. In Proceedings of the 15th Conference on Natural Language Processing (KONVENS 2019): Long Papers, pages 79–90, Erlangen, Germany. German Society for Computational Linguistics & Language Technology.
* [Clore and Ortony, 2013] Gerald. L. Clore and Andrew Ortony. 2013\. Psychological Construction in the OCC Model of Emotion. Emotion Review, 5(4):335–343.
* [Ekman, 1992] Paul Ekman. 1992\. An argument for basic emotions. Cognition & emotion, 6(3-4):169–200.
* [Ekman, 1999] Paul Ekman. 1999\. Basic emotions. In Tim Dalgleish and Mick J. Power, editors, Handbook of Cognition and Emotion. John Wiley & Sons, Sussex, UK.
* [Felbo et al., 2017] Bjarke Felbo, Alan Mislove, Anders Søgaard, Iyad Rahwan, and Sune Lehmann. 2017\. Using millions of emoji occurrences to learn any-domain representations for detecting sentiment, emotion and sarcasm. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pages 1615–1625, Copenhagen, Denmark. Association for Computational Linguistics.
* [Feldman Barrett, 2006] Lisa Feldman Barrett. 2006\. Solving the emotion paradox: categorization and the experience of emotion. Personality and Social Psychology Review, 10(1):20–46.
* [Gendron et al., 2014] Maria Gendron, Debi Roberson, Jacoba Marietta van der Vyver, and Lisa Feldman Barrett. 2014\. Perceptions of emotion from facial expressions are not culturally universal: evidence from a remote culture. Emotion, 14(2):251–262.
* [Izard, 1971] Carroll E. Izard. 1971\. The face of emotion. Appleton-Century-Crofts, New York.
* [Kim et al., 2017] Evgeny Kim, Sebastian Padó, and Roman Klinger. 2017\. Investigating the relationship between literary genres and emotional plot development. In Proceedings of the Joint SIGHUM Workshop on Computational Linguistics for Cultural Heritage, Social Sciences, Humanities and Literature, pages 17–26, Vancouver, Canada. Association for Computational Linguistics.
* [Kim, 2014] Yoon Kim. 2014\. Convolutional neural networks for sentence classification. In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 1746–1751, Doha, Qatar. Association for Computational Linguistics.
* [Klinger et al., 2018] Roman Klinger, Orphée De Clercq, Saif Mohammad, and Alexandra Balahur. 2018\. IEST: WASSA-2018 implicit emotions shared task. In Proceedings of the 9th Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis, pages 31–42, Brussels, Belgium. Association for Computational Linguistics.
* [Li et al., 2017] Yanran Li, Hui Su, Xiaoyu Shen, Wenjie Li, Ziqiang Cao, and Shuzi Niu. 2017\. DailyDialog: A manually labelled multi-turn dialogue dataset. In Proceedings of the Eighth International Joint Conference on Natural Language Processing (Volume 1: Long Papers), pages 986–995, Taipei, Taiwan. Asian Federation of Natural Language Processing.
* [Marsella and Gratch, 2009] Stacy C. Marsella and Jonathan Gratch. 2009\. EMA: A process model of appraisal dynamics. Cognitive Systems Research, 10(1):70 – 90. Modeling the Cognitive Antecedents and Consequences of Emotion.
* [Marsella et al., 2010] Stacy C. Marsella, Jonathan Gratch, and Paolo Petta. 2010\. Computational models of emotion. In Klaus R. Scherer, Tanja Bänziger, and Etienne Roesch, editors, A Blueprint for Affective Computing. Oxford University Press.
* [Mohammad and Bravo-Marquez, 2017] Saif Mohammad and Felipe Bravo-Marquez. 2017\. WASSA-2017 shared task on emotion intensity. In Proceedings of the 8th Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis, pages 34–49, Copenhagen, Denmark. Association for Computational Linguistics.
* [Mohammad and Turney, 2012] Saif M. Mohammad and Peter D. Turney. 2012\. Crowdsourcing a word-emotion association lexicon. Computational Intelligence, 29(3).
* [Mohammad et al., 2017] Saif M. Mohammad, Parinaz Sobhani, and Svetlana Kiritchenko. 2017\. Stance and sentiment in tweets. ACM Trans. Internet Technol., 17(3).
* [Mohammad et al., 2018] Saif Mohammad, Felipe Bravo-Marquez, Mohammad Salameh, and Svetlana Kiritchenko. 2018\. SemEval-2018 task 1: Affect in tweets. In Proceedings of The 12th International Workshop on Semantic Evaluation, pages 1–17, New Orleans, Louisiana. Association for Computational Linguistics.
* [Mohammad, 2012] Saif Mohammad. 2012\. #emotional tweets. In *SEM 2012: The First Joint Conference on Lexical and Computational Semantics – Volume 1: Proceedings of the main conference and the shared task, and Volume 2: Proceedings of the Sixth International Workshop on Semantic Evaluation (SemEval 2012), pages 246–255, Montréal, Canada. Association for Computational Linguistics.
* [Mohammad, 2018] Saif M. Mohammad. 2018\. Obtaining reliable human ratings of valence, arousal, and dominance for 20,000 english words. In Proceedings of The Annual Conference of the Association for Computational Linguistics (ACL), Melbourne, Australia.
* [Nair and Hinton, 2010] Vinod Nair and Geoffrey E. Hinton. 2010\. Rectified linear units improve restricted Boltzmann machines. In Johannes Fürnkranz and Thorsten Joachims, editors, Proceedings of the 27th International Conference on Machine Learning (ICML-10), pages 807–814, Haifa, Israel. Omnipress.
* [Oatley and Johnson-Laird, 1987] Keith Oatley and Philip N. Johnson-Laird. 1987\. Towards a cognitive theory of emotions. Cognition and emotion, 1(1):29–50.
* [Oatley, 1993] Keith Oatley. 1993\. Social construction in emotions. In M. Lewis and J. M. Haviland, editors, Handbook of emotions, pages 341–352. Guilford Press, New York.
* [Pennebaker et al., 2001] James W Pennebaker, Martha E Francis, and Roger J Booth. 2001\. Linguistic inquiry and word count: Liwc 2001. Mahway: Lawrence Erlbaum Associates, 71:2001.
* [Pennington et al., 2014] Jeffrey Pennington, Richard Socher, and Christopher Manning. 2014\. Glove: Global vectors for word representation. In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 1532–1543, Doha, Qatar. Association for Computational Linguistics.
* [Plutchik, 2001] Robert Plutchik. 2001\. The nature of emotions human emotions have deep evolutionary roots, a fact that may explain their complexity and provide tools for clinical practice. American Scientist, 89(4):344–350.
* [Posner et al., 2005] Jonathan Posner, James A. Russell, and Bradley S. Peterson. 2005\. The circumplex model of affect: an integrative approach to affective neuroscience, cognitive development, and psychopathology. Development and Psychopathology, 17(3):715–734.
* [Preoţiuc-Pietro et al., 2016] Daniel Preoţiuc-Pietro, H. Andrew Schwartz, Gregory Park, Johannes Eichstaedt, Margaret Kern, Lyle Ungar, and Elisabeth Shulman. 2016\. Modelling valence and arousal in Facebook posts. In Proceedings of the 7th Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis, pages 9–15, San Diego, California. Association for Computational Linguistics.
* [Scherer and Wallbott, 1997] Klaus R. Scherer and Harald G. Wallbott. 1997\. The ISEAR questionnaire and codebook. Geneva Emotion Research Group. https://www.unige.ch/cisa/research/materials-and-online-research/research-material/.
* [Scherer et al., 2001] Klaus R. Scherer, Angela Schorr, and Tom Johnstone. 2001\. Appraisal processes in emotion: Theory, methods, research. Oxford University Press.
* [Scherer et al., 2010] Klaus R. Scherer, Tanja Bänziger, and Etienne Roesch. 2010\. A Blueprint for Affective Computing. Oxford University Press.
* [Scherer et al., 2019] Klaus R. Scherer, Heiner Ellgring, Anja Dieckmann, Matthias Unfried, and Marcello Mortillaro. 2019\. Dynamic facial expression of emotion and observer inference. Frontiers in Psychology, 10:508.
* [Scherer, 1982] Klaus R. Scherer. 1982\. Emotion as a process: Function, origin and regulation. Social Science Information, 48:555–70.
* [Scherer, 2005] Klaus R. Scherer. 2005\. What are emotions? And how can they be measured? Social Science Information, 44(4):695–729.
* [Schuff et al., 2017] Hendrik Schuff, Jeremy Barnes, Julian Mohme, Sebastian Padó, and Roman Klinger. 2017\. Annotation, modelling and analysis of fine-grained emotions on a stance and sentiment detection corpus. In Proceedings of the 8th Workshop on Computational Approaches to Subjectivity, Sentiment and Social Media Analysis, pages 13–23, Copenhagen, Denmark. Association for Computational Linguistics.
* [Shaikh et al., 2009] Mostafa Al Masum Shaikh, Helmut Prendinger, and Mitsuru Ishizuka. 2009\. A linguistic interpretation of the occ emotion model for affect sensing from text. Affective Information Processing, pages 45–73.
* [Smith and Ellsworth, 1985] Craig. A. Smith and Phoebe. C. Ellsworth. 1985\. Patterns of cognitive appraisal in emotion. Journal of Personality and Social Psychology, 48(4):813–838.
* [Strapparava and Mihalcea, 2007] Carlo Strapparava and Rada Mihalcea. 2007\. SemEval-2007 task 14: Affective text. In Proceedings of the Fourth International Workshop on Semantic Evaluations (SemEval-2007), pages 70–74, Prague, Czech Republic, June. Association for Computational Linguistics.
* [Strapparava and Valitutti, 2004] Carlo Strapparava and Alessandro Valitutti. 2004\. WordNet affect: an affective extension of WordNet. In Proceedings of the Fourth International Conference on Language Resources and Evaluation (LREC’04), Lisbon, Portugal. European Language Resources Association (ELRA).
* [Tooby and Cosmides, 2008] John Tooby and Leda Cosmides. 2008\. The evolutionary psychology of the emotions and their relationship to internal regulatory variables. In J.M. Haviland-Jones M. Lewis and L. F. Barrett, editors, Handbook of emotions, pages 114–137. Guilford Press, New York.
* [Tracy and Robins, 2006] Jessica L. Tracy and Richard W. Robins. 2006\. Appraisal antecedents of shame and guilt: Support for a theoretical model. Personality and social psychology bulletin, 32(10):1339–1351.
* [Troiano et al., 2019] Enrica Troiano, Sebastian Padó, and Roman Klinger. 2019\. Crowdsourcing and validating event-focused emotion corpora for German and English. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 4005–4011, Florence, Italy. Association for Computational Linguistics.
## Appendix A Appraisal Prediction Performance of the Multitask Model
| T$\rightarrow$A/E
---|---
Appraisal | P | R | $\textrm{F}_{1}$
Attention | 82 | 78 | 80
Certainty | 85 | 76 | 80
Effort | 65 | 62 | 64
Pleasantness | 76 | 59 | 67
Responsibility | 71 | 69 | 70
Control | 60 | 42 | 49
Circumstance | 67 | 48 | 56
Macro avg. | 72 | 62 | 67
Micro avg. | 76 | 68 | 71
Table 8: Multi-task learning model performance on predicting appraisal
dimensions.
We focused in the discussion of the results in Section 4.2 on the goal to
improve the performance of the emotion prediction with appraisal information.
Though we evaluated the appraisal prediction model as an intermediate step in
the pipeline to emotion prediction, we did only evaluate the emotion
performance in the multitask learning setting due to our focus on emotion
categorization.
However, it would be meaningful to also experiment with a
T$\rightarrow$E,E$\rightarrow$A model when the goal would be to predict the
appraisal dimensions with or without knowledge of an emotion category. We
leave such evaluation for future work. However, our multitask model
T$\rightarrow$A/E also produces appraisal predictions in the context of
emotion predictions. We therefore provide the appraisal evaluation of the
T$\rightarrow$A/E model in Table 8. We see that the appraisal prediction does
not show any improvement over the T$\rightarrow$A model with the respective
results in Table 4.
## Appendix B Examples from the Multitask Model
We show examples from the T$\rightarrow$A/E model in Table 9, similarly to the
results from T$\rightarrow$A,A$\rightarrow$E examples in Table 6. Note that,
given the bidirectional interdependencies between emotion and appraisal taking
place in the multitask learning, in contrast to the unidirectional information
flow from appraisal to emotion in the pipeline, these results are more
challenging to interpret.
| Appraisal |
---|---|---
| Emotion (G/P) | A | Ce | Ex | Px | Rx | Co | Ci | Text
Appr+Emo corr. | Anger | 1 | 1 | 0 | 0 | 0 | 0 | 0 | when I heard someone gossip about a friend of mine.
Disgust | 0 | 1 | 0 | 0 | 0 | 0 | 1 | because the drain smelled.
Fear | 1 | 0 | 1 | 0 | 0 | 0 | 0 | when I thought my son could be in trouble when mountain climbing.
Guilt | 0 | 1 | 0 | 0 | 1 | 1 | 0 | because I drank way too much, knowing that I had things to do the next day.
Joy | 1 | 1 | 0 | 1 | 0 | 0 | 0 | because my son gave me a big hug when I got home from work.
Sadness | 1 | 1 | 0 | 0 | 0 | 0 | 0 | when I read about orangutans and palm oil
Shame | 0 | 1 | 0 | 0 | 1 | 1 | 0 | because I walked past a beggar and didn’t bother to stop and give her a pound.
Emo incorr. | Anger/Shame | 1 | 0 | 0 | 0 | 0 | 0 | 0 | because the parents dropping their kids off to school were driving and parking dangerously.
Anger/Disgust | 1 | 0 | 1 | 0 | 0 | 0 | 0 | someone stabbed a man on a train.
Disgust/Sadness | 1 | 1 | 0 | 0 | 0 | 0 | 0 | when I read that hunters had killed one of the world famous lions.
Disgust/Anger | 1 | 1 | 0 | 0 | 0 | 0 | 0 | when I watched videos of how some people treat animals.
Guilt/Shame | 0 | 1 | 0 | 0 | 1 | 1 | 0 | because I cheated on a previous girlfriend.
Sadness/Anger | 1 | 1 | 0 | 0 | 0 | 0 | 0 | when listening to the national news.
Shame/Guilt | 0 | 1 | 0 | 0 | 1 | 1 | 0 | when my daughter was rude to my wife.
Ap+Emo incorr. | Anger/Disgust | 1 | 1 | 0 | 0 | 0 | 0 | 0 | when a drunk man insulted my intelligence.
Disgust/Fear | 1 | 1 | 0 | 0 | 1 | 0 | 0 | towards my friend because he stole my wallet.
Fear/Shame | 0 | 1 | 0 | 0 | 1 | 0 | 0 | because a fight broke out in a bar where I was drinking
Guilt/Disgust | 0 | 1 | 0 | 0 | 0 | 0 | 0 | because I accidentally killed my pet fish by decorating his tank with untreated sand.
Joy/Sadness | 1 | 1 | 0 | 0 | 0 | 0 | 0 | when I rescued a dog at the local shelter and he became my best friend.
Sadness/Joy | 1 | 1 | 0 | 0 | 0 | 0 | 1 | when I was on holiday and it was the last day and had to come home.
Shame/Guilt | 0 | 1 | 0 | 0 | 1 | 0 | 1 | when I ate my whole Easter egg in one go.
Table 9: Examples for the prediction of the multi-task setting
(T$\rightarrow$A/E) . A: Attention, Ce: Certainty, E: Effort, P: Pleasantness,
R: Responsibility, Co: Control, Ci: Circumstance. First emotion mention is
gold, second is prediction. Appraisal shown is prediction with errors shown in
bold.
|
# MoRS: An Approximate Fault Modelling Framework for Reduced-Voltage SRAMs
İsmail Emir Yüksel3, Behzad Salami2, Oğuz Ergin3, Osman S. Ünsal2, Adrián
Cristal Kestelman24 3 TOBB University of Economics and Technology (TOBB ETÜ)
2 Barcelona Supercomputing Center (BSC) 4 Universitat Politècnica de
Catalunya (UPC)
###### Abstract
On-chip memory (usually based on Static RAMs-SRAMs) are crucial components for
various computing devices including heterogeneous devices, e.g, GPUs, FPGAs,
ASICs to achieve high performance. Modern workloads such as Deep Neural
Networks (DNNs) running on these heterogeneous fabrics are highly dependent on
the on-chip memory architecture for efficient acceleration. Hence, improving
the energy-efficiency of such memories directly leads to an efficient system.
One of the common methods to save energy is undervolting i.e., supply voltage
underscaling below the nominal level. Such systems can be safely undervolted
without incurring faults down to a certain voltage limit. This safe range is
also called voltage guardband. However, reducing voltage below the guardband
level without decreasing frequency causes timing-based faults.
In this paper, we propose MoRS, a framework that generates the first
approximate undervolting fault model using real faults extracted from
experimental undervolting studies on SRAMs to build the model. We inject the
faults generated by MoRS into the on-chip memory of the DNN accelerator to
evaluate the resilience of the system under the test. MoRS has the advantage
of simplicity without any need for high-time overhead experiments while being
accurate enough in comparison to a fully randomly-generated fault injection
approach. We evaluate our experiment in popular DNN workloads by mapping
weights to SRAMs and measure the accuracy difference between the output of the
MoRS and the real data. Our results show that the maximum difference between
real fault data and the output fault model of MoRS is 6.21%, whereas the
maximum difference between real data and random fault injection model is
23.2%. In terms of average proximity to the real data, the output of MoRS
outperforms the random fault injection approach by 3.21x.
###### Index Terms:
Modeling, Fault-injection, Neural Networks, Undervolting, SRAM.
## I Introduction
SRAMs are traditionally the building block of different components in
different computing systems such as branch predictor (in CPUs), register files
(in GPUs), on-chip buffer memory (in hardware accelerators like FPGAs and
ASICs), thanks to the low-latency access time of such memories. However, the
power consumption of SRAMs significantly contributes to the total system
power. For instance, prior works on GPUs[1] show that register file in GPUs
consumes 15-20% of the total power. Another work on modern out-of-order
cores[2] estimates that the SRAM-dominated front-end of the microprocessor
consumes up to 33% of the total power of the CPU. Other work on DNN
accelerators on FPGAs [3] shows that on-chip SRAMs consume 27% of the total
power.
Since the total power consumption of any underlying hardware is directly
related to its supply voltage, voltage underscaling is an effective solution
to save power [4, 5]. This technique is widely used in various devices such as
CPUs[6, 7, 8, 9, 10], GPUs[11] FPGAs[3, 12, 13, 14, 15], and ASICs[16, 17, 18]
as well as DRAMs[19, 20, 21], HBMs[22], SRAMs[23, 24], and Flash Disks[25, 26,
27, 28]. However, while performing voltage underscaling, reliability issues
can arise due to the increased circuit delay at reduced voltage levels. In
most commercial devices, there is a timing-fault free guardband between
nominal voltage and minimum safe voltage, i.e., $V_{min}$. Below this voltage
guardband, faults occur as a consequence of circuit delays.
Although further aggressive voltage underscaling below $V_{min}$ can achieve
more reduction in power consumption, it compromises system reliability through
undervolting faults. This particular region between $V_{min}$ and the lowest
operational voltage level, i.e., $V_{crash}$, still operational but with
faults is called the critical area[3]. Several prior fault mitigation
techniques were proposed[3, 16, 17, 29, 30]. However, these techniques need
either high-effort engineering[3, 30] or totally random fault injection
campaigns[16, 17]. These approaches are either impractical or not accurate.
Our solution offers the advantages of both random fault injection and
empirical experiments.
In this study, we propose MoRS, a framework that generates the first
approximate voltage underscaled model for SRAMs. MoRS consists of three steps:
1) Experiment, 2) Behavior Extraction, and 3) Model Generation. In the
Experiment step, MoRS uses publicly available undervolted fault map data[31]
of SRAMs based on our prior work[3]. In the Behavior Extraction step, we
extract the characteristic fault behavior features of undervolted SRAM blocks.
We establish and confirm that undervolting based faults do not occur randomly.
These faults are correlated with each other in space. We examine the faults of
row-based and column-based approaches and see the distance between
consecutively faulty bitcells, the number of each bit-fault in both rows and
columns, and the total number of faulty rows and columns per SRAM block are
not uniformly distributed. These fine-grained features show characteristic
behaviors. In this step, we extract characteristic features and categorize
them into two profiles: coarse-grained and fine-grained profiling. The reason
behind these categorizing is that random fault injection studies use only
coarse-grained features, the number of bit faults, and the number of faulty
SRAM blocks. The output of MoRS that we term Mixed Model uses both fine-
grained and coarse-grained features and applies probabilistic custom modeling
algorithm to achieve an approximate model as the last step.
In recent years, the power consumption of SRAMs on DNNs is increasing
drastically[16, 3, 17]. We evaluate MoRS for DNN accelerators where on-chip
SRAMs play an important role. MoRS generates fault models that are not limited
to a certain domain and can be potentially used in many domains such as branch
prediction units, register files, caches, and any other memories based on
SRAM, unlike the other prior studies. To evaluate MoRS, we generate a baseline
model by applying a uniform random distribution function to coarse-grained
features. This baseline, which we term as Random Model, is a standard fault
injection scheme and used in prior studies[32, 17, 16, 33]. Besides, we
process empirical data to see the difference in accuracy between each
artificial model (Random Model and Mixed Model) and real data.
In our evaluation methodology, we map weights to SRAM blocks. When we map
weights to SRAM blocks, there can be different mapping options such as MSB
mapping, LSB mapping, the first half of bits MSB the other half of bits LSB
mapping, and the first half of bits LSB the other half of bits MSB mapping.
Another method to save energy on DNNs is quantization. Our quantization method
is reducing the precision of weights from 32-bit to 16-bit, 8-bit, 4-bit, and
1-bit, respectively. While performing undervolting, some unwanted bits can be
flipped, and then the corresponding value becomes infinity or NaN. To avoid
these values masking techniques are used[34]. We mask infinity or NaN value to
1 or 0. We examine the behavior of different weight mappings, quantizations,
and masking options. We evaluate our experiments on trained LeNeT-5[35] and
cuda-convnet[36] DNNs in the classification stage. We examine the
classification accuracy for each voltage level. Our experiments show that
generated artificial model has similar behavior with the real data on
different DNN benchmarks with an accuracy of 96.4%. We also see that Random
Model is not close enough with real data with the difference in accuracy up to
23%. We find that on average our Mixed Model has 3.6% difference with real
data and on average 3x up to 7x closer than the baseline random model.
Our contributions are as follows:
* •
We propose a framework, MoRS that generates an artificial model that can
realistically emulate real undervolting fault data with a difference in the
accuracy of 3.6% on average. To the best of our knowledge, this study provides
the first reasonably accurate model for SRAM blocks under low-voltage
conditions.
* •
We evaluate our models and real data on state-of-the-art DNNs to see how
different weight mappings, quantization, and value maskings affect the
accuracy of inference. We find the similar observation with prior works[37,
38] that if we continue to reduce the precision of weights, DNNs become more
resilient to undervolting. At the lowest reduced voltage level 8-bit LeNeT
accuracy is 14% while 4-bit LeNeT accuracy is 60%. Even in this unlikely
situation, our Mixed Model shows similar behavior to real data.
The remainder of this paper is structured as follows. In Section II, we
introduce the most important concepts used in this paper. In Section III, we
propose our approximate fault modeling framework, MoRS. Section IV describes
the methodology that we perform our experiments. Section V detail the
experimental results. Related works are introduced in Section VI. Finally,
Section VII concludes this paper.
## II Background
### II-A Undervolting SRAM based on-chip Memories
CMOS is the dominant circuit technology for current computing devices. The
power consumption of CMOS-based SRAMs is the sum of two parts: the first is
dynamic power, also called active power, and the other one is leakage power,
also called static power. Earlier studies[39, 40, 41] show that the power
consumption of SRAMs is dominated by dynamic power.
Dynamic power i.e., $P_{dyn}$, is dissipated when there is switching activity
at some nodes in transistors. Leakage power i.e., $P_{leak}$, is dissipated by
the leakage currents flowing even when the device is not active.
Mathematically, these power equations are given by,
$P_{dyn}\propto f\times V_{dd}^{2}$ (1) $P_{leak}=k_{design}\times n\times
I_{leak}\times V_{dd}$ (2)
Here $V_{dd}$ shows the supply voltage, $f$ shows the operating frequency in
Equation 1. Further, $n$ denotes the number of transistors, $k_{design}$ is
the design-dependent parameter, and $I_{leak}$ shows the leakage current a
technology-dependent parameter in Equation 2. From Equation 1, dynamic power
consumption can be reduced by adjusting the supply voltage and operating
frequency. Likewise, from Equation 2, leakage power consumption can be reduced
by underscaling the supply voltage and reducing the total number of
transistors. Total power consumption can be reduced by underscaling the supply
voltage.
Voltage underscaling is a widely used technique for energy-efficient
applications. From Equation 1, dynamic power consumption is reduced
quadratically by underscaling the supply voltage. This technique can achieve
nominal voltage level i.e., $V_{nom}$ performance due to the operating
frequency not being changed. However, further aggressive undervolting below
the minimum safe voltage i.e., $V_{min}$ may cause reliability issues as the
result of timing faults. Between $V_{nom}$ and $V_{min}$ is called voltage
guardband. This guardband is to provide an assurance of correct functionality
even in the worst environmental case. Prior works achieve by applying
undervolting that power consumption reduces; 39% on FPGA on-chip memories [3],
20% in GPUs [42], and 16% in DRAMs [19] without any timing-related errors.
Also, recent studies show that undervolting internal components of FPGAs [12]
lead to around 3x power-efficiency and underscaling supply voltage of HBMs
[22] achieve a total of 2.3x power savings. In Minerva[16], lowering SRAM
voltages achieves a total of 2.7x power savings.
### II-B Fault Injection
The fault injection mechanism is a way to examine the behavior of systems
under different circumstances. Prior works on fault injection study for
reduced-voltage SRAMs are applied on branch prediction units [43, 32], caches
[44, 45], and FPGA on-chip memories [33]. There are several approaches for the
fault injection with having a trade-off on the engineering effort and
accuracy. i) The first one is applying random faults to random locations
without any information used from empirical experiments. ii) Another one is
directly using the empirical data as a fault map. iii) The last one is the
approximate modeling that is based on real data from empirical experiments and
close enough to empirical data. Table I gives a summary of comparison of these
three fault injection techniques in terms of effort and how close they are to
real data. The accuracy percentage is from the results of this study.
TABLE I: Comparison of fault injection techniques in terms of engineering
effort and accuracy of representing real data
Fault Injection Method | Engineering Effort | Accuracy (Min)
---|---|---
Random Fault Injection | Low | 77%
MoRS | Low | 94%
Empirical Data | High | 100%
### II-C Deep Neural Networks
Deep Neural Networks (DNNs) are widely used as an effective solution for
object recognition, classification and segmentation, and many other areas.
MNIST[46] and CIFAR-10 [47] datasets are widely used by the ML community to
showcase the latest technology advancements in DNNs. DNNs perform in two
phases: training and prediction (inference). The first process is training
that weights of DNNs are fitted to data. The training phase is mainly
performed in high-performance computing platforms like CPUs or GPUs. The
second phase is the inference phase, DNN models, trained in the first phase,
are used to extract unknown output from input data. Usually, the training
phase performs once whereas the prediction phase performs repeatedly. The
inference phase consists of mainly three elements: input, weights, and output.
Figure 1: A high level abstraction of one layer of DNNs.
Figure 1 shows a high-level abstraction of one inference layer. Each inference
layer of the network consists of weights, an input matrix, and an output
matrix. Weights of these layers are stored in on-chip memories. Due to their
large size and computational intensity, DNNs consume power; while they are
fault-tolerant to some extend. Some studies improve the power-efficiency and
reduce the total power consumption of DNNs by applying voltage underscaling
[3, 12], architectural techniques[48, 49, 50, 51, 52, 53, 54, 55], and
hardware-level techniques[56, 57, 58].
Quantization is an architectural optimization technique that reduces the
precision of data types. This technique can improve performance and energy-
efficiency. Recent studies[59, 16, 60, 61, 62] show that quantization to some
extent does not significantly affect the accuracy of DNN models. In our study,
we reduce weights precisions to 16-bit half-precision floating point,
8-bit($Q_{4.4}$), 4-bit($Q_{2.2}$) in fixed-point format, and 1-bit binary
values.
## III MoRS Framework
We propose MoRS, a framework to generate approximate artificial reduced-
voltage SRAM fault models. This mechanism is the first framework based on real
fault maps. MoRS stands between fully hardware and fully software fault
injection techniques. This framework generates an approximate model that is
close enough to real data compared to the fully software fault injection
mechanism. Also, MoRS does not require high-effort engineering in comparison
to a fully hardware approach that uses real data from empirical experiments.
As shown in Figure 2 MoRS consists of three steps: ❶ Experiment, ❷ Behavior
Extraction, and ❸ Model Generation. We first explain the first step existing
experiment that provides real fault maps from real SRAM blocks in Section 3.1.
In Section 3.2 we extract the behavior of fault maps as fine-grained and
coarse-grained profiles by using the output of the first step. In fine-grained
profiling extract the row and column behaviors of SRAM blocks in terms of
physical distance, the number of each bit-faults in rows and columns, and the
number of faulty rows and columns per block. Coarse-grained profiling is a
shallow approach generally used in prior fault injection studies. In coarse-
grained profiling, we extract only the total number of bit-faults and faulty
SRAM blocks. In Section 3.3 we generate an artificial model with outputs of
the second steps.
At the endpoint, we provide Mixed Model which is an approximate model to real
data. It is generated by applying probabilistic modeling on both fine-grained
and coarse-grained fault profiles. Figure 2 provides an overview of the three
steps of the MoRS.
Figure 2: Overview of the MoRS
### III-A Experiment
Voltage underscaling, i.e., undervolting is a widely used technique to save
energy. We study undervolting for modern SRAM blocks. Each SRAM block is a
matrix of bitcells formed of rows and columns. Also, each SRAM block is 16
Kbits with 1024 rows and 16 columns.
Board Name | VC707 | ZC702 | KC705
---|---|---|---
Technology Node | 28nm | 28nm | 28nm
Nominal Voltage | 1V | 1V | 1V
Operating Temperature | $50^{\circ}C$ | $50^{\circ}C$ | $50^{\circ}C$
Minimum Voltage Level | 0.54V | 0.53V | 0.54V
Number of SRAM Block | 2060 | 280 | 890
TABLE II: The summary of deployed FPGA boards
We perform this empirical study on SRAM blocks available in the off-the-shelf
Xilinx FPGAs, i.e., VC707, ZC702, and two identical samples of KC705 (referred
as KC705-A and KC705-B). Table II shows the detail of deployed FPGAs. This
experiment consists of two parts. The first part is the monitor SRAM blocks to
see undervolting faults. The second part is adjusting the supply voltage of
SRAM blocks by using a power management unit of FPGAs. FPGAs have a power
management bus (PMBUS) a voltage regulator that monitors and adjusts voltage
rails on FPGAs. This bus uses PMBUS standard and has an I2C protocol to
execute commands. The supply voltage of SRAM blocks of FPGAs is named
$V_{CCBRAM}$. By reducing $V_{CCBRAM}$ through PMBUS, only the power
consumption of SRAM blocks reduces. Since this situation does not affect any
other logic parts (DSPs, LUTs, etc..) the effect of undervolting on SRAM
blocks is seen clearly. The setup of this methodology is shown in Figure 3.
Figure 3: The methodology of empirical experiment
The method of this experiment follows an algorithm that as a first step writes
data to SRAMs after in the second step analyzes the faults in terms of rate
and location, in the third step reduces the supply voltage by 10$mV$, finally
repeat these steps until FPGA crashes.
When undervolting is applied below the voltage guardband i.e., $V_{min}$, the
fault rate exponentially increases. Voltage can be reduced until the voltage
that FPGA stops operating i.e., $V_{crash}$. Between $V_{min}$ and $V_{crash}$
faults occur due to timing failures while the power consumption significantly
reduces. SRAM-based real fault maps significantly vary even for different
FPGAs of the same platform as the result of the process variation. Besides,
the study shows that the pattern of faults is mostly permanent. Also, we
observe that the fault rate and location for different run are mostly the
same. Most importantly, undervolting faults are not uniformly distributed over
different SRAMs.
We use the publicly available data[31] of the prior work [3] as the output of
the first stage (Experimental Data) to generate artificial fault maps. We use
2000 SRAM blocks for extracting behavior and 950 SRAM blocks for testing our
framework, 2060 SRAM blocks from VC707 and 890 SRAM blocks from KC705. We make
the methodology of MoRS reliable by using different data set to create and
also to test our artificial model.
At $V_{crash}$ the fault rate, up to 0.06% and 0.005% per 1 Mbits for VC707
and KC705-B, respectively. It should be noted that faults appear in between
$V_{min}$ = 0.6V, $V_{crash}$ = 0.54V for VC707 and $V_{min}$ = 0.59V,
$V_{crash}$ = 0.53V for KC705-B. As the prior work[3] mentioned VC707 has the
most bit-faults among the other three boards. At the lowest voltage level,
VC707 has 10.2% faulty SRAMs with 23706 bit-faults. We refer [3] to more
detailed information about this prior work.
TABLE III: Features and which model they are used in
Profiling Type | Features | Models
---|---|---
Name | Percentage | Mixed-Model | Random-Model
Coarse-grained | Total number of bit-faults | $P_{F}$ | ✓ | ✓
Total number of faulty SRAM blocks | $P_{S}$ | ✓ | ✓
Fine-grained | Total number of faulty rows per SRAM block | $P_{SR_{0..1024}}$ | ✓ | ✗
Total number of faulty columns per SRAM block | $P_{SC_{0..16}}$ | ✓ | ✗
Number of each bit-fault in rows | $P_{FR_{0..15}}$ | ✓ | ✗
Number of each bit-fault in columns | $P_{FC_{0..1023}}$ | ✓ | ✗
Distance between consecutively faulty bitcells in rows | $P_{FDR_{1..15}}$ | ✓ | ✗
Distance between consecutively faulty bitcells in columns | $P_{FDC_{1..1023}}$ | ✓ | ✗
### III-B Behavior Extraction
As we mentioned in Section 3.1 voltage underscaling faults have patterns and
these fault patterns i.e., fault maps, are mostly the same. There are features
that affect the probability of bit-faults. In this step, we profile the
behavior of undervolting-related bit-faults to extract such important
features. We perform the profiling in two steps: coarse-grained profiling and
fine-grained profiling. A summary of all features, profiling types, and which
model they are used in can be found in Table III.
Coarse-grained profiling consists of two features. The first one is the
percentage of bit-faults in all SRAMs bitcell i.e., $P_{F}$. The second one is
the percentage of faulty SRAMs in all SRAM blocks i.e., $P_{S}$. For VC707, at
$V_{crash}$, total bit-faults are 0.07% and the percentage of faulty SRAMs is
10.2%.
Fine-grained profiling comprises two parts: row-based, column-based. Both row-
based and column-based have three features. The first one is the percentage of
faulty rows i.e., $P_{SR_{0..1024}}$ and faulty columns $P_{SC_{0..16}}$ in
faulty SRAM blocks. The second one is the percentage of each bit-faults in
rows i.e., $P_{FR_{0..16}}$, or columns i.e., $P_{FC_{0..1024}}$. The last one
is the percentage of physical distance between consecutively faulty bitcells
i.e., bitcell-distances, in the same row i.e., $P_{FDR_{1..15}}$, or column
i.e., $P_{FDC_{1..1023}}$. We discover that these two features for row and
column are not randomly or uniformly distributed.
At $V_{crash}$,
* •
The percentage of each bit-faults
* –
In terms of row concentrates at 2-bit faults ($P_{FR_{2}}$) and no faults
($P_{FR_{0}}$).
* –
In terms of column it concentrates no faults to 10-bit faults
($P_{FC_{0..10}}$).
* •
The percentage of bitcell-distances between consecutively faulty bitcells
* –
In terms of row concentrates at 8-bit distance ($P_{FDR_{8}}$).
* –
For rows that there is no such bitcell-distance is more than 8-bitcells
($P_{FDR_{9..15}}$).
* –
For the columns, it is concentrated in even numbers in decreasing
order$P_{FDC_{0..2..1024}}$, 2-bitcells distance ($P_{FDC_{2}}$) has the
highest percentage while 1022-bitcells distance ($P_{FDC_{1022}}$) has the
lowest in even numbers.
* –
Also, for columns, the bitcell-distances in odd numbers
($P_{FDC_{1..3..1023}}$) are stuck between 0 and 4.
Figure 4: Physical features on a faulty SRAM block as an example
As an illustration, we show the fault behavior of an $8*8$ SRAM block in
Figure 4. In this example, the column-based distance between cell A and cell B
is 6-bitcells. For cell C and cell D, the row-based distance is 5-bitcells.
When we examine this example in Figure 4 in terms of bit-fault for both row-
based and column-based, a column that contains cell C has 2-bit faults. Also,
cell A’s row has 4-bit faults. One of the coarse-grained profile features is
the percentage of total bit-faults ($P_{F}$). It can be calculated by dividing
the total number of faulty cells by all cells. Hence, for Figure 4’s SRAM
block $P_{F}$ is 28.125%.
As we extract the behavior of this example in Figure 4, we performed this
process for experimental data. When the profiling step is done, we start
generating models by using these coarse-grained and fine-grained profiles with
probabilistic modeling and uniform random distribution.
Algorithm 1 Generate Mixed Model
0: $n\leftarrow\\#ofSRAMblocks$
$bitfaults\leftarrow n\times 1024\times 16\times P_{F}$
$faultyS\leftarrow n\times P_{S}$
while $faultyS>0\ and\ bitfaults>0$ do
$block\leftarrow random$
$faultyBlock\leftarrow SRAMblocks[block]$
$faultyrows\leftarrow selectfrom(F_{SR_{0..16}})$
while $faultyrows>0$ do
$row\leftarrow random$
$column\leftarrow random$
$bitfaultsinrow\leftarrow selectfrom(F_{FR_{0..16}})$
while $bitfaultsinrow>0$ do
$distance\leftarrow selectfrom(F_{FDR_{1..15}})$
$column\leftarrow column+distance$
$faultyBlock[row][column]\leftarrow fault$
$bitfaultsinrow\leftarrow bitfaultsinrow-1$
end while
$faultyrows\leftarrow faultyrows-1$
end while
$ArtificialCF\leftarrow ColumnFeatures(faultyBlock)$
$RealCF\leftarrow ColumnFeatures(RealData)$
if $Similarity(ArtificialCF,RealCF)>80\%$ then
$SRAMblocks[block]\leftarrow faultyBlock$
$faultyS\leftarrow faultyS-1$
$bitfaults\leftarrow bitfaults-\\#offaults(faultyBlock)$
end if
end while
### III-C Model Generation
Probabilistic modeling [34] and uniform random distribution [34, 17, 63, 43]
are widely used in many modeling studies to generate fault maps and to inject
faults. For an approximate model i.e., Mixed Model, we use both fine-grained
and coarse-grained features with custom probabilistic modeling function. In
addition to these, we need the number of SRAM blocks that will generate.
To generate Mixed Model we follow the method shown in Algorithm 1. The input
of Algorithm 1 is the number of SRAM blocks and the output is faulty SRAM
data, also called fault map. First, in Algorithm 1, faulty SRAM blocks are
determined. After that, by using $P_{F}$ value, the number of faulty cells are
calculated. First, we randomly select faulty SRAMs in all SRAM blocks. Then,
we inject faults corresponding cells in faulty SRAMs. This injection algorithm
uses fine-grained features. This process is performed in two stages: row-based
fault injection and column-based control mechanism. The row-based fault
injection uses row-based features. We determine how many rows will be faulty
by using the probability of the number of faulty rows per SRAM block i.e.,
$F_{SR_{0..1024}}$ derivating from $P_{SR_{0..1024}}$. Then according to the
probability of the number of each bit-faults in rows i.e., $F_{FR_{0..16}}$
derivating from $P_{FR_{0..16}}$ we inject bit-faults to the corresponding
row. To inject more than one fault in a row we use the probability of physical
distance between consecutively faulty bitcells in rows $F_{FDR_{1..15}}$
derivating from $P_{FDR_{1..15}}$. In the second step of this algorithm,
first, we extract column-based features of each artificial fault map.
Figure 5: The running time overhead and accuracy difference between real data
and the generated artificial model for different similarity threshold levels
After extraction, we compare these features with fine-grained column-based
features extracted from the second step. If an artificial faulty SRAM has a
lower than 80% of similarity with experimental real data, we perform these
steps again until the similarity is 80% or higher. We select the 80%
similarity threshold as a good trade-off between the run-time (to generate the
fault models) and accuracy (of the generated fault models). As shown in Figure
5, if we increase the threshold level, the run time of MoRS increases
drastically. However, increasing the threshold is not achieve significant
accuracy compared to the optimal threshold. Below this threshold, the
approximate model converges to the Random Model that do not have an acceptable
accuracy.
## IV Experimental Methodology
MoRS is a general framework that generates approximate fault maps for
undervolted SRAM blocks. In this study, we test MoRS on state-of-the-art Deep
Neural Networks. Our experiments are based on injecting faults into weights of
trained DNNs. To evaluate how precise MoRS is we use Caffe[64]. Also, we
perform different quantizations (precisions), bit-mappings, and value masking
to diversify our experiments. The summary of these options is in Table IV.
TABLE IV: Different options for evaluation
Options | Name
---|---
Precision | 32-bit single-precision floating point
16-bit half-precision floating point
8-bit fixed point (Q4.4)
4-bit fixed point (Q2.2)
| 1-bit (Binary)
Bit-Mapping | MSB
LSB
First half MSB and other half LSB
First half LSB and other half MSB
Value Masking | Infinity or NaN to 1
Infinity or NaN to 0
The experiment is performed for each voltage level between $V_{min}$ and
$V_{crash}$, precision, mapping, and masking option. To compare artificial
models with real data we also process this methodology for real data.
Artificial Models are Random Model and Mixed Model. Random Model is a naive
random baseline used in prior works to inject faults. Mixed Model is an
approximate model, the output of the MoRS. Real Data is the experimental
data[31] extracted from the prior empirical study[3]. To evaluate real data,
we select the required amount of SRAM blocks randomly from real data. Instead
of Artificial Models, we process real data to evaluate in Figure 6.
Figure 6: Overall Methodology
The methodology consists of four parts as shown in Figure 6: ❶ MoRS &
Experiment, ❷ Bit-Precision & Bit-Mapping Unit, 3 ❸ Fault Injection Unit
(Weights to SRAM Mapping) and ❹ DNN Inference. In the first step, we acquire
artificial models from MoRS that are explained in Section 3. In the second
step, the Fault Injection Unit changes healthy weights into updated weights by
performing mapping and precision options. In the third step, with the outputs
of the first and second steps, we generate faulty weights. In the last step,
we obtain the final accuracy percentage.
1\. MoRS & Experiment. In this step, we choose which SRAM data is sent to
Fault Injection Unit. To evaluate MoRS, we generate a baseline model, i.e.,
Random Model. Random Model is generated by uniform random distribution with
coarse-grained features. We process every step in MoRS described in Section
II. However, we apply a directly random distribution function to only coarse-
grained features instead of applying custom probabilistic modeling and
algorithm to coarse-grained and fine-grained features.
In Random Model, first, with a given number of SRAM blocks we determine which
and how many blocks are faulty by using $P_{S}$ value. Then, using $P_{F}$
value we calculate how many cells are going to be faulty. When it is
calculated, we randomly inject faults in cells of randomly selected faulty
SRAM blocks. Because of the uniform random distribution, every cell in faulty
blocks has the same probability. The differences between Mixed Model and
Random Model are summarized in Table III.
In Figure 6, we call Mixed Model and Random Model Artificial Models. To
understand how accurate our approximate model we evaluate empirical data also
called Real Data. Since evaluated networks utilize 850 SRAM blocks at maximum,
we randomly choose the required amount of SRAM blocks.
2\. Bit-Precision & Bit-Mapping Unit. Quantization and undervolting are both
effective techniques to improve the energy efficiency of DNNs. However, they
may lead to accuracy loss with aggressive exploitation. MoRS enables us to
explore their correlation to find an optimal operating point.
In this step, we change weights according to precision and mapping options.
Caffe’s weights are 32-bit single-precision floating points. Since each row of
SRAM block has 16-bit, we store those weights in two rows when precision is
not reduced.
We use four fixed point precisions: 16-bit half-precision floating point,
8-bit (Q.4.4), 4-bit (Q2.2), and binary. To enable fixed-precision options we
use a prior study[65] an adapted version of the original Caffe with limited
numerical precision of weights. For 16-bit half-precision floating point, we
use NVCaffe [66], NVIDIA-maintained Caffe that supports 16-bit half-precision
floating point train and inference. When precision is reduced to $X-bit$, we
store $16/X$ weights in one row consecutively. Therefore, by reducing
precision, the usage of SRAM blocks and power consumption decrease with a cost
of accuracy loss.
In addition to precision options, we change the mapping of weights to SRAM
blocks. There are four mapping options: MSB, LSB, the first half of bit MSB,
and another half of bits LSB, the first half of bits LSB, and another half of
bit MSB. MSB means the most significant bit of weights maps to the first cell
of a row whereas LSB means the least significant bit of weights maps to the
first cell of a row.
3\. Fault Injection Unit (Weights to SRAM Mapping). After precision and
mapping options we obtain updated weights. In this step, we use artificial
models to inject faults in updated weights. Each cell of artificial models
contains either faulty or healthy information. If a bit of weight is mapped in
the faulty cell, we flip its value. Else, the value is not changed. When this
bit-flip operation performs, sometimes the value of weights could be infinity
or NaN. To prevent this situation we mask these values to either one or zero.
The masking operation is only performed for 32-bit single-precision and 16-bit
half-precision floating point. Because fixed point does not have any mantissa
or exponent parts to converge NaN or infinite value. In our study, the largest
fixed point representation is $Q_{4.4}$ and its maximum value is 15.
4\. DNN Inference. After injecting faults to the weights, we use Caffe
Framework to measure the accuracy of neural networks. Our most accurate
baseline is Real Data. We train our model based on part of real data and we
test it using another of that data. To diversify we have four different bit-
mapping, three different precision, and two different masking options for each
voltage level.
(a) $Infinity-NaN\ value\leftarrow 0$ $Mapping\leftarrow MSB$
(b) $Infinity-NaN\ value\leftarrow 1$ $Mapping\leftarrow MSB$
(c) $Infinity-NaN\ value\leftarrow 0$ $Mapping\leftarrow LSB$
(d) $Infinity-NaN\ value\leftarrow 1$ $Mapping\leftarrow LSB$
(e) $Infinity-NaN\ value\leftarrow 0$ $Mapping\leftarrow MSB\mid LSB$
(f) $Infinity-NaN\ value\leftarrow 1$ $Mapping\leftarrow MSB\mid LSB$
(g) $Infinity-NaN\ value\leftarrow 0$ $Mapping\leftarrow LSB\mid MSB$
(h) $Infinity-NaN\ value\leftarrow 1$ $Mapping\leftarrow LSB\mid MSB$
Figure 7: Voltage and resilience behavior of artificial models and
experimental (real) data on LeNeT-5[35] network for each bit-mapping and value
masking option when precision is not reduced (32-bit single-precision floating
point.)
## V Experimental Results
As we mentioned in the previous section we use Caffe[64], a deep learning
framework, and test the output of MoRS and random fault injection model on two
different neural network models: LeNeT-5[35] with MNIST dataset[46] and cuda-
convnet[36] with CIFAR-10 dataset[47]. We perform different bit-mappings and
value maskings for each neural network architecture. In addition to these
tests, we also perform reduced precision tests on LeNeT-5. Details of each
evaluated benchmark are summarized in Table V.
TABLE V: Details of evaluated neural networks
NN Model | LeNeT-5[35] | cuda-convnet[36]
---|---|---
Dataset Name | MNIST[46] | CIFAR-10[47]
# of Weights | 430500 | 89440
# of SRAM Blocks Utilized | 850 | 180
Inference Accuracy (%) | 99.05% | 79.59%
### V-A Overall Resilience
Figure 7 shows the accuracy of LeNeT-5[35] on the MNIST dataset[46] with
different bit-mapping and value masking options. We observe that in all
options Mixed Model is more precise than Random Model. Also, if the
application becomes less resilient, Mixed Model is closer to the real data and
the gap between the baseline and Mixed Model is increasing in terms of how
close they are to the real data.
We observe that that if we mask infinity and NaN value to 0, the LeNeT-5
network is more resilient than masking to 1. We think that the cause of this
situation is in the MNIST dataset hand-written digits are represented by one
and the rest of the background is represented by zero. Therefore, ones more
impact than zeros when it comes to classification.
We see that MSB and $MSB\mid LSB$ mapping cause more faults and are less
resilient than LSB and $LSB\mid MSB$ mapping. We find that the cause of this
is undervolting-based faults generally occur in first cells. Since MSB means
the most significant bit, when bit-flips happen it affects the corresponding
value more than others. Figure 7b and Figure 7f show more characteristic
behavior and are less resilient than others since both have MSB mappings and
masking to 1 option.
In Figure,7 we perform 300 iterations for each option (value masking, bit
mapping). Then we average these options for LeNeT-5 and cuda-convnet[36]
network architectures. In addition to these, we perform these experiments at
different precision levels for LeNeT-5 and for each precision level, we
average all options.
Figure 8 shows the accuracy of cuda-convnet on the CIFAR-10 dataset[47]
without reducing precision, i.e, the precision of weights is 32-bit floating
points. As we evaluate the LeNeT-5[35] network in Figure 9a, we average every
bit-mapping and value masking option for cuda-convnet [36] network.
We find that the cause of LeNeT-5 works better is cuda-convnet on CIFAR-10 is
rather smaller in terms of the number of weights and SRAM block utilization.
However, even in this network, Mixed Model is 1.47x more accurate than Random
Model in terms of accuracy difference with Real Data on average. While the
supply voltage of SRAM blocks is reducing, we observe more characteristic and
distinguishing behavior from real data. At $V_{crash}$ level, the accuracy
difference between Mixed Model and Real Data is only 6% whereas between Random
Model and Real Data is 10%.
### V-B Quantization
Figure 9 shows the accuracy of LeNeT-5 on the MNIST dataset at different
precision levels. We see that in each precision level our the Mixed Model has
more similar behavior than Random Model has to the real data. Figure 9a is the
average of all options demonstrated in Figure 7.
Figure 8: Average of all options on voltage and resilience behaviour of
artificial models and experimental (real) data on cuda-convnet network without
reducing precision.
(a) 32-bit single-precision floating point
(b) 16-bit half-precision floating point
(c) 8-bit fixed point (Q4.4)
(d) 4-bit fixed point (Q2.2)
(e) 1-bit (Binary)
Figure 9: Average of all options on voltage and resilience behaviour of
artificial models and experimental (real) data on LeNeT-5 network with
different precisions.
Figure 9a shows the resilience of both artificial models and real data under
nominal precision level (32-bit single-precision floating point) in terms of
accuracy. Mixed Model is 3.74x closer than Random Model to the real data on
average. At $V_{crash}$, the accuracy difference between Mixed Model and Real
Data is 2.46%, on the other hand, the difference between Random Model and Real
Data is 12.47%.
Figure 9b shows the accuracy of LeNeT for weights with the precision reduced
to 16-bit half-precision floating point. The behavior of all models are
similar to the 32-bit architecture, since 16-bit precision is mostly cover all
values in 32-bit [67, 68] without significant affect on accuracy. The highest
accuracy difference between real data and Mixed Model is 4.47% at 0.55V
whereas the highest accuracy difference between real data and Random Model is
9.05% at 0.57V. The average accuracy difference of all voltage level between
real data and Mixed Model is 2.25% whereas between real data and Random Model
is 4.84%, which is 2.15x worse than Mixed Model in terms of similarity of the
behavior of real data.
Figure 9c shows the accuracy of LeNeT with 8-bit precision for weights. We see
that compared to 32-bit, the network are less resilient to undervolting
faults. On average, Mixed Model is 7x better than Random Model in terms of
approximation to real data. At $560mV$, the accuracy gap between Mixed Model
and Real Data is 4% while the difference between Random Model and Real Data is
23%.
Figure 9d shows the accuracy of both artificial models and real data when the
precision of weights reduces to 4-bit. We see that reducing the precision of
weights 8-bit to 4-bit network become more resilient and more fault-tolerant
to errors based on voltage underscaling. Although, at $V_{nom}$, 8-bit LeNeT
is more accurate than 4-bit LeNeT, at $V_{crash}$, 8-bit accuracy is 14%
whereas 4-bit accuracy is 62%. Even in this unexpected situation, artificial
models of the MoRS framework have similar behavior to real data. On average,
Mixed Model is 2x closer to real data than Random Model. At $550mV$, the
difference in accuracy between Mixed Model and real data is around 1.5%, while
the difference in accuracy between Random Model and real data is 3%.
Figure 9e shows the accuracy of LeNeT when precision reduces to 1-bit. In
1-bit tests, we map weights to three different value sets. First one is
{-1,1}, second is {-1,0} and the last one is {0,1}. However, we have not
observed much difference between value sets. As we see in 4-bit, 1-bit LeNeT
network becomes more resilient to faults. The difference between $V_{nom}$
accuracy and $V_{crash}$ accuracy is 0.79%. Both artificial models have the
same behavior and do not have significant difference in accuracy. Mixed Model
has 0.03% difference in accuracy, whereas Random Model is 0.02%. Because of
these negligible statistics, we do not add 1-bit LeNeT to Figure 10.
To point out the resilience of reduced precision networks, we examine the
accuracy drop between $V_{nom}$ and $V_{crash}$. The drop is 68.4%, 12.9% and
0.79% for 8-bit, 4-bit and 1-bit LeNet, respectively.
LeNeT-5 Precision | SRAM Utilization | Accuracy
---|---|---
32-bit FP | 850 | 99.05%
16-bit Half FP | 425 | 97.03%
8-bit (Q4.4) | 213 | 82.79%
4-bit (Q2.2) | 107 | 74.75%
1-bit (Binary) | 27 | 49.59%
TABLE VI: LeNeT-5 SRAM Block Utilization and Inference Accuracy(at $V_{nom}$)
under different precisions
Table VI shows the SRAM utilization and accuracy in various precision levels
of weights. Inference accuracy represents the accuracy at nominal voltage
level ($V_{nom}$). 32-bit FP denotes the 32-bit single-precision floating
point, 16-bit Half FP stands for 16-bit half-precision floating points.
### V-C Comparison of Artificial Models
Figure 10 shows the accuracy gap between Real Data and both two artificial
models. We see that on the average of all benchmarks, Mixed Model is 3.21x
closer than Random Model on average. For most of the benchmarks, the maximum
difference in accuracy between Mixed Model and Real Model is under 5%.
However, for the Random Model, the maximum difference in accuracy is 23.2%.
We conclude that the proposed model not only has the same behavior with real
data against undervolting effects but also can imitate real data with a
tolerable difference in terms of accuracy. Most importantly, if the system is
not resilient to faults, randomized fault injection does not show the behavior
of real data. To be precise in how undervolting affects systems, it has to
profile the real data in fine-grained. Coarse-grained features are
insufficient to model real SRAM behavior when undervolting is performed.
Figure 10: The accuracy difference between Real Data and each artificial
model for all benchmarks.
## VI Related Work
To the best of our knowledge, this study provides the first approximate fault
modeling framework and injection in voltage underscaled SRAMs. In this
section, we discuss related work on fault injection and modeling on
undervolted systems and the resilience of DNNs.
Resilience of DNNs. DNNs are inherently reliable to faults. However, in harsh
environments, process variations, voltage underscaling can cause significant
accuracy loss. Reagen et al.[16] propose Minerva a fault mitigation mechanism
to mitigate low-voltage SRAM faults effects in DNN accelerators. Salami et
al.[3] study undervolting SRAM-based FPGA on-chip memories and present
intelligently-constrained BRAM placement to mitigate undervolting faults in
the NN classification layer. Torres-Huitzil et al.[29] present a comprehensive
review on fault and error tolerance in neural networks and mention mitigation
and fault injection techniques. They also mention that more realist/novel
fault models need to be developed to understand the effects of faults on
neural networks deeply as an open challenge. Deng et al.[30] present a
retraining-based mitigation technique for neural networks to become more
resilient.
Fault injection. Fault injection is a widely used technique in resilience
studies. Also, fault injection is used as bit-flip-based adversarial weight
attacks also called bit-flip attacks(BFA)[69, 70, 71] and fault-injection
attacks[72, 73]. Many studies focus on the reliability and resilience of
systems on soft-errors, noise[74, 75, 63, 76], and voltage underscaling by
injecting faults. Koppula et al.[34] propose a framework, EDEN, that proposes
combining training on profiled DRAM faults in terms of timing violations and
voltage underscaling with mitigation strategies and mappings. EDEN provides
four different error models since uniform random distribution does not cover
whole DRAMs. Chatzidimitrou et al.[43, 32] inject faults randomly to branch
prediction units to examine the effects of voltage underscaling.
Chandramoorthy et al.[17] study the undervolting faults in different layers of
networks by injecting faults to SRAMs randomly and do not take account of the
patterns or spatial distribution of bit errors. Stutz et al.[77] propose
random bit error training assumed voltage underscaled SRAMs faults distribute
randomly. Salami et al. [33] study the resilience of RTL NN Accelerators and
fault characterization and mitigation. To characterize and mitigate they
assume that each bitcell of SRAMs has the same probability. Yang et al. [23]
study energy-efficient CNNs by performing voltage scaling on SRAMs. To study
the effect of bit errors they hypothesize that errors in SRAM are roughly
uniformly distributed. The prior work[78] on Near-Threshold Voltage FinFET
SRAMs presents a fault model for SRAMs based on uniform random distribution.
Givaki et al.[79] study the resilience of DNNs under reduced voltage SRAM-
based FPGA on-chip memories by using directly the experimental data to examine
the training phase of DNNs.
All of these undervolted on-chip fault injection studies perform injection
randomly and do not take into account fine-grained profiling such as spatial
distances between cells, row-based and column-based approaches. Randomly
injecting faults approach can cause misleading to understand how the system
works under the low-voltage domain. As we mentioned in Section 4, if the
system does not have much resilience to voltage underscaling, the Mixed Model
of MoRS is 7x closer than the randomly injected model to the real data.
## VII Conclusion
In this paper, we propose MoRS, a framework that generates the first
approximate fault injection model i.e., artificial fault maps. The advantage
of the proposed framework is to inject errors into various systems including
heterogeneous computing devices. We evaluated the accuracy of the proposed
framework for state-of-the-art DNN applications. To evaluate our proposed
model, we measure the difference in accuracy between artificial error models
and real data. We show that compared to random model-based error injections,
the proposed model can provide 3.21x on average closer than to the real data.
## Acknowledgement
This work is partially funded by Open Transprecision Computing (OPRECOM)
project, Summer of Code 2020.
## References
* [1] H. Jeon, G. S. Ravi, N. S. Kim, and M. Annavaram, “Gpu register file virtualization,” in _Proceedings of the 48th International Symposium on Microarchitecture_ , ser. MICRO-48. New York, NY, USA: Association for Computing Machinery, 2015, p. 420–432. [Online]. Available: https://doi.org/10.1145/2830772.2830784
* [2] J. Haj-Yihia, A. Yasin, Y. B. Asher, and A. Mendelson, “Fine-grain power breakdown of modern out-of-order cores and its implications on skylake-based systems,” _ACM Trans. Archit. Code Optim._ , vol. 13, no. 4, Dec. 2016. [Online]. Available: https://doi.org/10.1145/3018112
* [3] B. Salami, O. S. Unsal, and A. C. Kestelman, “Comprehensive evaluation of supply voltage underscaling in fpga on-chip memories,” in _2018 51st Annual IEEE/ACM International Symposium on Microarchitecture (MICRO)_. IEEE, 2018, pp. 724–736.
* [4] G. Papadimitriou, A. Chatzidimitriou, D. Gizopoulos, V. J. Reddi, J. Leng, B. Salami, O. S. Unsal, and A. C. Kestelman, “Exceeding conservative limits: A consolidated analysis on modern hardware margins,” _IEEE Transactions on Device and Materials Reliability_ , vol. 20, no. 2, pp. 341–350, 2020.
* [5] D. Gizopoulos, G. Papadimitriou, A. Chatzidimitriou, V. J. Reddi, B. Salami, O. S. Unsal, A. C. Kestelman, and J. Leng, “Modern hardware margins: Cpus, gpus, fpgas recent system-level studies,” in _2019 IEEE 25th International Symposium on On-Line Testing and Robust System Design (IOLTS)_. IEEE, 2019, pp. 129–134.
* [6] A. Bacha and R. Teodorescu, “Using ecc feedback to guide voltage speculation in low-voltage processors,” in _2014 47th Annual IEEE/ACM International Symposium on Microarchitecture_ , 2014, pp. 306–318.
* [7] G. Papadimitriou, A. Chatzidimitriou, and D. Gizopoulos, “Adaptive voltage/frequency scaling and core allocation for balanced energy and performance on multicore cpus,” in _2019 IEEE International Symposium on High Performance Computer Architecture (HPCA)_ , 2019, pp. 133–146.
* [8] K. Parasyris, P. Koutsovasilis, V. Vassiliadis, C. D. Antonopoulos, N. Bellas, and S. Lalis, “A framework for evaluating software on reduced margins hardware,” in _2018 48th Annual IEEE/IFIP International Conference on Dependable Systems and Networks (DSN)_ , 2018, pp. 330–337.
* [9] G. Papadimitriou, M. Kaliorakis, A. Chatzidimitriou, D. Gizopoulos, P. Lawthers, and S. Das, “Harnessing voltage margins for energy efficiency in multicore cpus,” in _2017 50th Annual IEEE/ACM International Symposium on Microarchitecture (MICRO)_ , 2017, pp. 503–516.
* [10] A. Bacha and R. Teodorescu, “Dynamic reduction of voltage margins by leveraging on-chip ecc in itanium ii processors,” _SIGARCH Comput. Archit. News_ , vol. 41, no. 3, p. 297–307, Jun. 2013. [Online]. Available: https://doi.org/10.1145/2508148.2485948
* [11] A. Zou, J. Leng, X. He, Y. Zu, C. D. Gill, V. J. Reddi, and X. Zhang, “Voltage-stacked gpus: A control theory driven cross-layer solution for practical voltage stacking in gpus,” in _Proceedings of the 51st Annual IEEE/ACM International Symposium on Microarchitecture_ , ser. MICRO-51. IEEE Press, 2018, p. 390–402. [Online]. Available: https://doi.org/10.1109/MICRO.2018.00039
* [12] B. Salami, E. B. Onural, I. E. Yuksel, F. Koc, O. Ergin, A. C. Kestelman, O. S. Unsal, H. Sarbazi-Azad, and O. Mutlu, “An experimental study of reduced-voltage operation in modern fpgas for neural network acceleration,” 2020\.
* [13] B. Salami, O. S. Unsal, and A. C. Kestelman, “Evaluating built-in ecc of fpga on-chip memories for the mitigation of undervolting faults,” in _2019 27th Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP)_. IEEE, 2019, pp. 242–246.
* [14] B. Salami, O. Unsal, and A. Cristal, “Fault characterization through fpga undervolting,” in _2018 28th International Conference on Field Programmable Logic and Applications (FPL)_. IEEE, 2018, pp. 85–853.
* [15] B. Salami, “Aggressive undervolting of fpgas: power & reliability trade-offs,” 2018.
* [16] B. Reagen, P. Whatmough, R. Adolf, S. Rama, H. Lee, S. K. Lee, J. M. Hernández-Lobato, G. Wei, and D. Brooks, “Minerva: Enabling low-power, highly-accurate deep neural network accelerators,” in _2016 ACM/IEEE 43rd Annual International Symposium on Computer Architecture (ISCA)_ , 2016, pp. 267–278.
* [17] N. Chandramoorthy, K. Swaminathan, M. Cochet, A. Paidimarri, S. Eldridge, R. V. Joshi, M. M. Ziegler, A. Buyuktosunoglu, and P. Bose, “Resilient low voltage accelerators for high energy efficiency,” in _2019 IEEE International Symposium on High Performance Computer Architecture (HPCA)_ , 2019, pp. 147–158.
* [18] J. Zhang, K. Rangineni, Z. Ghodsi, and S. Garg, “Thundervolt: enabling aggressive voltage underscaling and timing error resilience for energy efficient deep learning accelerators,” in _Proceedings of the 55th Annual Design Automation Conference_ , 2018, pp. 1–6.
* [19] K. K. Chang, A. G. Yağlıkçı, S. Ghose, A. Agrawal, N. Chatterjee, A. Kashyap, D. Lee, M. O’Connor, H. Hassan, and O. Mutlu, “Understanding reduced-voltage operation in modern dram devices: Experimental characterization, analysis, and mechanisms,” _Proc. ACM Meas. Anal. Comput. Syst._ , vol. 1, no. 1, Jun. 2017. [Online]. Available: https://doi.org/10.1145/3084447
* [20] H. David, C. Fallin, E. Gorbatov, U. R. Hanebutte, and O. Mutlu, “Memory power management via dynamic voltage/frequency scaling,” in _Proceedings of the 8th ACM International Conference on Autonomic Computing_ , ser. ICAC ’11. New York, NY, USA: Association for Computing Machinery, 2011, p. 31–40. [Online]. Available: https://doi.org/10.1145/1998582.1998590
* [21] Q. Deng, D. Meisner, L. Ramos, T. F. Wenisch, and R. Bianchini, “Memscale: Active low-power modes for main memory,” in _Proceedings of the Sixteenth International Conference on Architectural Support for Programming Languages and Operating Systems_ , ser. ASPLOS XVI. New York, NY, USA: Association for Computing Machinery, 2011, p. 225–238. [Online]. Available: https://doi.org/10.1145/1950365.1950392
* [22] S. S. N. Larimi, B. Salami, O. S. Unsal, A. C. Kestelman, H. Sarbazi-Azad, and O. Mutlu, “Understanding power consumption and reliability of high-bandwidth memory with voltage underscaling,” 2020.
* [23] L. Yang and B. Murmann, “Sram voltage scaling for energy-efficient convolutional neural networks,” in _2017 18th International Symposium on Quality Electronic Design (ISQED)_ , 2017, pp. 7–12.
* [24] L. Yang and B. Murmann, “Approximate sram for energy-efficient, privacy-preserving convolutional neural networks,” _2017 IEEE Computer Society Annual Symposium on VLSI (ISVLSI)_ , pp. 689–694, 2017.
* [25] Y. Cai, G. Yalcin, O. Mutlu, E. F. Haratsch, A. Crista, O. S. Unsal, and K. Mai, “Error analysis and retention-aware error management for nand flash memory.” _Intel Technology Journal_ , vol. 17, no. 1, 2013.
* [26] Y. Cai, S. Ghose, E. F. Haratsch, Y. Luo, and O. Mutlu, “Error characterization, mitigation, and recovery in flash-memory-based solid-state drives,” _Proceedings of the IEEE_ , vol. 105, no. 9, pp. 1666–1704, 2017\.
* [27] Y. Cai, E. F. Haratsch, O. Mutlu, and K. Mai, “Error patterns in mlc nand flash memory: Measurement, characterization, and analysis,” in _2012 Design, Automation Test in Europe Conference Exhibition (DATE)_ , 2012, pp. 521–526.
* [28] Y. Cai, Y. Luo, S. Ghose, and O. Mutlu, “Read disturb errors in mlc nand flash memory: Characterization, mitigation, and recovery,” in _2015 45th Annual IEEE/IFIP International Conference on Dependable Systems and Networks_ , 2015, pp. 438–449.
* [29] C. Torres-Huitzil and B. Girau, “Fault and error tolerance in neural networks: A review,” _IEEE Access_ , vol. 5, pp. 17 322–17 341, 2017.
* [30] J. Deng, Y. Fang, Z. Du, Y. Wang, H. Li, O. Temam, P. Ienne, D. Novo, X. Li, Y. Chen, and C. Wu, “Retraining-based timing error mitigation for hardware neural networks,” in _2015 Design, Automation Test in Europe Conference Exhibition (DATE)_ , 2015, pp. 593–596.
* [31] B. Salami., “FPGA BRAMs Undervolting Study,” https://github.com/behzadsalami/FPGA-BRAMs-Undervoltig-Study, 2018.
* [32] A. Chatzidimitriou, G. Papadimitriou, D. Gizopoulos, S. Ganapathy, and J. Kalamatianos, “Analysis and characterization of ultra low power branch predictors,” in _2018 IEEE 36th International Conference on Computer Design (ICCD)_ , 2018, pp. 144–147.
* [33] B. Salami, O. S. Unsal, and A. C. Kestelman, “On the resilience of rtl nn accelerators: Fault characterization and mitigation,” in _2018 30th International Symposium on Computer Architecture and High Performance Computing (SBAC-PAD)_ , 2018, pp. 322–329.
* [34] S. Koppula, L. Orosa, A. G. Yağlıkçı, R. Azizi, T. Shahroodi, K. Kanellopoulos, and O. Mutlu, “Eden: Enabling energy-efficient, high-performance deep neural network inference using approximate dram,” in _Proceedings of the 52nd Annual IEEE/ACM International Symposium on Microarchitecture_ , ser. MICRO ’52. New York, NY, USA: Association for Computing Machinery, 2019, p. 166–181. [Online]. Available: https://doi.org/10.1145/3352460.3358280
* [35] Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” _Proceedings of the IEEE_ , vol. 86, no. 11, pp. 2278–2324, 1998.
* [36] A. Krizhevsky., “cuda-convnet,” https://github.com/akrizhevsky/cuda-convnet2, 2014.
* [37] M. Sabbagh, C. Gongye, Y. Fei, and Y. Wang, “Evaluating fault resiliency of compressed deep neural networks,” in _2019 IEEE International Conference on Embedded Software and Systems (ICESS)_ , 2019, pp. 1–7.
* [38] G. Li, S. K. S. Hari, M. Sullivan, T. Tsai, K. Pattabiraman, J. Emer, and S. W. Keckler, “Understanding error propagation in deep learning neural network (dnn) accelerators and applications,” in _Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis_ , ser. SC ’17. New York, NY, USA: Association for Computing Machinery, 2017. [Online]. Available: https://doi.org/10.1145/3126908.3126964
* [39] P. Upadhyay, S. Ghosh, R. Kar, D. Mandal, and S. P. Ghoshal, “Low static and dynamic power mtcmos based 12t sram cell for high speed memory system,” in _2014 11th International Joint Conference on Computer Science and Software Engineering (JCSSE)_ , 2014, pp. 212–217.
* [40] T. Azam, B. Cheng, and D. R. S. Cumming, “Variability resilient low-power 7t-sram design for nano-scaled technologies,” in _2010 11th International Symposium on Quality Electronic Design (ISQED)_ , 2010, pp. 9–14.
* [41] G. Chen, D. Sylvester, D. Blaauw, and T. Mudge, “Yield-driven near-threshold sram design,” _IEEE Transactions on Very Large Scale Integration (VLSI) Systems_ , vol. 18, no. 11, pp. 1590–1598, 2010.
* [42] J. Leng, A. Buyuktosunoglu, R. Bertran, P. Bose, and V. J. Reddi, “Safe limits on voltage reduction efficiency in gpus: A direct measurement approach,” in _2015 48th Annual IEEE/ACM International Symposium on Microarchitecture (MICRO)_ , 2015, pp. 294–307.
* [43] A. Chatzidimitriou, G. Panadimitriou, D. Gizopoulos, S. Ganapathy, and J. Kalamatianos, “Assessing the effects of low voltage in branch prediction units,” in _2019 IEEE International Symposium on Performance Analysis of Systems and Software (ISPASS)_ , 2019, pp. 127–136.
* [44] L. Rivière, Z. Najm, P. Rauzy, J. Danger, J. Bringer, and L. Sauvage, “High precision fault injections on the instruction cache of armv7-m architectures,” in _2015 IEEE International Symposium on Hardware Oriented Security and Trust (HOST)_ , 2015, pp. 62–67.
* [45] M. Kaliorakis, S. Tselonis, A. Chatzidimitriou, N. Foutris, and D. Gizopoulos, “Differential fault injection on microarchitectural simulators,” in _2015 IEEE International Symposium on Workload Characterization_ , 2015, pp. 172–182.
* [46] Y. Lecun and C. Cortes, “The MNIST database of handwritten digits.” http://yann.lecun.com/exdb/mnist/, 1999.
* [47] A. Krizhevsky, V. Nair, and G. Hinton, “he CIFAR-10 dataset,” https://www.cs.toronto.edu/ kriz/cifar.html.
* [48] A. Zhou, A. Yao, Y. Guo, L. Xu, and Y. Chen, “Incremental network quantization: Towards lossless cnns with low-precision weights,” 2017.
* [49] S. Han, X. Liu, H. Mao, J. Pu, A. Pedram, M. A. Horowitz, and W. J. Dally, “Eie: Efficient inference engine on compressed deep neural network,” 2016.
* [50] Z. Zhu, H. Sun, Y. Lin, G. Dai, L. Xia, S. Han, Y. Wang, and H. Yang, “A configurable multi-precision cnn computing framework based on single bit rram,” in _Proceedings of the 56th Annual Design Automation Conference 2019_ , ser. DAC ’19. New York, NY, USA: Association for Computing Machinery, 2019. [Online]. Available: https://doi.org/10.1145/3316781.3317739
* [51] P. Molchanov, S. Tyree, T. Karras, T. Aila, and J. Kautz, “Pruning convolutional neural networks for resource efficient inference,” 2017.
* [52] R. Yazdani, M. Riera, J. Arnau, and A. González, “The dark side of dnn pruning,” in _2018 ACM/IEEE 45th Annual International Symposium on Computer Architecture (ISCA)_ , 2018, pp. 790–801.
* [53] S. Han, J. Pool, J. Tran, and W. J. Dally, “Learning both weights and connections for efficient neural networks,” 2015.
* [54] Y. Shen, M. Ferdman, and P. Milder, “Escher: A cnn accelerator with flexible buffering to minimize off-chip transfer,” in _2017 IEEE 25th Annual International Symposium on Field-Programmable Custom Computing Machines (FCCM)_ , 2017, pp. 93–100.
* [55] C. Deng, S. Liao, Y. Xie, K. K. Parhi, X. Qian, and B. Yuan, “Permdnn: Efficient compressed dnn architecture with permuted diagonal matrices,” 2020\.
* [56] Y. Shen, M. Ferdman, and P. Milder, “Maximizing cnn accelerator efficiency through resource partitioning,” 2018.
* [57] C. Zhang, P. Li, G. Sun, Y. Guan, B. Xiao, and J. Cong, “Optimizing fpga-based accelerator design for deep convolutional neural networks,” in _Proceedings of the 2015 ACM/SIGDA International Symposium on Field-Programmable Gate Arrays_ , ser. FPGA ’15. New York, NY, USA: Association for Computing Machinery, 2015, p. 161–170. [Online]. Available: https://doi.org/10.1145/2684746.2689060
* [58] M. Riera, J. Arnau, and A. Gonzalez, “Computation reuse in dnns by exploiting input similarity,” in _2018 ACM/IEEE 45th Annual International Symposium on Computer Architecture (ISCA)_ , 2018, pp. 57–68.
* [59] M. Courbariaux and Y. Bengio, “Binarynet: Training deep neural networks with weights and activations constrained to +1 or -1,” _CoRR_ , vol. abs/1602.02830, 2016. [Online]. Available: http://arxiv.org/abs/1602.02830
* [60] B. Jacob, S. Kligys, B. Chen, M. Zhu, M. Tang, A. G. Howard, H. Adam, and D. Kalenichenko, “Quantization and training of neural networks for efficient integer-arithmetic-only inference,” _CoRR_ , vol. abs/1712.05877, 2017. [Online]. Available: http://arxiv.org/abs/1712.05877
* [61] J. Wu, C. Leng, Y. Wang, Q. Hu, and J. Cheng, “Quantized convolutional neural networks for mobile devices,” _CoRR_ , vol. abs/1512.06473, 2015. [Online]. Available: http://arxiv.org/abs/1512.06473
* [62] K. Ueyoshi, K. Ando, K. Hirose, S. Takamaeda-Yamazaki, J. Kadomoto, T. Miyata, M. Hamada, T. Kuroda, and M. Motomura, “Quest: A 7.49tops multi-purpose log-quantized dnn inference engine stacked on 96mb 3d sram using inductive-coupling technology in 40nm cmos,” in _2018 IEEE International Solid - State Circuits Conference - (ISSCC)_ , 2018, pp. 216–218.
* [63] B. Reagen, U. Gupta, L. Pentecost, P. Whatmough, S. K. Lee, N. Mulholland, D. Brooks, and G. Wei, “Ares: A framework for quantifying the resilience of deep neural networks,” in _2018 55th ACM/ESDA/IEEE Design Automation Conference (DAC)_ , 2018, pp. 1–6.
* [64] Y. Jia, E. Shelhamer, J. Donahue, S. Karayev, J. Long, R. Girshick, S. Guadarrama, and T. Darrell, “Caffe: Convolutional architecture for fast feature embedding,” in _Proceedings of the 22nd ACM International Conference on Multimedia_ , ser. MM ’14. New York, NY, USA: Association for Computing Machinery, 2014, p. 675–678. [Online]. Available: https://doi.org/10.1145/2647868.2654889
* [65] M. Milde, D. Neil, A. Aimar, T. Delbruck, and G. Indiveri, “Adaption: Toolbox and benchmark for training convolutional neural networks with reduced numerical precision weights and activation,” 11 2017.
* [66] NVIDIA., “NVIDIA-caffe extension,” https://github.com/NVIDIA/caffe, 2017\.
* [67] P. Micikevicius, S. Narang, J. Alben, G. Diamos, E. Elsen, D. Garcia, B. Ginsburg, M. Houston, O. Kuchaiev, G. Venkatesh _et al._ , “Mixed precision training,” _arXiv preprint arXiv:1710.03740_ , 2017.
* [68] S. Markidis, S. W. Der Chien, E. Laure, I. B. Peng, and J. S. Vetter, “Nvidia tensor core programmability, performance & precision,” in _2018 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW)_. IEEE, 2018, pp. 522–531.
* [69] A. S. Rakin, Z. He, J. Li, F. Yao, C. Chakrabarti, and D. Fan, “T-bfa: Targeted bit-flip adversarial weight attack,” 2021.
* [70] Z. He, A. S. Rakin, J. Li, C. Chakrabarti, and D. Fan, “Defending and harnessing the bit-flip based adversarial weight attack,” in _2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)_ , 2020, pp. 14 083–14 091.
* [71] A. S. Rakin, Z. He, and D. Fan, “Bit-flip attack: Crushing neural network with progressive bit search,” 2019.
* [72] Y. Liu, L. Wei, B. Luo, and Q. Xu, “Fault injection attack on deep neural network,” in _2017 IEEE/ACM International Conference on Computer-Aided Design (ICCAD)_ , 2017, pp. 131–138.
* [73] M. S. Kelly, K. Mayes, and J. F. Walker, “Characterising a cpu fault attack model via run-time data analysis,” in _2017 IEEE International Symposium on Hardware Oriented Security and Trust (HOST)_ , 2017, pp. 79–84.
* [74] G. Li, S. K. S. Hari, M. Sullivan, T. Tsai, K. Pattabiraman, J. Emer, and S. W. Keckler, “Understanding error propagation in deep learning neural network (dnn) accelerators and applications,” in _Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis_ , ser. SC ’17. New York, NY, USA: Association for Computing Machinery, 2017. [Online]. Available: https://doi.org/10.1145/3126908.3126964
* [75] M. A. Neggaz, I. Alouani, P. R. Lorenzo, and S. Niar, “A reliability study on cnns for critical embedded systems,” in _2018 IEEE 36th International Conference on Computer Design (ICCD)_ , 2018, pp. 476–479.
* [76] A. H. Salavati and A. Karbasi, “Multi-level error-resilient neural networks,” in _2012 IEEE International Symposium on Information Theory Proceedings_ , 2012, pp. 1064–1068.
* [77] D. Stutz, N. Chandramoorthy, M. Hein, and B. Schiele, “Bit error robustness for energy-efficient dnn accelerators,” 2020.
* [78] S. Ganapathy, J. Kalamatianos, K. Kasprak, and S. Raasch, “On characterizing near-threshold sram failures in finfet technology,” in _2017 54th ACM/EDAC/IEEE Design Automation Conference (DAC)_ , 2017, pp. 1–6.
* [79] K. Givaki, B. Salami, R. Hojabr, S. R. Tayaranian, A. Khonsari, D. Rahmati, S. Gorgin, A. Cristal, and O. S. Unsal, “On the resilience of deep learning for reduced-voltage fpgas,” in _2020 28th Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP)_. IEEE, 2020, pp. 110–117.
| İsmail Emir Yüksel is an MSc student and researcher in the Computer
Engineering Department of TOBB University of Economics and Technology (TOBB
ETÜ). He received his BSc in Electrical Electronics Engineering from TOBB
University of Economics and Technology in 2019. His research interests are
energy-efficient heterogeneous computing and low-power & fault-resilient
hardware accelerators.
---|---
| Behzad Salami is a post-doctoral researcher in the Computer Science (CS)
department of Barcelona Supercomputing Center (BSC) and an affiliated research
member of SAFARI Research Group at ETH Zurich. He received his Ph.D. with
honors in Computer Architecture from Universitat Politécnica de Catalunya
(UPC) in 2018. Also, he obtained MSc and BSc degrees in Computer Engineering
from Amirkabir University of Technology (AUT) and Iran University of Science
and Technology (IUST), respectively. He has received mutiple awards and grants
for his research. His research interests are heterogeneous systems, low-power
& fault-resilient hardware accelerators, and near-data processing systems.
Contact him at<EMAIL_ADDRESS>
---|---
| Oğuz Ergin is a professor in the department of computer engineering in TOBB
University of Economics and Technology. He received his BS in electrical and
electronics engineering from Middle East Technical University, MS, and Ph.D.
in computer science from the State University of New York at Binghamton. He
was a senior research scientist in Intel Barcelona Research Center prior to
joining TOBB ETÜ. He is currently leading a research group in TOBB ETÜ working
on energy-efficient, reliable, and high-performance computer architectures.
---|---
| Osman Sabri Ünsal is co-manager of the Parallel Paradigms for Computer
Architecture research group at Barcelona Supercomputing Center (BSC). He got
his B.S., M.S., and Ph.D. in Computer Engineering from Istanbul Technical
University, Brown University, and the University of Massachusetts, Amherst
respectively. His current research interests are in computer architecture,
fault-tolerance, energy-efficiency, and heterogeneous computing. He is
currently leading LEGaTO EU H2020 research project on heterogeneous energy-
efficiency computing.
---|---
| Adrián Cristal Kestelman received the Licenciatura degree in Computer
Science from the Faculty of Exact and Natural Sciences, Universidad de Buenos
Aires, Buenos Aires, Argentina, in 1995, and the Ph.D. degree in Computer
Science from the Universitat Politécnica de Catalunya (UPC), Barcelona, Spain.
Since 2006, he is a co-manager of the Computer Architecture for Parallel
Paradigms Research Group at Barcelona Supercomputing Center (BSC). His current
research interests include the areas of microarchitecture, multicore, and
heterogeneous architectures, and programming models for multicore
architectures. Currently, he is leading the architecture development of the
vector processor unit in the European Processor Initiative.
---|---
|
# Stochastic modulational instability in the nonlinear Schrödinger equation
with colored random dispersion
Andrea Armaroli Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes
et Molécules, F-59000 Lille, France Guillaume Dujardin Univ. Lille, Inria,
CNRS, UMR 8524 - Laboratoire Paul Painlevé, F-59000 Lille, France Alexandre
Kudlinski Univ. Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et
Molécules, F-59000 Lille, France Arnaud Mussot Univ. Lille, CNRS, UMR
8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000 Lille, France
Stefano Trillo Department of Engineering, University of Ferrara, I-44122
Ferrara, Italy Stephan De Bièvre Univ. Lille, CNRS, Inria, UMR 8524 -
Laboratoire Paul Painlevé, F-59000 Lille, France Matteo Conforti Univ.
Lille, CNRS, UMR 8523-PhLAM-Physique des Lasers Atomes et Molécules, F-59000
Lille, France
###### Abstract
We study modulational instability (MI) in optical fibers with random group-
velocity dispersion (GVD). We consider Gaussian and dichotomous colored
stochastic processes. We resort to different analytical methods (namely, the
cumulant expansion and the functional approach) and assess their reliability
in estimating the MI gain of stochastic origin. If the power spectral density
(PSD) of the GVD fluctuations is centered at null wavenumber, we obtain low-
frequency MI sidelobes which converge to those given by a white noise
perturbation when the correlation length tends to $0$. If instead the
stochastic processes are modulated in space, one or more MI sidelobe pairs
corresponding to the well-known parametric resonance (PR) condition can be
found. A transition from small and broad sidelobes to peaks nearly
indistinguishable from PR-MI is predicted, in the limit of large perturbation
amplitudes and correlation lengths of the random process. We find that the
cumulant expansion provides good analytical estimates for small PSD values and
small correlation lengths, when the MI gain is very small. The functional
approach is rigorous only for the dichotomous processes, but allows us to
model a wider range of parameters and to predict the existence of MI sidelobes
comparable to those observed in homogeneous fibers of anomalous GVD.
## I Introduction
A physical system exhibiting an interplay of weak nonlinearity and group
velocity dispersion (GVD) is subject to modulational instability (MI), i.e.,
the destabilization of a homogeneous state (plane or continuous waves), via
the exponential growth of small harmonic perturbations on a uniform background
[1]. After pioneering works in fluid mechanics [2, 3], MI was discovered in
electromagnetic waves [4] as well as in plasmas [5]; in the 80s the phenomenon
was observed in nonlinear fiber optics [6]. In uniform fibers, MI arises for
anomalous (negative) GVD, but it may also appear for normal GVD if
polarization [7], higher-order modes [8] or higher-order dispersion are
considered [9]. A different kind of MI related to a parametric resonance (PR)
mechanism emerges when the dispersion or the nonlinearity of the fiber are
periodically modulated [10, 11, 12, 13].
The impact of a random variation of GVD on MI was also the subject of a
considerable research effort. The particular case in which the GVD is
perturbed by a Gaussian white noise, i.e., a process exhibiting a vanishing
correlation length or equivalently a flat power spectral density (PSD), is
exactly solvable[14, 15, 16, 17, 18]. When the unperturbed fiber has an
anomalous GVD, the conventional MI gain profile is deformed due to the random
perturbation. In addition, MI sidebands of stochastic origin appear in the
case of normal GVD. A white noise, however, implies arbitrarily large
variations of GVD on arbitrarily small scales: an idealization that does not
always provide a relevant modeling of the randomness that may occur in
physical fibers. A non-conclusive theoretical study of parametric
amplification in the case of a GVD perturbed by a Gaussian process with a
finite correlation length was proposed in [19], and a numerical study can be
found in [20].
We aim at studying the MI problem in a class of random-GVD fibers that is both
experimentally accessible and theoretically tractable. In [21], we studied the
case of a GVD perturbed by randomly located sharp and large kicks. Two
different families of random processes were chosen to generate their mutual
spacing and amplitude. Different MI sidebands were predicted, including
multibump ones around zero detuning and others localized around PR
frequencies.
Here, we consider random fluctuations extended in space as in Ref. [14], but
focus on colored processes exhibiting an exponentially decaying
autocorrelation function. A lowpass (LP) and a bandpass (BP, modulated) case
are considered. We resort to cumulant expansion [22, 23, 24, 25, 26, 27, 28],
functional methods [29, 30, 31], and numerical simulations. The LP processes
converge to the white noise results for vanishing correlation lengths. The MI
sidelobes are located in the same detuning range and have amplitudes of the
same order of magnitude of those calculated from white noise. The BP processes
yield PR-like MI sidebands that are also comparable (in their maximal values)
to the white noise MI gain. For this reason, the white noise turns out to be a
reference for many crucial properties of the MI gain in a more generic
setting, eventhough there is little hope of an experimental realization of it.
Since the values of frequency detuning span a large range, the validity of the
different analytical methods has to be questioned, like in other physical
settings and systems [32, 33]. We thus have to rely on different approaches
and comparatively assess their soundness in describing the different features
of the MI sidelobes (position, height, width).
The rest of the paper is organized as follows. After presenting the model
equations (Sec. II), we study in Sec. III the lowpass random fluctuation by
means of the two different approaches mentioned above. We then analyze in Sec.
IV bandpass random fluctuations, for which we find convenient to introduce an
averaging approach to the stochastic equations. In each case we present a
thorough comparison of analytical approaches to numerical results. To close
the paper we report our conclusions and perspectives.
## II Model description
The propagation of optical pulses in a nonlinear optical fiber can be modeled
by the universal nonlinear Schrödinger equation (NLSE),
$i\partial_{z}U-\frac{1}{2}\beta_{2}(z)\partial_{tt}U+\gamma|U|^{2}U=0.$ (1)
Here $U(z,t)$ is the envelope of the optical pulse field in units of
$\sqrt{W}$, function of the propagation distance $z$ and time $t$ in the frame
of reference propagating at the group velocity of the mode propagating in the
fiber, $\beta_{2}$ is the GVD and $\gamma$ is the nonlinear coefficient [34].
We suppose that $\beta_{2}$ fluctuates randomly in $z$, while $\gamma$ stays
constant.
Let $\beta_{2}(z)=\beta_{2}^{0}+\delta\\!\beta(z)$, where $\beta_{2}^{0}>0$
(normal average GVD) and $\delta\\!\beta(z)$ is a stochastic process of zero
mean, that we specify below.
We observe that, for $\delta\\!\beta=0$, Eq. (1) has a continuous wave
($t$-independent) solution $U_{0}(z)=\sqrt{P}\exp\left(i\gamma Pz\right)$.
In order to study the stability of this continuous wave solution, we insert in
Eq. (1) the perturbed solution
${U}(z,t)=\left[\sqrt{P}+\check{x}_{1}(z,t)+i\check{x}_{2}(z,t)\right]\exp(i\gamma
Pz)$, linearize and Fourier-transform the resulting equation with respect to
$t$ ($\omega$ is used as the associated angular frequency detuning from the
carrier $U_{0}$), to obtain
$\frac{\mathrm{d}x}{\mathrm{d}z}=\begin{bmatrix}0&-g(z)\\\
h(z)&0\end{bmatrix}x,$ (2)
with $x\equiv(x_{1},x_{2})^{\mathrm{T}}$ (functions of $\omega$ and $z$),
$g(z)=g_{0}+\delta\\!g(z)$ and $h(z)=h_{0}+\delta\\!g(z)$, with
$g_{0}\equiv\beta_{2}^{0}\frac{\omega^{2}}{2}$, $h_{0}\equiv g_{0}+2\gamma P$,
and $\delta\\!g(z)\equiv{\delta\\!\beta(z)}\frac{\omega^{2}}{2}$. Eq. (2) is a
system of stochastic differential equations (SDEs) for each value $\omega$. In
the following sections we will discuss how to reduce it to a system of
ordinary differential equations (ODEs) for the first and second moments of the
probability density function of $x$ in order to estimate the MI gain.
We recall that we consider only normal average GVD, thus
$k^{2}\equiv g_{0}h_{0}>0$ (3)
and no conventional MI appears for $\delta\\!\beta(z)=0$. The MI sidebands we
predict below are therefore completely ascribed to random fluctuations. The
case of anomalous average GVD will be the subject of future work.
We focus here on two families of random processes, where the stochastic
process $\delta\\!\beta$ is characterized by two parameters $N_{0}>0$ and
$B>0$. The first family, for which we denote $\delta\\!\beta\equiv\chi$, is
characterized by an autocorrelation function of the form
$R_{\chi}(\zeta\equiv
z-z^{\prime})\equiv\langle\chi(z)\chi(z^{\prime})\rangle=\frac{N_{0}B}{4}\exp(-B|\zeta|).$
(4)
We recall that the variance of the process is
$\sigma_{\chi}^{2}=R_{\chi}(0)=\frac{N_{0}B}{4}$. For $B\to\infty$,
$R_{\chi}\to\frac{N_{0}}{2}\delta(\zeta)$, i.e., the white noise
autocorrelation function. Different stochastic processes exhibit this same
autocorrelation function. Here, we consider the Gaussian (often denoted in the
physics literature as Ornstein-Uhlenbeck [28]) and the dichotomous processes,
which find important applications in physics and allow us to obtain workable
approximations [33].
Both are stationary in $z$. By virtue of the Wiener-Khinchin theorem, the PSD
of $\chi$ coincides with the Fourier transform of $R_{\chi}$
$S_{\chi}(\kappa)=\int\limits_{-\infty}^{\infty}{\mathrm{d}\,\zeta}R_{\chi}(\zeta)\exp(i\kappa\zeta)=\frac{N_{0}}{2}\frac{B^{2}}{B^{2}+\kappa^{2}}.$
(5)
The Gaussian process is numerically generated in the $\kappa$-domain by
filtering an approximated white noise of PSD $N_{0}/2$ by means of a lowpass
filter of transfer function
$H_{\mathrm{LP}}(\kappa)=\frac{B}{B+i\kappa}.$ (6)
The dichotomous process is more conveniently obtained directly in the $z$
domain by switching the amplitude of the fluctuation between
$\pm\sigma_{\chi}$ with an exponentially distributed _waiting_ (i.e., between
switching points) length with mean $2/B$. Both are pertinent to fiber optics.
Indeed, the Gaussian process corresponds to a continuous variation of
dispersion as can be obtained by varying the fiber radius during the drawing
process. The dichotomous process corresponds to splicing together fibers with
different GVD $\beta_{2}=\beta_{2}^{0}(1\pm\sigma_{\chi})$ and random lengths.
We refer to the processes belonging to this class, with autocorrelation
function as in Eq. (4) and PSD as in Eq. (6) as LP processes.
The second family of stochastic processes, for which we denote
$\delta\\!\beta\equiv\xi$, is obtained as the modulated version of $\chi$ with
central wavenumber $\kappa_{0}=\frac{2\pi}{\Lambda_{0}}>0$ ($\Lambda_{0}$ is
the associated spatial period), that is written in phase-quadrature
representation
$\xi(z)=\psi_{1}(z)\cos\kappa_{0}z+\psi_{2}(z)\sin\kappa_{0}z.$ (7)
Here $\psi_{1,2}$ two stationary (in $z$) and independent random processes
with zero mean and autocorrelation functions
$R_{\psi_{i}}(\zeta)=\frac{N_{0}B}{2}\exp(-B|\zeta|),$ (8)
for $i=1,2$; moreover,
$\langle\psi_{i}\psi_{j}\rangle=\frac{N_{0}B}{2}\delta_{ij}$. In analogy to
$\chi$, we consider either two Gaussian or two dichotomous processes for
$\psi_{1,2}$. They are generated according to their distribution as is done
for $\chi$. The process $\xi$ thus exhibits an autocorrelation function of the
form
$R_{\xi}(\zeta)=\frac{N_{0}B}{2}\cos\kappa_{0}\zeta\exp(-B|\zeta|).$ (9)
The variance of the process is $\sigma^{2}_{\xi}=\frac{N_{0}B}{2}$, like for
$\psi_{1,2}$.
The PSD of $\xi$ reads
$S_{\xi}(\kappa)=\frac{N_{0}}{2}\left[\frac{B^{2}}{B^{2}+(\kappa-\kappa_{0})^{2}}+\frac{B^{2}}{B^{2}+(\kappa+\kappa_{0})^{2}}\right].$
(10)
We note that, for $B\ll\kappa_{0}$, $S_{\xi}$ is centered approximately around
the wavenumber $\pm\kappa_{0}$, with
$S_{\xi}(\pm\kappa_{0})\approx\frac{N_{0}}{2}=S_{\chi}(0)$ and bandwidth $B$
(in wavenumber units). We refer to this family with autocorrelation function
as in Eq. (9) and PSD as in Eq. (10) as BP processes.
For both LP and BP families, we employ the definition of correlation length
[35]
$\zeta^{\mathrm{c}}\equiv\frac{1}{R(0)}\int\limits_{0}^{\infty}{\mathrm{d}\zeta|R(\zeta)}|$,
which gives $\zeta_{\chi}^{\mathrm{c}}=1/B$ for $\chi$ and
$\zeta^{\mathrm{c}}_{\xi}\approx 2/(\pi B)$ for $\xi$, if $B\ll\kappa_{0}$.
In the next two sections we will study the effect $\chi$ and $\xi$,
respectively, on the MI predicted by Eq. (2).
## III Lowpass random dispersion
First we consider processes with the autocorrelation function given in Eq.
(4). We will discuss both the first and the second moment equations associated
to Eq. (2).
### III.1 Cumulant expansion (first moments)
The cumulant expansion yields a series development for the ODEs associated to
a SDE [28]. It is similar to the Dyson series of scattering theory [36] and
provides a solid base for more ad-hoc schemes [22]. See [26] for a systematic
derivation of terms to arbitrary order.
Let us rewrite Eq. (2) in the standard form
$\dot{{x}}=\left[{A_{1}}+\alpha\eta(z){C_{1}}\right]{x}$ with
${A_{1}}=\begin{bmatrix}0&-g_{0}\\\
h_{0}&0\end{bmatrix},\;{C_{1}}=\begin{bmatrix}0&-1\\\ 1&0\end{bmatrix},$
$\alpha=\frac{\omega^{2}}{4}\sqrt{N_{0}B}$ and $\eta=2\chi/\sqrt{N_{0}B}$ a
random process with unit variance and zero mean. The expansion is performed in
the formal parameter $\alpha$. To second order (the first-order term is
obviously 0), we write the ODE for the first moment
$\langle{x}\rangle=(\langle x_{1}\rangle,\langle x_{2}\rangle)^{\mathrm{T}}$
as
$\frac{\mathrm{d}}{\mathrm{d}z}\langle{x}\rangle=\left[\mathrm{A_{1}}+\alpha^{2}K_{2}^{1}\right]\langle{x}\rangle,$
(11)
with
$K_{2}^{1}=\int\limits_{0}^{\infty}\\!\mathrm{d}\zeta\,C_{1}e^{A_{1}\zeta}C_{1}e^{-A_{1}\zeta}R_{\eta}(\zeta).$
(12)
Other terms can be added in the expansion of Eq. (11): their contribution to
the solution rapidly decreases if
$\varepsilon\equiv\alpha\zeta^{\mathrm{c}},$ (13)
is small, i.e. $\frac{\omega^{2}}{4}\sqrt{\frac{N_{0}}{B}}\ll 1$. For large
detuning or small filter bandwidth $B$, the approximation may be invalid. A
fixed initial condition $x(0)$ can be considered and the solutions of Eq. (11)
do not keep memory of it for $z\gg\zeta_{\mathrm{c}}$. The long term dynamics
being our main focus, we set the limit of integration to infinity in Eq. (12),
see [25].
By tedious but straightforward algebra, we obtain
$\frac{\mathrm{d}}{\mathrm{d}z}\langle{x}\rangle=\begin{bmatrix}-\frac{\omega^{4}}{4}\frac{g_{0}}{2k^{2}}\left[(g_{0}+h_{0})c_{1}-(h_{0}-g_{0})c_{2}\right]&-g_{0}+\frac{\omega^{4}}{8k}(h_{0}-g_{0})c_{3}\\\
h_{0}+\frac{\omega^{4}}{8k}(h_{0}-g_{0})c_{3}&-\frac{\omega^{4}}{4}\frac{h_{0}}{2k^{2}}\left[(g_{0}+h_{0})c_{1}+(h_{0}-g_{0})c_{2}\right]\end{bmatrix}\langle{x}\rangle,$
(14)
with
$\displaystyle c_{1}$
$\displaystyle\equiv\int\limits_{0}^{\infty}\mathrm{d}\zeta
R_{\chi}(\zeta)=\frac{1}{2}S_{\chi}(0)=\frac{N_{0}}{4},$ (15) $\displaystyle
c_{2}$ $\displaystyle\equiv\int\limits_{0}^{\infty}\mathrm{d}\zeta
R_{\chi}(\zeta)\cos
2k\zeta=\frac{1}{2}S_{\chi}(2k)=\frac{N_{0}B^{2}}{4}\frac{1}{B^{2}+4k^{2}}$
$\displaystyle c_{3}$
$\displaystyle\equiv\int\limits_{0}^{\infty}\mathrm{d}\zeta
R_{\chi}(\zeta)\sin(2k\zeta)=\frac{N_{0}Bk}{2}\frac{1}{B^{2}+4k^{2}}.$
Since $c_{i}>0$, $i=1,2,3$, for all $\omega$, and $c_{1}>c_{2}$, it is easy to
verify that the eigenvalues of the matrix in Eq. (14) have both a negative
real part, so that the system Eq. (14) does not predict any MI gain. This
finding is analogous to the conventional harmonic oscillator with random
frequency [28, 33], for which the first moment undergoes only damping. For
this reason it is necessary to resort to the equations for the second moments.
We recall that, for white noise, the cumulant expansion at second order is
exact and in that limit, $B\to\infty$, $c_{1}=c_{2}$ and $c_{3}=0$, so the
eigenvalues of Eq. (14) reduce to Eq. (23) of Ref. [14].
### III.2 Cumulant expansion (second moments)
We now consider second moments. First we let $X_{1}=x_{1}^{2}$,
$X_{2}=x_{2}^{2}$, and $X_{3}=x_{1}x_{2}$ and derive from Eq. (2) a system for
their evolution, which reads
$\frac{\mathrm{d}}{\mathrm{d}z}{X}=\begin{bmatrix}0&0&-2g(z)\\\ 0&0&2h(z)\\\
h(z)&-g(z)&0\end{bmatrix}{X}.$ (16)
In order to perform the cumulant expansion, we write Eq. (16) in standard form
by letting
${A_{2}}=\begin{bmatrix}0&0&-2g_{0}\\\ 0&0&2h_{0}\\\
h_{0}&-g_{0}&0\end{bmatrix},\;{C_{2}}=\begin{bmatrix}0&0&-2\\\ 0&0&2\\\
1&-1&0\end{bmatrix},$
the other quantities $\alpha$ and $\eta$ being the same as in the previous
subsection.
Up to second order, the cumulant expansion reads
$\begin{gathered}\frac{\mathrm{d}}{\mathrm{d}z}\langle{X}\rangle=\left[\mathrm{A_{2}}+\alpha^{2}K_{2}^{2}\right]\langle{X}\rangle,\\\
\text{with
}K_{2}^{2}=\int\limits_{0}^{\infty}\\!\mathrm{d}\zeta\,C_{2}e^{A_{2}\zeta}C_{2}e^{-A_{2}\zeta}R_{\eta}(\zeta),\end{gathered}$
(17)
which gives
$\frac{\mathrm{d}}{\mathrm{d}z}\langle{X}\rangle=\begin{bmatrix}-\frac{\omega^{4}}{4g_{0}}\left[(g_{0}+h_{0})c_{1}+(g_{0}-h_{0})c_{2}\right]&\frac{\omega^{4}}{4h_{0}}\left[(g_{0}+h_{0})c_{1}-(g_{0}-h_{0})c_{2}\right]&-2g_{0}\\\
\frac{\omega^{4}}{4g_{0}}\left[(g_{0}+h_{0})c_{1}+(g_{0}-h_{0})c_{2}\right]&-\frac{\omega^{4}}{4h_{0}}\left[(g_{0}+h_{0})c_{1}-(g_{0}-h_{0})c_{2}\right]&2h_{0}\\\
h_{0}+\frac{\omega^{4}}{4k}(h_{0}-g_{0})c_{3}&-g_{0}+\frac{\omega^{4}}{4k}(h_{0}-g_{0})c_{3}&-\frac{\omega^{4}}{4k^{2}}\left[(g_{0}+h_{0})^{2}c_{1}-(g_{0}-h_{0})^{2}c_{2}\right]\end{bmatrix}\langle{X}\rangle,$
(18)
with the $c_{i}$s defined as above. The validity condition of the expansion is
the same as in the previous sub-section.
As in Ref. [21], the MI of stochastic origin is related to the growth rate of
the second moment. The eigenvalues of the matrix in Eq. (18) can be written
analytically. Their form is rather complicated: in general we have two complex
conjugate eigenvalues ($\lambda_{\pm}$) with negative real part and one
positive real eigenvalue, $\lambda_{0}$. The MI gain is thus defined as
$G_{2}(\omega)\equiv\frac{\lambda_{0}}{2}$. Since $G_{2}(\omega)$ is small for
small $N_{0}$, we proceed like in Ref. [14] to derive the following
approximation
$G_{2}(\omega)\approx\frac{4(\gamma
P)^{2}k\omega^{4}\left[8c_{2}k^{3}+c_{3}\omega^{4}\left(c_{1}j_{0}^{2}-4c_{2}(\gamma
P)^{2}\right)\right]}{64k^{6}-32c_{3}(\gamma
P)^{2}k^{3}\omega^{4}-\omega^{8}\left(c_{1}j_{0}^{2}-4c_{2}(\gamma
P)^{2}\right)},$ (19)
with $j_{0}\equiv g_{0}+h_{0}$. Nevertheless, this expression is still very
cumbersome and, below, we rely only on the numerically computed eigenvalue
$\lambda_{0}$.
We note that the cumulant expansion could be extended to fourth order (the
process being Gaussian, the third-order terms vanish), but the resulting terms
are very involved and do not clarify the behavior of gain at large $\omega$,
where the method breaks down (see below).
### III.3 Functional approach
An alternative approach follows [37, 33] and generalizes the use of Furustu-
Novikov-Shapiro-Loginov formulas, on which the treatment of white noise in
Ref. [14] is based.
Let us consider one of the second moments $X_{i}$, $i=1,2,3$. They are
functionals of $\delta\\!g$. Since $\delta\\!g\propto\chi$, its
autocorrelation function has the form of Eq. (4). According to Ref. [31], we
have
$\langle\delta\\!g\frac{\mathrm{d}X_{i}}{\mathrm{d}z}\rangle=\left(\frac{\mathrm{d}}{\mathrm{d}z}+B\right)\langle\delta\\!gX_{i}\rangle.$
(20)
Two steps are needed to write an averaged system: (i) average directly Eq.
(16); (ii) multiply each row by $\delta\\!g$ and average. We introduce three
new variables, $X_{3+i}\equiv\delta\\!gX_{i}$ and, in order to truncate the
system, we assume
$\langle\delta\\!g^{2}X_{i}\rangle=\sigma_{\delta\\!g}^{2}\langle
X_{i}\rangle$, where $\sigma_{\delta\\!g}^{2}\equiv
N_{0}B\frac{\omega^{4}}{16}$ is the variance of the process. This last
assumption is rigorously valid only for a dichotomous process [33] and in
general an infinite hierarchy of equations is obtained for a Gaussian one, see
[31].
We obtain a 6th-order system of ODEs
$\frac{\mathrm{d}}{\mathrm{d}z}\langle{X}\rangle=\begin{bmatrix}0&0&-2g_{0}&0&0&-2\\\
0&0&2h_{0}&0&0&2\\\ h_{0}&-g_{0}&0&1&-1&0\\\
0&0&-2\sigma_{\delta\\!g}^{2}&-B&0&-2g_{0}\\\
0&0&2\sigma_{\delta\\!g}^{2}&0&-B&2h_{0}\\\
\sigma_{\delta\\!g}^{2}&-\sigma_{\delta\\!g}^{2}&0&h_{0}&-g_{0}&-B\\\
\end{bmatrix}\langle{X}\rangle.$ (21)
The matrix in Eq. (21) has six eigenvalues. We observse numerically that one
is real and negative, one ($\lambda_{0}$) is real and positive, the last four
are two pairs of complex conjugate values with negative real part. The MI gain
is defined as above, $G_{2}(\omega)\equiv\frac{\lambda_{0}}{2}$. By numerical
inspection, we notice that the system in Eq. (21) generally gives different
eigenvalues with respect to Eq. (18). We stress that, in contrast to Sec.
III.2, the accuracy of the functional approach does not require any condition
on $\varepsilon$. However, in the white noise limit of $B\to\infty$, the
system reduces to three independent variables [33] and we obtain the Eqs.
(26-30) of [14], which as expected coincide with the cumulant expansion
results.
In the next subsection, we will show what are the limits of validity of the
two approximations and which fits better to numerical results.
### III.4 Results
Figure 1: MI gain as a function of detuning $\omega$ for a LP random
dispersion. Comparison of numerical values for a Gaussian (blue solid lines)
and dichotomous process (green dotted lines) obtained from Eq. (2) with the
estimates provided by Eq. (18) (red dashed lines), by Eq. (21) (dash-dotted
purple lines). We include also the gain corresponding to the white process
(yellow dotted line with pluses). In panel (a) $N_{0}=0.005$ and $B=4\pi$, so
that $\varepsilon<0.5$ over the considered $\omega$ range, (b) $N_{0}=0.4$ and
$B=4\pi$, so that $\varepsilon\approx 1$ around the maximum gain. The inset in
panel (a) shows a single realization of the two processes.
In order to validate our theoretical analysis, we resort to solving Eq. (2) by
generating a large number, $N$, of realizations of $\chi$ and studying the
moments of the resulting sample of solutions. The solution of Eq. (2) is known
exactly for piecewise constant $\beta_{2}(z)$ in terms of transfer matrices
[34]. For the Gaussian process, we choose the length $L$ of the domain
sufficiently large to ensure that the sampling rate in the spatial frequency,
i.e., $\Delta\\!\kappa=\frac{2\pi}{L}$, fairly represents the PSD of the
process, namely $B/\Delta\\!\kappa\ll 1$. An array of identically distributed
Gaussian random variables of zero mean and variance
$\sigma^{2}=\frac{N_{0}}{2\Delta\\!z}$, with $\Delta\\!z$ the sampling
distance in $z$ (over which the dispersion is assumed constant), is generated
numerically. This array is transformed to the $\kappa$ domain by means of FFT,
then multiplied by the Lorentzian filter of Eq. (6) and transformed back to
space domain. A realization of the dichotomous process is instead obtained in
the $z$ domain, by randomly generating the switching distances (where $\chi$
changes sign) from a randomly generated exponential distribution of mean
$2/B$. Between two switching points the GVD is assumed constant.
We solve Eq. (2) for a given initial condition
$(x_{1}(0),x_{2}(0))^{\mathrm{T}}=(1,0)$, corresponding to balanced sidebands,
over each random sequence of constant dispersion segments, to obtain a set of
output vectors $(x_{1}(L),x_{2}(L))^{\mathrm{T}}$ from which we estimate the
MI gain. Let $P_{\mathrm{out}}\equiv x_{1}^{2}(L)+x_{2}^{2}(L)$. We then
compute the mean gain, defined as [21, 20]
$G\equiv\frac{1}{2L}\log\langle\frac{P_{\mathrm{out}}}{P_{\mathrm{in}}}\rangle.$
(22)
In all our results we take $\gamma P=\beta_{2}^{0}=1$. This is equivalent to
putting Eq. (1) in the standard adimensional form.
In Fig. 1 we consider two examples of MI gain curves. We notice that the
sidelobes exhibit a single maximum $G_{\mathrm{max}}$ at detuning
$\omega_{\mathrm{max}}$.
We consider two different values of $N_{0}$. For small values, the gain is
generally small, see Fig. 1(a), where $N_{0}=0.005$. We choose a long domain
$L=5000$ to prevent finite-size effects that may appear at small $\omega$, and
$\Delta\\!z=0.01$ for the Gaussian case. Satisfactory statistics are obtained
for $N=2000$.
In this case $\varepsilon\approx 0.5$ at $\omega=10$ (see the definition of
$\varepsilon$ in (13)), the cumulant expansion is thus expected to be valid.
The functional approach gives very similar results, apart from some deviations
in the large $\omega$ tails. The numerical data show that the two processes
provide the same trend, that matches almost perfectly with the analytical
estimates (the functional approach proves more accurate in the tails, as
expected). The behavior near $\omega=0$ is due to the above-mentioned finite-
size effects. We notice that the LP process provide a much narrower and
smaller gain than the white noise (dotted yellow line with pluses). In this
regime, the correlations of the stochastic process suppress the MI gain.
Then we study the case of an intermediate value $N_{0}=0.4$, where
$\varepsilon>1$. We choose $L=50$, $\Delta\\!z=0.01$, and $N=1000$. We show in
Fig. 1(b) how the numerical results compare to the analytical estimates. The
functional approach proves very accurate for the dichotomous process over the
full range of $\omega$. The cumulant expansion models the numerical data quite
well, except in the large $\omega$ tails. We notice that the white noise gain
is smaller than the LP gain lobes, particularly for the dichotomous case.
Thus, for large $N_{0}$, correlations can improve the MI gain.
Figure 2: Detuning at the gain maximum as a function of $B$. Numerical results
of Gaussian (blue crosses) and dichotomous (green pluses) process are compared
to Eq. (18) (red dashed line), Eq. (21) (purple dash-dotted line). We include
$\omega_{\mathrm{max}}$ for the white noise, too, as reference (yellow dotted
line). The dashed vertical lines highlight the value of $B$ used in Fig. 1.
(a) $N_{0}=0.005$, (b) $N_{0}=0.4$.
Figure 3: Maximum gain as a function of $B$. Same convention as in Fig. 2.
In Figs. 2-3 we show the effect of variations of $B$ on
$\omega_{\mathrm{max}}$ and $G_{\mathrm{max}}$, respectively; $L$,
$\Delta\\!z$, and $N$ are chosen to guarantee a satisfactory statistical
sample for each point.
For $N_{0}=0.005$ [Figs. 2(a) and 3(a)] the position as well as the value of
the maximum gain are, for both stochastic processes, in the same range of
values and are, within the residual oscillation margins, well approximated by
both methods. Both $\omega_{\mathrm{max}}$ and $G_{\mathrm{max}}$ are monotone
increasing functions of $B$. As $B\to+\infty$, they converge from below to the
corresponding values for a white noise process. The limiting values are not
shown, because they are well beyond the axis scales.
The behavior for $N_{0}=0.4$ is very different [Figs. 2(b) and 3(b)]. First,
for the dichotomous case, the numerical points are well approximated by the
eigenvalues of Eq. (21). The maximum MI gain exhibits a sharp peak, see Fig.
3(b). We notice in Fig. 2(b) that $\omega_{\mathrm{max}}$ diverges for this
value of $B$: this means that the gain lobe never decays.
To understand this phenomenon, we study the eigenvalues of Eq. (21) at
$\omega\to\infty$ and we notice that, for $\sigma_{\delta\\!g}=g_{0}$, i.e.,
$B=4\left(\beta_{2}^{0}\right)^{2}/N_{0}$, $G_{2}(\omega)$ converges, for
small $N_{0}$, to
$G_{\mathrm{max}}\approx\frac{1}{4}\left[-B+\sqrt{\frac{B^{2}+B\sqrt{B^{2}+64(\gamma
P)^{2}}}{2}}\right]$. Notice also that for this value $\sigma^{2}_{\chi}=1$,
i.e., in dimensional units the GVD switches from $0$ to $2\beta_{2}^{0}$. In
the present case, $B=10$ and $G_{\mathrm{max}}\approx 0.17$, as observed in
Fig. 3(b). For $N_{0}>1$, the reasoning is still valid: the value of $B$ for
which $G_{2}(\omega)$ does not decay at $\omega\to\infty$ is well predicted,
while its value is generally larger than the approximated $G_{\mathrm{max}}$
above. From a physical point of view, this case is very pathological because
higher-order dispersion effects should be included in Eq. (1).
For the Gaussian process, $\omega_{\mathrm{max}}$ and $G_{\mathrm{max}}$ stay
close to the white noise limit (yellow dotted line) even for small $B$ and are
satisfactorily described by the cumulant expansion for large $B$. As expected,
the cumulant expansion is not accurate for small $B$ and is very different
from the estimate of the functional approach.
We conclude that the cumulant expansion provides an accurate approximation for
$G_{\mathrm{max}}$ and $\omega_{\mathrm{max}}$ for the Gaussian process. We
recall that, on the contrary, this is not the case for the behavior of the MI
gain at large $\omega$, see Fig. 1(b).
In conclusion, the LP processes yield a very small gain for $\varepsilon\ll
1$, always less than white noise limit. For $\varepsilon\sim 1$,
$G_{\mathrm{max}}$ can yield an MI gain larger than the white noise gain, but
is about one order of magnitude smaller than the conventional MI gain in
homogeneous fibers with anomalous GVD.
## IV Narrowband random dispersion
Here we consider the modulated process $\xi$ with the autocorrelation function
in Eq. (9), for which a new degree of freedom, $\kappa_{0}$, is included. We
start directly from Eq. (16).
### IV.1 Direct cumulant expansion
The cumulant expansion gives the same result as Eq. (18) with
$\displaystyle c_{1}$
$\displaystyle=\frac{N_{0}}{2}\frac{B^{2}}{B^{2}+\kappa_{0}^{2}},$ (23)
$\displaystyle c_{2}$
$\displaystyle=\frac{N_{0}B^{2}}{4}\left[\frac{1}{B^{2}+(2k-\kappa_{0})^{2}}+\frac{1}{B^{2}+(2k+\kappa_{0})^{2}}\right],$
$\displaystyle c_{3}$
$\displaystyle=N_{0}B\kappa\frac{B^{2}-\kappa_{0}^{2}+4k^{2}}{\left[B^{2}+(2k-\kappa_{0})^{2}\right]\left[B^{2}+(2k+\kappa_{0})^{2}\right]}.$
We observe that, while $c_{1}$ is constant, $c_{2}$ and $c_{3}$ exhibit a
resonant lineshape behavior close to $2k=\kappa_{0}$, i.e., the 1st PR
condition: the former exhibits a maximum at
$4k^{2}\approx\kappa_{0}^{2}-\frac{B^{4}}{4\kappa_{0}^{2}}$, while the latter
crosses zero at $4k^{2}=\kappa_{0}^{2}-B^{2}$. We recall that the $m$-th order
PR condition is in general $2k=m\kappa_{0}$, which gives
$\omega_{\mathrm{PR},m}^{2}=\sqrt{4\left(\frac{\gamma
P}{\beta_{2}^{0}}\right)^{2}+\left(\frac{m\kappa_{0}}{\beta_{2}^{0}}\right)^{2}}-\frac{2\gamma
P}{\beta_{2}^{0}}.$ (24)
For $B\ll\kappa_{0}$, we can set $c_{1}\approx 0$. Then around the first
parametric resonance detuning, tedious but straightforward calculations show
that the instability gain is approximated by
$G_{2}(\omega)\approx\frac{1}{2}\frac{(2\gamma
P)^{2}\omega^{4}}{\kappa_{0}^{2}}c_{2},$ (25)
that attains its maximum approximately at $\omega_{\mathrm{PR,1}}$,
$\displaystyle G_{2}^{\mathrm{max}}\approx G_{2}(\omega_{\mathrm{PR,1}})$
$\displaystyle\approx\frac{N_{0}(\gamma
P)^{2}}{2}\frac{\omega_{\mathrm{PR,1}}^{4}}{\kappa_{0}^{2}}$ (26)
$\displaystyle=\frac{N_{0}(\gamma
P)^{2}}{\left(2\beta_{2}^{0}\right)^{2}}\frac{\omega_{\mathrm{PR,1}}^{2}}{\omega_{\mathrm{PR,1}}^{2}+\frac{4\gamma
P}{\beta_{2}^{0}}}.$
We remark also that Eq. (25) is composed of a Lorentzian factor depending on
the process PSD ($c_{2}$) and a factor independent of the process. Moreover,
it turns out that the maximum MI gain, Eq. (26), coincides, for small $\omega$
and a given $N_{0}$ with the white noise MI gain evaluated at
$\omega=\omega_{{\rm PR},1}$, as hinted at in Ref. [32].
Notice that in Eq. (26), the MI gain is proportional to
$\frac{N_{0}\omega_{\mathrm{PR,1}}^{4}}{\kappa_{0}^{2}}$, while the PR-MI
scales like $\theta\frac{\omega_{\mathrm{PR,1}}^{2}}{\kappa_{0}}$ [12], with
$\theta$ the (constant) amplitude of the periodic variation. As in the low-
pass case, according to Eq. (26) the maximal MI gain depends mainly on
$N_{0}$. For $B\to 0$, the analogy to PR would suggest, instead, a dependence
on the amplitude of the fluctuations in real space, i.e., $N_{0}B$, as
discussed also in Ref. [32]. Below, we clarify this ostensible inconsistency.
### IV.2 Near-resonance reduction
In analogy to Ref. [32], a significant simplification of Eq. (2) can be
obtained by the conventional averaging method used for PR [38].
In order to average Eq. (2), we let
$\displaystyle x_{1}(z)$
$\displaystyle=y_{1}(z)\cos\left(\frac{\kappa_{0}}{2}z\right)+y_{2}(z)\sin\left(\frac{\kappa_{0}}{2}z\right)$
(27) $\displaystyle x_{2}(z)$
$\displaystyle=\frac{\kappa_{0}}{2g_{0}}\left[y_{1}(z)\sin\left(\frac{\kappa_{0}}{2}z\right)-y_{2}(z)\cos\left(\frac{\kappa_{0}}{2}z\right)\right],$
where $x_{2}$ is written assuming $g=g_{0}$ constant, in the spirit of the
variation of constants.
By averaging out oscillating terms, noticing that
$\frac{\delta\\!g}{g_{0}}=\xi$, and employing the phase-quadrature
representation of $\xi$, see Eq. (7), we obtain
$\displaystyle\kappa_{0}\dot{y}_{1}$
$\displaystyle=\Delta^{2}y_{2}+\Gamma\psi_{1}y_{2}-\Gamma\psi_{2}y_{1}$ (28)
$\displaystyle\kappa_{0}\dot{y}_{2}$
$\displaystyle=-\Delta^{2}y_{1}+\Gamma\psi_{1}y_{1}+\Gamma\psi_{2}y_{2},$
with $\Gamma\equiv\frac{1}{2}\left(\frac{\kappa_{0}^{2}}{4}-g_{0}^{2}\right)$
and $\Delta^{2}\equiv k^{2}-\frac{\kappa_{0}^{2}}{4}$ quantifies the detuning
from the PR condition; close to resonance,
$\Delta^{2}=(k+\kappa_{0}/2)(k-\kappa_{0}/2)\approx\kappa_{0}(k-\kappa_{0}/2)$.
Notice that the PR-MI gain of the first PR tongue is obtained from Eq. (28) by
putting $\psi_{1}=\psi_{2}=\theta$. It reads
$G_{\mathrm{PR,1}}=\frac{\sqrt{2(\Gamma\theta)^{2}-\Delta^{2}}}{\kappa_{0}}.$
(29)
Starting from Eq. (28), in this section we only study the evolution of second
moments. We introduce $Y_{1}\equiv y_{1}^{2}$, $Y_{2}\equiv y_{2}^{2}$, and
$Y_{3}\equiv y_{1}y_{2}$. It is easy to verify that
${Y}\equiv(Y_{1},Y_{2},Y_{3})^{\mathrm{T}}$ obeys
$\kappa_{0}\frac{\mathrm{d}}{\mathrm{d}z}{Y}=\begin{bmatrix}-2\Gamma\psi_{2}&0&2\Delta^{2}+2\Gamma\psi_{1}\\\
0&2\Gamma\psi_{2}&-2\Delta^{2}+2\Gamma\psi_{1}\\\
-\Delta^{2}+\Gamma\psi_{1}&\Delta^{2}+\Gamma\psi_{1}&0\end{bmatrix}{Y},$ (30)
where the parameters are the same as those used throughout this section.
Two random processes appear in Eq. (30) and both the cumulant expansion and
functional approach need generalizing accordingly. While we present the former
in App. A, because it provides very similar results to the direct cumulant
expansion of the previous subsection, the latter is reported below.
### IV.3 Functional approach near resonance
Following [39], we generalize the functional approach. We define
$Y_{3+i}\equiv\psi_{1}Y_{i}$, $Y_{6+i}\equiv\psi_{2}Y_{i}$, and
$Y_{9+i}\equiv\psi_{1}\psi_{2}Y_{i}$, $i=1,2,3$. We perform four different
averaging steps: (i) average directly Eq. (30), (ii) multiply each row of Eq.
(30) by $\psi_{1}$ and average, (iii) multiply by $\psi_{2}$ and average; (iv)
multiply by $\psi_{1}\psi_{2}$ and average. We employ the formula of
differentiation in Eq. (20) and its generalization
$\begin{gathered}\langle\psi_{1}\psi_{2}\frac{\mathrm{d}Y_{i}}{\mathrm{d}t}\rangle=\left(\frac{\mathrm{d}}{\mathrm{d}t}+2B\right)\langle\psi_{1}\psi_{2}Y_{i}\rangle.\end{gathered}\
$ (31)
If we assume as above that we can factor the variance out if the same process
occurs twice in an average, we obtain
$\kappa_{0}\frac{\mathrm{d}\langle{Y}\rangle}{\mathrm{d}z}=\left[\begin{array}[]{c|c|c|c}A_{2}&\Gamma
C_{2}^{\prime}&\Gamma C_{2}^{\prime\prime}&\mathbf{0}\\\
\hline\cr\sigma^{2}_{\xi}\Gamma
C_{2}^{\prime}&A_{2}-B\mathbf{I}&\mathbf{0}&\Gamma C_{2}^{\prime\prime}\\\
\hline\cr\sigma^{2}_{\xi}\Gamma
C_{2}^{\prime\prime}&\mathbf{0}&A_{2}-B\mathbf{I}&\Gamma C_{2}^{\prime}\\\
\hline\cr\mathbf{0}&\sigma^{2}_{\xi}\Gamma
C_{2}^{\prime\prime}&\sigma^{2}_{\xi}\Gamma
C_{2}^{\prime}&A_{2}-2B\mathbf{I}\end{array}\right]\langle{Y}\rangle,$ (32)
with
$\begin{gathered}{A_{2}}=\begin{bmatrix}0&0&2\Delta^{2}\\\ 0&0&-2\Delta^{2}\\\
-\Delta^{2}&\Delta^{2}&0\end{bmatrix},\\\
{C_{2}^{\prime}}=\begin{bmatrix}0&0&2\\\ 0&0&2\\\
1&1&0\end{bmatrix},\;{C_{2}^{\prime\prime}}=\begin{bmatrix}-2&0&0\\\ 0&2&0\\\
0&0&0\end{bmatrix},\end{gathered}$ (33)
and denoting $\mathbf{0}$ and $\mathbf{I}$ the null and identiy matrix,
respectively.
We compute numerically the eigenvalues of the matrix in Eq. (32) and look for
the dominant one, $\lambda^{*}$. The maximum MI gain, for $\Delta^{2}=0$, is
found analytically as
$(G_{2}^{\mathrm{PR}})^{\mathrm{max}}=\frac{1}{4\kappa_{0}}\left[\sqrt{B^{2}+4N_{0}B\omega_{\mathrm{PR,1}}^{4}}-B\right].$
(34)
We remark that the dependence on $\sigma_{\xi}$ is different from Eq. (26).
For small $B$, the gain is no longer proportional to
$\frac{\omega_{\mathrm{PR,1}}^{4}N_{0}}{\kappa_{0}^{2}}$, but to
$\frac{\omega_{\mathrm{PR,1}}^{2}\sqrt{N_{0}B}}{\kappa_{0}}$, as we would
formally obtain in the conventional periodic dispersion case, once we replace
the amplitude of parametric oscillation $\theta$ with $\sqrt{2}\sigma_{\xi}$.
Thus a very different result is found when compared to the cumulant expansions
above. We now assess which approximation works better by comparing them to
numerical solutions of Eq. (2).
### IV.4 Results
Figure 4: MI gain as a function of detuning $\omega$ for a BP random
dispersion, $\kappa_{0}=2\pi$. Comparison of numerical values for a Gaussian
(blue solid lines) and dichotomous process (green dotted lines) obtained from
Eq. (2) with the estimates provided by Eqs. (18) and (23) (red dashed lines),
and by Eq. (32) (dash-dotted purple lines). In panel (a) $N_{0}=0.005$ and
$B=\pi/4$, so that $\varepsilon<0.1$ around the PR resonance, (b) $N_{0}=3.2$
and $B=\pi/32$, so that $\varepsilon\gg 1$ around the $\omega_{\mathrm{max}}$.
In panel (b), we include the PR-MI gain of Eq. (29) with
$\theta=\sqrt{2}\sigma_{\xi}=\sqrt{\pi/5}$ (yellow dotted line with circles).
The insets show the numerical results for a Gaussian process on a larger
$\omega$ range.
In order to generate a realization of the process $\xi$ with Gaussian
distribution (resp. dichotomous), we employ the same approach in the spectral
(resp. spatial) domain as above, generate two independent LP processes and
modulate them according to Eq. (7). The numerical domain is obviously
discretized in both cases: for the Gaussian process a lower limit on $L$ is
required as in Sec. III.4, the dichotomous process requires a short
$\Delta\\!z$ to avoid spurious correlations. We do not forget to correctly
sample the period $\Lambda_{0}$ of the process spatial oscillations.
While in the previous section we were interested in the limit $B\to\infty$ to
contrast the LP processes to white noise, here we aim at understanding the
opposite limit, $B\to 0$, with fixed and finite $N_{0}B$. We expect that the
gain associated to the stochastic fluctuations converges to the one of the
periodically modulated fiber in this limit.
First, we consider a small perturbation, $N_{0}=0.005$, with a intermediate
bandwidth $B=\pi/4$ around $\kappa_{0}=2\pi$. We take $L=500$,
$\Delta\\!z=0.01$; a consistent statistical sample is collected for $N=2000$.
In Fig. 4(a) we compare the numerical mean gain $G(\omega)$ with the
analytical estimates obtained above. For such a small perturbation only a
single MI peak can be observed (see the inset for a larger detuning range). We
notice that the MI gain is centered about $\omega_{\mathrm{PR,1}}$ with gain
$G_{\mathrm{max}}$ and width $\Delta\\!\omega$. The Gausssian and dichotomous
processes give two almost identical results. The cumulant expansion provides a
very good approximation, because $\varepsilon\approx 0.1$. The near-resonance
functional approach proves instead very imprecise, as far as both
$G_{\mathrm{max}}$ and $\Delta\\!\omega$ are concerned. We explain this as
follows: it is easy to verify that the period of the modulation is comparable
to correlation length, $\Lambda_{0}\approx\zeta_{c}$. Thus, we apply
successively two distinct averaging procedures (near-resonant expansion and
functional approach) upon two perturbations occurring at the same scale as if
they were independent. This is a sure recipe for failure.
Moreover, we are not showing here the result of Eq. (37) that coincide with
the direct cumulant expansion around $\omega_{\mathrm{PR,1}}$, but is skewed
towards $\omega\to 0$, contrary to the numerical results.
As a second case, we consider a large $N_{0}=3.2$ and a small bandwidth
$B=\pi/32$. All the other parameters are the same as the previous. In Fig.
4(b) we observe, as above, that the MI lobe occurs around
$\omega_{\mathrm{PR,1}}$ and the Gaussian and dichotomous processes provide
two very close results. For such a large perturbation, several MI sidelobes
can be observed, in analogy to PR-MI, see the inset of Fig. 4(b). Now
$\varepsilon\approx 6.3\gg 1$; we thus expect that the cumulant expansion
fails to correctly describe the numerical results: indeed it overestimates the
peak MI while it underestimates its width (red dashed line). It captures
approximately the behavior of the MI gain in the tails (both left-hand and
right-hand sides). The functional method provides instead a good approximation
(purple dash-dotted line). For these parameters, Eq. (32) provides a result
very close to the conventional PR gain (yellow dotted line with circles) apart
from the tails. We can thus state that for $\zeta_{c}\gg\Lambda_{0}$, the
proposed functional approach gives a good estimate of the numerically
estimated mean gain, because the two independent approximations are performed
in the correct order on the two different scales, i.e., for $B\ll\kappa_{0}$,
the BP process can be considered a small perturbation to the PR-MI effect.
Figure 5: Maximum gain $G_{\mathrm{max}}$ as a function of $B$ at constant
$N_{0}B$. Same convention as in Fig. 2, dichotomous process excluded. We take
a constant $\kappa_{0}=2\pi$ and (a) $N_{0}B=0.0039$, (b)
$N_{0}B=\pi/10=0.314$. Notice the ordinate axis is in logarithmic scale in
panel (a). The dashed vertical lines highlight the values of $B$ used in Fig.
4(a)[(b)], respectively.
We show in Fig. 5 the effect of variations of $B$ on $G_{\mathrm{max}}$. We
keep $\kappa_{0}$ and $N_{0}B$ constant; $L$, $\Delta\\!z$, and $N$ are chosen
to guarantee a significant statistical sample for each value. Only the
Gaussian process is considered.
For small $N_{0}B=0.0039$ the MI gain follows very well the cumulant
expansion: $G_{\mathrm{max}}$ grows for $B\to 0$ (notice the logarithmic
scale). The functional approach always overestimate it. For larger
$N_{0}B=\pi/10$ we still observe a decreasing trend of $G_{\mathrm{max}}$. It
is apparent that the cumulant expansion is valid only for large $B>\pi/2$ and
completely loses its validity below $B=\pi/4$. The functional approach works
well in the PR limit, i.e., for $B<\pi/16$. Indeed, $\zeta_{c}$ depends
inversely on $B$ and the approximation is expected to become more faithful.
The residual discrepancies may depend on a systematic error of the averaging
procedure or on numerical inaccuracies. In brief, we show the transition from
the stochastic regime, where the MI gain scales with $N_{0}$, to the
parametric regime where it depends on $\sigma_{\xi}$, i.e. the fluctuation
amplitude.
Figure 6: MI gain curves $G(\omega)$ and their maxima $G_{\mathrm{max}}$
obtained for different values of
$\kappa_{0}\in\\{\pi,2\pi,4\pi,8\pi,16\pi,32\pi,64\pi\\}$. Comparisons of
numerical results [blue crosses and blue solid lines] with the peak of Eq.
(26) (red dashed line), of Eq. (34) [purple dash-dotted line, not shown in
panel (a), out of scale]. We include also $G_{2}(\omega)$ for the white noise
limit at the corresponding (yellow dotted line). The dashed vertical lines
highlight the value of $\kappa_{0}$ used in Figs. 4 and 5. (a) $N_{0}=0.005$,
$B=\pi/4$; (b) $N_{0}=3.2$, $B=\pi/32$.
Finally, we report in Fig. 6 different MI gain curves obtained from numerical
simultations of the Gaussian process by varying
$\kappa_{0}\in\\{\pi,2\pi,4\pi,8\pi,16\pi,32\pi,64\pi\\}$. The simulation
parameters are chosen for each value to ensure that the sample is statically
significant.
In Fig. 6(a), we choose $N_{0}=0.005$ and $B=\pi/4$ and show that the MI lobes
have a Lorentzian shape dominated by a simple envelope, as predicted by Eq.
(25). We explicitly show $G_{2}^{\mathrm{max}}$ obtained from Eq. (26) for the
values of $\omega_{\mathrm{PR,1}}$ corresponding to a continuous set of
$\kappa_{0}$ (red dashed line) and compare it to the values of
$G_{\mathrm{max}}$ (blue crosses). They match very well over the full range of
$\omega$. The result of the functional approach, Eq. (34), is not shown here,
because it always overestimates the MI gain. We also include the white noise
gain for the given $N_{0}$ (yellow dotted line). For such a small PSD value,
it exhibits a monotone growth up to $\omega\approx 9$, compare to Fig. 1(a).
The expression in Eq. (26) coincides with it up to this point and tends to
$N_{0}/2$ for $\kappa_{0},\omega_{\mathrm{PR,1}}\to\infty$. This strengthens
the idea that the MI gain of stochastic origin is ruled predominantly by
$N_{0}$, at least for small values.
If, instead, we choose $N_{0}=3.2$ and $B=\pi/32$, we observe, see Fig. 6(b),
that the MI lobes are sharper than a Lorentzian. Their maxima (blue crosses)
are well approximated by the functional approximation (purple dash-dotted
line), while the cumulant expansion (red dashed line) always overestimates the
MI gain by more than 3 times. For such a small $B$, Eq. (34) effectively
coincides with Eq. (29) (not shown) as expected. As in the previous case, the
gain monotonously increases with $\kappa_{0}$, consistently with the PR nature
of the phenomenon, see Eq. (29). If we consider the white noise, we see that
the MI gain exhibits a narrow lobe, that attains a maximum for $\omega\approx
1.25$ and coincides with (26) only for small detuning, as expected.
As $\kappa_{0}$ is increased, the modulation period of the process becomes
smaller, while $\zeta_{c}$ is kept constant; thus, the near-resonant
approximation performs better for large $\kappa_{0}$ and the discrepancies
observed 6(b) are probably due to a systematic error or residual numerical
inaccuracies.
## V Conclusions
We discuss the modulational instability in nonlinear optical fibers in which
the group-velocity dispersion is randomly modulated. In contrast to the
exactly solvable case of white noise or of random kicks, we consider the case
of stochastic processes with exponentially decaying autocorrelation function.
This is equivalent to a Lorentzian-shaped power spectral-density, i.e., the
process is colored. Two families of colored processes are studied: low-pass
and band-pass. For each family, we consider Gaussian and dichotomous
processes.
The distinction between the two stochastic processes turns out to be important
in the LP and marginal in the BP case.
While for very small perturbation the LP yields very small MI gain, for larger
power spectral densities it can yield MI sidelobes larger than the white
noise, sitting in the same range of detuning and exhibiting larger gain in the
tails. For small bandwidth, the gain disappears, while in the opposite limit
it converges to the white noise limit. The variance demanded for obtaining a
measurable MI gain is large, though, and the dichotomous process looks more
promising in view on an experimental characterization of the phenomenon,
because values of bandwidth (correlation length) exist where the gain is quite
larger than the white noise limit.
As far as a BP process is concerned, if the perturbation is large enough, we
may observe several MI sidelobes sitting around PR detunings. We focus on the
dominant first peak: it converges to the PR sidelobe for small bandwidth and
is generally broader and smaller for small correlation lengths.
We compare our numerical results to different analytical approximations, based
on the cumulant expansion (as formalized by van Kampen) or the functional
(Furutsu-Novikov-Loginov-Shapiro) formulas.
While the former is reliable only for small perturbations and small detuning
and provides some qualitative estimates elsewhere, the latter provides good
results for the dichotomous processes, for which the closure of the system is
rigorously obtained.
For both families of correlation functions, the functional method emerges as
more reliable and allows us to describe the transition from parametric to
stochastic resonances in the BP case. Notwithstanding, the cumulant expansion
provides good estimates of the tails of resonant peaks even for relatively
small bandwidth values (large correlation length), beyond the expected
validity range of the approximation.
Our results pave the way for tailoring MI gain sidebands in optical fibers by
means of stochastic GVD fluctuations and suggest the regimes to achieve that.
Such fluctuations can be implemented by continuous or discrete variations of
fiber specifications.
## Acknowledgments
The work was supported in part the by the French government through the
Programme Investissement d’Avenir with the Labex CEMPI (Grant
ANR-11-LABX-0007-01) and the I-SITE ULNE (Grant ANR-16-IDEX-0004 ULNE,
projects VERIFICO) managed by the Agence Nationale de la Recherche. The work
was also supported by the Nord-Pas-de-Calais Regional Council and the European
Regional Development Fund through the Contrat de Projets État-Région (CPER)
and IRCICA.
## Appendix A Cumulant expansion near resonance
In this section we will apply the cumulant expansion to Eq. (30).
The cumulant expansion to second order is built by decomposing Eq. (30) as
$\kappa_{0}\dot{Y}=\left[A_{2}+\Gamma\psi_{1}C_{2}^{\prime}+\Gamma\psi_{2}C_{2}^{\prime\prime}\right]Y$,
with $A_{2}$, $C_{2}$, and $C_{2}^{\prime}$ as in Eq. (33).
As $\psi_{1,2}$ are mutually independent, the expansion is obtained by
computing $K_{2}$ separately for $C_{2}^{\prime}$ and $C_{2}^{\prime\prime}$,
according to Eq. (17), and adding them up.
We obtain
$\kappa_{0}\frac{\mathrm{d}\langle{Y}\rangle}{\mathrm{d}z}=\begin{bmatrix}6d_{2}\Gamma^{2}&2d_{2}\beta^{2}&2\Delta^{2}-4\Gamma^{2}d_{3}\\\
2d_{2}\beta^{2}&6d_{2}\beta^{2}&-2\Delta^{2}+4\Gamma^{2}d_{3}\\\
2\Gamma^{2}d_{3}-\Delta^{2}&-2\Gamma^{2}d_{3}+\Delta^{2}&4\Gamma^{2}d_{2}\end{bmatrix}\langle{Y}\rangle,$
(35)
with
$\displaystyle d_{2}$
$\displaystyle\equiv\frac{1}{\kappa_{0}}\int\limits_{0}^{\infty}\mathrm{d}\zeta
R_{\psi_{1}}(\zeta)\cos\frac{2\Delta^{2}}{\kappa_{0}}\zeta=\frac{N_{0}B^{2}}{2\kappa_{0}}\frac{1}{B^{2}+\frac{4\Delta^{4}}{\kappa_{0}^{2}}}$
(36) $\displaystyle d_{3}$
$\displaystyle\equiv\frac{1}{\kappa_{0}}\int\limits_{0}^{\infty}\mathrm{d}\zeta
R_{\psi_{1}}(\zeta)\sin\frac{2\Delta^{2}}{\kappa_{0}}\zeta=\frac{N_{0}B}{2\kappa_{0}}\frac{\frac{2\Delta^{2}}{\kappa_{0}}}{B^{2}+\frac{4\Delta^{4}}{\kappa_{0}^{2}}}.$
We note that there is here no counterpart to $c_{1}$ and near PR
$\kappa_{0}d_{2}\approx 2c_{2}$.
The dominant eigenvalue of the matrix in Eq. (35) is exactly
$\lambda^{*}=8d_{2}\Gamma^{2}$, so the instability gain is
$G_{2}^{\mathrm{PR}}=\frac{8d_{2}\Gamma^{2}}{\kappa_{0}},$ (37)
which attains a maximum
$(G_{2}^{\mathrm{PR}})^{\mathrm{max}}\approx\frac{N_{0}\omega_{\mathrm{PR,1}}^{4}}{2\kappa_{0}^{2}}$,
identical to what found above in Eq. (26).
## References
* Zakharov and Ostrovsky [2009] V. Zakharov and L. Ostrovsky, Modulation instability: The beginning, Physica D 238, 540 (2009).
* Benjamin and Feir [1967] T. B. Benjamin and J. E. Feir, The disintegration of wave trains on deep water Part 1. Theory, Journal of Fluid Mechanics 27, 417 (1967).
* Zakharov [1968] V. E. Zakharov, Stability of periodic waves of finite amplitude on the surface of a deep fluid, J. Appl. Mech. Tech. Phys. 9, 86 (1968).
* Bespalov and Talanov [1966] V. Bespalov and V. Talanov, Filamentary Structure of Light Beams in Nonlinear Liquids, Soviet Journal of Experimental and Theoretical Physics Letters 3, 307 (1966).
* H. Ichikawa _et al._ [1973] Y. H. Ichikawa, T. Suzuki, and T. Taniuti, Modulation Instability of Electron Plasma Wave, J. Phys. Soc. Jpn 34, 1089 (1973).
* Tai _et al._ [1986] K. Tai, A. Hasegawa, and A. Tomita, Observation of modulational instability in optical fibers, Phys. Rev. Lett. 56, 135 (1986).
* Berkhoer and Zakharov [1970] A. L. Berkhoer and V. E. Zakharov, Self Excitation of Waves with Different Polarizations in Nonlinear Media, Soviet Journal of Experimental and Theoretical Physics 31, 486 (1970).
* Stolen _et al._ [1974] R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, Phase‐matched three‐wave mixing in silica fiber optical waveguides, Appl. Phys. Lett. 24, 308 (1974).
* Cavalcanti _et al._ [1991] S. B. Cavalcanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveia-Neto, Modulation instability in the region of minimum group-velocity dispersion of single-mode optical fibers via an extended nonlinear Schrödinger equation, Phys. Rev. A 43, 6162 (1991).
* Smith and Doran [1996] N. J. Smith and N. Doran, Modulational instabilities in fibers with periodic dispersion management., Opt. Lett. 21, 570 (1996).
* Droques _et al._ [2012] M. Droques, A. Kudlinski, G. Bouwmans, G. Martinelli, and A. Mussot, Experimental demonstration of modulation instability in an optical fiber with a periodic dispersion landscape., Opt. Lett. 37, 4832 (2012).
* Armaroli and Biancalana [2012] A. Armaroli and F. Biancalana, Tunable modulational instability sidebands via parametric resonance in periodically tapered optical fibers, Opt. Expr. 20, 25096 (2012).
* Mussot _et al._ [2018] A. Mussot, M. Conforti, S. Trillo, F. Copie, and A. Kudlinski, Modulation instability in dispersion oscillating fibers, Adv. Opt. Phot. 10, 1 (2018).
* Abdullaev _et al._ [1996] F. K. Abdullaev, S. A. Darmanyan, A. Kobyakov, and F. Lederer, Modulational instability in optical fibers with variable dispersion, Phys. Lett. A 220, 213 (1996).
* Abdullaev _et al._ [1997] F. K. Abdullaev, S. A. Darmanyan, S. Bischoff, and M. P. Sørensen, Modulational instability of electromagnetic waves in media with varying nonlinearity, J. Opt. Soc. Am. B 14, 27 (1997).
* Abdullaev and Garnier [1999] F. K. Abdullaev and J. Garnier, Modulational instability of electromagnetic waves in birefringent fibers with periodic and random dispersion, Physical Review E 60, 1042 (1999).
* Garnier and Abdullaev [2000] J. Garnier and F. K. Abdullaev, Modulational instability induced by randomly varying coefficients for the nonlinear Schrödinger equation, Phys. D 145, 65 (2000).
* Chertkov _et al._ [2001] M. Chertkov, I. Gabitov, and J. Moeser, Pulse confinement in optical fibers with random dispersion, Proceedings of the National Academy of Sciences 98, 14208 (2001).
* Karlsson [1998] M. Karlsson, Four-wave mixing in fibers with randomly varying zero-dispersion wavelength, J. Opt. Soc. Am. B 15, 2269 (1998).
* Farahmand and de Sterke [2004] M. Farahmand and M. de Sterke, Parametric amplification in presence of dispersion fluctuations, Opt. Expr. 12, 136 (2004).
* Dujardin _et al._ [2021] G. Dujardin, A. Armaroli, S. R. Nodari, A. Mussot, A. Kudlinski, S. Trillo, M. Conforti, and S. De Bievre, Modulational instability in optical fibers with randomly kicked normal dispersion, Phys. Rev. A 103, 1 (2021).
* Bourret [1962] R. C. Bourret, Propagation of Randomly Perturbed Fields, Canadian Journal of Physics 40, 782 (1962).
* Van Kampen [1976] N. G. Van Kampen, Stochastic differential equations, Phys. Rep. 24, 171 (1976).
* Van Kampen [1974a] N. G. Van Kampen, A cumulant expansion for stochastic linear differential equations. I, Physica 74, 215 (1974a).
* Van Kampen [1974b] N. G. Van Kampen, A cumulant expansion for stochastic linear differential equations. II, Physica 74, 239 (1974b).
* Terwiel [1974] R. H. Terwiel, Projection operator method applied to stochastic linear differential equations, Physica 74, 248 (1974).
* Fox [1986] R. F. Fox, Uniform convergence to an effective Fokker-Planck equation for weakly colored noise, Phys. Rev. A 34, 4525 (1986).
* Kampen [2007] V. N. Kampen, _Stochastic Processes in Physics and Chemistry_ (Elsevier, 2007).
* Furutsu [1963] K. Furutsu, On the statistical theory of electromagnetic waves in a fluctuating medium (I), J. Res. Nat. Bur. Stand., Sect. D 67D, 303 (1963).
* Novikov [1965] E. A. Novikov, Functionals and the Random-force Method in Turbulence Theory, Sov. Phys.—JETP 20, 1290 (1965).
* Shapiro and Loginov [1978] V. E. Shapiro and V. M. Loginov, ”Formulae of differentiation” and their use for solving stochastic equations, Physica A 91, 563 (1978).
* Zhang _et al._ [1992] W. Zhang, J. Casademunt, and J. Viñals, Study of the parametric oscillator driven by narrow-band noise to model the response of a fluid surface to time-dependent accelerations, Phys. Fluids A 5, 3147 (1992).
* Gitterman [2005a] M. Gitterman, _Noisy Oscillator, The: Random Mass, Frequency, Damping_ (World Scientific Publishing Co. Pte. Ltd., 2005).
* Agrawal [2012] G. P. Agrawal, _Nonlinear Fiber Optics_ , fifth edit ed. (Academic Press, Oxford, 2012) p. 648.
* Stratonovich [1963] R. L. Stratonovich, _Topics in the theory of random noise (Vol. 1)_ , 1st ed. (Gordon and Breach Science Publishers, New York, 1963).
* Dubkov and Muzychuk [1977] A. A. Dubkov and O. V. Muzychuk, Analysis of higher approximations of Dyson’s equation for the mean value of the Green function, Radiophys. Quant. Electr. 20, 623 (1977).
* Gitterman [2005b] M. Gitterman, Classical harmonic oscillator with multiplicative noise, Physica A 352, 309 (2005b).
* Landau [1976] L. D. Landau, _Mechanics Course of theoretical physics (Volume 1)_ , 3rd ed. (1976) p. 170.
* Burov and Gitterman [2016] S. Burov and M. Gitterman, Noisy oscillator: Random mass and random damping, Phys. Rev. E 94, 1 (2016).
|
20cm(1cm,1cm) Accepted by ACM Transactions on Software Engineering and
Methodology (TOSEM)
# Hierarchical Distribution-Aware Testing of Deep Learning
Wei Huang Purple Mountain LaboratoriesChina,University of LiverpoolU.K.
<EMAIL_ADDRESS>, Xingyu Zhao WMG, University of WarwickU.K.
<EMAIL_ADDRESS>, Alec Banks Defence Science and Technology
LaboratoryU.K<EMAIL_ADDRESS>, Victoria Cox Defence Science and
Technology LaboratoryU.K<EMAIL_ADDRESS>and Xiaowei Huang University of
LiverpoolU.K<EMAIL_ADDRESS>
(2023)
###### Abstract.
With its growing use in safety/security-critical applications, Deep Learning
(DL) has raised increasing concerns regarding its dependability. In
particular, DL has a notorious problem of lacking robustness. Input added with
adversarial perturbations, i.e. Adversarial Examples (AEs) are easily mis-
predicted by the DL model. Despite recent efforts made in detecting AEs via
state-of-the-art attack and testing methods, they are normally input
distribution agnostic and/or disregard the perceptual quality of adversarial
perturbations. Consequently, the detected AEs are irrelevant inputs in the
application context or unrealistic that can be easily noticed by humans. This
may lead to a limited effect on improving the DL model’s dependability, as the
testing budget is likely to be wasted on detecting AEs that are encountered
very rarely in its real-life operations.
In this paper, we propose a new robustness testing approach for detecting AEs
that considers both the feature level distribution and the pixel level
distribution, capturing the perceptual quality of adversarial perturbations.
The two considerations are encoded by a novel hierarchical mechanism. First,
we select test seeds based on the density of feature level distribution and
the vulnerability of adversarial robustness. The vulnerability of test seeds
are indicated by the auxiliary information, that are highly correlated with
local robustness. Given a test seed, we then develop a novel genetic algorithm
based local test case generation method, in which two fitness functions work
alternatively to control the perceptual quality of detected AEs. Finally,
extensive experiments confirm that our holistic approach considering
hierarchical distributions is superior to the state-of-the-arts that either
disregard any input distribution or only consider a single (non-hierarchical)
distribution, in terms of not only detecting imperceptible AEs but also
improving the overall robustness of the DL model under testing.
Deep learning robustness, adversarial examples detection, natural
perturbations, distribution-aware testing, robustness growth, safe AI
††copyright: acmcopyright††journalyear: 2023††doi: XXXXXXX.XXXXXXX††journal:
TOSEM††journalvolume: 0††journalnumber: 0††article: X††publicationmonth:
9††ccs: Software and its engineering Software testing and debugging††ccs:
Software and its engineering Software reliability††ccs: Computing
methodologies Machine learning
## 1\. Introduction
Deep Learning (DL) is being explored to provide transformational capabilities
to many industrial sectors including automotive, healthcare and finance. The
reality that DL is not as dependable as required now becomes a major
impediment. For instance, key industrial foresight reviews identified that the
biggest obstacle to gaining benefits of DL is its dependability (Lane et al.,
2016). There is an urgent need to develop methods to enable the dependable use
of DL, for which great efforts have been made in recent years in the field of
DL Verification and Validation (V&V) (Huang et al., 2020; Zhang et al., 2022).
DL robustness is arguably the property in the limelight. Informally,
robustness requires that the decision of the DL model is invariant against
small perturbations on inputs. That is, all inputs in a small input region
(e.g., a norm ball defined in some $L_{p}$-norm distance) should share the
same prediction label by the DL model. Inside that region, if an input is
predicted differently to the given label, then this input is normally called
an Adversarial Example (AE). Most V&V methods designed for DL robustness are
essentially about detecting AEs, e.g., adversarial attack based methods
(Goodfellow et al., 2015; Madry et al., 2018) and coverage-guided testing (Du
et al., 2019; Xie et al., 2022; Ma et al., 2018; Pei et al., 2017; Huang et
al., 2022; Sun et al., 2019).
As recently noticed by the software engineering community, emerging studies on
systematically evaluating AEs detected by aforementioned state-of-the-arts
have two major drawbacks: (i) they do not take the input data distribution
into consideration, therefore it is hard to judge whether the identified AEs
are meaningful to the DL application (Berend et al., 2020; Dola et al., 2021);
(ii) most detected AEs are of poor perceptual quality that are too
unnatural/unrealistic (Harel-Canada et al., 2020) to be seen in real-life
operations. That said, not all AEs are equal nor significantly contribute to
the robustness improvement, given limited resources. A wise strategy is to
detect those AEs that are both being “distribution-aware” and with
natural/realistic pixel-level perturbations, which motivates this work.
Prior to this work, a few notable attempts at distribution-aware testing for
DL have been made. Broadly speaking, the field has developed two types of
approaches: Out-Of-Distribution (OOD) detector based (Dola et al., 2021;
Berend, 2021) and feature-only based (Toledo et al., 2021; Byun et al., 2020).
The former can only detect anomalies/outliers, rather than being “fully-aware”
of the distribution. While the latter is indeed generating new test cases
according to the learnt distribution (in a latent space), it ignores the
pixel-level information due to the compression nature of generative models
used (Zhong et al., 2020). To this end, our approach is advancing in this
direction with the following novelties and contributions:
a) We provide a “divide and conquer” solution—Hierarchical Distribution-Aware
(HDA) testing—by decomposing the input distribution into two levels (named as
global and local) capturing how the feature-wise and pixel-wise information
are distributed, respectively. At the global level, isolated problems of
estimating the feature distribution and selecting best test seeds can be
solved by dedicated techniques. At the local level where features are fixed,
the clear objective is to precisely generate test cases considering perceptual
quality111While determining perceptual quality typically involves subjective
assessments from human observers, objective metrics such as peak signal-to-
noise ratio (PSNR) and structural similarity index (SSIM) can also be used as
measures of perceptual quality (Wang et al., 2004). Throughout the paper,
perceptual quality is defined as the metrics in Sec. 2.4 that are commonly
used in computer vision.. Our extensive experiments show that such
hierarchical consideration is more effective to detect high-quality AEs than
state-of-the-art that either disregards any data distribution or only
considers a single (non-hierarchical) distribution. Consequently, we also show
the DL model under testing exhibits higher robustness after “fixing” the high-
quality AEs detected.
b) At the global level, we propose novel methods to select test seeds based on
the approximated feature distribution of the training data and predictive
robustness indicators, so that the norm balls of the selected seeds are both
from the high-density area of the distribution and relatively unrobust (thus
more cost-effective to detect AEs in later stages). Notably, state-of-the-art
DL testing methods normally select test seeds randomly from the training
dataset without any principled rules. Thus, from a software engineering
perspective, our test seed selection is more practically useful in the given
application context.
c) Given a carefully selected test seed, we propose a novel two-step Genetic
Algorithm (GA) to generate test cases locally (i.e. within a norm ball) to
control the perceptual quality of detected AEs. At this local level, the
perceptual quality distribution of data-points inside a norm ball requires
pixel-level information that cannot be sufficiently obtained from the training
data alone. Thus, we innovatively use common perceptual metrics that quantify
image quality as an approximation of such local distribution. Our experiments
confirm that the proposed GA is not only effective after being integrated into
HDA (as a holistic testing framework), but also outperforms other pixel level
AE detectors in terms of perception quality when applied separately.
d) We investigate black-box (to the DL model under testing) methods for the
main tasks at both levels. Thus, to the best of our knowledge, our HDA
approach provides an end-to-end, black-box solution, which is the first of its
kind and more versatile in software engineering practice.
e) A publicly accessible tool of our HDA testing framework with all source
code, datasets, DL models and experimental results.
## 2\. Preliminaries and Related Work
In this section, we first introduce preliminaries and related work on DL
robustness, together with formal definitions of concepts adopted in our HDA
approach. Then existing works on distribution-aware testing are discussed.
Since our HDA testing also considers the naturalness of detected AEs, some
common perception quality metrics are introduced. In summary, we present Fig.
1 to show the stark contrast of our proposed HDA testing (the green route) to
other related works (the red and amber routes).
Figure 1. Comparison between our proposed Hierarchical Distribution-Aware
(HDA) testing and related works.
### 2.1. DL Robustness and Adversarial Examples
We denote the prediction output of DL model as the vector $f(x)$ with size
equal to the total number of labels. The predicted label
$\hat{f}(x)\\!=\\!\operatorname*{arg\,max}_{i}f_{i}(x)$ where $f_{i}(x)$ is
the $i^{th}$ attribute of vector $f(x)$.
DL robustness requires that the decision of the DL model $\hat{f}(x)$ is
invariant against small perturbations on input $x$. That is, all inputs in an
input region $\eta$ have the same prediction label, where $\eta$ is usually a
small norm ball (defined with some $L_{p}$-norm distance222$p=0,1,2$ and
$\infty$. $L_{\infty}$ norm is more commonly used.) around an input $x$. If an
input $x^{\prime}$ inside $\eta$ is predicted differently to $x$ by the DL
model, then $x^{\prime}$ is called an Adversarial Example (AE).
DL robustness V&V can be based on formal methods (Huang et al., 2017; Ruan et
al., 2018) or statistical approaches (Weng et al., 2018; Webb et al., 2019),
and normally aims at detecting AEs. In general, we may classify two types of
methods (the two branches in the red route of Fig. 1) depends on how the test
cases are generated: (i) Adversarial attack based methods normally optimise
the DL prediction loss to find AEs, which include white-box attack methods
like Fast Gradient Sign Method (FGSM) (Goodfellow et al., 2015) and Projected
Gradient Descent (PGD) (Madry et al., 2018), as well as black-box attacks
(Alzantot et al., 2019; Wu et al., 2021) using GA with gradient-free
optimisation. (ii) Coverage-guided testing optimises the certain coverage
metrics on the DL model’s internal structure, which is inspired by the
coverage testing for traditional software. Several popular test metrics, like
neuron coverage (Pei et al., 2017; Ma et al., 2018), modified
condition/decision coverage (Sun et al., 2019) for CNNs and temporal coverage
(Huang et al., 2022; Du et al., 2019) for RNNs are proposed. While it is
argued that coverage metrics are not strongly correlated with DL robustness
(Yan et al., 2020; Harel-Canada et al., 2020), they are seen as providing
insights into the internal behaviours of DL models and hence may guide test
selection to find more diverse AEs (Huang et al., 2022).
Without loss of generality, we reuse the formal definition of DL robustness in
(Webb et al., 2019; Weng et al., 2019) in this work:
###### Definition 0 (Local Robustness).
The local robustness of the DL model $f(x)$, w.r.t. a local region $\eta$ and
a target label $y$, is:
(1) $\mathcal{R}_{l}(\eta,y):=\int_{x\in\eta}I(x)p_{l}(x\mid
x\in\eta)\mathop{}\\!\mathrm{d}{x}$
where $p_{l}(x\mid x\in\eta)$ is the local distribution of region $\eta$ which
is precisely the “input model” used by both (Webb et al., 2019; Weng et al.,
2019). $I(x)$ is an indicator function, and $I(x)=1$ when $\hat{f}(x)=y$,
$I(x)=0$ otherwise.
To detect as many AEs as possible, normally the first question is—which local
region shall we search for those AEs? I.e. how to select test seeds? To be
cost-effective, we want to explore unrobust regions, rather than regions where
AEs are relatively rare. This requires the local robustness of a region to be
known a priori, which may imply a paradox (cf. Remark 4 later). In this
regard, we can only predict the local robustness of some regions before doing
the actual testing in those regions. We define:
###### Definition 0 (Local Robustness Indicator).
Auxiliary information that strongly correlated with $\mathcal{R}_{l}(\eta,y)$
(thus can be leveraged in its prediction) is named as a local robustness
indicator.
We later seek for such indicators (and empirically show their correlation with
the local robustness), which forms one of the two key factors considered in
selecting test seeds in our method.
Given a test seed, we search for AEs in a local region $\eta$ (around the test
seed) that produce different label from the test seed. This involves the
question on what size of $\eta$ should be, for which we later utilise the
property of:
###### Remark 1 ($r$-separation).
For real-world image datasets, any data-points with different ground truth
labels are at least distance $2r$ apart in the input (pixel) space, where $r$
is estimated case by case and depending on the dataset.
The $r$-separation property was first observed by (Yang et al., 2020):
intuitively it says, there is a minimum distance between two real-world
objects of different labels.
Finally, not all AEs are equal in terms of the “strength of being adversarial”
(stronger AEs may lead to greater robustness improvement in, e.g., adversarial
training (Wang et al., 2019)), for which we define:
###### Definition 0 (Prediction Loss).
Given a test seed $x$ with label $y$, the prediction loss of an input
$x^{\prime}$, which is obtained by adding perturbations to $x$, is defined as:
(2) $\mathcal{J}(f(x^{\prime}),y)=\max_{i\neq
y}(f_{i}(x^{\prime})-f_{y}(x^{\prime}))$
where $f_{i}(x^{\prime})$ returns the probability of label $i$ after input
$x^{\prime}$ being processed by the DL model $f$.
Note, $\mathcal{J}\geq 0$ implies $\operatorname*{arg\,max}_{i}f_{i}(x)\neq y$
and thus $x^{\prime}$ is an AE of $x$.
Next, to measure the DL models’ overall robustness across the whole input
domain, we introduce a notion of global robustness. Being different to some
existing definitions where robustness of local regions are treated equally
when calculating global robustness over several regions (Wang et al., 2021a;
Wang et al., 2021c), ours is essentially a “weighted sum” of the robustness of
local regions where each weight is the probability of the associated region on
the input data distribution. Defining global robustness in such a
“distribution-aware” manner aligns with our motivation—as revealed later by
empirically estimated global robustness, our HDA appears to be more effective
in supporting the growth of the overall robustness after “fixing” those
distribution-aware AEs.
###### Definition 0 (Global Robustness).
The global robustness of the DL model $f(x)$ is defined as:
(3)
$\mathcal{R}_{g}:=\sum_{\eta\in\mathcal{X}}p_{g}(x\in\eta)\mathcal{R}_{l}(\eta,y)$
where $p_{g}(x\mid x\in\eta)$ is the global distribution of region $\eta$
(i.e., a pooled probability of all inputs in the region $\eta_{z}$) and
$\mathcal{R}_{l}(\eta,y)$ is the local robustness of region $\eta$ to the
label $y$.
The estimation of $\mathcal{R}_{g}$, unfortunately, is very expensive that
requires to compute the local robustness $\mathcal{R}_{l}$ of a large number
of regions over the input domain $\mathcal{X}$. Thus, from a practical
standpoint, we adopt an empirical definition of the global robustness in our
later experiments, which has been commonly used for DL robustness evaluation
in the adversarial training (Wang et al., 2021b; Madry et al., 2018; Wang et
al., 2019; Zhang et al., 2019).
###### Definition 0 (Empirical Global Robustness).
Given a DL model $f$ and a validation dataset $D_{v}$, we define the empirical
global robustness as $\hat{\mathcal{R}_{g}}:(f,D_{v},T)\rightarrow[0,1]$ where
T denotes a given type of AE detection method and $\hat{\mathcal{R}_{g}}$ is
the weighted accuracy on AEs obtained by conducting T on $\langle
f,D_{v}\rangle$.
To be “distribution-aware”, the synthesis of $D_{v}$ should conform to the
global distribution. $D_{v}$ can be sampled from the train/test data according
to global distribution. The norm ball around each input data $x$ in $D_{v}$
represents a region $\eta$. Each region $\eta$ is explicitly assigned a weight
to indicate the density on global distribution. For each region $\eta$, we
calculate the prediction accuracy on AEs, detect by T according to local
distribution, to approximate the local robustness $\mathcal{R}_{l}$.
Consequently, the set of AEs for $D_{v}$ may represent the input distribution
and the weighted accuracy on these AEs approximate the global robustness.
### 2.2. Distribution-Aware Testing for DL
There are increasing amount of DL testing works developed towards being
distribution-aware (as summarised in the amber route of Fig. 1). Deep
generative models, such as Variational Auto-Encoders (VAE) and Generative
Adversarial Networks (GAN), are applied to approximate the training data
distribution, since the inputs (like images) to Deep Neural Network (DNN) are
usually in a high dimensional space. Previous works heavily rely on OOD
detection (Dola et al., 2021; Berend, 2021) or synthesising new test cases
directly from latent spaces (Byun and Rayadurgam, 2020; Toledo et al., 2021;
Dunn et al., 2020; Kang et al., 2020; Dunn et al., 2021). The former does not
comprehensively consider the whole distribution, rather flags outliers, thus a
more pertinent name of it should be out-of-distribution-aware (OODA) testing.
While for both types of methods, another problem arises that the distribution
encoded by generative models only contain the feature-wise information and
easily filter out the pixel-wise perturbations (Zhong et al., 2020). The
images added with pixel-wise perturbations should still fall into the local
region $\eta$, which is decided by $r$-separation property (ref. Remark 1).
Although, Dunn et al. (Dunn et al., 2020) propose to generate the fine-grained
perturbations by perturbing the output of last layers of GAN’s generator, the
fine-grained perturbations cannot be guaranteed to follow $r$-separation
property and thus may change the class of the image. Consequently, directly
searching and generating test cases from the latent space of generative models
may only perturb features, thus called Feature-Only Distribution-Aware (FODA)
in this paper (while also named as semantic AEs in some literature (Hosseini
and Poovendran, 2018; Zhao et al., 2018)). As an example in later experiment
results, i.e. Table 8(b), FODA will produce AEs, the perturbations of which
exceed the $r$-separation limit. Our approach, the green route in Fig. 1,
differs from aforementioned works by 1) considering both the global (feature
level) distribution in latent spaces and the local (pixel level) perceptual
quality distribution in the input space; 2) leveraging the robustness
indicator $\mathcal{R}$ to select test seeds which is more error-prone and
thus easier for detecting AEs; 3) proposing novel GA based test case
generation to detect AEs with high perceptual quality. Therefore, when HDA is
experimentally compared to the existing works, the selected test seeds of HDA
have higher probability density and lower robustness. The detected AEs by HDA
also have higher perceptual quality (in terms of those metrics encoded by GA)
than others.
### 2.3. Test Input Prioritisation and Generation
When testing DL based systems, in order to save computation and reduce the
cost on labelling data, test input prioritisation strategies are adopted.
(Weiss and Tonella, 2022) experimentally confirms that DeepGini outperforms
various types of surprise and neuron coverage metrics in terms of the
capability to detect misclassifications. (Attaoui et al., 2023) propose the
unsafe set selection algorithm, leveraging the density-based clustering of
error-inducing images. Both test selection methods aim at detecting
misclassified images for retraining and improving generalisation performance
of DNNs.
There are also some test input generation methods, the main idea of which is
to promote diversity of test cases in order to cover more faults of DL
systems. (Riccio and Tonella, 2020) develops a search-based tool to generate
frontier inputs for DL systems. DEEPMETIS (Riccio et al., 2021) augment the
test set for DL by increasing the mutation scores.
The above test input generation/prioritisation methods have different goals
from our HDA testing. They focus on detecting or generating more misclassified
test cases within certain test budget, while our HDA testing targets at
generating on-distribution AEs to improve operational robustness of DL
systems. Feature distribution learnt from VAE as well as the predicted
robustness indicator facilitate the generation of those on-distribution AEs.
### 2.4. Perceptual Quality of Images
Locally, data-points (around the selected seed) sharing the same feature
information may exhibit perceptual difference from the selected seed. To
capture such distribution, some perceptual quality metric can be utilised to
compare the perceptual difference between the original image $x$ and perturbed
image $x^{\prime}$. Some common metrics for perceptual quality include:
* •
Mean Squared Error (MSE): measures the mean squared difference between the
original image $x$ and perturbed image $x^{\prime}$,
(4) $MSE=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-x_{i}^{\prime})^{2}$
where $n$ is the image size, and $x_{i}$ is the value of image pixel $i$.
* •
Peak Signal-to-Noise Ratio (PSNR) (Gonzalez and Woods, 1992): measures the
quality of a signal’s representation after being subject to noise during
transmission or processing. PSNR is expressed in decibels (dB) and is
calculated by comparing the maximum possible power of the original signal to
the power of the noise that affects its fidelity,
(5) $PSNR=20*log_{10}\frac{MAX}{\sqrt{MSE}}$
where $MAX$ is the maximum possible pixel value of the image. When the pixels
are represented using 8 bits per sample, this is 255.
* •
Structural Similarity Index Measure (SSIM) (Wang et al., 2004): measures the
similarity between two images based on the idea that the human visual system
is highly sensitive to changes in structural information, such as edges,
textures, and patterns, rather than just changes in pixel values. It works by
comparing the luminance (l), contrast (c), and structural information (s) of
two images, and producing a score between 0 (completely dissimilar) and 1
(identical) that indicates their similarity,
(6) $\begin{split}SSIM(x,x^{\prime})=&l(x,x^{\prime})^{\alpha}\cdot
c(x,x^{\prime})^{\beta}\cdot s(x,x^{\prime})^{\gamma}\\\
l(x,x^{\prime})=\frac{2\mu_{x}\mu_{x^{\prime}}+c_{1}}{\mu_{x}^{2}+\mu_{x^{\prime}}^{2}+c_{1}},\;c(x,x^{\prime})&=\frac{2\sigma_{x}\sigma_{x^{\prime}}+c_{2}}{\sigma_{x}^{2}+\sigma_{x^{\prime}}^{2}+c_{2}},\;s(x,x^{\prime})=\frac{\sigma_{xx^{\prime}}+c_{3}}{\sigma_{x}\sigma_{x^{\prime}}+c_{3}}\end{split}$
where $\mu_{x}$ and $\mu_{x^{\prime}}$ are the mean values of $x$ and
$x^{\prime}$, respectively. $\sigma_{x}$ and $\sigma_{x^{\prime}}$ are the
standard deviations of $x$ and $x^{\prime}$, respectively.
$\sigma_{xx^{\prime}}$ is the covariance between $x$ and $x^{\prime}$.
$c_{1}$, $c_{2}$, and $c_{3}$ are constants that prevent division by zero. The
constants $\alpha$, $\beta$, and $\gamma$ are typically set to 1, 1, and 1,
respectively, although different values may be used depending on the
application. Unlike other similarity measures, such as MSE and PSNR, SSIM is
able to account for perceptual differences in image quality, and is often
considered a more accurate measure of image quality.
* •
Fréchet Inception Distance (FID) (Heusel et al., 2017): compares the
distribution between a set of original images and a set of perturbed images.
It is calculated by first using a pre-trained Inception-v3 neural network
(Szegedy et al., 2016) to extract features from both sets of images. Then, the
mean and covariance of these features are calculated for each set, and the
distance between these mean and covariance matrices is calculated using the
Fréchet distance,
(7)
$FID=\|\mu-\mu^{\prime}\|_{2}^{2}+Tr(\Sigma+\Sigma^{\prime}-2(\Sigma^{1/2}\cdot\Sigma^{\prime}\cdot\Sigma^{1/2})^{1/2})$
where $\mu$ and $\mu^{\prime}$ are the mean feature vectors of the original
and perturbed image sets, respectively. $\Sigma$ and $\Sigma^{\prime}$ are the
covariance matrices of the original and perturbed image sets, respectively.
$Tr()$ denotes the trace of a matrix. Lower FID scores indicate that the
perturbed images are closer in terms of perception to the original images,
while higher FID scores indicate greater differences between the perturbed and
original images.
Notably, all these metrics are current standards for assessing the quality of
images, as widely used in the experiments of aforementioned related works.
## 3\. The Proposed Method
Figure 2. An example of Hierarchical Distribution Aware Testing
We first present an overview of our HDA testing, cf. the green route in Fig.
1, and then dive into details of how we implement each stage by referring to
an illustrative example in Fig. 2.
### 3.1. Overview of HDA Testing
The core of HDA testing is the hierarchical structure of two distributions. We
formally define the following two levels of distributions:
###### Definition 0 (Global Distribution).
The global distribution captures how feature level information is distributed
in some (low-dimensional) latent space after data compression.
###### Definition 0 (Local Distribution).
Given a data-point sampled from the latent space, we consider its norm ball in
the input pixel space. The local distribution is a conditional distribution
capturing the perceptual quality of all data-points within the norm ball.
Due to the sparsity of data over the high dimensional input space, it is hard
to estimate the input distribution. Therefore, we turn to estimate the feature
level (global) distribution in latent space, which is the representation of
compressed data, in which data points with similar features are closer to each
other (Liu et al., 2019). DNNs, e.g. encoder of VAEs, map any data points in
the high dimensional input space to the low dimensional latent space. It
infers that the input space can be partitioned into well-defined regions,
where each region corresponds to a particular data point in the latent space.
latent space is an abstract, multidimensional space where each dimension
represents a feature or characteristic of the data (Liu et al., 2019). The
regions in the input space are determined based on the values of these
features and how they relate to each other. In other words, we can identify
distinct patterns or clusters of data in the input space, which share the same
set of features, and maps them to corresponding point in the latent space. By
fitting a global distribution in the latent space, we actually model the
distribution of distinct regions over the input space. The local distribution
is defined as a conditional distribution within each region. Thus, we propose
the following remark.
###### Remark 2 (Decompose one distribution into two levels).
Given the definitions of global and local distributions, denoted as $p_{g}$
and $p_{l}$ respectively, we may decompose a single distribution over the
entire input domain $\mathcal{X}$ as:
(8) $p(x)=\int
p_{l}(x|x\in\eta_{z})p_{g}(x\in\eta_{z})\mathop{}\\!\mathrm{d}{z}$
where variable $z$ represents a set of features while $\eta_{z}$ represents a
region in the input space that “maps” to the $z$ point in the latent space.
Intuitively, compared to modelling a single distribution, our hierarchical
structure of distributions is superior in that the global distribution guides
for which regions of the input space to test, while the local distribution can
be leveraged to precisely control the quality of test cases. Given the
definition of distribution, the goal of HDA testing can be formalised as
below:
###### Remark 3 (Goal of HDA testing).
The goal of HDA testing is to detect AE $x^{\prime}$, which is around the
local region of test seed $x$, such that $x^{\prime}$ lies within the high
probability density region of distribution $p(x^{\prime})$, and $x$ and
$x^{\prime}$ are classified differently.
To detect such on distribution AEs, HDA testing has the following process,
which can divided into three stages, depicted as the green route in Fig. 1:
##### Stage 1: Explicitly Approximate the Global Distribution
We first extract feature-level information from the given dataset by using
data compression techniques—the encoder of VAEs in our case, and then
explicitly approximate the global distribution in the latent-feature space,
using Kernel Density Estimator (KDE).
##### Stage 2: Select Test Seeds Based on the Global Distribution and Local
Robustness Indicators
Given the limited testing budget, we want to test in those local input regions
that are both more error-prone and representative of the input distribution.
Thus, when selecting test seeds, we consider two factors—the local robustness
indicators (cf. Definition 2) and the global distribution. For the former, we
propose several auxiliary information with empirical studies showing their
correlation with the local robustness, while the latter has already been
quantified in the first stage via KDE.
##### Stage 3: Generate Test Cases Around Test Seeds Considering the Local
Distribution and Prediction Loss of AEs
When searching for AEs locally around a test seed given by the 2nd stage, we
develop a two-step GA in which the objective function is defined as a fusion
of the prediction loss (cf. Definition 3) and the local distribution (modelled
by common perceptual quality metrics). Such fusion of two fitness functions
allows the trade-off between the “strength of being adversarial” and the
perceptual quality of the detected AEs. The optimisation is subject to the
constraint of only exploring in a norm ball whose central point is the test
seed and with a radius smaller than the $r$-separation distance (cf. Remark
1).
While there are some alternatives may also suffice for the purpose of each
stage, our chosen technical solutions are the most effective and popular in
consideration of the distribution awareness.
### 3.2. Approximation of the Global Distribution
Given the training dataset $\mathcal{D}$, the task of approximating the input
distribution is equivalent to estimating a Probability Density Function (PDF)
over the input domain $\mathcal{X}$ given $\mathcal{D}$. Despite this is a
common problem with many established solutions, it is hard to accurately
approximate the distribution due to the relatively sparse data of
$\mathcal{D}$, compared to the high dimensionality of the input domain
$\mathcal{X}$. So the practical solution is to do dimensionality reduction and
then estimate the global distribution, which indeed is the first step of all
existing methods of distribution-aware DL testing.
Specifically, we choose VAE-Encoder+KDE 333We only use the encoder of VAEs for
feature extraction, rather than generate new data from the decoder, which is
different to other methods mentioned in Section 2.2. for their simplicity and
popularity. To effectively train the VAE model, we use a combination of a
reconstruction loss and a KL divergence loss to optimise the model. The
reconstruction loss measures the difference between the input data and the
output of the decoder, while the KL divergence loss measures the difference
between the learned latent distribution and the prior distribution. During
training, the reconstruction loss and KL divergence loss are calculated and
minimised simultaneously. Regularisation techniques such as dropout and weight
decay can be used to prevent overfitting.
Assume $\mathcal{D}$ contains $n$ samples and each $x_{i}\in\mathcal{D}$ is
encoded by VAE-Encoder as a Gaussian distribution $z_{i}$ in the latent space,
we can estimate the PDF of $z$ (denoted as $Pr(z)$) based on the encoded
$\mathcal{D}$. The $Pr(z)$ conforms to the mixture of Gaussian distributions,
i.e., $z\sim\mathcal{N}(\mu_{z_{i}},\sigma_{z_{i}})$. Notably, this mixture of
Gaussian distributions nicely aligns with the gist of adaptive KDE (Lokerse et
al., 1995), which uses the following estimator:
(9) $p_{g}(x\in\eta_{z})\propto
Pr(z)\simeq\frac{1}{n}\sum_{i=1}^{n}K_{h_{i}}(z-\mu_{z_{i}})$
That is, when choosing a Gaussian kernel for $K$ in Eqn. (9) and adaptively
setting the bandwidth parameter $h_{i}=\sigma_{z_{i}}$ (i.e., the standard
deviation of the Gaussian distribution representing the compressed sample
$z_{i}$), the VAE-Encoder and KDE are combined “seamlessly”. Finally, our
global distribution $p_{g}(x\in\eta_{z})$ (a pooled probability of all inputs
in the region $\eta_{z}$ that corresponds to a point $z$ in the latent space)
is proportional to the approximated distribution of $z$ with the PDF $Pr(z)$.
Running Example: The left diagram in Fig. 2 depicts the global distribution
learnt by KDE, after projected to a two-dimensional space for visualisation.
The peaks444Most training data lie in this region or gather around the region.
are evaluated with highest probability density over the latent space by KDE.
### 3.3. Test Seeds Selection
Selecting test seeds is actually about choosing which norm balls (around the
test seeds) to test for AEs. To be cost-effective, we want to test those with
higher global probabilities and lower local robustness at the same time. For
the latter requirement, there is potentially a paradox:
###### Remark 4 (A Paradox of Selecting Unrobust Norm Balls).
To be informative on which norm balls to test for AEs, we need to estimate the
local robustness of candidate norm balls (by invoking robustness estimators to
quantify $\mathcal{R}_{l}(\eta,y)$, e.g., (Webb et al., 2019)). However, local
robustness evaluation itself is usually about sampling for AEs (then fed into
statistical estimators) that consumes the testing resources.
To this end, instead of directly evaluating the local robustness of a norm
ball, we can only indirectly predict it (i.e., without testing/searching for
AEs) via auxiliary information that we call local robustness indicators (cf.
Definition 2). In doing so, we save all the testing budget for the later stage
when generating local test cases.
Given a test seed $x$ with label $y$, we propose two robustness indicators
(both relate to the vulnerability of the test seed to adversarial attacks)—the
prediction gradient based score (denoted as ${S}_{\textit{grad}}$) and the
score based on separation distance of the output-layer activation (denoted as
$S_{\textit{sep}}$):
(10)
$\begin{split}&S_{\textit{grad}}=||\nabla_{x}\mathcal{J}(f(x),y)||_{\infty}\\\
&S_{\textit{sep}}=\min_{\hat{x}}||f(x)-f(\hat{x})||_{\infty}\;\quad\textrm{s.t.}\;y\neq\hat{y}\end{split}$
These allow prediction of a whole norm ball’s local robustness by the limited
information of its central point (the test seed). The gradient of a DNN’s
prediction with respect to the input is a white-box metric, that is widely
used in adversarial attacks, such as FGSM (Goodfellow et al., 2015) and PGD
(Madry et al., 2018) attacks. A greater gradient calculated at a test seed
implies that AEs are more likely to be found around it. The activation
separation distance is regarded as a black-box metric and refers to the
minimum $L_{\infty}$ norm between the output activations of the test seed and
any other data with different labels. Intuitively, a smaller separation
distance implies a greater vulnerability of the seed to adversarial attacks.
We later show empirically that indeed these two indicators are highly
correlated with the local robustness.
After quantifying the two required factors, we combine them in a way that was
inspired by (Zhao et al., 2021a). In (Zhao et al., 2021a), the DL reliability
metric is formalised as a weighted sum of local robustness where the weights
are operational probabilities of local regions. To align with that reliability
metric, we do the following steps to select test seeds:
(i) For each data-point $x_{i}$ in the test set, we calculate its global
probability (i.e., $Pr(z_{i})$ where $z_{i}$ is its compressed point in the
VAE latent space) and one of the local robustness indicators (either white-box
or black-box, depending on the available information).
(ii) Normalise both quantities to the same scale.
(iii) Rank all data-points by the product of their global probability and
local robustness indicator.
(iv) Finally we select top-$k$ data-points as our test seeds, and $k$ depends
on the testing budget.
Running Example: In the middle diagram of Fig. 2, we add in the local
robustness indicator results of the training data which are represented by a
scale of colours—darker means lower predicted local robustness while lighter
means higher predicated local robustness. By our method, test seeds selected
are both from the highest peak (high probability density area of the global
distribution) and relatively darker ones (lower predicated local robustness).
### 3.4. Local Test Cases Generation
Not all AEs are equal in terms of the “strength of being adversarial”, and
stronger AEs are associated with higher prediction loss (cf. Definition 3).
Detecting AEs with higher prediction loss may benefit more when considering
the future “debugging” step, e.g., by adversarial retraining (Wang et al.,
2019). Thus, at this stage, we want to search for AEs that exhibit a high
degree of adversarial behaviour while also conform to the local distribution.
That is, the local test case generation can be formulated as the following
optimisation given a seed $(x,y)$:
(11) $\displaystyle\max_{x^{\prime}}$
$\displaystyle\mathcal{J}(f(x^{\prime}),y)+\alpha\cdot
p_{l}(x^{\prime}|x^{\prime}\in\eta_{z_{x}})$ s.t.
$\displaystyle||x-x^{\prime}||_{\infty}\leq r$
where $\mathcal{J}$ is the prediction loss,
$p_{l}(x^{\prime}|x^{\prime}\in\eta_{z_{x}})$ is the local distribution (note,
$z_{x}$ represents the latent features of test seed $x$), $r$ is the
$r$-separation distance, and $\alpha$ is a coefficient to balance the two
terms. As what follows, we note two points on Eqn. (11): why we need the
constraint and how we quantify the local distribution.
The constraint in Eqn. (11) determines the right locality of local
robustness—the “neighbours” that should have the same ground truth label $y$
as the test seed. We notice the $r$-separation property of real-world image
datasets (cf. Remark 1) provides a sound basis to the question. Thus, it is
formalised as a constraint that the optimiser can only search in a norm ball
with a radius smaller than $r$, to guarantee the detected AEs are indeed
“adversarial” to label $y$.
While the feature level information is captured by the global distribution
over a latent space, we only consider how the pixel level information is
locally distributed in terms of perceptual quality. Three common quantitative
metrics—MSE, PSNR and SSIM introduced in Section 2.4—are investigated. We
note, those three metrics by no means are the true local distribution
representing perceptual quality, rather quantifiable indicators from different
aspects. Thus, in the optimisation problem of Eqn. (11), replacing the local
distribution term with them would suffice our purpose. So, we redefine the
optimisation problem as:
(12)
$\displaystyle\max_{x^{\prime}}\mathcal{J}(f(x^{\prime}),y)\\!+\\!\alpha\\!\cdot\\!\mathcal{L}(x,x^{\prime}),\quad\textrm{s.t.}\,||x\\!-\\!x^{\prime}||_{\infty}\\!\leq\\!r$
where $\mathcal{L}(x,x^{\prime})$ represents those perceptual quality metrics
correlated with the local distribution of the seed $x$. Certainly,
implementing $\mathcal{L}(x,x^{\prime})$ requires some prepossessing, e.g.,
normalisation and negation, depending on which metric is adopted.
Considering that the second term of the objective function in Eqn. (12) may
not be differentiable and/or the DL model’s parameters are not always
accessible, Non-dominated Sorting Genetic Algorithm II (NSGA-II) (Deb et al.,
2002) may be adopted here to solve the multi-objective optimisation. NSGA-II
is designed to search for a diverse set of solutions along the Pareto-optimal
front, rather than just a single solution. This means that NSGA-II produces a
set of solutions that span the entire Pareto-optimal front, providing
decision-makers with a range of options to choose from. Therefore, NSGA-II is
usually computationally intensive and requires a huge amount of queries of DL
model’s prediction. Since we are only concern about test cases which have high
perceptual quality and conditioned on AEs (prediction loss greater than 0), it
is not necessary to produce a range of options for the trades-off between
perceptual quality and prediction loss, which may waste a lot of computational
resource on generating high perceptual quality but no adversarial examples.
For this reason, we propose to scalarize the vector of objectives into one
objective by averaging the objectives with weight vector and reduce the weight
dependency with alternation mechanism. That is, we develop a GA with two
fitness functions to effectively and efficiently detect AEs, as shown in
Algorithm 1.
Algorithm 1 Two-Step GA Based Local Test Cases Generation
1:Test seed $(x,y)$, neural network function $f(x)$, local perceptual quality
metric $\mathcal{L}(x,x^{\prime})$, population size $N$, maximum iterations
$T$, norm ball radius $r$, weight parameter $\alpha$, number of generated test
cases $m$.
2:A set of $m$ test cases $\mathcal{T}$
3:$F_{1}=\mathcal{J}(f(x^{\prime}),y)$,
$F_{2}=\mathcal{J}(f(x^{\prime}),y)+\alpha\cdot\mathcal{L}(x,x^{\prime})$
4:for $i=1,...,N$ do
5: $\mathcal{T}[i]=x+\mathit{uniform}(-r,+r)$
6:end for
7:while $t<T$ or $\max(\mathit{fit\\_list_{2}})$ does not converge do
8: $\mathit{fit\\_list_{1}}=\mathit{cal\\_fitness}(F_{1},\mathcal{T})$
9: $\mathit{fit\\_list_{2}}=\mathit{cal\\_fitness}(F_{2},\mathcal{T})$
10: if $\mathit{majority}(\mathit{fit\\_list_{1}}<0)$ then
11: $\mathit{parents}=\mathit{selection}(\mathit{fit\\_list_{1}},\mathcal{T})$
12: else
13: $\mathit{parents}=\mathit{selection}(\mathit{fit\\_list_{2}},\mathcal{T})$
14: end if
15: $\mathcal{T}=\mathit{crossover}(\mathit{parents},N)$
16: $\mathcal{T}=\mathit{mutation}(\mathcal{T})\cup\mathit{parents}$
17: $t=t+1$
18:end while
19:$\mathit{fit\\_list_{2}}=\mathit{cal\\_fitness}(F_{2},\mathcal{T})$
20:$idx=\operatorname*{arg\,max}(fit\\_list_{2})[:m]$
21:$\mathcal{T}=\mathcal{T}[idx]$
22:return test set $\mathcal{T}$
Algorithm 1 presents the process of generating a set of $m$ test cases
$\mathcal{T}$ from a given seed $x$ with label $y$ (denoted as $(x,y)$). At
line 1, we define two fitness functions (the reason behind it will be
discussed later). GA based test case generation consists of 4 steps:
initialisation, selection, crossover, and mutation, the last three of which
are repeated until the convergence of fitness values or the maximum iterations
are reached.
##### Initialisation
The initialisation of population is crucial to the quick convergence.
Diversity of initial population could promise approximate global optimal
(Konak et al., 2006). We initialise the population by adding uniform noise in
range $(-r,+r)$ to the test seed, at line 2-4.
##### Selection
The fitness function is defined to select fitted individuals as parents for
the latter operations. We use the fitness proportionate selection (Lipowski
and Lipowska, 2012) for operator $selection()$.
(13)
$p_{i}=\frac{\mathcal{F}_{i}}{\sum_{i=1}^{n}\mathcal{F}_{i}},\;\mathcal{F}_{i}\in
fit\\_list$
The fitness value is used to associate a probability of selection $p_{i}$ for
each individuals to maintaining good diversity of population and avoid
premature convergence. The fitness function is the objective function to be
optimised. At line 6-12, we calculate two defined fitness function values on
the population and select individuals based on one of the fitness values
according to some judgement (the reason behind it will be discussed later).
Figure 3. Illustration of crossover and mutation in Two-Step GA Based Local
Test Cases Generation
##### Crossover
At line 13, the crossover operator will combine a pair of parents from last
step to generate a pair of children, which share many of the characteristics
from the parents. The half elements of parents are randomly exchanged as an
example shown in Fig. 3.
##### Mutation
At line 14, some elements of children are randomly altered to add variance in
the evolution. It should be noticed that the mutated samples should still fall
into the local region $\eta$ around test seed $x$. Finally, the children and
parents will be the individuals for the next generation.
##### Termination
At line 5, the termination condition of GA is either maximum number of
iterations is reached or the highest ranking of fitness reaches a plateau such
that successive iterations no longer produce better results.
As seen above, the main difference between our approach and common GA test
case generation is that we use two fitness functions, which work alternatively
to guide the selection of parents. The reason why we propose two fitness
functions is because, we notice that there is a trade-off between the two
objectives $\mathcal{J}$ and $\mathcal{L}$ in the optimisation. Prediction
loss $\mathcal{J}$ is related to the adversarial strength, while $\mathcal{L}$
indicates the local distribution. Intuitively, generating the test cases with
high local probability tends to add small amount of perturbations to the seed,
while a greater perturbation is more likely to induce high prediction loss. To
avoid the competition between the two terms that may finally leads to a
failure of detecting AEs, we define two fitness functions to precisely control
the preference at different stages:
(14) $\displaystyle F_{1}=\mathcal{J}(f(x^{\prime}),y),\quad
F_{2}=\mathcal{J}(f(x^{\prime}),y)+\alpha\cdot\mathcal{L}(x,x^{\prime})$
At early stage, $F_{1}$ is optimised to quickly direct the generation of AEs,
with $F_{1}>0$ meaning the detection of an AE. When most individuals in the
population are AEs, i.e., $\mathit{majority}(\mathit{fit\\_list_{1}}\geq 0)$,
the optimisation moves to the second stage, in which $F_{1}$ is replaced by
$F_{2}$ to optimise the local distribution indicator as well as the prediction
loss. It is possible555Especially when a large $\alpha$ is used, i.e., with
preference on detecting AEs with high local probability than with high
adversarial strength, cf. the aforementioned trade-off. that the prediction
loss of most individuals again becomes negative, then the optimisation will go
back to the first stage. With such a mechanism of alternatively using two
fitness functions in the optimisation, the proportion of AEs in the population
is effectively prevented from decreasing.
Algorithm 1 describes the process for generating $m$ local test cases given a
single test seed. Suppose $n$ test seeds are selected earlier and in total $M$
local test cases are affordable, we can allocate, for each test seed $x_{i}$,
the number of local test cases $m_{i}$, according to the $n$ (re-normalised)
global probabilities, which emphasises more on the role of distribution in our
detected set of AEs.
Running Example: The right diagram in Fig. 2 plots the local distribution
using MSE as its indicator, and visualises the detected AEs by different
testing methods. Unsurprisingly, all AEs detected by our proposed HDA testing
are located at the high density regions (and very close to the central test
seed), given it considers the perceptual quality metric as one of the
optimisation objectives in the two-step GA based test case generation. In
contrast, other methods (PGD and coverage-guided testing) are less effective.
## 4\. Evaluation
We evaluate the proposed HDA testing method by performing extensive
experiments to address the following research questions (RQs):
##### RQ1 (Effectiveness): How effective are the methods adopted in the three
main stages of HDA?
Namely, we conduct experiments to i) examine the accuracy of combining VAE-
Encoder+KDE to approximate the global distribution; ii) check the correlation
significance of the two proposed local robustness indicators with the local
robustness; iii) investigate the effectiveness of our two-step GA for local
test cases generation.
##### RQ2 (AE Quality): How is the quality of AEs detected by HDA?
Comparing to conventional adversarial attack and coverage-guided testing
methods and more recent distribution-aware testing methods, such as OODA and
FODA, we introduce a comprehensive set of metrics to evaluate the quality of
AEs detected by HDA and others.
##### RQ3 (Sensitivity): How sensitive is HDA to the DL models under testing?
We carry out experiments to assess the capability of HDA applied on DL models
(adversarially trained) with different levels of robustness.
##### RQ4 (Robustness Growth): How useful is HDA to support robustness growth
of the DL model under testing?
We examine the global robustness of DL models after “fixing” the AEs detected
by various testing methods.
### 4.1. Experiment Setup
We consider five popular benchmark datasets and five diverse model
architectures for evaluation. In order to obtain statistical results, we train
10 models for each benchmark dataset, initialising their weights according to
the normal distribution. We use Adam optimiser with learning rate $10^{-2}$
and weight decay $10^{-5}$, and train 100 epochs. Details of the datasets and
average accuracy (Mean $\pm$ SD) of trained DL models under testing are listed
in Table 1. The norm ball radius $r$ is calculated based on the $r$-separation
distance (cf. Remark 1) of each dataset. When comparing different AE detection
approaches on 10 models, the evaluation results are dependent on individual
model and the difference approximately follow the normal distribution by
visualisation and normality test666Some statistical tests for AE Prop. and %
of Valid AEs will output “NaN” due to the identical evaluation results between
different approaches.. Therefore, we perform paired two-sample T-test and
discuss the statistical significance in the experiments. The null hypothesis
is that different AE detection approaches have identical average values for
evaluation metrics. The calculated $p-value>0.05$ indicates that the null
hypothesis is true, otherwise it is false.
In RQ1, we validate the accuracy and effectiveness of HDA on detecting high
quality AEs from high probability density region and decide the hyper-
parameter settings for the following experiments. That is, for RQ2-RQ4, we use
the activation separation distance score ${S}_{\textit{sep}}$ as local
robustness indicator and MSE as perceptual quality metric for images. In RQ2
we compare the quality of AEs detected by HDA and others. In RQ3, we add the
comparison on DL models, enhanced by PGD-based adversarial training, for
sensitivity analysis. Table 1 also records the accuracy of these adversarially
trained models. Adversarial training trades the generalisation accuracy for
the robustness as expected (thus a noticeable decrement of the training and
testing accuracy) (Zhang et al., 2019). In RQ4, we firstly sample 10000 data
points from the global distribution as validation set and detect AEs around
them by different methods. Then, we fine-tune the normally trained models with
training dataset augmented by these AEs. 10 epochs are taken along with ‘slow
start, fast decay’ learning rate schedule (Jeddi et al., 2021) to reduce the
computational cost while improving the accuracy-drop and robustness. To
empirically estimate the global robustness on validation set, we find another
set of AEs according to local distribution, different from the fine-tuning
data. These AEs, crafted from validation datasets, are miss-classified by
normally trained models. Thus, empirical global robustness of normally trained
models is set to 0 as the baseline.
When training VAE models, the loss function is a combination of reconstruction
loss and KL divergence loss. Two losses are weighted equally and
simultaneously optimised. We also use Adam optimiser with learning rate
$10^{-2}$ and weight decay $10^{-5}$, and train 100 epochs to obtain VAE
models. To avoid the posterior collapse that VAE often converges to a
degenerated local optimum, we add Batch Normalisation (Zhu et al., 2020)
before the output of encoder. The latent dimensions and the reconstruction
loss are listed in Table 3.
For readers’ convenience, all the metrics used in RQ2, RQ3 and RQ4 for
comparisons are listed in Table 2. The metrics are introduced to
comprehensively evaluate the quality of detected AEs and the DL models from
different aspects. When comparing HDA testing with PGD attack, coverage guided
testing, OODA and FODA, we have the following settings for the tools.
Specifically, we use PGD attack with 10 steps for gradient ascent, and step
size 2/255; HDA testing with population size $N=1000$, maximum iterations
$T=500$, and weight parameter $\alpha=1$; neuron coverage (Sun et al., 2018)
with Gaussian noise $mean=0,std=1$ to generate perturbed data, and 100
iterations to increase the coverage rate by fuzzing; OODA testing (Dola et
al., 2021) with 10 steps for gradient ascent, step size 2/255, default
hyperparameter for balancing two goals and default reconstruction probability
threshold used in the released code 777https://github.com/swa112003/
DistributionAwareDNNTesting.; FODA testing (Byun et al., 2020) with the same
latent space encoded by VAE used in HDA, random sampling to search for AEs in
the latent space. To achieve the fair comparison, we set the same perturbation
radius $r$ for PGD attack, HDA testing and coverage guided testing, since they
have the restrictions for the validity of test cases. In addition, we set the
same number of steps and step size for PGD attack and OODA testing, and
utilise the same latent space across HDA and FODA.
Table 1. Details of the datasets and DL models under testing.
Dataset | Image Size | $r$ | DL Model | Normal Training | Adversarial Training
---|---|---|---|---|---
Avg. Train Acc. | Avg. Test Acc. | Avg. Train Acc. | Avg. Test Acc.
MNIST | $1\times 32\times 32$ | 0.1 | LeNet5 | $99.88\%\pm 0.06\%$ | $98.73\%\pm 0.12\%$ | $99.77\%\pm 0.01\%$ | $98.84\%\pm 0.01\%$
Fashion-MNIST | $1\times 32\times 32$ | 0.08 | AlexNet | $95.12\%\pm 0.85\%$ | $90.70\%\pm 0.25\%$ | $86.23\%\pm 0.05\%$ | $85.11\%\pm 0.05\%$
SVHN | $3\times 32\times 32$ | 0.03 | VGG11 | $96.12\%\pm 0.45\%$ | $94.86\%\pm 0.21\%$ | $89.89\%\pm 0.69\%$ | $90.32\%\pm 0.81\%$
CIFAR-10 | $3\times 32\times 32$ | 0.03 | ResNet20 | $97.81\%\pm 0.58\%$ | $88.28\%\pm 0.63\%$ | $79.48\%\pm 0.72\%$ | $76.42\%\pm 1.01\%$
CelebA | $3\times 64\times 64$ | 0.05 | MobileNetV1 | $94.19\%\pm 1.63\%$ | $90.79\%\pm 0.49\%$ | $77.74\%\pm 0.22\%$ | $79.72\%\pm 0.22\%$
Table 2. Evaluation metrics for the quality of detected AEs and DL models
Metrics | Meanings
---|---
AE Prop. | Proportion of test seeds from which AEs can be detected over the total number of test seeds
Pred. Loss | Adversarial strength of AEs as formally defined by Definition 3
$p_{g}$ | Normalised global probability density of test-seeds/AEs
$\mathcal{R}_{l}$ | Local robustness to the correct classification label, as formally defined by Definition 1
$\hat{\mathcal{R}_{g}}$ | Empirical global robustness of DL models over input domain as defined in Definition 5
FID | Distribution difference between original images (test seeds) and perturbed images (AEs)
$\epsilon$ | Average perturbation distance between test seeds and AEs
% of Valid AEs | Percentage of “in-distribution” AEs in all detected AEs
All experiments were run on a machine of Ubuntu 18.04.5 LTS x86_64 with Nvidia
A100 GPU and 40G RAM. The source code, DL models, datasets and all experiment
results are publicly available at https://github.com/havelhuang/HDA-Testing.
### 4.2. Evaluation Results and Discussions
#### 4.2.1. RQ1
There are 3 sets of experiments in RQ1 to examine the accuracy of technical
solutions in our tool-chain, corresponding to the 3 main stages respectively.
First, to approximate the global distribution, we essentially proceed in two
steps—dimensionality reduction and PDF fitting, for which we adopt the VAE-
Encoder+KDE solution. Notably, the VAE trained in this step is for data-
compression only (not for generating new data). To reflect the effectiveness
of both aforementioned steps, we (i) compare VAE-Encoder with the Principal
Component Analysis (PCA), and (ii) measure the FID between the training
dataset and a set of random samples drawn from the fitted global distribution
by KDE.
PCA is a common approach for dimensionality reduction. We use scikit-learn
(Pedregosa et al., 2011) to implement the PCA with ’auto’ solver for applying
Singular Value Decomposition (SVD). That is, if the input data is larger than
500x500 and the number of components to extract (latent dimensions) is lower
than 80% of the smallest dimension of the data, then the more efficient
‘randomised’ method (Pedregosa et al., 2011) is enabled. Otherwise the exact
full SVD is computed and optionally truncated afterwards. To achieve the fair
comparison, the latent dimensions of PCA and VAE-Encoder are set to be the
same. We compare the performance of VAE-Encoder and PCA from the following two
perspectives. The quality of latent representation can be measured by the
clustering and reconstruction accuracy.
To learn the feature level (global) distribution from latent data, we require
that latent representations should group together data points that share
similar semantic features. To evaluate this clustering ability, we apply
K-means clustering to the latent data, which partitions the data points into
clusters based on their similarity. Then, we calculate the Completeness Score
(CS), Homogeneity Score (HS) and V-measure Score (VS) (Rosenberg and
Hirschberg, 2007). These scores provide a measure of how well the resulting
clusters group together data points that share similar semantic features.
Specifically, the CS measures how well the clustering captures all data points
that belong to the same true class in a single cluster, while the HS measures
how well the clustering captures all data points within a cluster that belong
to the same true class. The VS is a harmonic mean of the CS and HS, providing
an overall measure of the quality of the clustering.
In addition to ensuring that latent representations group together similar
data points, we also require that the latent representations can be decoded to
reconstruct the original images with minimal information loss. The
reconstruction loss is calculated based on the MSE. As is shown in Table 3,
VAE-Encoder achieves higher CS, HS, VS scores and less reconstruction loss
than PCA. In other words, the latent representations encoded by VAE-Encoder is
better in terms of capturing feature information than that of PCA.
Table 3. Quality of Latent Representation in PCA & VAE-Encoder
Dataset | Latent Dimensions | PCA | VAE-Encoder
---|---|---|---
Clustering | Recon. Loss | Clustering | Recon. Loss
CS | HS | VS | CS | HS | VS
MNIST | 8 | 0.505 | 0.508 | 0.507 | 44.09 | 0.564 | 0.566 | 0.565 | 27.13
F.-MNIST | 4 | 0.497 | 0.520 | 0.508 | 55.56 | 0.586 | 0.601 | 0.594 | 23.72
SVHN | 4 | 0.007 | 0.007 | 0.007 | 65.75 | 0.013 | 0.011 | 0.015 | 62.38
CIFAR-10 | 8 | 0.084 | 0.085 | 0.085 | 188.22 | 0.105 | 0.105 | 0.105 | 168.44
CelebA | 32 | 0.112 | 0.092 | 0.101 | 764.94 | 0.185 | 0.150 | 0.166 | 590.54
Dataset | Global Dist. | Uni. Dist.
---|---|---
MNIST | 0.395 | 13.745
Fashion-MNIST | 0.936 | 90.235
SVHN | 0.961 | 141.134
CIFAR-10 | 0.285 | 12.053
CelebA | 0.231 | 8.907
[table]..
Figure 4. Samples drawn from the approximated global distribution by KDE and a
uniform distribution over the latent feature space (Figure); and FID to the
ground truth based on 1000 samples (Table).
To evaluate the accuracy of using KDE to fit the global distribution, we
calculate the FID between a new dataset (with 1000 samples) based on the
fitted global distribution by KDE and the training dataset. The new dataset is
sampled from the fitted global distribution over latent space and decoded by
VAE decoder into images. The FID scores are shown in Table 4. As a baseline,
we also present the results of using a uniform distribution over the latent
space. As expected, we observe that all FID scores based on approximated
distributions are significantly smaller (better). We further decode the newly
generated images for visualisation in Fig. 4, from which we can see that
generated images by KDE keep high fidelity while the uniformly sampled images
are more difficult to be recognised.
Answer to RQ1 on HDA stage 1: The combination of VAE-Encoder+KDE may
accurately approximate the global distribution, since the new sampled data
from approximated distribution keep high fidelity.
Move on to stage 2, we study the correlations between a norm ball’s local
robustness and its two indicators proposed earlier—the prediction gradient
based score and the score based on separation distance of output-layer
activation (cf. Eq. 10).
Figure 5. Scatter plots of the local robustness evaluation vs. its two
indicators, based on 1000 random norm balls. Table 4. Pearson correlation
coefficients (in absolute values) between the local robustness & its two
indicators.
Dataset | $S_{\textit{grad}}$ | $S_{\textit{sep}}$ | $T(S_{\textit{grad}},S_{\textit{sep}})$
---|---|---|---
$t$ | $p-value$
MNIST | $\mathbf{0.631\pm 0.025}$ | $0.564\pm 0.019$ | $5.488$ | $5.813\times 10^{-4}$
Fashion-MNIST | $0.717\pm 0.109$ | $\mathbf{0.789\pm 0.056}$ | $-1.793$ | $0.107$
SVHN | $\mathbf{0.747\pm 0.039}$ | $0.745\pm 0.030$ | $0.150$ | $0.884$
CIFAR-10 | $0.603\pm 0.038$ | $\mathbf{0.668\pm 0.065}$ | $-5.466$ | $3.975\times 10^{-4}$
CelebA | $0.639\pm 0.073$ | $\mathbf{0.728\pm 0.077}$ | $-3.872$ | $0.004$
We invoke the tool (Webb et al., 2019) for estimating the local robustness
$\mathcal{R}_{l}$ defined in Definition 1. Based on 1000 randomly selected
data-points from the test set as the central point of 1000 norm balls, we
calculate the local robustness of each norm ball888Radius $r$ is usually small
by definition (cf. Remark 1), yielding very small $log{(1-\mathcal{R}_{l})}$.
as well as the two proposed indicators. Then, we do the scatter plots (in log-
log scale999There are dots collapsed on the vertical line of $log{(1-R)}=-70$,
due to a limitation of the estimator (Webb et al., 2019)—it terminates with
the specified threshold when the estimation is lower than that value. Note,
the correlation calculated with such noise is not undermining our conclusion,
rather the real correlation would be even higher.), as shown in Fig. 5.
Apparently, for all 5 datasets, the indicator based on activation separation
distance is negatively correlated (1st row), while the gradient based
indicator is positively correlated with the estimated local robustness (2nd
row). We further quantify the correlation by calculating the Pearson
correlation coefficients, as recorded in Table 4. There is a rule of thumb
that Pearson correlation coefficients greater than 0.6 indicate strong
correlations (Akoglu, 2018). We observe, both indicators are highly correlated
with the local robustness, while the separation distance based indicator is
slightly better. The statistical test shows that the separation distance based
indicator is significantly better than gradient based indicator in CIFAR10 and
CelebA datasets. Therefore, for the latter experiment in RQ2-RQ4, we choose
separation distance based indicator $S_{\textit{sep}}$ to guide the selection
of test seeds when comparing HDA testing with other AE detection methods.
Answer to RQ1 on HDA stage 2: The two proposed local robustness indicators are
significantly correlated with the local robustness.
Figure 6. The prediction loss (red) and the three quantified local
distribution indicators (blue) of the best fitted test case during the
iterations of our two-step GA based local test case generation.
For the local test case generation in stage 3, by configuring the parameter
$\alpha$ in our two-step GA, we may do trade-off between the “strength of
being adversarial” (measured by prediction loss $\mathcal{J}$) and the local
distribution (measured by a specific perceptual quality metric $\mathcal{L}$,
they are MSE, PSNR and SSIM), so that the quality of detected AEs can be
optimised.
In Fig. 6, we visualise the changes of the two fitness values as the
iterations of the GA. As shown in the first plot, only the prediction loss
$\mathcal{J}$ is taken as the fitness function (i.e., $\alpha=0$) during the
whole iteration process. The GA can effectively find AEs with maximised
adversarial strength, as observed by the convergence of the prediction loss of
the best-fitted test case in the population after hundreds of iterations. From
the second to the last plot, the fitness function consists not only of
prediction loss $\mathcal{J}$, but also of a perceptual quality metric
$\mathcal{L}$, representing the local distribution information (i.e.,
$\alpha>0$). Intuitively, a smaller MSE or greater PSNR and SSIM implies
higher local probability density.
Thanks to the two-step setting of the fitness functions, the prediction loss
$\mathcal{J}$ of best-fitted test case goes over 0 quickly in less than 200
iterations, which means it detects a first AE in the population. The
$\mathcal{J}$ of the best fitted test case is always quite close to the rest
in the population, thus we may confidently claim that many AEs are efficiently
detected by the population not long after the first AE was detected. Then, the
optimisation goes to the second stage, in which the quantified local
distribution indicator $\mathcal{L}$ is pursued. The $\mathcal{J}$ and
$\mathcal{L}$ finally converge and achieve a balance between them. If we
configure the coefficient $\alpha$, the balance point will change
correspondingly. A greater $\alpha$ (e.g., $\alpha=1.1$ in the plots) detects
less perceptible AEs (i.e., with higher local probability density), and the
price paid is that the detected AEs are with weaker adversarial strength
(i.e., with smaller but still positive prediction loss).
Figure 7. Comparison between regular GA and two-step GA. Figure 8. AEs
detected by our two-step GA (last 3 columns) & other methods
We further investigate the advantages of our 2-step GA over the regular GA
(using $F_{2}$ as the objective function). In Fig. 7, as $\alpha$ increases,
the proportion of AEs in the population exhibits a sharp drop to 0 when using
the regular GA. In contrast, the two-step GA prevents such decreasing of the
AE proportion while preserving it at a high-level of 0.6, even when $\alpha$
is quite large. Moreover, larger $\alpha$ represents the situations when the
AEs are less perceptible in terms of perceptual quality metrics—as shown by
the blue curves101010The blue dashed line stops earlier as there is no AEs in
the population when $\alpha$ is big., the imperceptibility ( measured by SSIM
in this case) is only sufficiently high when $\alpha$ is big enough. Thus,
compared to the regular GA, we may claim our novel 2-step GA is more robust
(in detecting AEs) to the choices of $\alpha$ and more suitable in our
framework for detecting AEs with high perceptual quality.
Table 5. Perceptual quality measured by FID between a set of original images
and a set of AEs detected by two-step GA with different fitness functions.
Dataset | $J$ | $J+MSE$ | $J+PSNR$ | $J+SSIM$
---|---|---|---|---
MNIST | $1.185\pm 0.157$ | $\mathbf{0.667\pm 0.202}$ | $0.668\pm 0.203$ | $0.875\pm 0.184$
Fashion-MNIST | $2.736\pm 0.680$ | $0.138\pm 0.106$ | $\mathbf{0.131\pm 0.100}$ | $0.679\pm 0.182$
SVHN | $120.172\pm 7.529$ | $105.399\pm 9.009$ | $\mathbf{104.147\pm 9.934}$ | $111.139\pm 6.754$
CIFAR-10 | $96.449\pm 5.906$ | $\mathbf{65.727\pm 5.672}$ | $67.426\pm 7.398$ | $76.293\pm 6.493$
CelebA | $85.658\pm 3.334$ | $\mathbf{65.981\pm 4.976}$ | $66.599\pm 3.312$ | $71.878\pm 3.153$
(a) Results
Dataset | $T(J+MSE,J)$ | $T(J+MSE,J+PSNR)$ | $T(J+MSE,J+SSIM)$
---|---|---|---
$t$ | $p-value$ | $t$ | $p-value$ | $t$ | $p-value$
MNIST | $-13.631$ | $2.581\times 10^{-7}$ | $-0.912$ | $0.385$ | $-10.279$ | $2.844\times 10^{-6}$
Fashion-MNIST | $-13.362$ | $3.065\times 10^{-7}$ | $2.188$ | $0.056$ | $-11.957$ | $7.934\times 10^{-7}$
SVHN | $-9.214$ | $7.036\times 10^{-6}$ | $1.271$ | $0.235$ | $-4.443$ | $0.002$
CIFAR-10 | $-17.32$ | $3.215\times 10^{-8}$ | $-1.363$ | $0.206$ | $-8.125$ | $1.954\times 10^{-5}$
CelebA | $-23.491$ | $2.187\times 10^{-9}$ | $-0.511$ | $0.621$ | $-3.931$ | $0.003$
(b) Statistical Test
Fig. 8 displays some selected AEs from the five datasets. Same as the PGD
attack and the coverage-guided testing, if we only use the prediction loss
$\mathcal{J}$ as the objective function in the GA, the perturbations added to
the images can be easily recognised. In stark contrast, AEs generated by our
two-step GA (with the 3 perceptual quality metrics in the last 3 columns) are
of high quality and less distinguishable from the original images (first
column). We further calculate the FID to quantify the perceptual quality of
detected AEs in Table 5(b). Since the comparison with PGD attack, coverage
guided testing, FODA, and OODA are given in the subsequent experiments, i.e.
Table 7(b) and Table 8(b), we focus on comparing the performance of 3
perceptual quality metrics here. Results show that MSE is significantly better
than J and SSIM to guide the generation of high fidelity AEs for the
experiment dataset. While MSE has similar performance with PSNR. Therefore, we
decide to utilise the J+MSE in the following experiments for the comparison
with the state-of-the-art.
Answer to RQ1 on HDA stage 3: Two-step GA based local test case generation can
effectively detect AEs with high perception quality (in terms of those metrics
encoded by GA).
#### 4.2.2. RQ2
We compare HDA with the state-of-the-art AE detection methods in two sets of
experiments. In the first set, we focus on comparing with the adversarial
attack and coverage-guided testing (i.e., the typical PGD attack and neuron
coverage metric for brevity, while the conclusion can be generalised to other
attacks and coverage metrics, since they all lack the consideration of
distribution when detecting AEs). Then in the second set of experiments, we
show the advantages of our HDA testing over other distribution-aware testing
methods.
In fact, both PGD attack and coverage-guided testing do not contribute to test
seeds selection. They simply use randomly sampled data from the test set as
test seeds, by default. We also notice that a large amount of test seeds
prioritisation metrics are proposed, the typical ones among which are
Distance-Based Surprise Adequacy (DSA) (Weiss and Tonella, 2022), DeepGini
(Weiss and Tonella, 2022). Thus, we compare the randomly selected test seeds,
DSA guided test seeds, DeepGini guided test seeds with our “global
distribution probability111111Refer to Section 3.2 for the calculation. The
value of probability density is further normalised by training dataset for a
better presentation. plus local robustness indicated” test seeds, shown as
“‘$p_{g}+\mathcal{R}_{l}$” in Table 6(b). Specifically, for each test seed, we
calculate two metrics—the local robustness $\mathcal{R}_{l}$ of its norm ball
and its corresponding global probability $p_{g}$. We invoke the estimator of
(Webb et al., 2019) to calculate the former ($log{(1-\mathcal{R}_{l})}$, to be
exact). To reduce the sampling noise, we repeat the test seed selection 100
times and present the averaged results in Table 6(b).
Table 6. Comparison between randomly selected test seeds, DSA guided test
seeds, DeepGini guided test seeds and our “$p_{g}+\mathcal{R}_{l}$ indicated”
test seeds (based on 100 test seeds).
Dataset | Random Test Seeds | DSA Test Seeds | DeepGini Test Seeds | $p_{g}+\mathcal{R}_{l}$ Test Seeds
---|---|---|---|---
$log{(1-\mathcal{R}_{l})}$ | $p_{g}$ | $log{(1-\mathcal{R}_{l})}$ | $p_{g}$ | $log{(1-\mathcal{R}_{l})}$ | $p_{g}$ | $log{(1-\mathcal{R}_{l})}$ | $p_{g}$
MNIST | $-64.67\pm 1.27$ | $0.0034\pm 0.0005$ | $-64.59\pm 1.15$ | $0.0016\pm 0.0003$ | $-65.16\pm 1.31$ | $0.0018\pm 0.0002$ | $\mathbf{-22.12\pm 3.01}$ | $\mathbf{0.0293\pm 0.0072}$
Fashion-MNIST | $-14.98\pm 4.50$ | $0.0033\pm 0.0009$ | $-15.74\pm 4.82$ | $0.0016\pm 0.0003$ | $-15.29\pm 4.89$ | $0.0017\pm 0.0003$ | $\mathbf{-1.24\pm 0.48}$ | $\mathbf{0.0226\pm 0.0088}$
SVHN | $-53.34\pm 3.98$ | $0.0032\pm 0.0002$ | $-53.44\pm 2.10$ | $0.0013\pm 0.0002$ | $-53.62\pm 2.32$ | $0.0008\pm 0.0001$ | $\mathbf{-8.27\pm 2.43}$ | $\mathbf{0.0132\pm 0.0021}$
CIFAR-10 | $-13.77\pm 1.24$ | $0.0033\pm 0.0009$ | $-14.25\pm 2.26$ | $0.0018\pm 0.0003$ | $-14.82\pm 0.86$ | $0.0018\pm 0.0003$ | $\mathbf{-2.49\pm 0.88}$ | $\mathbf{0.0397\pm 0.0121}$
CelebA | $-17.68\pm 5.00$ | $0.0034\pm 0.0002$ | $-17.27\pm 5.44$ | $0.0010\pm 0.0001$ | $-17.28\pm 5.37$ | $0.0005\pm 0.0001$ | $\mathbf{-1.59\pm 0.54}$ | $\mathbf{0.0118\pm 0.0005}$
(a) Results
Dataset | Metric | $T(p_{g}+\mathcal{R}_{l},\text{Random})$ | $T(p_{g}+\mathcal{R}_{l},\text{DSA})$ | $T(p_{g}+\mathcal{R}_{l},\text{DeepGini})$
---|---|---|---|---
$t$ | $p-value$ | $t$ | $p-value$ | $t$ | $p-value$
MNIST | $log{(1-\mathcal{R}_{l})}$ | $41.186$ | $2.890\times 10^{-19}$ | $41.680$ | $2.337\times 10^{-19}$ | $41.461$ | $2.568\times 10^{-19}$
$p_{g}$ | $11.348$ | $1.234\times 10^{-9}$ | $12.155$ | $4.101\times{-10}$ | $12.073$ | $4.574\times 10^{-10}$
Fashion-MNIST | $log{(1-\mathcal{R}_{l})}$ | $9.601$ | $1.665\times 10^{-8}$ | $9.466$ | $2.063\times 10^{-8}$ | $9.042$ | $4.106\times 10^{-8}$
$p_{g}$ | $6.899$ | $1.885\times 10^{-6}$ | $7.542$ | $5.616\times 10^{-7}$ | $7.506$ | $6.001\times 10^{-7}$
SVHN | $log{(1-\mathcal{R}_{l})}$ | $30.564$ | $5.769\times 10^{-17}$ | $44.475$ | $7.345\times 10^{-20}$ | $42.686$ | $1.528\times 10^{-19}$
$p_{g}$ | $14.991$ | $1.303\times 10^{-11}$ | $17.839$ | $6.867\times 10^{-13}$ | $18.651$ | $3.203\times 10^{-13}$
CIFAR-10 | $log{(1-\mathcal{R}_{l})}$ | $23.459$ | $6.033\times 10^{-15}$ | $15.334$ | $8.916\times 10^{-12}$ | $31.688$ | $3.045\times 10^{-17}$
$p_{g}$ | $9.487$ | $1.997\times 10^{-8}$ | $9.902$ | $1.039\times 10^{-8}$ | $9.902$ | $1.039\times 10^{-8}$
CelebA | $log{(1-\mathcal{R}_{l})}$ | $10.117$ | $7.461\times 10^{-9}$ | $9.071$ | $3.922\times 10^{-8}$ | $9.193$ | $3.207\times 10^{-8}$
$p_{g}$ | $49.326$ | $1.156\times 10^{-20}$ | $66.979$ | $4.830\times 10^{-23}$ | $70.079$ | $2.145\times 10^{-23}$
(b) Statistical Test
From Table 6(b), we observe: (i) test seeds selected by our method have much
higher global probability density, meaning their norm balls are much more
representative of the data distribution; (ii) the norm balls of our test seeds
have worse local robustness, meaning it is more cost-effective to detect AEs
in them. Both metrics of HDA are significantly different from those of other
test seeds selection methods, as indicated by large t-scores and small
p-values. This is unsurprising because we have explicitly considered the
distribution and local robustness information in the test seed selection.
(iii) DSA and DeepGini guided test seeds are even worse than random test
seeds, since they target at prioritising the test seeds which are prone to be
misclassified. These misclassified test seeds are usually out of distribution.
Finally, the overall evaluation on the generated test cases and the detected
AEs by them are shown in Table 7(b). The results are presented in two
dimensions—3 types of testing methods versus 2 ways of test seeds selection,
yielding 6 combinations (although by default, PGD attack and coverage-guided
methods are using random seeds, while our method is using the
“$p_{g}+\mathcal{R}_{l}$” seeds). For each combination, we study 4 metrics
(cf. Table 2 for meanings behind them): (i) the AE proportion; (ii) the
average prediction loss; (iii) the FID121212To show how close the perturbed
test cases are to the test seeds in the latent space, we use the last
convolutional layer of InceptionV3 to extract the latent representations of
colour images for FID. InceptionV3 is a well-trained CNN and commonly used to
show FID that captures the perturbation levels, e.g., in (Heusel et al.,
2017). While InceptionV3 is used for colour images, VAE is used for grey-scale
datasets MNIST and Fashion-MNIST. of the test set quantifying the image
quality; and (iv) the computational time (and an additional coverage rate for
coverage-guided testing). We note the observations on these 4 metrics in the
following paragraphs.
Table 7. Evaluation of the generated test cases and detected AEs by PGD
Attack, coverage-guided testing and the proposed HDA testing (all results are
averaged over 100 seeds). HDA can detect more high perception quality AEs by
$p_{g}$+$\mathcal{R}_{l}$ seeds selection, which is evidenced by high AE Prop.
and low FID.
| AE Detection
---
Method
Test Seeds | Metric | MNIST | F.-MNIST | SVHN | CIFAR-10 | CelebA
PGD Attack | Random Seeds | AE Prop. | $\mathbf{0.422\pm 0.046}$ | $\mathbf{1.000\pm 0.000}$ | $\mathbf{0.951\pm 0.031}$ | $\mathbf{1.000\pm 0.000}$ | $0.989\pm 0.035$
Pred. Loss | $\mathbf{6.362\pm 1.089}$ | $\mathbf{29.200\pm 15.754}$ | $\mathbf{6.911\pm 0.571}$ | $\mathbf{46.099\pm 4.037}$ | $\mathbf{46.502\pm 48.154}$
FID | $0.705\pm 0.047$ | $2.938\pm 0.249$ | $106.581\pm 1.275$ | $96.149\pm 2.265$ | $88.401\pm 1.459$
Time(s) | $\mathbf{0.038\pm 0.070}$ | $\mathbf{0.035\pm 0.013}$ | $\mathbf{9.632\pm 0.274}$ | $\mathbf{9.294\pm 0.532}$ | $\mathbf{8.998\pm 0.646}$
$p_{g}$+$\mathcal{R}_{l}$ Seeds | AE Prop. | $\mathbf{0.785\pm 0.103}$ | $\mathbf{1.000\pm 0.000}$ | $\mathbf{0.989\pm 0.006}$ | $\mathbf{1.000\pm 0.000}$ | $\mathbf{1.000\pm 0.000}$
Pred. Loss | $\mathbf{10.184\pm 1.135}$ | $\mathbf{36.139\pm 18.205}$ | $\mathbf{9.385\pm 0.763}$ | $\mathbf{44.816\pm 3.221}$ | $\mathbf{52.136\pm 58.991}$
FID | $0.696\pm 0.051$ | $1.351\pm 0.208$ | $103.269\pm 1.153$ | $98.592\pm 1.187$ | $83.709\pm 1.095$
Time(s) | $\mathbf{0.031\pm 0.024}$ | $\mathbf{0.035\pm 0.014}$ | $\mathbf{9.541\pm 0.499}$ | $\mathbf{9.739\pm 0.763}$ | $\mathbf{9.473\pm 0.529}$
Cov. Guided Testing | Random Seeds | Cov. Rate | $0.980\pm 0.002$ | $0.946\pm 0.023$ | $0.966\pm 0.002$ | $0.979\pm 0.004$ | $0.985\pm 0.002$
AE Prop. | $0.0002\pm 0.0006$ | $0.153\pm 0.021$ | $0.014\pm 0.005$ | $0.150\pm 0.016$ | $0.080\pm 0.033$
Pred. Loss | $0.019\pm 0.001$ | $2.404\pm 1.512$ | $0.230\pm 0.045$ | $3.732\pm 1.098$ | $2.841\pm 4.311$
FID | $0.802\pm 0.026$ | $3.771\pm 0.357$ | $101.034\pm 1.014$ | $87.899\pm 1.807$ | $85.918\pm 1.109$
Time(s) | $248.902\pm 11.088$ | $1189.87\pm 223.93$ | $4416.95\pm 144.93$ | $6678.35\pm 690.49$ | $972.17\pm 83.57$
$p_{g}$+$\mathcal{R}_{l}$ Seeds | Cov. Rate | $0.977\pm 0.003$ | $0.915\pm 0.025$ | $0.921\pm 0.008$ | $0.978\pm 0.004$ | $0.978\pm 0.002$
AE Prop. | $0.030\pm 0.017$ | $0.485\pm 0.068$ | $0.138\pm 0.030$ | $0.449\pm 0.049$ | $0.286\pm 0.056$
Pred. Loss | $1.184\pm 0.632$ | $2.867\pm 1.244$ | $0.198\pm 0.039$ | $3.607\pm 0.459$ | $2.137\pm 2.283$
FID | $0.804\pm 0.043$ | $1.816\pm 0.285$ | $94.043\pm 1.786$ | $90.491\pm 1.071$ | $81.243\pm 1.162$
Time(s) | $81.33\pm 7.09$ | $216.59\pm 152.62$ | $3073.97\pm 598.92$ | $1822.64\pm 102.24$ | $2065.70\pm 32.68$
HDA | Random Seeds | AE Prop. | $0.208\pm 0.042$ | $0.999\pm 0.003$ | $0.843\pm 0.046$ | $1.000\pm 0.000$ | $\mathbf{1.000\pm 0.000}$
Pred. Loss | $1.076\pm 0.341$ | $5.214\pm 3.325$ | $2.784\pm 0.198$ | $32.251\pm 2.287$ | $16.020\pm 15.306$
FID | $\mathbf{0.094\pm 0.009}$ | $\mathbf{0.119\pm 0.043}$ | $\mathbf{91.431\pm 1.118}$ | $\mathbf{62.587\pm 2.144}$ | $\mathbf{59.162\pm 1.469}$
Time(s) | $74.211\pm 1.71$ | $136.652\pm 22.71$ | $153.956\pm 62.92$ | $351.992\pm 74.62$ | $191.313\pm 68.35$
$p_{g}$+$\mathcal{R}_{l}$ Seeds | AE Prop. | $0.676\pm 0.127$ | $\mathbf{1.000\pm 0.000}$ | $\mathbf{0.989\pm 0.006}$ | $\mathbf{1.000\pm 0.000}$ | $\mathbf{1.000\pm 0.000}$
Pred. Loss | $1.591\pm 0.306$ | $10.020\pm 5.961$ | $3.925\pm 0.433$ | $33.361\pm 1.873$ | $20.702\pm 21.941$
FID | $\mathbf{0.042\pm 0.011}$ | $\mathbf{0.016\pm 0.004}$ | $\mathbf{91.102\pm 1.342}$ | $\mathbf{65.829\pm 2.154}$ | $\mathbf{55.478\pm 1.399}$
Time(s) | $204.002\pm 85.96$ | $61.793\pm 1.69$ | $237.882\pm 98.26$ | $708.832\pm 269.81$ | $305.499\pm 82.47$
(a) Results
Dataset | Metric | Random Seeds | $p_{g}$+$\mathcal{R}_{l}$ Seeds
---|---|---|---
$T(\text{HDA},\text{PGD})$ | $T(\text{HDA},\text{Cov.})$ | $T(\text{HDA},\text{PGD})$ | $T(\text{HDA},\text{Cov.})$
$t$ | $p-value$ | $t$ | $p-value$ | $t$ | $p-value$ | $t$ | $p-value$
MNIST | AE Prop. | $-10.864$ | $2.458\times 10^{-9}$ | $15.644$ | $6.365\times 10^{-12}$ | $-2.108$ | $0.049$ | $15.943$ | $4.627\times 10^{-12}$
FID | $-40.376$ | $4.118\times 10^{-19}$ | $-81.374$ | $1.469\times 10^{-24}$ | $-39.640$ | $5.715\times 10^{-19}$ | $-54.290$ | $2.080\times 10^{-21}$
Fashion-MNIST | AE Prop. | $-1.054$ | $0.306$ | $126.114$ | $5.595\times 10^{-28}$ | $nan$ | $nan$ | $23.949$ | $4.204\times 10^{-15}$
FID | $-35.279$ | $4.541\times 10^{-18}$ | $-32.117$ | $2.400\times 10^{-17}$ | $-20.293$ | $7.495\times 10^{-14}$ | $-19.970$ | $9.882\times 10^{-14}$
SVHN | AE Prop. | $-6.157$ | $8.186\times 10^{-6}$ | $56.656$ | $9.695\times 10^{-22}$ | $nan$ | $nan$ | $87.961$ | $3.629\times 10^{-25}$
FID | $-28.252$ | $2.312\times 10^{-16}$ | $-20.119$ | $8.690\times 10^{-14}$ | $-21.746$ | $2.259\times 10^{-14}$ | $-4.163$ | $5.843\times 10^{-4}$
CIFAR-10 | AE Prop. | $nan$ | $nan$ | $167.996$ | $3.221\times 10^{-30}$ | $nan$ | $nan$ | $35.559$ | $3.945\times 10^{-18}$
FID | $-34.030$ | $8.609\times 10^{-18}$ | $-28.547$ | $1.925\times 10^{-16}$ | $-42.126$ | $1.933\times 10^{-19}$ | $-32.419$ | $2.033\times 10^{-17}$
CelebA | AE Prop. | $0.994$ | $0.333$ | $88.160$ | $3.485\times 10^{-25}$ | $nan$ | $nan$ | $40.319$ | $4.223\times 10^{-19}$
FID | $-44.658$ | $6.825\times 10^{-20}$ | $-45.968$ | $4.073\times 10^{-20}$ | $-50.251$ | $8.295\times 10^{-21}$ | $-44.801$ | $6.449\times 10^{-20}$
(b) Statistical Test
Regarding the AE proportion in the set of generated test cases, the default
setting of our proposed approach ($p_{g}$+$\mathcal{R}_{l}$ Seeds with HDA)
has comparable performance with PGD attack, as indicated by identical results
and $nan$ output by statistical test. Both our novel $p_{g}$+$\mathcal{R}_{l}$
test seed selection and two-step GA local test case generation methods
contribute to the comparable performance. This is evident from the decreased
AE proportion when using random seeds in our method, but still the result is
relatively higher than most combinations. PGD attack, as a white-box approach
using the gradient information, is definitely quite efficient in detecting
AEs, especially when paired with our new test seed selection method. On the
other hand, coverage-guided testing is comparatively less effective in
detecting AEs (even with high coverage rate), but our test seed selection
method can improve it.
As per the results of prediction loss, PGD attack, as a gradient based attack
method, unsurprisingly finds the AEs with the largest prediction loss. With
better test seeds considering local robustness information by our method, the
prediction loss of PGD attack can be even higher. Both coverage-guided testing
and our HDA testing can detect AEs with relatively lower prediction loss,
meaning the AEs are with “weaker adversarial strength”. The reason for the low
prediction loss of AEs detected by our approach is that two-step GA makes the
trade-off and sacrifices it for higher local probabilities (i.e., perceptual
quality). This can be seen through the significantly smaller FID of test set
generated by HDA, compared with PGD attack and coverage-guided testing. PGD
attack has relatively high FID scores, as well as coverage-guided testing.
On the computational overheads, we observe PGD is the most efficient, given it
is by nature a white-box approach using the gradient information. While, our
approach is an end-to-end black-box approach (if without using the gradient
based indicator when selecting test seeds) requiring less information and
being more generic, at the price of being relatively less efficient. That
said, the computational time of our approach is still acceptable and better
than coverage-guided testing.
Answer to RQ2 on comparing with adversarial attack and coverage-guided
testing: HDA can select more significant test seeds, which are from high
probability density region and lack of robustness. HDA can generate higher
perception quality AEs, which are measured with smaller FID values.
Table 8. Evaluation of AEs detected by OODA, FODA and our HDA testing methods
(based on 100 test seeds).
Dataset | | AE Detection
---
Method
$p_{g}$ | | % of
---
Valid AEs
$\epsilon$ | FID
MNIST | OODA | $0.0018\pm 0.0002$ | $22.5\pm 8.3$ | $0.917\pm 0.031$ | $3.463\pm 0.729$
FODA | $0.0032\pm 0.0006$ | $98.5\pm 0.4$ | $0.558\pm 0.009$ | $0.158\pm 0.017$
HDA | $\mathbf{0.0292\pm 0.0071}$ | $\mathbf{99.3\pm 0.5}$ | $\mathbf{0.076\pm 0.003}$ | $\mathbf{0.042\pm 0.011}$
SVHN | OODA | $0.0026\pm 0.0004$ | $11.3\pm 1.2$ | $0.811\pm 0.009$ | $125.126\pm 2.838$
FODA | $0.0034\pm 0.0002$ | $100\pm 0$ | $0.252\pm 0.012$ | $111.694\pm 1.461$
HDA | $\mathbf{0.0132\pm 0.0021}$ | $\mathbf{100\pm 0}$ | $\mathbf{0.029\pm 0.001}$ | $\mathbf{91.102\pm 1.342}$
(a) Results
Dataset | Metric | $T(\text{HDA},\text{OODA})$ | $T(\text{HDA},\text{FODA})$
---|---|---|---
$t$ | $p-value$ | $t$ | $p-value$
MNIST | $p_{g}$ | $12.199$ | $3.871\times 10^{-10}$ | $11.539$ | $9.456\times 10^{-10}$
% of Valid AEs | $29.208$ | $1.286\times 10^{-16}$ | $3.951$ | $9.368\times 10^{-4}$
$\epsilon$ | $-85.391$ | $6.183\times 10^{-25}$ | $-160.667$ | $7.186\times 10^{-30}$
FID | $-14.838$ | $1.546\times 10^{-11}$ | $-18.116$ | $5.275\times 10^{-13}$
SVHN | $p_{g}$ | $15.680$ | $6.124\times 10^{-12}$ | $14.691$ | $1.826\times 10^{-11}$
% of Valid AEs | $233.745$ | $8.456\times 10^{-33}$ | $nan$ | $nan$
$\epsilon$ | $-273.086$ | $5.146\times 10^{-34}$ | $-58.563$ | $5.359\times 10^{-22}$
FID | $-34.273$ | $7.588\times 10^{-18}$ | $-32.825$ | $1.632\times 10^{-17}$
(b) Statistical Test
Next, we try to answer the difference between HDA testing and other
distribution-aware testing as summarised earlier (the amber route of Fig. 1).
We not only study the common evaluation metrics in earlier RQs, but also the
input validation method in (Dola et al., 2021), which flags the validity of
AEs according to a user-defined reconstruction probability threshold. Overall,
HDA is significantly better than OODA and FODA in four evaluation metrics, as
observed from the results of statistical test.
As shown in Table 8(b), HDA can select test seeds from much higher density
region on the global distribution and find more valid AEs than OODA. The
reason behind this is that OODA aims at detecting outliers—only AEs with lower
reconstruction probabilities (from the test seed) than the given threshold
will be marked as invalid test cases. While, HDA explicitly explores the high
density meanwhile error-prone regions by combining the global distribution and
local robustness indicators. In other words, HDA performs priority ordering
(according to the global distribution and local robustness) and then selects
the best, while OODA rules out the worst. As expected, FODA performs similarly
poorly as OODA in terms of $p_{g}$, since both use randomly selected seeds.
However, FODA has high proportion of valid AEs since the test cases are
directly sampled from the distribution in latent space.
Regarding the perceptual quality of detected AEs, HDA can always find AEs with
small pixel-level perturbations ($\epsilon$) in consideration of the
$r$-separation constraint, and with small FID thanks to the use of perceptual
quality metrics (MSE in this case) as objective functions. While OODA only
utilises the reconstruction probability (from VAE) to choose AEs, and FODA
directly samples test cases from VAE without any restrictions (thus may suffer
from the oracle problem, cf. Remark 13 later). Due to the compression nature
of generative models—they are good at extracting feature level information but
ignore pixel level information (Zhong et al., 2020), AEs detected by OODA and
FODA are all distant to the original test seeds, yielding large $\epsilon$ and
FID scores. Notably, the average distance $\epsilon$ between test seeds and
AEs detected by OODA and FODA are much (7$\sim$28 times) greater than the
$r$-separation constraints (cf. Table 1), leading to the potential oracle
issues of those AEs, for which we have the following remark:
###### Remark 5 (Oracle Issues of AEs Detected by OODA and FODA).
AEs detected by OODA and FODA are normally distant to the test seeds with a
perturbation distance even greater than the $r$-separation constraint.
Consequently, there is the risk that the perturbed image may not share the
same ground truth label of the test seed, and thus hard to determine the
ground truth label of the “AE”131313In quotes, because the perturbed image
could be a “benign example” with a correct predicted label (but different to
the test seed)..
Figure 9. Example AEs detected by different distribution-aware testing
methods. AEs detected by our HDA are indistinguishable from the original
images, while AEs detected by FODA and OODA are of low perceptual quality and
subject to the oracle issues noted by Remark 13.
To visualise the difference between AEs detected by HDA, FODA and OODA, we
present 4 examples in Fig. 9. We may observe the AEs detected by HDA are
almost indistinguishable from the original images. Moreover, the AEs by FODA
is a set of concrete evidence for Remark 13—it is actually quite hard to tell
what is the ground truth label of some perturbed image (e.g., the bottom left
one), while others appear to have a different label of the seed (e.g., the
bottom right one should be with a label “1” instead of “7”).
Answer to RQ2 on comparing with other distribution-aware testing: Compared to
OODA and FODA, the proposed HDA testing can detect more valid AEs, free of
oracle issues, with higher global probabilities and perception quality.
#### 4.2.3. RQ3
In earlier RQs, we have varied the datasets and model architectures to check
the effectiveness of HDA. In this RQ3, we are concern about HDA’s sensitivity
to DL models with different levels of robustness. Adversarial training may
greatly improve the robustness of DL models and is widely used as the defence
to adversarial attack. To this end, we apply HDA on both normally and
adversarially trained models (by (Madry et al., 2018) to be exact), and then
compare with three most representative adversarial attack methods—the most
classic FGSM, the most popular PGD, and the most advanced AutoAttack (Croce
and Hein, 2020). Experimental results are presented in Table 9(b).
Table 9. Evaluation of AEs generated by FGSM, PGD, AutoAttack and HDA on
normally and adversarially trained DL models (all results are averaged over
100 test seeds).
Model | | AE Detection
---
Method
Eval. Metric | MNIST | F.-MNIST | SVHN | CIFAR-10 | CelebA
Normally Trained | FGSM | $p_{g}$ | $0.0045\pm 0.0005$ | $0.0034\pm 0.0008$ | $0.0031\pm 0.0002$ | $0.0028\pm 0.0004$ | $0.0034\pm 0.0002$
AE Prop. | $0.413\pm 0.093$ | $0.797\pm 0.039$ | $0.727\pm 0.069$ | $0.893\pm 0.016$ | $0.989\pm 0.035$
FID | $1.022\pm 0.053$ | $4.134\pm 0.322$ | $114.256\pm 2.931$ | $104.877\pm 1.832$ | $88.401\pm 1.459$
PGD | $p_{g}$ | $0.0040\pm 0.0005$ | $0.0034\pm 0.0008$ | $0.0032\pm 0.0002$ | $0.0028\pm 0.0004$ | $0.0034\pm 0.0002$
AE Prop. | $0.422\pm 0.046$ | $1.000\pm 0.000$ | $0.984\pm 0.011$ | $1.000\pm 0.000$ | $0.989\pm 0.035$
FID | $0.705\pm 0.047$ | $2.938\pm 0.248$ | $107.370\pm 1.206$ | $101.157\pm 1.774$ | $88.401\pm 1.459$
AutoAttack | $p_{g}$ | $0.0042\pm 0.0004$ | $0.0034\pm 0.0008$ | $0.0032\pm 0.0002$ | $0.0028\pm 0.0004$ | $0.0035\pm 0.0002$
AE Prop. | $\mathbf{0.787\pm 0.075}$ | $1.000\pm 0.000$ | $0.984\pm 0.011$ | $1.000\pm 0.000$ | $1.000\pm 0.000$
FID | $0.795\pm 0.099$ | $4.962\pm 0.359$ | $106.780\pm 1.297$ | $101.173\pm 1.543$ | $90.385\pm 1.305$
HDA | $p_{g}$ | $\mathbf{0.0292\pm 0.0071}$ | $\mathbf{0.0224\pm 0.0087}$ | $\mathbf{0.0132\pm 0.0021}$ | $\mathbf{0.0406\pm 0.012}$ | $\mathbf{0.0124\pm 0.001}$
AE Prop. | $0.676\pm 0.127$ | $\mathbf{1.000\pm 0.000}$ | $\mathbf{0.989\pm 0.006}$ | $\mathbf{1.000\pm 0.000}$ | $\mathbf{1.000\pm 0.000}$
FID | $\mathbf{0.042\pm 0.011}$ | $\mathbf{0.016\pm 0.004}$ | $\mathbf{91.102\pm 1.342}$ | $\mathbf{65.829\pm 2.154}$ | $\mathbf{55.478\pm 1.399}$
Adversarially Trained | FGSM | $p_{g}$ | $0.0042\pm 0.0005$ | $0.0033\pm 0.0004$ | $0.0036\pm 0.0004$ | $0.0028\pm 0.0005$ | $0.0031\pm 0.0003$
AE Prop. | $0.033\pm 0.021$ | $0.208\pm 0.037$ | $0.323\pm 0.061$ | $0.455\pm 0.039$ | $0.404\pm 0.045$
FID | $1.064\pm 0.055$ | $4.898\pm 0.503$ | $135.730\pm 4.321$ | $113.106\pm 3.989$ | $108.070\pm 4.368$
PGD | $p_{g}$ | $0.0038\pm 0.0005$ | $0.0032\pm 0.0004$ | $0.0034\pm 0.0004$ | $0.0028\pm 0.0005$ | $0.0032\pm 0.0003$
AE Prop. | $0.014\pm 0.017$ | $0.201\pm 0.039$ | $0.507\pm 0.049$ | $0.455\pm 0.039$ | $0.372\pm 0.036$
FID | $0.762\pm 0.040$ | $3.871\pm 0.407$ | $124.388\pm 2.613$ | $113.106\pm 3.989$ | $111.931\pm 2.539$
AutoAttack | $p_{g}$ | $0.0031\pm 0.0004$ | $0.0031\pm 0.0004$ | $0.0034\pm 0.0004$ | $0.0028\pm 0.0005$ | $0.0031\pm 0.0003$
AE Prop. | $0.049\pm 0.021$ | $0.268\pm 0.038$ | $0.505\pm 0.049$ | $0.419\pm 0.039$ | $0.405\pm 0.045$
FID | $0.035\pm 0.009$ | $0.128\pm 0.046$ | $125.382\pm 2.315$ | $117.257\pm 3.750$ | $107.742\pm 4.169$
HDA | $p_{g}$ | $\mathbf{0.0214\pm 0.0075}$ | $\mathbf{0.0118\pm 0.0032}$ | $\mathbf{0.0154\pm 0.0021}$ | $\mathbf{0.0261\pm 0.0112}$ | $\mathbf{0.0103\pm 0.0007}$
AE Prop. | $\mathbf{0.368\pm 0.071}$ | $\mathbf{0.903\pm 0.021}$ | $\mathbf{0.960\pm 0.029}$ | $\mathbf{0.865\pm 0.051}$ | $\mathbf{0.968\pm 0.012}$
FID | $\mathbf{0.029\pm 0.003}$ | $\mathbf{0.058\pm 0.010}$ | $\mathbf{92.389\pm 1.152}$ | $\mathbf{62.804\pm 2.398}$ | $\mathbf{58.558\pm 0.895}$
(a) Results
Model | Dataset | Metric | $T(\text{HDA},\text{FGSM})$ | $T(\text{HDA},\text{PGD})$ | $T(\text{HDA},\text{AutoAttack})$
---|---|---|---|---|---
$t$ | $p-value$ | $t$ | $p-value$ | $t$ | $p-value$
Normally Trained | MNIST | $p_{g}$ | $10.97$ | $2.10\times 10^{-9}$ | $11.20$ | $1.53\times 10^{-9}$ | $11.12$ | $1.71\times 10^{-9}$
AE Prop. | $5.28$ | $5.05\times 10^{-5}$ | $5.95$ | $1.26\times 10^{-5}$ | $-2.38$ | $0.03$
FID | $-57.25$ | $8.04\times 10^{-22}$ | $-43.43$ | $1.12\times 10^{-19}$ | $-23.91$ | $4.34\times 10^{-15}$
F.-MNIST | $p_{g}$ | $6.88$ | $1.97\times 10^{-6}$ | $6.88$ | $1.97\times 10^{-6}$ | $6.88$ | $1.97\times 10^{-6}$
AE Prop. | $16.46$ | $2.70\times 10^{-12}$ | $nan$ | $nan$ | $nan$ | $nan$
FID | $-40.44$ | $4.01\times 10^{-19}$ | $-37.25$ | $1.72\times 10^{-18}$ | $-43.56$ | $1.06\times 10^{-19}$
SVHN | $p_{g}$ | $15.14$ | $1.10\times 10^{-11}$ | $14.99$ | $1.30\times 10^{-11}$ | $14.99$ | $1.30\times 10^{-11}$
AE Prop. | $11.96$ | $5.31\times 10^{-10}$ | $1.26$ | $0.22$ | $1.26$ | $0.22$
FID | $-22.71$ | $1.05\times 10^{-14}$ | $-28.51$ | $1.97\times 10^{-16}$ | $-26.56$ | $6.84\times 10^{-16}$
CIFAR-10 | $p_{g}$ | $9.96$ | $9.56\times 10^{-9}$ | $9.96$ | $9.56\times 10^{-9}$ | $9.96$ | $9.56\times 10^{-9}$
AE Prop. | $21.15$ | $3.67\times 10^{-14}$ | $nan$ | $nan$ | $nan$ | $nan$
FID | $-43.67$ | $1.02\times 10^{-19}$ | $-40.03$ | $4.79\times 10^{-19}$ | $-42.18$ | $1.89\times 10^{-19}$
CelebA | $p_{g}$ | $27.91$ | $2.87\times 10^{-16}$ | $27.91$ | $2.87\times 10^{-16}$ | $27.59$ | $3.49\times 10^{-16}$
AE Prop. | $0.99$ | $0.33$ | $0.99$ | $0.33$ | $nan$ | $nan$
FID | $-51.51$ | $5.34\times 10^{-21}$ | $-51.51$ | $5.34\times 10^{-21}$ | $-57.69$ | $6.99\times 10^{-22}$
Adversarially Trained | MNIST | $p_{g}$ | $7.24$ | $9.92\times 10^{-7}$ | $7.40$ | $7.24\times 10^{-7}$ | $7.71$ | $4.17\times 10^{-7}$
AE Prop. | $14.31$ | $2.83\times 10^{-11}$ | $15.33$ | $8.9\times 10^{-12}$ | $13.62$ | $6.37\times 10^{-11}$
FID | $-59.42$ | $4.12\times 10^{-22}$ | $-57.79$ | $6.81\times 10^{-22}$ | $-2.00$ | $0.06$
F.-MNIST | $p_{g}$ | $8.33$ | $1.36\times 10^{-7}$ | $8.43$ | $1.15\times 10^{-7}$ | $8.53$ | $9.70\times 10^{-8}$
AE Prop. | $51.66$ | $5.06\times 10^{-21}$ | $50.12$ | $8.70\times 10^{-21}$ | $46.25$ | $3.65\times 10^{-20}$
FID | $-30.42$ | $6.26\times 10^{-17}$ | $-29.62$ | $1.06\times 10^{-16}$ | $-4.70$ | $1.78\times 10^{-4}$
SVHN | $p_{g}$ | $17.46$ | $9.95\times 10^{-13}$ | $17.75$ | $7.47\times 10^{-13}$ | $17.75$ | $7.47\times 10^{-13}$
AE Prop. | $29.82$ | $8.90\times 10^{-17}$ | $25.16$ | $1.77\times 10^{-15}$ | $25.27$ | $1.64\times 10^{-15}$
FID | $-30.65$ | $5.50\times 10^{-17}$ | $-35.43$ | $4.19\times 10^{-18}$ | $-40.35$ | $4.17\times 10^{-19}$
CIFAR-10 | $p_{g}$ | $6.57$ | $3.57\times 10^{-6}$ | $6.57$ | $3.57\times 10^{-6}$ | $6.57$ | $3.57\times 10^{-6}$
AE Prop. | $20.19$ | $8.15\times 10^{-14}$ | $20.19$ | $8.15\times 10^{-14}$ | $21.97$ | $1.89\times 10^{-14}$
FID | $-34.18$ | $7.98\times 10^{-18}$ | $-34.18$ | $7.98\times 10^{-18}$ | $-38.69$ | $8.82\times 10^{-19}$
CelebA | $p_{g}$ | $29.90$ | $8.52\times 10^{-17}$ | $29.48$ | $1.09\times 10^{-16}$ | $29.90$ | $8.52\times 10^{-17}$
AE Prop. | $38.30$ | $1.05\times 10^{-18}$ | $49.67$ | $1.02\times 10^{-20}$ | $38.23$ | $1.09\times 10^{-18}$
FID | $-35.12$ | $4.93\times 10^{-18}$ | $-62.69$ | $1.58\times 10^{-22}$ | $-36.48$ | $2.51\times 10^{-18}$
(b) Statistical Test
As expected, after the adversarial training by (Madry et al., 2018), the
robustness of all five DL models are greatly improved. This can be observed
from the metric of AE Prop.: For all four methods, the proportion of AEs
detected in the set of test case is sharply decreased for adversarially
trained models. Nevertheless, the AE detection performance of HDA is less
affected by the adversarial training. HDA testing can maintain significantly
higher proportion of AEs in the test set for adversarially trained models,
compared with the state of art adversarial attack. This is evident from the
fact that there is an insignificant difference in AE Prop. between HDA and
other adversarial attacks for normally trained models, while the difference is
significant for adversarially trained models.
In terms of the probability density $p_{g}$ and perception quality measured by
FID on generated test cases, HDA significantly outperforms others for both
normally and adversarially trained models, as observed from the large t-scores
and p-values $<<0.05$ . This is unsurprising, since the rationales behind the
three adversarial attacks disregard the consideration of input data
distribution and perception quality.
Finally, we find that the measured $p_{g}$ on test cases detected by HDA
changes due to the variations in local robustness before and after the
adversarial training, yet it remained much higher than all other attack
methods.
Answer to RQ3: HDA is shown to be capable and superior to common adversarial
attacks when applied on DL models with different levels of robustness.
#### 4.2.4. RQ4
The ultimate goal of developing HDA testing is to improve the global
robustness of DL models. To this end, we refer to a validation set of 10000
test seeds. We fine-tune (Jeddi et al., 2021) the DL models with AEs detected
for validation set from different methods. Then, we calculate the train
accuracy, test accuracy and empirical global robustness before and after the
adversarial fine-tuning. Empirical global robustness is measured on a new set
of on-distribution AEs for validation set, different from the fine-tuning
data. Results are presented in Table 10(b).
Table 10. Evaluation of DL models’ train accuracy, test accuracy, and
empirical global robustness (based on 10000 on-distribution AEs) after
adversarial fine-tuning, using different number (N) of test cases.
AE Detection Method | Metric | MNIST | SVHN
---|---|---|---
N = 500 | N = 5000 | N = 50000 | N = 500 | N = 5000 | N = 50000
PGD Attack | Train Acc. | $98.26\%\pm 0.05\%$ | $98.10\%\pm 0.07\%$ | $97.70\%\pm 0.05\%$ | $94.85\%\pm 0.43\%$ | $92.76\%\pm 0.39\%$ | $94.02\%\pm 0.40\%$
Test Acc. | $97.64\%\pm 0.11\%$ | $97.05\%\pm 0.09\%$ | $96.90\%\pm 0.08\%$ | $93.32\%\pm 0.21\%$ | $81.43\%\pm 0.31\%$ | $63.81\%\pm 0.28\%$
$\mathcal{R}_{g}$ | $46.27\%\pm 0.02\%$ | $84.09\%\pm 0.05\%$ | $90.52\%\pm 0.05\%$ | $46.43\%\pm 0.11\%$ | $72.88\%\pm 0.10\%$ | $70.09\%\pm 0.12\%$
Cov. Guided Testing | Train Acc. | $99.89\%\pm 0.05\%$ | $99.61\%\pm 0.07\%$ | $99.12\%\pm 0.05\%$ | $95.73\%\pm 0.44\%$ | $93.65\%\pm 0.42\%$ | $95.76\%\pm 0.43\%$
Test Acc. | $98.93\%\pm 0.08\%$ | $98.77\%\pm 0.08\%$ | $98.41\%\pm 0.07\%$ | $94.11\%\pm 0.19\%$ | $85.66\%\pm 0.19\%$ | $76.43\%\pm 0.19\%$
$\mathcal{R}_{g}$ | $36.71\%\pm 0.04\%$ | $48.91\%\pm 0.04\%$ | $71.12\%\pm 0.05\%$ | $16.21\%\pm 0.11\%$ | $38.59\%\pm 0.16\%$ | $56.58\%\pm 0.12\%$
OODA | Train Acc. | $98.45\%\pm 0.06\%$ | $98.01\%\pm 0.06\%$ | $97.66\%\pm 0.07\%$ | $94.12\%\pm 0.38\%$ | $92.11\%\pm 0.39\%$ | $93.21\%\pm 0.38\%$
Test Acc. | $98.12\%\pm 0.10\%$ | $97.87\%\pm 0.09\%$ | $97.12\%\pm 0.08\%$ | $94.75\%\pm 0.19\%$ | $80.23\%\pm 0.18\%$ | $72.19\%\pm 0.19\%$
$\mathcal{R}_{g}$ | $40.21\%\pm 0.05\%$ | $45.69\%\pm 0.04\%$ | $51.12\%\pm 0.05\%$ | $11.21\%\pm 0.11\%$ | $16.23\%\pm 0.12\%$ | $18.21\%\pm 0.11\%$
FODA | Train Acc. | $98.44\%\pm 0.05\%$ | $98.18\%\pm 0.05\%$ | $97.32\%\pm 0.06\%$ | $94.66\%\pm 0.42\%$ | $92.75\%\pm 0.41\%$ | $94.11\%\pm 0.41\%$
Test Acc. | $97.87\%\pm 0.10\%$ | $97.43\%\pm 0.10\%$ | $97.11\%\pm 0.11\%$ | $92.13\%\pm 0.22\%$ | $82.32\%\pm 0.21\%$ | $78.19\%\pm 0.22\%$
$\mathcal{R}_{g}$ | $37.71\%\pm 0.04\%$ | $47.26\%\pm 0.05\%$ | $55.37\%\pm 0.05\%$ | $12.21\%\pm 0.12\%$ | $20.56\%\pm 0.12\%$ | $23.98\%\pm 0.11\%$
HDA | Train Acc. | $99.56\%\pm 0.07\%$ | $99.10\%\pm 0.07\%$ | $98.89\%\pm 0.06\%$ | $95.00\%\pm 0.33\%$ | $92.83\%\pm 0.39\%$ | $94.40\%\pm 0.40\%$
Test Acc. | $98.52\%\pm 0.09\%$ | $98.42\%\pm 0.08\%$ | $98.21\%\pm 0.08\%$ | $93.86\%\pm 0.24\%$ | $88.67\%\pm 0.25\%$ | $80.60\%\pm 0.22\%$
$\mathcal{R}_{g}$ | $\mathbf{89.67\%\pm 0.03\%}$ | $\mathbf{96.71\%\pm 0.06\%}$ | $\mathbf{99.12\%\pm 0.05\%}$ | $\mathbf{51.15\%\pm 0.09\%}$ | $\mathbf{86.88\%\pm 0.09\%}$ | $\mathbf{91.26\%\pm 0.05\%}$
(a) Results
Dataset | No. of Test Cases | Metric | T(HDA, PGD) | T(HDA, Cov.) | T(HDA, OODA) | T(HDA, FODA)
---|---|---|---|---|---|---
$t$ | $p-value$ | $t$ | $p-value$ | $t$ | $p-value$ | $t$ | $p-value$
MNIST | N=500 | Train Acc. | $47.79$ | $2.04\times 10^{-20}$ | $-12.13$ | $4.24\times 10^{-10}$ | $38.07$ | $1.17\times 10^{-18}$ | $41.17$ | $2.91\times 10^{-19}$
Test Acc. | $19.58$ | $1.39\times 10^{-13}$ | $-10.77$ | $2.83\times 10^{-9}$ | $9.40$ | $2.29\times 10^{-8}$ | $15.28$ | $9.47\times 10^{-12}$
$\mathcal{R}_{g}$ | $3806.43$ | $1.31\times 10^{-54}$ | $3349.48$ | $1.31\times 10^{-53}$ | $2682.35$ | $7.12\times 10^{-52}$ | $3286.24$ | $1.84\times 10^{-53}$
N=5000 | Train Acc. | $31.94$ | $2.64\times 10^{-17}$ | $-16.29$ | $3.21\times 10^{-12}$ | $37.39$ | $1.62\times 10^{-18}$ | $33.82$ | $9.61\times 10^{-18}$
Test Acc. | $35.98$ | $3.20\times 10^{-18}$ | $-9.78$ | $1.25\times 10^{-8}$ | $14.44$ | $2.42\times 10^{-11}$ | $24.45$ | $2.94\times 10^{-15}$
$\mathcal{R}_{g}$ | $510.97$ | $6.52\times 10^{-39}$ | $2096.17$ | $6.03\times 10^{-50}$ | $2237.38$ | $1.86\times 10^{-50}$ | $2002.17$ | $1.38\times 10^{-49}$
N=50000 | Train Acc. | $48.18$ | $1.76\times 10^{-20}$ | $-9.31$ | $2.64\times 10^{-8}$ | $42.19$ | $1.88\times 10^{-19}$ | $58.51$ | $5.45\times 10^{-22}$
Test Acc. | $36.62$ | $2.35\times 10^{-18}$ | $-5.95$ | $1.25\times 10^{-5}$ | $30.47$ | $6.10\times 10^{-17}$ | $25.57$ | $1.33\times 10^{-15}$
$\mathcal{R}_{g}$ | $384.60$ | $1.08\times 10^{-36}$ | $1252.19$ | $6.42\times 10^{-46}$ | $2146.63$ | $3.93\times 10^{-50}$ | $1956.56$ | $2.08\times 10^{-49}$
SVHN | N=500 | Train Acc. | $0.88$ | $0.39$ | $-4.19$ | $5.42\times 10^{-4}$ | $5.53$ | $2.99\times 10^{-5}$ | $2.01$ | $0.06$
Test Acc. | $5.35$ | $4.34\times 10^{-5}$ | $-2.58$ | $0.02$ | $-9.19$ | $3.20\times 10^{-8}$ | $16.80$ | $1.90\times 10^{-12}$
$\mathcal{R}_{g}$ | $105.02$ | $1.50\times 10^{-26}$ | $777.40$ | $3.42\times 10^{-42}$ | $888.65$ | $3.08\times 10^{-43}$ | $820.93$ | $1.28\times 10^{-42}$
N=5000 | Train Acc. | $0.40$ | $0.69$ | $-4.52$ | $2.63\times 10^{-4}$ | $4.13$ | $6.31\times 10^{-4}$ | $0.45$ | $0.66$
Test Acc. | $57.49$ | $7.46\times 10^{-22}$ | $30.31$ | $6.67\times 10^{-17}$ | $86.64$ | $4.77\times 10^{-25}$ | $61.50$ | $2.23\times 10^{-22}$
$\mathcal{R}_{g}$ | $329.07$ | $1.79\times 10^{-35}$ | $831.84$ | $1.01\times 10^{-42}$ | $1489.43$ | $2.83\times 10^{-47}$ | $1398.15$ | $8.83\times 10^{-47}$
N=50000 | Train Acc. | $2.12$ | $0.05$ | $-7.32$ | $8.43\times 10^{-7}$ | $6.82$ | $2.19\times 10^{-6}$ | $1.60$ | $0.13$
Test Acc. | $149.10$ | $2.75\times 10^{-29}$ | $45.36$ | $5.16\times 10^{-20}$ | $91.49$ | $1.79\times 10^{-25}$ | $24.50$ | $1.83\times 10^{-15}$
$\mathcal{R}_{g}$ | $514.96$ | $5.67\times 10^{-39}$ | $843.60$ | $7.86\times 10^{-43}$ | $1911.81$ | $3.16\times 10^{-49}$ | $1760.80$ | $1.39\times 10^{-48}$
(b) Statistical Test
We first observe that adversarial fine-tuning is effective to improve the DL
models’ empirical global robustness, measured by the prediction accuracy on
AEs, detected from normally trained models ($\mathcal{R}_{g}$), while
compromising the train/test accuracy as expected (in contrast to normal
training in Table 1). In most cases, DL models enhanced by HDA testing suffer
from the least drop of generalisation, compared with PGD attack, OODA and
FODA. This can been from significant difference of Test Acc. between HDA and
others. The reason behind this is that HDA testing targets at AEs from high
density regions on distributions, usually with small prediction loss, shown in
Fig. 6. Thus, eliminating AEs detected by HDA testing requires relatively
minor adjustment to DL’s models, the generalisation of which can be easily
tampered during the fine-tuning with new samples.
In terms of empirical global robustness, HDA testing detects AEs around test
seeds from the high global distribution region, which are more significant to
the global robustness improvement. When comparing HDA with other methods, the
p-values of $\mathcal{R}_{g}$ is way smaller than 0.05, indicating HDA
contributes more to the global robustness improvement than others. It also can
be seen that with 5000 test cases generated by utilising 1000 test seeds, the
HDA testing can improve empirical global robustness to nearly or over $90\%$,
very closed to the fine-tuning with 50000 test cases from 10000 test seeds.
This means the distribution-based test seeds selection is more efficient than
random test seeds selection. Moreover, even fine-tuning with 50000 test cases,
leveraging all the test seeds in the validation set, HDA is still better than
others, due to the consideration of local distributions (approximated by
perceptual quality metrics). We notice that PGD-based adversarial fine-tuning
minimises the maximum prediction loss within the local region, which is also
effective to eliminate the high perceptual quality AEs, but sacrificing more
train/test accuracy. DL models fined-tuned with HDA testing achieve the best
balance between the generalisation and global robustness.
Answer to RQ4: Compared with adversarial attack and coverage-guide testing,
HDA contributes more to the growth of global robustness, while mitigating the
drop of train/test accuracy during adversarial fine-tuning.
## 5\. Threats to Validity
### 5.1. Internal Validity
Threats may arise due to bias in establishing cause-effect relationships,
simplifications and assumptions made in our experiments. In what follows, we
list the main threats of each research question and discuss how we mitigate
them.
#### 5.1.1. Threats from HDA Techniques
In RQ1, both the performance of the VAE-Encoder and KDE are threats. For the
former, it is mitigated by using four established quality metrics (in Table 3)
on evaluating dimensionality reduction techniques and compared to the common
PCA method. It is known that KDE performs poorly with high-dimensional data
and works well when the data dimension is modest (Scott, 1991; Liu et al.,
2007). The data dimensions in our experiments are relatively low given the
datasets have been compressed by VAE-Encoder, which mitigates the second
threat. When studying the local robustness indicators, quantifying both the
indicators and the local robustness may subject to errors, for which we reduce
them by carefully inspecting the correctness of the script on calculating the
indicators and invoking a reliable local robustness estimator (Webb et al.,
2019) with fine-tuned hyper-parameters. For using two-step GA to generate
local test cases, a threat arises by the calculation of norm ball radius,
which has been mitigated by $r$-separation distance presented in the paper
(Yang et al., 2020). Also, the threat related to estimating the local
distribution is mitigated by quantifying its three indicators (MSE, PSNR and
SSIM) that are typically used in representing image-quality by human-
perception.
#### 5.1.2. Threats from AEs’ Quality Measurement
A threat for RQ1, RQ2 and RQ3 (when examining how effective our method models
the global distribution and local distribution respectively) is the use of FID
as a metric, quantifying how “similar” two image datasets are. Given FID is
currently the standard metric for this purpose, this threat is sufficiently
mitigated now and can be further mitigated with new metrics in future. RQ2
includes the method of validating AEs developed in (Dola et al., 2021), which
utilises generative models and OOD techniques to flag valid AEs with
reconstruction probabilities greater than a threshold. The determination of
this threshold is critical, thus poses a threat to RQ2. To mitigate it, we use
same settings across all the experiments for fair comparisons.
#### 5.1.3. Threats from Adversarial Training and Fine-Tuning
In RQ3 and RQ4, the first threat rises from the fact that adversarial training
and adversarial fine-tuning will sacrifice the DL model’s generalisation for
robustness. Since the training process is data-driven and of black-box nature,
it is hard to know how the predication of a single data-point will be
affected, while it is meaningless to study the robustness of an incorrectly
predicted seed. To mitigate this threat when we compare the robustness before
and after adversarial training/fine-tuning, we select enough number of seeds
and check the prediction of each selected seed (filtering out incorrect ones
if necessary) to make sure test seeds are always predicted correctly. For the
global robustness computation in RQ4, we refer to a validation dataset, where
a threat may arise if the empirical result based on the validation dataset
cannot represent the global robustness. To mitigate it, we synthesise the
validation set with enough data—10000 inputs sampled from global distribution.
We further attack the validation dataset to find an AE per seed according to
the local distribution. Thus, DL models’ prediction accuracy on this dataset
empirically represents the global robustness as defined. For the
training/fine-tuning to be effective, we need a sufficient number of AEs to
augment the training dataset. A threat may arise due to a small proportion of
AEs in the augmented training dataset (the DL model will be dominated by the
original training data during the training/fine-tuning). To mitigate such a
threat, we generate a large proportion of AEs in our experiments.
### 5.2. External Validity
Threats might challenge the generalisability of our findings, e.g. the number
of models and datasets considered for experimentation; thus we mitigate these
threats as follows. All our experiments are conducted on 5 popular benchmark
datasets, covering 5 typical types of DL models, cf. Table 1. Experimental
results on the effectiveness of each stage in our framework are all based on
averaging a large number of samples, reducing the random noise in the
experiments. In two-step GA based test case generation, a wide range of the
$\alpha$ parameter has been studied showing converging trends. Finally, we
enable replication by making all experimental results publicly
available/reproducible on our project website to further mitigate the threat.
## 6\. Conclusion & Future Work
In this paper, we propose a HDA testing approach for detecting AEs that
considers both the data distribution (thus with higher operational impact
assuming the training data statistically representing the future inputs) and
perceptual quality (thus looks natural and realistic to humans). The key
novelty lies in the hierarchical consideration of two levels of distributions.
To the best of our knowledge, it is the first DL testing approach that
explicitly and collectively models both (i) the feature-level information when
selecting test seeds and (ii) pixel-level information when generating local
test cases. To this end, we have developed a tool chain that provides
technical solutions for each stage of our HDA testing. Our experiments not
only show the effectiveness of each testing stage, but also the overall
advantages of HDA testing over state-of-the-arts. From a software
engineering’s perspective, HDA is cost-effective (by focusing on practically
meaningful AEs), flexible (with end-to-end, black-box technical solutions) and
may effectively contribute to the robustness growth of the DL software under
testing.
The purpose of detecting AEs is to fix them. Although existing DL
retraining/repairing techniques (e.g. (Jeddi et al., 2021) used in RQ4 and
(Wang et al., 2019; Yu et al., 2022)) may satisfy the purpose to some extent,
bespoke “debugging” methods with more emphasise on the feature-distribution
and perceptual quality can be integrated into our framework in a more
efficient way. To this end, our important future work is to close the loop of
“detect-fix-assess” as depicted in (Zhao et al., 2021b) and then organise all
generated evidence as safety cases (Zhao et al., 2020; Dong et al., 2023).
Finally, same as other distribution-aware testing methods, we assume the input
data distribution is same as the training data distribution. To relax this
assumption, we plan to take distribution-shift into consideration in future
versions of HDA. Distribution-aware testing for systematically detecting
explanation AEs (Huang et al., 2023) will also be explored.
###### Acknowledgements.
This work is supported by the U.K. DSTL (through the project of Safety
Argument for Learning-enabled Autonomous Underwater Vehicles) and the U.K.
EPSRC (through End-to-End Conceptual Guarding of Neural Architectures
[EP/T026995/1]). Xingyu Zhao and Alec Banks’ contribution to the work is
partially supported through Fellowships at the Assuring Autonomy International
Programme. This project has received funding from the European Union’s Horizon
2020 research and innovation programme under grant agreement No 956123. We
thank all three anonymous reviewers whose comments helped improve the paper.
This document is an overview of U.K. MOD (part) sponsored research and is
released for informational purposes only. The contents of this document should
not be interpreted as representing the views of the U.K. MOD, nor should it be
assumed that they reflect any current or future U.K. MOD policy. The
information contained in this document cannot supersede any statutory or
contractual requirements or liabilities and is offered without prejudice or
commitment.
## References
* (1)
* Akoglu (2018) Haldun Akoglu. 2018\. User’s guide to correlation coefficients. _Turkish journal of emergency medicine_ 18, 3 (2018), 91–93.
* Alzantot et al. (2019) Moustafa Alzantot, Yash Sharma, Supriyo Chakraborty, Huan Zhang, Cho-Jui Hsieh, and Mani B Srivastava. 2019\. Genattack: Practical black-box attacks with gradient-free optimization. In _Proceedings of the Genetic and Evolutionary Computation Conference_. 1111–1119.
* Attaoui et al. (2023) Mohammed Attaoui, Hazem Fahmy, Fabrizio Pastore, and Lionel Briand. 2023. Black-Box Safety Analysis and Retraining of DNNs Based on Feature Extraction and Clustering. _ACM Trans. Softw. Eng. Methodol._ 32, 3, Article 79 (2023), 40 pages.
* Berend (2021) David Berend. 2021\. Distribution Awareness for AI System Testing. In _43rd IEEE/ACM Int. Conf. on Software Engineering: Companion Proceedings, ICSE Companion 2021, Madrid, Spain, May 25-28, 2021_. IEEE, 96–98.
* Berend et al. (2020) David Berend, Xiaofei Xie, Lei Ma, Lingjun Zhou, Yang Liu, Chi Xu, and Jianjun Zhao. 2020. Cats Are Not Fish: Deep Learning Testing Calls for out-of-Distribution Awareness. In _Proc. of the 35th IEEE/ACM Int. Conference on Automated Software Engineering_ _(ASE’20)_. ACM, New York, NY, USA, 1041–1052. https://doi.org/10.1145/3324884.3416609
* Byun and Rayadurgam (2020) Taejoon Byun and Sanjai Rayadurgam. 2020. Manifold-based Test Generation for Image Classifiers. In _ICSE ’20: 42nd Int. Conference on Software Engineering, Workshops_. ACM, 221\. https://doi.org/10.1145/3387940.3391460
* Byun et al. (2020) Taejoon Byun, Abhishek Vijayakumar, Sanjai Rayadurgam, and Darren Cofer. 2020. Manifold-based Test Generation for Image Classifiers. In _Int. Conf. On Artificial Intelligence Testing (AITest)_. IEEE, Oxford, UK, 15–22.
* Croce and Hein (2020) Francesco Croce and Matthias Hein. 2020. Reliable evaluation of adversarial robustness with an ensemble of diverse parameter-free attacks. In _Proc. of the 37th Int. Conf. on Machine Learning (ICML’20)_ , Vol. 119. PMLR, 2206–2216.
* Deb et al. (2002) Kalyanmoy Deb, Amrit Pratap, Sameer Agarwal, and TAMT Meyarivan. 2002. A fast and elitist multiobjective genetic algorithm: NSGA-II. _IEEE transactions on evolutionary computation_ 6, 2 (2002), 182–197.
* Dola et al. (2021) Swaroopa Dola, Matthew B. Dwyer, and Mary Lou Soffa. 2021\. Distribution-Aware Testing of Neural Networks Using Generative Models. In _IEEE/ACM 43rd Int. Conference on Software Engineering_ _(ICSE’21)_. IEEE, Madrid, Spain, 226–237.
* Dong et al. (2023) Yi Dong, Wei Huang, Vibhav Bharti, Victoria Cox, Alec Banks, Sen Wang, Xingyu Zhao, Sven Schewe, and Xiaowei Huang. 2023. Reliability Assessment and Safety Arguments for Machine Learning Components in System Assurance. _ACM Trans. Embed. Comput. Syst._ 22, 3 (2023).
* Du et al. (2019) Xiaoning Du, Xiaofei Xie, Yi Li, Lei Ma, Yang Liu, and Jianjun Zhao. 2019. Deepstellar: Model-based quantitative analysis of stateful deep learning systems. In _Proc. of the 27th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering_. 477–487.
* Dunn et al. (2020) Isaac Dunn, Laura Hanu, Hadrien Pouget, Daniel Kroening, and Tom Melham. 2020. Evaluating robustness to context-sensitive feature perturbations of different granularities. _arXiv preprint arXiv:2001.11055_ (2020).
* Dunn et al. (2021) Isaac Dunn, Hadrien Pouget, Daniel Kroening, and Tom Melham. 2021. Exposing Previously Undetectable Faults in Deep Neural Networks. In _ACM SIGSOFT Int. Symposium on Software Testing and Analysis (ISSTA’21)_. in press.
* Gonzalez and Woods (1992) Rafael C. Gonzalez and Richard E. Woods. 1992. _Digital image processing_. 793 pages.
* Goodfellow et al. (2015) Ian J. Goodfellow, Jonathon Shlens, and Christian Szegedy. 2015. Explaining and Harnessing Adversarial Examples. In _3rd Int. Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, Conference Track Proceedings_.
* Harel-Canada et al. (2020) Fabrice Harel-Canada, Lingxiao Wang, Muhammad Ali Gulzar, Quanquan Gu, and Miryung Kim. 2020\. Is Neuron Coverage a Meaningful Measure for Testing Deep Neural Networks?. In _Proc. of the 28th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering_. ACM, 851–862.
* Heusel et al. (2017) Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. 2017\. GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium. In _Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017_. 6626–6637.
* Hosseini and Poovendran (2018) Hossein Hosseini and Radha Poovendran. 2018. Semantic Adversarial Examples. In _2018 IEEE Conference on Computer Vision and Pattern Recognition Workshops, CVPR Workshops 2018_. 1614–1619.
* Huang et al. (2022) Wei Huang, Youcheng Sun, Xingyu Zhao, James Sharp, Wenjie Ruan, Jie Meng, and Xiaowei Huang. 2022. Coverage-Guided Testing for Recurrent Neural Networks. _IEEE Transactions on Reliability_ 71, 3 (2022), 1191–1206.
* Huang et al. (2023) Wei Huang, Xingyu Zhao, Gaojie Jin, and Xiaowei Huang. 2023\. SAFARI: Versatile and Efficient Evaluations for Robustness of Interpretability. In _IEEE/CVF Int. Conf. on Computer Vision (ICCV’23)_.
* Huang et al. (2020) Xiaowei Huang, Daniel Kroening, Wenjie Ruan, and et al. 2020\. A survey of safety and trustworthiness of deep neural networks: Verification, testing, adversarial attack and defence, and interpretability. _Computer Science Review_ 37 (2020), 100270.
* Huang et al. (2017) Xiaowei Huang, Marta Kwiatkowska, Sen Wang, and Min Wu. 2017\. Safety verification of deep neural networks. In _Computer Aided Verification_ _(LNCS, Vol. 10426)_. Springer International Publishing, Cham, 3–29.
* Jeddi et al. (2021) Ahmadreza Jeddi, Mohammad Javad Shafiee, and Alexander Wong. 2021. A simple fine-tuning is all you need: Towards robust deep learning via adversarial fine-tuning. In _Workshop on Adversarial Machine Learning in Real-World Computer Vision Systems and Online Challenges (AML-CV) @ CVPR’21_. 1–5.
* Kang et al. (2020) Sungmin Kang, Robert Feldt, and Shin Yoo. 2020. Sinvad: Search-based image space navigation for dnn image classifier test input generation. In _Proc. of the IEEE/ACM 42nd Int. Conf. on Software Engineering Workshops_. 521–528.
* Konak et al. (2006) Abdullah Konak, David W Coit, and Alice E Smith. 2006\. Multi-objective optimization using genetic algorithms: A tutorial. _Reliability engineering & system safety_ 91, 9 (2006), 992–1007.
* Lane et al. (2016) David Lane, David Bisset, Rob Buckingham, Geoff Pegman, and Tony Prescott. 2016. _New foresight review on robotics and autonomous systems_. Technical Report No. 2016.1. LRF. 65 pages.
* Lipowski and Lipowska (2012) Adam Lipowski and Dorota Lipowska. 2012. Roulette-wheel selection via stochastic acceptance. _Physica A: Statistical Mechanics and its Applications_ 391, 6 (2012), 2193–2196.
* Liu et al. (2007) Han Liu, John Lafferty, and Larry Wasserman. 2007. Sparse nonparametric density estimation in high dimensions using the rodeo. In _Artificial Intelligence and Statistics_. PMLR, 283–290.
* Liu et al. (2019) Yang Liu, Eunice Jun, Qisheng Li, and Jeffrey Heer. 2019\. Latent space cartography: Visual analysis of vector space embeddings. In _Computer graphics forum_ , Vol. 38. Wiley Online Library, 67–78.
* Lokerse et al. (1995) S. H. Lokerse, L. P. J. Veelenturf, and J. G. Beltman. 1995\. Density Estimation Using SOFM and Adaptive Kernels. In _Neural Networks: Artificial Intelligence and Industrial Applications - Proceedings of the Third Annual SNN Symposium on Neural Networks, Nijmegen, The Netherlands, September 14-15, 1995_. Springer, 203–206.
* Ma et al. (2018) Lei Ma, Felix Juefei-Xu, Fuyuan Zhang, Jiyuan Sun, Minhui Xue, Bo Li, Chunyang Chen, Ting Su, Li Li, Yang Liu, et al. 2018\. Deepgauge: Multi-granularity testing criteria for deep learning systems. In _Proce. of the 33rd ACM/IEEE Int. Conference on Automated Software Engineering (ASE’18)_. 120–131.
* Madry et al. (2018) Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. 2018\. Towards Deep Learning Models Resistant to Adversarial Attacks. In _6th Int. Conference on Learning Representations, ICLR 2018, Vancouver, BC, Canada, April 30 - May 3, 2018, Conference Track Proceedings_. OpenReview.net.
* Pedregosa et al. (2011) F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. 2011. Scikit-learn: Machine Learning in Python. _Journal of Machine Learning Research_ 12 (2011), 2825–2830.
* Pei et al. (2017) Kexin Pei, Yinzhi Cao, Junfeng Yang, and Suman Jana. 2017\. DeepXplore: Automated Whitebox Testing of Deep Learning Systems. In _Proceedings of the 26th Symposium on Operating Systems Principles_. ACM, 1–18.
* Riccio et al. (2021) Vincenzo Riccio, Nargiz Humbatova, Gunel Jahangirova, and Paolo Tonella. 2021. Deepmetis: Augmenting a deep learning test set to increase its mutation score. In _36th IEEE/ACM Int. Conf. on Automated Software Engineering_. 355–367.
* Riccio and Tonella (2020) Vincenzo Riccio and Paolo Tonella. 2020. Model-based exploration of the frontier of behaviours for deep learning system testing. In _Proceedings of the 28th ACM Joint Meeting on European Software Engineering Conference and Symposium on the Foundations of Software Engineering_. 876–888.
* Rosenberg and Hirschberg (2007) Andrew Rosenberg and Julia Hirschberg. 2007. V-Measure: A Conditional Entropy-Based External Cluster Evaluation Measure. In _EMNLP-CoNLL 2007, Proceedings of the 2007 Joint Conference on Empirical Methods in Natural Language Processing and Computational Natural Language Learning, June 28-30, 2007, Prague, Czech Republic_. ACL, 410–420.
* Ruan et al. (2018) Wenjie Ruan, Xiaowei Huang, and Marta Kwiatkowska. 2018\. Reachability Analysis of Deep Neural Networks with Provable Guarantees. In _Proceedings of the Twenty-Seventh Int. Joint Conference on Artificial Intelligence, IJCAI-18_. Int. Joint Conferences on Artificial Intelligence Organization, 2651–2659.
|
# Visibility Aware Human-Object Interaction Tracking from Single RGB Camera
Xianghui Xie Bharat Lal Bhatnagar Gerard Pons-Moll
University of Tübingen, Tübingen AI Center, Germany
Max Planck Institute for Informatics, Saarland Informatics Campus, Germany
{xxie<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Capturing the interactions between humans and their environment in 3D is
important for many applications in robotics, graphics, and vision. Recent
works to reconstruct the 3D human and object from a single RGB image do not
have consistent relative translation across frames because they assume a fixed
depth. Moreover, their performance drops significantly when the object is
occluded. In this work, we propose a novel method to track the 3D human,
object, contacts between them, and their relative translation across frames
from a single RGB camera, while being robust to heavy occlusions. Our method
is built on two key insights. First, we condition our neural field
reconstructions for human and object on per-frame SMPL model estimates
obtained by pre-fitting SMPL to a video sequence. This improves neural
reconstruction accuracy and produces coherent relative translation across
frames. Second, human and object motion from visible frames provides valuable
information to infer the occluded object. We propose a novel transformer-based
neural network that explicitly uses object visibility and human motion to
leverage neighbouring frames to make predictions for the occluded frames.
Building on these insights, our method is able to track both human and object
robustly even under occlusions. Experiments on two datasets show that our
method significantly improves over the state-of-the-art methods. Our code and
pretrained models are available at: https://virtualhumans.mpi-
inf.mpg.de/VisTracker.
## 1 Introduction
Figure 1: From a monocular RGB video, our method tracks the human, object and
contacts between them even under occlusions.
Perceiving and understanding human as well as their interaction with the
surroundings has lots of applications in robotics, gaming, animation and
virtual reality etc. Accurate interaction capture is however very hard. Early
works employ high-end systems such as dense camera arrays [37, 8, 17] that
allow accurate capture but are expensive to deploy. Recent works [6, 36, 34]
reduce the requirement to multi-view RGBD cameras but it is still complicated
to setup the full capture system hence is not friendly for consumer-level
usage. This calls for methods that can capture human-object interaction from a
single RGB camera, which is more convenient and user-friendly.
However, reasoning about the 3D human and object from monocular RGB images is
very challenging. The lack of depth information makes the predictions
susceptible to depth-scale ambiguity, leading to temporally incoherent
tracking. Furthermore, the object or human can get heavily occluded, making
inference very hard. Prior work PHOSA [91] relies on hand-crafted heuristics
to reduce the ambiguity but such heuristic-based method is neither very
accurate nor scalable. More recently, CHORE [80] combines neural field
reconstructions with model based fitting obtaining promising results. However,
CHORE, assumes humans are at a fixed depth from the camera and predicts scale
alone, thereby losing the important _relative translation_ across frames.
Another limitation of CHORE is that it is not robust under occlusions as
little information is available from single-frame when the object is barely
visible. Hence CHORE often fails in these cases, see Fig. 3.
In this work, we propose the first method that can track both human and object
accurately from monocular RGB videos. Our approach combines neural field
predictions and model fitting, which has been consistently shown to be more
effective than directly regressing pose [4, 6, 80, 5]. In contrast to existing
neural field based reconstruction methods [61, 80], we can do tracking
including inference of relative translation. Instead of assuming a fixed
depth, we condition the neural field reconstructions (for object and human) on
per frame SMPL estimates (SMPL-T) including translation in camera space
obtained by pre-fitting SMPL to the video sequence. This results in coherent
translation and improved neural reconstruction. In addition, we argue that
during human-object interaction, the object motion is highly correlated with
the human motion, which provides us valuable information to recover the object
pose even when it is occluded (see Fig. 1 column 3-4). To this end, we propose
a novel transformer based network that leverages the human motion and object
motion from nearby visible frames to predict the object pose under heavy
occlusions.
We evaluate our method on the BEHAVE [6] and InterCap dataset [35].
Experiments show that our method can robustly track human, object and
realistic contacts between them even under heavy occlusions and significantly
outperforms the currrent state of the art method, CHORE [80]. We further
ablate the proposed _SMPL-T conditioning_ and _human and visibility aware_
object pose prediction network and demonstrate that they are key for accurate
human-object interaction tracking.
In summary, our key contributions include:
* •
We propose the first method that can jointly track full-body human interacting
with a movable object from a monocular RGB camera.
* •
We propose _SMPL-T conditioned interaction fields_ , predicted by a neural
network that allows consistent 4D tracking of human and object.
* •
We introduce a novel _human and visibility aware_ object pose prediction
network along with an object visibility prediction network that can recover
object poses even under heavy occlusions.
* •
Our code and pretrained models are publicly available to foster future
research in this direction.
Figure 2: Given an input RGB sequence of a human interacting with an object
and their corresponding human-object masks (A), we aim to reconstruct and
track the 3D human, object and the contacts between them (E). Our first key
idea is a SMPL-T conditioned interaction field network (SIF-Net, details in
Sec. 3.3) that predicts neural fields conditioned on estimated SMPL meshes in
camera space (col. B, SMPL-T, details in Sec. 3.2). SMPL-T conditioning
provides us temporally consistent relative translation, which is important for
coherent 4D tracking. Our second key insight is to predict object pose under
occlusions (D) leveraging human motion and object visibility information
(HVOP-Net, details in Sec. 3.4). This prediction provides robust object
tracking for frames with heavy occlusion. We then jointly optimize human and
object (details in Sec. 3.5) to satisfy image observations and contact
constraints.
## 2 Related Work
In this section, we first review recent works that deal with human or object
pose estimation and tracking separately. We then discuss recent progresses
that model human and object interactions and works that deal with occlusions
explicitly.
Human or object pose estimation and tracking. After the introduction of SMPL
[49] body model, tremendous progress has been made in human mesh recovery
(HMR) from single images [55, 7, 52, 3, 2, 43, 18] or videos [89, 56, 57, 40,
20]. We refer readers to a recent review of HMR methods in [70]. On the other
hand, deep learning method has also significantly improved object 6D pose
estimation from single RGB images [73, 50, 25, 45, 22, 53, 33]. However,
object pose tracking has received less attention and most works focus on RGBD
inputs [76, 65, 94, 21, 75]. Two works explore the camera localization ideas
from SLAM communities and can track object from RGB videos [48, 68].
Nevertheless, they heavily rely on visual evidence and the performance is
unknown under heavy occlusions. They also do not track human-object
interactions.
Human-object interaction. Modelling human object interaction is an emerging
research topic in recent years. Hand-object interaction is studied with works
modelling hand-object interaction from RGB [38, 19, 30, 24, 85], RGBD [9, 11,
28] or 3D inputs [69, 10, 95, 54]. There are also works that model human
interacting with a static scene [29, 34, 63, 77, 87, 27, 86] or deformable
surface [46]. More recently, the release of BEHAVE[6] and InterCap[35]
datasets allows bench-marking of full-body interacting with a movable object.
However, human-object interaction capture usually deploys multi-view RGB [67]
or RGBD[6, 34, 35, 93, 28, 9, 28, 20] cameras. Only a few works [91, 80, 74]
reconstruct dynamic human and object from monocular RGB input and our
experiments show that they are not suitable for tracking.
Pose estimation under occlusion. Most existing methods assume occlusion-free
input images hence are not robust under occlusions. Only a few methods address
human pose estimation under partial occlusions [41, 92, 58, 59, 26, 42] or
long term occlusions [89]. For object pose estimation, pixel-wise voting [53]
and self-occlusion [22] are explored for more robust prediction under
occlusions. More recently, TP-AE [94] predicts object occlusion ratio to guide
pose estimation but relies on depth input. Although being impressive on
separate human or object pose estimation, these methods do not reason about
human-object interaction. Our method is the first one that takes both human
and object visibility into account for interaction tracking.
## 3 Method
We present a novel method for jointly tracking the human, the object and the
contacts between them, in 3D, from a monocular RGB video. The first main
challenge in monocular tracking is the estimation of human and object
translations in camera space due to the depth-scale ambiguity problem.
Existing method, CHORE [80], reconstructs human and object at a fixed depth to
the camera, leading to inconsistent 3D translation across frames. Our key idea
is to fit a SMPL model with single shape parameters to a video sequence to
obtain consistent relative translation across frames. We call the estimated
SMPL as SMPL-T and describe it in more details in Sec. 3.2. Based on the
estimated SMPL-T, we then jointly model the 3D human, object and the
interactions using our proposed _SMPL-T conditioned Interaction Fields_
network (SIF-Net, Sec. 3.3)
Object tracking from a single-frame only, is difficult when the object is
barely visible. Hence we introduce a _Human and Visibility aware Object Pose
Network_ (HVOP-Net) that leverages human and object motion from visible frames
to recover the object under occlusion (Sec. 3.4). We then use the SIF-Net and
HVOP-Net outputs to optimize SMPL model and object pose parameters of this
sequence to satisfy neural prediction and image observations (Sec. 3.5). An
overview of our approach can be found in Fig. 2.
### 3.1 Preliminaries
In this work, we focus on a single human interacting with an object, which is
a common setting in other hand-object interaction [85, 69, 95] and full body-
object interaction works [80, 6, 35]. We represent human using the SMPL [49]
body model $H(\boldsymbol{\theta},\boldsymbol{\beta})$ that parameterises the
3D human mesh using pose $\boldsymbol{\theta}$ (including global translation)
and shape $\boldsymbol{\beta}$ parameters. The object is represented by a
known mesh template and we estimate the rotation $\mathbf{R}^{o}\in SO(3)$ and
translation $\mathbf{t}^{o}\in\mathbb{R}^{3}$ parameters. Given a video
sequence $\\{\mathbf{I}_{1},...,\mathbf{I}_{T}\\}$ where
$\mathbf{I}_{i}\in\mathbb{R}^{H\times W\times 5}$ (RGB, human and object
masks), our goal is to estimate the SMPL shape $\boldsymbol{\beta}$, a
sequence of SMPL pose
$\mathbf{\Theta}=\\{\boldsymbol{\theta}_{1},...,\boldsymbol{\theta}_{T}\\}$,
object rotation
$\mathbf{\mathcal{R}}^{o}=\\{\mathbf{R}_{1}^{o},...,\mathbf{R}_{T}^{o}\\}$ and
translation
$\mathbf{\mathcal{T}}^{o}=\\{\mathbf{t}^{o}_{1},...,\mathbf{t}^{o}_{T}\\}$
parameters that satisfy 2D image observation and realistic interaction
constraints between the human and the object.
### 3.2 SMPL-T: Temporally consistent SMPL meshes in camera space
Our first step is to obtain SMPL meshes in camera space that have consistent
translation in a video sequence. We leverage 2D body keypoint predictions from
openpose[12] and natural motion smoothness for this. Specifically, we use
FrankMocap [60] to initialise SMPL-T pose
$\mathbf{\Theta}=\\{\boldsymbol{\theta}_{1},...,\boldsymbol{\theta}_{T}\\}$
and shape
$\mathbf{\mathcal{B}}=\\{\boldsymbol{\beta}_{1},...,\boldsymbol{\beta}_{T}\\}$
parameters for a sequence of images. Note here the original SMPL meshes from
FrankMocap are centered at origin. We average the SMPL-T shape parameters over
the sequence as the shape for the person and optimize the SMPL-T global
translation and body poses to minimize 2D reprojection and temporal smoothness
error:
$E(\mathbf{\Theta})=\lambda_{\text{J2D}}L_{\text{J2D}}+\lambda_{\text{reg}}L_{\text{reg}}+\lambda_{a}L_{\text{accel}}+\lambda_{\text{pi}}L_{\text{pi}}$
(1)
where $L_{\text{J2D}}$ is the sum of body keypoint reprojection losses[7] over
all frames and $L_{\text{reg}}$ is a regularization on body poses using priors
learned from data[49, 52, 71]. $L_{\text{accel}}$ is a temporal smoothness
term that penalizes large accelerations over SMPL-T vertices $H_{i}$:
$L_{\text{accel}}=\sum_{i=0}^{T-2}||H_{i}-2H_{i+1}+H_{i+2}||_{2}^{2}$.
$L_{\text{pi}}$, an L2 loss between optimized and initial body pose, prevents
the pose from deviating too much from initialization. $\lambda_{*}$ denotes
the loss weights detailed in Supp.
Note that this optimization does not guarantee that we get the absolute
translation in the world coordinates, it just ensures that our predictions
will be consistent with the SMPL model over a sequence, i.e. we will be off by
one rigid transformation for the entire sequence.
### 3.3 SIF-Net: SMPL-T conditioned interaction field
Our SMPL-T provides translation about the human but does not reason about the
object and the interaction between them. Existing method CHORE [80] can
jointly reason human and object but their humans are predicted at fixed depth.
Our key idea is to leverage our SMPL-T meshes to jointly reason human, object
and interaction while having consistent relative translation. We model this
using a single neural network which we call SIF-Net. The input to SIF-Net
consists of RGB image, human and object masks, and our estimated SMPL-T. With
features from SMPL-T and input images, it then predicts interaction field
which consists of human and object distance fields, SMPL part correspondence
field, object pose and visibility field.
SIF-Net feature encoding. Existing neural implicit methods [62, 80, 61] rely
mainly on features from input image, a main reason that limits their human
prediction at fixed depth. Instead, we extract features from both our
estimated SMPL-T meshes and input image, providing more distinct features for
the query points along the same ray. Inspired by EG3D [13], we use the
triplane representation for SMPL-T feature learning due to its efficiency.
Specifically, we use orthographic camera $\pi^{o}(\cdot)$, to render the
SMPL-T mesh silhouette from right, back and top-down views and obtain three
images $\mathbf{S}^{r}_{i},\mathbf{S}^{b}_{i},\mathbf{S}^{t}_{i}$
respectively, where $\mathbf{S}^{i}\in\mathbb{R}^{H\times W}$, see
supplementary for more visualization. Note here the triplane origin is placed
at our SMPL-T mesh center (Fig. 2 B). We then train an image encoder
$f^{\text{tri}}(\cdot)$ that extracts a pixel aligned feature grid
$\mathbf{D}^{j}_{i}\in\mathbb{R}^{H_{c}\times W_{c}\times C}$ from each
rendered view $\mathbf{S}^{j}_{i}$, where $j\in\\{r,b,t\\}$ and
$H_{c},W_{c},C$ are the feature grid dimensions. To extract features for a
query point $\mathbf{p}\in\mathbb{R}^{3}$, we project $\mathbf{p}$ into the
three planes using the same orthographic projection
$\pi_{\mathbf{p}}^{o}=\pi^{o}(\mathbf{p})$ and extract local features
$\mathbf{D}_{i}^{\mathbf{p}}=(\mathbf{D}^{r}_{i}(\pi_{\mathbf{p}}^{o}),\mathbf{D}^{b}_{i}(\pi_{\mathbf{p}}^{o}),\mathbf{D}^{t}_{i}(\pi_{\mathbf{p}}^{o}))$
using bilinear interpolation.
In addition to the SMPL-T features, SIF-Net also extracts information from
input images. More specifically, we train an image encoder
$f^{\text{enc}}(\cdot)$ to extract feature grid
$\mathbf{Z}_{i}\in\mathbb{R}^{H_{f}\times W_{f}\times C_{f}}$ from input image
$\mathbf{I}_{i}\in\mathbb{R}^{H\times W\times 5}$, here $H_{f},W_{f},C_{f}$
and $H,W$ are feature grid and input image dimensions respectively. Given
query point $\mathbf{p}\in\mathbb{R}^{3}$, we project it to 2D image using
full perspective projection $\pi_{\mathbf{p}}=\pi(\mathbf{p})$ and extract
pixel-aligned features
$\mathbf{Z}_{i}^{\mathbf{p}}=\mathbf{Z}_{i}(\pi_{\mathbf{p}})$. The input
image feature is concatenated with the SMPL-T feature to form an input and
translation aware point feature:
$\mathbf{F}_{i}^{\mathbf{p}}=(\mathbf{Z}_{i}^{\mathbf{p}},\mathbf{D}_{i}^{\mathbf{p}})$.
SIF-Net predictions. From the point feature $\mathbf{F}^{\mathbf{p}}_{i}$
discussed above, we predict our interaction fields that jointly model human,
object and their contacts, similar to CHORE [80]. Specifically, we predict the
unsigned distances to human and object surfaces using
$f^{u}:\mathbf{F}_{i}^{\mathbf{p}}\mapsto\mathbb{R}^{2}_{\geq 0}$. This allows
fitting the SMPL mesh and object template by minimizing the predicted
distances at mesh vertices. We can also infer contacts, as the points having
small distances to both human and object surfaces. For more robust SMPL
fitting [4] and modelling which body part the object point is in contact with,
we predict SMPL part correspondence using
$f^{p}:\mathbf{F}_{i}^{\mathbf{p}}\mapsto\\{1,2,...,K\\}$ where $K$ is the
number of SMPL parts. For more accurate object fitting, we additionally
predict object rotation with
$f^{R}:\mathbf{F}_{i}^{\mathbf{p}}\mapsto\mathbb{R}^{3\times 3}$ and
translation with $f^{c}:\mathbf{F}_{i}^{\mathbf{p}}\mapsto\mathbb{R}^{3}$. The
predicted $3\times 3$ matrix is projected to SO(3) using symmetric
orthogonalization[44]. At test time, we first use $f^{u}$ to find points on
the object surface[16] and take the average rotation and translation
predictions of these points as the object pose. To handle occlusions, we also
predict the object visibility field using
$f^{\text{vis}}:\mathbf{F}_{i}^{\mathbf{p}}\mapsto[0,1]$. The visibility is
useful to recover the object pose under occlusion, see more details in Sec.
3.4.
Why use triplane to encode SMPL-T? A direct alternative to triplane based
SMPL-T encoding is to find the closest point in SMPL-T mesh and concatenate
that coordinate to the point features. But such a method is slow (computing
point to surface distance) and does not allow flexible learning of local and
global features. Another choice is to voxelize the SMPL-T mesh and extract
point local features using IF-Nets [15] but such a method is still expensive.
Therefore we chose the more efficient triplane representation to encode our
estimated SMPL-T meshes.
Implementation. Our SMPL-T feature extractor $f^{\text{tri}}$ is shared for
three views and trained end to end with image encoder $f^{\text{enc}}$ and
other neural field predictors. At training time, we input the renderings from
ground truth SMPL meshes and train the network to predict GT labels. At test
time, we obtain the SMPL-T meshes using Eq. 1. In order to have smoother
SMPL-T feature in a sequence, we use SmoothNet [90] to smooth the optimized
SMPL parameters. We evaluate this component and provide more implementation
details in supplementary.
### 3.4 HVOP-Net: Human and Visibility aware Object Pose under occlusions
Our SIF-Net recovers translation and more accurate object pose. However, the
object pose prediction under very heavy occlusions remains challenging because
no image evidence from single frame is available for accurate prediction, see
Fig. 5. Our key idea is to use the SMPL-T and object pose from other visible
frames to predict the object of occluded frames. To this end, we first predict
object visibility scores in each image, which are then leveraged together with
the human evidence from neighbouring frames to predict the object poses of the
occluded frames.
Object visibility score. Our visibility score denotes how much the object is
visible in the input image. We train a visibility decoder
$f^{\text{vis}}(\cdot)$, a prediction head of SIF-Net, that takes a point
feature $\mathbf{F}_{i}^{\mathbf{p}}$ as input and predicts visibility score
$v_{i}\in[0,1]$ for frame $i$. At test time, we first use the neural object
distance predictor $f^{u}$ to find object surface points [16] and then take
the average visibility predictions of these points as the object visibility
score for this image.
Object pose prediction under heavy occlusion. Our goal now is to predict
accurate object pose for heavily occluded frames. We consider frames whose
visibility score $v_{i}$ is smaller than $\delta=0.5$ as the occluded frames.
Inspired by works from motion infill [39, 23] and synthesis [88, 93], we
design our HVOP-Net that leverages transformer [72] and explicitly takes the
human motion and object visibility into account to recover object pose under
heavy occlusions.
More specifically, we first use a transformer $f^{s}(\cdot)$ to aggregate
temporal information of the SMPL-T poses:
$f^{s}:\mathbb{R}^{T\times|\theta_{i}|}\mapsto\mathbb{R}^{T\times D_{hs}}$,
where $|\theta_{i}|$ is the SMPL pose dimension and $D_{hs}$ is the hidden
feature dimension. Similarly, we use a transformer $f^{o}(\cdot)$ to aggregate
temporal information of the object poses: $f^{o}:\mathbb{R}^{T\times
D_{o}}\mapsto\mathbb{R}^{T\times D_{ho}}$. Note here the SMPL-T transformer
$f^{s}$ attends to all frames while the object transformer $f^{o}$ only
attends to frames where object is visible ($v_{i}\geq\delta$). We then
concatenate the SMPL-T and object features and use another transformer
$f^{\text{comb}}$ to aggregate both human and object information and predict
the object poses:
$f^{\text{comb}}:\mathbb{R}^{T\times(D_{hs}+D_{ho})}\mapsto\mathbb{R}^{T\times
D_{o}}$. The joint transformer $f^{\text{comb}}$ attends to all frames. Our
experiments show that our HVOP-Net is important to accurately predict object
pose under heavy occlusions, see Fig. 5.
Implementation. The visibility decoder $f^{\text{vis}}$ is trained end to end
with other SIF-Net components using L2 loss. The GT visibility score is
computed as the number of visible object pixels (from object mask) divided by
total number of object pixels (from GT object rendering). To train our HVOP-
Net, we randomly zero out the object pose for a small clip of the input
sequence and provide ground truth SMPL and object poses for other frames as
input. We train our network to accept input sequence of a fixed length but at
test time, the sequence can have various length and object occlusion can last
for a longer time. To this end, we use an auto-regressive algorithm to recover
the object pose of a full video, similar to [89]. At test time, we use SIF-Net
object pose predictions and zero out highly occluded frames based on the
predicted visibility scores. We empirically find that having smooth object
pose as input is helpful for more accurate prediction. Hence we use SmoothNet
[90] to smooth SIF-Net object pose predictions before inputing them to HVOP-
Net. The evaluation of this component and more training details are described
in our supplementary.
### 3.5 Visibility aware joint optimization
To obtain SMPL and object meshes that align with input images and satisfy
contact constraints, we leverage our network predictions from Sec. 3.3 and
Sec. 3.4 to formulate a robust joint optimization objective. Our goal is to
obtain an optimal set of parameters
$\mathbf{\Phi}=\\{\mathbf{\Theta},\boldsymbol{\beta},\mathbf{\mathcal{R}}^{o},\mathbf{\mathcal{T}}^{o}\\}$
for SMPL pose, shape, object rotation and translation respectively. We
initialize the SMPL parameters from our estimated SMPL-T (Eq. 1) and object
parameters from our HVOP-Net predictions (Sec. 3.4). Inspired by CHORE[80],
our energy function consists of human data $E_{data}^{h}$, object data
$E_{data}^{o}$, contact data $E_{data}^{c}$ and SMPL pose prior term
$E_{reg}$:
$E_{(}\mathbf{\Phi})=E_{data}^{h}+E_{data}^{o}+E_{data}^{c}+E_{reg}.$ (2)
here $E_{reg}$ is a body pose and shape prior loss[49]. We explain other loss
data terms next.
Human data term. $E_{data}^{h}$ minimizes the discrepancy between SIF-Net
prediction and SMPL meshes as well as a temporal smoothness error:
$E_{data}^{h}(\mathbf{\Theta},\boldsymbol{\beta})=\sum_{i=1}^{T}L_{\text{neural}}^{h}(\boldsymbol{\theta}_{i},\boldsymbol{\beta})+\lambda_{\text{ah}}L_{\text{accel}}(\mathbf{\Theta})$,
where $L_{\text{accel}}$ is the same used in Eq. 1. $L_{\text{neural}}^{h}$
pushes the SMPL vertices to the zero-level set of the human distance field
represented by neural predictor $f^{u,h}_{i}$ and forces correct SMPL part
locations predicted by $f^{p}_{i}$:
$E^{h}_{\text{neural}}(\mathbf{\Theta},\boldsymbol{\beta})=\sum_{i=1}^{T}(\sum_{\mathbf{p}\in
H(\boldsymbol{\theta}_{i},\boldsymbol{\beta})}(\lambda_{h}\min(f^{u,h}_{i}(\mathbf{F}^{\mathbf{p}}_{i}),\delta_{h})+\\\
\lambda_{p}L_{p}(l_{\mathbf{p}},f^{p}_{i}(\mathbf{F}^{\mathbf{p}}_{i}))))$ (3)
here $l_{\mathbf{p}}$ is the predefined SMPL part label [4] of SMPL vertex
$\mathbf{p}$ and $L_{p}$ is the categorical cross entropy loss function.
$\delta_{h}$ is a small clamping value.
Object data term. We transform the object template vertices
$\mathbf{O}\in\mathbb{R}^{3\times N}$ using object pose parameters of frame
$i$ by:
$\mathbf{O}^{\prime}_{i}=\mathbf{R}^{o}_{i}\mathbf{O}+\mathbf{t}^{o}_{i}$.
Intuitively, the object vertices should lie on the zero-level set of the
object distance field represented by $f^{u,o}_{i}$ and the rendered silhouette
should match the 2D object mask $\mathbf{M}^{o}_{i}$. Hence we formulate the
loss as:
$E^{o}_{data}(\mathbf{\mathcal{R}}^{o},\mathbf{\mathcal{T}}^{o})=\sum_{i=1}^{T}v_{i}(\sum_{\mathbf{p}\in\mathbf{O}^{\prime}_{i}}\lambda_{o}\min(f^{u,o}_{i}(\mathbf{F}^{\mathbf{p}}_{i}),\delta_{o})\\\
+\lambda_{\text{occ}}L_{\text{occ-
sil}}(\mathbf{O}_{i}^{\prime},\mathbf{M}^{o}_{i}))+\lambda_{\text{ao}}L_{\text{ao}}$
(4)
where $v_{i}$ is the predicted object visibility score described in Sec. 3.4.
This down-weights the loss values of network predictions for frames where
object is occluded and allows more temporal regularization. $L_{\text{occ-
sil}}$ is an occlusion-aware silhouette loss[91] and $L_{\text{ao}}$ is a
temporal smoothness loss applied to object vertices $\mathbf{O}^{\prime}_{i}$,
similar to $L_{\text{accel}}$ in Eq. 1.
Contact data term. The contact data term [80] minimizes the distance between
human and object points that are predicted to be in contact:
$E_{\text{data}}^{c}(\mathbf{\mathcal{R}}^{o},\mathbf{\mathcal{T}}^{o})=\lambda_{c}\sum_{i=1}^{T}(\sum_{j=1}^{K}d(H_{j}^{c}(\boldsymbol{\theta}_{i},\boldsymbol{\beta}),\mathbf{O}_{ij}^{c}))$
(5)
here $d(\cdot,\cdot)$ is chamfer distance. We consider human points on the
$j^{\text{th}}$ body part of SMPL mesh $H_{i}$ of frame $i$ (denoted as
$H_{ij}$) are in contact when their predicted distance to the object is
smaller than a threshold:
$H^{c}_{j}(\boldsymbol{\theta}_{i},\boldsymbol{\beta})=\\{\mathbf{p}|\mathbf{p}\in
H_{ij}\,\text{and}\,f^{u,o}_{i}(\mathbf{F}^{\mathbf{p}}_{i})\leq\epsilon\\}$.
Similarly, we find contact points on object meshes with
$\mathbf{O}_{ij}^{c}=\\{\mathbf{p}|\mathbf{p}\in\mathbf{O}^{\prime}_{i}\,\text{and}\,f^{u,h}_{i}(\mathbf{F}^{\mathbf{p}}_{i})\leq\epsilon\,\text{and}\,f^{p}_{i}(\mathbf{F}^{\mathbf{p}}_{i})=j\\}$.
Please see Supp. for more details about loss weights $\lambda_{*}$.
## 4 Experiments
Figure 3: Comparison with PHOSA [91] and CHORE[80] on BEHAVE[6] (row 1-2) and
InterCap [35] (row 3-4). PHOSA’s object pose optimization often gets stuck in
local minima due to heavy occlusions. CHORE also fails to predict accurate
object pose as it does not fully explore the human, temporal and visibility
information while our method can robustly track human and object in these
challenging cases.
In this section, we first compare our method against existing approaches on
tracking human and object and then evaluate the key components of our methods.
Our experiments show that our method clearly outperforms existing joint human
object reconstruction method and our novel _SMPL-T conditioned interaction
fields_ (SIF-Net) as well as _human and visibility aware_ object pose
prediction (HVOP-Net) works better than existing state of the art methods.
Baselines. (1) Joint human and object tracking. We compare against PHOSA [91]
and CHORE [80] in the joint human-object reconstruction task. (2) Object pose
prediction. Our HVOP-Net leverages nearby (un-occluded) frames to predict the
object pose of occluded frames. We compare this with a simple baseline that
linearly interpolates the object pose between visible frames to recover
occluded poses. We also find similarity between our task and motion
smoothing/infilling. Hence we compare our HVOP-Net with SoTA smoothing [90]
and infill method [39].
Datasets. We conduct experiments on the BEHAVE [6] and InterCap [35] dataset.
(1) BEHAVE [6] captures 7 subjects interacting with 20 different objects in
natural environments and contains SMPL and object registrations annotated at
1fps. We use the extended BEHAVE dataset, which registers SMPL and object for
BEHAVE sequences at 30 fps. We follow the official split [80] with $217$
sequences for training and $82$ for testing. (2) InterCap [35] is a similar
dataset that captures 10 subjects interacting with 10 different objects. The
dataset comes with pseudo ground truth SMPL and object registrations at 30fps.
We train our model on sequences from subject 01-08 (173 sequences) and test on
sequences from subject 09-10 (38 sequences).
We compare with CHORE [80] on the full test set of both datasets. PHOSA [91]
optimizes hundreds of random object pose initialization per image hence the
inference speed is very slow (2min/image), which makes it infeasible to run on
full video sequences. Hence we compare with PHOSA on key frames only, denoted
as $\text{BEHAVE}^{*}$(3.9k images) and $\text{InterCap}^{*}$(1.1k images)
respectively. Due to the large number of frames in the full BEHAVE test set
(127k frames), we conduct other ablation experiments in a sub test set (42k
frames) of BEHAVE.
Evaluation metrics. (1) Joint human-object tracking. We evaluate the
performance of SMPL and object reconstruction using Chamfer distance between
predicted SMPL and object meshes, and the ground truth. CHORE [80] uses
Procrustes alignment on combined SMPL and object meshes for _each frame_
before computing errors. However, this does not reflect the real accuracy in
terms of the relative translation between nearby frames in a video. Inspired
by the world space errors proposed by SPEC[42] and GLAMR[89], we propose to
perform joint Procrustes alignment in a sliding window, as also used in SLAM
evaluations [31, 66]. More specifically, we combine all SMPL and object
vertices within a sliding window and compute a single optimal Procrustes
alignment to the ground truth vertices. This alignment is then applied to all
SMPL and object vertices within this window and Chamfer distance of SMPL and
object meshes are computed respectively. We report both the errors using per-
frame alignment (w=1) and alignment with a sliding window of 10s (w=10).
(2) Object only evaluation. For experiments evaluating object pose only, we
evaluate the rotation accuracy using rotation angle [32]. The object
translation error is computed as the distance between reconstructed and GT
translation. We report all errors in centimetre in our experiments.
### 4.1 Evaluation of tracking results
Dateset | Methods | Align w=1 | Align w=10
---|---|---|---
SMPL $\downarrow$ | Obj. $\downarrow$ | SMPL $\downarrow$ | Obj. $\downarrow$
BEHAVE | CHORE | 5.55 | 10.02 | 18.33 | 20.32
Ours | 5.25 | 8.04 | 7.81 | 8.49
$\text{BEHAVE}^{*}$ | PHOSA | 12.86 | 26.90 | 27.01 | 59.08
CHORE | 5.54 | 10.12 | 21.28 | 22.39
Ours | 5.24 | 7.89 | 8.24 | 8.49
InterCap | CHORE | 7.12 | 12.59 | 16.11 | 21.05
Ours | 6.76 | 10.32 | 9.35 | 11.38
$\text{InterCap}^{*}$ | PHOSA | 11.20 | 20.57 | 24.16 | 43.06
CHORE | 7.01 | 12.81 | 16.10 | 21.08
Ours | 6.78 | 10.34 | 9.35 | 11.54
Table 1: Human and object tracking results on BEHAVE [6] and InterCap [35]
datasets (unit: cm). * denotes key frames only. w is the temporal window size
used for Procrustes alignment where w=1 means per-frame Procrustes and w=10
means alignment over a sliding window of 10s. We can see our method clearly
outperforms baseline PHOSA[91] and CHORE[80] in all metrics.
We compare our human and object tracking results against baseline PHOSA [91]
and CHORE [80] and report the errors in Tab. 1. Note that the comparison with
PHOSA and CHORE is not strictly fair as they do not use temporal information.
Nevertheless they are our closest baselines and we show that our method
outperforms them using per-frame alignment and is significantly better under a
more relevant evaluation metric (align w=10) for video tracking. We also show
qualitative comparisons in Fig. 3. It can be seen that both PHOSA and CHORE
are not able to accurately capture the object under heavy occlusions while our
method is more robust under these challenging occlusion cases.
### 4.2 Importance of SMPL-T conditioning
We propose to condition our interaction field predictions on SMPL-T meshes, we
evaluate this for the joint tracking task in Tab. 2. Our SMPL-T conditioning
allows us to obtain consistent relative translation instead of predicting the
human at fixed depth. This significantly reduces the error when evaluating
using alignment of temporal sliding window, see Tab. 2 and qualitative
examples in Supp.
Note that the SIF-Net prediction relies on the estimated SMPL-T discussed in
Sec. 3.2. We also evaluate how the noisy SMPL-T predictions can affect the
joint tracking performance. We input GT SMPL to our SIF-Net and HVOP-Net to
predict the object pose and perform joint optimization. The object errors
(Chamfer distance in cm) are: 17.29 (w/o SMPL-T cond.), 8.23 (w/ SMPL-T cond.,
ours), 6.50 (GT SMPL). Our method is robust and is close to the ideal lower
bound with GT SMPL.
Method | w/o SMPL-T | w/o HVOP-Net | w/o joint opt. | Ours
---|---|---|---|---
SMPL $\downarrow$ | 14.40 | 8.05 | 8.20 | 8.03
Obj. $\downarrow$ | 17.29 | 9.36 | 16.02 | 8.23
Table 2: Ablation studies. We report the joint tracking error (cm) after an
alignment window of 10s. It can be seen that our proposed SMPL-T conditioning,
HVOP-Net and joint optimization are important to achieve the best results.
### 4.3 Importance of HVOP-Net
Figure 4: Object rotation(left) and translation(right) error vs. occlusion
(1-fully occluded) for variants of our method. Our full model with HVOP-Net
predicts more robust rotation in occlusions. Figure 5: Importance of our HVOP-
Net object pose prediction. We can see that our HVOP-Net corrects erroneous
object pose under heavy occlusions. Better viewed after zoom in.
We propose a _human and visibility aware_ object pose prediction network
(HVOP-Net) to reason about the object under heavy occlusions. Without this
component, the object error is much higher (Tab. 2 column 3), which suggests
the importance of our HVOP-Net.
For the object pose prediction task, there are other similar alternatives to
HVOP-Net: 1). Linear interpolation (SLERP), infill the object pose of
invisible frames using spherical linear interpolation between two visible
frames. 2). SmoothNet [90], a fully connected neural network trained to smooth
the object motion. 3). CMIB [39], a SoTA method for human motion infilling. To
evaluate the effectiveness of our HVOP-Net, we replace HVOP-Net with these
methods and run the full tracking pipeline respectively. The SMPL errors are
similar (deviate $<$ 0.1cm) as HVOP-Net only affects objects. We separate
object error into rotation (angle distance) and translation, and further
analyse the error under varying occlusion for CHORE, our method (SMPL-T +
HVOP-Net) and its variants (SMPL-T + SmoothNet/SLERP/CMIB) in Fig. 4.
All methods except CHORE predict similar translation (due to SMPL-T cond.) but
our HVOP-Net obtains clearly more robust rotation under heavy occlusions.
SmoothNet smooths the object motion but cannot correct errors from long-term
occlusions. SLERP and CMIB [39] are able to correct some pose errors but do
not take the human context into account hence cannot handle heavy occlusions
very well. Our method leverages the human motion and object pose from visible
frames hence achieves the best result. We show one example where our HVOP-Net
corrects the erroneous raw prediction in Fig. 5. Please see our supplementary
for more examples.
### 4.4 Generalization
Figure 6: Results in NTU-RGBD dataset [47]. It can be seen that our method
generalizes well and wormks better than CHORE[80].
To verify the generalization ability of our method, we apply our model trained
on BEHAVE to the NTU-RGBD [47] dataset. We leverage Detectron [79],
interactive [64] and video [14] segmentation to obtain the input human and
object masks. Two example comparisons with CHORE [80] are shown in Fig. 6. We
can see our method generalizes well to NTU-RGBD and works better than CHORE.
Please see Supp. for evaluation details and more comparisons.
## 5 Limitations
Although our method works robustly under heavy occlusions, there are still
some limitations. For instance, we assume known object templates for tracking.
An interesting direction is to build such a template from videos as
demonstrated by recent works [82, 83, 84, 78]. Our method may fail under
challenging cases such as significant object pose change under heavy
occlusions. It can also make noisy pose predictions when the object is
symmetric or the pose is uncommon. Example failure cases are shown in Supp.
## 6 Conclusion
We present a novel method for tracking human, object and realistic contacts
between them from monocular RGB cameras. Our first contribution is a SMPL-T
conditioned neural field network that allows consistent and more accurate 3D
reconstruction in a video sequence. Our second contribution is a human motion
and object visibility aware network that can recover the object pose under
heavy occlusions. Our experiments show that our method significantly
outperforms state of the art methods in two datasets. Our extensive ablations
show that our SMPL-T conditioning and HVOP-Net are important for accurate
tracking. We also show that our method generalizes well to another dataset it
is not trained on. Our code and model are released to promote future research
in this direction.
Acknowledgements. We thank RVH group members [1] for their helpful
discussions. We also thank reviewers for their feedback which improves the
manuscript. This work is funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) - 409792180 (Emmy Noether Programme, project: Real
Virtual Humans), and German Federal Ministry of Education and Research (BMBF):
Tübingen AI Center, FKZ: 01IS18039A. Gerard Pons-Moll is a member of the
Machine Learning Cluster of Excellence, EXC number 2064/1 – Project number
390727645.
In this supplementary, we first list all the implementation details of our
method and then show more ablation study results as well as comparison with
CHORE[80] on NTU-RGBD[47] dataset. We end with discussions of failure cases
and future works.
## Appendix A Implementation details
### A.1 Obtaining SMPL-T meshes
To obtain the image-aligned SMPL meshes that have consistent translation
(SMPL-T) we keep the SMPL shape parameters and optimize the body pose and
global translation values. The loss weights for this optimization are:
$\lambda_{\text{J2D}}=0.09,\lambda_{\text{reg}}=1.0\times
10^{-5},\lambda_{a}=25,\lambda_{\text{pi}}=900$. We optimize the parameters
until convergence with a maximum iteration of 1000.
### A.2 SIF-Net: SMPL-T conditioned interaction field
A visualization of our SMPL-T triplane rendering and query point projection
can be found in Fig. 7. We discuss our network architecture and training
details next.
#### Network architecture.
We use the stacked hourglass network [51] for both RGB image encoder
$f^{\text{enc}}$ and SMPL rendering encoder $f^{\text{tri}}$. We use 3 stacks
for $f^{\text{tri}}$ and the output feature dimension is
$d^{\text{tri}}_{\text{o}}=64$. Hence $f^{\text{tri}}:\mathbb{R}^{H\times
W}\mapsto\mathbb{R}^{H/4\times W/4\times 64}$ where $H=W=512$. We also use 3
stacks for $f^{\text{enc}}$ but the feature dimension is
$d^{\text{enc}}_{\text{o}}=256$. Hence $f^{\text{enc}}:\mathbb{R}^{H\times
W\times 5}\mapsto\mathbb{R}^{H/4\times W/4\times 256}$. We also concatenate
the image features extracted from the first convolution layer and query point
coordinate to the features. Thus the total feature dimension to our decoders
is: $d=(d^{\text{tri}}_{1}+d^{\text{tri}}_{\text{o}})\times
3+d^{\text{enc}}_{1}+d^{\text{enc}}_{\text{o}}+3=611$, here
$d^{\text{tri}}_{1}=32,d^{\text{enc}}_{1}=64$. All decoders consist of three
FC layers with ReLU activation and one output FC layer with hidden dimension
of 128 for the intermediate features. The visibility decoder $f^{v}$
additionally has a sigmoid output activation layer. The output shape is
$2,14,9,3,1$ for $f^{u},f^{p},f^{R},f^{c},f^{v}$ respectively.
Figure 7: Visualization of our SMPL-T triplane feature extraction and
rendering. The triplane origin is placed at the SMPL-T body center and we
render the mesh from three views using orthographic projection: righ-left (A),
back-front (B) and top-down (C). The query point $p$ is projected into the
three planes using same projection for rendering and we extract pixel aligned
features $\mathbf{F}^{A},\mathbf{F}^{B},\mathbf{F}^{C}$ from the feature
planes respectively. Note that we render the SMPL-T with color here for
visualization, the actual input to our network are silhouette images only.
#### Training.
All feature encoders and decoders are trained end to end with the loss:
$L=\lambda_{u}(L_{u_{h}}+L_{u_{o}})+\lambda_{p}L_{p}+\lambda_{R}L_{R}+\lambda_{c}L_{c}+\lambda_{v}L_{v}$.
Here $L_{u_{i}}$ is the $L_{1}$ distance between ground truth and predicted
unsigned distance to human or object surface [80]. $L_{p}$ is a standard
categorical cross entropy loss for SMPL part correspondence prediction.
$L_{R},L_{c},L_{v}$ are mean square losses between ground truth and predicted
values for rotation matrix, translation vector and visibility score
respectively. The loss weights are:
$\lambda_{u}=1.0,\lambda_{R}=0.006,\lambda_{c}=500,\lambda_{c}=\lambda_{v}=1000$.
The model is trained for 18 epochs and it takes 25h to converge on a machine
with 4 RTX8000 GPUs each with 48GB memory. The training batch size is 8.
### A.3 HVOP-Net: object pose under occlusion
We use three transformers $f^{s},f^{o},f^{\text{comb}}$ to aggregate features
from SMPL-T, object pose and joint human object information respectively. We
use the 6D vector [96] to represent the ration matrix of SMPL-T and object
pose parameters. Hence the SMPL-T pose dimension is $24\times 6+3=147$, where
3 denotes the global translation. We predict the object rotation only thus the
object data dimension is 6. The SMPL-T transformer $f^{s}$ consists of an MLP:
$\mathbb{R}^{T\times 147}\mapsto\mathbb{R}^{T\times 128}$ and two layers of
multi-head self-attention (MHSA) module [72] with 4 heads. Similarly, the
object transformer $f^{o}$ consists of an MLP: $\mathbb{R}^{T\times
6}\mapsto\mathbb{R}^{T\times 32}$ and two layers of MHSA module with 2 heads.
The joint transformer $f^{\text{comb}}$ consists of 4 layers of MHSA module
with 1 head only. GeLU activation is used in all MHSA modules. We finally
predict the object pose using two MLP layers with an intermediate feature
dimension of 32 and LeakyReLU activation.
The model is trained to minimize the $L_{1}$ losses of pose value and
accelerations:
$L=\lambda_{\text{pose}}L_{\text{pose}}+\lambda_{\text{accel}}L_{\text{accel}}$,
where $\lambda_{\text{pose}}=1.0,\lambda_{\text{accel}}=0.1$. It is trained on
a server with 2 RTX8000 GPUs, each GPU has 48GB memory capacity. It takes
around 7h to converge (64 epochs).
### A.4 SmoothNet for SMPL-T and object
We use SmoothNet[90] to smooth our SMPL-T and SIF-Net object pose predictions.
We use exactly the same model and training strategy proposed by the original
paper. The input to the SMPL-T SmoothNet is our estimated SMPL-T pose and
translation (relative to the first frame). The input to the object SmoothNet
is the object rotation (6D vector). Following the standard practice of
SmoothNet[90], we train both models on the predictions from the BEHAVE [6]
training set. Note that we do not fine-tune them on InterCap[35] dataset. We
evaluate this component in Sec. B.3.
### A.5 Visibility aware joint optimization
The objective function defined in Eq. 2 is highly non-convex thus we solve
this optimization problem in two stages. We first optimize the SMPL pose and
shape parameters using human data term only. We then optimize the object
parameters using the object and contact data terms. The loss weights are set
to: $\lambda_{\text{reg}}=2.5\times
10^{-4},\lambda_{\text{ah}}=10^{4},\lambda_{h}=10^{4},\lambda_{p}=t\times
10^{-4},\lambda_{o}=900,\lambda_{\text{occ}}=9\times
10^{-4},\lambda_{\text{ao}}=225,\lambda_{c}=900$, where $\lambda_{c}$ is the
loss weight for the contact data term defined in Eq. 5.
## Appendix B Additional ablation results
### B.1 Further evaluation of SMPL-T conditioning
We show some example images from one sequence in Fig. 8 to evaluate the
importance of our SMPL-T conditioning. It can be seen that without this
conditioning, the human is reconstructed at fixed depth, leading to
inconsistent relative translation across time. Our method predicts more
coherent relative human translation and more accurate object pose.
To further evaluate SMPL-T conditioning, we compute the object pose error from
the raw network predictions and compare it with the object pose of CHORE which
is also the raw prediction from the network. The pose error is computed as
Chamfer distance (CD) and vertex to vertex (v2v) error after centring the
prediction and GT mesh at origin. We also report the translation error
(transl.) as the distance between predicted and GT translation. The results
are shown in Tab. 3. We can clearly see that our SMPL feature improves both
the raw object pose prediction and distance fields (results after optimization
are also improved).
Method | Raw prediction | After opt. w=10
---|---|---
CD$\downarrow$ | v2v$\downarrow$ | transl.$\downarrow$ | SMPL$\downarrow$ | obj.$\downarrow$
w/o SMPL-T | 5.56 | 16.10 | 14.28 | 14.40 | 17.29
Ours | 3.98 | 12.34 | 9.53 | 8.03 | 8.23
Table 3: Importance of SMPL-T conditioning (errors in cm). We can see that our
SMPL-T feature improves both the raw object pose prediction and distance
fields (after opt.). Without our SMPL-T conditioning, the reconstructed
translation is not consistent across frames, leading to large errors after
alignment of temporal window of 10s (w=10). Figure 8: Evaluating SMPL-T
conditioning for neural field prediction. We can see that without conditioning
on SMPL-T meshes, the object pose prediction is worse and human is
reconstructed at fixed depth, leading to inconsistent relative location across
frames. Our method recovers the relative translation more faithfully and
obtain better object pose predictions.
### B.2 Comparing different pose prediction methods
We show some example comparisons of different object pose prediction methods
under heavy occlusions in Fig. 10. We compare our method against: 1). Raw
prediction from our SIF-Net. 2). Linearly interpolate the occluded poses from
visible frames (SLERP). 3). CMIB [39], a transformer based model trained to
infill the object motion using visible frames. Note here the evaluation is
based on the final tracking results and we report the object errors only as
the difference of SMPL error is very small. Similar to Sec. B.1, the object
errors are computed as Chamfer distance, v2v error and translation error.
It can be seen that the raw pose prediction is noisy due to occlusion. SLERP
and CMIB corrects some pose errors but is not robust as they do not leverage
the human information. Our method is more accurate as it takes the human
context and object pose into account.
### B.3 Evaluating SmoothNet
Method | Chamfer | v2v | Acceleration
---|---|---|---
w/o SmoothNet | 8.71 | 9.84 | 1.38
w/ SmoothNet | 8.01 | 9.12 | 1.18
Table 4: Ablate SMPL SmoothNet (errors in cm). We can see that SmoothNet [90] improves the overall smoothness and slightly reduces the pose errors. Method | Chamfer | v2v | Translation
---|---|---|---
a. Raw prediction | 5.03 | 10.39 | 10.01
b. Raw + SmoothNet | 4.22 | 8.60 | 10.16
c. Raw + our pose pred. | 4.09 | 8.02 | 10.20
d. Our full model | 3.62 | 7.20 | 9.96
Table 5: Ablate SmoothNet for object pose prediction (errors in cm). We can
see our pose prediction (c) is better than SmoothNet [90] (b). Combing both we
obtain the best result (d).
SmoothNet [90] is used to smooth the SMPL-T parameters after 2D keypoint based
optimization. We evaluate this step by computing the SMPL errors, shown in
Tab. 4. We can see that SmoothNet reduces the SMPL error slightly.
We also use SmoothNet to smooth the object pose before sending it to our human
and visibility aware object pose prediction network. SmoothNet cannot correct
errors under long-term occlusions. However, it provides smoother object motion
for visible frames which can benefit our pose prediction network. We evaluate
this using object pose errors and report the results in Tab. 5. It can be seen
that our method (Tab. 5c) works better than SmoothNet (Tab. 5b) on raw
predictions. Nevertheless, with smoothed pose after SmoothNet, our method
achieves the best result (Tab. 5 d).
### B.4 Runtime cost
SMPL-T pre-fitting and joint optimization can be run in batches hence the
average runtime per frame is not long: SMPL-T pre-fitting: 6.38s, SIF-Net
object pose prediction: 0.89s, HVOP-Net: 1.3ms, joint optimization: 9.26s,
total: 16.53s. Compared to CHORE ($\sim$12s/frame) [80], the additional cost
is mainly from the SMPL-T pre-fitting. Yet, SMPL-T conditioning allows faster
convergence of joint optimization and much better reconstruction. Since we use
efficient 2D encoder instead of 3D encoder, it takes only 1.05GB GPU memory to
load the SIF-Net model. This allows us to do joint optimization with batch
size up to 128 on a GPU with 48GB memory.
## Appendix C Generalization to NTU-RGBD dataset
#### Obtaining input masks.
Unlike BEHAVE and InterCap where the human and object masks are provided by
the dataset, there are no masks in NTU-RGBD. To this end, we run DetectronV2
[79] to obtain the human masks. We manually segment the object in the first
frame using interactive segmentation [64] ($<$1min/image) and then use video
segmentation [14] to propagate the masks. The overhead of 1min/video manual
label is small.
We show more results from our method on NTU-RGBD dataset[47] and compare
against CHORE[80] in Fig. 11. It can be seen that CHORE may predict some
reasonable object pose but it fails quite often to capture the fine-grained
contacts between the human and object. Our method obtains more coherent
reconstruction for different subjects, human-backpack interactions, camera
view points and backgrounds. Please see our project website for comparison in
full sequences.
## Appendix D Limitations and future works
Figure 9: Failure cases analysis. We show three typical failure cases of our
method: A. The occluded object pose (T=156) changes significantly between two
visible frames (T=0 and T=319) and it is difficult to accurate track the
contact changes. B. The object pose is not commonly seen during interaction
and it is difficult to predict for this rare pose. C. The object is symmetric.
The joint optimization satisfies the object mask and contacts but is not
semantically correct.
Although our method works robustly under heavy occlusions, there are still
some limitations. Firstly, we assume known object templates for tracking, an
interesting direction is to build such a template from videos as demonstrated
by recent works [82, 83, 84, 78]. Secondly, it would be interesting to model
multi-person or even multi-object interactions which is a more realistic
setting in real-life applications. In addition, the backpack can also deform
non-rigidly which is not modelled in our method. Further works can incorporate
the surface deformation [46] or object articulation [81] into the human object
interaction. We leave these for future works.
We identify three typical failure cases of our method, some examples are shown
in Fig. 9. The first typical failure case comes from heavy occlusion when the
object undergoes significant changes (object pose and contact locations)
between two visible frames. In this case, it is very difficult to track the
pose and contact changes accurately (Fig. 9 A). Second typical failure is due
to the difficulty of pose prediction itself even the object is fully visible.
In this case the object pose is uncommon and the network failed to predict it
correctly (Fig. 9 B). Another failure is caused by symmetric objects. Our
optimization minimizes the 2D mask loss and contact constraints but the
network is confused by the symmetry and the initial pose prediction is not
semantically correct (Fig. 9 C). In addition, the training data for these
objects is very limited (only $1/3$ of other objects). More training data or
explicitly reasoning about the symmetry [94] can be helpful.
Figure 10: Comparing different object pose prediction method under heavy
occlusions. Raw prediction is from our SIF-Net output, SLERP denotes linear
interpolation and CMIB is from [39]. We can see SLERP and CMIB can correct
some errors (row 5) but they do not take the human motion into account hence
often fail in more challenging cases. Our method is more robust as it
leverages information from both human motion and object pose from visible
frames. Figure 11: Comparing our method with CHORE[80] on NTU-RGBD [47]
dataset. It can be seen that CHORE does not capture the realistic contacts
between the person and the backpack. Our method recovers the 3D human, the
object and contacts more faithfully in different interaction types, camera
view points and backgrounds.
## References
* [1] http://virtualhumans.mpi-inf.mpg.de/people.html.
* [2] Thiemo Alldieck, Marcus Magnor, Weipeng Xu, Christian Theobalt, and Gerard Pons-Moll. Detailed human avatars from monocular video. In International Conf. on 3D Vision, sep 2018.
* [3] Thiemo Alldieck, Marcus Magnor, Weipeng Xu, Christian Theobalt, and Gerard Pons-Moll. Video based reconstruction of 3D people models. In IEEE Conf. on Computer Vision and Pattern Recognition, 2018\.
* [4] Bharat Lal Bhatnagar, Cristian Sminchisescu, Christian Theobalt, and Gerard Pons-Moll. Combining implicit function learning and parametric models for 3d human reconstruction. In European Conference on Computer Vision (ECCV). Springer, August 2020.
* [5] Bharat Lal Bhatnagar, Cristian Sminchisescu, Christian Theobalt, and Gerard Pons-Moll. Loopreg: Self-supervised learning of implicit surface correspondences, pose and shape for 3d human mesh registration. In Advances in Neural Information Processing Systems (NeurIPS), December 2020.
* [6] Bharat Lal Bhatnagar, Xianghui Xie, Ilya Petrov, Cristian Sminchisescu, Christian Theobalt, and Gerard Pons-Moll. Behave: Dataset and method for tracking human object interactions. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2022.
* [7] Federica Bogo, Angjoo Kanazawa, Christoph Lassner, Peter Gehler, Javier Romero, and Michael J Black. Keep it SMPL: Automatic estimation of 3D human pose and shape from a single image. In European Conf. on Computer Vision. Springer International Publishing, 2016.
* [8] Derek Bradley, Tiberiu Popa, Alla Sheffer, Wolfgang Heidrich, and Tamy Boubekeur. Markerless garment capture. ACM Trans. Graph., 27(3):1–9, aug 2008.
* [9] Samarth Brahmbhatt, Cusuh Ham, Charles C. Kemp, and James Hays. ContactDB: Analyzing and predicting grasp contact via thermal imaging. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2019.
* [10] Samarth Brahmbhatt, Ankur Handa, James Hays, and Dieter Fox. Contactgrasp: Functional multi-finger grasp synthesis from contact. In IROS, 04 2019.
* [11] Samarth Brahmbhatt, Chengcheng Tang, Christopher D. Twigg, Charles C. Kemp, and James Hays. ContactPose: A dataset of grasps with object contact and hand pose. In The European Conference on Computer Vision (ECCV), August 2020\.
* [12] Z. Cao, G. Hidalgo Martinez, T. Simon, S. Wei, and Y. A. Sheikh. Openpose: Realtime multi-person 2d pose estimation using part affinity fields. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2019\.
* [13] Eric R. Chan, Connor Z. Lin, Matthew A. Chan, Koki Nagano, Boxiao Pan, Shalini De Mello, Orazio Gallo, Leonidas Guibas, Jonathan Tremblay, Sameh Khamis, Tero Karras, and Gordon Wetzstein. Efficient geometry-aware 3D generative adversarial networks. In CVPR, 2022.
* [14] Ho Kei Cheng, Yu-Wing Tai, and Chi-Keung Tang. Modular interactive video object segmentation: Interaction-to-mask, propagation and difference-aware fusion. In CVPR, 2021.
* [15] Julian Chibane, Thiemo Alldieck, and Gerard Pons-Moll. Implicit functions in feature space for 3d shape reconstruction and completion. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, jun 2020.
* [16] Julian Chibane, Aymen Mir, and Gerard Pons-Moll. Neural unsigned distance fields for implicit function learning. In Neural Information Processing Systems (NeurIPS). , December 2020.
* [17] Alvaro Collet, Ming Chuang, Pat Sweeney, Don Gillett, Dennis Evseev, David Calabrese, Hugues Hoppe, Adam Kirk, and Steve Sullivan. High-quality streamable free-viewpoint video. ACM Trans. Graph., 34(4), jul 2015.
* [18] Enric Corona, Gerard Pons-Moll, Guillem Alenya, and Francesc Moreno-Noguer. Learned vertex descent: A new direction for 3d human model fitting. In European Conference on Computer Vision (ECCV). Springer, October 2022.
* [19] Enric Corona, Albert Pumarola, Guillem Alenya, Francesc Moreno-Noguer, and Gregory Rogez. Ganhand: Predicting human grasp affordances in multi-object scenes. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.
* [20] Yudi Dai, YiTai Lin, XiPing Lin, Chenglu Wen, Lan Xu, Hongwei Yi, Siqi Shen, Yuexin Ma, and Cheng Wang. Sloper4d: A scene-aware dataset for global 4d human pose estimation in urban environments. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2023.
* [21] Xinke Deng, Arsalan Mousavian, Yu Xiang, Fei Xia, Timothy Bretl, and Dieter Fox. PoseRBPF: A Rao–Blackwellized Particle Filter for 6-D Object Pose Tracking. IEEE Transactions on Robotics, 37(5):1328–1342, Oct. 2021. Conference Name: IEEE Transactions on Robotics.
* [22] Yan Di, Fabian Manhardt, Gu Wang, Xiangyang Ji, Nassir Navab, and Federico Tombari. SO-Pose: Exploiting Self-Occlusion for Direct 6D Pose Estimation. In 2021 IEEE/CVF International Conference on Computer Vision (ICCV), pages 12376–12385, Montreal, QC, Canada, Oct. 2021. IEEE.
* [23] Yinglin Duan, Yue Lin, Zhengxia Zou, Yi Yuan, Zhehui Qian, and Bohan Zhang. A unified framework for real time motion completion. Proceedings of the AAAI Conference on Artificial Intelligence, 36(4):4459–4467, Jun. 2022.
* [24] Kiana Ehsani, Shubham Tulsiani, Saurabh Gupta, Ali Farhadi, and Abhinav Gupta. Use the force, luke! learning to predict physical forces by simulating effects. In CVPR, 2020.
* [25] Zhaoxin Fan, Yazhi Zhu, Yulin He, Qi Sun, Hongyan Liu, and Jun He. Deep Learning on Monocular Object Pose Detection and Tracking: A Comprehensive Overview, Apr. 2022. arXiv:2105.14291 [cs].
* [26] Mihai Fieraru, Mihai Zanfir, Elisabeta Oneata, Alin-Ionut Popa, Vlad Olaru, and Cristian Sminchisescu. Three-dimensional reconstruction of human interactions. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 7212–7221, 2020.
* [27] Vladimir Guzov, Aymen Mir, Torsten Sattler, and Gerard Pons-Moll. Human poseitioning system (hps): 3d human pose estimation and self-localization in large scenes from body-mounted sensors. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR). IEEE, jun 2021.
* [28] Shreyas Hampali, Mahdi Rad, Markus Oberweger, and Vincent Lepetit. Honnotate: A method for 3d annotation of hand and object poses. In CVPR, 2020.
* [29] Mohamed Hassan, Vasileios Choutas, Dimitrios Tzionas, and Michael J. Black. Resolving 3d human pose ambiguities with 3d scene constraints. In International Conference on Computer Vision, 2019.
* [30] Yana Hasson, Gül Varol, Dimitrios Tzionas, Igor Kalevatykh, Michael J. Black, Ivan Laptev, and Cordelia Schmid. Learning joint reconstruction of hands and manipulated objects. In CVPR, 2019.
* [31] Sachini Herath, Hang Yan, and Yasutaka Furukawa. Ronin: Robust neural inertial navigation in the wild: Benchmark, evaluations, new methods. In 2020 IEEE International Conference on Robotics and Automation (ICRA), pages 3146–3152, 2020.
* [32] Tomáš Hodaň, Jiří Matas, and Štěpán Obdržálek. On evaluation of 6d object pose estimation. In Gang Hua and Hervé Jégou, editors, Computer Vision – ECCV 2016 Workshops, pages 606–619, Cham, 2016. Springer International Publishing.
* [33] Yinlin Hu, Pascal Fua, Wei Wang, and Mathieu Salzmann. Single-stage 6d object pose estimation. In CVPR, 2020.
* [34] Chun-Hao P. Huang, Hongwei Yi, Markus Höschle, Matvey Safroshkin, Tsvetelina Alexiadis, Senya Polikovsky, Daniel Scharstein, and Michael J. Black. Capturing and inferring dense full-body human-scene contact. In IEEE/CVF Conf. on Computer Vision and Pattern Recognition (CVPR), pages 13274–13285, June 2022.
* [35] Yinghao Huang, Omid Taheri, Michael J. Black, and Dimitrios Tzionas. InterCap: Joint markerless 3D tracking of humans and objects in interaction. In German Conference on Pattern Recognition (GCPR), volume 13485 of Lecture Notes in Computer Science, pages 281–299. Springer, 2022\.
* [36] Yuheng Jiang, Suyi Jiang, Guoxing Sun, Zhuo Su, Kaiwen Guo, Minye Wu, Jingyi Yu, and Lan Xu. Neuralhofusion: Neural volumetric rendering under human-object interactions. arXiv preprint arXiv:2202.12825, 2022.
* [37] Hanbyul Joo, Tomas Simon, and Yaser Sheikh. Total capture: A 3d deformation model for tracking faces, hands, and bodies. In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 8320–8329, 2018.
* [38] Korrawe Karunratanakul, Jinlong Yang, Yan Zhang, Michael Black, Krikamol Muandet, and Siyu Tang. Grasping field: Learning implicit representations for human grasps. In 8th International Conference on 3D Vision, pages 333–344. IEEE, Nov. 2020.
* [39] Jihoon Kim, Taehyun Byun, Seungyoun Shin, Jungdam Won, and Sungjoon Choi. Conditional motion in-betweening. Pattern Recognition, page 108894, 2022.
* [40] Muhammed Kocabas, Nikos Athanasiou, and Michael J. Black. Vibe: Video inference for human body pose and shape estimation. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2020.
* [41] Muhammed Kocabas, Chun-Hao P. Huang, Otmar Hilliges, and Michael J. Black. PARE: Part attention regressor for 3D human body estimation. In Proceedings International Conference on Computer Vision (ICCV), pages 11127–11137. IEEE, Oct. 2021.
* [42] Muhammed Kocabas, Chun-Hao P. Huang, Joachim Tesch, Lea Müller, Otmar Hilliges, and Michael J. Black. SPEC: Seeing people in the wild with an estimated camera. In Proceedings International Conference on Computer Vision (ICCV), pages 11035–11045. IEEE, Oct. 2021.
* [43] Nikos Kolotouros, Georgios Pavlakos, Michael J Black, and Kostas Daniilidis. Learning to reconstruct 3d human pose and shape via model-fitting in the loop. In ICCV, 2019.
* [44] Jake Levinson, Carlos Esteves, Kefan Chen, Noah Snavely, Angjoo Kanazawa, Afshin Rostamizadeh, and Ameesh Makadia. An analysis of svd for deep rotation estimation. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 22554–22565. Curran Associates, Inc., 2020.
* [45] Yi Li, Gu Wang, Xiangyang Ji, Yu Xiang, and Dieter Fox. Deepim: Deep iterative matching for 6d pose estimation. In European Conference on Computer Vision (ECCV), 2018.
* [46] Zhi Li, Soshi Shimada, Bernt Schiele, Christian Theobalt, and Vladislav Golyanik. Mocapdeform: Monocular 3d human motion capture in deformable scenes. In International Conference on 3D Vision (3DV), 2022.
* [47] Jun Liu, Amir Shahroudy, Mauricio Perez, Gang Wang, Ling-Yu Duan, and Alex C. Kot. Ntu rgb+d 120: A large-scale benchmark for 3d human activity understanding. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2019\.
* [48] Yuan Liu, Yilin Wen, Sida Peng, Cheng Lin, Xiaoxiao Long, Taku Komura, and Wenping Wang. Gen6d: Generalizable model-free 6-dof object pose estimation from rgb images. In ECCV, 2022.
* [49] Matthew Loper, Naureen Mahmood, Javier Romero, Gerard Pons-Moll, and Michael J. Black. SMPL: A skinned multi-person linear model. In ACM Transactions on Graphics. ACM, Oct. 2015.
* [50] Mateusz Majcher and Bogdan Kwolek. Shape enhanced keypoints learning with geometric prior for 6d object pose tracking. In 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), pages 2985–2991, 2022.
* [51] Alejandro Newell, Kaiyu Yang, and Jia Deng. Stacked hourglass networks for human pose estimation. In Bastian Leibe, Jiri Matas, Nicu Sebe, and Max Welling, editors, Computer Vision – ECCV 2016, pages 483–499, Cham, 2016. Springer International Publishing.
* [52] Georgios Pavlakos, Vasileios Choutas, Nima Ghorbani, Timo Bolkart, Ahmed A. A. Osman, Dimitrios Tzionas, and Michael J. Black. Expressive body capture: 3d hands, face, and body from a single image. In Proceedings IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), 2019.
* [53] Sida Peng, Yuan Liu, Qixing Huang, Xiaowei Zhou, and Hujun Bao. PVNet: Pixel-Wise Voting Network for 6DoF Pose Estimation. In 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 4556–4565, Long Beach, CA, USA, June 2019. IEEE.
* [54] Ilya A Petrov, Riccardo Marin, Julian Chibane, and Gerard Pons-Moll. Object pop-up: Can we infer 3d objects and their poses from human interactions alone? In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2023.
* [55] Gerard Pons-Moll and Bodo Rosenhahn. Model-Based Pose Estimation, chapter 9, pages 139–170. Springer, 2011.
* [56] Jathushan Rajasegaran, Georgios Pavlakos, Angjoo Kanazawa, and Jitendra Malik. Tracking people with 3d representations. Advances in Neural Information Processing Systems, 34:23703–23713, 2021.
* [57] Jathushan Rajasegaran, Georgios Pavlakos, Angjoo Kanazawa, and Jitendra Malik. Tracking people by predicting 3d appearance, location and pose. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 2740–2749, 2022.
* [58] Davis Rempe, Tolga Birdal, Aaron Hertzmann, Jimei Yang, Srinath Sridhar, and Leonidas J. Guibas. Humor: 3d human motion model for robust pose estimation. In International Conference on Computer Vision (ICCV), 2021.
* [59] Chris Rockwell and David F. Fouhey. Full-body awareness from partial observations. In ECCV, 2020.
* [60] Yu Rong, Takaaki Shiratori, and Hanbyul Joo. Frankmocap: A monocular 3d whole-body pose estimation system via regression and integration. In IEEE International Conference on Computer Vision Workshops, 2021\.
* [61] Shunsuke Saito, , Zeng Huang, Ryota Natsume, Shigeo Morishima, Angjoo Kanazawa, and Hao Li. Pifu: Pixel-aligned implicit function for high-resolution clothed human digitization. In IEEE International Conference on Computer Vision (ICCV). IEEE, oct 2019.
* [62] Shunsuke Saito, Tomas Simon, Jason Saragih, and Hanbyul Joo. Pifuhd: Multi-level pixel-aligned implicit function for high-resolution 3d human digitization. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, June 2020.
* [63] Soshi Shimada, Vladislav Golyanik, Zhi Li, Patrick Pérez, Weipeng Xu, and Christian Theobalt. Hulc: 3d human motion capture with pose manifold sampling and dense contact guidance. In European Conference on Computer Vision (ECCV), pages 516–533, 2022.
* [64] Konstantin Sofiiuk, Ilia Petrov, Olga Barinova, and Anton Konushin. f-brs: Rethinking backpropagating refinement for interactive segmentation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pages 8623–8632, 2020.
* [65] Manuel Stoiber, Martin Sundermeyer, and Rudolph Triebel. Iterative Corresponding Geometry: Fusing Region and Depth for Highly Efficient 3D Tracking of Textureless Objects. In 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 6845–6855, New Orleans, LA, USA, June 2022. IEEE.
* [66] Jürgen Sturm, Stéphane Magnenat, Nikolas Engelhard, François Pomerleau, Francis Colas, Daniel Cremers, Roland Siegwart, and Wolfram Burgard. Towards a benchmark for RGB-D SLAM evaluation. In RGB-D Workshop on Advanced Reasoning with Depth Cameras at Robotics: Science and Systems Conf.(RSS), Los Angeles, United States, 2011.
* [67] Guoxing Sun, Xin Chen, Yizhang Chen, Anqi Pang, Pei Lin, Yuheng Jiang, Lan Xu, Jingya Wang, and Jingyi Yu. Neural free-viewpoint performance rendering under complex human-object interactions. In Proceedings of the 29th ACM International Conference on Multimedia, 2021.
* [68] Jiaming Sun, Zihao Wang, Siyu Zhang, Xingyi He, Hongcheng Zhao, Guofeng Zhang, and Xiaowei Zhou. OnePose: One-shot object pose estimation without CAD models. CVPR, 2022.
* [69] Omid Taheri, Nima Ghorbani, Michael J. Black, and Dimitrios Tzionas. GRAB: A dataset of whole-body human grasping of objects. In European Conference on Computer Vision (ECCV), 2020.
* [70] Yating Tian, Hongwen Zhang, Yebin Liu, and Limin Wang. Recovering 3d human mesh from monocular images: A survey. arXiv preprint arXiv:2203.01923, 2022.
* [71] Garvita Tiwari, Dimitrije Antic, Jan Eric Lenssen, Nikolaos Sarafianos, Tony Tung, and Gerard Pons-Moll. Pose-ndf: Modeling human pose manifolds with neural distance fields. In European Conference on Computer Vision (ECCV). Springer, October 2022.
* [72] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Ł ukasz Kaiser, and Illia Polosukhin. Attention is all you need. In I. Guyon, U. Von Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vishwanathan, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 30. Curran Associates, Inc., 2017.
* [73] Gu Wang, Fabian Manhardt, Federico Tombari, and Xiangyang Ji. GDR-Net: Geometry-guided direct regression network for monocular 6d object pose estimation. In IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 16611–16621, June 2021.
* [74] Xi Wang, Gen Li, Yen-Ling Kuo, Muhammed Kocabas, Emre Aksan, and Otmar Hilliges. Reconstructing action-conditioned human-object interactions using commonsense knowledge priors. In International Conference on 3D Vision (3DV), 2022.
* [75] Bowen Wen and Kostas Bekris. BundleTrack: 6D Pose Tracking for Novel Objects without Instance or Category-Level 3D Models. In 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 8067–8074, Prague, Czech Republic, Sept. 2021. IEEE Press.
* [76] Bowen Wen, Chaitanya Mitash, Baozhang Ren, and Kostas E. Bekris. se(3)-tracknet: Data-driven 6d pose tracking by calibrating image residuals in synthetic domains. 2020 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Oct 2020.
* [77] Zhenzhen Weng and Serena Yeung. Holistic 3d human and scene mesh estimation from single view images. arXiv preprint arXiv:2012.01591, 2020.
* [78] Shangzhe Wu, Tomas Jakab, Christian Rupprecht, and Andrea Vedaldi. DOVE: Learning deformable 3d objects by watching videos. arXiv preprint arXiv:2107.10844, 2021.
* [79] Yuxin Wu, Alexander Kirillov, Francisco Massa, Wan-Yen Lo, and Ross Girshick. Detectron2. https://github.com/facebookresearch/detectron2, 2019.
* [80] Xianghui Xie, Bharat Lal Bhatnagar, and Gerard Pons-Moll. Chore: Contact, human and object reconstruction from a single rgb image. In European Conference on Computer Vision (ECCV). Springer, October 2022.
* [81] Xiang Xu, Hanbyul Joo, Greg Mori, and Manolis Savva. D3d-hoi: Dynamic 3d human-object interactions from videos. arXiv preprint arXiv:2108.08420, 2021.
* [82] Gengshan Yang, Deqing Sun, Varun Jampani, Daniel Vlasic, Forrester Cole, Huiwen Chang, Deva Ramanan, William T Freeman, and Ce Liu. Lasr: Learning articulated shape reconstruction from a monocular video. In CVPR, 2021.
* [83] Gengshan Yang, Deqing Sun, Varun Jampani, Daniel Vlasic, Forrester Cole, Ce Liu, and Deva Ramanan. Viser: Video-specific surface embeddings for articulated 3d shape reconstruction. In NeurIPS, 2021.
* [84] Gengshan Yang, Minh Vo, Natalia Neverova, Deva Ramanan, Andrea Vedaldi, and Hanbyul Joo. Banmo: Building animatable 3d neural models from many casual videos. In CVPR, 2022.
* [85] Lixin Yang, Xinyu Zhan, Kailin Li, Wenqiang Xu, Jiefeng Li, and Cewu Lu. CPF: Learning a contact potential field to model the hand-object interaction. In ICCV, 2021.
* [86] Hongwei Yi, Chun-Hao P. Huang, Shashank Tripathi, Lea Hering, Justus Thies, and Michael J. Black. MIME: Human-aware 3D scene generation. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2023.
* [87] Hongwei Yi, Chun-Hao P. Huang, Dimitrios Tzionas, Muhammed Kocabas, Mohamed Hassan, Siyu Tang, Justus Thies, and Michael J. Black. Human-aware object placement for visual environment reconstruction. In IEEE/CVF Conf. on Computer Vision and Pattern Recognition (CVPR), pages 3959–3970, June 2022.
* [88] Hongwei Yi, Hualin Liang, Yifei Liu, Qiong Cao, Yandong Wen, Timo Bolkart, Dacheng Tao, and Michael J Black. Generating holistic 3d human motion from speech. In IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2023.
* [89] Ye Yuan, Umar Iqbal, Pavlo Molchanov, Kris Kitani, and Jan Kautz. Glamr: Global occlusion-aware human mesh recovery with dynamic cameras. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2022.
* [90] Ailing Zeng, Lei Yang, Xuan Ju, Jiefeng Li, Jianyi Wang, and Qiang Xu. Smoothnet: A plug-and-play network for refining human poses in videos. In European Conference on Computer Vision. Springer, 2022.
* [91] Jason Y. Zhang, Sam Pepose, Hanbyul Joo, Deva Ramanan, Jitendra Malik, and Angjoo Kanazawa. Perceiving 3d human-object spatial arrangements from a single image in the wild. In European Conference on Computer Vision (ECCV), 2020.
* [92] Tianshu Zhang, Buzhen Huang, and Yangang Wang. Object-occluded human shape and pose estimation from a single color image. In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), pages 7374–7383, 2020.
* [93] Xiaohan Zhang, Bharat Lal Bhatnagar, Sebastian Starke, Vladimir Guzov, and Gerard Pons-Moll. Couch: Towards controllable human-chair interactions. In European Conference on Computer Vision (ECCV). Springer, October 2022.
* [94] Linfang Zheng, Aleš Leonardis, Tze Ho Elden Tse, Nora Horanyi, Hua Chen, Wei Zhang, and Hyung Jin Chang. TP-AE: Temporally Primed 6D Object Pose Tracking with Auto-Encoders. In 2022 International Conference on Robotics and Automation (ICRA), pages 10616–10623, Philadelphia, PA, USA, May 2022. IEEE Press.
* [95] Keyang Zhou, Bharat Lal Bhatnagar, Jan Eric Lenssen, and Gerard Pons-Moll. Toch: Spatio-temporal object correspondence to hand for motion refinement. In European Conference on Computer Vision (ECCV). Springer, October 2022.
* [96] Yi Zhou, Connelly Barnes, Lu Jingwan, Yang Jimei, and Li Hao. On the continuity of rotation representations in neural networks. In The IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2019.
|
# Gradient estimates of nonlinear equation on complete noncompact metric
measure space with compact boundary
Xiang-Zhi Cao School of information engineering, Nanjing Xiaozhuang
University, Nanjing 211171, ChinaEmail<EMAIL_ADDRESS>
|
11institutetext: Delft University of Technology, Delft, The Netherlands
11email: {L.Lyu, N.Roy<EMAIL_ADDRESS>22institutetext: Radboud
University, Nijmegen, The Netherlands
22email<EMAIL_ADDRESS>
# Is Interpretable Machine Learning Effective at Feature Selection for Neural
Learning-to-Rank?
Lijun Lyu 11 0000-0002-7268-4902 Nirmal Roy 11 0000-0003-0860-5269 Harrie
Oosterhuis 22 0000-0002-0458-9233 Avishek Anand 11 0000-0002-0163-0739
###### Abstract
Neural ranking models have become increasingly popular for real-world search
and recommendation systems in recent years. Unlike their tree-based
counterparts, neural models are much less interpretable. That is, it is very
difficult to understand their inner workings and answer questions like how do
they make their ranking decisions? or what document features do they find
important? This is particularly disadvantageous since interpretability is
highly important for real-world systems. In this work, we explore feature
selection for neural learning-to-rank (LTR). In particular, we investigate six
widely-used methods from the field of interpretable machine learning (ML) and
introduce our own modification, to select the input features that are most
important to the ranking behavior. To understand whether these methods are
useful for practitioners, we further study whether they contribute to
efficiency enhancement. Our experimental results reveal a large feature
redundancy in several LTR benchmarks: the local selection method TabNet can
achieve optimal ranking performance with less than 10 features; the global
methods, particularly our G-L2x, require slightly more selected features, but
exhibit higher potential in improving efficiency. We hope that our analysis of
these feature selection methods will bring the fields of interpretable ML and
LTR closer together.
## 1 Introduction
Learning-to-rank (LTR) is at the core of many information retrieval (IR) and
recommendation tasks [38]. The defining characteristic of LTR, and what
differentiates it from other machine learning (ML) areas, is that LTR methods
aim to predict the optimal ordering of items. This means that LTR methods are
not trying to estimate the exact relevance of an item, but instead predict
relative relevance differences, i.e., whether it is more or less relevant than
other items. Traditionally, the most widely adopted and prevalent LTR methods
were based on Gradient Boosted Decision Trees (GBDT) [12, 65, 29]. However, in
recent years, neural LTR methods have become increasingly popular [21, 46,
47]. Recently, [51] have shown that neural models can provide ranking
performance that is comparable, and sometimes better, than that of state-of-
the-art GBDT LTR models on established LTR benchmark datasets [49, 14, 17]. It
thus seems likely that the prevalence of neural LTR models will only continue
to grow in the foreseeable future.
Besides the quality of the results that ranking systems return, there is an
increasing interest in building trustworthy systems through interpretability,
e.g., by understanding which features contribute the most to ranking results.
Additionally, the speed at which results are provided is also highly important
[5, 7, 3]. Users expect ranking systems to be highly responsive and previous
work indicates that even half-second increases in latency can contribute to a
negative user experience [7]. A large part of ranking latency stems from the
retrieval and computation of input features for the ranking model.
Consequently, _feature selection_ for ranking systems has been an important
topic in the LTR field [22, 23, 59, 52, 45, 68, 48]. These methods reduce the
number of features used, thereby helping users understand and greatly reduce
latency and infrastructure costs, while maintaining ranking quality as much as
possible. In line with the history of the LTR field, existing work on feature
selection has predominantly focused on GBDT and support-vector-machine (SVM)
ranking models [26, 21], but has overlooked neural ranking models. To the best
of our knowledge, only two existing works have looked at feature selection for
neural LTR [48, 52]. This scarcity is in stark contrast with the importance of
feature selection and the increasing prevalence of neural models in LTR.
Outside of the LTR field, feature selection for neural models has received
much more attention, for the sake of efficiency [35, 36], and also to better
understand the model behaviours [72, 4]. Those methods mainly come from the
_interpretable_ ML field [18, 44], where the idea is that the so-called
_concrete_ feature selection can give insights into what input information a
ML model uses to make decisions. This tactic has already been successfully
applied to natural language processing [71], computer vision [6], and tabular
data [69, 4]. Accordingly, there is a potential for these methods to also
enable embedded feature selection for neural LTR models, where the selection
and prediction are optimized simultaneously. However, the effectiveness of
these interpretable ML methods for LTR tasks is currently unexplored, and
thus, it remains unclear whether their application can translate into useful
insights for LTR practitioners.
The goal of this work is to investigate whether six prevalent feature
selection methods – each representing one of the main branches of
interpretable ML field – can be applied effectively to neural LTR. In
addition, we also propose a novel method with minor modifications. Our aim is
to bridge the gap between the two fields by translating the important concepts
of the interpretable ML field to the LTR setting, and by demonstrating how
interpretable ML methods can be adapted for the LTR task. Moreover, our
experiments consider whether these methods can bring efficiency into the
practical application by reducing irrelevant input features for neural ranking
models.
Our results reveal a large feature redundancy in LTR benchmark datasets, but
this redundancy can be understood differently for interpretability and for
efficiency: For understanding the model, feature selection can vary per
document and less than 10 features are required to approximate optimal ranking
behavior. In contrast, for practical efficiency purposes, the selection should
be static, and then 30% of features are needed. We conclude that – when
adapted for the LTR task – not all, but a few interpretable ML methods lead to
effective and practical feature selection for neural LTR.
To the best of our knowledge, this is the first work that extensively studies
embedded feature selection for neural LTR. We hope our contributions bring
more attention to the potential of interpretable ML for IR field. To stimulate
future work and enable reproducibility, we have made our implementation
publicly available at:
https://github.com/GarfieldLyu/NeuralFeatureSelectionLTR (MIT license).
## 2 Related Work
Learning-to-Rank (LTR). Traditional LTR algorithms mainly rely on ML models,
such as SVMs and decision trees to learn the correlation between numerical
input features and human-annotated relevance labels [37, 15, 20, 70, 67, 65,
31, 27]. Neural approaches [11, 10, 13, 66, 54, 61] have also been proposed,
but did not show significant improvements over traditional non-neural models.
Inspired by the transformer architecture [63], recent works have also adapted
self-attention [46, 47, 51] and produced the neural LTR methods that
outperform LambdaMART [65], albeit with a relatively small difference. It
shows that neural rankers can provide competitive performance, consequently,
the interest and effort towards neural models for LTR are expected to increase
considerably in the near future.
Efficiency is crucial in real-world systems since users expect them to be
highly responsive [5, 7, 3]. Aside from model execution, the latency of
ranking system is largely due to feature construction, as it happens _on-the-
fly_ for incoming queries. Thus, efficiency is often reached by reducing
(expensive) features. Previous works [21, 64] apply a cascading setup to
reduce the usage of expensive features. Another growing trend in LTR is to
design interpretable models. Existing methods rely on specific architecture
design, such as general additive model (GAM) [73] or a feature-interaction
constrained and depth-reduced tree model [39].
Feature Selection for LTR. Feature selection can achieve both efficiency and
interpretability [4, 35, 36, 72]. By selecting a subset of input features, the
input complexity of models is reduced while maintaining competitive
performance. This helps with (1) efficiency as it avoids unnecessary
construction of features [35, 36], and (2) interpretability as fewer input
features are involved in prediction [72, 4].
Existing feature selection methods in LTR are classified commonly as filter,
wrapper and embedded methods [22, 23, 48]. Filter and wrapper methods are
applied to given static ranking models which are not updated in any way;
filter methods are model-agnostic [22] while wrapper methods are designed for
a particular type of model [23]. In this work we will focus on the third
category, embedded methods, where feature selection is performed
simultaneously with model optimization. Most embedded methods are limited to
particular model designs such as SVMs [32, 33, 34] or decision trees [45, 68,
40]. To the best of our knowledge, only two methods are designed for neural
LTR [52, 48]: one applies group regularization methods [52] to reduce both
input and other model parameters; the other [48] uses the gradients of a
static ranking model to infer feature importance, and thus it belongs to the
_filter_ category. We do not investigate these two methods further, as the
focus of this work is on _embedded_ input feature selection methods.
Interpretable Machine Learning. The earliest work in interpretable ML
attempted to explain a trained model in _post-hoc_ manner, mainly relying on
input perturbations [53, 41], gradients [58, 60] and so on [57]. In parallel,
more recent works advocated intrinsically interpretable models, that are
categorized as _interpretable-by-design_ methods [56, 2, 1]. For neural
networks, explaining the decision path is challenging due to the large set of
parameters. Therefore, the more prevalent choice for intrinsic interpretable
neural models is to shift the transparency to the input features. Namely, the
final prediction comes from a subset selection of input elements, e.g., words
or pixels and the rest irrelevant features are masked out [71, 16, 36].
Importantly, this selection decision can be learned jointly with the
predictive accuracy of a model. Thereby, we limit our research focus in
intrinsic interpretable ML models.
Due to the discrete nature of selection, many approaches such as L2X [16],
Concrete AutoEncoders (CAE) [6], Instance-wise Feature grouping (IFG) [43]
apply Gumbel-Softmax sampling [24] to enable backpropagation through the
feature selection process. Alternatively, regularization is also a commonly-
used feature selection approach in traditional ML algorithms [62, 31], and is
applicable to neural models, i.e., with INVASE [69] or LassoNet [35].
Moreover, TabNet [4] applies both regularization and the sparsemax activation
function [42] to realize sparse selection. These approaches have been
successfully applied in language, vision and tabular domains, and suggested
that the resulting feature selections substantially improved the user
understanding of models and datasets [28, 55].
Despite their success in other domains, we find that the above-mentioned
feature selection methods for neural models (L2X, CAE, IFG, INVASE, LassoNet
and TabNet) have not been studied in the LTR setting. In response, we hope to
bridge this gap between the interpretable ML and the LTR field by adapting and
applying these methods to neural ranking models.
## 3 Background
### 3.1 Learning-to-Rank (LTR)
The LTR task can be formulated as optimizing a scoring function $f$ that given
item features $x$ predicts an item score $f(x)\in\mathbb{R}$, so that ordering
items according to their scores corresponds to the optimal ranking [38].
Generally, there are relevance labels $y$ available for each item, often these
are labels provided by experts where $y\in\\{0,1,2,3,4\\}$ [14, 50, 49]. Given
a training set $\mathcal{D}_{q}=\\{(x_{i},y_{i})\\}^{N_{q}}_{i=1}$ for a
single query $q$, optimization is done by minimizing a LTR loss, for instance,
the _softmax cross entropy loss_ [9, 13]:
$\mathcal{L}(f\mid\mathcal{D}_{q})=-\frac{1}{|\mathcal{D}_{q}|}\underset{(x,y)\in\mathcal{D}_{q}}{\sum}\sum_{i=1}^{N_{q}}y_{i}\log\sigma(x_{i}\mid
f,\mathcal{D}_{q}),$ (1)
where $\sigma$ is the softmax activation function:
$\sigma(x\mid
f,\mathcal{D}_{q})=\frac{\exp(f(x))}{\sum_{x^{\prime}\in\mathcal{D}_{q}}\exp(f(x^{\prime}))}.$
(2)
The resulting $f$ is then commonly evaluated with a ranking metric, for
instance, the widely-used normalized discounted cumulative gain metric (NDCG)
[25].
### 3.2 Properties of Feature Selection Methods
Table 1: Properties of feature selection methods from the interpretable ML field as discussed in Section 3. Method | Global | Local | Sampling | Regularization | Fixed-Budget | Composable
---|---|---|---|---|---|---
L2x [16] | | ✓ | ✓ | | ✓ | ✓
Invase [69] | | ✓ | ✓ | ✓ | | ✓
Cae [6] | ✓ | | ✓ | | ✓ | ✓
Ifg [43] | | ✓ | ✓ | | | ✓
LassoNet [35] | ✓ | | | ✓ | | ✓
TabNet [4] | | ✓ | | ✓ | |
G-L2x (ours) | ✓ | | ✓ | | ✓ | ✓
As discussed before, feature selection is used in the interpretable ML field
to better understand which input features ML models use to make their
predictions. Furthermore, feature selection is also important to LTR for
increasing the efficiency of ranking systems. However, selecting a subset of
input features without compromising the model performance is an NP-hard
problem, since the number of possible subsets grows exponentially with the
number of available features [23, 48]. As a solution, the interpretable ML
field has proposed several methods that approach feature selection as an
optimization problem. We will now lay out several important properties that
can be used to categorize these methods, which will be elaborated in next
section.
Global vs. local. Global methods select a single subset of features for the
entire dataset, whereas local methods can vary their selection over different
items.
Composable vs. non-composable. Non-composable methods are designed for a
specific model architecture, and therefore, they can only perform feature
selection for those models. Conversely, composable methods are not constrained
to specific architectures, and thus, they work for any (differentiable) model.
Fixed-budget vs. budget-agnostic. Fixed-budget methods work with a pre-defined
selection budget, i.e., what number of features should be selected in total,
or a cost per feature and a maximum total cost for the selection. Their
counterparts are budget-agnostic methods that do not use an explicit budget,
consequently, one has to carefully fine-tune their hyper-parameters to achieve
a desired performance-sparsity trade-off.
Sampling-based vs. regularization-based. As their names imply, sampling-based
methods optimize a sampling procedure to perform the feature selection,
whereas regularization-based methods use an added regularization loss to
stimulate sparsity in feature selection. While these groups apply very
different approaches, whether one is significantly more useful for LTR
purposes than the other remains unknown.
## 4 Feature Selection from Interpretable ML for LTR
Figure 1: Methods overview, as described in Section 4.
In this section, we present a brief technical overview of our selection of six
interpretable ML methods and their adaption to neural LTR models, and propose
our G-L2x method based on a minor modification. While any ranking loss can be
chosen, we use a listwise softmax cross entropy (Eq. 1) with all methods, for
the sake of simplicity. Therefore, the training of each query is conducted
after generating the output of all documents associated with the query. Table
1 highlights the properties of all methods and Figure 1 provides a visual
overview to accompany this section.
### 4.1 Sampling-based Feature Selection
Sampling-based approaches use a two-stage architecture consisting of a
_selector_ that generates a sparse selection over the input features; and a
_ranker_ that only takes selected features as its input, in the form of a
masked vector $\hat{x}$.
The training of a ranker follows conventional LTR, i.e., Eq 1 with $x_{i}$
replaced by $\hat{x_{i}}$. But the optimization of a selector ($\zeta$) is not
as straightforward; Usually, $\zeta$ constructs a probability distribution
$\mathbf{p}=[p_{1},p_{2},\cdots p_{d}]$, indicating a selection probability
per feature. However, the ranker uses a concrete selection $m\in\\{0,1\\}^{d}$
from the probability distribution, and this concrete operation does not allow
optimization of the selector via backpropagation. The common solution is to
generate a differentiable approximation $\tilde{m}$, by _concrete relaxation_
or the Gumbel-Softmax trick [24]. Namely, the selection of $p_{i}$ can be
approximated with the differentiable $c_{i}$ as:
$c_{i}=\frac{exp\\{(\log p_{i}+g_{i})/\tau\\}}{\sum_{j=1}^{d}exp\\{(\log
p_{j}+g_{i})/\tau\\}},$ (3)
where $g$ is the Gumbel noise and $\tau\in\mathbb{R}^{>0}$ is the temperature
parameter. Now, the selector $\zeta$ can be optimized with stochastic gradient
descent by using $\tilde{m}$. The following four sampling-based methods apply
this overall procedure, but differ in how they generate $\mathbf{p}$ and
$\tilde{m}$.
Learning to explain (L2x). L2x [16] is a local selection method since its
neural selector generates a probability distribution $\mathbf{p}$ for each
individual input instance. To generate $\tilde{m}$, L2x repeats the sampling
procedure $k$ times (Eq. 3), and subsequently, uses the maximum $c_{i}$ out of
the $k$ repeats for the $i_{th}$ element in $\tilde{m}$. The intention behind
this maximization step is to make the top-$k$ important features more likely
to have high probability scores (ideally close to 1).
Global learning to explain (G-L2x, ours). As a counterpart, we propose a
global method G-L2xbased on L2x. Our change is straightforward, where L2x
generates a different distribution $\mathbf{p}$ for each item, we apply the
same $\mathbf{p}$ to all items. In other words, G-L2x includes a global
selector layer $\zeta$ ($\zeta\in\mathbb{R}^{d}$) to simulate $\mathbf{p}$,
and sampling is conducted in the same way as L2x on the selector weights.
Thereby, G-L2x will select the same features for all items in the dataset.
Concrete autoencoder (Cae). Cae [6] is a global method where the selector is
the encoder part of an auto-encoder model [30]. Specifically, the selector
compresses the input into a smaller representation $\hat{x}$, by linearly
combining selected features, i.e. $x^{\top}\tilde{m}$, where
$\tilde{m}\in\mathbb{R}^{k\times d}$ can be viewed as approximated k-hot
concrete selection, sampled from the selector weights
($\zeta\in\mathbb{R}^{k\times d}$). Therefore, Cae might result in repetitive
selection, and the input dimension to the predictor is reduced to $k$.
Instance-wise Feature Grouping (Ifg). Ifg [43] applies a similar approach as
L2x, but clusters features into groups and then selects $k$ feature groups for
prediction. Ifg first assigns a group for each feature via Gumbel-sampling,
and then makes a feature selection by Gumbel-sampling $k$ out of the resulting
groups. This grouping decision is also guided by how rich the selected
features are to recover the original input. Therefore, apart from the ranking
objective, Ifg jointly optimizes an additional input restoring objective as
well (similar to auto-encoders [30]). Ifg is agnostic to the number of
selected features and the group sizes, it can produce oversized groups and
very large selections.
### 4.2 Regularization-based Feature Selection
Instead of the budget-explicit feature selection, regularization-based methods
induce sparsity through implicit constraints enforced by regularization terms
in the training objective. We propose modifications to three existing methods
to make them applicable to the LTR setting.
Invase. We consider Invase [69] to be a hybrid approach involving both
sampling and regularization. Built on the same structure as L2x, the selector
of Invase generates a boolean/hard mask $m$ (instead of the approximation
$\tilde{m}$) via Bernoulli sampling to train the predictor. Since this
disables backpropagation, Invase uses a customized loss function that does not
need the gradients from the predictor to train the selector. The idea is to
apply another individual baseline predictor model that takes the full-feature
input, simultaneously with the predictor that takes the masked input. The loss
difference between the two predictors is used as a scale to train the
selector. Meanwhile, the L1 regularization is applied to the selector output
to enforce selection sparsity. Ultimately, Invase will push the selector to
output a small set of selections which leads to the most similar predictions
as using all features.
LassoNet. As the name suggests, LassoNet [35] adapts a traditional Lasso (L1
regularization) [62] on the first layer of a neural model to eliminate
unnecessary features. The challenge with neural models is, all weights
corresponding to a particular feature entry in the layer have to be zero in
order to mask out the feature. Towards this, LassoNet adds a residual layer
with one weight per input feature to the original model to perform as the
traditional Lasso. Then, after every optimization step, LassoNet develops a
proximal optimization algorithm to adjust the weights of the first layer, so
that all absolute elements of each row are smaller than the respective weight
of residual layer corresponding to a specific feature. Thereby, LassoNet
performs global selection and the sparsity scale is adjusted by the L1
regularization on the residual layer weights.
TabNet. Unlike the previous methods, TabNet [4] is non-composable and tied to
a specific tree-style neural architecture. It imitates a step-wise selection
process before it outputs the final prediction based only on the selected
features. Each step has the same neural component/block but with its own
parameters, thus the model complexity and selection budget grow linearly with
the number of steps. At each step, the full input is transformed by a feature
transformer block first, and then an attentive transformer block conducts
feature selection by sparsemax activation [42], as the weights in the
resulting distribution corresponding to unselected features are zeros. The
final prediction is aggregated from all steps to simulate ensemble models. A
final mask $m$ is a union of selections from all steps, and the entropy of the
selection probabilities is used as the sparsity regularization.
## 5 Experimental Setup
Since feature selection can be applied in various manners and situations, we
structure our experiments around three scenarios:
* •
Scenario 1: Simultaneously train and select. Both the ranking model and the
feature selection are learned once and jointly. The methods are evaluated by
the performance-sparsity trade-off. It is the standard setup for evaluating
embedded feature selection methods in the interpretable ML field [52, 43].
* •
Scenario 2: Train then select with an enforced budget. Practitioners generally
set hard limits to the computational costs a system may incur and the
efficiency of the system can be greatly enhanced if it only requires a much
smaller amount of features to reach competitive performance. Following the
previous scenario, we evaluate the trained model with test instances where
only a fixed amount of features (which the method deems important and selects
frequently during training) are presented and the rest are masked out. The
resulting ranking performance and the costs of computing the required features
indicate how practical the method is in efficiency improvements.
Datasets and preprocessing. We choose three public benchmark datasets: MQ2008
(46 features) [50], Web30k (136 features) [49] and Yahoo (699 features) [14],
to cover varying numbers of available features. We apply a $\log_{1p}$
transformation to the features of Web30k, as suggested in [51]. Yahoo contains
cost labels for each feature, for Web30k we use cost estimates suggested by
previous work [21].111MQ2008 is omitted from cost analysis since no associated
cost information is available. All reported results are evaluated on the held-
out test set partitions of the datasets.
Models. We use a standard feed-forward neural network with batch
normalization, three fully-connected layers and tanh activation as the ranking
model, denoted as Dnn. According to the findings in [51], this simple model
performs closely to the most effective transformer-based models, but requires
much less resources to run. The selector models of L2x and Invase have the
same architecture, and as the only exception, TabNet is applied with its own
unalterable model (see Section 4).
Implementation. Our experimental implementation is done in PyTorch Lightning
[19]. For TabNet and LassoNet existing implementations were
used.222https://github.com/dreamquark-ai/tabnet; https://github.com/lasso-
net/lassonet We created our own implementations for the rest of the methods.
Table 2: Results of ranking performance and feature sparsity for methods
applied in Scenario 1. For comparison, we also include Gbdt [29, 8] and Dnn
baselines without feature selection as upper bound. #F denotes the number of
selected features. Reported results are averaged over $5$ random seeds (std in
parentheses). Bold font indicates the highest performing selection method; the
$\star$ and underlines denote scores that are _not_ significantly outperformed
by Gbdt and the bold-score method, respectively ($p>0.05$, paired t-tests
using Bonferonni’s correction).
Listwise | MQ2008 NDCG@k (%) | Web30k NDCG@k (%) | Yahoo NDCG@k (%)
---|---|---|---
loss | @1 | @10 | #F | @1 | @10 | #F | @1 | @10 | #F
Without feature selection.
Gbdt | 69.3 (2.5) | 80.8 (1.7) | 46 | 50.4 (0.1) | 52 (0.1) | 136 | 72.2 (0.1) | 79.2 (0.1) | 699
Dnn | 66.2⋆ (2) | 80.2⋆ (0.6) | 46 | 46.1 (0.6) | 47.7 (0.2) | 136 | 69.4 (0.3) | 76.9 (0.1) | 699
Fixed-budget feature selection using the Dnn ranking model. |
Cae | 63.0 (1.1) | 78.7 (0.5) | 4 | 32.9 (2.9) | 36.6 (2.2) | 13 | 59.2 (0.2) | 69.5 (0.1) | 6
G-L2x | 63.8⋆ (1.3) | 79.1 (0.4) | 4 | 41.1 (0.9) | 44.4 (0.3) | 13 | 65.4 (0.1) | 74.0 (0.0) | 6
L2x | 63.0 (2.1) | 78.7 (0.7) | 4 | 34.5 (2.4) | 39.7 (1.9) | 13 | 61.9 (1.1) | 73.2 (0.3) | 6
Budget-agnostic feature selection using the Dnn ranking model. |
Invase | 62.5 (2.2) | 77.5 (2.1) | 5 (2) | 15.1 (0.0) | 22.1 (0.0) | 0 | 38.7 (0.0) | 57.8 (0.0) | 0
Ifg | 66.4⋆ (0.9) | 80.4⋆ (0.5) | 20 (2) | 32.5 (5.3) | 37.5 (5.3) | 72 (30) | 69.6 (0.2) | 77.1 (0.2) | 58 (3)
LassoNet | 64.7⋆ (2.2) | 79.3 (1.2) | 6 (3) | 39.4 (0.8) | 42.1 (0.3) | 8 (2) | 63.1 (2.3) | 72.4 (1.5) | 12 (4)
Budget-agnostic feature selection using a method-specific ranking model. |
TabNet | 64.7⋆ (2.7) | 78.2 (1.2) | 7 (3) | 47.0 (0.4) | 49.2 (0.1) | 8 (1) | 70.2 (0.4) | 77.7 (0.1) | 6 (1)
| MQ2008 | Web30k | Yahoo
---|---|---|---
NDCG@10 | | |
Figure 2: Results of three fixed-budget methods applied to scenario 1. The
x-axis indicates the pre-specified percentile of selected features ($k$). The
shaded area shows the standard deviation over 5 random seeds.
## 6 Results
We report the findings in this section, aiming to answer two questions: (1)
how effective are investigated methods in the ranking setup? and (2) can those
methods improve efficiency? Each question corresponds to one of the scenarios
described in Section 5.
Simultaneous Optimization and Selection. We begin by investigating the
effectiveness of the feature selection methods when applied to Scenario 1,
where feature selection and model optimization are performed simultaneously.
The results for this scenario are displayed in Table 2 and Figure 2.
Table 2 shows the ranking performance and the respective feature sparsity of
all feature selection methods and two baselines without any selection as the
upper-bound reference. For fixed-budget methods, the budgets were set to 10%
of the total number of features for MQ2008 and Web30k, and 1% for Yahoo (the
results with varying budgets are displayed in Figure 2). Since the sparsity of
budget-agnostic methods is more difficult to control, we performed an
extensive grid search and used the hyper-parameters that produced the highest
ranking performance with a comparable feature sparsity as the other methods.
The results in Table 2 show that not all feature selection methods are equally
effective, and their performance can vary greatly over datasets. For instance,
on MQ2008 all methods perform closely to the baselines, with only a fraction
of the features. However, this is not the case for bigger datasets like Web30k
and Yahoo. In particular, Invase selects no features at all due to big
uncertainty in selection (for this reason, we omit Invase from all further
comparisons). On the other hand, Ifg performs poorly in inducing sparsity,
mainly because of its input reconstruction objective, whereas the ranking
performance is not substantially better than the rest of methods.
Additionally, Cae does not seem effective either, and furthermore, increasing
the selected features does not always result in better ranking performance
(cf. Figure 2). This is most-likely because Cae samples with replacement, and
thus the same features can be selected repeatedly.
In contrast, the other two sampling-based methods L2x and G-L2x are designed
to avoid the repetitive selection issue. Overall, the global selection G-L2x
outperforms the local counterpart L2x, possibly because global selection
generalizes better to unseen data. Another global method LassoNet is also
inferior to G-L2x, mainly due to the difficulties in sparsity weight tuning
and manually adjusting weights in the input layer.
Lastly, TabNet shows the best performance-sparsity balance across all
datasets, and even outperforms the Dnn baseline. Although, the comparison
between TabNet and Dnn is not completely fair, as they optimize different
neural architectures. It does reveal large feature redundancies in these
datasets: TabNet uses <10% of features on Web30k and 1% on Yahoo, yet still
beats the Dnn baseline with all features.
To summarize, we find that the local method TabNet is the most effective at
balancing ranking performance and sparsity. Slightly inferior but competitive
enough is the global method G-L2x, which reached $>95\%$ of baseline
performance with only 1% features on Yahoo and $>93\%$ with 10% on Web30k.
Feature Cost | Ranking Performance (NDCG@10)
---|---
Web30k | Yahoo | Web30k | Yahoo
| | |
Figure 3: Scenario 2. Feature cost (left two) and ranking performance (right
two) under incomplete input. The x-axis indicates how many percentages of
features are present in the input, to test the trained ranking model. Note
this differs from specifying $k$ during training for fixed-budget methods in
scenario 1.
Feature Selection for Trained Ranking Models. Next, we evaluate the methods in
Scenario 2, where only a specified budget (i.e., a given number of features)
of features are present in the test input. Figure 3 displays both the ranking
performance and the total feature cost for varying degrees of sparsity. The
costs represent the time it requires to retrieve the selected feature sets,
and allow us to estimate the actual efficiency improvements they provide.
Unlike previous scenario, all local methods including TabNet, are no longer
able to maintain superior performance. This is because for local methods, the
selection is made conditioned on full input information, and an incomplete
input could affect the selection and thus disrupt its prediction performance.
In contrast, global methods are immune to input changes. Therefore, Cae is
still not performing well as it did in Scenario 1; G-L2x and LassoNet provide
the best overall performance under small costs. LassoNet maintains baseline
performance with less than 40% of features on both datasets, while G-L2x
outperforms LassoNet when selected features are less than 30%. Meanwhile, it
also shows LassoNet tends to select more costly features than G-L2x.
To conclude, we find that global methods G-L2x and LassoNet perform the best
in Scenario 2, where upcoming query inputs are masked under enforced feature
budgets. Particularly, G-L2x is superior in both ranking and computing cost
when the feature budget is small. This translates to substantial efficiency
improvements in practical terms, as ranking performance is maintained by
selected features only.
## 7 Conclusion
The main goal of this work is to bring the interpretable ML and the LTR fields
closer together. To this end, we have studied whether feature selection
methods from the interpretable ML are effective for neural LTR, for both
interpretability and efficiency purposes.
Inspired by the scarcity of feature selection methods for neural ranking
models in previous work, we adapted six existing methods from the
interpretable ML for the neural LTR setting, and also proposed our own G-L2x
approach. We discussed different properties of these methods and their
relevance to the LTR task. Lastly, we performed extensive experiments to
evaluate the methods in terms of their trade-offs between ranking performance
and sparsity, in addition, their efficiency improvements through feature cost
reductions. Our results have shown that several methods from interpretable ML
are highly effective at embedded feature selection for neural LTR. In
particular, the local method TabNet can reach the upper bound with less than
10 features; the global methods, in particular G-L2x, can reduce feature
retrieval costs by more than 70%, while maintaining 96% of performance
relative to a full feature model.
We hope our investigation can bridge the gap between the LTR and interpretable
ML fields. The future work can be developing more interpretable and efficient
ranking systems, and how that interpretability could support both
practitioners and the users of ranking systems.
#### 7.0.1 Acknowledgements
This work is partially supported by German Research Foundation (DFG), under
the Project IREM with grant No. AN 996/1-1, and by the Netherlands
Organisation for Scientific Research (NWO) under grant No. VI.Veni.222.269.
## References
* [1] Abdul, A.M., von der Weth, C., Kankanhalli, M.S., Lim, B.Y.: COGAM: measuring and moderating cognitive load in machine learning model explanations. In: Bernhaupt, R., Mueller, F.F., Verweij, D., Andres, J., McGrenere, J., Cockburn, A., Avellino, I., Goguey, A., Bjøn, P., Zhao, S., Samson, B.P., Kocielnik, R. (eds.) CHI ’20: CHI Conference on Human Factors in Computing Systems, Honolulu, HI, USA, April 25-30, 2020. pp. 1–14. ACM (2020). https://doi.org/10.1145/3313831.3376615, https://doi.org/10.1145/3313831.3376615
* [2] Alvarez-Melis, D., Jaakkola, T.S.: Towards robust interpretability with self-explaining neural networks. In: Bengio, S., Wallach, H.M., Larochelle, H., Grauman, K., Cesa-Bianchi, N., Garnett, R. (eds.) Advances in Neural Information Processing Systems 31: Annual Conference on Neural Information Processing Systems 2018, NeurIPS 2018, December 3-8, 2018, Montréal, Canada. pp. 7786–7795 (2018), https://proceedings.neurips.cc/paper/2018/hash/3e9f0fc9b2f89e043bc6233994dfcf76-Abstract.html
* [3] Arapakis, I., Bai, X., Cambazoglu, B.B.: Impact of response latency on user behavior in web search. In: Geva, S., Trotman, A., Bruza, P., Clarke, C.L.A., Järvelin, K. (eds.) The 37th International ACM SIGIR Conference on Research and Development in Information Retrieval, SIGIR ’14, Gold Coast , QLD, Australia - July 06 - 11, 2014. pp. 103–112. ACM (2014). https://doi.org/10.1145/2600428.2609627, https://doi.org/10.1145/2600428.2609627
* [4] Arik, S.Ö., Pfister, T.: Tabnet: Attentive interpretable tabular learning. In: Thirty-Fifth AAAI Conference on Artificial Intelligence, AAAI 2021, Thirty-Third Conference on Innovative Applications of Artificial Intelligence, IAAI 2021, The Eleventh Symposium on Educational Advances in Artificial Intelligence, EAAI 2021, Virtual Event, February 2-9, 2021. pp. 6679–6687. AAAI Press (2021). https://doi.org/10.1609/AAAI.V35I8.16826, https://doi.org/10.1609/aaai.v35i8.16826
* [5] Bai, X., Cambazoglu, B.B.: Impact of response latency on sponsored search. Inf. Process. Manag. 56(1), 110–129 (2019). https://doi.org/10.1016/J.IPM.2018.10.005, https://doi.org/10.1016/j.ipm.2018.10.005
* [6] Balin, M.F., Abid, A., Zou, J.Y.: Concrete autoencoders: Differentiable feature selection and reconstruction. In: Chaudhuri, K., Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA. Proceedings of Machine Learning Research, vol. 97, pp. 444–453. PMLR (2019), http://proceedings.mlr.press/v97/balin19a.html
* [7] Barreda-Ángeles, M., Arapakis, I., Bai, X., Cambazoglu, B.B., Pereda-Baños, A.: Unconscious physiological effects of search latency on users and their click behaviour. In: Baeza-Yates, R., Lalmas, M., Moffat, A., Ribeiro-Neto, B.A. (eds.) Proceedings of the 38th International ACM SIGIR Conference on Research and Development in Information Retrieval, Santiago, Chile, August 9-13, 2015. pp. 203–212. ACM (2015). https://doi.org/10.1145/2766462.2767719, https://doi.org/10.1145/2766462.2767719
* [8] Bruch, S.: An alternative cross entropy loss for learning-to-rank. In: Leskovec, J., Grobelnik, M., Najork, M., Tang, J., Zia, L. (eds.) WWW ’21: The Web Conference 2021, Virtual Event / Ljubljana, Slovenia, April 19-23, 2021. pp. 118–126. ACM / IW3C2 (2021). https://doi.org/10.1145/3442381.3449794, https://doi.org/10.1145/3442381.3449794
* [9] Bruch, S., Wang, X., Bendersky, M., Najork, M.: An analysis of the softmax cross entropy loss for learning-to-rank with binary relevance. In: Fang, Y., Zhang, Y., Allan, J., Balog, K., Carterette, B., Guo, J. (eds.) Proceedings of the 2019 ACM SIGIR International Conference on Theory of Information Retrieval, ICTIR 2019, Santa Clara, CA, USA, October 2-5, 2019. pp. 75–78. ACM (2019). https://doi.org/10.1145/3341981.3344221, https://doi.org/10.1145/3341981.3344221
* [10] Burges, C.J.C., Ragno, R., Le, Q.V.: Learning to rank with nonsmooth cost functions. In: Schölkopf, B., Platt, J.C., Hofmann, T. (eds.) Advances in Neural Information Processing Systems 19, Proceedings of the Twentieth Annual Conference on Neural Information Processing Systems, Vancouver, British Columbia, Canada, December 4-7, 2006. pp. 193–200. MIT Press (2006), https://proceedings.neurips.cc/paper/2006/hash/af44c4c56f385c43f2529f9b1b018f6a-Abstract.html
* [11] Burges, C.J.C., Shaked, T., Renshaw, E., Lazier, A., Deeds, M., Hamilton, N., Hullender, G.N.: Learning to rank using gradient descent. In: Raedt, L.D., Wrobel, S. (eds.) Machine Learning, Proceedings of the Twenty-Second International Conference (ICML 2005), Bonn, Germany, August 7-11, 2005. ACM International Conference Proceeding Series, vol. 119, pp. 89–96. ACM (2005). https://doi.org/10.1145/1102351.1102363, https://doi.org/10.1145/1102351.1102363
* [12] Burges, C.J.: From ranknet to lambdarank to lambdamart: An overview. Learning 11(23-581), 81 (2010)
* [13] Cao, Z., Qin, T., Liu, T., Tsai, M., Li, H.: Learning to rank: from pairwise approach to listwise approach. In: Ghahramani, Z. (ed.) Machine Learning, Proceedings of the Twenty-Fourth International Conference (ICML 2007), Corvallis, Oregon, USA, June 20-24, 2007. ACM International Conference Proceeding Series, vol. 227, pp. 129–136. ACM (2007). https://doi.org/10.1145/1273496.1273513, https://doi.org/10.1145/1273496.1273513
* [14] Chapelle, O., Chang, Y.: Yahoo! learning to rank challenge overview. In: Chapelle, O., Chang, Y., Liu, T. (eds.) Proceedings of the Yahoo! Learning to Rank Challenge, held at ICML 2010, Haifa, Israel, June 25, 2010. JMLR Proceedings, vol. 14, pp. 1–24. JMLR.org (2011), http://proceedings.mlr.press/v14/chapelle11a.html
* [15] Chapelle, O., Keerthi, S.S.: Efficient algorithms for ranking with svms. Inf. Retr. 13(3), 201–215 (2010). https://doi.org/10.1007/S10791-009-9109-9, https://doi.org/10.1007/s10791-009-9109-9
* [16] Chen, J., Song, L., Wainwright, M.J., Jordan, M.I.: Learning to explain: An information-theoretic perspective on model interpretation. In: Dy, J.G., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning, ICML 2018, Stockholmsmässan, Stockholm, Sweden, July 10-15, 2018. Proceedings of Machine Learning Research, vol. 80, pp. 882–891. PMLR (2018), http://proceedings.mlr.press/v80/chen18j.html
* [17] Dato, D., Lucchese, C., Nardini, F.M., Orlando, S., Perego, R., Tonellotto, N., Venturini, R.: Fast ranking with additive ensembles of oblivious and non-oblivious regression trees. ACM Trans. Inf. Syst. 35(2), 15:1–15:31 (2016). https://doi.org/10.1145/2987380, https://doi.org/10.1145/2987380
* [18] Du, M., Liu, N., Hu, X.: Techniques for interpretable machine learning. Commun. ACM 63(1), 68–77 (2020). https://doi.org/10.1145/3359786, https://doi.org/10.1145/3359786
* [19] Falcon, W., et al.: Pytorch lightning. GitHub. Note: https://github. com/PyTorchLightning/pytorch-lightning 3, 6 (2019)
* [20] Freund, Y., Iyer, R.D., Schapire, R.E., Singer, Y.: An efficient boosting algorithm for combining preferences. In: Shavlik, J.W. (ed.) Proceedings of the Fifteenth International Conference on Machine Learning (ICML 1998), Madison, Wisconsin, USA, July 24-27, 1998. pp. 170–178. Morgan Kaufmann (1998)
* [21] Gallagher, L., Chen, R., Blanco, R., Culpepper, J.S.: Joint optimization of cascade ranking models. In: Culpepper, J.S., Moffat, A., Bennett, P.N., Lerman, K. (eds.) Proceedings of the Twelfth ACM International Conference on Web Search and Data Mining, WSDM 2019, Melbourne, VIC, Australia, February 11-15, 2019. pp. 15–23. ACM (2019). https://doi.org/10.1145/3289600.3290986, https://doi.org/10.1145/3289600.3290986
* [22] Geng, X., Liu, T., Qin, T., Li, H.: Feature selection for ranking. In: Kraaij, W., de Vries, A.P., Clarke, C.L.A., Fuhr, N., Kando, N. (eds.) SIGIR 2007: Proceedings of the 30th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, Amsterdam, The Netherlands, July 23-27, 2007. pp. 407–414. ACM (2007). https://doi.org/10.1145/1277741.1277811, https://doi.org/10.1145/1277741.1277811
* [23] Gigli, A., Lucchese, C., Nardini, F.M., Perego, R.: Fast feature selection for learning to rank. In: Carterette, B., Fang, H., Lalmas, M., Nie, J. (eds.) Proceedings of the 2016 ACM on International Conference on the Theory of Information Retrieval, ICTIR 2016, Newark, DE, USA, September 12- 6, 2016. pp. 167–170. ACM (2016). https://doi.org/10.1145/2970398.2970433, https://doi.org/10.1145/2970398.2970433
* [24] Jang, E., Gu, S., Poole, B.: Categorical reparametrization with gumble-softmax (2017)
* [25] Järvelin, K., Kekäläinen, J.: Cumulated gain-based evaluation of IR techniques. ACM Trans. Inf. Syst. 20(4), 422–446 (2002). https://doi.org/10.1145/582415.582418, http://doi.acm.org/10.1145/582415.582418
* [26] Joachims, T.: Optimizing search engines using clickthrough data. In: Proceedings of the Eighth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, July 23-26, 2002, Edmonton, Alberta, Canada. pp. 133–142. ACM (2002). https://doi.org/10.1145/775047.775067, https://doi.org/10.1145/775047.775067
* [27] Joachims, T.: Training linear svms in linear time. In: Eliassi-Rad, T., Ungar, L.H., Craven, M., Gunopulos, D. (eds.) Proceedings of the Twelfth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, Philadelphia, PA, USA, August 20-23, 2006. pp. 217–226. ACM (2006). https://doi.org/10.1145/1150402.1150429, https://doi.org/10.1145/1150402.1150429
* [28] Kaur, H., Nori, H., Jenkins, S., Caruana, R., Wallach, H.M., Vaughan, J.W.: Interpreting interpretability: Understanding data scientists’ use of interpretability tools for machine learning. In: Bernhaupt, R., Mueller, F.F., Verweij, D., Andres, J., McGrenere, J., Cockburn, A., Avellino, I., Goguey, A., Bjøn, P., Zhao, S., Samson, B.P., Kocielnik, R. (eds.) CHI ’20: CHI Conference on Human Factors in Computing Systems, Honolulu, HI, USA, April 25-30, 2020. pp. 1–14. ACM (2020). https://doi.org/10.1145/3313831.3376219, https://doi.org/10.1145/3313831.3376219
* [29] Ke, G., Meng, Q., Finley, T., Wang, T., Chen, W., Ma, W., Ye, Q., Liu, T.: Lightgbm: A highly efficient gradient boosting decision tree. In: Guyon, I., von Luxburg, U., Bengio, S., Wallach, H.M., Fergus, R., Vishwanathan, S.V.N., Garnett, R. (eds.) Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA. pp. 3146–3154 (2017), https://proceedings.neurips.cc/paper/2017/hash/6449f44a102fde848669bdd9eb6b76fa-Abstract.html
* [30] Kingma, D.P., Welling, M.: Auto-encoding variational bayes. In: Bengio, Y., LeCun, Y. (eds.) 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings (2014), http://arxiv.org/abs/1312.6114
* [31] Lai, H., Pan, Y., Liu, C., Lin, L., Wu, J.: Sparse learning-to-rank via an efficient primal-dual algorithm. IEEE Trans. Computers 62(6), 1221–1233 (2013). https://doi.org/10.1109/TC.2012.62, https://doi.org/10.1109/TC.2012.62
* [32] Lai, H., Pan, Y., Liu, C., Lin, L., Wu, J.: Sparse learning-to-rank via an efficient primal-dual algorithm. IEEE Trans. Computers 62(6), 1221–1233 (2013). https://doi.org/10.1109/TC.2012.62, https://doi.org/10.1109/TC.2012.62
* [33] Lai, H., Pan, Y., Tang, Y., Yu, R.: Fsmrank: Feature selection algorithm for learning to rank. IEEE Trans. Neural Networks Learn. Syst. 24(6), 940–952 (2013). https://doi.org/10.1109/TNNLS.2013.2247628, https://doi.org/10.1109/TNNLS.2013.2247628
* [34] Laporte, L., Flamary, R., Canu, S., Déjean, S., Mothe, J.: Nonconvex regularizations for feature selection in ranking with sparse SVM. IEEE Trans. Neural Networks Learn. Syst. 25(6), 1118–1130 (2014). https://doi.org/10.1109/TNNLS.2013.2286696, https://doi.org/10.1109/TNNLS.2013.2286696
* [35] Lemhadri, I., Ruan, F., Abraham, L., Tibshirani, R.: Lassonet: A neural network with feature sparsity. J. Mach. Learn. Res. 22, 127:1–127:29 (2021), http://jmlr.org/papers/v22/20-848.html
* [36] Leonhardt, J., Rudra, K., Anand, A.: Extractive explanations for interpretable text ranking. ACM Trans. Inf. Syst. (dec 2022). https://doi.org/10.1145/3576924, https://doi.org/10.1145/3576924
* [37] Li, P., Burges, C.J.C., Wu, Q.: Mcrank: Learning to rank using multiple classification and gradient boosting. In: Platt, J.C., Koller, D., Singer, Y., Roweis, S.T. (eds.) Advances in Neural Information Processing Systems 20, Proceedings of the Twenty-First Annual Conference on Neural Information Processing Systems, Vancouver, British Columbia, Canada, December 3-6, 2007. pp. 897–904. Curran Associates, Inc. (2007), https://proceedings.neurips.cc/paper/2007/hash/b86e8d03fe992d1b0e19656875ee557c-Abstract.html
* [38] Liu, T.: Learning to rank for information retrieval. Found. Trends Inf. Retr. 3(3), 225–331 (2009). https://doi.org/10.1561/1500000016, https://doi.org/10.1561/1500000016
* [39] Lucchese, C., Nardini, F.M., Orlando, S., Perego, R., Veneri, A.: ILMART: interpretable ranking with constrained lambdamart. In: SIGIR ’22: The 45th International ACM SIGIR Conference on Research and Development in Information Retrieval, Madrid, Spain, July 11 - 15, 2022. pp. 2255–2259. ACM (2022). https://doi.org/10.1145/3477495.3531840, https://doi.org/10.1145/3477495.3531840
* [40] Lucchese, C., Nardini, F.M., Orlando, S., Perego, R., Veneri, A.: ILMART: interpretable ranking with constrained lambdamart. In: Amigó, E., Castells, P., Gonzalo, J., Carterette, B., Culpepper, J.S., Kazai, G. (eds.) SIGIR ’22: The 45th International ACM SIGIR Conference on Research and Development in Information Retrieval, Madrid, Spain, July 11 - 15, 2022. pp. 2255–2259. ACM (2022). https://doi.org/10.1145/3477495.3531840, https://doi.org/10.1145/3477495.3531840
* [41] Lundberg, S.M., Lee, S.: A unified approach to interpreting model predictions. In: Guyon, I., von Luxburg, U., Bengio, S., Wallach, H.M., Fergus, R., Vishwanathan, S.V.N., Garnett, R. (eds.) Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA. pp. 4765–4774 (2017), https://proceedings.neurips.cc/paper/2017/hash/8a20a8621978632d76c43dfd28b67767-Abstract.html
* [42] Martins, A.F.T., Astudillo, R.F.: From softmax to sparsemax: A sparse model of attention and multi-label classification. In: Balcan, M., Weinberger, K.Q. (eds.) Proceedings of the 33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016. JMLR Workshop and Conference Proceedings, vol. 48, pp. 1614–1623. JMLR.org (2016), http://proceedings.mlr.press/v48/martins16.html
* [43] Masoomi, A., Wu, C., Zhao, T., Wang, Z., Castaldi, P.J., Dy, J.G.: Instance-wise feature grouping. In: Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M., Lin, H. (eds.) Advances in Neural Information Processing Systems 33: Annual Conference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual (2020), https://proceedings.neurips.cc/paper/2020/hash/9b10a919ddeb07e103dc05ff523afe38-Abstract.html
* [44] Molnar, C.: Interpretable machine learning. Lulu. com (2020)
* [45] Pan, F., Converse, T., Ahn, D., Salvetti, F., Donato, G.: Feature selection for ranking using boosted trees. In: Cheung, D.W., Song, I., Chu, W.W., Hu, X., Lin, J. (eds.) Proceedings of the 18th ACM Conference on Information and Knowledge Management, CIKM 2009, Hong Kong, China, November 2-6, 2009. pp. 2025–2028. ACM (2009). https://doi.org/10.1145/1645953.1646292, https://doi.org/10.1145/1645953.1646292
* [46] Pang, L., Xu, J., Ai, Q., Lan, Y., Cheng, X., Wen, J.: Setrank: Learning a permutation-invariant ranking model for information retrieval. In: Huang, J.X., Chang, Y., Cheng, X., Kamps, J., Murdock, V., Wen, J., Liu, Y. (eds.) Proceedings of the 43rd International ACM SIGIR conference on research and development in Information Retrieval, SIGIR 2020, Virtual Event, China, July 25-30, 2020. pp. 499–508. ACM (2020). https://doi.org/10.1145/3397271.3401104, https://doi.org/10.1145/3397271.3401104
* [47] Pobrotyn, P., Bartczak, T., Synowiec, M., Bialobrzeski, R., Bojar, J.: Context-aware learning to rank with self-attention. CoRR abs/2005.10084 (2020), https://arxiv.org/abs/2005.10084
* [48] Purpura, A., Buchner, K., Silvello, G., Susto, G.A.: Neural feature selection for learning to rank. In: Hiemstra, D., Moens, M., Mothe, J., Perego, R., Potthast, M., Sebastiani, F. (eds.) Advances in Information Retrieval - 43rd European Conference on IR Research, ECIR 2021, Virtual Event, March 28 - April 1, 2021, Proceedings, Part II. Lecture Notes in Computer Science, vol. 12657, pp. 342–349. Springer (2021). https://doi.org/10.1007/978-3-030-72240-1_34, https://doi.org/10.1007/978-3-030-72240-1_34
* [49] Qin, T., Liu, T.: Introducing LETOR 4.0 datasets. CoRR abs/1306.2597 (2013), http://arxiv.org/abs/1306.2597
* [50] Qin, T., Liu, T., Xu, J., Li, H.: LETOR: A benchmark collection for research on learning to rank for information retrieval. Inf. Retr. 13(4), 346–374 (2010). https://doi.org/10.1007/S10791-009-9123-Y, https://doi.org/10.1007/s10791-009-9123-y
* [51] Qin, Z., Yan, L., Zhuang, H., Tay, Y., Pasumarthi, R.K., Wang, X., Bendersky, M., Najork, M.: Are neural rankers still outperformed by gradient boosted decision trees? In: 9th International Conference on Learning Representations, ICLR 2021, Virtual Event, Austria, May 3-7, 2021. OpenReview.net (2021), https://openreview.net/forum?id=Ut1vF_q_vC
* [52] Rahangdale, A., Raut, S.A.: Deep neural network regularization for feature selection in learning-to-rank. IEEE Access 7, 53988–54006 (2019). https://doi.org/10.1109/ACCESS.2019.2902640, https://doi.org/10.1109/ACCESS.2019.2902640
* [53] Ribeiro, M.T., Singh, S., Guestrin, C.: "why should I trust you?": Explaining the predictions of any classifier. In: Krishnapuram, B., Shah, M., Smola, A.J., Aggarwal, C.C., Shen, D., Rastogi, R. (eds.) Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA, August 13-17, 2016. pp. 1135–1144. ACM (2016). https://doi.org/10.1145/2939672.2939778, https://doi.org/10.1145/2939672.2939778
* [54] Rigutini, L., Papini, T., Maggini, M., Scarselli, F.: Sortnet: Learning to rank by a neural-based sorting algorithm. CoRR abs/2311.01864 (2023). https://doi.org/10.48550/ARXIV.2311.01864, https://doi.org/10.48550/arXiv.2311.01864
* [55] Rong, Y., Leemann, T., Nguyen, T., Fiedler, L., Seidel, T., Kasneci, G., Kasneci, E.: Towards human-centered explainable AI: user studies for model explanations. CoRR abs/2210.11584 (2022). https://doi.org/10.48550/arXiv.2210.11584, https://doi.org/10.48550/arXiv.2210.11584
* [56] Rudin, C.: Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nat. Mach. Intell. 1(5), 206–215 (2019). https://doi.org/10.1038/s42256-019-0048-x, https://doi.org/10.1038/s42256-019-0048-x
* [57] Shrikumar, A., Greenside, P., Kundaje, A.: Learning important features through propagating activation differences. In: Precup, D., Teh, Y.W. (eds.) Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017. Proceedings of Machine Learning Research, vol. 70, pp. 3145–3153. PMLR (2017), http://proceedings.mlr.press/v70/shrikumar17a.html
* [58] Simonyan, K., Vedaldi, A., Zisserman, A.: Deep inside convolutional networks: Visualising image classification models and saliency maps. In: Bengio, Y., LeCun, Y. (eds.) 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Workshop Track Proceedings (2014), http://arxiv.org/abs/1312.6034
* [59] Sun, Z., Qin, T., Tao, Q., Wang, J.: Robust sparse rank learning for non-smooth ranking measures. In: Allan, J., Aslam, J.A., Sanderson, M., Zhai, C., Zobel, J. (eds.) Proceedings of the 32nd Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, SIGIR 2009, Boston, MA, USA, July 19-23, 2009. pp. 259–266. ACM (2009). https://doi.org/10.1145/1571941.1571987, https://doi.org/10.1145/1571941.1571987
* [60] Sundararajan, M., Taly, A., Yan, Q.: Axiomatic attribution for deep networks. In: Precup, D., Teh, Y.W. (eds.) Proceedings of the 34th International Conference on Machine Learning, ICML 2017, Sydney, NSW, Australia, 6-11 August 2017. Proceedings of Machine Learning Research, vol. 70, pp. 3319–3328. PMLR (2017), http://proceedings.mlr.press/v70/sundararajan17a.html
* [61] Taylor, M.J., Guiver, J., Robertson, S., Minka, T.: Softrank: optimizing non-smooth rank metrics. In: Najork, M., Broder, A.Z., Chakrabarti, S. (eds.) Proceedings of the International Conference on Web Search and Web Data Mining, WSDM 2008, Palo Alto, California, USA, February 11-12, 2008. pp. 77–86. ACM (2008). https://doi.org/10.1145/1341531.1341544, https://doi.org/10.1145/1341531.1341544
* [62] Tibshirani, R.: Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society: Series B (Methodological) 58(1), 267–288 (1996)
* [63] Vaswani, A., Shazeer, N., Parmar, N., Uszkoreit, J., Jones, L., Gomez, A.N., Kaiser, L., Polosukhin, I.: Attention is all you need. In: Guyon, I., von Luxburg, U., Bengio, S., Wallach, H.M., Fergus, R., Vishwanathan, S.V.N., Garnett, R. (eds.) Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA. pp. 5998–6008 (2017), https://proceedings.neurips.cc/paper/2017/hash/3f5ee243547dee91fbd053c1c4a845aa-Abstract.html
* [64] Wang, L., Lin, J., Metzler, D.: A cascade ranking model for efficient ranked retrieval. In: Ma, W., Nie, J., Baeza-Yates, R., Chua, T., Croft, W.B. (eds.) Proceeding of the 34th International ACM SIGIR Conference on Research and Development in Information Retrieval, SIGIR 2011, Beijing, China, July 25-29, 2011. pp. 105–114. ACM (2011). https://doi.org/10.1145/2009916.2009934, https://doi.org/10.1145/2009916.2009934
* [65] Wu, Q., Burges, C.J.C., Svore, K.M., Gao, J.: Adapting boosting for information retrieval measures. Inf. Retr. 13(3), 254–270 (2010). https://doi.org/10.1007/S10791-009-9112-1, https://doi.org/10.1007/s10791-009-9112-1
* [66] Xia, F., Liu, T., Wang, J., Zhang, W., Li, H.: Listwise approach to learning to rank: theory and algorithm. In: Cohen, W.W., McCallum, A., Roweis, S.T. (eds.) Machine Learning, Proceedings of the Twenty-Fifth International Conference (ICML 2008), Helsinki, Finland, June 5-9, 2008. ACM International Conference Proceeding Series, vol. 307, pp. 1192–1199. ACM (2008). https://doi.org/10.1145/1390156.1390306, https://doi.org/10.1145/1390156.1390306
* [67] Xu, J., Li, H.: Adarank: a boosting algorithm for information retrieval. In: Kraaij, W., de Vries, A.P., Clarke, C.L.A., Fuhr, N., Kando, N. (eds.) SIGIR 2007: Proceedings of the 30th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, Amsterdam, The Netherlands, July 23-27, 2007. pp. 391–398. ACM (2007). https://doi.org/10.1145/1277741.1277809, https://doi.org/10.1145/1277741.1277809
* [68] Xu, Z.E., Huang, G., Weinberger, K.Q., Zheng, A.X.: Gradient boosted feature selection. In: Macskassy, S.A., Perlich, C., Leskovec, J., Wang, W., Ghani, R. (eds.) The 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’14, New York, NY, USA - August 24 - 27, 2014. pp. 522–531. ACM (2014). https://doi.org/10.1145/2623330.2623635, https://doi.org/10.1145/2623330.2623635
* [69] Yoon, J., Jordon, J., van der Schaar, M.: INVASE: instance-wise variable selection using neural networks. In: 7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019. OpenReview.net (2019), https://openreview.net/forum?id=BJg_roAcK7
* [70] Yue, Y., Finley, T., Radlinski, F., Joachims, T.: A support vector method for optimizing average precision. In: Kraaij, W., de Vries, A.P., Clarke, C.L.A., Fuhr, N., Kando, N. (eds.) SIGIR 2007: Proceedings of the 30th Annual International ACM SIGIR Conference on Research and Development in Information Retrieval, Amsterdam, The Netherlands, July 23-27, 2007. pp. 271–278. ACM (2007). https://doi.org/10.1145/1277741.1277790, https://doi.org/10.1145/1277741.1277790
* [71] Zhang, Z., Rudra, K., Anand, A.: Explain and predict, and then predict again. In: WSDM ’21, The Fourteenth ACM International Conference on Web Search and Data Mining, Virtual Event, Israel, March 8-12, 2021. pp. 418–426. ACM (2021). https://doi.org/10.1145/3437963.3441758, https://doi.org/10.1145/3437963.3441758
* [72] Zhang, Z., Setty, V., Anand, A.: Sparcassist: A model risk assessment assistant based on sparse generated counterfactuals. In: Amigó, E., Castells, P., Gonzalo, J., Carterette, B., Culpepper, J.S., Kazai, G. (eds.) SIGIR ’22: The 45th International ACM SIGIR Conference on Research and Development in Information Retrieval, Madrid, Spain, July 11 - 15, 2022. pp. 3219–3223. ACM (2022). https://doi.org/10.1145/3477495.3531677, https://doi.org/10.1145/3477495.3531677
* [73] Zhuang, H., Wang, X., Bendersky, M., Grushetsky, A., Wu, Y., Mitrichev, P., Sterling, E., Bell, N., Ravina, W., Qian, H.: Interpretable ranking with generalized additive models. In: WSDM ’21, The Fourteenth ACM International Conference on Web Search and Data Mining, Virtual Event, Israel, March 8-12, 2021. pp. 499–507. ACM (2021). https://doi.org/10.1145/3437963.3441796, https://doi.org/10.1145/3437963.3441796
|
# Existence of Invariant Measures for Stochastic Inviscid Multi-Layer Quasi-
Geostrophic Equations
Federico Butori Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126,
Pisa, Italy federico.butori at sns.it , Francesco Grotto Università di
Pisa, Dipartimento di Matematica, 5 Largo Bruno Pontecorvo, 56127 Pisa, Italia
francesco.grotto at unipi.it , Eliseo Luongo Scuola Normale Superiore,
Piazza dei Cavalieri 7, 56126, Pisa, Italy eliseo.luongo at sns.it and
Leonardo Roveri Università di Pisa, Dipartimento di Matematica, 5 Largo Bruno
Pontecorvo, 56127 Pisa, Italia leonardo.roveri at phd.unipi.it
###### Abstract.
We consider an inviscid 3-layer quasi-geostrophic model with stochastic
forcing in a 2D bounded domain. After establishing well-posedness of such
system under natural regularity assumptions on the initial condition and the
(additive) noise, we prove the existence of an invariant measure supported on
bounded functions by means of the Krylov-Bogoliubov approach developed by
Ferrario and Bessaih (Comm. Math. Phys. 377, 2020).
###### Acknowledgements.
F. G. and L. R. were both supported by the project _Mathematical methods for
climate science_ , funded by he Ministry of University and Research (MUR) as
part of the PON 2014-2020 ”Research and Innovation” resources - Green Action -
DM MUR 1061/2022
###### 1991 Mathematics Subject Classification:
35R60, 37N10, 60H15, 76U60, 86A10
## 1\. Introduction
Multi-layer quasi-geostrophic equations are a standard model in the
theoretical study of atmospheric and oceanic phenomena. The equations are
derived from general Navier-Stokes (shallow water) models [34, Section 5.3],
and they describe the dynamics of stratified fluids characterized by an
approximate balance between the pressure gradient and the Coriolis force.
Our study is focused on an inviscid and stochastically driven 3-layer quasi-
geostrophic system:
(1.1) $\displaystyle\frac{\partial}{\partial
t}q^{i}+(u^{i}\cdot\nabla)q^{i}=f^{i},\quad
u^{i}=\operatorname{\nabla}^{\perp}\psi^{i},\quad i=1,2,3,$
in which dynamics is expressed in terms of potential vorticity $q^{i}$ of the
$i$-th layer and the associated stream function $\psi^{i}$, with $f^{i}$ being
a random forcing term. The interaction between different layers is encoded in
the linear equations linking potential vorticity and stream functions,
(1.2) $\begin{cases}q^{1}=\Delta\psi^{1}+\beta
y+\frac{c^{2}}{H_{1}g_{1}}(\psi^{2}-\psi^{1})+\frac{c^{2}}{gH_{2}}\psi^{1},\\\
q^{2}=\Delta\psi 2+\beta
y+\frac{c^{2}}{H_{2}g_{1}}(\psi^{1}-\psi^{2})+\frac{c^{2}}{H_{2}g_{2}}(\psi^{3}-\psi^{2}),\\\
q^{3}=\Delta\psi^{3}+\beta
y+\frac{c^{2}}{H_{3}g_{2}}(\psi^{2}-\psi^{3})+\frac{c}{H_{3}}\eta,\end{cases}$
the space domain is a bounded region $D\subset\mathbb{R}^{2}$ and the latter
elliptic equations for potential vorticity are solved under Dirichlet boundary
conditions. In (1.2), parameters $H_{i}>0$ describe the basic-state thickness
of single layers, $g_{i}=g\left(\rho_{i+1}-\right.$
$\left.\rho_{i}\right)/\rho_{1}$ the gravitational acceleration due to density
differences between layers, $\eta$ is the height of the bottom topography, and
finally $c$ is a variable Coriolis parameter. There is no variation in the
fluid densities $\rho_{i}$ (Boussinesq approximation). Moreover, viscosity is
absent and the fluid motion includes stochastic forcing modelling the
interaction with small scales, coherently with the setup for the observation
of geostrophic turbulence (cf. [34, Chapter 9]). The parameter $\beta>0$ rules
the $\beta$-plane approximation of Coriolis force. All the forthcoming
arguments are easily generalized to an arbitrary number of interacting layers,
and we restrict to the case of three layers for the sake of a clear
exposition.
From a mathematical standpoint, when the $f^{i}$’s are deterministic or null
the well-posedness theory of the PDE system described above is completely
analogous to that of 2D Euler’s equations, and there exists a unique solution
for potential vorticity of class $L^{\infty}$. The proof follows by the
classical argument of Yudovich [35], and we refer to [11] for details. In
order to preserve well-posedness, stochastic forcing terms must satisfy
suitable regularity assumptions. Moreover, in order to observe stationary
states of the system, friction terms must be included in the dynamics in order
to compensate the input energy (see (3.4) below for a proper formulation of
the stochastic PDE problem we consider). Both regularity of noise and the
presence of friction are physically sensible (cf. [28, p. 212]) and allow to
consider long-time evolution of the system towards equilibrium states.
The aim of the present contribution is to rigorously establish an existence
result for invariant measures, i.e. equilibrium states, supported by the well-
posedness class $L^{\infty}$ of potential vorticity. Invariant measures and
convergence to equilibrium in quasi-geostrophic and related models have been
the object of numerous works in the viscous case: we mention for instance the
recent works [10, 9] and refer to [8] for an overview. Equilibrium states for
geophysical models in low regularity regimes were considered in [19, 20]
(mirroring analogous techniques from the theory of Euler equations [1, 16, 17,
22, 21]), but to the best of our knowledge our result is the first to study
invariant measures in inviscid geophysical model within the well-posedness
regularity regime. Let us stress that in this context we can not rely on
parabolic regularity theory, and instead need to resort to the averaging
effect of stochastic forcing.
We will establish well-posedness of a stochastic 3-layer quasi-geostrophic
system by means of an inviscid limit approach, using non-physical viscous
approximations that prevent vorticity creation at the boundary, thus avoiding
a critical issue of viscous models in bounded domains (cf. [12, 18]).
Concerning invariant measures, our arguments will be based on the infinite-
dimensional Krylov-Bogoliubov approach developed in [3] for 2D Euler equations
on a periodic domain, allowing to obtain existence of invariant measures
supported by function spaces suited for the study of turbulent phenomena. In
generalizing the techniques of [3] to quasi-geostrophic systems, we are also
able to improve their results in terms of Sobolev regularity both for well-
posedness of the stochastic PDE system and for the support of invariant
measures.
The paper is organized as follows. Section 2 recalls basic notions to be used
in the remainder of the paper, and with which we can rigorously present the
main results of the paper and outline their proofs in Section 3. Sections 4
and 4.4 establish well-posedness of the stochastic PDE under consideration and
properties of solutions, proving Theorem 3.3 Finally, Section 5 proves the
main theorem of the paper, Theorem 3.6.
## 2\. Functional Analytic Setup
Assume from now on that $D\subset\mathbb{R}^{2}$ is a simply connected, smooth
bounded domain. We denote $\mathcal{D}=C^{\infty}_{c}(D)$ and by
$\mathcal{D}^{\prime}=C^{\infty}_{c}(D)^{\prime}$ its topological dual, i.e.
the space of distributions. Brackets $\left\langle\cdot,\cdot\right\rangle$
will denote duality couplings in $\mathcal{D}\times\mathcal{D}^{\prime}$, to
be understood as $L^{2}$ products whenever both arguments belong to the
latter.
We will also denote by $\left\langle\cdot,\cdot\right\rangle$ the $L^{2}$
scalar product for time-space functions, i.e. elements of spaces
$L^{p}(0,T;L^{q}(D))$, whose norm will be denoted by
$\left\|\cdot\right\|_{L^{p}_{t}L^{q}_{x}}$.
### 2.1. Functional spaces
We will rely on fractional Sobolev spaces and fractional powers of the
Dirichlet Laplacian, which have various possible definitions, often non-
equivalent. We choose the approach base on the _spectral_ fractional Laplace
operator. We refer to [7],[26, Chapter 1], [33, Chapter 3] for details on the
material covered in this paragraph.
By $W^{s,p}(D)$, $s\in\mathbb{N},\ p\in[1,+\infty]$, we denote Sobolev spaces
obtained as the closure of $C^{s}(\bar{D})$ under the usual $L^{p}$-based
Sobolev norm, and we set
$H^{s}(D)=W^{s,2}(D),\quad W^{0,p}(D)=L^{p}(D).$
Fractional Sobolev spaces $H^{s}(D)$, $0<s<1$, are the completion of
$C^{1}(\bar{D})$) under the Sobolev-Slobodeckij norm
$\displaystyle\left(\lVert
u\rVert^{2}_{L^{2}(D)}+\int_{D}\int_{D}\frac{(u(x)-u(y))^{2}}{|x-y|^{2+2s}}dxdy\right)^{1/2};$
the definition is extended to any $s=m+\sigma$, $m\in\mathbb{N}$,
$\sigma\in(0,1)$, defining the norm
$\displaystyle\lVert u\rVert_{H^{s}(D)}^{2}=\lVert
u\rVert_{H^{m}(D)}^{2}+\sum_{|\alpha|=m}\lVert
D^{\alpha}u\rVert_{H^{s}(D)}^{2}.$
Considering the closure of $\mathcal{D}$ instead of $C^{s}(\bar{D})$, we can
define as above the spaces $W_{0}^{s,p}(D)$ and $H_{0}^{s}(D)$ (again, we set
$W_{0}^{0,p}(D)=L^{p}(D)$). In particular, there exist an explicit
relationship between these different spaces: for $0\leq s\leq 1/2$ it holds
$H^{s}_{0}(D)=H^{s}(D)$, while for $s>1/2$ one has
$H^{s}_{0}(D)\hookrightarrow H^{s}(D)$, the trace operator is well defined and
$H^{s}_{0}(D)$ contains the null-trace elements.
For each $s>0$ the topological dual of $H^{s}_{0}(D)$, denoted by $H^{-s}(D)$,
is a subspace of the space $\mathcal{D}^{\prime}$ of distributions on $D$,
since $H^{s}_{0}(D)$ was obtained as the closure of $\mathcal{D}$ with respect
to a suitable norm. Moreover, embeddings $H^{s}_{0}(D)\hookrightarrow
L^{2}(D)\hookrightarrow H^{-s}(D)$ are continuous and dense.
The following proposition gathers some classical statements on the relations
between the spaces defined until now, intended in a two-dimensional setting.
###### Proposition 2.1.
The following hold:
* •
(Sobolev embeddings) given $1<q<\infty$ and $\alpha>0$,
$H^{\alpha}(D)\hookrightarrow L^{q}(D),\qquad H_{0}^{\alpha}(D)\hookrightarrow
L^{q}(D)$
whenever $\alpha-1\geq-\frac{2}{q}$. These embeddings are also compact if the
inequality is strict;
* •
(Rellich Theorem) given $\alpha>\beta\geq 0$,
$H^{\alpha}(D)\stackrel{{\scriptstyle c}}{{\hookrightarrow}}H^{\beta}(D),\quad
H_{0}^{\alpha}(D)\stackrel{{\scriptstyle
c}}{{\hookrightarrow}}H_{0}^{\beta}(D),\quad
H^{-\beta}(D)\stackrel{{\scriptstyle c}}{{\hookrightarrow}}H^{-\alpha}(D);$
* •
(Morrey’s inequality) given $l\in\mathbb{N}$ and $\lambda\in[0,1)$, for every
$\alpha$ such that $\alpha-1>l+\lambda$
$H^{\alpha}(D)\stackrel{{\scriptstyle
c}}{{\hookrightarrow}}C^{l,\lambda}\left(\bar{D}\right),\qquad
H_{0}^{\alpha}(D)\stackrel{{\scriptstyle
c}}{{\hookrightarrow}}C^{l,\lambda}\left(\bar{D}\right),$
and for every $p$ such that $1-\frac{2}{p}>l+\lambda$
$W^{1,p}(D)\stackrel{{\scriptstyle
c}}{{\hookrightarrow}}C^{l,\lambda}(\bar{D});$
* •
(Interpolation) given $\theta\in[0,1]$ and $(1-\theta)s+\theta
s^{\prime}\notin(\mathbb{N}+1/2)$,
$\displaystyle\left[H^{s}_{0}(D),H^{s^{\prime}}_{0}(D)\right]_{\theta}$
$\displaystyle=H^{(1-\theta)s+\theta s^{\prime}}_{0}(D),$
$\displaystyle\left[H^{-s}(D),H^{-s^{\prime}}(D)\right]_{\theta}$
$\displaystyle=H^{-(1-\theta)s+\theta s^{\prime}}(D).$
We are also going to use a different definition of fractional Sobolev space,
based on Fourier series expansions. There exists an orthonormal basis of
$L^{2}(D)$ of eigenfunctions $\left\\{e_{n}\right\\}_{n\in\mathbb{N}}\subset
H^{1}_{0}(D)$ of the Dirichlet Laplacian $-\Delta$, i.e. $-\Delta
e_{n}=\lambda_{n}e_{n}$ for some $\lambda_{n}\in\mathbb{R}$. For $s\geq 0$,
spectral fractional Laplacian is defined as a densely defined, positive self-
adjoint operator on $L^{2}(D)$ by
$\displaystyle(-\Delta)^{s}u=\sum_{n=1}^{\infty}\lambda_{n}^{s}\hat{u}_{n}e_{n}\in
L^{2}(D),\quad\hat{u}_{n}=\left\langle u,e_{n}\right\rangle,\quad
u\in\mathsf{D}\big{(}(-\Delta)^{s}\big{)},$
$\displaystyle\mathsf{D}\big{(}(-\Delta)^{s}\big{)}=\left\\{u=\sum_{n=1}^{\infty}\hat{u}_{n}e_{n}:\hat{u}_{n}\in\mathbb{R},\sum_{n=1}^{\infty}\lambda_{n}^{2s}|\hat{u}_{n}|^{2}<\infty\right\\}:=\mathcal{H}^{s}.$
It is easy to check that $\mathcal{H}^{s}$, endowed with the norm
$\lVert{u}\rVert_{\mathcal{H}^{s}}^{2}=\left\langle(-\Delta)^{s/2}u,(-\Delta)^{s/2}u\right\rangle,$
is a Hilbert space. Moreover, $\mathcal{H}^{s}=H^{s}(D)$ for $s\in(0,1/2)$,
$\mathcal{H}^{s}=H^{s}_{0}(D)$ for $s\in(1/2,3/2)$, and
$\mathcal{H}^{s}=H^{1}_{0}(D)\cap H^{s}(D)$ for $s\in[3/2,2]$ in the sense
that they coincide as function spaces and their norms are equivalent (cf.
[7]). For completeness, in the case $s=1/2$, $\mathcal{H}^{1/2}$ coincides
with the so-called Lions-Magenes space $H^{1/2}_{00}(D)$ [26], but we are not
going to use it.
### 2.2. Weak$\star$ and bounded weak$\star$ topologies
In the last part of this work we will make use of some notions on weak
topologies that we now recall. Consider the space $L^{\infty}$ with the norm
topology $\tau_{n}$. The weak$\star$ topology $\tau_{\star}$ on $L^{\infty}$
is the coarsest topology among those making the duality pairing
$\left\langle\cdot,\psi\right\rangle:L^{\infty}\rightarrow\mathbb{R}$
continuous, for all $\psi\in L^{1}$.
A sequence of elements $\phi_{n}\in L^{\infty}$ is said to converge to some
$\phi\in L^{\infty}$ in the weak$\star$ sense,
$\phi_{n}\overset{\star}{\rightharpoonup}\phi$, if
$\left\langle\phi_{n},\psi\right\rangle\rightarrow\left\langle\phi,\psi\right\rangle$
for all $\psi\in L^{1}$. Due to the impossibility of metrizing $\tau_{\star}$,
such notion of convergence does not completely identify the topology. However,
we can circumvent this issue by only considering bounded sets as, being
$L^{1}$ separable, finite-radius balls are weakly$\star$ metrizable (cf. [5]
for an extended discussion). Since we will mostly be working with sequences,
it is then convenient to introduce the notion of bounded weak$\star$ topology
$\tau^{b}_{\star}$, that is, the finest topology on $L^{\infty}$ coinciding
with the weak$\star$ topology on all norm-bounded sets. We have the strict
inclusions
$\tau_{\star}\varsubsetneq\tau^{b}_{\star}\varsubsetneq\tau_{n}.$
As a consequence of the weak$\star$ metrizability of balls, a map
$F:L^{\infty}\rightarrow\mathbb{R}$ is sequentially $\tau_{\star}$-continuous
(i.e. $F(\phi_{n})\rightarrow F(\phi)$ whenever
$\phi_{n}\overset{\star}{\rightharpoonup}\phi$) if and only if $F$ is
$\tau^{b}_{\star}$-continuous (see [3]).
Concerning measurability, Borel sets generated by $\tau_{\star}$ and
$\tau_{\star}^{b}$ coincide, but they are a proper subset of those generated
by $\tau_{n}$, as opposed to the case where the space is separable in the
strong topology, in which $\tau_{\star}^{b}$, $\tau_{\star}$ and $\tau_{n}$
coincide, cf. [31]. We will therefore always specify with respect to which
norm measurability is intended.
## 3\. Overview of Main Results and Methodology
### 3.1. Notation for the multi-layer system
The quasi-geostrophic system (1.1), (1.2) can be written as
(3.1) $\left(\frac{\partial}{\partial
t}+u^{i}\cdot\operatorname{\nabla}\right)\bigg{(}\zeta^{i}+\sum_{j=1}^{3}\tilde{l}_{ij}\psi^{j}\bigg{)}=\tilde{f}^{i},\quad
i=1,2,3,$
where $u^{i}$ is the velocity field of the $i$-th layer,
$\zeta^{i}=\operatorname{\nabla}\times
u^{i}=\partial_{2}u^{i}_{1}-\partial_{1}u^{i}_{2}$ is a scalar function and
$\psi^{i}$ is the (scalar) stream function associated to $u^{i}$, i.e.
$\psi^{i}$ satisfies $\operatorname{\nabla}^{\perp}\psi^{i}=u^{i}$, and
$\Delta\psi^{i}=\zeta^{i}$. The vector $\tilde{f}^{i}$ accounts for external
forces, while by $\tilde{L}=(\tilde{l}_{ij})_{i,j=1,2,3}$ we denote the
layers’ interaction matrix: in the case of (1.2) it has the form
$\tilde{L}=\begin{pmatrix}-\lambda_{1}&\lambda_{1}&0\\\
\lambda_{2}&-2\lambda_{2}&\lambda_{2}\\\
0&\lambda_{3}&-\lambda_{3}\end{pmatrix},$
where $\lambda_{i}$’s are combinations of physical constants of the problem.
Setting $u=(u^{1},u^{2},u^{3})$, $\psi=(\psi^{1},\psi^{2},\psi^{3})$,
$f=(f^{1},f^{2},f^{3})$ and denoting by $\tilde{q}$ the potential vorticity
$\tilde{q}=\Delta\psi+\tilde{L}\psi$
(where $\Delta$ stands for $\operatorname{diag}(\Delta,\Delta,\Delta)$ abusing
notation), the system (3.1) can be conveniently rewritten in the form
(3.2) $\frac{\partial}{\partial
t}\tilde{q}+u\cdot\operatorname{\nabla}\tilde{q}=\tilde{f},$
in which the velocity field $u$ can be formally reconstructed from $\tilde{q}$
via $u=\operatorname{\nabla}^{\perp}(\Delta+\tilde{L})^{-1}\tilde{q}$. In the
following, it may be convenient to assume that the matrix $\tilde{L}$ is
symmetric and negative definite. This is not consistent with the physical
reality that the model aims at describing, but we can circumvent the problem
by introducing a scaling factor in the equation. Indeed, if
$D=\operatorname{diag}(h_{1},h_{2},h_{3})$ such that $h_{i}>0$ and
$h_{i}\lambda_{i}=\lambda$ for all $i=1,2,3$, then $D\tilde{L}$ is symmetric
and negative semidefinite (since for example $(1,1,1)^{\top}\in\ker(L)$). By
multiplying (3.2) by $D$, we can thus define
$q=(D\Delta+D\tilde{L})\psi,\quad
u=\operatorname{\nabla}^{\perp}(D\Delta+D\tilde{L})^{-1}\tilde{q}=\tilde{u},\quad
f=D\tilde{f},$
and obtain an equation of the same form,
$\frac{\partial}{\partial t}q+u\cdot\operatorname{\nabla}q=f,$
where now the operator $(D\Delta+D\tilde{L})$ is symmetric and negative
definite. For the sake of notation, in the following we write it as $(A+L)$,
with
$A\coloneqq D\Delta\quad\text{and}\quad L\coloneqq D\tilde{L},$
now assuming that $L$ is symmetric and negative semidefinite.
In order to properly define the PDE problem and to give meaning to the inverse
of $(A+L)$ we need to specify boundary conditions, and we will choose
Dirichlet conditions,
$\psi(t,\cdot)_{|\partial D}=0.$
As for function spaces we will be working with, dealing with a multi-layer-
problem we are interested in vector fields having some Sobolev regularity. Let
us therefore define, for $p\in[1,+\infty]$, $s\in\mathbb{N}$,
$\displaystyle\mathbf{W}^{s,p}=\\{u=(u^{1},u^{2},u^{3})^{t}:\
u^{1},u^{2},u^{3}\in
W^{s,p}\\},\quad\left\|u\right\|_{\mathbf{W}^{s,p}}=\sum_{i=1}^{3}\left\|u^{i}\right\|_{W^{s,p}}.$
and for $s\in\mathbb{R}$
$\displaystyle\mathbf{H}^{s}=\\{u=(u^{1},u^{2},u^{3})^{t}:\
u^{1},u^{2},u^{3}\in
H^{s}\\},\quad\left\|u\right\|_{\mathbf{H}^{s}}^{2}=\sum_{i=1}^{3}\left\|u^{i}\right\|_{H^{s}}^{2}.$
In an analogous way we can define $\mathbf{W}_{0}^{s,p}$ and
$\mathbf{H}_{0}^{s}$ (for $s\geq 0$). When $s=0$, we set
$\mathbf{W}^{s,p}_{0}(D)=\mathbf{W}^{s,p}(D)=\mathbf{L}^{p}(D)$.
### 3.2. Solutions of Quasi-Geostrophic Equations
We will make use of the following classic result for elliptic boundary value
problems, for which we refer to [2].
###### Lemma 3.1.
For every $q\in\mathbf{L}^{\infty}(D)$ the boundary value problem
(3.3) $\begin{cases}q=A\psi+L\psi,&x\in D,\\\ \psi(x)=0,&x\in\partial
D,\end{cases}$
has a unique weak solution $\psi\in\mathbf{W}^{2,p}(D)$. Moreover, for every
$i=1,\ldots,3$,
$\|\operatorname{\nabla}^{2}\psi^{i}\|_{\mathbf{L}^{p}}\leq
Cp\|q^{i}\|_{\mathbf{L}^{\infty}}\qquad\forall p\in[1,\infty).$
We are interested in a stochastic version of (3.2). Consider a filtered
probability space $(\Omega,\mathcal{F},\mathcal{F}_{t},\mathbb{P})$, let
$\\{\rho_{k}\\}_{k\geq 0}$ be an orthonormal basis of $\mathbf{L}^{2}$ made of
eigenvectors of the operator $(A+L)$ and define the Wiener process
$W_{t}(x)\coloneqq\sum_{k=0}^{+\infty}c_{k}\rho_{k}(x)W^{k}_{t},$
where $\\{c_{k}\\}_{k}\subset\mathbb{R}$ and $\\{W^{k}\\}_{k}$ are independent
standard $\mathbb{R}^{3}$-valued Brownian motions. Assume that
$\\{\mathcal{F}_{t}\\}_{t\geq 0}$ is the filtration generated by this family
of Brownian motions. This choice implies that $W_{t}|_{\partial D}\equiv 0$.
Notice also that one can consider $t\in\mathbb{R}$ just by taking
$W^{k}_{t}\coloneqq\tilde{W}^{k}_{-t}$ for $t<0$ where
$\\{\tilde{W}^{k}\\}_{k}$ are independent copies of $\\{W^{k}\\}_{k}$. The
space regularity for $W$ is encoded in its Fourier coefficients. From now on
we will assume that $W_{t}\in\mathbf{H}^{5/2}$ for all $t\in\mathbb{R}$, hence
$\sum_{k}c_{k}^{2}\|\rho_{k}\|^{2}_{\mathbf{H}^{5/2}}<\infty.$
The problem under consideration can now be formulated as
(3.4) $\begin{cases}dq_{t}+\left[(u_{t}\cdot\operatorname{\nabla})q_{t}+\gamma
q_{t}\right]dt=dW_{t},\qquad&\text{on }D,\\\ \operatorname{\nabla}\cdot
u_{t}=0,\,\,u_{t}=\operatorname{\nabla}^{\bot}(A+L)^{-1}q_{t},&\text{on }D,\\\
q_{t}=0,&\text{on }\partial D,\end{cases}$
where $\gamma>0$ is a damping coefficient ruling the friction term that
prevents the accumulation of energy introduced by the random forcing (we refer
again to [28] for a physical motivation). Recall that, given a filtered
probability space $(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t\geq
0},\mathbb{P})$, a collection of random variables $X_{t}(\omega),\ t\in[0,T],\
\omega\in\Omega$ taking values in a Banach space $B$ is an adapted stochastic
process if for every $t\in\mathbb{R}$ $X_{t}$ if $\mathcal{F}_{t}$-measurable.
It is said progressively measurable if for every $t\leq T$, the mapping
$[0,t]\times\Omega\ni(s,\omega)\rightarrow X_{s}(\omega)\in B$ is measurable
with respect to $\mathcal{B}([0,t])\otimes\mathcal{F}_{t}$.
###### Definition 3.2.
Given $q_{0}\in\mathbf{L}^{2}$, a stochastic process $q_{t}$ such that
$q\in C_{w}(0,T;\mathbf{L}^{2})\qquad\mathbb{P}\text{-a.s.}$
is said to be a weak solution of (3.4) if it is adapted with respect to
$\\{\mathcal{F}_{t}\\}_{t\leq 0}$ and for all
$\phi\in\mathbf{H}^{1}_{0}\cap\mathbf{H}^{2}$ and for all $t\in[0,T]$, it
holds $\mathbb{P}$-almost surely
(3.5) $\left\langle q_{t},\phi\right\rangle=\left\langle
q_{0},\phi\right\rangle+\int_{0}^{t}\left\langle
q_{s},u_{s}\cdot\operatorname{\nabla}\phi\right\rangle
ds-\gamma\int_{0}^{t}\left\langle q_{s},\phi\right\rangle
ds+\sum_{k=0}^{\infty}c_{k}\left\langle\rho_{k},\phi\right\rangle W^{k}_{t}.$
The regularity requirement on test functions is necessary to make sense of the
nonlinear term. We will be able to loosen it when assuming better
integrability of the initial datum. The following is the first main result of
the present paper:
###### Theorem 3.3 (Well-posedness).
For every $q_{0}\in\mathbf{L}^{\infty}$ there exists a pathwise unique
solution to equation (3.4) in the sense of Definition 3.2. Moreover, $q\in
C_{w}(0,T;\mathbf{L}^{\infty})$ $\mathbb{P}$-almost surely, $q_{t}$ is
progressively measurable as a process with values in
$(\mathbf{L}^{\infty},\tau_{\star})$ and the formulation (3.5) is valid for
all $\phi\in\mathbf{H}^{1}_{0}$. It also holds that the difference $q-W\in
W^{1,2}(0,T;\mathbf{H}^{-1})$ $\mathbb{P}$-almost surely.
In addition, if $q_{0}\in\mathbf{W}^{1,4}$, then $q\in
C_{w}(0,T;\mathbf{W}^{1,4})$ $\mathbb{P}$-almost surely and $q_{t}$ is
progressively measurable as a process in $\mathbf{W}^{1,4}$.
Existence of solutions will be established by means of a vanishing viscosity
argument in Proposition 4.8, while uniqueness follows from the standard
argument of Yudovich [35], cf. Proposition 4.10.
The following extends the notion of weak solution to time-dependent test
functions, and it will prove to be a convenient formulation in the forthcoming
arguments. Definitions 3.2 and 3.4 are in fact equivalent, as one can prove
along the lines of [27, Lemma 3].
###### Definition 3.4.
Given $q_{0}\in\mathbf{L}^{2}$, a stochastic process $q^{\varepsilon}_{t}$
such that
$q\in C_{w}(0,T;\mathbf{L}^{2})\qquad\mathbb{P}\text{-a.s.}$
is said to be a weak solution of (3.4) if it is adapted with respect to
$\\{\mathcal{F}_{t}\\}_{t\leq 0}$ and, for all $\phi\in
C(0,T;\mathbf{H}^{1}_{0}\cap\mathbf{H}^{2})\cap C^{1}(0,T;\mathbf{L}^{2})$ and
for all $t\in[0,T]$, $\mathbb{P}$-almost surely
(3.6) $\left\langle q_{t},\phi_{t}\right\rangle=\left\langle
q_{0},\phi_{0}\right\rangle+\int_{0}^{t}\left\langle
q_{s},\partial_{s}\phi_{s}\right\rangle ds+\int_{0}^{t}\left\langle
q_{s},u_{s}\cdot\operatorname{\nabla}\phi_{s}\right\rangle
ds-\gamma\int_{0}^{t}\left\langle q_{s},\phi_{s}\right\rangle ds\\\
+\sum_{k=0}^{\infty}c_{k}\left\langle\rho_{k},\phi_{t}\right\rangle
W_{t}^{k}-\sum_{k=0}^{\infty}\int_{0}^{t}c_{k}\left\langle\rho_{k},\partial_{s}\phi_{s}\right\rangle
W_{s}^{k}ds$
### 3.3. Properties of Solutions and Existence of Invariant Measures
Consider the solution $q_{t}$ of (3.4) obtained in Theorem 3.3. Let
$B_{b}(\mathbf{L}^{\infty},\tau_{\star})$ be the set of bounded real-valued
functions on $\mathbf{L}^{\infty}$ that are measurable with respect to the
weak$\star$ topology. For $\phi\in B_{b}(\mathbf{L}^{\infty},\tau_{\star})$ we
denote
$P_{t}\phi(\chi)\coloneqq\mathbb{E}\left[\phi(q(t,\chi))\right].$
If $q_{t}$ possesses the Markov property, the latter is the associated Markov
semigroup of operators. This is in fact the case, but the functional setting
in $\mathbf{L}^{\infty}$ requires a careful argument, to which we devote
Section 5.1. In particular, Lemma 5.1 provides sufficient conditions for the
Markov property of stochastic flows taking values in topological vector
spaces, under the assumption that a separable Banach space embedded in the
latter is preserved by the evolution. In order to show that such condition is
satisfied in our particular case, that is for the solution of Theorem 3.3, we
can rely on the stability of solutions in $\mathbf{W}^{1,4}$, to be proved in
Section 4.4.
Once established the Markov property of the stochastic process under
consideration, we can adopt the following standard definition of invariant
measure:
###### Definition 3.5.
An invariant measure for the system (3.4) is a probability measure $\mu$ on
$(\mathbf{L}^{\infty}(D),\mathcal{B}(\tau^{b}_{\star}))$ such that
$\int P_{t}fd\mu=\int fd\mu,\qquad\forall t\geq 0,\forall\phi\in
C_{b}(L^{\infty},\mathcal{B}(\tau^{b}_{\star})).$
Section 5 will be devoted to the proof of the anticipated existence theorem
for invariant measures:
###### Theorem 3.6.
Under the hypothesis of Theorem 3.3, there exist at least one invariant
measure for the problem (3.4).
The argument consists in averaging over larger and larger time intervals the
law of the stochastic process $q_{t}$, in the classical Krylov-Bogoliubov
fashion. The crucial step is thus to prove tightness of the time-averaged law
of $q_{t}$: as in [3] we rely on an idea introduced by Flandoli [15] in order
to obtain the necessary uniform bounds, to be detailed in Proposition 5.3.
###### Remark 3.7.
Our regularity assumption on the Wiener process $W_{t}$ slightly improves the
ones in [4, 3], where the authors assumed regularity in $H^{3+}$ for the
stochastic forcing (at the level of vorticity). In particular, less regularity
on the noise is required both to perform the vanishing viscosity argument,
thanks to the introduction of an additional stochastic term which vanishes in
the limit, and to ensure the stability of the solution in $W^{1,4}$ thanks to
a more careful energy estimate. As a consequence, either Theorem 3.3 and
Theorem 3.6 are improved, as they hold under weaker assumption on the space
regularity of the noise.
## 4\. Well-posedness of Stochastic Quasi-Geostrophic Systems
In order to study the existence of solutions to (3.4) it is convenient to
focus on the quantity $\eta=q-W$, and to study a vanishing viscosity limit for
the dynamics of $\eta$. Consider the Navier-Stokes system
(4.1)
$\begin{cases}dq_{t}^{\varepsilon}+(u_{t}^{\varepsilon}\cdot\operatorname{\nabla}q_{t}^{\varepsilon}+\gamma
q_{t}^{\varepsilon})dt=\varepsilon^{2}\Delta
q_{t}^{\varepsilon}dt+-\varepsilon^{2}\Delta W_{t}dt+dW_{t},\\\
u^{\varepsilon}=\operatorname{\nabla}^{\bot}(A+L)^{-1}q^{\varepsilon},\\\
q^{\varepsilon}_{t}|_{\partial D}=0,\end{cases}$
including both a viscosity term $\varepsilon^{2}\Delta q_{t}dt$ and a further
stochastic term $-\varepsilon^{2}\Delta W_{t}dt$ that will be instrumental in
dealing with the quantity $\eta=q-W$ (see Remark 4.5 below). Notice that by
assumption such additional stochastic forcing takes values in
$C(0,T;\mathbf{H}^{1/2})$.
The condition $u^{\varepsilon}\cdot n=0$ on $\partial D$ is implicit in the
very definition of $u^{\varepsilon}$, as reconstructed from the operator
$\operatorname{\nabla}^{\bot}(A+L)^{-1}$ with zero boundary condition. The
notion of solution for (4.1) is similar to the one already given for the
problem with $\varepsilon=0$, except for the fact that a second order term in
the equation allows to consider more regular solutions.
###### Definition 4.1.
Given $q_{0}\in\mathbf{L}^{2}$, a stochastic process $q^{\varepsilon}_{t}$
such that
$q^{\varepsilon}(\omega)\in C(0,T;\mathbf{L}^{2})\cap
L^{2}(0,T;\mathbf{H}^{1}_{0})\qquad a.s.$
and is progressively measurable with respect to the norm-induced topologies is
said to be a weak solution of (4.1) if, for all
$\phi\in\mathbf{H}^{1}_{0}\cap\mathbf{H}^{2}$ and every $t\in[0,T]$, it holds
$\mathbb{P}$-almost surely
$\left\langle q^{\varepsilon}_{t},\phi\right\rangle=\left\langle
q^{\varepsilon}_{0},\phi\right\rangle+\int_{0}^{t}\left\langle
q^{\varepsilon}_{s},u^{\varepsilon}_{s}\cdot\operatorname{\nabla}\phi\right\rangle
ds-\gamma\int_{0}^{t}\left\langle q^{\varepsilon}_{s},\phi\right\rangle ds\\\
+\varepsilon^{2}\int_{0}^{t}{\left\langle\operatorname{\nabla}W_{s}-\operatorname{\nabla}q^{\varepsilon}_{s},\operatorname{\nabla}\phi_{s}\right\rangle}ds+\sum_{k\geq
0}c_{k}\left\langle\rho_{k},\phi\right\rangle W^{k}_{t},$
where
$u^{\varepsilon}_{t}\coloneqq\operatorname{\nabla}^{\perp}(A+L)^{-1}q^{\varepsilon}_{t}$.
As in the case of Definitions 3.2 and 3.4, we can use an equivalent time-
dependent definition for (4.1):
###### Lemma 4.2.
Let $q^{\varepsilon}$ be solution of the system (4.1). Then for every
$\phi\in C(0,T;\mathbf{L}^{2})\cap L^{2}(0,T;\mathbf{H}_{0}^{1})\cap
H^{1}(0,T;\mathbf{H}^{-1})$
it holds
$\left\langle q^{\varepsilon}_{t},\phi_{t}\right\rangle=\left\langle
q_{0},\phi_{0}\right\rangle+\int_{0}^{t}\left\langle
q^{\varepsilon}_{s},\partial_{s}\phi_{s}\right\rangle
ds+\int_{0}^{t}\left\langle
q^{\varepsilon}_{s},u^{\varepsilon}_{s}\cdot\operatorname{\nabla}\phi_{s}\right\rangle
ds-\gamma\int_{0}^{t}\left\langle q_{s},\phi_{s}\right\rangle ds\\\
-\varepsilon^{2}\int_{0}^{t}\left\langle\operatorname{\nabla}q^{\varepsilon}_{s}-\operatorname{\nabla}W_{s},\operatorname{\nabla}\phi_{s}\right\rangle
ds\\\ +\sum_{k=0}^{\infty}c_{k}\left\langle\rho_{k},\phi_{t}\right\rangle
W_{t}^{k}-\sum_{k=0}^{\infty}\int_{0}^{t}c_{k}\left\langle\rho_{k},\partial_{s}\phi_{s}\right\rangle
W^{k}_{s}ds.$
Since we are considering additive noise, no explicit stochastic integral
appears in the definition of a solution. This allows to adopt a pathwise
approach by focusing on $\eta^{\varepsilon}\coloneqq q^{\varepsilon}-W$, which
solves
(4.2)
$\begin{cases}\partial_{t}\eta^{\varepsilon}+({u}^{\varepsilon}\cdot\operatorname{\nabla})\eta^{\varepsilon}+\gamma\eta^{\varepsilon}=\varepsilon^{2}\Delta\eta^{\varepsilon}-({u}^{\varepsilon}\cdot\operatorname{\nabla})W-\gamma
W,\\\
{{u}^{\varepsilon}=\operatorname{\nabla}^{\bot}(A+L)^{-1}(\eta^{\varepsilon}+W)},\\\
\eta^{\varepsilon}_{t}|_{\partial D}=0.\end{cases}$
The above is equivalent to the equation for $q^{\varepsilon}$, as we
understand it in the analogous weak sense. Since $W_{t}$ is regular enough, it
is easy to prove that if $q^{\varepsilon}$ is a solution of (4.1) then
$\frac{d}{dt}\eta^{\varepsilon}$ belongs to the space
$L^{2}(0,T;\mathbf{H}^{-1})$ almost surely. Its
$L^{2}(0,T;\mathbf{H}^{-1})$-norm however diverges as $\varepsilon\rightarrow
0$, unless we prove higher regularity of $\eta$ (cf. computations in Lemma
4.4).
###### Remark 4.3.
The viscous approximation we consider is non-physical: it is motivated by
mathematical convenience and, as mentioned above, it avoids the confrontation
with the (physical) creation of vorticity at the boundary $\partial D$.
Physical viscous approximations satisfying no-slip boundary conditions do not
allow to prove inviscid limit results, either in the deterministic [25] or the
stochastic framework [27, 6].
### 4.1. Galerkin approximation
The following provides an a priori bound on the solution of (4.1) in $L^{p}$
norm, independent of $p$ and $\varepsilon$.
###### Lemma 4.4.
Let $q_{0}\in\mathbf{L}^{\infty}$ and assume $q^{\varepsilon}_{t}$ to be a
weak solution of (4.1). Then $q^{\varepsilon}\in
L^{\infty}(0,T;\mathbf{L}^{\infty})$ $\mathbb{P}$-almost surely and, for all
$k\in\mathbb{N}$,
$\left\|q^{\varepsilon}\right\|_{L_{t}^{\infty}\mathbf{L}_{x}^{2k}}\leq
C_{1}\left(\gamma,T,\left\|q_{0}\right\|_{\mathbf{L}_{x}^{\infty}},\left\|W(\omega)\right\|_{L_{t}^{\infty}\mathbf{H}_{x}^{5/2}}\right)$
and
(4.3)
$\left\|q^{\varepsilon}\right\|_{L^{2}_{t}\mathbf{H}^{1}_{x}}\leq\frac{1}{\varepsilon}C_{2}\left(T,\left\|q_{0}\right\|_{\mathbf{L}_{x}^{\infty}},\left\|W(\omega)\right\|_{L_{t}^{\infty}\mathbf{H}_{x}^{5/2}}\right),$
where constants $C_{1},C_{2}$ are in fact random variables as they depend on
the noise sample $\omega\in\Omega$.
###### Remark 4.5.
We will actually prove something more, namely that the bound still holds with
the $\mathbf{H}^{2+\delta}$ norm of $W_{t}$ for each $\delta>0$. Since we will
assume more regularity on the noise term in later arguments, we state a sub-
optimal result.
Let us also remark that the addition of the term $-\varepsilon^{2}\Delta
W_{t}dt$ in (4.1) is essential for the latter estimates in the case $k\geq 1$.
Without that additional term and the subsequent cancellation in the dynamics
of $\eta^{\varepsilon}$, the estimate would require better regularity of the
noise $W$.
###### Proof.
We take a pathwise approach and assume that $\omega\in\Omega$ belongs to a set
of full probability such that $W(\omega,t)\in
L^{\infty}(0,T;\mathbf{H}^{5/2})$. Let
$\eta^{\varepsilon}_{t}=q^{\varepsilon}_{t}-W_{t}$ solve (4.2). Since
$\eta^{\varepsilon}\in L^{2}(0,T;\mathbf{H}^{1})$, it is possible to use
$\eta^{\varepsilon,i}|\eta^{\varepsilon,i}|^{2k-2}$, $i=1,2,3$ as test
functions. Considering the equation term by term,
$\left\langle\frac{d}{dt}\eta^{\varepsilon,i}_{t},\eta^{\varepsilon,i}|\eta^{\varepsilon,i}|^{2k-2}\right\rangle=\left\langle|\eta^{\varepsilon,i}_{t}|^{k-1}{\frac{d}{dt}\eta^{\varepsilon,i}_{t},|\eta^{\varepsilon,i}_{t}|^{k}}\right\rangle\\\
=\frac{1}{2k}\frac{d}{dt}\left\|\eta^{\varepsilon,i}\right\|_{L^{2k}}^{2k}=\left\|\eta^{\varepsilon,i}\right\|_{L^{2k}}^{2k-1}\frac{d}{dt}\left\|\eta^{\varepsilon,i}\right\|_{L^{2k}};$
also, by the incompressibility of $u^{\varepsilon}$
$\left\langle\eta^{\varepsilon,i}_{t},u^{\varepsilon,i}_{t}\cdot\operatorname{\nabla}[(\eta^{\varepsilon,i}|\eta^{\varepsilon,i}|^{2k-2})]\right\rangle=\frac{2k-1}{2k}\left\langle
u^{\varepsilon,i}_{t},\operatorname{\nabla}[(\eta^{\varepsilon,i}_{t})^{2k}]\right\rangle=0,$
while the dissipative term yields
$\left\langle\operatorname{\nabla}\eta^{\varepsilon,i}_{t},\operatorname{\nabla}[(\eta^{\varepsilon,i}|\eta^{\varepsilon,i}|^{2k-2})]\right\rangle=(2k-1)\left\|(\eta^{\varepsilon,i}_{t})^{k-1}\operatorname{\nabla}\eta^{\varepsilon,i}_{t}\right\|_{L^{2}}^{2}.$
By Hölder inequality with $p=2k$ and $p^{\prime}=\frac{2k}{2k-1}$ on the terms
involving $W$ and $|\eta^{\varepsilon,i}|^{2k-1}$, and combining the
contributions of all layers,
(4.4)
$\left\|\eta^{\varepsilon}_{t}\right\|_{\mathbf{L}^{2k}}^{2k-1}\frac{d}{dt}\left\|\eta^{\varepsilon}_{t}\right\|_{\mathbf{L}^{2k}}+\varepsilon^{2}(2k-1)\left\|(\eta^{\varepsilon}_{t})^{k-1}\operatorname{\nabla}\eta^{\varepsilon}_{t}\right\|_{\mathbf{L}^{2}}^{2}\\\
\leq-\gamma\left\|\eta^{\varepsilon}_{t}\right\|_{\mathbf{L}^{2k}}^{2k}+\left\|\eta^{\varepsilon}_{t}\right\|_{\mathbf{L}^{2k}}^{2k-1}\Big{(}\left\|(u^{\varepsilon}_{t}\cdot\operatorname{\nabla})W_{t}\right\|_{\mathbf{L}^{2k}}+\gamma\left\|W_{t}\right\|_{\mathbf{L}^{2k}}\Big{)}.$
Neglecting positive terms on the left-hand side, the latter simplifies to
$\frac{d}{dt}\left\|\eta^{\varepsilon}_{t}\right\|_{\mathbf{L}^{2k}}\leq-\gamma\left\|\eta^{\varepsilon}_{t}\right\|_{\mathbf{L}^{2k}}+\left\|u_{t}^{\varepsilon}\right\|_{\mathbf{L}^{2k}}\left\|\operatorname{\nabla}W_{t}\right\|_{\mathbf{L}^{\infty}}+\gamma\left\|W_{t}\right\|_{\mathbf{L}^{2k}}.$
In order to close the estimate, we only need to treat carefully the nonlinear
term involving $\left\|u^{\varepsilon}\right\|_{\mathbf{L}^{2k}}$, applying
the regularizing property of the elliptic problem 3.3 stated in Lemma 3.1.
Since
$u^{\varepsilon}=\operatorname{\nabla}^{\bot}(A+L)^{-1}(\eta^{\varepsilon}+W)$,
there are two cases: using Sobolev embeddings one gets, for $k=1,2$,
$\|u^{\varepsilon}\|_{\mathbf{L}^{k}}\leq
c\|u^{\varepsilon}\|_{\mathbf{H}^{1}}\leq
C(\|\eta^{\varepsilon}\|_{\mathbf{L}^{2}}+\|W_{t}\|_{\mathbf{L}^{2}}),$
while for $k>2$
$\|u^{\varepsilon}\|_{\mathbf{L}^{\infty}}\leq
c\|u^{\varepsilon}\|_{\mathbf{W}^{1,4}}\leq
C(\|\eta^{\varepsilon}\|_{\mathbf{L}^{4}}+\|W_{t}\|_{\mathbf{L}^{4}}),$
where the constants are independent of $\varepsilon$ and $k$.
We conclude by applying Grönwall’s Lemma. Consider first the case $k=1$: we
deduce a uniform bound for the $\mathbf{L}^{2}$ norm of $\eta^{\varepsilon}$
of the form
$\sup_{[0,T]}\left\|\eta^{\varepsilon}_{t}\right\|_{\mathbf{L}^{2}}\leq
C_{1}\left(\gamma,T,\left\|q_{0}\right\|_{\mathbf{L}_{x}^{\infty}},\left\|W(\omega)\right\|_{L_{t}^{\infty}\mathbf{H}_{x}^{5/2}}\right).$
We then integrate (4.4) in time, getting rid of unnecessary terms, and obtain
$\displaystyle{\varepsilon}^{2}\int_{0}^{t}\left\|\operatorname{\nabla}\eta^{\varepsilon}\right\|_{\mathbf{L}^{2}}^{2}\leq\frac{1}{2}$
$\displaystyle\left\|\eta_{0}\right\|_{\mathbf{L}^{2}}^{2}+\int_{0}^{t}\left\|\eta^{\varepsilon}_{s}\right\|_{\mathbf{L}^{2}}\Big{(}\left\|(u_{s}^{\varepsilon}\cdot\operatorname{\nabla})W_{s}\right\|_{\mathbf{L}^{2}}+\gamma\left\|W_{s}\right\|_{\mathbf{L}^{2}}\Big{)}ds,$
which implies the second statement. Taking $k=2$, we use the bound we just
proved to obtain a similar one for the $\mathbf{L}^{4}$ norm, again by
Grönwall’s Lemma. Now, using the $\mathbf{L}^{4}$ estimate and knowing that
$u^{\varepsilon}$ only depends on
$\left\|\eta^{{\varepsilon}}\right\|_{L_{t}^{\infty}\mathbf{L}_{x}^{4}}$, we
can estimate all other $2k$-norms uniformly in $k$. In conclusion we get, for
all $k\geq 1$,
$\sup_{[0,T]}\left\|\eta^{\varepsilon}_{t}\right\|_{\mathbf{L}^{2k}}\leq
C_{1}\left(\gamma,T,\left\|q_{0}\right\|_{\mathbf{L}_{x}^{\infty}},\left\|W(\omega)\right\|_{L_{t}^{\infty}\mathbf{H}_{x}^{5/2}}\right),$
with the constant $C_{1}$ independent of $k$. This implies that
$\eta^{\varepsilon}\in L^{\infty}(0,T;\mathbf{L}^{p})\cap
L^{2}(0,T;\mathbf{H}^{1})$ uniformly for all $p$, hence
$\eta^{\varepsilon}\in L^{\infty}(0,T;\mathbf{L}^{\infty})\cap
L^{2}(0,T;\mathbf{H}^{1}).\qed$
###### Remark 4.6.
The family of bounds in $L^{\infty}(0,T;\mathbf{L}^{2k})$ is stable under the
weak$\star$ convergence of $\eta^{\varepsilon}\rightharpoonup\eta$ in
$L^{\infty}(0,T;\mathbf{L}^{2k})$. Indeed, the weak$\star$ convergence implies
a uniform bound on the $L^{\infty}(0,T;\mathbf{L}^{2})$ norms, which in turn
gives a uniform bound in $L^{\infty}(0,T;\mathbf{L}^{4})$, by what was showed
in the lemma. As we have proved, this gives a uniform bound for all other
norms, meaning that, up to passing to a subsequence which converges
weakly$\star$ in all $L^{\infty}(0,T;\mathbf{L}^{2k})$, the bounds hold for
any limit function $\eta$ due to the lower-semicontinuity of the
$\mathbf{L}^{2k}$-norms.
###### Proposition 4.7.
Let $\varepsilon>0$, $q_{0}\in\mathbf{L}^{2}$ and $W\in
C([0,T];\mathbf{H}^{5/2})$ $\mathbb{P}$-almost surely Then the problem (4.2)
admits a unique solution $\eta^{\varepsilon}\coloneqq q^{\varepsilon}-W$. In
particular,
$\eta^{\varepsilon}\in C(0,T;\mathbf{\mathbf{L}}^{2})\cap
L^{2}(0,T;\mathbf{H}^{1})\cap W^{1,2}(0,T;\mathbf{H}^{-1}).$
###### Proof.
Consider a complete linearly independent system $\\{e_{k}\\}_{k\geq 0}$ in
$\mathbf{H}^{1}(\Omega)$, made of eigenfunctions of $(A+L)$, and
$\mathbf{H}_{N}=\operatorname{Span}(e_{1},\dots,e_{N})$. Let
$\eta^{\varepsilon,N}(x,t)=\sum_{k=1}^{N}c_{k}(t)\operatorname{e}_{k}(x)$,
with coefficients $c_{k}$ defined in such a way that $\eta^{\varepsilon,N}$
solves the equation on $\mathbf{H}_{N}$, meaning that for every $k=1\ldots N$
$\left\langle\partial_{t}\eta^{\varepsilon,N},e_{k}\right\rangle-\left\langle\eta^{\varepsilon,N},u^{\varepsilon,N}\operatorname{\nabla}e_{k}\right\rangle+\gamma\left\langle\eta^{\varepsilon,N},e_{k}\right\rangle\\\
=-\varepsilon^{2}\left\langle\operatorname{\nabla}\eta^{\varepsilon,N},\operatorname{\nabla}e_{k}\right\rangle-\left\langle(u^{\varepsilon,N}\cdot\operatorname{\nabla})W,e_{k}\right\rangle-\gamma\left\langle
W,e_{k}\right\rangle.$
Local well-posedness of the system above follows from classical results about
stochastic differential equations with locally Lipshitz coefficients, cf. for
example [24, 30]. As for global well-posedness and the possibility to pass to
the limit in $N$, we need some estimates on the solutions independent from $N$
and from time $t$. We give now an energy estimate: by testing
$\eta^{\varepsilon,N}$ against itself we obtain, arguing as in Lemma 4.4,
$\frac{1}{2}\frac{d}{dt}\left\|\eta^{\varepsilon,N}_{t}\right\|_{\mathbf{L}^{2}}^{2}+\varepsilon^{2}\left\|\operatorname{\nabla}\eta^{\varepsilon,N}_{t}\right\|_{\mathbf{L}^{2}}^{2}\\\
\leq-\gamma\left\|\eta^{\varepsilon,N}_{t}\right\|_{\mathbf{L}^{2}}^{2}+\Big{(}\left\|(u^{\varepsilon,N}_{t}\cdot\operatorname{\nabla})W_{t}\right\|_{\mathbf{L}^{2}}+\gamma\left\|W_{t}\right\|_{\mathbf{L}^{2}}\Big{)}\left\|\eta^{\varepsilon,N}_{t}\right\|_{\mathbf{L}^{2}},$
from which we obtain
(4.5)
$\sup_{[0,T]}\left\|\eta^{\varepsilon,N}_{t}\right\|_{\mathbf{L}^{2}}\leq
C_{1}\left(\gamma,T,\left\|q^{N}_{0}\right\|_{\mathbf{L}_{x}^{2}},\left\|W(\omega)\right\|_{L_{t}^{\infty}\mathbf{H}_{x}^{5/2}}\right)\\\
\leq
C_{1}\left(\gamma,T,\left\|q_{0}\right\|_{\mathbf{L}_{x}^{2}},\left\|W(\omega)\right\|_{L_{t}^{\infty}\mathbf{H}_{x}^{5/2}}\right).$
and consequently
(4.6)
$\left\|q^{\varepsilon,N}\right\|_{L^{2}_{t}\mathbf{H}^{1}_{x}}\leq\frac{1}{\varepsilon}C_{2}\left(T,\left\|q_{0}\right\|_{\mathbf{L}_{x}^{2}},\left\|W(\omega)\right\|_{L_{t}^{\infty}\mathbf{H}_{x}^{5/2}}\right).$
We can also say something about the regularity in time of
$\eta^{\varepsilon,N}$. Indeed, testing against a function
$\phi\in\mathbf{H}^{1}$ yields
$\left\langle\frac{d}{dt}\eta^{\varepsilon,N},\phi\right\rangle+\left\langle(u^{\varepsilon,N}\cdot\operatorname{\nabla})\eta^{\varepsilon,N},\phi\right\rangle+\gamma\left\langle\eta^{\varepsilon,N},\phi\right\rangle=\\\
-\varepsilon^{2}\left\langle\operatorname{\nabla}\eta^{\varepsilon,N},\operatorname{\nabla}\phi\right\rangle-\left\langle(u^{\varepsilon,N}\cdot\operatorname{\nabla})W,\phi\right\rangle-\gamma\left\langle
W,\phi\right\rangle.$
By Hölder inequality, Sobolev embedding $H^{1}\hookrightarrow L^{4}$, Lemma
3.1 we obtain
$\displaystyle\int_{0}^{T}\Big{|}\Big{\langle}\frac{d}{dt}\eta^{\varepsilon,N}_{t},\phi\Big{\rangle}\Big{|}^{2}dt$
$\displaystyle\qquad\leq\int_{0}^{T}\left(\big{\|}u_{t}^{\varepsilon,N}\big{\|}^{2}_{\mathbf{L}^{2}}\|\phi\|^{2}_{\mathbf{H}^{1}}\big{\|}\eta_{t}^{\varepsilon,N}\big{\|}^{2}_{\mathbf{L}^{4}}+\gamma\big{\|}\eta_{t}^{\varepsilon,N}\big{\|}_{\mathbf{L}^{2}}+\|\phi\|^{2}_{\mathbf{H}^{1}}\|\operatorname{\nabla}\eta_{t}^{\varepsilon,N}\|_{\mathbf{L}^{2}}\right)dt$
$\displaystyle\qquad\quad+\int_{0}^{T}\left(\|\phi\|^{2}_{\mathbf{L}^{2}}\|W_{t}\|^{2}_{\mathbf{L}^{2}}+\|u_{t}^{\varepsilon,N}\|^{2}_{\mathbf{L}^{2}}\|\phi\|_{\mathbf{H}^{1}}\|W_{t}\|^{2}_{\mathbf{L}^{4}}\right)dt$
$\displaystyle\qquad\leq
C\left(\gamma+\left\|W\right\|_{L^{\infty}_{t}\mathbf{L}_{x}^{2}}+\left\|\eta^{\varepsilon,N}\right\|_{L^{\infty}_{t}\mathbf{L}_{x}^{2}}\right)^{2}\left(\left\|W\right\|_{L^{\infty}_{t}\mathbf{L}_{x}^{4}}+\left\|\eta^{\varepsilon,N}\right\|_{L^{\infty}_{t}\mathbf{H}_{x}^{1}}\right)^{2}\left\|\phi\right\|_{H_{x}^{1}}^{2}$
$\displaystyle\qquad\quad+\varepsilon^{2}M^{2}\left\|\phi\right\|_{H_{x}^{1}}^{2},$
therefore, by Eq. 4.5 and
$\|\operatorname{\nabla}\eta^{\varepsilon,N}\|_{L^{2}_{t}\mathbf{L}^{2}_{x}}\leq
M/\varepsilon$ (following from (4.6)),
(4.7)
$\int_{0}^{T}\Big{|}\Big{\langle}\frac{d}{dt}\eta^{\varepsilon,N}_{t},\phi\Big{\rangle}\Big{|}^{2}dt\leq\frac{1}{\varepsilon}C(\gamma,\|q_{0}\|_{L}^{2},\omega)\|\phi\|_{H_{x}^{1}}^{2}.$
This shows that $\frac{d}{dt}\eta^{\varepsilon,N}$ defines an operator in
$L^{2}(0,T;\mathbf{H}^{-1})$ whose norm is independent of $N$ (but diverges
for $\varepsilon\rightarrow 0$) and, together with (4.3) and Aubin-Lions-Simon
lemma (cf. [29]), we obtain that the sequence
$\\{\eta^{\varepsilon,N}\\}_{N\geq 0}$ is precompact in
$C(0,T;\mathbf{\mathbf{L}}^{2})$. We can then extract a subsequence converging
in $C(0,T;\mathbf{\mathbf{L}}^{2})$ to some limit point $\eta^{\varepsilon}$
satisfying (4.2) and assume weak convergence in $L^{2}(0,T;\mathbf{H}^{1})$
and weak$\star$ convergence in $L^{\infty}(0,T;\mathbf{L}^{2})$ . Considering
the weak formulation for $\eta^{\varepsilon,N}$,
$\left\langle\eta_{t}^{\varepsilon,N},\phi\right\rangle-\left\langle
q_{0}^{N},\phi\right\rangle+\int_{0}^{t}\big{[}\left\langle(u_{s}^{\varepsilon,N}\cdot{}\nabla)\eta_{s}^{\varepsilon,N},\phi\right\rangle+\gamma\left\langle\eta_{s}^{\varepsilon,N},\phi\right\rangle\big{]}ds\\\
=-\int_{0}^{t}\big{[}\varepsilon^{2}\left\langle\nabla\eta_{s}^{\varepsilon,N},\nabla\phi\right\rangle-\left\langle(u_{s}^{\varepsilon,N}\cdot\nabla)W_{s},\phi\right\rangle-\gamma\left\langle
W_{s},\phi\right\rangle\big{]}ds.$
we are able to pass the equation to the limit $N\rightarrow+\infty$: weak
convergence in $L^{2}(0,T;\mathbf{H}^{1})$ yields convergence for the linear
terms on the right hand side and convergence in $C(0,T,\mathbf{L}^{2})$
assures the convergence of
$\left\langle\eta_{t}^{\varepsilon,N},\phi\right\rangle$, while the non-linear
term converges due to the additional strong convergence of $u^{\varepsilon,N}$
in $C(0,T;\mathbf{H}^{1})$. We deduce that $\eta_{t}$ solves (4.1) and that
$\eta^{\varepsilon}\in C(0,T;\mathbf{L}^{2})\cap L^{2}(0,T;\mathbf{H}^{1}).$
Finally, uniqueness follows by very similar arguments based on energy
estimates that we omit. Let us notice that uniqueness implies the possibility
of repeating existence proof without using subsequences. This gives for free
the progressive measurability of $\eta$ as a process in $\mathbf{L}^{2}$,
since any finite-dimensional pathwise approximation is measurable and
converges (in the appropriate sense) to the limit, which is a continuous
process in $\mathbf{L}^{2}$, without passing to $\omega$-dependent
subsequences. ∎
### 4.2. Vanishing viscosity
Fix $q_{0}\in\mathbf{L}^{\infty}$. The bound of
$\frac{d}{dt}\eta^{\varepsilon}$ in $L^{2}(0,T;\mathbf{H}^{-1})$ can be
improved and made independent of $\varepsilon$: it is sufficient to repeat the
computations in (4.7) estimating directly the $\mathbf{L}^{4}$ norm of
$\eta^{\varepsilon}$, thanks to Lemma 4.4, instead of using the Sobolev
embedding $\mathbf{H}^{1}\hookrightarrow\mathbf{L}^{4}$. We are then ready to
prove an existence result.
###### Proposition 4.8.
Let $\eta^{\varepsilon}$ be a solution of (4.2). For every sequence
$\varepsilon_{n}\rightarrow 0$ there exist a subsequence $\varepsilon_{k_{n}}$
and a function $\eta$ such that $\eta^{\varepsilon_{k_{n}}}\rightarrow\eta$
strongly in $C(0,T;\mathbf{H}^{-\delta})$ and weakly$\star$ in
$L^{\infty}(0,T;\mathbf{L}^{\infty})$. Moreover, $q_{t}=\eta_{t}+W_{t}$ is a
solution of
$\begin{cases}dq_{t}+([U_{t}\cdot\operatorname{\nabla}]q_{t}+\gamma
q_{t})dt=dW_{t},\\\ U=\operatorname{\nabla}^{\bot}(A+L)^{-1}q,\\\
\operatorname{\nabla}\cdot U=0\end{cases}$
in the sense of Definition 3.2 and $q_{t}\in C_{w}(0,T;\mathbf{L}^{\infty}).$
###### Proof.
By Lemma 4.4, the family $\\{\eta^{\varepsilon}\\}_{\varepsilon\geq 0}$ is
bounded in $L^{\infty}(0,T;\mathbf{L}^{2})$, uniformly in $\varepsilon$, and
since the bound in $L^{2}(0,T;\mathbf{H}^{-1})$ is now uniform in
$\varepsilon$ as well, by Aubin-Lions-Simon lemma
$\\{\eta^{\varepsilon}\\}_{\varepsilon\geq 0}$ is compactly embedded in
$C(0,T;\mathbf{H}^{-\delta})$. Therefore, for every sequence
$\varepsilon_{n}\rightarrow 0$ there exists a subsequence
$\varepsilon_{k_{n}}$ such that $\eta^{\varepsilon_{k_{n}}}$ converges
strongly to some $\eta$ in $C(0,T;\mathbf{H}^{-\delta})$, and weakly$\star$ in
$L^{\infty}(0,T;\mathbf{L}^{\infty})$. With these convergences, we are able to
pass to the limit in the weak formulation of (4.2) exactly as in the proof of
Proposition 4.7 and obtain that $\eta$ solves
$\left\langle\eta_{t},\phi\right\rangle+\int_{0}^{t}\big{[}\left\langle(u_{s}\cdot\operatorname{\nabla})\eta_{s},\phi\right\rangle+\gamma\left\langle\eta_{s},\phi\right\rangle\big{]}ds\\\
=\left\langle
q_{0},\phi\right\rangle+\int_{0}^{t}[\left\langle(u_{s}\cdot\operatorname{\nabla})W_{s},\phi\right\rangle+\gamma\left\langle
W_{s},\phi\right\rangle\big{]}ds.$
Now, since $\eta\in C(0,T;\mathbf{H}^{-\delta})\cap
L^{\infty}(0,T;\mathbf{L}^{\infty})$, by [32, Ch. 3, Lemma 1.2] it holds that
$\eta\in C_{w}(0,T;\mathbf{L}^{\infty})$. Clearly, the same also holds for
$q=\eta+W$ thanks to the regularity of $W$. ∎
###### Remark 4.9.
We have not proved measurability of the process $\omega\mapsto q_{t}(\omega)$.
This will follow, as in the case of $q^{\varepsilon}$, by the uniqueness
result we prove in the following section, so we postpone the issue. However,
by standard theory of Banach spaces the measurability of the process $q_{t}$
can be proved also without uniqueness of the solutions, arguing as in [4,
Section 3.1].
### 4.3. Uniqueness
The following establishes the uniqueness statement in Theorem 3.3.
###### Proposition 4.10.
Given $q_{0}\in L^{\infty}$ and a stochastic basis
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{t},\mathbb{P},\\{W_{t}\\}_{t})$,
there is at most one pathwise solution $q$ to problem (3.4) in the space
$L^{\infty}(0,T;\mathbf{L}^{\infty})$.
In fact, we prove a stronger result than probabilistically strong well-
posedness: for $\mathbb{P}$-a.e. $\omega\in\Omega$, existence and uniqueness
hold for the deterministic equation with rough forcing $W(\omega)$. This
notion is often referred to as “path-by-path” uniqueness.
###### Proof.
Assume there exist two solutions $q$ and $q^{\prime}$ and let $u$,
$u^{\prime}$ be the associated velocities, $\psi$, $\psi^{\prime}$ the stream
functions. We denote their difference by
$\bar{\psi}=\psi-\psi^{\prime}.$
By Definition 3.4, for all test functions $\phi\in\mathbf{C}^{1}_{t,x}$ with
compact support it holds
$\displaystyle\left\langle(A+L)\bar{\psi}_{t},\phi(t)\right\rangle={}$
$\displaystyle\int_{0}^{t}\left\langle(A+L)\bar{\psi}_{s},\partial_{s}\phi(s)\right\rangle
ds-\gamma\int_{0}^{t}\left\langle(A+L)\bar{\psi}_{s},\phi(s)\right\rangle ds$
$\displaystyle+\int_{0}^{t}\left\langle(A+L)\psi^{\prime}_{s},\operatorname{\nabla}^{\perp}\bar{\psi}_{s}\cdot\operatorname{\nabla}\phi(s)\right\rangle
ds$
$\displaystyle+\int_{0}^{t}\left\langle(A+L)\bar{\psi}_{s},u_{s}\cdot\operatorname{\nabla}\phi(s)\right\rangle
ds.$
If $q$ is a solution of (3.4) in $L^{\infty}(0,T;\mathbf{L}^{\infty})$, then
$\frac{d}{dt}(q-W)\in L^{2}(0,T;\mathbf{H}^{-1})$, as it is easily derived
like in (4.7). Therefore $q-q^{\prime}$ has the same regularity, and we can
integrate by parts in time to obtain
$\displaystyle\int_{0}^{t}\left\langle\partial_{s}(A+L)\bar{\psi}_{s},\phi(s)\right\rangle
ds={}$
$\displaystyle-\gamma\int_{0}^{t}\left\langle(A+L)\bar{\psi}_{s},\phi(s)\right\rangle
ds$
$\displaystyle+\int_{0}^{t}\left\langle(A+L)\psi^{\prime}_{s},\operatorname{\nabla}^{\perp}\bar{\psi}_{s}\cdot\operatorname{\nabla}\phi(s)\right\rangle
ds$
$\displaystyle+\int_{0}^{t}\left\langle(A+L)\bar{\psi}_{s},u_{s}\cdot\operatorname{\nabla}\phi(s)\right\rangle
ds.$
By a density argument this is extended to $\phi\in
L^{\infty}(0,T;\mathbf{H}^{1})$, hence we can choose $\phi=\bar{\psi}$ and
obtain
$\frac{1}{2}\left\|{D^{1/2}}\operatorname{\nabla}\bar{\psi}_{t}\right\|_{\mathbf{L}^{2}}^{2}=\frac{1}{2}\left\langle
L\bar{\psi}_{t},\bar{\psi}_{t}\right\rangle-\int_{0}^{t}\left\langle(A+L)\psi^{\prime}_{s},\operatorname{\nabla}^{\perp}\bar{\psi}_{s}\cdot\operatorname{\nabla}\bar{\psi}_{s}\right\rangle
ds\\\
-\int_{0}^{t}\left\langle(A+L)\bar{\psi}_{s},u_{s}\cdot\operatorname{\nabla}\bar{\psi}_{s}\right\rangle
ds+\gamma\int_{0}^{t}\left\langle(A+L)\bar{\psi}_{s},\bar{\psi}_{s}\right\rangle
ds\\\ =:I+II+III+IV.$
Since we are assuming $L$ to be negative semi-definite the term $I$ is
negative, and $II$ vanishes because
$\operatorname{\nabla}^{\perp}\bar{\psi}_{s}\cdot\operatorname{\nabla}\bar{\psi}_{s}=0$.
We now need estimates on the remaining terms. As for $III$, the idea is to
integrate by parts as in [35]: the equality
$u=\operatorname{\nabla}^{\perp}\psi$ leads, up to a constant depending on
$A$, to
$\int_{0}^{t}\int_{D}\frac{\partial^{2}\psi_{s}}{\partial x_{1}\partial
x_{2}}\Big{[}\big{(}\frac{\partial\bar{\psi}_{s}}{\partial
x_{1}}\big{)}^{2}-\big{(}\frac{\partial\bar{\psi}_{s}}{\partial
x_{2}}\big{)}^{2}\Big{]}+\Big{(}\frac{\partial^{2}\psi_{s}}{\partial
x_{2}^{2}}-\frac{\partial^{2}\psi_{s}}{\partial
x_{1}^{2}}\Big{)}\frac{\partial\bar{\psi}_{s}}{\partial
x_{1}}\frac{\partial\bar{\psi}_{s}}{\partial x_{2}}dxds;$
taking absolute values we reduce ourselves to estimate the quantity
(4.8)
$\int_{0}^{t}\int_{D}|\operatorname{\nabla}^{2}\psi_{s}||\operatorname{\nabla}\bar{\psi}_{s}|^{2}dxds.$
By assumption $\psi\in L^{\infty}(0,T;B^{\infty})$, where $B^{\infty}$ is the
space of those $\psi$ solving (3.3) for some $q_{0}\in\mathbf{L}^{\infty}$ (it
is a Banach space if endowed with the norm
$\left\|\psi\right\|_{B^{\infty}}\coloneqq\left\|(A+L)\psi\right\|_{\mathbf{L}^{\infty}}$),
so we do not have bounds on the $\mathbf{L}^{\infty}_{t,x}$ norm of the first
factor, but we have a control on any $\mathbf{L}^{p}$ norm, for $p<\infty$: in
fact, we know that
$\operatorname{\nabla}\psi,\operatorname{\nabla}\psi^{\prime}\in
L^{\infty}_{t}(W^{1,p}_{0,x})^{3}\subset L^{\infty}_{t,x}$. Let
$M^{\varepsilon}\coloneqq\|\operatorname{\nabla}\bar{\psi}\|^{\varepsilon}_{L^{\infty}_{t}\mathbf{L}^{\infty}_{x}}$,
$\forall\varepsilon>0$. By Hölder inequality with conjugated exponents
$2/(2-\varepsilon)$ and $2/\varepsilon$, and since by Lemma 3.1
$\left\|\operatorname{\nabla}^{2}\psi\right\|_{\mathbf{L}^{2/\varepsilon}}\leq\frac{C}{\varepsilon}\left\|q\right\|_{\mathbf{L}^{\infty}},$
(4.8) can be bounded by
$\frac{CM^{\varepsilon}}{\varepsilon}\left\|q\right\|_{L^{\infty}_{t}\mathbf{L}^{\infty}_{x}}\int_{0}^{t}\left\|\operatorname{\nabla}\bar{\psi}_{s}\right\|_{\mathbf{L}_{x}^{2}}^{2-\varepsilon}ds.$
We are thus left with $IV$: we use again Hölder inequality with exponents $2$,
$\infty$, and $2$, then we estimate
$\left\|L\bar{\psi}_{s}\right\|_{\mathbf{L}^{2}}\leq\left\|L\right\|_{HS}\left\|\bar{\psi}_{s}\right\|_{\mathbf{L}^{2}}\leq
C\left\|\operatorname{\nabla}\bar{\psi}_{s}\right\|_{\mathbf{L}^{2}}$, where
$\left\|\cdot\right\|_{HS}$ is the Hilbert-Schmidt norm and the last
inequality follows from the Poincaré inequality (since
$\bar{\psi}\in\mathbf{W}^{1,2}_{0}$). Hence, it holds
$\int_{0}^{t}\left\langle
L\bar{\psi}_{s},u_{s}\cdot\operatorname{\nabla}\bar{\psi}_{s}\right\rangle
ds\leq
C\int_{0}^{t}\left\|\operatorname{\nabla}\bar{\psi}_{s}\right\|_{\mathbf{L}^{2}}^{2}ds$
Gathering all estimates together and setting
$z_{t}\coloneqq|\operatorname{\nabla}\bar{\psi}_{t}|_{\mathbf{L}_{x}^{2}}$ we
obtain
$\frac{1}{2}z^{2}_{t}\leq\frac{C^{\varepsilon}_{1}}{\varepsilon}\int_{0}^{t}z^{2-\varepsilon}_{s}ds+C_{2}\int_{0}^{t}z^{2}_{s}ds.$
Differentiation in time then yields
$\frac{d}{dt}z_{t}\leq\frac{C^{\varepsilon}_{1}}{\varepsilon}z^{1-\varepsilon}_{t}+C_{2}z_{t}$
from which, by Grönwall inequality,
$z_{t}\leq(C_{1}^{\varepsilon}t)^{\frac{1}{\varepsilon}}e^{tC_{2}}.$
Recalling that $C_{1}^{\varepsilon}$ is bounded in $\varepsilon$, it is
possible choose some $t_{0}\leq T$ such that $C_{1}^{\varepsilon}t<1$ whenever
$t\leq t_{0}$. Eventually, sending $\varepsilon$ to zero we obtain $z\equiv 0$
in $[0,t_{0}]$. We can then repeat the same argument starting from $t_{0}$ and
see that $z\equiv 0$ on $[0,T]$. This, together with $\bar{\psi}=0$ on
$\partial D$, implies $\bar{\psi}\equiv 0$ and consequently $q\equiv
q^{\prime}.$ ∎
### 4.4. Stability Properties of Solutions
In order to complete the proof of Theorem 3.3 we need some further stability
results, which lay the ground for the proof of the existence of an invariant
measure, Theorem 3.6. We will denote by $q(t_{0},t,\chi)$ the solution to
(3.4) at time $t$, started at time $t_{0}$ with initial datum $\chi$. Whenever
the initial time is $t_{0}=0$, it is omitted.
###### Proposition 4.11.
Let $\chi^{n}\overset{\star}{\rightharpoonup}\chi$ in $\mathbf{L}^{\infty}(D)$
and let $q^{n}_{t}\coloneqq q(t_{0},t,\chi^{n})$ be the solution of (3.4) with
initial datum $\chi^{n}$ at some fixed initial time $t_{0}$. Then
$\mathbb{P}$-almost surely
$q_{t}^{n}\overset{\star}{\rightharpoonup}q_{t}\coloneqq
q(t_{0},t,\chi)\in\mathbf{L}^{\infty}(D)$ for every $t>t_{0}$.
###### Proof.
By the a priori estimates in Lemma 4.4 and by Remark 4.6,
$\eta^{n}_{t}=q^{n}_{t}-W_{t}$ is uniformly bounded in
$L^{\infty}(0,T;\mathbf{L}^{\infty})$ and $\frac{d}{dt}\eta^{n}_{t}$ is
uniformly bounded in $L^{2}(0,T;\mathbf{H}^{-1})$, hence $\eta^{n}$ is
relatively compact in $C(0,T;\mathbf{H}^{-\delta})$. Consequently we can
extract a subsequence (still denoted as $\eta^{n}$) such that
$\displaystyle\eta^{n}\rightarrow\eta\qquad$ $\displaystyle\in
C(0,T;\mathbf{H}^{-\delta})$
$\displaystyle\eta^{n}\overset{\star}{\rightharpoonup}\eta\qquad$
$\displaystyle\in L^{\infty}(0,T;\mathbf{L}^{\infty})$
for some $\eta\in C(0,T;\mathbf{H}^{-\delta})\cap
L^{\infty}(0,T;\mathbf{L}^{\infty})$. The same holds for $q^{n}$, with limit
$q=\eta+W_{t}$. Then, for any $g\in\mathbf{L}^{1}(D)$
$\int_{t_{0}}^{T}\left\langle q_{t}^{n},g\right\rangle
dt\rightarrow\int_{t_{0}}^{T}\left\langle q_{t},g\right\rangle dt$
and, testing against $\chi^{\delta}_{t}g$ for some $\chi^{\delta}_{t}$
localized around $t$ at scale $\delta$, we get, sending $\delta\rightarrow 0$,
(4.9) $\left\langle q_{t}^{n},g\right\rangle\rightarrow\left\langle
q_{t},g\right\rangle$
$\mathbb{P}$-almost surely for almost every $t\in[t_{0},T]$. Now we are ready
to see that $q_{t}$ solves (3.4). In fact, arguing as in the proof of
Proposition 4.8 and using what we just showed for the first term, we are able
to pass to the limit the weak formulation for $\eta^{n}$ for every
$g\in\mathbf{C}_{c}^{1}(D)$,
$\left\langle\eta^{n}_{t},g\right\rangle=\left\langle\chi^{n},g\right\rangle+\int_{t_{0}}^{t}\left(\left\langle\eta^{n}_{s},u^{n}\cdot\operatorname{\nabla}g\right\rangle-\gamma\left\langle\eta^{n}_{s},g\right\rangle+\left\langle
W_{s},(u^{n}_{s})\cdot\operatorname{\nabla}g\right\rangle-\gamma\left\langle
W_{s},\eta^{n}_{s}\right\rangle\right)ds.$
from which it follows that $q_{t}$ is the unique solution of (3.4) with
initial data $\chi$, that $q_{t}\in C_{w}(0,T,\mathbf{L}^{\infty})$ and
consequently that (4.9) holds for all $t\in[0,T]$, but only if
$g\in\mathbf{C}_{c}^{1}(D)$. We can conclude by considering $q_{t}^{n}-q_{t}$
as a linear operator on $\mathbf{L}^{1}$, for every $t>t_{0}$: it converges
pointwise to zero on the dense subspace $\mathbf{C}^{1}_{c}(D)$ and, since
$\\{q_{t}^{n}\\}$ and $q_{t}$ are uniformely bounded in $\mathbf{L}^{\infty}$,
a simple density argument yields $\left\langle
q_{t}^{n}-q_{t},g\right\rangle\rightarrow 0$ for all $t\geq t_{0}$,
$g\in\mathbf{L}^{1}$. ∎
Proposition 4.11 allows to derive a weak Feller property for the semigroup
associated to the solution of our problem. Recall (cf. also Section 5) that,
for every $\phi\in B_{b}(\mathbf{L}^{\infty},\tau_{\star})$, such semigroup is
defined as
(4.10) $P_{t}\phi(\chi)\coloneqq\mathbb{E}\left[\phi(q(t,\chi))\right]$
###### Corollary 4.12.
The solution map $(t,\chi)\mapsto q(t_{0},t,\chi)$ defines a sequentially
weakly$\star$ Feller semigroup in $\mathbf{L}^{\infty}$, i.e. $P_{t}$ is
continuous as an operator
$P_{t}:SC_{b}(\mathbf{L}^{\infty},\tau_{\star})\rightarrow
SC_{b}(\mathbf{L}^{\infty},\tau_{\star})$
on the space of bounded sequentially weakly$\star$ continuous functions ($f\in
SC_{b}(\mathbf{L}^{\infty},\tau_{\star})$ if $f(\xi^{n})\rightarrow f(\xi)$
whenever $\xi\overset{\star}{\rightharpoonup}\xi$ in $\mathbf{L}^{\infty}$).
Since the weak$\star$ topology is in general non-metrizable, continuity and
sequential continuity do not coincide. Considering instead the bounded
weak$\star$ topology, continuity with respect to $\tau^{b}_{\star}$ is
equivalent to sequential continuity with respect to $\tau_{\star}$, as
recalled in Section 2. Hence, $P_{t}$ is continuous as an operator
$P_{t}:C_{b}(\mathbf{L}^{\infty},\tau^{b}_{\star})\rightarrow
C_{b}(\mathbf{L}^{\infty},\tau^{b}_{\star}).$
The forthcoming Proposition concerns regularity of solutions when the initial
datum is in $\mathbf{W}^{1,4}$, a separable proper subspace of
$\mathbf{L}^{\infty}$. Together with the dense embedding of $\mathbf{W}^{1,4}$
into $(\mathbf{L}^{\infty},\tau_{\star})$, this will be intstrumental in
establishing the Markov property of the solution semigroup.
###### Proposition 4.13.
Let $q_{0}\in\mathbf{W}^{1,4}$. Then the solution $q(t_{0},t,q_{0})$ to (3.4)
started at $q_{0}$ belongs to $C_{w}(t_{0},T;\mathbf{W}^{1,4})$
$\mathbb{P}$-almost surely and is adapted as a process in $\mathbf{W}^{1,4}$.
The proof consists in a Grönwall argument and closely resembles that of [3,
Theorem 5]. It is however possible to weaken the regularity hypothesis on
$W_{t}$ required in the cited work: in order to highlight how so, we provide a
sketch of the proof. Let us state first a preliminary result, adapting a
classic bound from elliptic PDE theory.
###### Lemma 4.14.
Given a solution $q$ of (3.4), if $u$ is the associated velocity it holds
$\left\|\operatorname{\nabla}u\right\|_{\mathbf{L}^{\infty}}\leq
C\left\|q\right\|_{\mathbf{L}^{\infty}}\big{(}1+\log_{+}\big{(}\left\|\operatorname{\nabla}q\right\|_{\mathbf{L}^{4}_{x}}\big{)}\big{)}$
for some constant $C>0$.
###### Proof.
The estimate is a consequence of the one contained in the work of A. Ferrari,
[14, Proposition 1]. By definition, each component $u^{i}$ of $u$ solves the
elliptic problem
$h_{i}\operatorname{\nabla}\times u^{i}=q^{i}-\sum_{j}L_{ij}\psi^{j}$
where both the coefficients $h_{j}$ of $D$ and the boundary conditions are as
discussed in Section 3. Set for simplicity $Q^{i}\coloneqq
q^{i}-\sum_{j}L_{ij}\psi^{j}$ By the estimate in [14], we then obtain
$h_{i}\|\operatorname{\nabla}u^{i}\|_{\mathbf{L}^{\infty}}\leq\bar{C}\left(1+\log\left(1+\frac{\left\|Q^{i}\right\|_{\mathbf{W}_{x}^{1,4}}}{\left\|Q^{i}\right\|_{\mathbf{L}^{\infty}}}\right)\right)\Big{\|}Q^{i}\Big{\|}_{\mathbf{L}^{\infty}}.$
In fact, the regularity requirements in the statement of Ferrari are stricter,
due to the fact that he works in a $3$-dimensional setting; since we only work
in $2$ dimensions, reproducing his result requires weaker assumptions due to
lower regularity loss in Sobolev embeddings.
In order to recover the estimate we need, we can work a little more on the
right-hand side. Using that $\log(x+y)\leq\log 2+\log_{+}x+\log_{+}y$, as well
as $-x\log x\leq 1/e$, we get to
$h_{i}\left\|\operatorname{\nabla}u^{i}\right\|_{\mathbf{L}^{\infty}}\leq
C\left(1+\left\|Q^{i}\right\|_{\mathbf{L}^{\infty}}\left(1+\log_{+}\left(\left\|Q^{i}\right\|_{\mathbf{W}_{x}^{1,4}}\right)\right)\right).$
We then take the maximum over $i=1,2,3$, use the triangular inequality and
again the properties of the logarithm again. This gives the estimate
$\|L_{ij}\psi^{j}\|_{\mathbf{W}_{x}^{1,4}}\leq
C\|q^{j}\|_{\mathbf{W}_{x}^{1,4}}$ (same for the $\mathbf{L}^{\infty}$ norm,
by Lemma 3.1). In the end, we get
$\|\operatorname{\nabla}u\|_{\mathbf{L}^{\infty}}\leq\bar{C}\|q\|_{\mathbf{L}^{\infty}}\left(C+\log_{+}\left(\|q\|_{\mathbf{W}^{1,4}_{x}}\right)\right).$
Finally, a few more computations and Poincaré inequality allow to substitute
$\|\operatorname{\nabla}q\|_{\mathbf{L}^{4}}$ to
$\|q\|_{\mathbf{W}^{1,4}_{x}}$, up to introducing new constants. ∎
###### Proof of Proposition 4.13 (sketch)..
By Proposition 4.11, we only need a bound on
$\left\|\operatorname{\nabla}q\right\|_{\mathbf{L}^{4}}$. We begin by taking
the gradient of (3.4), then set $\eta=q-W$: for each layer $j$ and component
$i$ we formally obtain the equation
$\frac{d}{dt}\partial_{i}\eta^{j}+\partial_{i}(u^{j}\cdot\operatorname{\nabla}(\eta^{j}+W^{j}))+\gamma\partial_{i}(\eta^{j}+W^{j})=0\qquad
j=1,\ldots,3.$
The following steps, as usual, should be performed first on the Galerkin’s
approximants of $\eta$ and then passed to the limit. We omit such passage for
brevity. By multiplying the equation by
$\partial_{i}\eta^{j}|\operatorname{\nabla}\eta^{j}|^{2}$, summing over $i$
and integrating over the spacial domain $D$,
$\frac{1}{4}\frac{d}{dt}\left\|\operatorname{\nabla}\eta^{j}\right\|_{\mathbf{L}^{2}}^{4}+\gamma\left\|\operatorname{\nabla}\eta^{j}\right\|_{\mathbf{L}^{2}}^{4}=-\gamma\sum_{i}^{3}\left\langle\partial_{i}W^{j},\partial_{i}\eta^{j}|\operatorname{\nabla}\eta^{j}|^{2}\right\rangle\\\
=-\sum_{i,k}^{3}\left\langle\partial_{i}u^{j}_{k}\partial_{k}\eta^{j},\partial_{i}\eta^{j}|\operatorname{\nabla}\eta^{j}|^{2}\right\rangle-\sum_{i,k}^{3}\left\langle\partial_{i}(u^{j}_{k}\partial_{k}W^{j}),\partial_{i}\eta^{j}|\operatorname{\nabla}\eta^{j}|^{2}\right\rangle\\\
=:I+II+III.$
The terms on the right-hand side are estimated by Hölder and Young
inequalities:
$\displaystyle|I|$
$\displaystyle\leq\frac{\gamma}{2}\left\|\operatorname{\nabla}\eta^{j}\right\|_{L^{4}}^{4}+C_{\gamma}\left\|W^{j}\right\|_{W^{1,4}},$
$\displaystyle|II|$ $\displaystyle\leq
C\left\|\operatorname{\nabla}u^{j}\right\|_{L^{\infty}}\left\|\operatorname{\nabla}\eta^{j}\right\|_{L^{4}}^{4},$
$\displaystyle|III|$
$\displaystyle\leq\sum_{i,k}^{3}\left\langle\partial_{i}u^{j}_{k}\partial_{k}W^{j}+u^{j}_{k}\partial^{2}_{ik}W^{j},\partial_{i}\eta^{j}|\operatorname{\nabla}\eta^{j}|^{2}\right\rangle$
$\displaystyle\leq\left\|\operatorname{\nabla}\eta^{j}\right\|_{L^{4}}^{4}+C\left\|\operatorname{\nabla}u^{j}\right\|_{L^{4}}^{4}\left\|\operatorname{\nabla}W^{j}\right\|_{L^{4}}^{4}+C^{\prime}\left\|u^{j}\right\|_{L^{\infty}}^{4}\left\|W^{j}\right\|_{W^{2,4}}.$
This way it becomes evident that the most regularity we need on the Brownian
motion $W^{j}$ is for it to be in $W^{2,4}$. We can therefore limit ourselves,
by Sobolev embedding, to requiring $W_{t}\in\mathbf{H}^{5/2}$. From the above
computations it emerges that there still are three terms to estimate:
$\left\|\operatorname{\nabla}u^{j}\right\|_{L^{\infty}}$,
$\left\|\operatorname{\nabla}u^{j}\right\|_{L^{4}}$ and
$\left\|u^{j}\right\|_{L^{\infty}}$. The first one is easily done via Lemma
4.14. As for the second one, either Lemma 4.14 or Lemma 3.1 and Lemma 4.4 can
be used. Finally, the third term is simply bounded by the second one.
Thanks to the estimates on $I$, $II$ and $III$ we have that, summing over $j$,
$\frac{1}{4}\frac{d}{dt}\left\|\operatorname{\nabla}\eta(t)\right\|_{\mathbf{L}^{4}}^{4}\leq\left\|\operatorname{\nabla}\eta\right\|_{\mathbf{L}^{4}}^{4}+C\left\|q\right\|_{\mathbf{L}^{\infty}}\left(C+\log_{+}\left(\|\operatorname{\nabla}q\|_{\mathbf{L}^{4}_{x}}\right)\right)\left\|\operatorname{\nabla}\eta\right\|_{\mathbf{L}^{4}}^{4}\\\
+C\left(\gamma,t_{0},T,\left\|q_{0}\right\|_{\mathbf{L}^{\infty}},\left\|W\right\|_{C\left(0,T;\mathbf{H}^{5/2}\right)}\right).$
At this point, the proof goes along exactly as in [3, Theorem 5], coming to
$\sup_{t_{0}\leq t\leq T}\|\operatorname{\nabla}\eta\|_{\mathbf{L}^{4}}\leq
C\left(\gamma,t_{0},T,\left\|q_{0}\right\|_{\mathbf{L}^{\infty}},\left\|W\right\|_{C\left(0,T;\mathbf{H}^{5/2}\right)}\right).$
Weak continuity in $\mathbf{W}^{1,4}$ is obtained as usual observing that it
has been proved $\frac{d}{dt}\eta\in L^{2}(0,T;\mathbf{H}^{-1})$ and
$\operatorname{\nabla}\eta\in L^{\infty}(0,T;\mathbf{L}^{4})$[32].
Measurability is obtained by a standard argument, see for instance [4]. ∎
###### Proof of Theorem 3.3.
We have already proved existence (Proposition 4.8) and uniqueness (Proposition
4.10). Exploiting the additional integrability of $q$ and the density of
$\mathbf{H}^{1}_{0}\cap\mathbf{H}^{2}$ into $\mathbf{H}^{1}_{0}$, we can relax
the regularity requirements for test functions in the definition of weak
solution and simply ask $\phi\in\mathbf{H}^{1}_{0}$, as
$\int_{0}^{t}\left\langle
q_{s},U_{s}\cdot\operatorname{\nabla}\phi\right\rangle
ds\leq\left\|q\right\|_{L^{\infty}_{t}\mathbf{L}^{\infty}_{x}}\left\|U\right\|_{L^{\infty}_{t}\mathbf{L}^{2}_{x}}\left\|\phi\right\|_{\mathbf{H}^{1}}$
now gives a continuous operator on $\mathbf{H}^{1}_{0}$.
As for measurability, it follows by uniqueness and weak continuity in
$\mathbf{L}^{2}$ . Indeed, we know from Proposition 4.7 that the Navier-Stokes
approximants are measurable and, thanks to the well-posedness of the limiting
problem, the whole sequence converges (in the appropriate sense) to the
solution of the inviscid problem. Thus, since there is no dependence on
$\omega\in\Omega$ in the choice of a suitable subsequence, solutions of the
limit problem are also measurable.
Finally, stability in $\mathbf{W}^{1,4}$ is the content of Proposition 4.13 we
just proved. ∎
## 5\. Invariant measure
### 5.1. Markov property
The following technical result establishes in our context a well-known general
principle, i.e. that time-homogeneous stochastic flows with independent
increments are Markov processes. In order to be able to treat non-metric
spaces, following [3] we exploit a further assumption, that is the invariance
of an embedded Banach space under the stochastic flow.
###### Lemma 5.1 (Markov Property for Stochastic Flows).
Let $\big{(}\Omega,\mathcal{F},\mathcal{F}_{t},\mathbb{P}\big{)}$ be a
filtered probability space and let $(H,\mathcal{T})$ be a topological vector
space.
Consider an $H$-valued stochastic flow $\xi_{s,t}^{\chi}(\omega)$, i.e. a
random field indexed by $0\leq s\leq t\leq T$, $\chi\in H$ such that:
* •
$\xi_{s,s}$ is the identity map $\mathbb{P}$-almost surely;
* •
$\mathbb{P}$-almost surely, for all $s\leq r\leq t$,
$\xi_{s,t}^{\chi}=\xi_{r,t}^{\xi_{s,r}^{\chi}}$;
* •
if $\chi_{n}\to\chi$ in $H$ then $\mathbb{P}$-almost surely for all
$s,t\in[0,T]$ it holds $\xi^{\chi_{n}}_{s,t}\to\xi^{\chi}_{s,t}$ in $H$.
Assume that
* •
(adaptedness) $\omega\mapsto\xi_{0,t}^{\chi}(\omega)$ is
$\mathcal{F}_{t}$-measurable for all $t\in[0,T]$;
* •
(independence of increments) for all $s\leq t$, $\xi_{s,t}^{\chi}$ is
independent of $\mathcal{F}_{s}$;
* •
(time homogeneity) for all $\chi\in H$, $\xi_{t,t+h}^{\chi}$ and
$\xi_{s,s+h}^{\chi}$ have the same law for all $s,t\in[0,T]$ and admissible
$h>0$;
* •
(continuous Banach embedding) there exists a separable Banach space $Y\subset
H$ such that the embedding is dense and continuous. Moreover, for all $\chi\in
Y$, $\mathbb{P}$-almost surely $\xi_{s,t}^{\chi}\in Y$ for all $s,t\in[0,T]$.
Then the evolution operator
$P_{t}:SC_{b}(H,\mathcal{T})\to
SC_{b}(H,\mathcal{T}),\quad(P_{t}\phi)(\chi)=\mathbb{E}[\phi(\xi_{0,t}^{\chi})].$
satisfies the Chapman-Kolmogorov equation: for all $\chi\in H$,
$P_{t+s}\phi(\chi)=P_{t}P_{s}\phi(\chi).$
###### Proof.
It is enough to prove that
$\mathbb{E}\left[\phi\left(\xi_{t+s}^{\chi}\right)|\mathcal{F}_{t}\right]=P_{s}\phi\left(\xi_{t}^{\chi}\right),$
as taking the expected value on both sides yields the thesis; in particular,
the above equality is equivalent to
$\mathbb{E}\left[\phi\left(\xi_{t+s}^{\chi}\right)Z\right]=\mathbb{E}\left[P_{s}\phi\left(\xi_{t}^{\chi}\right)Z\right]$
for every bounded $\mathcal{F}_{t}$-measurable random variable $Z$. Since
$\xi_{t+s}^{\chi}=\xi_{t,t+s}^{\xi_{t}^{\chi}}$ almost surely, it is
sufficient to prove
(5.1)
$\mathbb{E}\left[\phi\left(\xi_{t,t+s}^{\zeta}\right)Z\right]=\mathbb{E}\left[P_{s}\phi\left(\zeta\right)Z\right]$
for every $H$-valued $\mathcal{F}_{t}$-measurable random variable $\zeta$.
Let us start by choosing $\zeta$ taking values in $Y$. Since $Y$ is a
separable metric space, there exists a sequence of $Y$-valued,
$\mathcal{F}_{t}$-measurable random variables
$\\{\zeta_{n}\\}_{n}\overset{Y}{\rightarrow}\zeta$ of the form
$\zeta_{n}=\sum_{i=1}^{k_{n}}\zeta_{n}^{(i)}\mathbbm{1}_{A_{n}^{(i)}},$
with $\zeta_{n}^{(i)}\in Y$ and
$\\{A_{n}^{(1)},\dots,A_{n}^{(k_{n})}\\}\subset\mathcal{F}_{t}$ a partition of
$\Omega$. Convergence in $Y$ implies convergence in $(H,\mathcal{T})$ and, due
to the continuity of $\xi$ with respect to initial conditions and to the
sequential continuity of $\phi$ and $P_{s}\phi$, it is enough to prove
(5.2)
$\mathbb{E}\left[\phi\left(\xi_{t,t+s}^{\zeta_{n}}\right)Z\right]=\mathbb{E}\left[P_{s}\phi\left(\zeta_{n}\right)Z\right]\quad\forall
n.$
Eq. 5.1 then follows by dominated convergence (whenever $\zeta(\omega)\in Y$
$\mathbb{P}$-almost surely).
Given that
$P_{s}\phi\left(\zeta_{n}\right)=\sum_{i=1}^{k_{n}}P_{s}\phi\left(\zeta_{n}^{(i)}\right)\mathbbm{1}_{A_{n}^{(i)}}\quad\mathbb{P}-a.s.$
and that
$\phi\left(\xi_{t,t+s}^{\zeta_{n}}\right)=\sum_{i=1}^{k_{n}}\phi\left(\xi_{t,t+s}^{\zeta_{n}^{(i)}}\right)\mathbbm{1}_{A_{n}^{(i)}}$
(because
$\xi_{t,t+s}^{\zeta_{n}}=\sum_{i=1}^{k_{n}}\xi_{t,t+s}^{\zeta_{n}^{(i)}}\mathbbm{1}_{A_{n}^{(i)}}$),
we can reduce to proving that
$\mathbb{E}\left[\phi\left(\xi_{t,t+s}^{\eta}\right)\mathbbm{1}_{A}Z\right]=\mathbb{E}\left[P_{s}\phi\left(\eta\right)\mathbbm{1}_{A}Z\right]$
for all $Y$-valued, $\mathcal{F}_{t}$-measurable random variables $\eta$ and
for all $A\in\mathcal{F}_{t}$. Moreover, we can neglect the term
$\mathbbm{1}_{A}$, as it is itself a bounded $\mathcal{F}_{t}$-measurable
random variable. In sight of this, we have
$\mathbb{E}\left[\phi\left(\xi_{t,t+s}^{\eta}\right)Z\right]=\mathbb{E}\left[\mathbb{E}\left[\phi\left(\xi_{t,t+s}^{\eta}\right)Z|\mathcal{F}_{t}\right]\right]=\mathbb{E}\left[\mathbb{E}\left[\phi\left(\xi_{t,t+s}^{\eta}\right)\right]Z\right]\\\
=\mathbb{E}\left[\phi\left(\xi_{t,t+s}^{\eta}\right)\right]\mathbb{E}\left[Z\right]=\mathbb{E}\left[\phi\left(\xi_{s}^{\eta}\right)x\right]\mathbb{E}\left[Z\right]=P_{s}\phi\left(\eta\right)\mathbb{E}\left[Z\right]=\mathbb{E}\left[P_{s}\phi\left(\eta\right)Z\right],$
which proves (5.2).
We are thus left to verify that (5.1) holds for every $H$-valued,
$\mathcal{F}_{t}$-measurable random variable we exploit the dense embedding of
$Y$ into $H$. Let $\chi(\omega)\in H$ $\mathbb{P}$-almost surely and
$\\{\chi_{n}\\}_{n\in\mathbb{N}}$ be a sequence of $Y$-valued random variables
converging to $\chi$ with respect to $\mathcal{T}$. We have already proved
that for all $n$ it holds
$\mathbb{E}\left[\phi\left(\xi_{t+s}^{\chi_{n}}\right)Z\right]=\mathbb{E}\left[P_{s}\phi\left(\xi_{t}^{\chi_{n}}\right)Z\right]$
whenever $\phi\in SC_{b}(H,\mathcal{T})$ and $Z$ is a bounded,
$\mathcal{F}_{t}$-measurable random variable. By continuity of $\xi$ with
respect to $\chi_{n}$ we have that both
$\phi\left(\xi_{t+s}^{\chi_{n}}\right)\rightarrow\phi\left(\xi_{t+s}^{\chi}\right)$
and $P_{s}\phi\left(\xi_{t}^{\chi_{n}}\right)\rightarrow
P_{s}\phi\left(\xi_{t}^{\chi}\right)$, hence by dominated convergence their
expected values converge as well, yielding (5.1). ∎
###### Corollary 5.2.
Let $P_{t}$ be the solution semigroup for the problem (3.4), as defined in
(4.10). Then $P_{t}$ enjoys the Markov property, i.e. for any $s,t\geq 0$,
$\phi\in SC_{b}(\mathbf{L}^{\infty},\tau_{\star})$
$P_{t+s}\phi=P_{t}P_{s}\phi.$
###### Proof.
It is easy to verify that $P_{t}$ meets the conditions of Lemma 5.1. In
particular, properties of the stochastic flow can be derived from Theorem 3.3
and from the properties of the Wiener process we chose as stochastic forcing.
As for the continuous Banach embedding, we can use the dense embedding of the
Banach space $\mathbf{W}^{1,4}$ into $(\mathbf{L}^{\infty},\tau_{\star})$. ∎
### 5.2. Construction of the invariant measure
We are finally able to show the existence of an invariant measure for our
problem.
###### Proof of Theorem 3.6.
Let $q$ be the pseudo-vorticity solution to (3.4) as in Theorem 3.3, started
at $t_{0}=0$, and let
$m_{t}\coloneqq\operatorname{\mathcal{L}}(q_{t})$
be its law on $\mathcal{B}(\tau^{b}_{\star})$. Consider the averaged measures
$\mu_{n}\coloneqq\frac{1}{n}\int_{0}^{n}m_{t}dt.$
This integration is allowed since we have proved joint measurability of the
process $q_{t}$, with respect to the corresponding topologies, in Theorem 3.3.
The family $\\{\mu_{n}\\}_{n\in\mathbb{N}}$ is tight (see next section for the
proof). On a separable metric space one would thus use Prokhorov’s theorem to
extract a subsequence converging in the sense of measures. However, the space
$(\mathbf{L}^{\infty},\mathcal{B}(\tau_{\star}^{b}))$ being non metrizable, we
have to resort to the variant given in [23, Theorem 3]. What we have to
verify, in order to apply such variant, is the existence of a countable family
of functionals $\\{f_{i}\\}_{i\in\mathbb{N}}$ on $\mathbf{L}^{\infty}$ taking
values in $[-1,1]$, $\tau^{b}_{\star}$-continuous and point-separating in
$\mathbf{L}^{\infty}$. The condition is satisfied: since all elements of the
separable space $\mathbf{L}^{1}$ are $\tau^{b}_{\star}$-continuous (they are
$\tau_{\star}$-continuous), any countable dense subset of $\mathbf{L}^{1}$ has
the required property. Hence, there exists a subsequence $\mu_{n_{k}}$
converging to some probability measure $\mu$ in the following sense:
$\int fd\mu_{n_{k}}\rightarrow\int fd\mu\qquad\forall f\in
C_{b}((\mathbf{L}^{\infty},\tau^{b}_{\star});\mathbb{R}).$
It is then immediate to conclude that $\mu$ is an invariant measure, since for
every $\phi\in C_{b}((\mathbf{L}^{\infty},\tau^{b}_{\star});\mathbb{R})$
$\left\langle
P_{t}\phi,\mu_{n_{k}}\right\rangle=\frac{1}{n_{k}}\int_{n_{k}}^{t+n_{k}}\left\langle
m_{r},\phi\right\rangle
dr+\left\langle\phi,\mu\right\rangle-\frac{1}{n_{k}}\int_{0}^{t}\left\langle
m_{r},\phi\right\rangle dr$
and the right-hand side converges to $\left\langle\phi,\mu\right\rangle$. By
Proposition 4.11 $P_{t}\phi\in
C_{b}((\mathbf{L}^{\infty},\tau^{b}_{\star});\mathbb{R})$, thus it also holds
that $\left\langle P_{t}\phi,\mu_{n_{k}}\right\rangle\rightarrow\left\langle
P_{t}\phi,\mu\right\rangle$ and by uniqueness of the limit
$\left\langle
P_{t}\phi,\mu\right\rangle=\left\langle\phi,\mu\right\rangle.\qed$
### 5.3. Tightness of averaged measures
###### Proposition 5.3.
Let $q(t_{0},t,\chi)$ be solution of (3.4). There exists a real-valued,
$\mathbb{P}$-almost surely finite random variable $r$ such that
$\sup_{t_{0}\leq
0}\left\|q(t_{0},0,q_{t_{0}}=0)\right\|_{\mathbf{L}^{\infty}}\leq
r\qquad\mathbb{P}\text{-a.s}.$
###### Proof.
Consider, for $\lambda>0$, the stochastic convolution
$\zeta_{\lambda}(t)\coloneqq\int_{-\infty}^{t}e^{-\lambda(t-s)}dW_{s},$
where $\zeta_{\lambda}$ has one component per layer. It is known that
$\zeta_{\lambda}$ is the unique solution of the linear equation
$d\zeta_{\lambda}(t)=-\lambda\zeta_{\lambda}(t)+dW(t)$ and has the same
regularity as $W_{t}$. Set $\theta_{\lambda}(t)\coloneqq
q(t_{0},t,0)-\zeta_{\lambda}(t)$. $\theta_{\lambda}$ solves the deterministic
equation
$\frac{d}{dt}\theta_{\lambda}=-\text{BS}(\theta_{\lambda}+\zeta_{\lambda})\cdot\operatorname{\nabla}\theta_{\lambda}-\text{BS}(\theta_{\lambda}+\zeta_{\lambda})\cdot\operatorname{\nabla}\zeta_{\lambda}-\gamma\theta_{\lambda}+(\lambda-\gamma)\zeta_{\lambda}$
where BS formally stands for the operator
$\operatorname{\nabla}^{\perp}(A+L)^{-1}$, in analogy with the Biot-Savart
kernel usually found in the formulation of the standard Euler equation.
Proceeding as in Section 4, i.e. by employing a vanishing viscosity
approximation (cf. also [3]) and with computations analogous to those used to
prove Lemma 4.4, we can test $\theta_{\lambda}$ against
$|\theta_{\lambda}|^{p-2}\theta_{\lambda}$ and obtain
$\frac{d}{dt}\left\|\theta_{\lambda}\right\|_{\mathbf{L}^{p}}\leq
C\left\|\operatorname{\nabla}\zeta_{\lambda}\right\|_{\mathbf{L}^{\infty}}\left\|\zeta_{\lambda}\right\|_{\mathbf{L}^{p}}+|\lambda-\gamma|\left\|\zeta_{\lambda}\right\|_{\mathbf{L}^{p}}+\left(C\left\|\operatorname{\nabla}\zeta_{\lambda}\right\|_{\mathbf{L}^{\infty}}-\gamma\right)\left\|\theta_{\lambda}\right\|_{\mathbf{L}^{p}},$
which yields, by Grönwall inequality on $[t_{0},0]$,
$\left\|\theta_{\lambda}(0)\right\|_{\mathbf{L}^{p}}\leq\left\|\theta_{\lambda}(t_{0})\right\|_{\mathbf{L}^{p}}e^{\int_{t_{0}}^{0}(C\left\|\operatorname{\nabla}\zeta_{\lambda}(s)\right\|_{\mathbf{L}^{\infty}}-\gamma)ds}\\\
+\int_{t_{0}}^{0}\left(C\left\|\operatorname{\nabla}\zeta_{\lambda}(r)\right\|_{\mathbf{L}^{\infty}}+|\lambda-\gamma|\right)\left\|\zeta_{\lambda}(s)\right\|_{\mathbf{L}^{p}}e^{\int_{s}^{0}(C\left\|\operatorname{\nabla}\zeta_{\lambda}(r)\right\|_{\mathbf{L}^{\infty}}-\gamma)dr}ds.$
Taking the limit $p\rightarrow\infty$ we obtain, since $H^{\alpha-1}\subset
L^{\infty}$ for $\alpha>2$ and
$\theta_{\lambda}(t_{0})=\zeta_{\lambda}(t_{0})$,
(5.3)
$\left\|\theta_{\lambda}(0)\right\|_{\mathbf{L}^{\infty}}\leq\left\|\zeta_{\lambda}(t_{0})\right\|_{\mathbf{H}^{\alpha}}e^{\int_{t_{0}}^{0}(C\left\|\zeta_{\lambda}(s)\right\|_{\mathbf{H}^{\alpha}}-\gamma)ds}\\\
+\int_{t_{0}}^{0}C\left(\left\|\zeta_{\lambda}(s)\right\|_{\mathbf{H}^{\alpha}}+|\lambda-\gamma|\right)\left\|\zeta_{\lambda}(s)\right\|_{\mathbf{H}^{\alpha}}e^{\int_{s}^{0}(C\left\|\zeta_{\lambda}(r)\right\|_{\mathbf{H}^{\alpha}}-\gamma)dr}ds,$
where we allowed the constant $C$ to change due to Sobolev embedding.
A uniform bound on the right-hand side of (5.3) can now be obtained working
term-by-term as follows. First, notice that
$\mathbb{E}\left[\left\|\zeta_{\lambda}(t)\right\|_{\mathbf{H}^{\alpha}}^{2}\right]=\frac{1}{2\lambda}\mathbb{E}\left[\left\|W(1)\right\|_{\mathbf{H}^{\alpha}}^{2}\right]\qquad\forall
t\in\mathbb{R}.$
By ergodicity of $\zeta_{\lambda}$ (cf. [13]),
$\lim_{t_{0}\rightarrow-\infty}\frac{1}{-t_{0}}\int_{t_{0}}^{0}\left\|\zeta_{\lambda}(s)\right\|_{\mathbf{H}^{\alpha}}ds=\mathbb{E}\left[\left\|\zeta_{\lambda}(0)\right\|_{\mathbf{H}^{\alpha}}\right]\qquad\mathbb{P}\text{-a.s.}$
and we can choose $\lambda$ in such a way that
$\lim_{t_{0}\rightarrow-\infty}\frac{1}{-t_{0}}\int_{t_{0}}^{0}\left\|\zeta_{\lambda}(s)\right\|_{\mathbf{H}^{\alpha}}ds<\frac{\gamma}{2C}\qquad\mathbb{P}\text{-a.s.},$
$C$ coming from (5.3). This means that there exists a random time
$\tau(\omega)\leq 0$ $\mathbb{P}$-almost surely such that, for every
$t_{0}<\tau(\omega)$,
$\int_{t_{0}}^{0}C\left\|\zeta_{\lambda}(s)\right\|_{\mathbf{H}^{\alpha}}ds\leq\frac{1}{2}\gamma(-t_{0})\leq
0.$
Hence, for $-t_{0}$ and $\lambda$ large enough,
$\exp\left(\int_{t_{0}}^{0}(C\|\zeta_{\lambda}(s)\|_{H^{\alpha}}-\gamma)ds\right)\leq\exp\left(\frac{\gamma}{2}t_{0}\right)$.
On the other hand, by continuity of $\zeta_{\lambda}(t)$ in
$\mathbf{H}^{\alpha}$ there exists a $\mathbb{P}$-almost surely finite random
variable $r_{1}$ such that
$\sup_{\tau(\omega)\leq t_{0}\leq
0}\int_{t_{0}}^{0}\|\zeta_{\lambda}(s,\omega)\|_{\mathbf{H}^{\alpha}}ds\leq
r_{1}(\omega)\qquad\mathbb{P}\text{-a.e. }\omega\in\Omega,$
and this yields a uniform bound for the exponential term, which moreover
vanishes as $t_{0}\rightarrow-\infty$. In a similar way one can also obtain,
for $t<0$, a uniform bound on
$\left\|\zeta_{\lambda}\right\|_{\mathbf{H}^{\alpha}}$ of the form
$\left\|\zeta_{\lambda}(t,\omega)\right\|_{\mathbf{H}^{\alpha}}\leq
r_{2}(1+|t|)$ for some $\mathbb{P}$-almost surely finite random variable
$r_{2}$. We have thus proved that
$\sup_{t_{0}\leq
0}\left\|\theta_{\lambda}\left(0,\theta_{\lambda}(t_{0})=-\zeta_{\lambda}(t_{0})\right)\right\|_{\mathbf{L}^{\infty}}\leq
r_{3}(\omega)$
for some $\mathbb{P}$-almost surely finite random variable $r_{3}$. The thesis
now follows recalling that $q=\theta_{\lambda}+\zeta_{\lambda}$. ∎
Tightness of averaged measures is a direct corollary of the above proposition.
###### Corollary 5.4.
For every $\varepsilon>0$ there exist a real number $R_{\varepsilon}>0$ such
that
$\inf_{t\geq
0}\mathbb{P}\left(\left\|q(0,t,q_{0}=0)\right\|_{\mathbf{L}^{\infty}}<R_{\varepsilon}\right)>1-\varepsilon.$
In particular, the family of measures $\\{\mu_{n}\\}_{n\in\mathbb{N}}$ is
tight in $\left(\mathbf{L}^{\infty},\tau^{b}_{\star}\right).$
###### Proof.
By time homogeneity $q(t_{0},0,q_{t_{0}}=0)\sim q(0,-t_{0},q_{0}=0)$ $\forall
t_{0}<0$. Since for all random variables $r$ $\mathbb{P}$-almost surely finite
and $\varepsilon>0$ there exists $R_{\varepsilon}>0$ such that
$\mathbb{P}(r<R_{\varepsilon})>1-\varepsilon$, by Proposition 5.3 we have the
following uniform estimate:
$\mathbb{P}\left(\left\|q(0,t,q_{0}=0)\right\|_{\mathbf{L}^{\infty}}<R_{\varepsilon}\right)=\mathbb{P}\left(\left\|q(-t,0,q_{-t}=0)\right\|_{\mathbf{L}^{\infty}}<R_{\varepsilon}\right)\\\
=\mathbb{P}(r<R_{\varepsilon})>1-\varepsilon.\qed$
## References
* [1] Sergio Albeverio and Ana Bela Cruzeiro. Global flows with invariant (Gibbs) measures for Euler and Navier-Stokes two-dimensional fluids. Comm. Math. Phys., 129(3):431–444, 1990.
* [2] Luigi Ambrosio, Alessandro Carlotto, and Annalisa Massaccesi. Lectures on elliptic partial differential equations, volume 18 of Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]. Edizioni della Normale, Pisa, 2018.
* [3] Hakima Bessaih and Benedetta Ferrario. Invariant measures for stochastic damped 2D Euler equations. Comm. Math. Phys., 377(1):531–549, 2020.
* [4] Hakima Bessaih and Franco Flandoli. $2$-D Euler equation perturbed by noise. NoDEA Nonlinear Differential Equations Appl., 6(1):35–54, 1999\.
* [5] Haim Brezis. Function Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2010.
* [6] Federico Butori and Eliseo Luongo. Large deviations principle for the inviscid limit of fluid dynamic systems in 2d bounded domains. arXiv preprint arXiv:2305.11148, 2023.
* [7] Luis Ángel Caffarelli and Pablo Raúl Stinga. Fractional elliptic equations, Caccioppoli estimates and regularity. Annales de l’Institut Henri Poincaré C. Analyse Non Linéaire, 33(3):767–807, 2016.
* [8] Giulia Carigi. Ergodic properties and response theory for a stochastic two-layer model of geophysical fluid dynamics. PhD thesis, 06 2021.
* [9] Giulia Carigi, Jochen Bröcker, and Tobias Kuna. Exponential ergodicity for a stochastic two-layer quasi-geostrophic model. Stoch. Dyn., 23(2):Paper No. 2350011, 33, 2023.
* [10] Giulia Carigi and Eliseo Luongo. Dissipation properties of transport noise in the two-layer quasi-geostrophic model. J. Math. Fluid Mech., 25(2):Paper No. 28, 27, 2023.
* [11] Qingshan Chen. On the well-posedness of the inviscid multi-layer quasi-geostrophic equations. Discrete Contin. Dyn. Syst., 39(6):3215–3237, 2019.
* [12] Georges-Henri Cottet. Boundary conditions and deterministic vortex methods for the Navier-Stokes equations. In Mathematical aspects of vortex dynamics (Leesburg, VA, 1988), pages 128–143. SIAM, Philadelphia, PA, 1989.
* [13] Giuseppe Da Prato and Jerzy Zabczyk. Stochastic equations in infinite dimensions, volume 152 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, 2014.
* [14] Andrew Becton Ferrari. On the blow-up of solutions of the $3$-D Euler equations in a bounded domain. Comm. Math. Phys., 155(2):277–294, 1993.
* [15] Franco Flandoli. Dissipativity and invariant measures for stochastic Navier-Stokes equations. NoDEA Nonlinear Differential Equations Appl., 1(4):403–423, 1994\.
* [16] Franco Flandoli, Francesco Grotto, and Dejun Luo. Fokker-Planck equation for dissipative 2D Euler equations with cylindrical noise. Theory Probab. Math. Statist., (102):117–143, 2020.
* [17] Francesco Grotto. Stationary solutions of damped stochastic 2-dimensional Euler’s equation. Electron. J. Probab., 25:Paper No. 69, 24, 2020.
* [18] Francesco Grotto, Eliseo Luongo, and Mario Maurelli. Uniform Approximation of 2D Navier-Stokes Equations with Vorticity Creation by Stochastic Interacting Particle Systems. arXiv e-prints, page arXiv:2212.12753, December 2022.
* [19] Francesco Grotto and Umberto Pappalettera. Equilibrium statistical mechanics of barotropic quasi-geostrophic equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 24(1):Paper No. 2150007, 23, 2021.
* [20] Francesco Grotto and Umberto Pappalettera. Gaussian invariant measures and stationary solutions of 2D primitive equations. Discrete Contin. Dyn. Syst. Ser. B, 27(5):2683–2699, 2022.
* [21] Francesco Grotto and Giovanni Peccati. Infinitesimal invariance of completely random measures for 2D Euler equations. Theory Probab. Math. Statist., (107):15–35, 2022.
* [22] Francesco Grotto and Marco Romito. A central limit theorem for Gibbsian invariant measures of 2D Euler equations. Comm. Math. Phys., 376(3):2197–2228, 2020.
* [23] Adam Jakubowski. The almost sure skorokhod representation for subsequences in nonmetric spaces. Theory of Probability & Its Applications, 42(1):167–174, 1998\.
* [24] Ioannis Karatzas and Steven Eugene Shreve. Brownian motion and stochastic calculus, volume 113 of Graduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1991.
* [25] Tosio Kato. Remarks on zero viscosity limit for nonstationary Navier-Stokes flows with boundary. In Seminar on nonlinear partial differential equations (Berkeley, Calif., 1983), volume 2 of Math. Sci. Res. Inst. Publ., pages 85–98. Springer, New York, 1984.
* [26] Jacques-Louis Lions and Enrico Magenes. Non-homogeneous boundary value problems and applications. Vol. I. Springer-Verlag, New York-Heidelberg, 1972. Translated from the French by P. Kenneth.
* [27] Eliseo Luongo. Inviscid limit for stochastic navier-stokes equations under general initial conditions. arXiv preprint arXiv:2111.14189, 2021.
* [28] Joseph Pedlosky. Geophysical fluid dynamics. Springer Science & Business Media, 2013.
* [29] Jacques Simon. Compact sets in the space $l^{p}(0,t;b)$. Annals of pure and applied mathematics, CXLVI(IV):65–96, 1987.
* [30] Anatoliy Volodymyrovych Skorokhod. Studies in the theory of random processes. Addison-Wesley Publishing Co., 1965. Translated from the Russian by Scripta Technica, Inc.
* [31] Michel Talagrand. Comparaison des boréliens pour les topologies fortes et faibles. Indiana Univ. Math. J., 21:1001–1004, 1978.
* [32] Roger Temam. Navier-Stokes equations. North-Holland publishing Company, 1979. Studies In Mathematics and Applications.
* [33] Hans Triebel. Theory of function spaces. Modern Birkhäuser Classics. Birkhäuser/Springer Basel AG, Basel, 2010. Reprint of 1983 edition [MR0730762], Also published in 1983 by Birkhäuser Verlag [MR0781540].
* [34] Geoffrey K. Vallis. Atmospheric and oceanic fluid dynamics. Cambridge University Press, 2017.
* [35] Victor Iosifovich Yudovich. Non-stationary flow of an ideal incompressible liquid. USSR Computational Mathematics and Mathematical Physics, 3(6):1407–1456, 1963.
|
# Algorithmic Improvements for Deep Reinforcement Learning applied to
Interactive Fiction
Vishal Jain,1, 3 William Fedus,1, 2 Hugo Larochelle,1, 2, 5 Doina Precup,1, 3,
4, 5 Marc G. Bellemare1, 2, 3, 5
1Mila, 2Google Brain, 3McGill University, 4DeepMind, 5CIFAR Fellow
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Text-based games are a natural challenge domain for deep reinforcement
learning algorithms. Their state and action spaces are combinatorially large,
their reward function is sparse, and they are partially observable: the agent
is informed of the consequences of its actions through textual feedback. In
this paper we emphasize this latter point and consider the design of a deep
reinforcement learning agent that can play from feedback alone. Our design
recognizes and takes advantage of the structural characteristics of text-based
games. We first propose a contextualisation mechanism, based on accumulated
reward, which simplifies the learning problem and mitigates partial
observability. We then study different methods that rely on the notion that
most actions are ineffectual in any given situation, following Zahavy et al.’s
idea of an admissible action. We evaluate these techniques in a series of
text-based games of increasing difficulty based on the TextWorld framework, as
well as the iconic game Zork. Empirically, we find that these techniques
improve the performance of a baseline deep reinforcement learning agent
applied to text-based games.
## 1 Introduction
In a text-based game, also called _interactive fiction_ (IF), an agent
interacts with its environment through a natural language interface. Actions
consist of short textual commands, while observations are paragraphs
describing the outcome of these actions (Figure 1). Recently, interactive
fiction has emerged as an important challenge for AI techniques (? ?), in
great part because the genre combines natural language with sequential
decision-making.
Figure 1: The introductory gameplay from Zork.
From a reinforcement learning perspective, IF domains pose a number of
challenges. First, the state space is typically combinatorial in nature, due
to the presence of objects and characters with which the player can interact.
Since any natural language sentence may be given as a valid command, the
action space is similarly combinatorial. The player observes its environment
through feedback in natural language, making this a partially observable
problem. The reward structure is usually sparse, with non-zero rewards only
received when the agent accomplishes something meaningful, such as retrieving
an important object or unlocking a new part of the domain.
We are particularly interested in bringing deep reinforcement learning
techniques to bear on this problem. In this paper, we consider how to design
an agent architecture that can learn to play text adventure games from
feedback alone. Despite the inherent challenges of the domain, we identify
three structural aspects that make progress possible:
* •
Rewards from subtasks. The optimal behaviour completes a series of subtasks
towards the eventual game end;
* •
Transition structure. Most actions have no effect in a given state;
* •
Memory as state. Remembering key past events is often sufficient to deal with
partial observability.
While these properties have been remarked on in previous work (?; ?), here we
relax some of the assumptions previously made and provide fresh tools to more
tractably solve IF domains. More generally, we believe these tools to be
useful in partially observable domains with similar structure.
Our first contribution takes advantage of the special reward structure of IF
domains. In IF, the accumulated reward within an episode correlates with the
number of completed subtasks and provides a good proxy for an agent’s
progress. Our _score contextualisation_ architecture makes use of this fact by
defining a piecewise value function composed of different deep network heads,
where each piece corresponds to a particular level of cumulative reward. This
separation allows the network to learn separate value functions for different
portions of the complete task; in particular, when the problem is linear
(i.e., there is a fixed ordering in which subtasks must be completed), our
method can be used to learn a separate value function for each subtask.
Our second contribution extends the work of ? (?) on action elimination. We
make exploration and action selection more tractable by determining which
actions are _admissible_ in the current state. Formally, we say that an action
is admissible if it leads to a change in the underlying game state. While the
set of available actions is typically large in IF domains, there are usually
few commands that are actually admissible in any particular context. Since the
state is not directly observable, we first learn an LSTM-based _auxiliary
classifier_ that predicts which actions are admissible given the agent’s
history of recent feedback. We use the predicted probability of an action
being admissible to modulate or _gate_ which actions are available to the
agent at each time step. We propose and compare three simple modulation
methods: masking, drop out, and finally consistent Q-learning (? ?). Compared
to ?’s algorithm, our techniques are simpler in spirit and can be learned from
feedback alone.
We show the effectiveness of our methods on a suite of seven IF problems of
increasing difficulty generated using the TextWorld platform (? ?). We find
that combining the score contextualisation approach to an otherwise standard
recurrent deep RL architecture leads to faster learning than when using a
single value function. Furthermore, our action gating mechanism enables the
learning agent to progress on the harder levels of our suite of problems.
## 2 Problem Setting
We represent an interactive fiction environment as a partially observable
Markov decision process (POMDP) with deterministic observations. This POMDP is
summarized by the tuple
$(\mathcal{S},\mathcal{A},P,r,\mathcal{O},\psi,\gamma)$, where $\mathcal{S}$
is the state space, $\mathcal{A}$ the action space, $P$ is the transition
function, $r:\mathcal{S}\times\mathcal{A}\to\mathbb{R}$ is the reward
function, and $\gamma\in[0,1)$ is the discount factor. The function
$\psi:\mathcal{S}\times\mathcal{A}\times\mathcal{S}\to\mathcal{O}$ describes
the observation $o=\psi(s,a,s^{\prime})$ provided to the agent when action $a$
is taken in state $s$ and leads to state $s^{\prime}$.
Throughout we will make use of standard notions from reinforcement learning (?
?) as adapted to the POMDP literature (?; ?). At time step $t$, the agent
selects an action according to a policy $\pi$ which maps a _history_
$h_{t}:=o_{1},a_{1},\dots,o_{t}$ to a distribution over actions, denoted
$\pi(\cdot\,|\,h_{t})$. This history is a sequence of observations and actions
which, from the agent’s perspective, replaces the unobserved environment state
$s_{t}$. We denote by $B(s\,|\,h_{t})$ the probability or _belief_ of being in
state $s$ after observing $h_{t}$. Finally, we will find it convenient to rely
on time indices to indicate the relationship between a history $h_{t}$ and its
successor, and denote by $h_{t+1}$ the history resulting from taking action
$a_{t}$ in $h_{t}$ and observing $o_{t+1}$ as emitted by the hidden state
$s_{t+1}$.
The action-value function $Q^{\pi}$ describes the expected discounted sum of
rewards when choosing action $a$ after observing history $h_{t}$, and
subsequently following policy $\pi$:
$Q^{\pi}(h_{t},a)=\operatorname*{{\mathbb{E}}}\big{[}\sum_{i\geq
0}\gamma^{i}r(s_{t+i},a_{t+i})\,|\,h_{t},a\big{]},$
where we assume that the action at time $t+j$ is drawn from
$\pi(\cdot\,|\,h_{t+j})$; note that the reward depends on the sequence of
hidden states $s_{t+1},s_{t+2},\dots$ implied by the belief state
$B(\cdot\,|\,h_{t})$. The action-value function satisfies the Bellman equation
over histories
$Q^{\pi}(h_{t},a)=\operatorname*{{\mathbb{E}}}_{s_{t},s_{t+1}}\big{[}r(s_{t},a)+\gamma\max_{a^{\prime}\in\mathcal{A}}Q^{\pi}(h_{t+1},a^{\prime})\big{]}.$
When the state is observed at each step ($\mathcal{O}=\mathcal{S}$), this
simplifies to the usual Bellman equation for Markov decision processes:
$Q^{\pi}(s_{t},a)=r(s_{t},a)+\gamma\operatorname*{{\mathbb{E}}}_{s_{t+1}\sim
P}\max_{a^{\prime}\in\mathcal{A}}Q^{\pi}(s_{t+1},a^{\prime}).$ (1)
In the fully observable case we will conflate $s_{t}$ and $h_{t}$.
The Q-learning algorithm (? ?) over histories maintains an approximate action-
value function $Q$ which is updated from samples $h_{t},a_{t},r_{t},o_{t+1}$
using a step-size parameter $\alpha\in[0,1)$:
$\displaystyle Q(h_{t},a_{t})$ $\displaystyle\leftarrow
Q(h_{t},a_{t})+\alpha\delta_{t}$ $\displaystyle\delta_{t}$
$\displaystyle=r_{t}+\gamma\max_{a\in\mathcal{A}}Q(h_{t+1},a)-Q(h_{t},a_{t}).$
(2)
Q-learning is used to estimate the _optimal action-value function_ attained by
a policy which maximizes $Q^{\pi}$ for all histories. In the context of our
work, we will assume that this policy exists. Storing this action-value
function in a lookup table is impractical, as there are in general an
exponential number of histories to consider. Instead, we use recurrent neural
networks approximate the Q-learning process.
### 2.1 Consistent Q-Learning
Consistent Q-learning (? ?) learns a value function which is consistent with
respect to a local form of policy stationarity. Defined for a Markov decision
process, it replaces the term $\delta_{t}$ in (2) by
$\delta^{\textsc{cql}}_{t}=r_{t}+\left\\{\begin{array}[]{cc}\gamma\max_{a\in\mathcal{A}}Q(s_{t+1},a)-Q(s_{t},a_{t})&s_{t+1}\neq
s_{t}\\\ (\gamma-1)Q(s_{t},a_{t})&s_{t+1}=s_{t}.\end{array}\right.$ (3)
Consistent Q-learning can be shown to decrease the action-value of suboptimal
actions while maintaining the action-value of the optimal action, leading to
larger _action gaps_ and a potentially easier value estimation problem.
Observe that consistent Q-learning is not immediately adaptable to the
history-based formulation, since $h_{t+1}$ and $h_{t}$ are sequences of
different lengths (and therefore not comparable). One of our contributions in
this paper is to derive a related algorithm suited to the history-based
setting.
### 2.2 Admissible Actions
We will make use of the notion of an _admissible action_ , following
terminology by ? (?).111Note that our definition technically differs from ?
(?)’s, who define an admissible action as one that is not ruled out by the
learning algorithm.
###### Definition 1.
An action $a$ is _admissible in state $s$_ if
$P(s\,|\,s,a)<1.$
That is, $a$ is admissible in $s$ if its application _may_ result in a change
in the environment state. When $P(s\,|\,s,a)=1$, we say that an action is
_inadmissible_.
We extend the notion of admissibility to histories as follows. We say that an
action $a$ is admissible given a history $h$ if it is admissible in _some_
state that is possible given $h$, or equivalently:
$\sum_{s\in\mathcal{S}}B(s\,|\,h)P(s\,|\,s,a)<1.$
We denote by $\xi(s)\subseteq\mathcal{A}$ the set of admissible actions in
state $s$. Abusing notation, we define the _admissibility function_
$\displaystyle\xi(s,a)$ $\displaystyle:=\mathbb{I}_{\left[a\in\xi(s)\right]}$
$\displaystyle\xi(h,a)$ $\displaystyle:=\Pr\\{a\in\xi(S)\\},S\sim
B(\cdot\,|\,h).$
We write $\mathcal{A}_{t}$ for the set of admissible actions given history
$h_{t}$, i.e. the actions whose admissibility in $h_{t}$ is strictly greater
than zero. In IF domains, inadmissible actions are usually dominated, and we
will deprioritize or altogether rule them out based on our estimate of
$\xi(h,a)$.
## 3 More Efficient Learning for IF Domains
Figure 2: Our IF architecture consists of three modules: a representation
generator $\Phi_{R}$ that learns an embedding for a sentence, an action scorer
$\Phi_{A}$ that chooses a network head $i$ (a feed-forward network)
conditional on score $u_{t}$, learns its Q-values and outputs
$Q(h_{t},:,u_{t})$ and finally, an auxilliary classifier $\Phi_{C}$ that
learns an approximate admissibility function $\hat{\xi}(h_{t},:)$. The
architecture is trained end-to-end.
We are interested in learning an action-value function which is close to
optimal and from which can be derived a near-optimal policy. We would also
like learning to proceed in a sample-efficient manner. In the context of IF
domains, this is hindered by both the partially observable nature of the
environment and the size of the action space. In this paper we propose two
complementary ideas that alleviate some of the issues caused by partial
observability and large action sets. The first idea contextualizes the action-
value function on a surrogate notion of progress based on total reward so far,
while the second seeks to eliminate inadmissible actions from the exploration
and learning process.
Although our ideas are broadly applicable, for concreteness we describe their
implementation in a deep reinforcement learning framework. Our agent
architecture (Figure 2) is derived from the LSTM-DRQN agent (? ?) and the work
of ? (?).
### 3.1 Score Contextualisation
In applying reinforcement learning to games, it is by now customary to
translate the player’s score differential into rewards (?; ?). Our setting is
similar to Arcade Learning Environment in the sense that the environment
provides the score. In IF, the player is awarded points for acquiring an
important object, or completing some task relevant to progressing through the
game. These awards occur in a linear, or almost linear structure, reflecting
the agent’s progression through the story, and are relatively sparse. We
emphasize that this is in contrast to the more general reinforcement learning
setting, which may provide reward for surviving, or achieving something at a
certain rate. In the video game Space Invaders, for example, the notion of
“finishing the game” is ill-defined: the player’s objective is to keep
increasing their score until they run out of lives.
We make use of the IF reward structure as follows. We call _score_ the agent’s
total (undiscounted) reward since the beginning of an episode, remarking that
the term extends beyond game-like domains. At time step $t$, the score $u_{t}$
is
$u_{t}:=\sum_{i=0}^{t-1}r_{i}.$
In IF domains, where the score reflects the agent’s progress, it is reasonable
to treat it as a state variable. We propose maintaining a separate action-
value function for each possible score. This action-value function is denoted
$Q(h_{t},a_{t},u_{t})$. We call this approach _score contextualisation_. The
use of additional context variables has by now been demonstrated in a number
of settings (? (?); ? (?); ? (?)). First, credit assignment becomes easier
since the score provides clues as to the hidden state. Second, in settings
with function approximation we expect optimization to be simpler since for
each $u$, the function $Q(\cdot,\cdot,u)$ needs only be trained on a subset of
the data, and hence can focus on features relevant to this part of the
environment.
In a deep network, we implement score contextualisation using $K$ network
heads and a map $\mathcal{J}:\mathbb{N}\to\\{1,\dots,K\\}$ such that the
$\mathcal{J}(u_{t})^{th}$ head is used when the agent has received a score of
$u_{t}$ at time $t$. This provides the flexibility to either map each score to
a separate network head, or multiple scores to one head. Taking $K=1$ uses one
monolothic network for all subtasks, and fully relies on this network to
identify state from feedback. In our experiments, we assign scores to networks
heads using a round-robin scheme with a fixed $K$. Using ? (?)’s terminology,
our architecture consists of a shared _representation generator_ $\Phi_{R}$
with $K$ independent LSTM heads, followed by a feed-forward _action scorer_
$\Phi_{A}(i)$ which outputs the action-values (Figure 2).
### 3.2 Action Gating Based on Admissibility
In this section we revisit the idea of using the admissibility function to
eliminate or more generally _gate_ actions. Consider an action $a$ which is
inadmissible in state $s$. By definition, taking this action does not affect
the state. We further assume that inadmissible actions produce a constant
level of reward, which we take to be 0 without loss of generality:
$a\text{ inadmissible in }s\implies r(s,a)=0.$
This assumption is reasonable in IF domains, and more generally holds true in
domains that exhibit subtask structure, such as the video game Montezuma’s
Revenge (? ?). We can combine knowledge of $P$ and $r$ for inadmissible
actions with Bellman’s equation (1) to deduce that for any policy $\pi$,
$a\text{ inadmissible in }s\implies
Q^{\pi}(s,a)\leq\max_{a^{\prime}\in\mathcal{A}}Q^{\pi}(s,a^{\prime})$ (4)
If we know that $a$ is inadmissible, then we do not need to learn its action-
value.
We propose learning a classifier whose purpose is to predict the admissibility
function. Given a history $h$, this classifier outputs, for each action $a$,
the probability $\hat{\xi}(h,a)$ that this action is admissible. Because of
state aliasing, this probability is in general strictly between 0 and 1;
furthermore, it may be inaccurate due to approximation error. We therefore
consider action gating schemes that are sensitive to intermediate values of
$\hat{\xi}(h,a)$. The first two schemes produce an approximately admissible
set $\hat{\mathcal{A}}_{t}$ which varies from time step to time step; the
third directly uses the definition of admissibility in a history-based
implementation of the consistent Bellman operator.
Dropout. The _dropout_ method randomly adds each action $a$ to
$\hat{\mathcal{A}}_{t}$ with probability $\hat{\xi}(h_{t},a)$.
Masking. The _masking_ method uses an elimination threshold $c\in[0,1)$. The
set $\hat{\mathcal{A}}_{t}$ contains all actions $a$ whose estimated
admissibility is at least $c$:
$\hat{\mathcal{A}}_{t}:=\\{a:\hat{\xi}(h_{t},a)\geq c\\}.$
The masking method is a simplified version of ? (?)’s action elimination
algorithm, whose threshold is adaptively determined from a confidence
interval, itself derived from assuming a value function and admissibility
functions that can be expressed linearly in terms of some feature vector.
In both the dropout and masking methods, we use the action set
$\hat{\mathcal{A}}_{t}$ in lieu of the the full action set $\mathcal{A}$ when
selecting exploratory actions.
Consistent Q-learning for histories (CQLH). The third method leaves the action
set unchanged, but instead drives the action-values of purportedly
inadmissible actions to 0. This is done by adapting the consistent Bellman
operator (3) to the history-based setting. First, we replace the indicator
$\mathbb{I}_{\left[s_{t+1}\neq s_{t}\right]}$ by the probability
$\hat{\xi}_{t}:=\hat{\xi}(h_{t},a_{t})$. Second, we drive $Q(s_{t},a_{t})$ to
$0$ in the case when we believe the state is unchanged, following the
argumentation of (4). This yields a version of consistent Q-learning which is
adapted to histories, and makes use of the predicted admissibility:
$\displaystyle\delta^{\textsc{cqlh}}_{t}=\;$ $\displaystyle
r_{t}+\gamma\max_{a\in\mathcal{A}}Q(h_{t+1},a)\hat{\xi}_{t}$
$\displaystyle+\gamma Q(h_{t},a_{t})(1-\hat{\xi}_{t})-Q(h_{t},a_{t}).$ (5)
One may ask whether this method is equivalent to a belief-state average of
consistent Q-learning when $\hat{\xi}(h_{t},a_{t})$ is accurate, i.e. equals
$\xi(h_{t},a_{t})$. In general, this is not the case: the admissibility of an
action depends on the hidden state, which in turns influences the action-value
at the next step. As a result, the above method may underestimate action-
values when there is state aliasing (e.g., $\hat{\xi}(h_{t},a_{t})\approx
0.5$), and yields smaller action gaps than the state-based version when
$\hat{\xi}(h_{t},a_{t})=1$. However, when $a_{t}$ is known to be inadmissible
($\hat{\xi}(h_{t},a_{t})=0$), the methods do coincide, justifying its use as
an action gating scheme.
We implement these ideas using an auxiliary classifier $\Phi_{C}$. For each
action $a$, this classifier outputs the estimated probability
$\hat{\xi}(h_{t},a)$, parametrized as a sigmoid function. These probabilities
are learned from bandit feedback: after choosing $a$ from history $h_{t}$, the
agent receives a binary signal $e_{t}$ as to whether $a$ was admissible or
not. In our setting, learning this classifier is particularly challenging
because the agent must predict admissibility solely based on the history
$h_{t}$. As a point of comparison, using the information-gathering commands
look and inventory to establish the state, as proposed by ? (?), leads to a
simpler learning problem, but one which does not consider the full history.
The need to learn $\hat{\xi}(h_{t},a)$ from bandit feedback also encourages
methods that generalize across histories and textual descriptions.
## 4 A Synthetic IF Benchmark
Both score contextualisation and action gating are tailored to domains that
exhibit the structure typical of interactive fiction. To assess how useful
these methods are, we will make use of a synthetic benchmark based on the
TextWorld framework (? ?). TextWorld provides a reinforcement learning
interface to text-based games along with an environment specification language
for designing new environments. Environments provide a set of locations, or
_rooms_ , objects that can picked up and carried between locations, and a
reward function based on interacting with these objects. Following the genre,
special _key_ objects are used to access parts of the environment.
Our benchmark provides seven environments of increasing complexity, which we
call _levels_. We control complexity by adding new rooms and/or objects to
each successive level. Each level also requires the agent to complete a number
of subtasks (Table 1), most of which involve carrying one or more items to a
particular location. Reward is provided only when the agent completes one of
these subtasks. Thematically, each level involves collecting food items to
make a salad, inspired by the first TextWorld competition. Example objects
include an apple and a head of lettuce, while example actions include get
apple and slice lettuce with knife. Accordingly we call our benchmark
SaladWorld.
SaladWorld provides a graded measure of an agent architecture’s ability to
deal with both partial observability and large action spaces. Indeed,
completing each subtasks requires memory of what has previously been
accomplished, along with where different objects are. Together with this, each
level in the SaladWorld involves some amount of history-dependent
admissibility i.e the admissibility of the action depends on the history
rather than the state. For example, put lettuce on counter can only be
accomplished once take lettuce (in a different room) has happened. Keys pose
an additional difficulty as they do not themselves provide reward. As shown in
Table 1, the number of possible actions rapidly increases with the number of
objects in a given level. Even the small number of rooms and objects
considered here preclude the use of tabular representations, as the state
space for a given level is the exponentially-sized cross-product of possible
object and agent locations. In fact, we have purposefully designed SaladWorld
as a small challenge for IF agents, and even our best method falls short of
solving the harder levels within the allotted training time. Full details are
given in Table 2 in the appendix.
Table 1: Main characteristics of each level in our synthetic benchmark. Level | # Rooms | # Objects | # Sub-tasks | $|\mathcal{A}|$
---|---|---|---|---
1 | 4 | 2 | 2 | 8
2 | 7 | 4 | 3 | 15
3 | 7 | 4 | 3 | 15
4 | 9 | 8 | 4 | 50
5 | 11 | 15 | 5 | 141
6 | 12 | 20 | 6 | 283
7 | 12 | 20 | 7 | 295
Table 2: Subtasks information and scores possible for each level of the suite. Level | Subtasks | Possible Scores
---|---|---
1 | Following subtasks with reward and fulfilling condition: • 10 points when the agent first enters the vegetable market. • 5 points when the agent gets lettuce from the vegetable market and puts lettuce on the counter. | 10, 15
2 | All subtasks from previous level plus this subtask: • 5 points when the agent takes the blue key from open space, opens the blue door, gets tomato from the supermarket and puts it on the counter in the kitchen. | 5, 10, 15, 20
3 | All subtasks from level 1 plus this subtask: • 5 points when the agent takes the blue key from open space, goes to the garden, opens the blue door with the blue key, gets tomato from the supermarket and puts it on the counter in the kitchen. Remark: Level 3 game differs from Level 2 game in terms of number of steps required to complete the additional sub-task (which is greater in case of Level 3) | 5, 10, 15, 20
4 | All subtasks from previous level plus this subtask: • 5 points when the agent takes parsley from the backyard and knife from the cutlery shop to the kitchen, puts parsley into fridge and knife on the counter. | 5, 10, 15, 20, 25
5 | All subtasks from previous level plus this subtask: • 5 points when the agent goes to fruit shop, takes chest key, opens container with chest key, takes the banana from the chest and puts it into the fridge in the kitchen. | 5, 10, 15, 20, 25, 30
6 | All subtasks from previous level plus this subtask: • 5 points when the agent takes the red key from the supermarket, goes to the playroom, opens the red door with the red key, gets the apple from cookhouse and puts it into the fridge in the kitchen. | 5, 10, 15, 20, 25, 30, 35
7 | All subtasks from previous level plus this subtask: • 5 points when the agent prepares the meal. | 5, 10, 15, 20, 25, 30, 35, 40
## 5 Empirical Analysis
In the first set of experiments, we use SaladWorld to establish that both
score contextualisation and action gating provide positive benefits in the
context of IF domain. We then validate these findings on the celebrated text-
based game Zork used in prior work (?; ?).
Our baseline agent is the LSTM-DRQN agent (? ?) but with a different action
representation. We augment this baseline with either or both score
contextualisation and action gating, and observe the resulting effect on agent
performance in SaladWorld. We measure this performance as the fraction of
subtasks completed during an episode, averaged over time. In all cases, our
results are generated from 5 independent trials of each condition. To smooth
the results, we use moving average with a window of 20,000 training steps. The
graphs and the histograms report average $\pm$ std. deviation across the
trials.
Score contextualisation uses $K=5$ network heads; the baseline corresponds to
$K=1$. Each head is trained using the Adam optimizer (? ?) with a learning
rate $\alpha=0.001$ to minimize a Q-learning loss (? ?) with a discount factor
of $\gamma=0.9$. The auxiliary classifier $\Phi_{C}$ is trained with the
binary cross-entropy loss over the selected action’s admissibility (recall
that our agent only observes the admissibility function for the selected
action). Training is done using a balanced form of prioritized replay which we
found improves baseline performance appreciably. Specifically, we use the
sampling mechanism described in ? (?) with prioritization i.e we sample
$\tau_{p}$ fraction of episodes that had atleast one positive reward,
$\tau_{n}$ fraction with atleast one negative reward and $1-\tau_{p}-\tau_{n}$
from whole episodic memory $\mathcal{D}$. Section C.1 in the Appendix compares
the baseline agent with and without prioritization. For prioritization,
$\tau_{p}=\tau_{n}=0.25$.
Actions are chosen from the estimated admissible set $\hat{\mathcal{A}}_{t}$
according to an $\epsilon$-greedy rule, with $\epsilon$ annealed linearly from
1.0 to 0.1 over the first million training steps. To simplify exploration, our
agent further takes a forced look action every 20 steps. Each episode lasts
for a maximum $T$ steps. For Level $1$ game, $T=100$, whereas for rest of the
levels $T=200$.To simplify exploration, our agent further takes a forced look
action every 20 steps (Section C.1).Full details are given in the Appendix
(Section A).
Figure 3: Fraction of tasks solved by each method at the end of training for
1.3 million steps. The tabular agents, which do not take history into account,
perform quite poorly. LI stands for “look, inventory” (see text for details).
### 5.1 Score Contextualisation
Figure 4: Comparing whether score contextualisation as an architecture
provides a useful representation for learning to act optimally. Row 1 and 2
correspond to Level 1 and 2 respectively.
We first consider the effect of score contextualisation on our agents’ ability
to complete tasks in SaladWorld. We ask,
> Does score contextualisation mitigate the negative effects of partial
> observability?
We begin in a simplified setting where the agent knows the admissible set
$\mathcal{A}_{t}$. We call this setting _oracle gating_. This setting lets us
focus on the impact of contextualisation alone. We compare our score
contextualisation (SC) to the baseline and also to two “tabular” agents. The
first tabular agent treats the most recent feedback as state, and hashes each
unique description-action pair to a Q-value. This results in a memoryless
scheme that ignores partial observability. The second tabular agent performs
the information-gathering actions look and inventory to construct its state
description, and also hashes these to unique Q-values. Accordingly, we call
this the “LI-tabular” agent. This latter scheme has proved to be a successful
heuristic in the design of IF agents (? ?), but can be problematic in domains
where taking information-gathering actions can have negative consequences (as
is the case in Zork).
Figure 3 shows the performance of the four methods across SaladWorld levels,
after 1.3 million training steps. We observe that the tabular agents’
performance suffers as soon as there are multiple subtasks, as expected. The
baseline agent performs well up to the third level, but then shows
significantly reduced performance. We hypothesize that this occurs because the
baseline agent must estimate the hidden state from longer history sequences
and effectively learn an implicit contextualisation. Beyond the fourth level,
the performance of all agents suffers, suggesting the need for a better
exploration strategy, for example using expert data (?).
We find that score contextualisation performs better than the baseline when
the admissible set is unknown. Figure 4 compares learning curves of the SC and
baseline agents with oracle gating and using the full action set,
respectively, in the simplest of levels (Level 1 and 2). We find that score
contextualisation can learn to solve these levels even without access to
$\mathcal{A}_{t}$, whereas the baseline cannot. Our results also show that
oracle gating simplifies the problem, and illustrate the value in handling
inadmissible actions differently.
We hypothesize that score contextualisation results in a simpler learning
problem in which the agent can more easily learn to distinguish which actions
are relevant to the task, and hence facilitate credit assignment. Our result
indicates that it might be unreasonable to expect contextualisation to arise
naturally (or easily) in partially observable domains with large actions sets.
We conclude that score contextualisation mitigates the negative effects of
partial observability.
Figure 5: Fraction of tasks solved by each method at the end of training for
1.3 million steps. Except in Level 1, action gating by itself does not improve
end performance. Figure 6: Effectiveness of action gating with score
contextualisation in Level 3. Of the three methods, masking performs best.
### 5.2 Score Contextualisation with Learned Action Gating
The previous experiment (in particular, Figure 4) shows the value of
restricting action selection to admissible actions. With the goal in mind of
designing an agent that can operate from feedback alone, we now ask:
> Can an agent learn more efficiently when given bandit feedback about the
> admissibility of its chosen actions?
We address this question by comparing our three action gating mechanisms. As
discussed in Section 3.2, the output of the auxiliary classifier describes our
estimate of an action’s admissibility for a given history.
As an initial point of comparison, we tested the performance of the baseline
agent when using the auxiliary classifier’s output to gate actions. For the
masking method, we selected $c=0.001$ from a larger initial parameter sweep.
The results are summarized in Figure 5. While action gating alone provides
some benefits in the first level, performance is equivalent for the rest of
the levels.
However, when combined with score contextualisation (see Fig 6, 7), we observe
some performance gains. In Level 3 in particular, we almost recover the
performance of the SC agent with oracle gating. From our results we conclude
that masking with the right threshold works best, but leave as an open
question whether the other action gating schemes can be improved.
Figure 8 shows the final comparison between the baseline LSTM-DRQN and our new
agent architecture which incorporates action gating and score
contextualisation (full learning curves are provided in the appendix, Figure
14). Our results show that the augmented method significantly outperforms the
baseline, and is able to handle more complex IF domains. From level 4 onwards,
the learning curves in the appendix show that combining score
contextualisation with masking results in faster learning, even though final
performance is unchanged. We posit that better exploration schemes are
required for further progress in SaladWorld.
Figure 7: Fraction of tasks solved by each method at the end of training for
1.3 million steps. For first 3 levels, SC + Masking is better or equivalent to
SC. For levels 4 and beyond, better exploration strategies are required.
Figure 8: Score contextualisation and masking compared to the baseline agent.
We show the fraction of tasks solved by each method at the end of training for
1.3 million steps.
### 5.3 Zork
As a final experiment, we evaluate our agent architecture on the interactive
fiction Zork I, the first installment of the popular trilogy. Zork provides an
interesting point of comparison for our methods, as it is designed by and for
humans – following the ontology of ? (?), it is a domain which is both
_interesting_ and _independent_. Our main objective is to compare the
different methods studied with ? (?)’s AE-DQN agent. Following their
experimental setup, we take $\gamma=0.8$ and train for 2 million steps. All
agents use the smaller action set (131 actions). Unlike AE-DQN, however, our
agent does not use information-gathering actions (look and inventory) to
establish the state.
Figure 9 shows the corresponding learning curves. Despite operating in a
harder regime than AE-DQN, the score contextualizing agent reaches a score
comparable to AE-DQN, in about half of the training steps. All agents
eventually fail to pass the 35-point benchmark, which corresponds to a
particularly difficult in-game task (the “troll quest”) which involves a
timing element, and we hypothesize requires a more intelligent exploration
strategy.
Figure 9: Learning curves for different agents in Zork.
## 6 Related Work
RL applied to Text Adventure games: LSTM-DQN by ? (?) deals with parser-based
text adventure games and uses an LSTM to generate feedback representation. The
representation is then used by an action scorer to generate scores for the
action verb and objects. The two scores are then averaged to determine Q-value
for the state-action pair. In the realm of choice-based games, ? (?) uses two
separate deep neural nets to generate representation for feedback and action
respectively. Q-values are calculated by dot-product of these representations.
None of the above approaches deals with partial observability in text
adventure games.
Admissible action set learning: ? (?) approach the issue of learning
admissible set given context as a supervised learning one. They train their
model on (input, label) pairs where input is context (concatenation of
feedbacks by look and inventory) and label is the list of admissible commands
given this input. AE-DQN (? ?) employs an additional neural network to prune
in-admissible actions from action set given a state. Although the paper
doesn’t deal with partial observability in text adventure games, authors show
that having a tractable admissible action set led to faster convergence. ? (?)
work on bounding the action set through affordances. Their agent is trained
through tabular Q-Learning.
Partial Observability: ? (?) replace the shared MLP in ? (?) with an LSTM cell
to calculate context representation. However, they use concatenation of
feedbacks by look and inventory as the given state to make the game more
observable. Their work also doesn’t focus on pruning in-admissible actions
given a context. Finally, ? (?) deal with partial observability by
representing state as a knowledge graph and continuously updating it after
every game step. However, the graph update rules are hand-coded; it would be
interesting to see they can be learned during gameplay.
## 7 Conclusions and Future work
We introduced two algorithmic improvements for deep reinforcement learning
applied to interactive fiction (IF). While naturally rooted in IF, we believe
our ideas extend more generally to partially observable domains and large
discrete action spaces. Our results on SaladWorld and Zork show the usefulness
of these improvements. Going forward, we believe better contextualisation
mechanisms should yield further gains. In Zork, in particular, we hypothesize
that going beyond the 35-point limit will require more tightly coupling
exploration with representation learning.
## 8 Acknowledgments
This work was funded by the CIFAR Learning in Machines and Brains program.
Authors thank Compute Canada for providing the computational resources.
## References
* [2019] Ammanabrolu, P., and Riedl, M. 2019\. Playing text-adventure games with graph-based deep reinforcement learning. In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), 3557–3565. Minneapolis, Minnesota: Association for Computational Linguistics.
* [2018] Atkinson, T.; Baier, H.; Copplestone, T.; Devlin, S.; and Swan, J. 2018\. The text-based adventure AI competition. IEEE Transactions on Games.
* [2013] Bellemare, M. G.; Naddaf, Y.; Veness, J.; and Bowling, M. 2013\. The arcade learning environment: An evaluation platform for general agents. Journal of Artificial Intelligence Research 47:253–279.
* [2016a] Bellemare, M.; Srinivasan, S.; Ostrovski, G.; Schaul, T.; Saxton, D.; and Munos, R. 2016a. Unifying count-based exploration and intrinsic motivation. In Lee, D. D.; Sugiyama, M.; Luxburg, U. V.; Guyon, I.; and Garnett, R., eds., Advances in Neural Information Processing Systems 29. Curran Associates, Inc. 1471–1479.
* [2016b] Bellemare, M. G.; Ostrovski, G.; Guez, A.; Thomas, P. S.; and Munos, R. 2016b. Increasing the action gap: New operators for reinforcement learning. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, AAAI’16, 1476–1483. AAAI Press.
* [2018] Côté, M.-A.; Kádár, A.; Yuan, X.; Kybartas, B.; Barnes, T.; Fine, E.; Moore, J.; Hausknecht, M.; Asri, L. E.; Adada, M.; Tay, W.; and Trischler, A. 2018\. Textworld: A learning environment for text-based games. CoRR abs/1806.11532.
* [2017] Fulda, N.; Ricks, D.; Murdoch, B.; and Wingate, D. 2017\. What can you do with a rock? affordance extraction via word embeddings. Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence.
* [2018] Ghosh, D.; Singh, A.; Rajeswaran, A.; Kumar, V.; and Levine, S. 2018\. Divide-and-conquer reinforcement learning. ICLR 2018.
* [2015] Hausknecht, M., and Stone, P. 2015\. Deep recurrent q-learning for partially observable mdps. In AAAI Fall Symposium on Sequential Decision Making for Intelligent Agents (AAAI-SDMIA15).
* [2016] He, J.; Chen, J.; He, X.; Gao, J.; Li, L.; Deng, L.; and Ostendorf, M. 2016\. Deep reinforcement learning with a natural language action space. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), 1621–1630. Berlin, Germany: Association for Computational Linguistics.
* [2018] Icarte, R. T.; Klassen, T. Q.; Valenzano, R.; and McIlraith, S. A. 2018\. Using reward machines for high-level task specification and decomposition in reinforcement learning. In Proceedings of the International Conference on Machine Learning.
* [2015] Kingma, D. P., and Ba, J. 2015\. Adam: A method for stochastic optimization. In International Conference on Learning Representations (ICLR).
* [2017] Lample, G., and Chaplot, D. S. 2017\. Playing fps games with deep reinforcement learning. In Thirty-First AAAI Conference on Artificial Intelligence.
* [1995] McCallum, A. K. 1995\. Reinforcement learning with selective perception and hidden state. Ph.D. Dissertation, University of Rochester.
* [2015] Mnih, V.; Kavukcuoglu, K.; Silver, D.; Rusu, A. A.; Veness, J.; Bellemare, M. G.; Graves, A.; Riedmiller, M.; Fidjeland, A. K.; Ostrovski, G.; et al. 2015\. Human-level control through deep reinforcement learning. Nature 518(7540):529.
* [2015] Narasimhan, K.; Kulkarni, T.; and Barzilay, R. 2015\. Language understanding for text-based games using deep reinforcement learning. Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing.
* [2018] OpenAI. 2018\. Gym retro. https://github.com/openai/retro.
* [2019] Rakelly, K.; Zhou, A.; Quillen, D.; Finn, C.; and Levine, S. 2019\. Efficient off-policy meta-reinforcement learning via probabilistic context variables. ICML 2019.
* [2010] Silver, D., and Veness, J. 2010\. Monte-carlo planning in large pomdps. In Advances in Neural Information Processing Systems.
* [1998] Sutton, R. S., and Barto, A. G. 1998\. Reinforcement learning: An introduction. MIT Press.
* [2018] Tao, R. Y.; Côté, M.-A.; Yuan, X.; and Asri, L. E. 2018\. Towards solving text-based games by producing adaptive action spaces.
* [2019] Tessler, C.; Zahavy, T.; Cohen, D.; Mankowitz, D. J.; and Mannor, S. 2019\. Sparse imitation learning for text based games with combinatorial action spaces. The Multi-disciplinary Conference on Reinforcement Learning and Decision Making (RLDM) 2019.
* [1989] Watkins, C. J. C. H. 1989\. Learning from delayed rewards. Ph.D. Dissertation, Cambridge University, Cambridge, England.
* [2018] Yuan, X.; Côté, M.-A.; Sordoni, A.; Laroche, R.; des Combes, R. T.; Hausknecht, M.; and Trischler, A. 2018\. Counting to explore and generalize in text-based games.
* [2018] Zahavy, T.; Haroush, M.; Merlis, N.; Mankowitz, D. J.; and Mannor, S. 2018\. Learn what not to learn: Action elimination with deep reinforcement learning. In Proceedings of the 32Nd International Conference on Neural Information Processing Systems, NIPS’18, 3566–3577. USA: Curran Associates Inc.
## Appendix A Training Details
### A.1 Hyper-parameters
Training hyper-parameters: For all the experiments unless specified,
$\gamma=0.9$. Weights for the learning agents are updated every $4$ steps.
Agents with score contextualisation architecture have $K=5$ network heads.
Parameters of score contextualisation architecture are learned end to end with
Adam optimiser (? ?) with learning rate $\alpha=0.001$. To prevent imprecise
updates for the initial states in the transition sequence due to in-sufficient
history, we use updating mechanism proposed by ? (?). In this mechanism,
considering the transition sequence of length $l,o_{1},o_{2},\dots,o_{l}$,
errors from $o_{1},o_{2},\dots,o_{n}$ aren’t back-propagated through the
network. In our case, the sequence length $l=15$ and minimum history size for
a state to be updated $n=6$ for all experiments. Score contextualisation heads
are trained to minimise the Q-learning loss over the whole transition
sequence. On the other hand, $\Phi_{C}$ minimises the BCE (binary cross-
entropy) loss over the predicted admissibility probability and the actual
admissibility signal for every transition in the transition sequence. The
behavior policy during training is $\epsilon-$greedy over the admissible set
$\hat{\mathcal{A}}_{t}$. Each episode lasts for a maximum $T$ steps. For Level
$1$ game, we anneal $\epsilon=1$ to $0.1$ over $1000000$ steps and $T=100$.
For rest of the games in the suite, we anneal $\epsilon=1$ to $0.1$ over
$1000000$ steps and $T=200$.
Architectural hyper-parameters: In $\Phi_{R}$, word embedding size is $20$ and
the number of hidden units in encoder LSTM is $64$. For a network head $k$,
the number of hidden units in context LSTM is $512$; $\Phi_{A}(k)$ is a two
layer MLP: sizes of first and second layer are 128 and $|\mathcal{A}|$
respectively. $\Phi_{C}$ has the same configuration as $\Phi_{A}(k)$.
Figure 10: Comparing the effect of oracle gating versus learning admissibility
from bandit feedback. Learning is faster in case of oracle gating since agent
is given admissible action set resulting in overall better credit assignment.
Figure 11: Final comparison shows that our algorithmic enhancements improve
the baseline. We show number of tasks solved by each method at the end of
training for 1.3 million steps.
### A.2 Action Gating Implementation
For _dropout_ and _masking_ when selecting actions, we set
$Q(h_{t},a_{t})=-\infty$ for $a\notin\hat{\mathcal{A}}_{t}$. Since
$\hat{\xi}_{t}$ is basically an estimate for admissibility for action $a$
given history $h_{t}$, we use (5) to implement consistent Q value backups:
$Q(h_{t},a_{t})\leftarrow Q(h_{t},a_{t})+\alpha\delta^{\prime\prime}_{t}$
where
$\delta^{\prime\prime}_{t}=r_{t}+\gamma\big{[}\max_{a\in\mathcal{A}}Q(h_{t+1},a)\hat{\xi}_{t}+\\\
Q(h_{t+1},a_{t})(1-\hat{\xi}_{t})\big{]}-Q(h_{t},a_{t})$
We notice by using the above equation, that for an action $a$ inadmissible in
$s$, it’s value indeed reduces to $0$ over time.
### A.3 Baseline Modifications
We modify LSTM-DRQN (? ?) in two ways. First, we concatenate the
representations $\Phi_{R}(o_{t})$ and $\Phi_{R}(a_{t-1})$ before sending it to
the history LSTM, in contrast ? (?) concatenates the inputs $o_{t}$ and
$a_{t-1}$ first and then generates $\Phi_{R}([o_{t};a_{t-1}])$. Second, we
modify the action scorer as action scorer in the LSTM-DRQN could only handle
commands with two words.
## Appendix B Notations and Algorithm
Following are the notations important to understand the algorithm:
* •
$o_{t},r_{t},e_{t}:$ observation (i.e. feedback), reward and admissibility
signal received at time $t$.
* •
$a_{t}:$ command executed in game-play at time $t$.
* •
$u_{t}:$ cumulative rewared/score at time $t$.
* •
$\Phi_{R}:$ representation generator.
* •
$\Phi_{C}:$ auxiliary classifier.
* •
$K$ : number of network heads in score contextualisation architecture.
* •
$\mathcal{J}$: dictionary mapping cumulative rewards to network heads.
* •
$H(k):$ LSTM corresponding to network head $k$.
* •
$\Phi_{A}(k):$ Action scorer corresponding to network head $k$.
* •
$h_{t}:$ agent’s context/history state at time $t$.
* •
$T$: maximum steps for an episode.
* •
$p_{i}:$ boolean that determines whether +ve reward was received in episode
$i$.
* •
$q_{i}:$ boolean that determines whether -ve reward was received in episode
$i$.
* •
$\tau_{p}:$ fraction of episodes where $\exists t<T:r_{t}>0$
* •
$\tau_{n}:$ fraction of episodes where $\exists t<T:r_{t}<0$
* •
$l:$ sequence length.
* •
$n:$ minimum history size for a state to be updated.
* •
$\mathcal{A}:$ action set.
* •
$\hat{\mathcal{A}}_{t}:$ admissible set generated at time $t$.
* •
$I_{target}:$ update interval for target network
* •
$\epsilon:$ parameter for $\epsilon-$greedy exploration strategy.
* •
$\epsilon_{1}:$ softness parameter i.e. $\epsilon_{1}$ fraction of times
$\hat{\mathcal{A}}_{t}=\mathcal{A}$.
* •
$c:$ threshold parameter for action elimination strategy Masking.
* •
$G_{\textrm{max}}:$ maximum steps till which training is performed.
Full training procedure is listed in Algorithm 1.
Algorithm 1 General training procedure
1:function
ACT$(o_{t},a_{t-1},u_{t},h_{t-1},\mathcal{J},\epsilon,\epsilon_{1},c,\theta)$
2: Get network head $k=\mathcal{J}(u_{t})$.
3: $h_{t}\leftarrow$ LSTM $H(k)[w_{t},h_{t-1}]$.
4:
${Q}(h_{t},:,u_{t};\theta)\leftarrow\Phi_{A}(k)(h_{t});\hat{\xi}(h_{t},a;\theta)\leftarrow\Phi_{C}(h_{t})$.
5: Generate $\hat{\mathcal{A}}_{t}$ (see Section 3.2).
6: With probability
$\epsilon,a_{t}\leftarrow\textrm{Uniform}(\hat{\mathcal{A}}_{t})$, else
$a_{t}\leftarrow\textrm{argmax}_{a\in\hat{\mathcal{A}}_{t}}{Q}(h_{t},a,u_{t};\theta)$
7: return $a_{t},h_{t}$
8:end function
9:function TARGETS$(f,\gamma,\theta^{-})$
10:
$(a_{0},o_{1},a_{1},r_{2},u_{2},e_{2},o_{2},\dots,o_{l},a_{l},r_{l+1},e_{l+1},u_{l+1})\leftarrow
f;h_{b,0}\leftarrow 0$
11: Pass transition sequence through $H$ to get
$h_{b,1},h_{b,2},\dots,h_{b,l}$
12: $E_{b,i}\leftarrow\xi(h_{b,i},a_{i};\theta^{-})$
13:
$y_{b,i}\leftarrow\textrm{max}_{a\in\mathcal{A}}{Q}(h_{b,i+1},a,u_{b,i+1};\theta^{-})$
14: $y_{b,i}\leftarrow E_{b,i}y_{b,i}+(1-E_{b,i})\
{Q}(h_{b,i+1},a_{i},u_{b,i+1};\theta^{-})$ if using CQLH.
15: $y_{b,i}\leftarrow r_{i+1}$ if $o_{i}$ is terminal else $y_{b,i}\leftarrow
r_{i+1}+\gamma y_{b,i}$
16: return $y_{b,:},E_{b,:}$
17:end function
18:
19:Input:
$G_{\textrm{max}},I_{\text{look}},I_{\text{update}},\gamma,\epsilon_{1},\epsilon,c,K,\mathbb{I}_{\left[{}_{\text{using}\Phi_{C}}\right]},n$
20:Initialize episodic replay memory $\mathcal{D}$, global step counter
$G\leftarrow 0$, dictionary $\mathcal{J}=\\{\\}$.
21:Initialize parameters $\theta$ of the network, target network parameter
$\theta^{-}\leftarrow\theta$.
22:while $G<G_{\textrm{max}}$ do
23: Initialize score $u_{1}=0$, hidden State of $H$, $h_{0}=0$ and get start
textual description $o_{1}$ and initial command $a_{0}=\textrm{'look'}$. Set
$p_{k}\leftarrow 0,q_{k}\leftarrow 0$.
24: for $t\leftarrow 1$ to $T$ do
25: $a_{t},h_{t}\leftarrow$
ACT$(o_{t},a_{t-1},u_{t},h_{t-1},\mathcal{J},\epsilon,\epsilon_{1},c,\theta)$
26: $a_{t}\leftarrow$ ’look’ if $t\bmod 20==0$
27: Execute action $a_{t}$, observe $\\{r_{t+1},o_{t+1},e_{t+1}\\}$.
28: $p_{k}\leftarrow 1$ if $r_{t}>0$; $q_{k}\leftarrow 1$ if $r_{t}<0$;
$u_{t+1}\leftarrow u_{t}+r_{t}$
29: Sample minibatch of transition sequences $f$
30: $y_{b,:},E_{b,:}\leftarrow$ TARGETS$(f,\gamma,\theta^{-})$
31: Perform gradient descent on
$\mathcal{L}(\theta)=\sum_{i=j+n-1}^{j+l}[y_{b,i}-{Q}(h_{b,i},a_{i},u_{b,i};\theta)^{2}+\mathbb{I}_{\left[{}_{using\Phi_{C}}\right]}\
\textrm{BCE}(e_{i},E_{b,i})]$
32: $\theta^{-}\leftarrow\theta$ if $t\bmod I_{update}==0$
33: $G\leftarrow G+1$
34: End episode if $o_{t+1}$ is terminal.
35: end for
36: Store episode in $\mathcal{D}$.
37:end while
## Appendix C More Empirical Analysis
### C.1 Prioritised Sampling & Infrequent look
Our algorithm uses prioritised sampling and executes a look action every
$I_{look}=20$ steps. The baseline agent LSTM-DRQN follows this algorithm. We
now ask,
> Does prioritised sampling and an infrequent look play a significant role in
> the baseline’s performance?
Figure 12: Learning curves for baseline ablation study.
For this experiment, we compare the Baseline to two agents. The first agent is
the Baseline without prioritised sampling and the second is the one without an
infrequent look. Accordingly, we call them “No-priority (NP)” and “No-look
(NL)” respectively. We use Zork as the testing domain.
From Fig 12, we observe that the Baseline performs better than the NP agent.
This is because prioritised sampling helps the baseline agent to choose the
episodes in which rewards are received in, thus assigning credit to the
relevant states faster and overall better learning. In the same figure, the
Baseline performs slightly better than the NL agent. We hypothesise that even
though look command is executed infrequently, it helps the agent in
exploration and do credit assignment better.
### C.2 CQLH
Our algorithm uses CQLH implementation as described in Section A.2. An
important case that CQLH considers is $s_{t+1}=s_{t}$. This manifests in
$(1-\hat{\xi})$ term in equation (5). We now ask whether ignoring the case
$s_{t+1}=s_{t}$ worsen the agent’s performance?
Figure 13: Learning curves for CQLH ablation study.
For this experiment, we compare CQLH agent with the agent which uses this
error for update:
$\displaystyle\delta^{\prime\prime}_{t}=$ $\displaystyle
r_{t}+\gamma\max_{a\in\mathcal{A}}Q(h_{t+1},a)\hat{\xi}_{t}-Q(h_{t},a_{t}).$
Accordingly, we call this new agent as “alternate CQLH (ACQLH)” agent. We use
Zork as testing domain. From Fig 13, we observe that although ACQLH has a
simpler update rule, its performance seems more unstable compared to the CQLH
agent.
Figure 14: Learning curves for Score contextualisation (SC : red), Score
contextualisation + Masking (SC + Masking : blue) and Baseline (grey) for all
the levels of the SaladWorld. For the simpler levels i.e. level 1 and 2, SC
and SC + Masking perform better than Baseline. With difficult level 3, only SC
+ Masking solves the game. For levels 4 and beyond, we posit that better
exploration strategies are required. Figure 15: Game map shows Level 1 and 2.
Simpler levels help us test the effectiveness of score contextualisation
architecture. Figure 16: Game map shows the _progression_ from Level 3 to
Level 7 in terms of number of rooms and objects. Besides every object in the
map, there is a tuple which shows in which levels the object is available. We
observe that with successive levels, the number of rooms and objects increase
making it more difficult to solve these levels.
|
# Turbulence in Milky Way Star-Forming Regions Traced by Young Stars and Gas
Trung Ha Department of Physics, University of North Texas, Denton, TX 76203,
USA<EMAIL_ADDRESS>Yuan Li Department of Physics, University of North
Texas, Denton, TX 76203, USA Marina Kounkel Physics & Astronomy Department,
Vanderbilt University, PMB 401807, 2301 Vanderbilt Place, Nashville, TN 37235
Siyao Xu NASA Hubble Fellow Institute for Advanced Study, 1 Einstein Drive,
Princeton, NJ 08540, USA Hui Li NASA Hubble Fellow Department of Astronomy,
Columbia University, 550 West 120th Street, New York, NY 10027, USA Yong
Zheng Department of Astronomy, University of California, Berkeley, CA 94720,
USA
###### Abstract
The interstellar medium (ISM) is turbulent on all scales and in all phases. In
this paper, we study turbulence with different tracers in four nearby star-
forming regions: Orion, Ophiuchus, Perseus, and Taurus. We combine the
APOGEE-2 and Gaia surveys to obtain the full 6-dimensional measurements of
positions and velocities of young stars in these regions. The velocity
structure functions (VSFs) of the stars show a universal scaling of
turbulence. We also obtain H$\alpha$ gas kinematics in these four regions from
the Wisconsin H-Alpha Mapper. The VSFs of the H$\alpha$ are more diverse
compared to the stars. In regions with recent supernova activities, they show
characteristics of local energy injections and higher amplitudes compared to
the VSFs of stars and of CO from the literature. Such difference in amplitude
of the VSFs can be explained by the different energy and momentum transport
from supernovae into different phases of the ISM, thus resulting in higher
levels of turbulence in the warm ionized phase traced by H$\alpha$. In regions
without recent supernova activities, the VSFs of young stars, H$\alpha$, and
CO are generally consistent, indicating well-coupled turbulence between
different phases. Within individual regions, the brighter parts of the
H$\alpha$ gas tend to have a higher level of turbulence than the low-emission
parts. Our findings support a complex picture of the Milky Way ISM, where
turbulence can be driven at different scales and inject energy unevenly into
different phases.
††facilities: Gaia, Sloan, WHAM††software: astropy (Astropy Collaboration et
al., 2013, 2018), matplotlib (Hunter, 2007), numpy (van der Walt et al., 2011;
Harris et al., 2020), pandas (pandas development team, 2020; Wes McKinney,
2010), scipy (Virtanen et al., 2020), Sagitta (McBride et al., 2021)
## 1 Introduction
The interstellar medium (ISM) is multi-phase and turbulent. Interstellar
turbulence affects the velocity statistics on all scales from the diffuse
ionized gas to the dense molecular gas. Larson (1981) compiled the internal
velocity dispersion of nearby molecular clouds from the width of various
emission lines, and found a power-law relation between the velocity dispersion
and cloud size, with slope 0.38, between the 1/3 slope expected for turbulence
dominated by solenoidal motions (Kolmogorov, 1941) and the 1/2 slope expected
for shock-dominated turbulence (e.g., Galtier & Banerjee, 2011; Federrath,
2013).
Inside molecular clouds, massive O- and B-type stars create H II regions by
ionizing and heating the ISM with UV radiation (see, e.g., Spitzer, 1978).
Compression from supernova (SN) explosions can trigger further star formation
(Kennicutt & Evans, 2012). Interstellar processes such as SNe, stellar winds,
ionizing radiation, and cosmic rays all play a role in maintaining the
structure of the ISM, as well as in regulating star formation (Hennebelle &
Falgarone, 2012; Rathjen et al., 2021; Hu et al., 2022; Xu & Lazarian, 2022).
Of these mechanisms, SN explosions are perhaps the most significant source of
energy injection into the local ISM (Mac Low & Klessen, 2004; Padoan et al.,
2016).
Many numerical simulations have been carried out to study how SN explosions
affect the surrounding ISM and star formation in the last decade. In general,
they find that most of the kinetic energy from an SN explosion is transferred
to the low-density region of the ISM in both single- (see, e.g., Kim &
Ostriker, 2015; Martizzi et al., 2015; Li et al., 2015) and multiple-SN events
(Padoan et al., 2016), suggesting that turbulence may not be evenly driven in
different phases of the ISM.
Observational studies of ISM turbulence have been done with various gas
tracers. Measurements of turbulence through density spectrum were carried out
using the Wisconsin H-Alpha Mapper (WHAM; Haffner et al., 2003, 2010), which
revealed a turbulent cascade consistent with Kolmogorov theory (Chepurnov &
Lazarian, 2010). Burkhart et al. (2010) found that most of the H I in the
Small Magellanic Cloud (SMC) is subsonic or transonic, and concluded that
turbulence in the SMC is dominated by hydrodynamical pressure over magnetic
pressure.
Statistical measurements of velocity fluctuations in the neutral ISM can be
done using the Velocity Channel Analysis, the Velocity Coordinate Spectrum
(Lazarian & Pogosyan, 2000, 2006), the Delta-variance technique (Stutzki et
al., 1998; Ossenkopf & Mac Low, 2002), or the Principal Component Analysis
(Heyer & Brunt, 2004; Roman-Duval et al., 2011) based on spectroscopic data.
These analyses generally reveal power-law spectra of turbulent velocities
(Lazarian, 2009; Chepurnov et al., 2010).
Additionally, kinematics of Galactic H II regions have been studied using the
observations of ionized gas. Fabry-Perot interferometry was used to probe the
H$\alpha$ line in emission nebulae. Louise & Monnet (1970) found a velocity-
size relation that agrees with the 1/3 slope of Kolmogorov turbulence in M8,
while Roy & Joncas (1985) and Joncas & Roy (1986) found slopes of the
structure functions closer to 1/2, indicating turbulence in the supersonic
regime in both S142 and M17, respectively.
Closer to our solar neighborhood, studies into the nature of turbulence in the
Orion Nebula is an ongoing effort since the 1950s. von Hoerner (1951) observed
a root-mean-square difference in emission line velocities that increases with
larger angular separations. He reasoned that such relation could be explained
by the Kolmogorov theory if the effective depth of the nebula was sufficiently
small as to neglect projection effects. Subsequent studies, carried out by
Münch (1958) and Castaneda (1988), disputes this conclusion. Münch (1958)
found an optical depth smaller than the value postulated by von Hoerner
(1951), and attributed the failure of Kolmogorov theory to the compressibility
of turbulence in the Orion Nebula, while Castaneda (1988) argued against a
simple power-law relation, instead suggested energy injections at different
scales.
ISM turbulence has also been probed using “point sources” such as molecular
cores. Qian et al. (2012) developed statistical measurements of core-to-core
velocity dispersion over a range of length scales , i.e., the core velocity
dispersion technique. Qian et al. (2018) find signatures for a turbulent
power-law spectrum in the Taurus Molecular Cloud, and demonstrate that the
statistics of turbulent velocities are imprinted in the velocities of dense
cores (Xu, 2020).
One limitation of using gas tracers is they lack three-dimensional information
on the position and the velocity information is also limited to only the line-
of-sight. In Ha et al. (2021, hereafter Paper I), we proposed a new method to
measure turbulence in the ISM with young stars. Since stars are formed out of
turbulent molecular clouds, we expected these young stars to inherit the
turbulent kinematics from their natal clouds. Using six-dimensional position
and velocity information of $>1400$ stars in the Orion Complex, we found that
the first-order velocity structure function (VSF) of young stars in loose
groups exhibit characteristics of turbulence. Our method can therefore be used
to probe the turbulent properties of molecular clouds in addition to analysis
of gas kinematics and has been used in recent work in Krolikowski et al.
(2021); Zhou et al. (2022).
In this work, we apply a similar analysis to a larger sample of molecular
complexes, which include three of the nearest massive star-forming regions:
Ophiuchus, Perseus, and Taurus. Additionally, we analyze the kinematics of
H$\alpha$ gas from the WHAM survey to study the connection between stars and
the multi-phase ISM.
In Section 2, we describe the data used in this analysis, the reduction
pipeline for the H$\alpha$ kinematic survey, and how we compute the VSFs using
stars and H$\alpha$. In Section 3, we present the VSF of young stars and
multi-phase gas of all four regions. In Section 4, we discuss sources of
uncertainty associated with our analysis and the nature of turbulence in the
Milky Way ISM. We summarize out results in Section 5.
Figure 1: Top panels: positions of young stars color-coded by their distances
in Orion, Ophiuchus, Taurus-Perseus (from left to right). Background shows
H$\alpha$ flux from the WHAM survey (lighter is brighter). The orange and
green boxes denote the areas where H$\alpha$ is analyzed for Taurus and
Perseus, respectively. Bottom panels show the line-of-sight velocities of the
young stars (in the LSR frame) in these regions. The background shows the
H$\alpha$ line-of-sight centroid velocities. Table 1: Properties of Star-
Forming Regions
Region | Number | Number of | Average | GLON | GLAT | Distance | $v_{\rm LSR}$ range | $v_{\rm LSR,~{}err}$ cut
---|---|---|---|---|---|---|---|---
| of stars | stars used | distance (pc) | (°) | (°) | range (pc) | ($\mathrm{km~{}s}^{-1}$) | ($\mathrm{km~{}s}^{-1}$)
Orion | 2647 | 2468 | 410 | $190,217$ | $-28,-5$ | $300,500$ | $-10,40$ | $10$
Ophiuchus | 113 | 107 | 140 | $342,360$ | $12,28$ | $120,170$ | $-30,30$ | $2$
Perseus | 159 | 138 | 320 | $156,164$ | $-23,-15$ | $250,400$ | $-6,30$ | $10$
Taurus | 157 | 139 | 140 | $165,177$ | $-23,-11$ | $97,200$ | $-5,25$ | $5$
Note. — Number of stars are all the stars with 6D information. Number of stars
used are the stars in our analysis after excluding outliers. GLON (Galactic
Longitude) and GLAT (Galactic Latitude) ranges denote the borders of windows
used to compute the H$\alpha$ VSF. The distance ranges, $v_{\rm LSR}$ ranges,
and $v_{\rm LSR,~{}err}$ cuts are applied to the stars to exclude outliers.
## 2 Observations, Data reduction, and Methods
### 2.1 Observations and Data Analysis
#### 2.1.1 Stellar observations
We obtain the locations, parallax distances, and proper motions of stars from
the Gaia mission (Gaia Collaboration et al., 2016) in four nearby star-forming
regions: the Orion Molecular Cloud complex (Orion), the $\rho$ Oph dark cloud
(Ophiuchus), the Perseus molecular cloud traced by two clusters, NGC 1333 and
IC 348 (Perseus), and the Taurus molecular clouds (Taurus). The Gaia catalog
includes five-parameter astrometric measurements (plane-of-sky coordinates,
proper motions, and parallax distances) of over 1.46 billion stars in the
Milky Way. Only a very limited subset ($\sim 7$ million) has line-of-sight
velocities measured by Gaia. The five-dimensional astrometric data for our
four regions are obtained from Gaia’s third early data release (EDR3; Gaia
Collaboration et al., 2021).
The line-of-sight velocities of the stars in our study were observed in the
near infrared with the Apache Point Observatory Galactic Evolution Experiment
(APOGEE) spectrograph, mounted on the 2.5 m Sloan Foundation Telescope of the
Sloan Digital Sky Survey. APOGEE-2 Data Release 17 (DR17) collected high
resolution spectral image of over 657,000 stars in the Milky Way (Gunn et al.,
2006; Blanton et al., 2017; Abdurro’uf et al., 2022). Combining these two
surveys provides six-dimensional (6D) information (3D positions and 3D
velocities) of stars for the four molecular clouds.
The stars in our study were assigned to each of the four region by Kounkel &
Covey (2019). They performed a clustering analysis on Gaia DR2 sources in the
Milky Way using Python implementation of HDBSCAN (Hierarchical Density-Based
Spatial Clustering of Applications with Noise; McInnes et al., 2017). We
crossmatch this catalog with Gaia EDR3 and APOGEE-2 to obtain 6D astrometries
of the stars.
Then, we exclude a small number of stars whose parallax distances $d$, line-
of-sight velocities $v_{\rm LSR}$, and line-of-sight velocity errors $v_{\rm
LSR,~{}err}$ are more than 4$\sigma$ deviated from the mean of the
distribution. In Taurus, one star with very high reported proper motion
($v_{\rm LSR,~{}RA}>100\mathrm{~{}mas~{}yr}^{-1}$) is excluded. The ranges of
$d$ and $v_{\rm LSR}$, and the threshold of $v_{\rm LSR,~{}err}$ that we use,
as well as the total number of stars before and after the cut are specified in
Table 1. We note, however, that the exact cut of these ranges do not affect
the conclusion of this work.
#### 2.1.2 H$\alpha$ observations
The H$\alpha$ data in our study comes from the WHAM survey (Haffner et al.,
2003, 2010). We obtained the data cubes from the “DR1-v161116-170912” release,
which comprises more than 49000 spectra divided into an irregular grid with a
1° seeing resolution and a 0.25° sampling resolution. The intensity of the
H$\alpha$ emission is recorded in a 3-dimensional array encoding its Galactic
longitude, Galactic latitude, and line-of-sight velocity ($-170$ km
s${}^{-1}<v_{\rm LSR}<160$ km s-1). We fit a Gaussian distribution to the flux
profile along individual lines of sight. Then we take the velocity at the peak
of the distribution as the centroid velocity and the uncertainty of the fit as
the uncertainty. We also correct for Galactic rotation following the procedure
in Qu et al. (2020).
Figure 1 shows the H$\alpha$ map of the four regions in our analysis, with
stars overplotted as foreground dots. The top panels show the integrated line-
of-sight fluxes in log scale. In the foreground, each star is color-coded by
their distance from Earth. We note that although Perseus and Taurus are very
close to each other on the sky, they are physically separated by $\sim 150$ pc
along the line of sight. In the bottom panels, we plot the centroid velocity
of the H$\alpha$ in the background, and the line-of-sight velocities of stars
in the foreground.
### 2.2 Data analysis
We compute the first-order VSF of stars in each region. The VSF is related to
the kinetic energy power spectrum of a velocity field, and is expressed as a
two-point correlation function between the absolute velocity differences
$|\delta v|$ versus the physical separation $\ell$. For a turbulent velocity
field, we expect the VSF to have a characteristic power-law scaling with slope
1/3 for incompressible turbulence dominated by solenoidal motions (Kolmogorov,
1941), or a slope 1/2 for supersonic turbulence dominated by compressive
motions. For the Milky Way molecular clouds, Larson (1981) found a
relationship between cloud size and velocity dispersion to be
$\sigma=1.1~{}R^{0.38}$.
The 3D VSF of stars is computed as follows. First, the positions, distances,
radial velocities, and proper motions of stars are translated into
galactocentric Cartesian coordinates and velocities. Then, for each pair of
stars, the absolute velocity vector difference $|\delta\vec{v}|$ and the
distance $d$ between them is calculated. For individual clouds, these
distances are sorted into logarithmic bins of $\ell$, and $|\delta\vec{v}|$
values within a bin are averaged out to obtain
$\langle|\delta\vec{v}|\rangle$. To estimate uncertainties, we first generate
1000 random samples of stellar parameters following a Gaussian distribution
around the reported measurements and with a 1$\sigma$ scatter equal to the
reported uncertainties. Then at each $\ell$, we take the 1$\sigma$ value of
the 1000 $\langle|\delta\vec{v}|\rangle$ to be the uncertainty.
For the WHAM H$\alpha$ data, we first define a rectangular box containing the
stars in each of the four regions as specified in Table 1 and shown in Figure
1. Our results are not sensitive to the exact size of the box (see Appendix C
for detailed discussions). We compute the first-order VSF of each region using
the projected position-position-velocity information. The resultant VSF is
$\langle|\delta v|\rangle_{\rm proj}$ versus angular separations $\theta$.
Then, we assume a constant distance equal to the mean distance of stars in
each region (see Table 1) and translate $\theta$ (°) to $\ell_{\rm proj}$
(pc). Since three degrees of freedom are missing in this projected VSF
calculation (the proper motions of gas and its exact distance), we scale
$\langle|\delta v|\rangle_{\rm proj}$ and $\ell_{\rm proj}$ by a constant
factor, assuming an isotropic turbulent velocity field.
$\langle|\delta\vec{v}|\rangle=\sqrt{3}~{}\langle|\delta v|\rangle_{\rm
proj},$ (1) $\ell=\sqrt{\frac{3}{2}}~{}\ell_{\rm proj}.$ (2)
## 3 Results
Figure 2: The first-order VSFs of the stars in Orion (light blue), Ophiuchus
(magenta), Perseus (green), and Taurus (orange). For reference, we plot a
solid blue line for Larson’s relation, a black dotted-dashed line for
Kolmogorov turbulence, and a pink dashed line for supersonic turbulence. The
vertical lines represent the uncertainties in $\langle|\delta\vec{v}|\rangle$
at each $\ell$.
### 3.1 The Universality of ISM Turbulence Traced by Young Stars
We first examine the velocity statistics of the four star-forming regions:
Orion, Ophiuchus, Perseus, and Taurus. Figure 2 shows the VSFs traced by stars
of these four regions. For reference, we also plot the Larson’s law of
velocity dispersion vs. cloud size, the 1/3 slope of Kolmogorov turbulence,
and the 1/2 slope of supersonic turbulence. Immediately, we recognize a
remarkable universality among the VSFs of different regions. Over a large
dynamical range, the magnitude and slopes of the VSFs show a general agreement
with Larson’s law (see Table 2 for the best-fit slopes of the VSFs). This
suggests that the motion of young stars in the Milky Way show characteristics
of turbulence, and supports our previous findings in Paper I that turbulence
in the ISM can be traced with young stars.
At $\ell\lesssim 10$ pc, the slopes of the VSF of all four regions are flatter
than Larson’s law, as well as Kolmogorov (slope 1/3) turbulence. We discuss
the possible cause of this flattening in Section 4.1.
### 3.2 The VSFs of Individual Regions
Figure 3: The first-order 3D VSF of stars (red) and H$\alpha$ (olive) for
Orion, Ophiuchus, Perseus, and Taurus. The shaded region around the H$\alpha$
VSF is its associated uncertainty. At small $\ell$, the dots showing the VSF
traced by H$\alpha$ are not connected to indicate the seeing limit of the WHAM
survey ($\theta<1$°). The slopes of the VSFs of stars and of H$\alpha$ are
specified in Table 2. The VSF of H$\alpha$ in Ophiuchus can also be fitted
with a broken power-law, with slope $0.58\pm 0.24$ for $3~{}\mathrm{pc}<l<9$
pc and slope $0.29\pm 0.08$ for $9~{}\mathrm{pc}<l<30$ pc. For comparison, we
also include CO (violet) analysis from Qian et al. (2015, 2018) for three of
these regions.
Figure 3 shows the VSFs of young stars as well as gas in each of the four
regions: Orion, Ophiuchus, Perseus, and Taurus. The best-fit slopes of these
VSFs are lists in Table 2.
#### 3.2.1 Orion
The first-order VSF of Orion traced by stars has been studied in detail in
Paper I. We only reiterate some of the main points here and address the
differences caused by the addition of $\sim 1000$ stars compared with Paper I.
Throughout the inertial range, the magnitude of
$\langle|\delta\vec{v}|\rangle$ is higher than Larson’s law. At small scales
($1~{}\mathrm{pc}\lesssim\ell\lesssim 20$ pc), the number statistics is
dominated by stars in the Orion Nebula Cluster (ONC), which flattens the VSF
(see Section 4.1 for more discussion on the small-scale flattening). At ranges
from $20$ pc to $\sim 70$ pc, Orion D, which is a loose stellar association at
the front of the Orion Complex along our line of sight, dominates the
statistics, and it exhibits a power-law scaling, which indicates an imprint of
turbulence from their natal molecular clouds. The VSF of Orion without
considering the ONC would be steeper at small scales, which renders it more
in-line with the VSF of other regions (see Figure 2). At $\sim 70$ pc, there
is a bump in the structure function, and we proposed that this is an evidence
supporting the existence of a SN explosion at the heart of Orion 6 Myr ago
(Kounkel, 2020). Paper I used Gaia DR2 only. The newly added stars in this
study from Gaia EDR3 have larger overall uncertainty in their parametric
solutions. Therefore, the noise floor for Orion at small scales is boosted
from $\sim 3$ km s-1 to $\sim 5$ km s-1. This slightly elevates the overall
VSF compared to Paper I.
H$\alpha$ emission in Orion mainly comes from the Barnard’s loop and the
$\lambda$ Ori bubble (see Figure 1), which are thought to be created by SN
explosions (Kounkel, 2020). We see the effect of such energy injection in the
VSF traced by H$\alpha$. At $\ell\gtrsim 20$ pc, the amplitude of the
H$\alpha$ VSF is higher than the VSF of young stars. Over the whole dynamical
range we probe here, the H$\alpha$ VSF has a roughly constant slope of 1/2.
The VSF traced by H$\alpha$ shows a bump at $\ell\sim 150$ pc. This is
indicative of a local energy injection, similar to those seen in the VSF
traced by young stars. However, we caution that this scale is very close to
the size of the H$\alpha$ box, and the bump can also be an artifact (see,
e.g., Mohapatra et al., 2022). We discuss this effect in more details in
Appendix C. At very small scales ($\theta=1^{\rm o}$, corresponding to
$\ell\sim 7$ pc), our analysis is limited by the resolution of the WHAM survey
(Haffner et al., 2003). We discuss the seeing effect and other uncertainties
related to H$\alpha$ VSF in Section 4.2.
#### 3.2.2 Ophiuchus
The young stars in Ophiuchus has a VSF similar to that of the Orion Complex.
At small $\ell$, the structure function is flatter than 1/3, and then steepens
to $\sim$ 1/3 beyond 10 pc up to $\sim 25$ pc. The H$\alpha$ VSF in Ophiuchus
has slope close to 1/3 and a higher amplitude than the VSF of the stars. Both
VSFs show clear bumps at $\sim 25$ pc, which we interpret as evidence of local
energy injection.
In Orion, such an energy injection is associated with a past SN explosion as
we discuss in detail in Paper I. If the energy injection in Ophiuchus is also
caused by a SN, we estimate that the event took place around
$25\,\mathrm{pc}/v_{\rm region}\sim 2.5$ Myr ago, where we use the
characteristic H$\alpha$ velocity $v_{\rm region}\sim 10$ km s-1. Such an
estimate is broadly consistent with observations of runaway stars which trace
back to a SN explosion in the region $\sim 2$ Myr ago (Neuhäuser et al.,
2020).
We note that the scale of the bump is close to the size of the region
containing the young stars in our analysis. Therefore, it is technically
possible that the bump in the VSF of young stars is only an artifact caused by
the limited sample size. In Appendix A, we analyze the entire Upper Sco
region. We find that as long as we apply our analysis to young stars ($<2$
Myr), the results are consistent, while the VSFs of older stars do not show
this feature. In addition, the SN scenario is also supported by the bump in
the H$\alpha$ VSF.
The cold and dense molecular cores, where stars are born out of, emits CO.
Qian et al. (2015, 2018) applied the core-velocity-dispersion (CVD; Qian et
al., 2012) method to analyze turbulence in star-forming regions. We translate
their best-fit CVD to a VSF of CO by
$\mathrm{VSF}=\sqrt{\frac{2}{\pi}}~{}\mathrm{CVD}$.
In Ophiuchus, the amplitude of CO VSF (pink solid lines in Figure 3) is much
lower than the VSF of young stars and H$\alpha$. It is possible that the
recent energy injection has affected the warm phase of the ISM more than the
cold phase (see discussion in Section 4.3). It is also possible that the low
amplitude is due to spacial variation of ISM turbulence (see discussion in
Appendix C). Qian et al. (2015) only examined the smaller scale velocity
structures ($\ell\lesssim 10$ pc), where there is no reliable component that
we can compare against (due to relaxation of stars and resolution limit of
H$\alpha$).
#### 3.2.3 Perseus
At length scales below 10 pc, the VSF of young stars in Perseus is very
similar to that of Ophiuchus (see Figure 2). Since we include two clusters in
our analysis of Perseus: NGC 1333 and IC 348, the VSF is flattened from
internal dynamical relaxation. The existence of subgroups and sub-clusters are
likely the cause of the small wiggles in the VSF. At larger scales, the VSF is
generally consistent with Larson’s law.
The VSF traced by H$\alpha$ here has an amplitude similar to the VSF traced by
young stars but a steeper slope ($\sim$ 1/2). VSF traced by CO matches the
H$\alpha$ VSF at $\ell>6$ pc, but flattens on smaller scales due to projection
effects for thick clouds (Qian et al., 2015; Xu, 2020).
#### 3.2.4 Taurus
Taurus is overall very similar to Perseus. The VSFs of young stars, H$\alpha$
and CO generally agree with each other, as well as Larson’s Law.
There is no prominent bump detected in the VSF in both Perseus and Taurus,
which is consistent with the findings of Krolikowski et al. (2021) that there
has been no SN activities in the last 5 Myr in these regions. However, there
are wiggles scattered throughout the length scales, likely because Taurus
contains several small dense groups that are primarily homogeneously
distributed throughout the region.
Table 2: Slopes of the VSFs
Region | Slope of | Fitting | Slope of | Fitting
---|---|---|---|---
| stars VSF | range (pc) | H$\alpha$ VSF | range (pc)
Orion | $0.25\pm 0.01$ | $12-75$ | $0.55\pm 0.02$ | $9-150$
Ophiuchus | $0.34\pm 0.03$ | $7-28$ | $0.36\pm 0.03$ | $3-30$
Perseus | $0.22\pm 0.03$ | $5-80$ | $0.61\pm 0.08$ | $7-55$
Taurus | $0.34\pm 0.03$ | $7-70$ | $0.42\pm 0.12$ | $3-25$
Note. — Inside the fitting ranges, the VSFs are fitted with an equation:
$\langle|\delta\vec{v}|\rangle$ $=A\cdot\ell^{B}$, where $B$ is the slope. The
uncertainties reported with each slope is obtained from the covariance matrix,
taking into account the uncertainties in $\langle|\delta\vec{v}|\rangle$ shown
in Figure 3. We choose the fitting ranges such that the uncertainties are low
and there is a consistent trend in the slope of the VSF.
## 4 Discussion
### 4.1 Uncertainties and biases in turbulence traced by young stars
We discussed various biases and uncertainties associated with measuring
turbulence from stars in Paper I, including dynamical relaxation, binary
contamination, and drifting. In this section, we reiterate the main points and
discuss additional mechanisms that can affect the shape of the VSF.
Stellar interactions can cause dynamical relaxation, which erases the memory
of turbulence and flattens the VSF. Similar to Orion’s loose groups discussed
in Paper I, the relaxation times of Ophiuchus, Perseus, and Taurus are
estimated to be tens of Myrs, much longer than their age (for a detailed
discussion of the age of each region, see Wilking et al., 2008; Krolikowski et
al., 2021; Pavlidou et al., 2021). Thus dynamical relaxation has not affected
the overall shape of the VSF for the three regions. However, there are some
small, compact star clusters in these regions, which can contribute to the
flattening at small scales ($\ell\lesssim 10$ pc, see Figure 2).
Another source of uncertainty originates from the drifting of stars along
their natural trajectories over time. Stars born with high velocity
differences drift apart at faster rates than stars born with low velocity
differences, which steepens the VSF roughly over the the crossing time
($t_{\rm cross}$) of the cloud. Similar to the star groups in Orion that we
discussed in Paper I, $t_{cross}$ of Ophiuchus, Perseus, and Taurus are all
larger than their current age, and thus this effect is unlikely to have
significantly altered the shape of the structure functions.
Low-velocity binary systems and contamination from field stars can act like
noise in our analysis, which also tend to flatten the VSF as is discussed in
Paper I.
Additionally, other stellar feedback processes such as winds, jets, and
radiation can also inject energy on small scales (e.g., Gritschneder et al.,
2009; Offner & Arce, 2015; Li et al., 2019; Rosen et al., 2021; Grudić et al.,
2022). This can contribute to the flattening of the VSFs at small scales,
provided that these stars are massive ($>8~{}M_{\odot}$) and are within $\sim
1$ pc of each other (Rosen et al., 2021).
Stutz & Gould (2016) quantified an ejection mechanism through which protostars
decouple from the molecular gas. They found that the undulation of star-
forming filaments in Orion A “slingshots” protostars out of the filament at
$\sim 2.5$ km s-1, which is also at the level where the VSF flattens in
Ophiuchus, Perseus, and Taurus.
All sources of uncertainties listed above can contribute to the flattening of
the VSF at small scales ($\ell\lesssim 10$ pc). Therefore, we only consider
young stars to be reliable tracers of ISM turbulence above $\sim 10$ pc.
### 4.2 ISM Turbulence traced by H$\alpha$
In Figure 2 and Figure 3, we find that while the VSFs of stars are broadly
consistent across the regions, the VSFs of H$\alpha$ are more diverse.
In regions with recent supernova explosions (Orion and Ophiuchus), the VSF of
H$\alpha$ gas has a higher amplitude than that of stars. Whereas in Perseus
and Taurus, the H$\alpha$ flux is lower (Figure 1) and the H$\alpha$ VSF is
more in line with the VSFs of stars and CO.
Figure 4: VSF of four regions traced by H$\alpha$. Solid lines denote the VSF
computed for pixels with low emission, and dotted-dashed lines denote the VSF
computed for pixels with high emission.
Within each individual region, we divide the pixels in half based on the
H$\alpha$ brightness– the high-emission half with all the pixels brighter than
the median and the low-emission half with all the pixels dimmer than the
median. Figure 4 shows the VSFs of high-emission H$\alpha$ (dotted-dashed
line) and low-emission H$\alpha$ (solid line) within each region. In general,
the VSF of high-emission H$\alpha$ has a higher amplitude compared to that of
the low-emission H$\alpha$.
One possible explanation for this is that bright H$\alpha$ comes from photo-
ionization by young massive stars (Tacchella et al., 2022). Therefore, it
traces the most intense stellar feedback regions and shows a higher VSF
amplitude. Whereas the low-emission H$\alpha$ comes from primarily collisional
ionization. Note that Orion has its high- and low-emission H$\alpha$ VSFs
essentially overlapping for the most part. This is likely because star
formation in Orion is the most intense among all the regions studied here (see
Table 2 of Lada et al., 2010), and thus most of the H$\alpha$ is photo-ionized
by massive stars.
There are many uncertainties and potential biases associated with the
H$\alpha$ analysis. One uncertainty comes from atmospheric seeing. Although
the WHAM data cubes provide us with H$\alpha$ line-of-sight measurements every
0.25°, the seeing limit of the survey is reported as 1°. At separations
$\theta\leq 1$° (or $\ell_{\rm proj}\leq d\cdot 1$°), we treat the VSF traced
by H$\alpha$ as unresolved (represented by the unconnected dots in Figure 3).
We note that seeing might not significantly alter the overall VSF, as
suggested by Li et al. (2020). Future studies with mock simulated H$\alpha$
observations may help properly evaluate the effects of seeing.
There are also uncertainties with the distance to the H$\alpha$ gas. The
bright H$\alpha$ emissions in Orion and Ophiuchus are likely associated with
the star forming regions. However, since Perseus and Taurus do not emit
intense H$\alpha$ (see top right panel of Figure 1), it is uncertain whether
the centroid velocities found in these two regions are at the exact same
distance as the stars.
Another bias can come from the projection effects. When analysing the VSF of
H$\alpha$, we assume that all the gas is located at the same distance,
effectively treating it as a 2D structure. This can flatten the VSF for thick
clouds, as is discussed in detail in Qian et al. (2015); Xu (2020); Mohapatra
et al. (2022). All the H$\alpha$ VSFs in our analysis is either as steep as
1/3 or steeper. This suggests that either most of the H$\alpha$ comes from a
thin sheet of gas, or the true VSF has a even steeper slope.
Another potential bias related to projection is that we are trying to infer 3D
turbulence using projected 2D information. This can introduce uncertainties as
the ISM is magnetized. In the presence of magnetic fields, turbulence is no
longer isotropic (Seifried & Walch, 2015; Hu et al., 2021).
We find that our results are not sensitive to the rotation model (see detailed
discussions in Appendix B).
### 4.3 The Complex Nature of ISM Turbulence
The ISM is multi-phase in nature. The kinematic coupling between these phases
is not well understood (for a review of the multi-phase ISM model, see Cox,
2005). More specifically, it is suggested that each phase possesses varying
levels of turbulence, where ionized gas has been observed to have higher
velocity dispersion than molecular gas in both high- and low-redshift galaxies
(Girard et al., 2021; Ejdetjärn et al., 2021). In our study, we observe a
similar trend comparing the VSF of H$\alpha$ and other tracers of turbulence.
In Orion and Ophiuchus, there is a large deviation between the level of
turbulence seen by different tracers. The VSFs traced by H$\alpha$ show the
largest amplitude at large scales, followed by the VSFs traced by stars, and
finally by CO. One possible explanation is the difference in the transport of
energy and momentum from SNe into different phases in these regions, as
mentioned in Section 3.2. In an inhomogeneous medium such as the Milky Way
ISM, SNe carry shocked matter radially outward following the path of least
resistance. As a consequence, the cool dense phase traced by CO receives less
energy injection than the warm diffuse phase traced by H$\alpha$, resulting in
lower levels of turbulence in the corresponding VSF. Such behavior is also
seen in hydrodynamical simulations of single SN remnants (e.g., Martizzi et
al., 2015; Li et al., 2015).
In Perseus and Taurus, the VSFs traced by H$\alpha$ and CO show a good
agreement in both the amplitudes and slopes in these regions, except for the
range $\ell\lesssim 7$ pc in Perseus, which suffers from projection effect due
to the thickness of the cloud (see Section 4.2 of Qian et al., 2015, for a
detailed discussion). At larger $\ell$, the VSF traced by H$\alpha$ and stars
also closely follow each other. Such trend is consistent with the evolutionary
history of these regions with no clear imprint of recent SN explosions.
Without this important local driver of turbulence, different phases of gas are
well-coupled, and the young stars formed out of these molecular clouds also
largely retain the same memory of turbulence.
The difference of H$\alpha$ VSFs in different regions show that ISM turbulence
can be unevenly distributed spatially as a result of local drivers. Turbulence
can also be unevenly distributed between different phases of the ISM (e.g.,
Orion and Ophiuchus). Section 4.2 and Figure 4 further demonstrate that even
with the same gas tracer H$\alpha$, the level of turbulence can vary depending
on its brightness. In addition, we note that local energy injection from
supernova explosions is also intermittent temporally. Using the H$\alpha$ VSF
in Ophiuchus, we estimate the eddy turnover time at the driving scale to be:
$\frac{\ell}{v}\sim\frac{20-30\,\rm{pc}}{5\,\rm{km/s}}\sim 4-6\,\rm{Myr}.$ (3)
The H$\alpha$ VSF in Orion gives a similar estimate. Typically, it takes a few
eddy turnover times to establish a steady-state turbulent flow. On the other
hand, given the size of these regions, the average time interval between
supernovae is on the order of a few Myr (Narayan & Ostriker, 1990; Li et al.,
2015). Each event also injects energy as a pulse with a very short duration.
Therefore, both the “on” and “off” states of the driver are short compared
with the timescale required to achieve a steady state. Thus ISM turbulence in
molecular complexes is often not in a steady state.
## 5 Conclusions
In this work, we analyze the kinematics of young stars and gas in four nearby
star-forming regions: Orion, Ophiuchus, Perseus, and Taurus. For each region,
we compute the first-order VSF of young stars and H$\alpha$ gas, and compare
the results against each other, as well as analysis of CO molecular cores from
the literature.
We find that the VSFs traced by young stars in all four regions exhibit
similar properties largely in agreement with Larson’s law and reveal a
universal scaling of turbulence in the ISM. This also confirms our previous
finding in Ha et al. (2021) that young stars retain the memory of turbulence
in their natal molecular clouds, and thus can be used to measure the turbulent
kinematics of molecular clouds.
The VSFs of H$\alpha$ are more diverse, likely as a result of local energy
injection. In regions with recent SN activities such as Orion and Ophiuchus,
the VSFs of H$\alpha$ are higher in amplitude compared to those without. The
H$\alpha$ VSFs are also higher than the VSFs of stars and CO in the these
regions, while regions without recent SN explosions (Perseus and Taurus) show
well-coupled turbulence between different phases. Within each individual
region, the high-emission H$\alpha$ tends to show higher VSFs than the low-
emission H$\alpha$.
Our findings support a complex picture of the Milky Way ISM, where turbulence
can be driven at difference scales and inject energy unevenly into different
phases. Future observations with more tracers of turbulence and on more star
forming regions will help us better understand the diversity and complexity of
ISM turbulence. In addition, high-resolution numerical studies can help
further our understanding of the coupling between different phases and
relevant timescales.
## Acknowledgments
We thank Shmuel Bialy, John Forbes, Di Li, Mordecai-Mark Mac Low, Lei Qian,
Amiel Sternberg, and Yuan-Sen Ting for the helpful discussions. We thank
Zhijie Qu for sharing the Galactic rotation model. H.L. and S.X. are supported
by NASA through the NASA Hubble Fellowship grant HST-HF2-51438.001-A and HST-
HF2-51473.001-A, respectively, awarded by the Space Telescope Science
Institute, which is operated by the Association of Universities for Research
in Astronomy, Incorporated, under NASA contract NAS5-26555. Y.L. acknowledges
financial support from NSF grant AST-2107735 and the College of Science and
College of Engineering at UNT. Y.Z. is supported by NASA through a grant (HST-
AR-16640.001-A) from the Space Telescope Science Institute, which is operated
by the Association of Universities for Research in Astronomy, Incorporated,
under NASA contract NAS5-26555. The Wisconsin H$\alpha$ Mapper and its
H$\alpha$ Sky Survey have been funded primarily by the National Science
Foundation. The facility was designed and built with the help of the
University of Wisconsin Graduate School, Physical Sciences Lab, and Space
Astronomy Lab. NOAO staff at Kitt Peak and Cerro Tololo provided on-site
support for its remote operation.
## References
* Abdurro’uf et al. (2022) Abdurro’uf, Accetta, K., Aerts, C., et al. 2022, ApJS, 259, 35, doi: 10.3847/1538-4365/ac4414
* Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
* Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
* Blanton et al. (2017) Blanton, M. R., Bershady, M. A., Abolfathi, B., et al. 2017, AJ, 154, 28, doi: 10.3847/1538-3881/aa7567
* Burkhart et al. (2010) Burkhart, B., Stanimirović, S., Lazarian, A., & Kowal, G. 2010, ApJ, 708, 1204, doi: 10.1088/0004-637X/708/2/1204
* Castaneda (1988) Castaneda, H. O. 1988, ApJS, 67, 93, doi: 10.1086/191268
* Chepurnov & Lazarian (2010) Chepurnov, A., & Lazarian, A. 2010, ApJ, 710, 853, doi: 10.1088/0004-637X/710/1/853
* Chepurnov et al. (2010) Chepurnov, A., Lazarian, A., Stanimirović, S., Heiles, C., & Peek, J. E. G. 2010, ApJ, 714, 1398, doi: 10.1088/0004-637X/714/2/1398
* Cox (2005) Cox, D. P. 2005, ARA&A, 43, 337, doi: 10.1146/annurev.astro.43.072103.150615
* Ejdetjärn et al. (2021) Ejdetjärn, T., Agertz, O., Östlin, G., Renaud, F., & Romeo, A. B. 2021, arXiv e-prints, arXiv:2111.09322. https://arxiv.org/abs/2111.09322
* Federrath (2013) Federrath, C. 2013, MNRAS, 436, 1245, doi: 10.1093/mnras/stt1644
* Gaia Collaboration et al. (2016) Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. 2016, A&A, 595, A1, doi: 10.1051/0004-6361/201629272
* Gaia Collaboration et al. (2021) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2021, A&A, 649, A1, doi: 10.1051/0004-6361/202039657
* Galtier & Banerjee (2011) Galtier, S., & Banerjee, S. 2011, Phys. Rev. Lett., 107, 134501, doi: 10.1103/PhysRevLett.107.134501
* Girard et al. (2021) Girard, M., Fisher, D. B., Bolatto, A. D., et al. 2021, ApJ, 909, 12, doi: 10.3847/1538-4357/abd5b9
* Gritschneder et al. (2009) Gritschneder, M., Naab, T., Walch, S., Burkert, A., & Heitsch, F. 2009, ApJ, 694, L26, doi: 10.1088/0004-637X/694/1/L26
* Grudić et al. (2022) Grudić, M. Y., Guszejnov, D., Offner, S. S. R., et al. 2022, MNRAS, 512, 216, doi: 10.1093/mnras/stac526
* Gunn et al. (2006) Gunn, J. E., Siegmund, W. A., Mannery, E. J., et al. 2006, AJ, 131, 2332, doi: 10.1086/500975
* Ha et al. (2021) Ha, T., Li, Y., Xu, S., Kounkel, M., & Li, H. 2021, ApJ, 907, L40, doi: 10.3847/2041-8213/abd8c9
* Haffner et al. (2003) Haffner, L. M., Reynolds, R. J., Tufte, S. L., et al. 2003, ApJS, 149, 405, doi: 10.1086/378850
* Haffner et al. (2010) Haffner, L. M., Reynolds, R. J., Madsen, G. J., et al. 2010, in Astronomical Society of the Pacific Conference Series, Vol. 438, The Dynamic Interstellar Medium: A Celebration of the Canadian Galactic Plane Survey, ed. R. Kothes, T. L. Landecker, & A. G. Willis, 388. https://arxiv.org/abs/1008.0612
* Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2
* Hennebelle & Falgarone (2012) Hennebelle, P., & Falgarone, E. 2012, A&A Rev., 20, 55, doi: 10.1007/s00159-012-0055-y
* Heyer & Brunt (2004) Heyer, M. H., & Brunt, C. M. 2004, ApJ, 615, L45, doi: 10.1086/425978
* Hu et al. (2022) Hu, Y., Federrath, C., Xu, S., & Mathew, S. S. 2022, MNRAS, doi: 10.1093/mnras/stac972
* Hu et al. (2021) Hu, Y., Lazarian, A., & Xu, S. 2021, ApJ, 915, 67, doi: 10.3847/1538-4357/ac00ab
* Hunter (2007) Hunter, J. D. 2007, Computing in Science Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
* Joncas & Roy (1986) Joncas, G., & Roy, J. R. 1986, ApJ, 307, 649, doi: 10.1086/164450
* Kennicutt & Evans (2012) Kennicutt, R. C., & Evans, N. J. 2012, ARA&A, 50, 531, doi: 10.1146/annurev-astro-081811-125610
* Kim & Ostriker (2015) Kim, C.-G., & Ostriker, E. C. 2015, ApJ, 802, 99, doi: 10.1088/0004-637X/802/2/99
* Kolmogorov (1941) Kolmogorov, A. 1941, Akademiia Nauk SSSR Doklady, 30, 301
* Kounkel (2020) Kounkel, M. 2020, The Astrophysical Journal, 902, 122, doi: 10.3847/1538-4357/abb6e8
* Kounkel & Covey (2019) Kounkel, M., & Covey, K. 2019, AJ, 158, 122, doi: 10.3847/1538-3881/ab339a
* Krolikowski et al. (2021) Krolikowski, D. M., Kraus, A. L., & Rizzuto, A. C. 2021, AJ, 162, 110, doi: 10.3847/1538-3881/ac0632
* Lada et al. (2010) Lada, C. J., Lombardi, M., & Alves, J. F. 2010, ApJ, 724, 687, doi: 10.1088/0004-637X/724/1/687
* Larson (1981) Larson, R. B. 1981, MNRAS, 194, 809, doi: 10.1093/mnras/194.4.809
* Lazarian (2009) Lazarian, A. 2009, Space Science Reviews, 143, 357, doi: 10.1007/s11214-008-9460-y
* Lazarian & Pogosyan (2000) Lazarian, A., & Pogosyan, D. 2000, ApJ, 537, 720, doi: 10.1086/309040
* Lazarian & Pogosyan (2006) —. 2006, ApJ, 652, 1348, doi: 10.1086/508012
* Li et al. (2019) Li, H., Vogelsberger, M., Marinacci, F., & Gnedin, O. Y. 2019, MNRAS, 487, 364, doi: 10.1093/mnras/stz1271
* Li et al. (2015) Li, M., Ostriker, J. P., Cen, R., Bryan, G. L., & Naab, T. 2015, ApJ, 814, 4, doi: 10.1088/0004-637X/814/1/4
* Li et al. (2020) Li, Y., Gendron-Marsolais, M.-L., Zhuravleva, I., et al. 2020, ApJ, 889, L1, doi: 10.3847/2041-8213/ab65c7
* Louise & Monnet (1970) Louise, R., & Monnet, G. 1970, A&A, 8, 486
* Mac Low & Klessen (2004) Mac Low, M.-M., & Klessen, R. S. 2004, Reviews of Modern Physics, 76, 125, doi: 10.1103/RevModPhys.76.125
* Martizzi et al. (2015) Martizzi, D., Faucher-Giguère, C.-A., & Quataert, E. 2015, MNRAS, 450, 504, doi: 10.1093/mnras/stv562
* McBride et al. (2021) McBride, A., Lingg, R., Kounkel, M., Covey, K., & Hutchinson, B. 2021, AJ, 162, 282, doi: 10.3847/1538-3881/ac2432
* McInnes et al. (2017) McInnes, L., Healy, J., & Astels, S. 2017, The Journal of Open Source Software, 2, 205, doi: 10.21105/joss.00205
* Mohapatra et al. (2022) Mohapatra, R., Jetti, M., Sharma, P., & Federrath, C. 2022, MNRAS, 510, 2327, doi: 10.1093/mnras/stab3429
* Münch (1958) Münch, G. 1958, Reviews of Modern Physics, 30, 1035, doi: 10.1103/RevModPhys.30.1035
* Narayan & Ostriker (1990) Narayan, R., & Ostriker, J. P. 1990, ApJ, 352, 222, doi: 10.1086/168529
* Neuhäuser et al. (2020) Neuhäuser, R., Gießler, F., & Hambaryan, V. V. 2020, MNRAS, 498, 899, doi: 10.1093/mnras/stz2629
* Offner & Arce (2015) Offner, S. S. R., & Arce, H. G. 2015, ApJ, 811, 146, doi: 10.1088/0004-637X/811/2/146
* Ossenkopf & Mac Low (2002) Ossenkopf, V., & Mac Low, M. M. 2002, A&A, 390, 307, doi: 10.1051/0004-6361:20020629
* Padoan et al. (2016) Padoan, P., Pan, L., Haugbølle, T., & Nordlund, Å. 2016, ApJ, 822, 11, doi: 10.3847/0004-637X/822/1/11
* pandas development team (2020) pandas development team, T. 2020, pandas-dev/pandas: Pandas, latest, Zenodo, doi: 10.5281/zenodo.3509134
* Pavlidou et al. (2021) Pavlidou, T., Scholz, A., & Teixeira, P. S. 2021, MNRAS, 503, 3232, doi: 10.1093/mnras/stab352
* Qian et al. (2018) Qian, L., Li, D., Gao, Y., Xu, H., & Pan, Z. 2018, ApJ, 864, 116, doi: 10.3847/1538-4357/aad780
* Qian et al. (2012) Qian, L., Li, D., & Goldsmith, P. F. 2012, ApJ, 760, 147, doi: 10.1088/0004-637X/760/2/147
* Qian et al. (2015) Qian, L., Li, D., Offner, S., & Pan, Z. 2015, ApJ, 811, 71, doi: 10.1088/0004-637X/811/1/71
* Qu et al. (2020) Qu, Z., Bregman, J. N., Hodges-Kluck, E., Li, J.-T., & Lindley, R. 2020, ApJ, 894, 142, doi: 10.3847/1538-4357/ab774e
* Rathjen et al. (2021) Rathjen, T.-E., Naab, T., Girichidis, P., et al. 2021, MNRAS, 504, 1039, doi: 10.1093/mnras/stab900
* Roman-Duval et al. (2011) Roman-Duval, J., Federrath, C., Brunt, C., et al. 2011, ApJ, 740, 120, doi: 10.1088/0004-637X/740/2/120
* Rosen et al. (2021) Rosen, A. L., Offner, S. S. R., Foley, M. M., & Lopez, L. A. 2021, arXiv e-prints, arXiv:2107.12397. https://arxiv.org/abs/2107.12397
* Roy & Joncas (1985) Roy, J. R., & Joncas, G. 1985, ApJ, 288, 142, doi: 10.1086/162772
* Seifried & Walch (2015) Seifried, D., & Walch, S. 2015, MNRAS, 452, 2410, doi: 10.1093/mnras/stv1458
* Spitzer (1978) Spitzer, L. 1978, Physical processes in the interstellar medium, doi: 10.1002/9783527617722
* Stutz & Gould (2016) Stutz, A. M., & Gould, A. 2016, A&A, 590, A2, doi: 10.1051/0004-6361/201527979
* Stutzki et al. (1998) Stutzki, J., Bensch, F., Heithausen, A., Ossenkopf, V., & Zielinsky, M. 1998, A&A, 336, 697
* Tacchella et al. (2022) Tacchella, S., Smith, A., Kannan, R., et al. 2022, MNRAS, doi: 10.1093/mnras/stac818
* van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science Engineering, 13, 22, doi: 10.1109/MCSE.2011.37
* Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
* von Hoerner (1951) von Hoerner, S. 1951, ZAp, 30, 17
* Wes McKinney (2010) Wes McKinney. 2010, in Proceedings of the 9th Python in Science Conference, ed. Stéfan van der Walt & Jarrod Millman, 56 – 61, doi: 10.25080/Majora-92bf1922-00a
* Wilking et al. (2008) Wilking, B. A., Gagné, M., & Allen, L. E. 2008, Star Formation in the $\rho$ Ophiuchi Molecular Cloud, ed. B. Reipurth, Vol. 5, 351
* Xu (2020) Xu, S. 2020, MNRAS, 492, 1044, doi: 10.1093/mnras/stz3092
* Xu & Lazarian (2022) Xu, S., & Lazarian, A. 2022, ApJ, 927, 94, doi: 10.3847/1538-4357/ac4dfd
* Zhou et al. (2022) Zhou, J.-X., Li, G.-X., & Chen, B.-Q. 2022, MNRAS, doi: 10.1093/mnras/stac900
## Appendix A Selection of the $\rho$ Oph dark cloud within the Upper Sco
region
Figure A.1: Top panel: positions of young stars color-coded by their age.
Stellar age is estimated using the Sagitta package developed by McBride et al.
(2021). Background shows H$\alpha$ flux from the WHAM survey. Bottom panel:
the VSF of stars in Upper Sco within three age brackets. Only stars younger
than 2 Myr show characteristic of SN energy injection.
In the main body of this paper, we select only stars within the actively star-
forming region Ophiuchus. Here, we show the analysis results when selecting
the entire Upper Sco region and discuss how the VSF of the stars depend on the
stellar age.
The top panel of Figure A.1 shows the spatial distribution of stars in Upper
Sco and their age. Most of the stars outside Ophiuchus are older than 4 Myr,
while stars inside Ophiuchus are much younger, primarily younger than 2 Myr.
The SN explosion in this region traced by runaway stars took place $\sim 2$
Myr ago (Neuhäuser et al., 2020), so the majority of stars in Upper Sco did
not receive energy injection from this event, and thus would not exhibit
characteristics of such energy injection in their VSF. Indeed, we see this
trend in the lower panel of Figure A.1. With the entire Upper Sco region taken
into account, the VSF shows only a slight bump at $\sim 25$ pc, while both the
VSF of stars older than 5 Myr and of stars between 2 and 5 Myr do not show any
discernible energy injection scale. Meanwhile, the VSF of stars younger than 2
Myr shows a clear energy injection scale at $\sim 25$ pc, similar to what we
obtain in the main paper. Because of these results, we elect to use only stars
categorized as members of Ophiuchus a majority of which are younger than 1
Myr, to analyze in the main body of this paper.
## Appendix B Effect of Galactic Rotation on the VSF of H$\alpha$
Figure B.2: The VSFs of H$\alpha$, with (solid lines) and without (dotted-
dashed lines) a correction to the Galactic rotation (Qu et al., 2020).
Figure B.3: Top panel: illustration of our H$\alpha$ window tests. The red box
is the smallest window within which all stars in Taurus belong. The two boxes
in green (the window used in the main analysis) and light-blue are each 4 deg2
bigger than the former. Inside the red box, we also test the variation of the
VSF within the region by dividing the window into four equal-sized boxes.
Bottom left panel: The VSF of Taurus traced by H$\alpha$, plotted across three
different window sizes. Bottom right panel: The H$\alpha$ VSFs of Taurus in
the four quadrants within the red window.
Figure B.2 shows the VSF traced by H$\alpha$ in four regions with and without
Galactic rotation correction. In three out of the four regions, when a
rotation profile is applied, the VSF does not significantly differ from the
VSF without a rotation correction up to $\sim 20$ pc. This behavior is
expected, since at small $\ell$ the rotation correction between two points are
very similar. Thus, $\delta\vec{v}$ between two points essentially cancel out
the rotation correction, and the shape of the VSF is preserved. For two points
separated by a large distance, the rotation correction applied to them can
differ more, and hence results in a shift in the VSF, as Figure B.2 shows.
However, the overall shifts in the VSFs due to rotation correction are
moderate and do not affect the conclusions of our work.
## Appendix C Effect of the size and location of the H$\alpha$ window
Another potential source of bias comes from the exact size and location of the
window chosen to calculate the VSF of H$\alpha$ within each region. In Section
3.2, we caution that the bumps seen in the VSF of H$\alpha$ is a potential
artifact since it is close to the size of the window. Such effect is noted as
the apparent driving scale in simulations of multi-phase turbulence, and is
equal to 1/2 of the box size (e.g., Mohapatra et al., 2022). To test whether
this effect also presents in the VSF of H$\alpha$, we select a solid angle
that encompasses all the stars in a region, and compute the VSF within this
window. We then compute the VSF for a window that is 4 deg2 larger, and repeat
this a second time (see top panel in Figure B.3).
In all four regions, the VSFs of H$\alpha$ are independent of the window size
up to $\ell\sim L/2$, where $L$ is the window size. In Taurus and Perseus, the
bump in the VSF shifts to larger $\ell$ when the window size increases. Thus
we conclude that in these regions, the VSF bump is likely caused by the box
limit rather than a real physical driving scale. As an example, the bottom
left panel of Figure B.3 shows the VSF of H$\alpha$ in Taurus as a function of
window size.
We also use Taurus as an example to illustrate the spatial variation of
turbulence on small scales. We divide the smallest box (8 deg2) in the
previous experiment into four equal-sized quadrants. The bottom right panel of
Figure B.3 shows the H$\alpha$ VSFs of these four quadrants in Taurus.
Although the slopes of these four VSFs are roughly consistent with one
another, their amplitudes vary by up to a factor of three, indicating
potentially large variations in turbulence between local patches of the ISM.
The VSF of CO is in the best agreement with the bottom-left window
($172^{\circ}<l<176^{\circ}$ and $-20^{\circ}<b<-16^{\circ}$), where many CO
cores were also identified (see Figure 2 of Qian et al., 2012, for detailed
locations of CO cores in Taurus).
|
# Toward Open Integrated Access and Backhaul
with O-RAN
Eugenio Moro1, Gabriele Gemmi2, Michele Polese3, Leonardo Maccari2, Antonio
Capone1, Tommaso Melodia3
This work was partially supported by NGIAtlantic.eu project within the
EUHorizon 2020 programme under Grant No. 871582, by the U.S. National Science
Foundation under Grant CNS-1925601, and by OUSD(R&E) through Army Research
Laboratory Cooperative Agreement Number W911NF-19-2-0221. The views and
conclusions contained in this document are those of the authors and should not
be interpreted as representing the official policies, either expressed or
implied, of the Army Research Laboratory or the U.S. Government. The U.S.
Government is authorized to reproduce and distribute reprints for Government
purposes notwithstanding any copyright notation herein. 1Department of
Electronics, Information and Bioengineering,
Polytechnic University of Milan, Italy
<EMAIL_ADDRESS>2Department of Environmental Sciences, Informatics
and Statistics,
Ca’ Foscari University of Venice, Italy.
<EMAIL_ADDRESS>3Institute for the Wireless Internet of Things,
Northeastern University, Boston, MA, U.S.A.
<EMAIL_ADDRESS>
###### Abstract
Millimeter wave (mmWave) communications has been recently standardized for use
in the fifth generation (5G) of cellular networks, fulfilling the promise of
multi-gigabit mobile throughput of current and future mobile radio network
generations. In this context, the network densification required to overcome
the difficult mmWave propagation will result in increased deployment costs.
Integrated Access and Backhaul (IAB) has been proposed as an effective mean of
reducing densification costs by deploying a wireless mesh network of base
stations, where backhaul and access transmissions share the same radio
technology. However, IAB requires sophisticated control mechanisms to operate
efficiently and address the increased complexity. The Open RAN paradigm
represents the ideal enabler of RAN intelligent control, but its current
specifications are not compatible with IAB. In this work, we discuss the
challenges of integrating IAB into the Open RAN ecosystem, detailing the
required architectural extensions that will enable dynamic control of 5G IAB
networks. We implement the proposed integrated architecture into the first
publicly-available Open-RAN-enabled experimental framework, which allows
prototyping and testing Open-RAN-based solutions over end-to-end 5G IAB
networks. Finally, we validate the framework with both ideal and realistic
deployment scenarios exploiting the large-scale testing capabilities of
publicly available experimental platforms.
###### Index Terms:
IAB, O-RAN, 5G, Colosseum
## I Introduction
Radio Access Network (RAN) densification is a key technique to boost the
coverage and performance metrics of current and future generations of mobile
radio networks [1]. However, these ultra-dense deployments come with increased
costs and complexity for provisioning wired backhaul to each base station [2].
To address this, the 3rd Generation Partnership Project (3GPP) has introduced
Integrated Access and Backhaul (IAB) in its Release 16 for NR [3]. With IAB,
the backhaul traffic is multiplexed on the air interface together with regular
User Equipments (UEs) access traffic. This effectively creates a wireless mesh
network of Base Stations (BSs) where only a few require an expensive wired
connection to the Core Network (CN) (i.e., the IAB-Donors). Hence the cost-
reduction potential through wireless relays (i.e., the IAB-Nodes) [4].
Additionally, IAB is especially relevant for millimeter wave (mmWave)-based
radio access, where inexpensive network densification is a fundamental
necessity [5].
While the standardization process has reached a sufficient maturity level, the
open challenges brought about by integrating access and backhaul are still
open. Consequently, IAB offers optimization opportunities at all layers of
communication abstraction. At the lowest levels, specialized IAB-aware
techniques are required to ensure a fair and effective resource allocation
among UEs and Mobile Terminations (MTs) [6, 7]. At the same time, backhaul and
access transmission multiplexing must be managed to minimize interference [8].
Furthermore, adaptive topology reconfiguration mechanisms must be provisioned
to maintain resiliency against link failures, traffic unbalances and anomalous
user distribution [9]. Overall, these sophisticated management procedures
require control primitives that go beyond what has been specified by 3GPP.
The unprecedented paradigm shift brought about by the O-RAN architecture,
developed by the O-RAN Alliance, promises to enable programmatic control of
RAN components through open interfaces and centralized control loops [10]. As
such, it is the ideal candidate to unlock the potential optimization and
management gains awaiting in IAB. However, the current O-RAN architecture is
tailored to traditional RAN deployments, and an extension to enable IAB
control is required. The first contribution of this work resides in a
discussion on how the O-RAN architecture, interfaces, and control loops can be
extended to IAB scenarios, with the ultimate goal of allowing large-scale,
data-driven control and management of 5th generation (5G) IAB networks.
Additionally, to foster prototyping and testing with IAB and O-RAN, we propose
a comprehensive framework where researchers can easily deploy an end-to-end
O-RAN-enabled IAB network with Over-The-Air (OTA) and hardware-in-the-loop
emulation capabilities. In line with O-RAN core concepts, our framework is
designed to be open, accessible and flexible by leveraging on open-source
software and Commercial Off-the-Shelf (COTS) hardware. The framework builds on
IABEST, the first large-scale accessible and open IAB testbed presented in
[11]. This testbed has been enriched to produce a complete O-RAN IAB
experimental solution, effectively replicating the proposed O-RAN-IAB
integrated architecture. In particular, IAB-Donors and IAB-Nodes have been
equipped with custom-developed agents for the so-called E2 and O1 standard
interfaces. These additions enable the controllers introduced by the O-RAN
architecture to manage IAB-Nodes, effectively representing the first publicly
available O-RAN-enabled IAB prototyping and testing solution.
To further facilitate experimental research activities, we have packaged and
integrated the entire framework into OpenRAN Gym, a publicly-available
research platform for data-driven O-RAN experimentation at scale [12]. Through
OpenRAN Gym, researchers can swiftly deploy and test the proposed framework
over large-scale and publicly available hardware experimental platforms, such
as the PAWR testbeds and Colosseum [13, 14]. Notably, we showcase how
Colosseum can be leveraged for large-scale IAB testing through hardware-in-
the-loop channel emulation to create sophisticated deployment scenarios. A
tutorial on how to deploy an O-RAN-driven IAB network, together with the
source code of all the framework, is available on the OpenRAN Gym
website.111https://openrangym.com/tutorials/iab-tutorial Finally, we use
Colosseum to validate the proposed framework numerically. In particular, we
test the attainable performance in a controlled radio scenario and in a more
realistic deployment in which we reconstruct a part of Florence, Italy.
The remainder of this paper is organized as follows. Section II analyses the
challenges of extending O-RAN to 5G IAB networks. Section III contains a
description of the proposed frameworks, focusing on the O-RAN extensions that
have been included in [11]. Section IV contains the results of the experiments
we performed to validate our framework by exploiting the large-scale testing
capabilities of Colosseum. Finally, Section V concludes the paper and
discusses future extensions.
Figure 1: IAB and O-RAN integrated architectures.
## II Integrating IAB in Open RAN
As discussed in Section I, IAB represents a scalable solution to the need for
backhaul in ultra-dense 5G and 6G deployments. At the same time, however, the
wireless backhaul introduces additional complexity to the network deployments:
new parameters and configurations that need to be tuned—and possibly, adapted
dynamically—to get the best performance out of the network and to seamlessly
adjust to updated conditions in the scenario and in the equipment status. For
example, it is possible to optimize the IAB network performance by properly
selecting the connectivity of IAB-Nodes to their parents [9], or by
appropriately allocating resources to backhaul and access flows sharing the
same air interface [6].
As for traditional RAN deployments with fiber-based backhaul [15], there is a
case to be made for providing IAB RAN equipment with primitives for flexible,
dynamic, data-driven programmatic control. This requires providing endpoints
to expose telemetry, measurements, and analytics from IAB-Nodes, as well as
parameters and control knobs to enable the optimization. So far, the Open RAN
paradigm has been successfully applied to non-IAB networks to achieve the same
goals, thanks to interfaces that give access to 3GPP Key Performance
Measurements (KPMs) and control parameters in the RAN nodes [16, 17]. The Open
RAN vision, which is being developed into technical specifications by the
O-RAN Alliance, includes controllers that run custom control loops, i.e., the
RAN Intelligent Controllers (RICs). The O-RAN Alliance has defined control
loops and related RICs that can operate at a time scale of 10 ms to 1 s (i.e.,
_near-real-time_) or more than 1 s (i.e., _non-real-time_) [18]. The near-
real-time, or near-RT, RIC is connected to the RAN nodes through the E2
interface, while the non-real-time RIC, which is part of the network Service
Management and Orchestration (SMO), interacts with the RAN through the O1
interface, as shown in the left part of Figure 1. Other interfaces from the
non-RT RIC/SMO include A1 to the near-RT RIC, for policy guidance and
Artificial Intelligence (AI)/Machine Learning (ML) model management, and the
O2 interface to the O-Cloud, which is an abstraction of the virtualization
infrastructure that can support the deployment of O-RAN functions. The use of
standard interfaces makes it possible to run even third-party applications in
the controllers, the so-called xApps and rApps for the Near Real-time RAN
Intelligent Controller (near-RT RIC) and Non-Real-Time Ran Intelligent
Controller (non-RT RIC), respectively.
The 3GPP already provides control and adaptation capabilities through the IAB
Backhaul Adaptation Protocol (BAP) layer, the F1 interface, and the Radio
Resource Control (RRC) layer across the IAB-Donor Central Unit (CU) and the
IAB-Node Distributed Unit (DU). How and when control and adaptation of such
configurations could be performed, however, is left to the vendor
implementation. This is where an extension of the O-RAN architecture to IAB
networks can play a role, exposing IAB-Donor and IAB-Node functions to the
RICs. These can leverage a centralized point of view on the RAN and a wealth
of analytics and information usually unavailable in the individual IAB-Donors
and Nodes. For IAB, this could translate into effective multi-donor
coordination with reduced interference and agile topology adaptation across
different IAB-Donor domains, and dynamic resource allocation with—for
example—data-driven proactive congestion identification and resolution across
access and backhaul links.
### II-A Extensions to Open RAN
Extending the O-RAN architecture and interfaces to IAB deployments, however,
presents some design and architectural challenges. Primarily, supporting O-RAN
interfaces in IAB-Nodes means either (i) terminating the interfaces at the
IAB-Donor; or (ii) transporting their data over the wireless backhaul. The
first option is simpler, does not require architectural updates, but at the
same time limits the control and reconfiguration to what is available in the
IAB-Donor, without insight on the IAB-Nodes. The second option, instead,
provides more granular access at the cost of additional complexity and
tunneling of data over the wireless backhaul.
The 3GPP already foresees performing SMO-like operations through the wireless
backhaul interface [19]. Therefore, in this paper and in the architecture
described in Figure 1 we consider the second option, which would provide
tighter and more effective integration between O-RAN and IAB deployments. In
general, the tunneling can be performed by encapsulating the O-RAN interfaces
payloads into dedicated bearers. Note that this requires some interaction
between functions of the control plane of the network and the transport in the
user plane, e.g., through a dedicated Packet Data Unit (PDU) session between a
local User Plane Function (UPF) in the IAB-Donor and in the IAB-Node MT. Then,
a local interface termination can be installed in the IAB-Node, as it would in
a traditional, fiber-equipped RAN node. The O-RAN traffic, in this case, would
be multiplexed with user data on the wireless backhaul resources, and it needs
to be properly prioritized to achieve the control goals while not harming
users’ performance or creating congestion.
Figure 2: Overview of the RAN architecture deployed over white-box hardware.
E2 extension for IAB. The extension of the E2 interface likely requires one or
multiple new, dedicated E2 Service Models (E2SMs). The E2SM represents the
semantic of the E2 interface, i.e., the RAN function with which an xApp in the
near-RT RIC interacts. For IAB, an extension of E2SM KPM [20] can be used to
expose performance metrics related to the MT, besides the DU. Another near-
real-time control target over E2 can include, for example, resource
partitioning between backhaul and access traffic, or dynamic Time Division
Duplexing (TDD) slot configuration to adapt to varying traffic on the access
and backhaul.
O1 extension for IAB. The O1 interface would connect the SMO to the IAB-Node,
e.g., to perform maintenance and updates of the components (MT and DU) of the
IAB-Node. Compared to E2 near-real-time control, the O1 interface would run
control loops at 1 s or more. Thus its traffic can be transported with lower
priority than the E2 traffic. This makes a case for dedicated bearers and
tunnels on the backhaul interface for _each_ of the O-RAN interfaces.
O2 extension for IAB. This interface can be used to integrate the IAB-Nodes as
resources in the O-Cloud. Compared to traditional virtualization
infrastructure for the O-Cloud, the IAB-Nodes are available—and reachable over
O2—only when a session is established from one IAB-Donor to the IAB-Node
itself.
## III An Experimental Framework for
IAB and O-RAN
Our proposed experimental framework packages the entire software chain
required to run the O-RAN-enabled IAB network described in Section II in a
multi-layer architecture. At the hardware level, our framework does not
present any specific requirement. Indeed, every software component can run on
COTS hardware like generic x86 machines and USRP Software-defined Radio (SDR).
On the other hand, some software components are customized or designed from
scratch to reproduce and support a 5G IAB network. In particular, we have
adapted OpenAirInterface (OAI), an open source 5G RAN framework [21], to
implement IAB-Donors, IAB-Nodes, and IAB-capable core functions. Additionally,
we have integrated agents for the E2 and O1 interfaces in the IAB-Donor and
IAB-Node, effectively implementing the architectural integration proposed in
Section II. These interfaces are used by the non-real-time and real-time RICs
packaged in our framework to control all the components of the deployed IAB
network. We now describe the aforementioned components, separating them into
the RAN and O-RAN domains.
### III-A RAN and Core Network Components
Figure 2 represents an overview of the radio access functional components that
enable end-to-end communication in our framework. In particular, we provide
the following: a minimal yet functional deployment of 5G CN functions,
software-defined IAB-Nodes and IAB-Donors and software-defined UEs.
IAB-Nodes and IAB-Donors. According to 3GPP specifications [4], an IAB-Donor
hosts a CU and multiple DUs. Similarly, IAB-Node is split into a DU and an MT.
Functionally, these have the task of enabling downstream and upstream
connectivity, respectively. At the time of writing, OAI’s main branch
implementation of the CU/DU functional split does not support multiple DUs
connected to a single CU [22]. This limitation is incompatible with the IAB
architecture. Consequently, we employ a full OAI Next Generation Node Base
(gNB) in place of both CU and DU. In other words, the IAB-Nodes and IAB-Donors
in our framework do not follow 3GPP split 2. Instead, these components are
deployed as monolithic gNBs. As for the MT, an open-source implementation is
currently unavailable. However, this component is functionally equivalent to a
UE, as it connects to upstream nodes using the same resources and protocols.
Consequently, we have selected OAI’s software-defined UE to act as MTs in the
proposed framework. This results in a system where a single IAB-Node is made
up of two concurrently running instances: an OAI gNB—acting as a DU—and an OAI
UE—acting as a MT. In the resulting architecture, IAB-Nodes are naturally
deployed over two separate machines, hosting the gNB and the UE, and connected
out-of-band as it is shown in Figure 2. Alternatively, the two software
components can run on a single x86 machine, provided that sufficient computing
power is available. While this architecture does not require any particular
modification to OAI’s implementations, we have added a signaling functionality
through which the IAB-Nodes or IAB-Donors can discern connected MTs from
connected UEs. This has been achieved through proper manipulation of the UE
Capability messages. Such information can be exploited to enable IAB-aware
optimization solutions in the gNB.
Core Network Functions. A minimal set of 5G CN functions have been included in
our framework: Network Repository Function (NRF), Access and Mobility
Management Function (AMF), Slicing Magangement Framework (SMF) and User Plane
Function (UPF), all based on the OAI 5G core implementation. All these
functions run as containers on a single x86 machine, as shown in Figure 2. Due
to the selected IAB system design, the UPF required modifications to enable
IAB operations. As previously mentioned, UEs acts as MTs in IAB-Nodes,
connecting to upstream nodes. The established GPRS Tunneling Protocol (GTP)
tunnels are then used to provide direct connectivity between the DU of the
node and the CN functions. In other words, MT-acting UEs relay the backhaul
traffic of the IAB-Nodes. However, OAI’s UPF implementation lacks support for
the required forwarding capability,222To the best of the authors’ knowledge,
there is no available open source implementation that supports this operating
mode. as any packet whose destination is not a UE is dropped. Therefore, we
have implemented a minimal version of framed routing [23] in OAI UPF, enabling
UEs to act as intermediate nodes.
User Equipment From the perspective of the UE, an IAB network deployed using
the components described above is entirely standard-compliant. As such, both
software-defined UEs (as shown in Figure 2) and COTS UEs can be used in the
proposed framework.
### III-B O-RAN Components
As mentioned in Section II, O-RAN defines a set of standardized and open
interfaces with which the RAN exposes data collection and control primitives
to the RICs. In the proposed framework, we have enabled IAB-Nodes and IAB-
Donors to be O-RAN-compatible by integrating software agents for the E2 and O1
interfaces into the codebase of OAI. Furthermore, our framework comprises a
near-RT RIC and a non-RT RIC.
E2 interface integration. The E2 interface is functionally split into two
protocols: E2AP—tasked with establishing a connection with the near-RT RIC—and
E2SM—which implements specific monitoring and control functionalities, namely
Service Models (SMs), as discussed in Section II. In the software
implementation we provide, E2AP has been adapted from O-RAN Alliance Software
Community reference implementation and, as such, it is entirely compliant with
O-RAN. On the other hand, the SMs provided by the O-RAN alliance are defined
using ASN.1: a powerful production-ready abstract description language which
is, however, cumbersome and challenging to use in the fast-paced research and
development environments targeted by our framework. In light of this, we
employ custom SM that are defined through Protocol Buffers (protobuf)—an
abstract definition language that is easier to handle and allows for fast
prototyping and testing, facilitating the development of IAB-aware control
solutions. Since the E2 interface is such that the E2SM messages are encoded
and decoded only in the RAN and xApp, the custom SM definitions are
transparent to the RIC, allowing our proposed solution to retain generic O-RAN
compliance. At the time of this writing, we have implemented a set of protobuf
messages that can be used to reproduce both the KPM and RAN Control (RC) SMs
[10]. These can be used to develop data collection and control xApps,
respectively.
(a) Linear IAB topology. $1$$2$$3$$4$$5$$10$$20$$30$$40$$50$Hops Throughput
[Mbps] DownlinkUplink (b) Throughput measurements for the linear topology.
$1$$2$$3$$4$$5$$20$$40$$60$$80$Hops RTT [ms] (c) RTT measures for the linear
topology.
Figure 3: Topology and results for the linear chain.
O1 interface integration. In order to properly manage all the different
aspects of networked elements, the O1 interface defines various Management
Services (MnS), which can be used either from the managed entities (the gNBs)
to report information back to the RIC or from the managing entity (the SMO and
the rApps running on it) to deploy configurations changes, transfer files or
update the software on the managed entities [24, 10]. Among all the different
MnS, we have focused our contribution on implementing the Heartbeat MnS, which
periodically transmits heartbeats; the Fault Supervision MnS, which reports
errors and events; and the Performance Assurance MnS, which streams
performance data. Those MnS have been integrated into the OAI codebase by
implementing a scheduler that, running on a dedicated thread, periodically
sends Virtual Network Function (VNF) Event Stream (VES) notifications in JSON
format over HTTP. This format and communication protocol has been chosen among
the different options defined in the standard, as it is widely known and
easily extendable by other researchers. As of now, our implementation reports
performance metrics, such as the throughput and information on the channel
quality between IAB-Nodes, and failure events, such as RRC or Uplink Shared
Channel (UL-SCH) failures, which can be used in rApps to monitor and optimize
the backhaul network. Provisioning MnS, which can be used by the rApps to
deploy configuration changes (e.g., topology optimizations), have not been
implemented by following the O1 specifications, as it would have needed major
reworks in the OAI codebase. Instead, we have taken advantage of IAB-Manager,
a software component we developed to orchestrate IAB experiments, as discussed
next.
IAB-Manager. IAB networks are generally expected to include several IAB-Nodes,
and the proposed framework can scale to such numbers. However, managing
experiments with tens or more RAN components can take time and effort. Indeed,
each component is potentially hosted by a dedicated machine, and setting up an
IAB deployment requires each one to be activated and configured according to a
sequence that starts from the CN functions and ends with the terminal IAB-
Nodes. To facilitate experimenting at such a large scale, we have developed
IAB-Manager [11]: a software component that can automate the IAB network
deployment and testing through a command line interface and an Application
Programming Interface (API). In particular, IAB-Manager is a single entrypoint
for controlling the entire experiment: network components and radio
environment setup (in case of wireless channel emulation), topology and
routing management and reconfiguration, automated testing, and result
collection. From a functional perspective, the manager connects to the
machines involved in the experimentation and configures them according to the
assigned roles. In particular, once the user specifies the roles, the manager
sequentially activates each network component until the final deployment is
ready for experimentation, greatly simplifying the setup phase. Additionally,
as previously mentioned, IAB-Manager executes the network configuration
changes mandated by the rApps.
RAN Intelligent Controllers. The proposed framework packages a near-RT RIC and
a non-RT RIC. Both are compliant with the standard and based on O-RAN Software
Community reference implementations.
## IV Validation and Results
Parameter | Value
---|---
Area Size for realistic deployment | 0.627 km2
gNB Density | 45 gNB/km2
IAB-donors/ IAB-nodes ratio | 1/10
Emulated center frequency | 28 GHz
Bandwidth | 40 MHz
Scheduler | 7 2 1
Subcarrier Spacing | 30khz
Colosseum Base loss | 50 dB
3GPP Channel Model | Urban Micro
MIMO layers | 1
TABLE I: Table of System Settings
This section focuses on validating our proposed framework from an experimental
perspective. In particular, we are interested in giving an initial
characterization of some fundamental Key Performance Indicators (KPIs) of the
deployments allowed by our IAB framework while validating its correct
functioning.
While the openness and flexibility of the software components are such that
the framework can run on generic hardware, we chose to base our validation
campaign on Colosseum [14]. The Colosseum testbed is a publicly available
large-scale testing platform with hardware-in-the-loop capabilities. It
comprises 128 Standard Radio Nodes (SRNs), each composed of a powerful x86
computing node and an USRP X310 SDR. All the components in the proposed
framework can be easily deployed on SRNs. Every SRN radio is interconnected by
an FPGA mesh that emulates arbitrary radio channels defined through tapered
delay models. With the capability of emulating complex scenarios of tens of
entities, Colosseum makes it possible to deploy large IAB networks over
complex propagation scenarios. As such, it represents an ideal validation
platform for our framework. Furthermore, Colosseum is open to the research
community, and the validation tools are made available, allowing interested
parties to start experimenting with a minimal initial effort.
### IV-A Experiments with a linear chain
We start by evaluating the performance of an IAB network deployed in a tightly
controlled scenario. To this end, we consider a 5-hop linear topology, as
shown in Figure 3(a). As detailed in Section II, each IAB-Node comprises an MT
and a DU, bringing this experiment’s overall radio node count to 10. In order
to characterize the upper-bound performance of the proposed framework, we
employ an ideal propagation scenario. Through properly manipulating
Colosseum’s channel emulator, a 0 dB pathloss model is selected for nodes
connected in the linear topology, and an infinite pathloss is set for all the
other channels, effectively suppressing any possible interference. In other
words, this radio scenario is equivalent to connecting the SDRs with coaxial
cables.333 Transmissions occur on band n78 with 106 Physical Resource Blocks
(PRBs) available, for a total of 40MHz bandwidth.
123$0$$10$$20$$30$$40$Hops Throughput [Mbps] (a) Downlink.
123$5$$10$$15$$20$Hops Throughput [Mbps] (b) Uplink. 123$20$$40$$60$$80$Hops
RTT [ms] (c) Round Trip Time.
Figure 5: Measurements for the realistic scenario.
Figure 3(b) shows the downlink and uplink TCP throughput against the number of
hops, as measured between the core network and the specific MT/UE. The first-
hop values of 47 Mbps in DL and 21 Mbps in UL represent the maximum throughput
attainable in the testing settings. This upper bound is far from the
theoretical maximum allowed by the available bandwidth. It is limited by
several factors that depend on the experimental platform, OAI software
implementation, and system design. Most notably, the strongest detractor to
the final throughput performance is given by the OAI implementation of the
software-defined UE, which is employed to build the MT. In particular, the OAI
UE is inefficient in reception and transmission, thus becoming a bottleneck
for the entire communication chain. Efforts are ongoing to improve the
performance and stability of this software framework. Furthermore, the
frameworks’ system design is such that each IP packet is encapsulated into as
many GTP packets as the number of hops. This increased overhead can cause
packet fragmentation with a further negative impact on the overall
performance. Furthermore, even if the emulated channel is set to a 0 dB
pathloss, Colosseum’s architecture includes an unavoidable base loss of 50 dB
[25] due to characteristics of the hardware architecture. This, together with
the aforementioned inefficiencies, make such that packet drops and subsequent
retransmissions happen also in this ideal scenario.
As the number of hops increase, the downlink throughput experiences a sharp
decrease before stabilizing on a per-hop loss of around 6 Mbps. The notable
throughput loss experienced at the second hop can be explained by observing
the standard deviation of the throughput, represented by the whiskers in
Figure 3(b). This value is at its maximum for the first hop, suggesting that
the first radio link is unstable due to the RX pipeline of the MT being
overwhelmed. This substantial variability is caused by packet loss and
retransmissions and internal buffer overflow, which negatively affect the
performance of the second hop, as it is noticeable in the numerical results.
At the same time, the second hop’s throughput standard deviation is lower, as
the decreased traffic volume causes less drops in the involved MTs. This
stabilizing effect propagates down the topology, as both the decreasing
standard deviation and the linear per-hop loss testify. On the other hand, the
uplink throughput is relatively stable and close to the upper bound, even at
the fourth hop. This is because the limited OAI UE performance and BS
scheduling process limits the uplink traffic volume, and the gNBs are far from
being overwhelmed. On the other hand, since the uplink throughput does not
significantly decrease from the maximum, the UE’s congestion level remains
relatively stable and high, as proven by the constant standard deviation
values.
Figure 4: Realistic deployment scenario in Florence, Italy. Donors are
represented in red, while IAB-Nodes are represented in yellow.
RTT is measured when the network is unloaded, that is when there is no traffic
flowing through the IAB network. As shown in Figure 3(c), the first hop
latency is around 11 ms. This value represents the base processing delay plus
a small fixed propagation delay that is, however, the same for each hop. As
the number of hops increases, the RTT experiences a linear increase comparable
with the first hop latency, as expected. This shows how the system does not
introduce any spurious latency when the network is unloaded. Finally, the
relatively higher RTT standard deviation of the last hop (as represented by
the whiskers in Figure 3(c)) suggests that multiple packet retransmissions are
required.
### IV-B Validation over realistic RF scenarios
After having validated the system performance in a controlled environment, we
move to more realistic urban scenarios, representing the typical deployment
environment of an IAB network. We start by selecting a densely populated area
of a city, from which we extract a precise 3D model of the environment. On top
of this, we apply a coverage planning heuristic to find the optimal locations
for the IAB-Nodes [26]. We then take advantage of a viewshed algorithm
implemented on GPU and the above-mentioned 3D models to evaluate the Line of
Sight (LoS) between each pair of locations and to produce the so-called
visibility graph [27]. Then, we characterize the propagation according to the
3GPP Urban Micro parametric model [28], and we produce a tapered-delay
representation of the communication channels, which Colosseum can then
emulate. This process has been carried out for several European cities and
four different scenarios are made
available.444https://colosseumneu.freshdesk.com/support/solutions/articles/61000303373-integrated-
access-and-backhaul-scenarios
Motivated by the fact that IAB is unanimously considered as a key enabler of
mmWave RAN [29], we are interested in providing an experimental solution that
enables testing in such conditions. While Colosseum is not directly capable of
operating at frequencies higher than 6 GHz, we can approximate these radio
scenarios by reproducing the most relevant propagation characteristics of
mmWaves, namely the extremely directive transmissions through beamforming and
the increased pathloss [30]. In particular, the pathloss between nodes that
are not directly connected in the provided topologies has been set to
infinite. The resulting suppression of inter-node interference might appear
too ideal at first. However, this is compatible with the highly directive
transmissions typical of mmWave, where interference in static conditions
(i.e., as in a backhaul IAB topology) can be practically neglected [31]. A
more refined mmWave channel emulation will be subject of future extensions. In
addition, since Colosseum’s channel emulation happens in base-band, we can
apply arbitrary pathloss independently of the radio frequency employed during
the experiments. Thanks to this flexibility, we could compute pathloss for a
carrier frequency of 28 GHz and apply them to LoS links. Nonetheless, the
scenarios made available to the Colosseum community are available for both 3.6
GHz and 28 GHz, both with and without inter-node interference suppression.
For the experimental evaluation presented in this work, we have selected a
scenario based on the city of Florence, Italy. Figure 4 shows both the urban
layout and the IAB deployment, which is extended over $0.7~{}\text{km}^{2}$
and comprises 21 nodes (3 of which are IAB-Donors). To determine which nodes
are going to become IAB-Donors, we have applied the group closeness centrality
metric [32] to the visibility graph. This centrality metric selects $k$ nodes
such that their distance to all the other nodes is minimized. Then, we have
determined the IAB topology as a Shortest-Path Forest computed over the
visibility graph of the area with the well-known Dijktra’s Algorithm. Similar
to what has been done for the previous analysis, we characterize the
throughput and latency at each hop in the network. In this case, however, the
different link lengths cause performance variations in the per-hop throughput
and latency. As such, we employ box plots to synthetically describe the
network performance statistics in Figure 5. In particular, consider Figure
5(a). Here the bottom and top edges of each box represent the first and third
quartile of the downlink throughput measurements taken at all the different
hops in the scenario. Similarly, the central marks indicate the median, and
the whiskers represent the extreme data points. The plotted values indicate
how the realistic pathloss introduced in the study scenario causes lower
performance than the ideal case previously analyzed, independently of the
considered hop. The same can be noted for the uplink throughput, as shown in
Figure 5(b). In both cases, the decreasing per-hop throughput trend is
conserved. However, the throughput variability is the same for the two
transmission directions. This is because, as opposed to the ideal scenario,
the link length now represents the main performance-determining factor. This
is testified by the significant distance between the first and third quartile
of the first hop in both downlink and uplink throughput, which is consistent
with the high variance of the first hop length in the topology of study. As
for the second and third hop, the relatively closer quartiles are motivated by
lower link length variations for these hops in the considered topology.
Finally, the upper whiskers represent the performance of the shortest links,
giving a further characterization of the system performance in this realistic
scenario.
Figure 5(c) shows the RTT statistic through the same plotting technique.
Differently from the throughput, the latency is not affected by the link
length variations in the considered scenario for the first two hops.
Additionally, the RTT increase at hops 1 and 2 is consistent with the one
experienced in the controlled scenario. On the other hand, the high RTT
variance of the third and last hop suggests a high probability of requiring
retransmissions along the IAB path.
## V Conclusions
In this work, we have discussed the motivations and challenges of integrating
IAB with O-RAN. On this matter, we have proposed possible architecture
extensions that enable dynamic control and data collection over 5G IAB
networks through O-RAN intelligent controllers. We have implemented the
proposed integrated architecture and packaged it into the first publicly
available experimental framework enabling at-scale testing and prototyping of
O-RAN-based solutions applied to IAB networks. The system comprises all the
software components required to establish end-to-end connectivity, plus
custom-developed E2 and O1 agents that allow software-defined IAB-Nodes to be
O-RAN-compliant. The framework is designed to be open and accessible and can
be deployed over COTS hardware. We numerically validated the framework
exploiting the large-scale testing capabilities of Colosseum, showing the
system’s performance over both an ideal linear topology and more sophisticated
realistic deployments. Finally, the framework has been packaged and released
into OpenRAN Gym and is available to the research community.
## References
* [1] S. Dang, O. Amin, B. Shihada, and M.-S. Alouini, “What should 6G be?” _Nature Electronics_ , vol. 3, no. 1, pp. 20–29, 2020.
* [2] N. Bhushan, J. Li, D. Malladi, R. Gilmore, D. Brenner, A. Damnjanovic, R. T. Sukhavasi, C. Patel, and S. Geirhofer, “Network densification: the dominant theme for wireless evolution into 5G,” _IEEE Communications Magazine_ , vol. 52, no. 2, pp. 82–89, 2014.
* [3] T. Inoue, “5G NR Release 16 and Millimeter Wave Integrated Access and Backhaul,” in _2020 IEEE Radio and Wireless Symposium (RWS)_ , 2020.
* [4] M. Polese, M. Giordani, T. Zugno, A. Roy, S. Goyal, D. Castor, and M. Zorzi, “Integrated Access and Backhaul in 5G mmWave Networks: Potential and Challenges,” _IEEE Communications Magazine_ , vol. 58, no. 3, pp. 62–68, 2020.
* [5] W. Feng, Y. Wang, D. Lin, N. Ge, J. Lu, and S. Li, “When mmWave Communications Meet Network Densification: A Scalable Interference Coordination Perspective,” _IEEE Journal on Selected Areas in Communications_ , vol. 35, no. 7, pp. 1459–1471, 2017.
* [6] Y. Zhang, V. Ramamurthi, Z. Huang, and D. Ghosal, “Co-optimizing performance and fairness using weighted pf scheduling and iab-aware flow control,” in _2020 IEEE Wireless Communications and Networking Conference (WCNC)_ , 2020\.
* [7] B. Zhang, F. Devoti, I. Filippini, and D. De Donno, “Resource allocation in mmWave 5G IAB networks: a reinforcement learning approach based on column generation,” _Computer Networks_ , vol. 196, p. 108248, 2021.
* [8] “Coordinated parallel resource allocation for integrated access and backhaul networks,” _Computer Networks_ , vol. 222, p. 109533, 2023.
* [9] S. Ranjan, P. Chaporkar, P. Jha, and A. Karandikar, “Backhaul-Aware Cell Selection Policies in 5G IAB Networks,” in _2021 IEEE Wireless Communications and Networking Conference (WCNC)_ , 2021.
* [10] M. Polese, L. Bonati, S. D’Oro, S. Basagni, and T. Melodia, “Understanding O-RAN: Architecture, Interfaces, Algorithms, Security, and Research Challenges,” _IEEE Communications Surveys & Tutorials_, pp. 1–1, 2023\.
* [11] E. Moro, M. Polese, I. Filippini, S. Basagni, A. Capone, and T. Melodia, “IABEST: An Integrated Access and Backhaul 5G Testbed for Large-Scale Experimentation,” in _Proceedings of the 28th Annual International Conference on Mobile Computing And Networking_ , ser. MobiCom ’22. Sydney, NSW, Australia: Association for Computing Machinery, 2022.
* [12] L. Bonati, M. Polese, S. D’Oro, S. Basagni, and T. Melodia, “OpenRAN Gym: AI/ML Development, Data Collection, and Testing for O-RAN on PAWR Platforms,” _Computer Networks_ , vol. 220, pp. 1–11, January 2023.
* [13] A. Gosain, “Platforms for Advanced Wireless Research: Helping Define a New Edge Computing Paradigm,” in _Proceedings of the 2018 on Technologies for the Wireless Edge Workshop_ , ser. WirelessEdge ’18. New Delhi, India: Association for Computing Machinery, 2018\.
* [14] L. Bonati, P. Johari, M. Polese, S. D’Oro, S. Mohanti, M. Tehrani-Moayyed, D. Villa, S. Shrivastava, C. Tassie, K. Yoder _et al._ , “Colosseum: Large-scale wireless experimentation through hardware-in-the-loop network emulation,” in _2021 IEEE International Symposium on Dynamic Spectrum Access Networks (DySPAN)_. IEEE, 2021.
* [15] S. Niknam, A. Roy, H. S. Dhillon, S. Singh, R. Banerji, J. H. Reed, N. Saxena, and S. Yoon, “Intelligent O-RAN for Beyond 5G and 6G Wireless Networks,” in _2022 IEEE Globecom Workshops (GC Wkshps)_ , 2022.
* [16] H. Lee, J. Cha, D. Kwon, M. Jeong, and I. Park, “Hosting AI/ML Workflows on O-RAN RIC Platform,” in _2020 IEEE Globecom Workshops (GC Wkshps)_ , 2020\.
* [17] B. Brik, K. Boutiba, and A. Ksentini, “Deep learning for B5G open radio access network: Evolution, survey, case studies, and challenges,” _IEEE Open Journal of the Communications Society_ , vol. 3, pp. 228–250, 2022\.
* [18] 3GPP, “NR; NR and NG-RAN Overall description; Stage-2,” 3rd Generation Partnership Project (3GPP), Technical Report (TR) 38.300, 09 2022, version 17.2.0.
* [19] ——, “NG-RAN; Architecture description,” 3rd Generation Partnership Project (3GPP), Technical Specification (TS) 38.401, 09 2022, version 17.2.0.
* [20] O.-R. WG3, “O-RAN Near-Real-time RAN Intelligent Controller E2 Service Model (E2SM) KPM 2.0,” Open-RAN Alliance, Technical Specification (TS), 7 2021, version 2.0.0.
* [21] F. Kaltenberger, A. P. Silva, A. Gosain, L. Wang, and T.-T. Nguyen, “OpenAirInterface: Democratizing innovation in the 5G Era,” _Computer Networks_ , vol. 176, p. 107284, 2020.
* [22] (2021) F1 interface - Data plane status. [Online]. Available: https://gitlab.eurecom.fr/oai/openairinterface5g/-/wikis/f1-interface
* [23] 3GPP, “Interface between the Control Plane and the User Plane Nodes,” 3rd Generation Partnership Project (3GPP), Technical Specification (TS) 29.144, 6 2022, version 16.4.0. [Online]. Available: https://portal.3gpp.org/desktopmodules/Specifications/SpecificationDetails.aspx?specificationId=3111
* [24] O-RAN Working Group 10, “O-RAN Operations and Maintenance Interface Specification 8.0,” O-RAN.WG10.O1-Interface.0-v08.00 Technical Specification, Oct 2022.
* [25] D. Villa, M. Tehrani-Moayyed, P. Johari, S. Basagni, and T. Melodia, “CaST: A Toolchain for Creating and Characterizing Realistic Wireless Network Emulation Scenarios,” in _Proceedings of the 16th ACM Workshop on Wireless Network Testbeds, Experimental Evaluation & Characterization_, ser. WiNTECH ’22. Sydney, NSW, Australia: Association for Computing Machinery, 2022.
* [26] G. Gemmi, R. Lo Cigno, and L. Maccari, “On Cost-Effective, Reliable Coverage for LoS Communications in Urban Areas,” _IEEE Transactions on Network and Service Management_ , vol. 19, no. 3, pp. 2767–2779, 2022.
* [27] G. Gemmi, R. L. Cigno, and L. Maccari, “On the Properties of Next Generation Wireless Backhaul,” _IEEE Transactions on Network Science and Engineering_ , vol. 10, no. 1, pp. 166–177, 2023.
* [28] 3GPP, “Study on channel model for frequencies from 0.5 to 100 GHz,” 3rd Generation Partnership Project (3GPP), Technical Specification (TS) 38.901, 01 2020, version 16.1.0. [Online]. Available: https://portal.3gpp.org/desktopmodules/Specifications/SpecificationDetails.aspx?specificationId=3173
* [29] M. Cudak, A. Ghosh, A. Ghosh, and J. Andrews, “Integrated Access and Backhaul: A Key Enabler for 5G Millimeter-Wave Deployments,” _IEEE Communications Magazine_ , vol. 59, no. 4, pp. 88–94, 2021.
* [30] S. Kutty and D. Sen, “Beamforming for millimeter wave communications: An inclusive survey,” _IEEE communications surveys & tutorials_, vol. 18, no. 2, pp. 949–973, 2015.
* [31] P. Fiore, E. Moro, I. Filippini, A. Capone, and D. D. Donno, “Boosting 5G mm-Wave IAB Reliability with Reconfigurable Intelligent Surfaces,” in _2022 IEEE Wireless Communications and Networking Conference (WCNC)_ , 2022\.
* [32] A. Bavelas, “Communication patterns in task-oriented groups,” _The journal of the acoustical society of America_ , vol. 22, no. 6, pp. 725–730, 1950\.
|
# Cluster analysis of homicide rates in the Brazilian state of Goiás from 2002
to 2014
Samuel Bruno da Silva Sousa Institute of Science and Technology
Federal University of São Paulo
São José dos Campos, SP - Brazil
<EMAIL_ADDRESS>Ronaldo de Castro Del-Fiaco Campus Henrique
Santillo
State University of Goiás
Anápolis, GO - Brazil
<EMAIL_ADDRESS>Lilian Berton Institute of Science and Technology
Federal University of São Paulo
São José dos Campos, SP - Brazil
<EMAIL_ADDRESS>
###### Abstract
Homicide mortality is a worldwide concern and has occupied the agenda of
researchers and public managers. In Brazil, homicide is the third leading
cause of death in the general population and the first in the 15-39 age group.
In South America, Brazil has the third highest homicide mortality, behind
Venezuela and Colombia. To measure the impacts of violence it is important to
assess health systems and criminal justice, as well as other areas. In this
paper, we analyze the spatial distribution of homicide mortality in the state
of Goiás, Center-West of Brazil, since the homicide rate increased from 24.5
per 100,000 in 2002 to 42.6 per 100,000 in 2014 in this location. Moreover,
this state had the fifth position of homicides in Brazil in 2014. We
considered socio-demographic variables for the state, performed analysis about
correlation and employed three clustering algorithms: K-means, Density-based
and Hierarchical. The results indicate the homicide rates are higher in cities
neighbors of large urban centers, although these cities have the best
socioeconomic indicators.
###### Index Terms:
Homicide, Violence, Spatial analysis, K-means clustering, Density-based
clustering, Hierarchical clustering
## I Introduction
Violence appears in the 21st century as one of the major social problems in
urban centers[15]. This phenomenon is noticed in several ways in different
spaces. It is not unusual to see news from all around the world reporting
violent acts every day. Both physical and psychological forms of violence may
change the relationships among people and the space they live[15]. We notice
violence in households, groups, communities, neighborhoods, cities, and so on.
Homicide is one of the most frightful forms of violence, and its occurrences
vary from a place to another.
In the most part of the world, the homicide rates are low, excluding war
zones. The homicide count in 2012[13] was around 437,000 victims in the world.
In 2017, the global homicide rate per 100,000 inhabitants was 6.2. 78% of the
victims were male, and the countries with the highest homicide rates were
Honduras (85.5), Venezuela (53.7), and Belize (44.7). On the other hand,
Monaco, Liechtenstein, and Andorra did not register any case of homicide which
made their rates per 100,000 inhabitants reach the value 0. Most of the
countries on the top of the list of homicide rates[13] were located in Latin
America or the Caribbean. These regions historically presented high homicide
rates, which is related to drug trafficking, corruption in governments, and
cartels operations[23].
In Brazil, the homicide rate per 100,000 inhabitants increased between 2002 to
2014. In 2002, the rate was 27.9 per 100,000, while in 2014 reached 29.7 per
100,000, according to data from Igarapé Institute[13]. The main reasons for
this increase are related to drug trafficking, corruption, and lack of
policing. Only in 2014, 59,681 people were murdered in Brazil. The country is
also responsible for more than 10% of the homicides in the world[23].
In some Brazilian states, the increase in homicide rates has been even higher.
For instance, in the state of Goiás, the homicide rate increased from 24.5 per
100,000 in 2002 to 42.6 per 100,000 in 2014, totalizing 24,300 victims in the
period[13]. The state of Goiás is located in the Center-West region of Brazil,
surrounding the Brazilian capital, Brasília. Its population was estimated at
around 6,7 million people in 2017[4], and its larger city is the capital,
Goiânia, with 1,4 million inhabitants. The state is formed by 246
municipalities whose social-economic features vary a lot from one to another.
Although the state concentrated the fifth highest homicide rate amongst the
Brazilian states in 2014, this theme is still under-researched.
Brazil covers 8.5 million $km^{2}$ (47% of South America), with an estimated
population of 190 million inhabitants in 2010[5]. The country is divided into
27 States with differing socioeconomic and health conditions. Recently
published studies in Brazil analyzed homicide and suicide mortality rates in
different states[24, 10, 11, 9, 20]. Despite the beliefs that homicides
cluster in specific regions[2], no papers using machine learning techniques
such as clustering algorithms to analyze homicide rates were found. Thus, this
paper introduces a new approach in this field.
Organizing data into groupings is an important mode of understanding and
learning. Cluster analysis can be defined as “the formal study of methods and
algorithms for grouping objects according to measured or perceived intrinsic
characteristics or similarity, without prior knowledge of the number of
clusters or any other information about their composition”[3]. Cluster
analysis does not use category labels that tag objects with prior identifiers,
such as class labels. The absence of category information distinguishes data
clustering (unsupervised learning) from classification (supervised
learning)[3].
The main contributions of this paper are: 1) evaluation of correlations
between homicide and socio-demographic variables for the state of Goiás,
Brazil; 2) employment of three clustering algorithms (K-means, Hierarchical
and Density-based clustering) to cluster the cities and to identify critical
areas for homicides; 3) pattern identification suggesting that, in Goiás, the
homicide rates are higher in cities neighbors of large urban centers, like
Brasília and Goiânia, despite these cities having the highest per capita
income in the country.
The rest of the paper is organized as follows: Section II presents related
work about homicide rate analysis in the world and in Brazil. Section III
presents the materials and methods employed in this work. The objectives,
datasets, and algorithms are described with details. Section IV presents the
results of our analysis. Finally, Section V presents the final remarks and
future works.
## II Related works
A plenty of works has been developed on homicide rate analysis since the last
half of the 20th century. The fields of sociology and criminology were the
first to start researching this theme. The main purposes of those works were
related to investigating whether demographic, economic, ecological, and social
variables maintained some correlations to the variation in homicide rates
across time and space[17]. Variables as resident racial segregation, racial
inequality, extreme poverty, social capital, and unemployment rate were used
for some well-succeeded findings[21].
In the United States of America, gunshot violence is responsible for about
34,000 deaths annually[8]. A paper published in 2017 used scan spatial
statistics to analyze clusters of the gunshot occurrences within the city of
Syracuse, New York. Amongst the results, it was noticed that the higher
violence rate was related to environmental and economic disparities[8].
In Central America, where the homicide rates are historically elevated, some
works have been developed on the analysis of possible causes. In El Salvador,
the clusters of homicides may be related to drug trafficking and organized
crime[7]. And in Mexico, the spatial variation of homicides was explained as
being linked to firearms possession, drug trafficking and social
exclusion[12].
Some papers analyzed mortality by homicide in Brazilian states. Souza et
al.[24] takes an ecological approach to the situation of homicides in all the
municipalities of Bahia for the male population aged 15 to 39, considering the
health macro-region. Lozada et al.[10] analyzed the homicide mortality trend
from 1979 to 2005 for males aged 15 to 49, living in the State of Paraná.
Souza et al.[11] made an ecological study which analyzed the violence-related
death records of women aged 10 years and older, in the Brazilian geographic
regions, between 1980 and 2014. Bando and Lester[9] evaluated correlations
between suicide, homicide and socio-demographic variables by an ecological
study for the Brazilian states. Peres et al.[20] described homicide mortality
in the municipality of São Paulo according to social-economic features of the
victims and type of weapon.
As far as we know no paper studied the homicide rates focused on the Goiás
state. Furthermore, we did not find any work using clustering algorithms to
analyze the spatial distribution of homicides. The techniques and tools
commonly used on this topic are: estimation of regression coefficients[17,
21]; spatial scan statistics[8]; SaTScan software methods[2]; Bayesian
approach with Monte Carlo Markov Chain algorithm[7]; Moran’s Global index[24];
descriptive statistics or correlation techniques[10, 9]; estimable functions
and negative binomial regression[11]; SSPS software tools[20]; and multiple
regression analysis (stepwise method)[12].
This paper aims to analyze spatial patterns of mortality by homicide for
better understanding violence in the Goiás state and to identify risky areas.
Due to the lack of use of clustering algorithms on this theme, this paper
proposes the use of K-means, Hierarchical and Density-based algorithms for
clustering, in an exploratory approach.
## III Materials and methods
This is an exploratory study of homicide mortality in the State of Goiás for
the period from 2002 to 2014. This period was chosen in function of data
availability. Goiás had an estimated population of 6,778,772 inhabitants in
2017 and has 246 municipalities, being recognized as the 12th most populated
state in Brazil[4]. The following sections present the objectives (III-A), the
methodology (III-B), the datasets used (III-C) and the evaluation measure
employed in the experiments (III-D).
### III-A Objectives
The general objective of this study is to analyze the spatial distribution of
mortality by homicide in the State of Goiás. The specific objectives can be
divided as follows:
* •
Collect data about mortality from homicide rates (MHR) in Goiás from 2002 to
2014.
* •
Collect demographic and socioeconomic characteristics of the various
territorial areas for the purpose of correlated with homicide rates.
* •
Apply cluster algorithms to group similar data, specifically, K-means,
Hierarchical and Density-based algorithms.
* •
Identify patterns in the clustering results to identify risky areas.
### III-B Methodology
The presence of clusters was evaluated by K-means, Hierarchical and Density-
based algorithms. The software used for building the database and for
computing the clusters was R.111https://www.r-project.org/
K-means clustering aims to partition $n$ data instances into K clusters in
which each instance belongs to the cluster with the nearest mean, serving as a
cluster’s prototype[16]. This is a very popular algorithm in data mining. It
proceeds by alternating between two steps, given an initial set of K centroids
$m_{1}$, $m_{2}$… $m_{n}$. The first step comprehends the assignment of each
instance to the cluster with the least squared Euclidean Distance. In the
second step, new means are calculated to be the centroids of the new clusters.
During the iterative process, it is common that some instances skip from one
cluster to another. The ending of the iterations occurs when the instances are
established and, at this time, the K-means are truly the means of the groups
formed[16].
Let $\mu_{k}$ be the centroid of cluster $c_{k}$. The squared error between
$\mu_{k}$ and the points in cluster $c_{k}$ is defined as in Equation 1.
$G(c_{k})=\sum_{\vec{x_{i}}\in c_{k}}||\vec{x_{i}}-\mu_{k}||^{2}$ (1)
The goal of K-means is to minimize the sum of the squared error overall K
clusters, as defined by Equation 2.
$G(C)=\sum_{k=1}^{K}\sum_{x_{i}\in c_{k}}||x_{i}-\mu_{k}||^{2}$ (2)
Automatically determining the number of clusters is a difficult problem in
data clustering. K-means is run independently for different values of K and
the partition that appears the most meaningful to the domain expert is
selected.
Hierarchical clustering is a method of cluster analysis which seeks to build a
hierarchy of clusters[22]. Its representation consists of a hierarchical
system (dendrogram) like the taxonomic hierarchy in biology. Strategies for
hierarchical clustering generally fall into two types:
* •
Agglomerative: This is a “bottom up” approach where each observation starts in
its own cluster and pairs of clusters are merged as one moves up the
hierarchy.
* •
Divisive: This is a “top-down” approach where all observations start in one
cluster, and splits are performed recursively as one moves down the hierarchy.
Formally, it is assumed that when there is a sequence of $n$ elements,
represented by values from 1 to $n$, there is also a sequence of clusterings
with length $m+1$ ($C_{0}$, $C_{1}$, $C_{2}$, …, $C_{m}$)[22]. Each cluster
has a number $\alpha_{i}$ with its value. The cluster $C_{0}$ is considered
the weakest clustering of $n$ elements ($\alpha_{0}$ = 0), and the cluster
$C_{m}$ is considered the strongest clustering. The numbers $\alpha_{i}$
increase ($\alpha_{i}$ $\leq$ $\alpha_{i+1}$) as well as the clusters $C_{i}$
($C_{i}$ $\leq$ $C_{i+1}$), which means each cluster $C_{i}$ is the merging
(or union) of clusters $C_{i-1}$[22]. In general, the merges and splits are
determined in a greedy manner. The results of hierarchical clustering are
usually presented in a dendrogram. The most common algorithms are Single-
linkage and Complete-linkage clustering.
Density-based clustering (DBSCAN) is one of the most common density-based
clustering algorithms. It was designed to discover clusters and noise in a
spatial dataset, requiring only one parameter for input whose value is
determined by the user[19]. This algorithm works as follows: given a set of
points in some space, it groups together points that are closely packed
together (points with many nearby neighbors), marking as outliers points that
lie alone in low-density regions (whose nearest neighbors are too far away).
The DBSCAN algorithm can be abstracted into the following steps:
* •
Find the $\epsilon$ neighbors of every point and identify the core points with
more than a predefined number of neighbors.
* •
Find the connected components of core points on the neighbor graph, ignoring
all non-core points.
* •
Assign each non-core point to a nearby cluster if the cluster is an $\epsilon$
neighbor, otherwise, signalize it as noise.
DBSCAN is an effective algorithm to discover clusters that present unusual
shapes even in large spatial databases[19].
### III-C Dataset
Data from 246 municipalities of the state of Goiás (Figure 1) were used as
analysis units.
Figure 1: Municipalities of the State of Goiás.
Source: http://www.sieg.go.gov.br/rgg/atlas/index.html/.
The data were collected between January and March of 2018 by downloads from
official websites of organizations maintained by the Brazilian government and
the United Nations. All data used in this analysis are publicly available for
download on the Internet. They are highly reliable once they are published by
governmental agencies and by internationally recognized organizations.
The data on demographics consist of the population estimate for each
municipality and its demographic density. They were extracted from the website
of the Brazilian Institute of Geography and Statistics (IBGE)[4]. The values
used in the query consisted of the municipalities’ names.
The data on mortality were extracted from the Mortality Information System
(SIM)[6]. This system is maintained and updated by the Brazilian Ministry of
Health with data from mortality in general. In the query, we aimed to extract
only the MHR, that included deaths by aggression, under the CID-10 codes X85
and Y09, and by legal intervention, under the codes Y35 and Y36. The selected
values in the query were: the area of influence of the State of Goiás, the
municipalities’ names, the period in the extraction was between the years 2002
and 2014, and the codes of CID-10 mentioned above.
The data about the Basic Education Development Index (IDEB)[14] were extracted
from the website of Instituto Nacional de Estudos e Pesquisas Educacionais
Anísio Teixeira (INEP), an agency from the Brazilian Ministry of Education.
This index measures the quality of basic education in Brazil, according to
international standards. The values for the index range from 0 to 10. In the
query, we selected to consult by: municipality, public (federal, state or
municipal) administrative dependency, series corresponding to the final years
of Secondary School (8th and 9th years in Brazil).
The data on social and economic features were collected from the website of
the United Nations Development Program (UNDP). In Brazil, UNDP develops an
atlas of development in the country. This atlas, named Human Development Atlas
in Brazil, covers all the Brazilian States and Federal District and its
municipalities. In this study, we collected the data that comprehended the
Municipal Human Development Index (MHDI)[25] and its variables, as life
expectancy, education index, the percentage of total income appropriated by
the 10% richest, and Gini index. These data were available to be downloaded in
a Microsoft Excel file.
Table I contains the name and description of all variables selected to compose
the dataset.
TABLE I: Description of the social-economic variables. Variable | Description
---|---
MHR | Total number of MHR since 2002 until 2014.
POPULATION | Population counting in IBGE 2010’s census.
DEMOGDENSITY | Municipality population by its total area.
IDEB2005 | Basic Education Development Index in 2005.
IDEB2007 | Basic Education Development Index in 2007.
IDEB2009 | Basic Education Development Index in 2009.
IDEB2011 | Basic Education Development Index in 2011.
IDEB2013 | Basic Education Development Index in 2013.
LIFEEXPECT | Life expectation in 2010.
GINI | Gini coefficient in 2010.
INRICHEST10 | Rate of the overall income held by richest 10%.
EDUCLEVEL | Education level of adult population in 2010.
MHDI | MHDI in 2010.
MHDIE | MHDI (Education) in 2010.
MHDIL | MHDI (Longevity) in 2010.
MHDII | MHDI (Income) in 2010.
For this analysis, the dataset was preprocessed and analyzed in R. We have
made use of Factoextra, Amap, FPC, Cluster, and DBSCAN packages.
### III-D Evaluation of results
The process of evaluating the results obtained from a clustering algorithm is
commonly called validation. There are two main types of grouping validation
indexes: i) external indexes, which compare the group structure discovered
with a previously known group structure; ii) internal indexes, which analyzes
the structure of groups discovered in relation to some criterion, such as, for
example, compactness or separability.
We employ the external index Silhouette, and the internal measures GAP
statistic and Sum of Squares Within (SSW) to validate the clustering results.
The Silhouette index (SIL) is calculated per data and the SIL of a group is
the average of the SIL of all the data in the group. And the clustering SIL is
the mean of the SIL of the groups. The higher the index value the better. It
is described by Equation 3.
$SIL=\frac{a_{i}-b_{i}}{max\\{a_{i},b_{i}\\}}$ (3)
where $a_{i}$ is the average distance from the data $i$ to all other data in
its group and $b_{i}$ is the minimum distance of the data $i$ to all other
data that do not belong to its group.
The GAP statistic compares the overall intra-cluster variation for a different
number of clusters with their expected values under a null reference data
distribution. The GAP statistic for a given value $k$ is defined as;
$GAP_{n}(k)=E^{*}_{n}log(W_{k})-log(w_{k}),$ (4)
where $E^{*}_{n}$ is an expectation in a sample of size $n$ from the reference
distribution, obtained by bootstrapping by computing the average $log(W_{k})$,
and by generating $B$ copies of the dataset. The GAP statistic measures the
deviation of the observed $W_{k}$ value from its expected value under a null
hypothesis.
SSW is a measure of compactness of a cluster structure, given by the distance
between the centroid of a cluster and the instances assigned to it. The SSW
equation has the form:
$SSW_{M}=\sum^{N}_{i=1}{||x_{i}-c_{Pi}||}^{2},$ (5)
where $x_{i}$ is an instance in each cluster whose centroid is $c_{Pi}$.
## IV Results and discussion
Section IV-A presents the analysis of the variables and their correlation with
homicide rate. Pearson’s, Spearman’s and Kendall’s indexes were employed to
measure correlation. Section IV-B presents the cluster analysis. Algorithms
Hierarchical, Density-based and K-means were employed to cluster the
municipalities. Section IV-C presents the discussion of the results.
### IV-A Social-economic variables analysis
From 2002 to 2014, 24,300 people were murdered in the State of Goiás[13]. The
number of MHR did not vary a lot between 2002 and 2007, but from 2007 to 2013
it increased year-to-year and reached the highest record in 2013 (Figure 2).
In 2014, it was noticed a slight decrease in the number of MHR, but this was
still the second highest record for the period.
Figure 2: MHR in the State of Goiás by year.
The Gini coefficient provides a measure of income distribution among
individuals or households within a territory, such as a country, a state, a
municipality, etc. A value 0 represents perfect equality, and a value 100
represents a total inequality. In our analysis, we used a normalized range of
Gini coefficient from 0 to 1, in the way it was presented on the data source.
Figure 3 presents the distribution of Gini coefficient among the
municipalities in the State of Goiás. Fewer than 20 municipalities have
presented values for Gini index under 0.4. Thus, it is possible to notice that
the State of Goiás presents high-income disparities among its municipalities
and within most of the municipalities too. For instance, the state’s largest
city, Goiânia, presented a Gini value of 0.58.
Figure 3: Distribution of Gini coefficient in Goiás (2010).
The educational variable EDUCLEVEL is related to the percent of the adult
population that has completed the last year of Secondary School. In most of
the municipalities, fewer than 60% of the adult population completed Secondary
School (Figure 4). This situation is due to the fact of the most of the
population lived in rural areas until the decade of 1970[18], when schools and
universities were concentrated in the largest cities. So the rural population
was aside from formal education.
Figure 4: Distribution of education level of adult population (2010).
The variable IDEB consists of the Basic Education Development Index. In this
study, we analyzed the values for IDEB in the last years of Secondary School
for public schools. The means for IDEB from 2003 to 2013, are shown in Figure
5. It is possible to notice that the values of IDEB were increasing through
the years, what means improvements in the quality of public basic education.
In 2013, the IDEB mean for the state reached 4.5. In this year, the
municipalities with the highest IDEB means were Perolândia (6.2), Córrego do
Ouro (5.9), and Ivolândia (5.9), that are all small cities. On the other hand,
the cities with the lowest IDEB means in 2013 were Águas Lindas de Goiás
(3.4), Água Fria de Goiás (3.5), and Cavalcante (3.6), where the first two are
located in the Surroundings of Federal District.
Figure 5: IDEB variation through the years.
In 2012 UNDP started calculating MHDI for all municipalities in Brazil. This
index is calculated the same way as HDI, which considers three dimensions
(income, education, and longevity). Despite having the seventh highest HDI in
Brazil (0.735) in 2010[4], the State of Goiás presents a large disparity in
the values of HDI among its municipalities (Figure 6). More than half of the
municipalities of Goiás have values for MHDI under 0.70, i.e., medium and low
human development.
Figure 6: Distribution of Municipal Human Development Index in Goiás (2010).
Figure 7: Regression lines between socio-economic variables and MHR (Color
online).
(a) Population
(b) Demographic density
(c) Life expectation
(d) IDEB index
(e) Educational level
(f) Gini index
(g) Income held by richest 10%
(h) MHDI
(i) MHDI (Education)
We also have calculated simple linear regression and local regression (LOWESS)
between the dataset variables. The results are shown in Figure 8(i). Simple
linear regression is a statistical approach to model a relationship in the
values of a dependent and one independent variable. This approach is commonly
used in forecasting, error reduction, and relationships explanation. LOWESS is
a non-parametric approach which makes use of a multiple linear method in a
model based in k-Nearest Neighbor, popularly employed in trends forecasting of
variables relationship.
The variables population (Sub-figure 7(a)) and demographic density (Sub-figure
7(b)) presented strong relationships with MHR values. In these sub-figures, it
is possible to notice that both simple regression lines (red lines) and LOWESS
lines (blue lines) assume rising values. Otherwise, the other variables are
shown in the Sub-figures 7(c) to 7(i) presented weak relationships with MHR
values. Both regression lines in these sub-figures assumed values equal or
close to 0. The lines for the variables MHDI (Longevity) and MHDI (Income)
also presented the values of simple and LOWESS regression close to 0.
Table II presents the Pearson’s, Spearman’s and Kendall’s coefficient
correlations between the social-economic variables considered and the MHR. The
values of the correlations are between -1 and +1. The signal indicates the
direction, whether the correlation is positive or negative, and the absolute
value of the variable indicates the strength of the correlation. Values
between 0.7 and 0.9 indicate a strong correlation. Notice that Population has
a strong correlation with MHR as indicated by the three coefficients.
Demographic Density also has a strong correlation, as indicated by the
Pearson’s coefficient.
TABLE II: Correlations between the total of MHR and social-economic variables. Variable | Pearson’s c. | Spearman’s c. | Kendall’s c.
---|---|---|---
POPULATION | 0.9915637 | 0.8932006 | 0.7385185
DEMOGDENSITY | 0.7262442 | 0.3446707 | 0.2389337
IDEB | -0.1440371 | -0.2019128 | -0.1417871
LIFEEXPECT | 0.09612802 | 0.1669409 | 0.1119476
GINI | 0.1134491 | 0.2435379 | 0.1748277
INRICHEST10 | 0.1039698 | 0.2740375 | 0.1939559
EDUCLEVEL | 0.3994967 | 0.4042891 | 0.287134
MHDI | 0.248962 | 0.1948997 | 0.133404
MHDIE | 0.206129 | 0.08632882 | 0.05902461
MHDIL | 0.096134 | 0.1656033 | 0.1120611
MHDII | 0.2649425 | 0.2717906 | 0.1852401
Before applying the cluster algorithms, we computed the distance between the
dataset columns (Figure 8), to visualize the (dis)similarity within the
dataset. It is possible to notice that values for demographic density,
population, and MHR are close to each other. At the same time, these variables
present values far from the educational, economic, and MHDI variables values,
showing to us that in the state of Goiás population growth was not accompanied
by educational and human development growth.
Figure 8: Dataset distance matrix (Color online).
### IV-B Cluster analysis
This section presents the results of applying the Hierarchical, Density-based
and K-means algorithms in the municipalities of Goiás. We validate the
clustering results with the Silhouette measure, GAP statistic, and SSW.
#### IV-B1 Hierarchical clustering
The hierarchical clustering algorithm, with single linkage method and
Manhattan distance, generated clusters with few municipalities. We have cut
the dendrogram into 3 partitions, which is shown in the Figure 9. The optimal
number of clusters to cut the tree was decided by the results of Silhouette
index and GAP statistic. We used the value which presented the biggest
difference to its successor, and the optimum GAP statistic is given by
$GAP_{(k)}>GAP_{(k+1)}-S_{k+1}$, being $S_{k+1}$ the standard deviation
multiplied by the square of 1 plus the inverse of B copies of the reference
dataset done by bootstrapping.
This algorithm assigned to the first cluster (in red) the largest city of the
state, Goiânia; assigned to the second cluster (in yellow) the city with the
highest demographic density, Valparaíso de Goiás; and assigned to the third
cluster (in purple) the other cities in the state, despite of the
dissimilarity between them.
Figure 9: Results of Agglomerative Hierarchical Algorithm (Color online).
#### IV-B2 Density-based clustering
The DBSCAN algorithm generates just one cluster and assigned as outliers 20
municipalities, amongst these outliers, there are: the largest cities of the
state, the municipality with the highest demographic density, and small cities
whose values for MHR are 0 or close to it. This algorithm did not generate a
meaningful result, because of the dissimilarity between the instances of the
dataset. Most of the municipalities assigned as outliers hold the highest
values for the variables population, demographic density and MHR. The results
of DBSCAN are shown in the Figure 10.
Figure 10: Results of DBSCAN (Color online). Figure 11: Results of K-means
clustering with K = 4 (Color online).
#### IV-B3 K-means clustering
We employed different K values (from 1 to 5) in the K-means clustering
algorithm. We also employed different distance measures, such as Euclidean,
Manhattan, Pearson, and Canberra. The one that provided the most meaningful
set of clusters was Canberra, once this measure is not sensible to outliers
presence. It was decided to start K with 5 because the state has 5 regions and
maybe this should influence the distribution of MHR. With K = 2, 3, and 5, the
results were not interesting. In every application, the algorithm assigned
Goiânia to the same cluster as small cities. The K = 4 value generated groups
more interesting as shown in Figure 11. The results signal different dynamics
in the distribution of MHR among the municipalities of Goiás. The results of
K-means algorithm are shown in Figure 11.
In cluster 1 (beige), there are 96 municipalities predominantly small cities.
The cluster presented the second lowest mean for the number of MHR and
variables, such as population, demographic density, life expectation, Gini
index, educational level of adult population, MHDI and its tree dimensions
(Education, Longevity, and Income). Otherwise, this cluster has the second
highest mean for income held by the richest 10% and the highest mean for IDEB
index amongst all the clusters.
In cluster 2 (pink), there are 108 municipalities, which are also small cities
close to the lowest means for population, demographic density, income held by
the richest 10%, educational level of adult population, MHDI and its three
dimensions and MHR amongst all the clusters. Furthermore, this cluster has the
second lowest IDEB and Gini index means. As the cities in this cluster are
predominantly small, far from the two metropolis in the region, the
educational level of the adult population tends to be low because, as cited in
Sec. IV-A, most of the state population was aside from formal education until
the decade of 1970.
In cluster 3 (black), there are 33 municipalities mid-sized like Rio Verde,
Cidade Ocidental, Catalão, Trindade, and Itumbiara; and some small cities.
This cluster presented the highest means of life expectation, Gini index,
income held by the richest 10%, MHDI, MHDI (Longevity), and MHDI (Income).
Furthermore, it was noticed in this cluster the second highest means for the
variables population, demographic density, educational level, MHDI
(Education), IDEB, and MHR.
The cluster 4 (purple) aggregates only 8 municipalities, all of them big or
mid-sized cities. In this cluster, there are the municipalities of Goiânia,
Aparecida de Goiânia, Anápolis, Valparaíso de Goiás, Águas Lindas de Goiás,
Senador Canedo, Novo Gama, and Luziânia. These cities compose the region
between the capital of the state of Goiás, Goiânia, and the capital of Brazil,
Brasília. The means of the population, demographic density, educational level,
MHDI (Education), and MHR were the highest amongst the 4 clusters. At the same
time, the cluster presented the second highest means for life expectation,
Gini index, MHDI, MHDI (Longevity), and MHDI (Income). The mean income held by
the richest 10% was the third highest. Although having the highest value for
the educational level of the adult population, the index which measures the
quality of basic education – IDEB – presented the lowest mean amongst all the
clusters. Furthermore, this cluster held 59.59% of the MHR in the state from
2002 until 2014.
In the Fig. 12, we plotted the results of K-means clustering. The small black
points represent the cluster 1 aforementioned. The yellow points represent the
cluster 2. The blue points represent the cluster 3, and the big red points
represent the cluster 4. It is possible to see that all the cities within the
fourth cluster are located close to the region largest cities, Goiânia and
Brasília, and Goiânia is also in this cluster.
Figure 12: Results of K-means clustering.
### IV-C Discussion
Applying clustering algorithms it was possible to identify clusters of
homicides in the state that are not merely random. Such clusters were
considered as critical areas for homicides. It was also possible to notice
different dynamics in the distribution of MHR in the state. The largest cities
presented most of the homicide cases, despite its income and human development
levels being higher than most of the other municipalities. On the other hand,
these large cities present low values for IDEB index, suggesting that although
having most people completing Elementary and Secondary School degrees, the
quality of teaching in public schools is still low.
We also computed Silhouette measure, GAP statistic, and SSW to validate the
clustering results (Table III). The values in bold represent the optimum
values for these measures. The optimum Silhouette value for each algorithm was
assumed as the value with the biggest difference to its successor in the
table. Optimum GAP statistic values were determined by selecting the lowest
which satisfied the condition: $GAP_{(k)}>GAP_{(k+1)}-S_{k+1}$. The optimal
values for SSW were assumed by analyzing the bend (knee) in the curves of SSW,
i.e., the SSW values in bold were the bends (knees) in their respective curves
of SSW.
TABLE III: Validation measures. Clustering algorithm | Silhouette | GAP | SSW
---|---|---|---
K-means (2) | 0.97 | 0.717 | 22.55
K-means (3) | 0.89 | 0.748 | 26.10
K-means (4) | 0.84 | 0.765 | 23.96
K-means (5) | 0.72 | 0.769 | 22.44
Hierarchical (2) | 0.97 | 0.722 | 50.00
Hierarchical (3) | 0.94 | 0.742 | 44.30
Hierarchical (4) | 0.86 | 0.731 | 40.00
Hierarchical (5) | 0.85 | 0.743 | 36.80
Density-based | 0.97 | 0.705 | 66.71
Measures such as Silhouette, GAP statistic, and SSW helps us to measure the
clustering quality under a mathematical perspective. As clustering is an
unsupervised task of data mining and Machine Learning, there is no label to be
predicted/classified, and no training and test phases, to lately measure the
error rate. So a meaningful clustering result is that one whose interpretation
makes sense for a human being [1].
Hierarchical and K-means clustering generates groups separating the large
cities from small ones. The Silhouette measure indicates high similarity
between the instances clustered. However, when the number of clusters was
increased, the Silhouette measure decreased. The value 4 for K in K-means
clustering was assumed as optimal because of GAP statistic value for K = 4 is
bigger than the value for K = 5, i.e., $GAP_{(k=4)}>GAP_{(k=5)}-S_{k=5}$.
K-means with K = 2 value achieves the highest Silhouette and the lowest SSW,
but this clustering generates one group with 147 small cities and another
group with 99 large cities. However, K = 4 gives a more interesting separation
splitting the outliers cities, and K = 5 presents a low value for Silhouette,
which suggests a weak clustering.
Density-based clustering resulted in just one cluster, and assigned as
outliers 20 municipalities, amongst them the state largest cities. The
Silhouette value for this results was also high, as well as the values for GAP
statistic and SSW.
Few studies bring information about all the municipalities from the State of
Goiás, especially in the interior regions. Our contribution emphasizes the
importance of the use of spatial analysis to understand violence indicators
and geographic distribution.
## V Conclusion
In this paper, we employed three clustering algorithms: Hierarchical, Density-
based and K-means to analyze spatial patterns of mortality by homicide in the
Brazilian state of Goiás.
The state’s three largest cities, Goiânia, Aparecida de Goiânia, and Anápolis,
had the highest numbers of MHR. The municipalities in the region known as
Surroundings of the Federal District also had elevated numbers of MHR. In the
other municipalities, the numbers were low. All clustering algorithms
separated these regions with high MHR from the others. However, K-means with K
= 4 results in a better separation.
The means for population, Gini index, MHDI, IDEB, income concentration, and
educational level of adult population varied among the municipalities as well
as the number of MHR. The highest IDEB values were found in the smallest
cities, which presented low homicide rates. In general, we noticed three
scenarios for the occurrence of MHR in the state of Goiás. The first is
characterized by a low occurrence of MHR in the small cities, as shown by the
clusters 1 and 2 in the Subsection IV-B. The second scenario refers to the
occurrence of MHR in mid-sized cities in municipalities with low demographic
density and high-income concentration, as shown by the cluster 3. The last
scenario is related to the occurrence of MHR in the state’s municipalities
with highest values of demographic density, low values of IDEB index, although
having high MHDI and income, and comprising the metropolitan areas of Goiânia
and Brasília, as noticed in the cluster 4.
The use of clustering algorithms in the analysis of the distribution of MHR
was an experimental approach, once we did not find studies that did it. In
general, the behavior of the algorithms was satisfactory. Hierarchical
clustering and DBSCAM provided results influenced principally by the
population size, although K-means, with K = 4, showed an interesting
separation whose quality was confirmed by its Silhouette measure, GAP
statistic, and SSW.
We expect this study to be useful to improve the knowledge about the
distribution of MHR in the State of Goiás and to aid the state’s government to
define risky regions, where policing should be more effective. This study is
also a source to know the variation of social-economic variables among the
state’s municipalities.
As future work other analysis employing different variables to explain the
rates of homicides in certain regions could be done, like precarious
socioeconomic conditions, social inequality, etc.
## VI Acknolowdgments
This work was conducted during a scholarship at ICT-UNIFESP – Institute of
Science and Technology of Federal University of São Paulo. Financed by CAPES –
Brazilian Federal Agency for Support and Evaluation of Graduate Education
within the Ministry of Education of Brazil.
## References
* [1] A. C. Lorena, J. Gama, K. Facelli, and A. C. P. L. F. de Carvalho, “Inteligência Artificial: uma abordagem de aprendizado de máquina,” Grupo Gen-LTC, 2011.
* [2] A. M. Zeoli, S. Grady, J.M. Pizarro, and C. Melde, “Modeling the movement of homicide by type to inform public health prevention efforts,” American journal of public health, Vol. 105, No. 10, pp. 2035-2041, 2015.
* [3] A. K. Jain, “Data clustering: 50 years beyond K-means,” Pattern recognition letters, Vol. 31, No. 8, pp. 651-666, 2010.
* [4] Brazilian Institute of Geography and Statistics. (2018, Mar.) Goiás - IBGE Cidade. [Online]. Available: https://cidades.ibge.gov.br/brasil/go.
* [5] Brazilian Institute of Geography and Statistics. (2018, Mar.) IBGE Censo 2010. [Online]. Available: https://censo2010.ibge.gov.br/sinopse/index.php?dados=4&uf=00.
* [6] Brazilian Ministry of Health. (2018, Mar.) TabbNet Win32 3.0: Mortalidade - Goiás. [Online]. Available: http://tabnet.datasus.gov.br/cgi/deftohtm.exe?sim/cnv/obt10GO.def.
* [7] C. Carcach, “A spatio-temporal model of homicide in El Salvador,” Crime science, Vol. 4, No. 1, pp. 20, 2015.
* [8] D. A. Larsen, S. Lane, T. Jennings-Bey, A. Haygood-El, K. Brundage, and R. A. Rubinstein, “Spatio-temporal patterns of gun violence in Syracuse, New York 2009-2015,” PLoS ONE, Vol. 12, No. 3, pp. e0173001, 2017.
* [9] D. H. Bando, and D. Lester, “An ecological study on suicide and homicide in Brazil,” Ciência & saúde coletiva, Vol. 19, pp. 1179-1189, 2014.
* [10] E. M. K Lozada, T. A. D. F. Mathias, S. M. D. Andrade, and T. Aidar, “Homicide mortality trend in the state of Paraná, Brazil, per Health District, 1979 to 2005,” Revista Brasileira de Epidemiologia, Vol. 12, No. 2, pp. 258-269, 2009.
* [11] E. R. de Souza, et al., “Homicides among women in the different Brazilian regions in the last 35 years: an analysis of age-period-birth cohort effects,” Ciência & Saúde Coletiva, Vol. 22, No. 9, pp. 2949-2962, 2017.
* [12] G. J. Gonzalez-Perez, M. G. Vega-lopez, C. E. Cabrera-pivaral, and A. Vega-Lopez, “P1-432 Violence and health: an epidemiological analysis of homicides in México, 1979–2008,” Journal of Epidemiology & Community Health, Vol. 65, No. 1, 2011.
* [13] Igarapé Institute. (2016, Mar.) Homicide Monitor. [Online]. Available: http://homicide.igarape.org.br.
* [14] Instituto Nacional de Estudos e Pesquisas Educacionais Anísio Teixeira. (2016, Mar.) IDEB - Resultados e Metas. [Online]. Available: http://ideb.inep.gov.br/resultado/home.seam?cid=2098801.
* [15] J. Laplante, “La violence, la peur et le crime,” University of Ottawa Press, 2001.
* [16] J. MacQueen, “Some methods for classification and analysis of multivariate observations,” Proceedings of the fifth Berkeley symposium on mathematical statistics and probability, Vol. 1, No. 14, 1967.
* [17] K. C. Land, P. L. McCall, and L. E. Cohen, “Structural covariates of homicide rates: Are there any invariances across time and social space?,” American journal of sociology, Vol. 95, No. 4, pp. 922-963, 1990.
* [18] Mauro Borges Institute. (2018, Mar.) Tabela 3 - Estado de Goiás, Centro-Oeste e Brasil: População por situação de domicílio – 1950, 1960, 1970, 1980, 1991, 2000, 2008 – 12. [Online]. Available: http://www.imb.go.gov.br/pub/Godados/2013/01-tab03.htm.
* [19] M. Ester, H. P. Kriegel, J. Sander, and X. Xu, “A density-based algorithm for discovering clusters in large spatial databases with noise,” Kdd. Vol. 96, No. 34, 1996.
* [20] M. F. T. Peres, D. Vicentin, M. B. Nery, et al., “Queda dos homicídios em São Paulo, Brasil: uma análise descritiva,” Revista Panamericana De Salud Publica, Vol. 29, No. 1, pp. 17-26, 2011.
* [21] P. L. McCall, K. C. Land, and K. F. Parker, “An empirical assessment of what we know about structural covariates of homicide rates: a return to a classic 20 years later,” Homicide Studies, Vol. 14, No. 3, pp. 219-243, 2010.
* [22] S. C. Johnson, “Hierarchical clustering schemes,” Psychometrika, Vol. 32, No. 3, pp. 241-254, 1967.
* [23] The Economist. (2016, Mar.) Daily chart: Revisiting the world’s most violent cities. [Online]. Available: https://www.economist.com/blogs/graphicdetail/2016/03/daily-chart-18.
* [24] T. O. de Souza, L. W. Pinto, and E. R. de Souza, “Spatial study of homicide rates in the state of Bahia, Brazil, 1996-2010,” Revista de saúde pública, Vol. 48, pp. 468-477, 2014.
* [25] United Nations Development Program. (2016, Mar.) Atlas do desenvolvimento humano no Brasil 2013. [Online]. Available: http://www.atlasbrasil.org.br/.
|
# All A-board: Sharing Educational Data Science Research With School Districts
Nabeel Gillani Northeastern UniversityBoston, MAUSA
<EMAIL_ADDRESS>, Doug Beeferman Massachusetts Institute of
TechnologyCambridge, MAUSA<EMAIL_ADDRESS>, Cassandra Overney Massachusetts
Institute of TechnologyCambridge, MAUSA<EMAIL_ADDRESS>, Christine Vega-
Pourheydarian Wellesley CollegeWellesley, MAUSA<EMAIL_ADDRESS>and
Deb Roy Massachusetts Institute of TechnologyCambridge, MAUSA
<EMAIL_ADDRESS>
(2023)
###### Abstract.
Educational data scientists often conduct research with the hopes of
translating findings into lasting change through policy, civil society, or
other channels. However, the bridge from research to practice can be fraught
with sociopolitical frictions that impede, or altogether block, such
translations—especially when they are contentious or otherwise difficult to
achieve. Focusing on one entrenched educational equity issue in US public
schools—racial and ethnic segregation—we conduct randomized email outreach
experiments and surveys to explore how local school districts respond to
algorithmically-generated school catchment areas (“attendance boundaries”)
designed to foster more diverse and integrated schools. Cold email outreach to
approximately 4,320 elected school board members across over 800 school
districts informing them of potential boundary changes reveals a large average
open rate of nearly 40%, but a relatively small click-through rate of 2.5% to
an interactive dashboard depicting such changes. Board members, however,
appear responsive to different messaging techniques—particularly those that
dovetail issues of racial and ethnic diversity with other top-of-mind issues
(like school capacity planning). On the other hand, media coverage of the
research drives more dashboard engagement, especially in more segregated
districts. A small but rich set of survey responses from school board and
community members across several districts identify data and operational
bottlenecks to implementing boundary changes to foster more diverse schools,
but also share affirmative comments on the potential viability of such
changes. Together, our findings may support educational data scientists in
more effectively disseminating research that aims to bridge educational
inequalities through systems-level change.
Education; Inequality; Segregation; Randomized Experiments
††journalyear: 2023††copyright: acmlicensed††conference: Proceedings of the
Tenth ACM Conference on Learning @ Scale; July 20–22, 2023; Copenhagen,
Denmark.††booktitle: Proceedings of the Tenth ACM Conference on Learning @
Scale (L@S ’23), July 20–22, 2023, Copenhagen, Denmark††price: 15.00††isbn:
979-8-4007-0025-5/23/07††doi: 10.1145/3573051.3593377††ccs: Applied computing
Education††ccs: Applied computing Sociology††ccs: Applied computing Decision
analysis††ccs: Applied computing Psychology
## 1\. Introduction
There is perhaps no place in America where failures of pluralism and democracy
are manifesting more clearly than in public schools. In recent years, US
school districts have become battlegrounds for political debates about in-
person schooling (Sawchuck, 2021), mask mandates (Harris, 2022), and critical
race theory, along with discussions about gender and sexuality (Natanson,
2021). Despite this climate, public schools have the capacity to help
cultivate common ground, more evenly distribute social capital, spark exposure
to and appreciation for divergent viewpoints, and foster other aspects of a
healthy, pluralistic society by bringing together students and families from
different backgrounds.
Yet across the US, school segregation along racial, cultural, and
socioeconomic lines remains rampant (Office, 2022) and continues to adversely
affect the academic performance and attainment of lower socioeconomic status
(SES) students and students of color (Reardon et al., 2018; Owens, 2018;
Reardon and Owens, 2014; Mayer, 2014). One reason for this is that segregation
may exacerbate inequities in how resources like experienced teachers and
advanced course offerings are distributed across schools (EdBuild, 2020;
Frankenberg et al., 2003; Orfield and Lee, 2005). Segregated schools may also
impede access to social networks, which may in turn limit access to the kinds
of “bridging social capital” that can help them access new resources, jobs,
and other quality-of-life-enhancing opportunities (Bell et al., 2017; Small,
2009; Burton and Welsh, 2015; Chetty et al., 2022). There is evidence that
integration can reduce inequalities in academic outcomes (Yeung, 2022; Quick,
2016; Johnson, 2011), especially when academic environments are re-designed to
be inclusive of an increasingly diverse student body (Burton and Welsh, 2015;
Bridges, 2016).
Despite such evidence, actual socioeconomic integration remains elusive across
many districts, largely because of racialized schooling preferences among
predominantly White and affluent families (Billingham and Hunt, 2016; Hailey,
2021; Hall and Hibel, 2017) and residential/school selection based on factors
related to income (Reardon and Bischoff, 2011; Hasan and Kumar, 2019). The
correlation between place of residence and place of schooling is reinforced by
“school attendance boundaries”, or school catchment areas that local education
agencies (“school districts”) draw to determine which neighborhoods are
assigned to (“zoned for”) which schools. Despite the growing popularity of
school choice policies, neighborhood-based assignment still accounts for the
vast majority of student assignment decisions across the US (of Education,
2021). Furthermore, choice policies can also exacerbate patterns of racial and
ethnic segregation (Monarrez et al., 2022; Whitehurt, 2017; Candipan,
2019)—despite how they, at least in theory, place of residence and schools.
Many choice systems also factor in geography in some form or fashion—for
example, through the priorities they assign students (Monarrez and Chien,
2021) or how they create zones that constrain choice sets for families (Campos
and Kearns, 2022). Boundaries, therefore, matter, and play an important role
in determining which children go to which schools and what the quality of
their ensuing educational experiences are.
Educational data scientists—an emerging class of researchers and practitioners
wielding computational methods to explore outstanding issues in education
(McFarland et al., 2021)—have expended great effort exploring how such methods
can improve online learning (Kizilcec et al., 2013; Gillani and Eynon, 2014),
personalized and adaptive tutoring (Ritter et al., 2007; Piech et al., 2015),
and other learner and teacher-centric practices. There appears to be fewer
studies, however, investigating how such methods might help address challenges
that are “upstream” to these: namely, social, organizational and contextual
barriers that might impede access to quality-of-life enhancing educational
opportunities. Attendance boundaries that exacerbate segregation often act as
such barriers. Yet they are malleable: school districts could, in theory,
redraw attendance boundaries in ways that seek to minimize racial and ethnic
segregation across schools. Doing so, however, is a politically challenging
task, and often runs the risk of increasing travel times or producing other
unpopular outcomes that face opposition from parents and other stakeholders
(Delmont, 2016; McMillan, 2018). It is also technically challenging: manual
map redrawing can be highly time-consuming given the large space of possible
configurations, and from the political redistricting literature, it is clear
that automated methods are challenging to develop and scale in their own right
(Gurnee and Shmoys, 2021; Becker and Solomon, 2020). There is an opportunity,
then, for educational data scientists to explore how computational methods
might help education policymakers—a stakeholder group not usually directly
targeted as an audience for pedagogy-focused research—foster more diverse and
integrated schools. This is, arguably, especially relevant for educational
data scientists interested in “Learning at Scale”, since the school assignment
policies that districts design and implement continue to shape the learning
experiences of millions of children across the US.
In this vein, a recent study developed computational redistricting algorithms
to explore how much school districts across the US might reduce racial and
ethnic segregation in elementary schools through boundary changes (Gillani et
al., 2023). The results, surprisingly, revealed the possibility of boundary
changes that might reduce segregation while also slightly reducing travel
times. These findings motivate this particular study: a follow-on effort to
explore to what extent US school districts might engage with the findings from
this research and take them into consideration as they conduct their own
boundary planning efforts. To investigate this, we adopt a multi-pronged
outreach strategy, combining cold email outreach to 4,320 elected school board
members across 803 districts with media coverage in order to invite leaders
and members of school districts to explore an online dashboard depicting
algorithmically-generated, diversity-promoting elementary school attendance
boundaries. Our focus on outreach seeks to address another gap, not just in
educational data science research, but academic research more broadly: better
understanding the conditions under which academic findings might be translated
into practice. The emerging field of “implementation science” is increasingly
exploring this question (Bauer et al., 2015), and recent studies in the
education domain have used school district outreach as a channel (Delevoye,
2021; Nakajima, 2021) for investigating how district leaders interpret and put
policy-relevant research to use. We aim to build on these bodies of work by
asking the following questions:
1. (1)
Which messaging and outreach strategies are more or less effective in
prompting school district leaders and community members in exploring potential
boundary changes that could help foster more racially/ethnically integrated
schools?
2. (2)
How do school district leaders and community members perceive these boundary
changes—namely, what potential merits and pitfalls do they see in both how
these boundaries were produced and their potential viability in practice?
We find that email open and click rates are highest when campaign content is
framed in terms of existing district objectives—like school capacity planning
and family satisfaction—which were suggested by district leaders through our
informal conversations with them. Furthermore, media coverage appears to be a
much more effective channel than email outreach in engaging audiences around
potential boundary changes in their school districts—especially districts that
exhibit higher levels of White/non-White segregation. Through an optional
survey available on the data dashboard, we obtain a more nuanced view of how
communities perceive such hypothetical, algorithmically-informed attendance
policy changes: many participants highlight aspects of their unique contexts
that would need to be considered in order for the research findings to be
adapted and eventually implemented in practice, like rapid shifts in
population due to immigration; school capacity constraints; and other factors.
Nevertheless, most respondents indicate more interest in the topic of
attendance boundary changes to foster diverse schools than they had prior to
learning about the study, and several express an interest in learning more
about how to implement some of the proposed changes in practice. While our
study focuses on the issue of changing attendance boundaries to foster more
racial and ethnic integration, we believe our findings may be relevant more
broadly to educational data scientists who wish to explore how findings from
research can be effectively communicated with key education stakeholders, and
eventually translated into effective socio-technical artifacts and practices
in learning settings.
## 2\. Related work
Our work builds upon work in school assignment planning for measuring and
promoting more diverse schools. It also builds upon recent efforts exploring
how to engage school district leaders and other policymakers to help them use
findings from academic research in their work. Most broadly, it is motivated
by and rooted in the emerging discipline of “implementation science” (Bauer et
al., 2015), which seeks to more precisely identify the contexts and conditions
under which research findings can be scaled and generalized to populations
beyond those included in the research. We briefly review these areas below.
### 2.1. School assignment for diversity
Across school districts, the lines between districts have been shown to
perpetuate racial and ethnic segregation across schools even more than those
within them (i.e., attendance boundaries) (Fiel, 2013). Unfortunately,
changing school district boundaries require collaboration across multiple
districts, and/or state-level intervention—both of which are subject to many
of the political frictions and challenges that accompany policy changes
requiring coordination across multiple governing bodies. Changing attendance
boundaries, on the other hand, generally falls under the purview of school
district leaders. Yet within-district boundary changes continue to be highly
contentious: parents may worry about impacts on values (Kane et al., 2005;
Bridges, 2016; Black, 1999), commute times (Frankenberg and Jacobsen, 2011;
Delmont, 2016), community cohesion (Bridges, 2016; Staff, 2019), and other
factors. These concerns can block boundary changes (Castro et al., 2022)
and/or spark families to move to other schools and districts (Reber, 2005;
Macartney and Singleton, 2018).
The present study does not explore how to address these parent/ community-
level factors, which are critical topics for future research on this issue.
Instead, it explores to what extent school district leaders are willing even
to engage with research on this contentious topic that suggests there might be
potentially palatable solutions within reach—and what their reactions to such
findings are. Given the positions of power that board members occupy, we
believe this is important to investigate, as identifying effective ways of
discussing the topic of boundary changes for diversity with district leaders
may illuminate key leverage points for supporting systemic policy change.
### 2.2. Engaging education policymakers in research findings
While still a relatively nascent area of inquiry, recent studies have started
to explore how to design and execute effective communications with school
districts to foster more engagement with, and policy decision-making based on,
evidence-based research. Two recent studies in particular come to mind. The
first conducted randomized outreach to over 2,000 education professionals
working across both state and local education agencies to assess both the
preferences these policymakers have for research evidence, and the extent to
which they update their beliefs about the effectiveness of various education
policies as they are presented with new evidence (Nakajima, 2021). The study
found that policymakers do indeed value research evidence, particularly when
they contain findings from multiple sites, large samples, and contexts similar
to their own—though they do not appear to prefer experimental studies over
observational ones. The study also found that policymakers are more likely to
update their beliefs about the effectiveness of education policies when
research evidence is presented with clear, accessible descriptions of its
research design. Together, these results offer valuable insights into how
clarity of communication and local contextualization of findings might help
increase policymakers’ engagement with education research.
A second study, conducted during the COVID-19 pandemic, explored the potential
agenda-setting role of cold email outreach to school district board members
and staff (Delevoye, 2021). The authors randomly assigned a subset of nearly
600 school districts leaders to receive an email containing a link to a memo
with research on emerging best practices around transportation in education
settings during the pandemic. They then analyzed video recordings and other
outputs from school board meetings to analyze whether those district leaders
who received emails were more likely to discuss transportation-related issues.
While the study did not report a positive treatment effect, it did find that
some of the words from the memo made it into some of the school boards’
discourse. It is possible that with a larger sample of districts, there may
have been more of an agenda-setting impact of such research-based outreach.
Our study builds on these efforts by exploring the types of cold email
outreach that are more or less effective at garnering engagement from school
district leaders vis-a-vis a particularly contentious and systemic issue: how
school attendance boundaries are drawn, and the impact these boundaries have
on issues of school segregation and diversity in their districts.
### 2.3. Implementation science
Yet another emerging area of inquiry is implementation science, or “the
scientific study of methods to promote the systematic uptake of research
findings and other evidence-based practices into routine practice …” (Bauer et
al., 2015). For example, development economists are increasingly exploring
implementation science through the lens of “scaling”, or investigating how
promising interventions conducted in relatively sand-boxed settings might be
effectively scaled across different contexts and environments (Mobarak, 2022).
Scaling successfully, of course, requires anticipating the complexities that
may come with scale—which, crucially, include potentially unanticipated or
indirect consequences (Mobarak, 2022).
Health researchers are also interested in implementation science; indeed, the
health domain has been a driver of this new discipline, perhaps because
evidence-based practices in healthcare take, on average, 17 years to be
implemented into routine general practice (Bauer et al., 2015). One
interesting emerging trend from this domain is the design of “hybrid research
designs”, or research projects that seek to both further understanding about
evidence-based practices (e.g., how well some practice improves a health
outcome—generally, the type of finding one might typically expect to uncover
by running a randomized control trial), as well as a
documentation/investigation of which practices and policies are most effective
in enabling such interventions to have this intended impact (Bauer et al.,
2015). Such hybrid designs offer the hope of both advancing basic scientific
inquiry as well as theories and knowledge about how the outputs of such
inquiry can actually serve people in different real-world contexts.
Many of these ideas have started to take hold in education research (Moir,
2018), as partly demonstrated by the district outreach studies in the prior
subsection. They are also beginning to spark new field-building and funding
priorities in education research (Foundation, 2022). For an issue as complex
as school assignment planning, particularly when intersected with a topic as
controversial as racial and ethnic segregation in schools, modeling
hypothetical boundary changes is just one small part—and perhaps the easiest
part—of investigating how to foster more diverse and integrated schools.
Deeply understanding the sociopolitical contexts of these districts, how to
capture the attention and interest of district leaders and community members,
and eliciting the conditions under which alternative boundaries may or may not
be viable are all critical pieces of translating such research into practice.
On their own, these activities do not actually identify conditions for
successful translations in the field, but rather, help identify potential
starting points for iterating on existing research to eventually help make it
more useful in practice. Our study, then, may offer helpful insights to
educational data scientists, public policy researchers, and other
practitioners interested in learning more about how to begin translating
potentially controversial, equity-promoting research findings into practice in
complex settings like school districts.
## 3\. Methods
### 3.1. Modeling and data dashboard
We begin with the datasets and rezoning algorithms used in (Gillani et al.,
2023). This preliminary study focused on elementary school segregation,
combining 2019/2020 student enrollment data by race and ethnicity across US
schools with 2020 Census demographics and 2021/2022 school attendance
boundaries to estimate counts of students living in each Census block and
attending each zoned elementary school (see (Gillani et al., 2023) for a more
thorough discussion of the datasets and some of the challenges involved in
making these estimations). It then developed and presented a redistricting
algorithm, building on top of open source tools from the operations research
community (OR-Tools, 2022; Perron and Didier, 2020), that produced
hypothetical attendance boundary redrawings to help foster more racially and
ethnically integrated schools. The redrawing problem was framed as an
assignment problem, namely, trying to find an alternative assignment of Census
blocks to schools that would minimize White/non-White segregation (as defined
by the dissimilarity index (Massey and Denton, 1988)), subject to constraints
around the maximum amount any family’s travel time might increase under the
new rezoning, as well as a cap on how much school capacities might increase.
Such redistricting problems are NP-hard and notoriously difficult to solve
given the large space of possible assignments; nevertheless, recent advances
in the operations research community, e.g. through innovations in methods like
constraint programming (Van Hentenryck, 1989; Perron and Didier, 2020), have
created powerful off-the-shelf solvers that can often produce high-quality
solutions to such problems.
(Gillani et al., 2023) simulated changes across 98 large school districts and
reported a median relative decrease in segregation of 12%, which would require
nearly 20% of students to switch schools. Perhaps most surprisingly, the study
showed that such reductions may be achieved with slight average decreases in
travel times—suggesting there may be modest changes that produce modest gains
in integration without requiring large travel-related sacrifices by families.
Using the datasets and algorithms from this study, we began by re-running the
authors’ redistricting algorithms for a larger set of over 4,000 school
districts—namely, those school districts that have at least two elementary
schools (and hence, different neighborhoods within the district zoned to
different schools). In total, we produced 4 different configurations across
each district by varying different parameter values for the aforementioned
travel time and school size constraints. Simulations were run on a university
computing cluster with a cutoff time of 5.5 hours each. We then produced a
dashboard similar to the one in (Gillani et al., 2023) to enable district
leaders and community members to explore hypothetical rezonings. The dashboard
was built using the Python-based Streamlit111https://streamlit.io/., a
lightweight framework for building and deploying data dashboards. Using
Streamlit limited some of our data tracking, logging, and customization
possibilities; on the other hand, it enabled us to rapidly prototype and
deploy a dashboard without requiring full-stack design and engineering.
Figure 1 shows screenshots of the dashboard, which served as the main
exploration platform for this study. To make the description of the project’s
aims and key findings as accessible as possible on the dashboard, we
informally sought feedback from a past school district superintendent involved
in boundary planning for integration; a current (at the time) school
assignment and planning director; and a data journalist who frequently writes
about issues relating to educational equity. We conducted Zoom calls with
these individuals, inviting them to explore the dashboard and share their
observations and questions (similar to a think-aloud protocol (Ericsson and
Simon, 1984)). Their feedback on clarity of wording (how we describe changes
in school-level demographics—and avoid esoteric segregation formulas in order
to make research findings more accessible (Nakajima, 2021)), framing of
diversity (making sure to not lose the focus on within-classroom learning
experiences) and data visualizations was invaluable and led us to make several
changes that we believe made the dashboard more accessible to a general
audience.
Figure 1. Key screenshots of rezoning dashboard, which can be found at
https://www.schooldiversity.org/: a) shows the view “above the fold” when
users land on the page and includes a description of why attendance boundaries
matter and what the objective of the dashboard is. It also provides dropdown
boxes for users to explore hypothetical boundary changes for different school
districts, and includes a verbal description of what the depicted changes for
any given district would mean in terms of changes in diversity across its
constituent elementary schools. b) shows a map view, where users can toggle
between boundaries proposed by the redistricting algorithm and current
“status-quo” boundaries. Further down the page and hidden by default is c),
which allows users to drill into specific schools to understand how their
demographics would change under the depicted rezoning—including cases where
rezonings move racial/ethnic proportions at some schools away from district-
levels instead of towards them, perhaps to achieve more balance at other
schools. Finally, further down is d), which shows how travel times would
change (on average) for different racial/ethnic groups under the rezoning.
### 3.2. Email campaign
Since school board members across many districts are elected officials, their
names and contact information are available through public ballot databases.
We collaborate with the XQ Institute222https://xqsuperschool.org/. to obtain
this information for over 115k elected officials across 13,233 school
districts—which constitute nearly all regular school districts across the
US333https://nces.ed.gov/programs/digest/d20/tables/dt20_214.10.asp.. Next,
via the elementary school attendance boundaries and school enrollment data
used in (Gillani et al., 2023), we identify the 1,000 school districts with
the largest elementary school enrollment that are also listed as having closed
enrollment (i.e., entirely boundary-assigned) elementary schools as our main
sample for outreach. Merging the district board member data onto this dataset
yields approximately 6,664 board members, 4,831 of which we have a valid email
address for. We then design and send emails using the email marketing platform
Mailchimp. Emails contain links unique to each recipient, and clicking the
link takes the recipient to a preliminary page on the dashboard that informs
them of the research study. The recipient is then invited to click through to
the main dashboard, where they are shown the dashboard depicted in Figure 1
with data specific to their district pre-selected.
As described in the Introduction by Research Question 1, we are interested in
better understanding which messaging techniques are more or less effective in
prompting school district leaders (board members) to explore and share
feedback on potential attendance boundary changes for school diversity. One
way to do this is to randomly assign board members to receive different
emails—e.g., emails with different subject lines—and then observe which ones
generate more engagement and eventual traffic to the data dashboard (we use
“email” and “subject line” below interchangeably, since email previews and
bodies were generally similar across runs and only adapted to reflect their
corresponding subject lines).
Randomly assigning board members within the same district to different email
conditions, however, runs the risk of treatment “spillover”—for example, if
one board member forwards an email to one of their colleagues—which would
limit our ability to identify the causal effect of any particular email on
engagement (Gerber and Green, 2012). To try and mitigate against this, we
design a series of cluster-randomized control trials, where we randomly assign
email conditions at the cluster (school district) level. There is still the
possibility that school board members across districts who are assigned to
different conditions forward received emails to one another, but we expect
this to be small, given their specific contexts (boundaries) are likely to be
different. To account for the fact that board members might respond
differently depending on the extent to which demographic imbalances are
actually an issue in their particular districts, we stratify districts into
quintiles according to their White/non-White dissimilarity indices, and
randomly assign those within each quintile to each possible email condition.
This increases the likelihood that districts receiving any of the email
conditions are well-balanced across the levels of segregation manifesting
across their schools.
We conduct these cluster randomized control trials in eight successive waves.
At the start of each wave, we begin with a set of hypotheses for which types
of subject lines would be most effective in driving open and click-through
rates. After each wave, we analyze engagement data (opens and click-throughs
to the dashboard) for each email. The next wave then contains emails informed
by well-performing ones in prior waves, plus those evaluating new hypotheses.
We also generate new hypotheses—based on observations and informal interviews
conducted with school board members who engaged in previous waves—to inform
the development of new subject lines. In the results section, we discuss some
of these interviews and the insights that emerged from them. Table 1 contains
some of our hypotheses and corresponding subject lines, with a full list
deferred to Appendix A. In total, we explored 14 subject lines across 17
campaigns, which were sent through 8 iterative waves. The median number of
recipients across campaigns was 205. Our final email included a subject line
that included aspects of well-performing emails in prior waves; we sent this
to 1,383 participants that remained in our total sample after running all
previous subject line experiments.
We note that due to the exploratory and iterative nature of this study, we did
not formally pre-register it in an Open Science repository. We acknowledge
this as a limitation of this work, as even exploratory studies and sequential
experimentation can be accommodated in existing pre-registration frameworks
(Nosek et al., 2018), and intend to rectify this in future studies of this
type.
Email subject line | Hypothesis
---|---
$20 for your views on diversifying attendance boundaries | A small incentive that board members can allocate to a teacher of their choice may increase engagement
Data science to improve learning experiences | Using a popular term (“data science”) and focusing on learning instead of diversity may increase engagement
More diversity with shorter commutes? | Stating the main finding of (Gillani et al., 2023) up front may increase engagement
Data science to decide which schools to open or close | Framing in terms of a problem school districts often face that triggers boundary planning may increase engagement
Table 1. Example email subject lines and corresponding hypotheses motivating
their inclusion across different email waves.
We use logistic regression with cluster robust standard errors to analyze the
causal effect of each subject line on open and click-through rates. The use of
cluster robust standard errors is typical in such types of experimental
designs and generally meant to account for correlations between participants
(board members) in the same cluster (district) that may impact results
(Cameron and Miller, 2015). We control for several variables that may also
affect the propensity for board members to engage with such outreach. The
regression specification is as follows:
$y^{o}_{i}=\beta_{0}+\beta_{1}\cdot S_{i}+\beta_{2}\cdot
R_{i}+\Sigma^{8}_{j=3}~{}\beta_{j}\cdot
D^{j}_{i}+\Sigma^{10}_{j=9}~{}\beta{j}\cdot C^{j}_{i}+\epsilon_{i}$
Where $i$ represents each recipient; $y^{o}_{i}$ represents the binary result
for each outcome $o\in\\{opened,clicked\\}$; $S_{i}$ represents the subject
line (campaign) randomly assigned to $i$; $R_{i}$ is an estimate of $i$’s
race/ethnicity using the rethnicity package in R (Xie, 2022); $D_{i}^{3:8}$
represent a set of district-level characteristics for the district $i$ belongs
to; $C_{i}^{9:10}$ represent campaign-specific variables—namely, the day of
the week and month (May or June) that the campaign was sent during, given
potential impacts of week and school year-level seasonalities on engagement;
$\beta_{0}$ is the intercept; and $\epsilon_{i}$ is the error term. In
particular, the variables $D_{i}^{3:8}$ represent: an indicator describing
whether the school board is whiter than the district, suggesting a lack of
racial representativeness in the district leadership; the district’s
White/non-White dissimilarity across elementary schools; the percentage of
students in the district who are White; the district’s urbanicity (urban,
suburban, small city, rural); the number of elementary students enrolled
across its schools; and its total number of elementary schools. Once the model
is fit, the coefficient $\beta_{1}$ indicates the causal effect of the
particular email subject line on open and click-through rates, and the other
coefficients indicate associations between various recipient, district, and
campaign-level variables and these outcomes measures.
### 3.3. Survey
We included an optional survey linked off of the data dashboard to invite
leaders and community members to share their feedback on the depicted
rezonings. The main objective of the survey was to help answer Research
Question 2, namely: “How do school district leaders and community members
perceive these boundary changes—namely, what potential merits and pitfalls do
they see in both how these boundaries were produced and their potential
viability in practice?”. Survey questions assessed respondents’ interest in
the topic of boundary planning; their views of the algorithms that produced
the hypothetical boundary scenarios; what they found interesting about the
depicted scenarios; what they found concerning about the depicted scenarios;
and space to leave any other questions or comments. A full list of questions
can be found in Appendix B.
## 4\. Results
### 4.1. Email campaign
We begin with a few descriptive statistics about the recipients and email
campaigns before exploring results from our regressions. Out of 4,831 board
members included in our sample, emails were delivered to nearly 90% (4,320);
approximately 10% of emails bounced (due to email address inaccuracies, being
classified as spam, or other reasons) and were not delivered to the intended
inbox. Two-thirds of school districts in our sample had school boards with a
Whiter racial composition than their elementary school student body. We caveat
this finding given potential biases and inaccuracies in race classification
(Lockhart et al., 2023) (the rethnicity package has an overall f1 score of
0.78 (Xie, 2022)). To the extent it is correct, however, it reflects a broader
trend of school boards failing to reflect the racial and ethnic compositions
of ever-diversifying student and family compositions (Samuels, 2020).
In terms of email engagement, 39% of recipients (1,686) opened the email they
received, but only 2.5% (106) clicked through to the dashboard. Analyzing log
data on the dashboard revealed that the vast majority of these click-throughs
did not make it past the IRB-required text included on the landing page that
describes the research study and data (see Appendix C for this text). Finally,
just under 1.5% of recipients (64) unsubscribed from receiving future emails.
Benchmarking open, click-through, and unsubscribe rates is generally difficult
as they may vary across industries, purpose/type of campaign, and many other
variables. Nevertheless, using average estimates produced by Mailchimp—the
platform we used for our email campaigns—an open rate of 39% vastly exceeds
their reported average of 23% 444https://mailchimp.com/resources/email-
marketing-benchmarks/.. Our observed 2.5% click-through rate, however, is
slightly below their reported average of 2.9%. Finally, while our 1.5%
unsubscribe rate is small in an absolute sense, it is an order of magnitude
higher than their average of 0.25%. We caveat these Mailchimp benchmark
numbers as only rough guideposts, given they represent averages over a wide
range of campaigns and audiences. Interestingly, our observed open rate is
nearly identical to the 40% open rate observed in (Delevoye, 2021), which
involved outreach to school board members during COVID, though their click-
through rate of 7% is much larger than ours. This may be due to the fact that
their call to action involved reading a memo sharing information about school
transportation during an evolving pandemic schooling context, while ours
contained an invitation to explore findings pertaining to a much more
politically contentious, and temporally less-pressing, issue.
Next, we analyze the results of our regressions to investigate Research
Question 1 (“Which messaging and outreach strategies are more or less
effective?”). First, we analyze the causal effect of different email subject
lines on the likelihood that recipients open our outreach emails, controlling
for the additional individual, district, and campaign-level variables
described in the Methods section. Figure 2(a) shows that campaigns with the
subject line “Data Science To Decide Which Schools To Open Or Close” and “Data
Science To Diversify Learning and Reduce Commuting Times” are significantly
more likely to be opened than a control email subject line that reads
“Diversify Learning”—selected for its simplicity. This is particularly
interesting because these subject lines were informed by conversations with
school board members in between campaign waves. After observing low dashboard
engagement following a few email campaign waves, we reached out to some of the
board members who were included in the outreach to see if they might be
willing to have a short conversation to share their views and feedback on our
project. We informally spoke with three board members across three different
districts in two states. The conversations were enlightening in many ways, but
a few key takeaways emerged. In one conversation, a school board member seemed
surprised that it might be possible to redraw attendance boundaries in ways
that might reduce segregation and travel times simultaneously, and believed
others may find this to be surprising as well. This motivated us to design the
“Data Science To Diversify Learning and Reduce Commuting Times” subject line
in an effort to create more of a “hook” to engage future recipients. In
another conversation, a school board member described the capacity planning
challenges they faced as their district’s population rapidly changed with
immigration. The board member described the need to determine which schools to
open or close (and where) as the main impetus for their ongoing boundary
planning efforts. Even though their board cared about school diversity, on its
own, the member highlighted it probably would not spark new boundary analyses.
The board member mentioned, however, that diversity might be an added
consideration as a part of such ongoing boundary planning efforts, and that
our tools and algorithms may help support such planning efforts by dovetailing
issues of diversity with issues that are of more pressing concern for
districts. This prompted us to design the subject line “Data Science To Decide
Which Schools To Open Or Close”, in an attempt to foreground and lead with a
concern that may be more top-of-mind (and less controversial) for school board
members than issues of segregation and diversity.
The results in Figure 2(a) also show that the higher the percentage of
elementary school student body in the district that is white, the more likely
they are to open our emails (regardless of campaign), and that suburban
district board members are slightly less likely to open our emails than those
in urban districts—in both cases, after controlling for all other variables in
the regression.
Figure 2. Associations between different regression variables and likelihood
of board members opening and clicking emails, respectively. The “Subject: …”
variables represent different email campaigns and the depicted values can be
interpreted as the magnitude of the causal effect of each campaign on open and
click outcomes, since participants were randomly assigned to campaigns.
Circles represent average associations (i.e., regression coefficient values,
which indicate log odds), and lines represent 95% confidence intervals.
Intervals that do not intersect zero indicate a statistically significant
association between that variable and its depicted outcome at an $\alpha$
level of 0.05. The results for categorical variables indicate the magnitude of
associations relative to a particular reference category: for BM Race, this is
“White”; for Urbanicity, it is “urban”; for day of week email was sent, it is
“Monday”; for month email was sent, it is “May”; and for the campaigns, it is
the control subject line of “Diversify Learning”.
Figure 2(b) shows results for email click-throughs to the dashboard (N=4,320;
results look virtually identical when conditioning on those who opened the
email). Despite the low number of click-throughs, the results suggest several
associations between email campaigns, individual/ district/ campaign-level
variables and click activity. For one, after controlling for the same
recipient and district-level characteristics as before, two campaigns offering
$20 to board members for their input on depicted boundaries generate lower
click-through rates than the control campaign with a subject line of
“Diversify Learning” and no incentive. While the body of the email specifies
that the $20 is meant to be donated to a school/teacher of their choosing, it
is possible that board members are weary of clicking through for fear of being
perceived as accepting a “gift”—a politically precarious practice—for their
participation. Mentioning “attendance boundaries” in the subject line and body
of the email also tends to generate lower click-through rates—perhaps because
of how controversial and unpopular the topic is555As we sought his feedback on
the dashboard, one former superintendent of a large suburban school district
shared that changing attendance boundaries is so unpopular that it often costs
school board members their jobs / opportunities for re-election. On the other
hand, the emails that produce higher open rates (as described earlier) also
appear to produce higher click-through rates.
Importantly, we highlight these results as exploratory and directional, and
far from conclusive. Given the number of variables in our regressions, there
is the possibility of an inflated Type I error rate (i.e., the likelihood of
detecting significant associations purely by chance) due to multiple
comparisons. Indeed, conservatively applying Bonferonni correction to correct
for the number of inferred coefficients in the email opens model renders all
predictors insignificant at $\alpha=0.05$. Predictors remain significant in
the email clicks model even after correction, but noting the small number of
overall clicks, we still encourage readers to interpret these results as
preliminary. With these caveats in mind, we observe district leaders’
engagement to be sensitive to how our outreach is framed and contextualized:
district leaders appear more likely to open and click through on emails that
use less controversial language that also appears to align with their a priori
student assignment and planning objectives. While these findings are
interesting and may inform future outreach efforts for this and other
projects, we note that the limited click-through rates coupled with even lower
rates of dashboard exploration due to large dropoff at the IRB study
description text render these cold email campaigns as ineffective channels for
engaging school districts in considering educational data science-informed
policies for fostering more diverse schools.
### 4.2. Media coverage
Observing low click-through rates and engagement with the dashboard, we turned
to another channel for reaching school districts: media coverage. There is a
vast literature on media effects, including how media coverage can shape
public discourse and opinion across a range of topics (King et al., 2017). We
reached out to an education journalist at the Hechinger Report, an outlet
covering Educational Inequality and Innovation, for advice on how to gain more
attention and engagement from district leaders. Our outreach was “almost
cold”: we had only previously spoken with the journalist once, almost 18
months earlier, about a separate research project. After demoing our dashboard
to the journalist, they decided to write a story describing the tool and how
the boundary scenarios were produced (Barshay, 2022). Stumbling across this
piece one week later, another journalist—a host of Georgia NPR’s All Things
Considered show—also published a piece highlighting the tool and how districts
might use it to inform their own diversity planning efforts (Biello, 2022).
We analyzed engagement with the dashboard in the 5 weeks following the first
story’s release to better understand how levels of exploration differ across
different types of districts represented on the dashboard. During this time,
the dashboard received over 3,700 visits, which contributed to the exploration
of boundary results for over 500 districts across 42 states. Given extremely
low levels of engagement with the dashboard prior to these stories, we believe
it is safe to attribute increased engagement with the dashboard during this
time period to the traffic that the stories likely drove.
Figure 3. Outputs of a negative binomial regression depicting associations
between the number of times a particular district is explored on the dashboard
and various district-level characteristics.
Figure 3 shows the associations inferred from a negative binomial regression
modeling the number of explorations per district on the dashboard as a
function of various district-level characteristics. Interestingly, urban
districts were more likely to be explored than suburban, small city, or rural
ones. Furthermore, larger districts (defined in terms of the number of
elementary schools in our data for them) were also more likely to be
explored—likely because they have larger populations overall. Perhaps most
notably, the strongest predictor of whether or not a district would be
explored, after controlling for all others, was its level of White/non-White
segregation: more segregated districts were more likely to be explored. This
suggests that media coverage was effective in drawing attention to those
districts where there is, arguably, great scope for change.
Given the broad nature of media coverage, it is likely that both district
leaders and community members visited and engaged with the dashboard during
the period following the articles’ publishing. Furthermore, given the
anonymous nature of the media-driven dashboard visits, it is impossible to
know who explored the dashboard in which ways, or from which districts.
Nevertheless, as the next section describes, a small set of people from both
sets of stakeholders took time to explore and share their reflections and
feedback on the premise and outputs of the boundary simulations.
### 4.3. Survey responses
In the months following the email campaigns and media coverage (from May
through October 2022), we collected approximately 11 responses via the
feedback survey linked off of the dashboard. Two survey responses came from
email recipients; the remainder, we believe, came from both district leaders
and community members who learned about the dashboard via the published
articles. Given that thousands of individuals explored the dashboard, 11
survey responses constitutes an extremely small—and hence, likely very
biased—sample of total possible respondents. Noting this selection bias, we
caution against making generalized claims from the input. Nevertheless, like
interviews, we believe the qualitative insights provided by these survey
responses shed invaluable light onto Research Question 2—“How do school
district leaders and community members perceive these boundary changes—namely,
what potential merits and pitfalls do they see in both how these boundaries
were produced and their potential viability in practice?”.
When asked how happy they would be with the depicted rezonings, approximately
an equal share of respondents expressed they would be somewhat/very happy and
somewhat/very unhappy (three each), with two responding “neutral”.
Interestingly, when asked how happy they believed other families would be with
the depicted rezonings, only one respondent indicated “somewhat happy”; eight
indicated believing families would be somewhat or very unhappy. This reflects
potential selection biases in who filled out the survey (i.e., those who may
be more open to systemic, equity-promoting changes in schools) and underscores
the controversial nature of boundary changes across many school districts. It
may also point to potential clashes between student assignment models and the
values of certain subsets of families (Robertson et al., 2021). When asked how
much they trusted the computer program that produced the rezonings, five
respondents indicated they sort of trusted it or trusted it completely, while
three indicated they don’t trust it, and two were unsure. While these
responses are mixed, it is interesting—and perhaps concerning—to see such high
levels of trust, especially since the dashboard provided no insights into how
the algorithm was working or which computational methods were applied to
produce the rezonings (though the media coverage offered a high-level
description). Only one out of 9 respondents disagreed that changing attendance
boundaries is a valuable strategy for promoting more diverse schools. Finally,
when asked how they felt about the idea of redrawing attendance boundaries to
increase diversity in their district’s schools after learning about this
project, only one respondent said they were less interested in the idea than
before. An approximately equal number indicated the same or more interest as
before (five and four, respectively), while one expressing same interest wrote
in that they felt more “…hopeful and inspired”.
Respondents’ open-ended feedback offered a number of rich insights. When asked
what they found surprising, interesting, or novel about the depicted
rezonings, several respondents shared interest in the possibility of fostering
more diverse schools without significant travel disruptions. For example, one
respondent commented that they felt “…Surprise that this could be done.
Surprise that the transportation impact is so small.” Similarly, another
respondent—a school district official–said that they “Love that it didn’t move
Children to Far from their old School.” Another school board member shared
that “I am interested to see that…there are improvements that could be made
without major changes in time/distance to schools”. One school board member
shared an optimistic view, commenting that “This is helpful information and a
worthy project, and I don’t want perfect to be the enemy of improvement. I
think there are practical solutions to improving racial diversity here.”
Yet respondents also had consistent suggestions for improvement. When asked
what they found troubling or concerning about the depicted rezonings, many
commented on the fact that the depicted changes to school demographics were
quite small. One school board member mentioned “The computer program only had
1 school out of 13 with a significant change which is not worth the headaches
rezoning causes. The other 12 barely changed if at all.” Another respondent
similarly questioned “Is it worth disrupting families for a small change.”
Several other respondents highlighted the dashboard’s failure to take into
account the nuances and details of their unique local context when depicting
hypothetical rezonings. One respondent described the complexity of redrawing
boundaries in the context of schools with special programs, sharing the
following: “I think the idea is great, but in a city like [redacted] there are
other complexities. We have several schools that are language immersion,
serving the local community. They are a 50/50 DLI [Dual Language Instruction]
model. This really influences how boundaries can move and what the actual
neighborhood space looks like. Additionally we have both k-5 and k-8 and this
changes how boundaries can move.” Another, a board member, commented on the
algorithm’s unrealistic school capacity constraints, highlighting that many of
the schools that the algorithm allocated more students to were already at or
over capacity (which is unknowable from the Department of Education’s (DOE)
Common Core of Data). Some described other shortcomings, like missing schools
(e.g. closed enrollment or newly built ones), a failure to account for
measures of community cohesion (like elementary-middle-high feeder patterns),
and other inaccuracies due to aggregated and dated DOE data.
Finally, we found it encouraging that several respondents shared their contact
information to continue the conversation and share additional feedback. One
board member wrote that they “…would like to chat with [us] about updating
[our] maps to include more accurate information about the schools”, ending
their note with the following question: “Are you able to talk with me
sometime???” One district official expressed interest in seeing results for
other schools, asking “When will the middle and high school data update”?
Finally, a community member wrote asking “How can I work to show this to my
county board and persuade them?” Despite the myriad of possible improvements
and iterations that would be required to make the depicted rezonings
applicable in practice, there appeared to be interest from at least a handful
of leaders and members across districts. We have followed up with respondents
to learn more about their unique contexts and are currently collaborating with
districts to translate the tools and methods from this research into practice.
## 5\. Discussion
Recall our original research questions:
1. (1)
Which messaging and outreach strategies are more or less effective in
prompting school district leaders and community members in exploring potential
boundary changes that could help foster more racially/ethnically integrated
schools?
2. (2)
How do school district leaders and community members perceive these boundary
changes—namely, what potential merits and pitfalls do they see in both how
these boundaries were produced and their potential viability in practice?
With respect to question 1, while we find that email campaigns using subject
lines that align diversity goals with other perennial district-level concerns
(like school capacity planning or travel times) yield larger open and click-
through rates, cold email outreach proves largely ineffective as a means for
prompting district leaders to explore hypothetical diversity-promoting
boundaries. Media coverage, however, generates much more
exploration—particularly among scenarios for districts that are more
segregated. We cannot say that media coverage is a more efficient method for
sharing educational data science research findings with districts; indeed, the
greater volume of exploration it prompts may simply be due to its wider reach.
Nevertheless, it is clear that in this particular setting, media coverage
resulting from (almost cold) outreach to an education journalist produced a
level of engagement that emails alone could not accomplish. We acknowledge
several limitations in our email campaign and analysis. For one, given the low
engagement volume (especially click-through rates) and differences in
engagement rates for emails with similar subject lines, it is important to
interpret all results as exploratory and suggestive. Replication efforts may
help illuminate the robustness of results, as might follow-on analyses that
account for other variables that could have introduced noise or otherwise
affected outcomes. While click-through rates were low, the high open rates
offer an interesting direction for future work. For example, instead of asking
education stakeholders to click through to a dashboard, the dashboard and/or
its most salient findings may be brought into the body of the email—increasing
the likelihood that recipients actually see it.
With respect to question 2, an important caveat of our largely survey
response-driven findings to this question is the extremely small number of
responses (11), especially when compared to the thousands of district data
explorations that occurred through the dashboard. We did not hear from the
vast majority of those who explored the dashboard, and so, cannot make general
claims about their perceptions of specific boundary changes. Nevertheless,
those who did respond generally indicated an interest in the idea of redrawing
school attendance boundaries to foster more diverse schools, and several asked
to have follow-on conversations to explore how these preliminary research
findings might be applied in their local settings. Many also had constructive
suggestions for improving the algorithms to make their outputs more useful in
practice. Admittedly, even though the response pool was small, given how
contentious and unpopular boundary changes are, we were surprised and
encouraged that anyone responded affirmatively at all. Our findings thus
suggest that there is hope for making progress on a problem as entrenched as
racial and ethnic segregation in schools, through means as controversial as
attendance boundary changes—even if just in a few school districts to begin
with. Indeed, sparking change in a few districts serving thousands or tens of
thousands of students can translate into scalable impacts on learning and
future life outcomes for students.
Looking ahead, we believe there are two under-explored opportunities for
educational data scientists to consider pursuing as they shape their research
agendas in the months and years ahead. The first involves conducting research
at the intersection of computation and education policy—particularly focusing
on systemic issues like segregation—in addition to the myriad of efforts
underway exploring the intersection of computation and learning. The growth of
educational data science practices like “learning engineering” (Thille, 2016)
offer new and exciting opportunities for using computational methods to
advance academic mastery, socioemotional understanding, and various other
cognitive and metacognitive skills. Yet learning is often shaped by political
and sociological forces before students even step foot in a school or
classroom. Through computational research projects like boundary modeling and
other policy-related efforts, educational data scientists can help study and
perhaps even shape these “upstream”, systemic forces that can silently, yet
powerfully, affect the opportunities afforded to students and families. The
second opportunity involves exploring which communication and engagement
strategies are most effective for translating research findings into practice.
Too often, the fruits of great research remain unrealized because the paths to
successful implementation are unclear. We believe there is a rich opportunity
for educational data scientists to explore intersections between their work
and “implementation science” in order to ensure that promising early research
is translated and further evaluated across different contexts. We hope our
study offers a useful, albeit preliminary, reference point for both of these
exciting future directions.
## 6\. Acknowledgements
Funding for this project was provided by the MIT Center for Constructive
Communication. The authors declare no competing interests. The research was
approved by MIT’s Institutional Review Board. We are grateful to Peter
Bergman, Kumar Chandra, Alvin Chang, Akeshia Craven-Howell, Rebecca Eynon, Tad
Hirsch, Melissa Krull, Justin Reich, Todd Rogers, Joshua Starr, district
leaders who responded to our outreach, and the anonymous reviewers for their
helpful comments and guidance on this project. We also thank Jill Barshay and
Peter Biello for their support in sharing the research through press channels.
Finally, we thank Lauren J. Bierbaum and Max Vargas at the XQ Institute for
sharing their input and enabling us to access public records for elected
school board members.
## References
* (1)
* Barshay (2022) Jill Barshay. 2022\. PROOF POINTS: Computer scientists create tool that can desegregate schools – and shorten bus routes. The Hechinger Report. https://hechingerreport.org/proof-points-computer-scientists-create-tool-that-can-desegregate-schools-and-shorten-bus-routes/
* Bauer et al. (2015) Mark S. Bauer, Laura Damschroder, Hildi Hagedorn, Jeffrey Smith, and Amy M. Kilbourne. 2015\. An introduction to implementation science for the non-specialist. _BMC Psychology_ 3, 32 (2015).
* Becker and Solomon (2020) Amariah Becker and Justin Solomon. 2020. Redistricting Algorithms. _arXiv: 2011.09504_ (2020).
* Bell et al. (2017) Alex M. Bell, Raj Chetty, Xavier Jaravel, Neviana Petkova, and John Van Reenen. 2017. Who Becomes an Inventor in America? The Importance of Exposure to Innovation. _NBER Working Paper No. 24062_ (2017).
* Biello (2022) Peter Biello. 2022\. Georgia Tech, MIT researchers develop tool for increasing racial integration in schools. All Things Considered, Georgia Public Broadcasting. https://www.gpb.org/news/2022/07/12/georgia-tech-mit-researchers-develop-tool-for-increasing-racial-integration-in
* Billingham and Hunt (2016) Chase M. Billingham and Matthew O. Hunt. 2016. School Racial Composition and Parental Choice: New Evidence on the Preferences of White Parents in the United States. _Sociology of Education_ 89, 2 (2016).
* Black (1999) Sandra E. Black. 1999\. Do Better Schools Matter? Parental Valuation of Elementary Education. _The Quarterly Journal of Economics_ 114, 2 (1999), 577–599.
* Bridges (2016) Kim Bridges. 2016\. Eden Prairie Public Schools: Adapting to Demographic Change in the Suburbs. The Century Foundation. https://tcf.org/content/report/eden-prairie-public-schools/
* Burton and Welsh (2015) Linda M. Burton and Whitney Welsh. 2015. Inequality and Opportunity: The Role of Exclusion, Social Capital, and Generic Social Processes in Upward Mobility. WT Grant Foundation. https://wtgrantfoundation.org/library/uploads/2016/01/Inequality-and-Opportunity-Burton-and-Welsh-William-T.-Grant-Foundation.pdf
* Cameron and Miller (2015) A. Colin Cameron and Douglas L. Miller. 2015. A Practitioner’s Guide to Cluster-Robust Inference. _The Journal of Human Resources_ 50, 2 (2015), 317–372.
* Campos and Kearns (2022) Christopher Campos and Caitlin Kearns. 2022. The Impact of Neighborhood School Choice: Evidence from Los Angeles’ Zones of Choice. _SSRN Working Paper_ (2022).
* Candipan (2019) Jennifer Candipan. 2019\. Neighbourhood change and the neighbourhood-school gap. _Urban Studies_ 56, 15 (2019).
* Castro et al. (2022) Andrene J. Castro, Genevieve Siegel-Hawley, Kim Bridges, and Shenita E. Williams. 2022\. Narratives of Race in School Rezoning: How the Politics of Whiteness Shape Belonging, Leadership Decisions, and School Attendance Boundaries. _AERA Open_ (2022).
* Chetty et al. (2022) Raj Chetty, Matthew O. Jackson, Theresa Kuchler, and Johannes et al. Stroebel. 2022. Social capital II: determinants of economic connectedness. _Nature_ 608 (2022), 122–134.
* Delevoye (2021) Angele Delevoye. 2021\. School boards in 2020: deliberation and decision-making in a crisis. _Working Paper_ (2021). https://drive.google.com/file/d/1-2rcwRetEJ6J8QeSb8JWF4hb4noWUuN5/view
* Delmont (2016) Matthew F. Delmont. 2016\. _Why Busing Failed: Race, Media, and the National Resistance to School Desegregation_. University of California Press.
* EdBuild (2020) EdBuild. 2020. Fault Lines: America’s Most Segregating School District Borders. EdBuild. https://edbuild.org/content/fault-lines/full-report.pdf
* Ericsson and Simon (1984) K. Anders Ericsson and Herbert A. Simon. 1984. _Protocol analysis: Verbal reports as data_. MIT Press.
* Fiel (2013) Jeremy E. Fiel. 2013\. Decomposing School Resegregation: Social Closure, Racial Imbalance, and Racial Isolation. _American Sociological Review_ 78, 5 (2013).
* Foundation (2022) WT Grant Foundation. 2022\. Improving the Use of Research Evidence. WT Grant Foundation. https://wtgrantfoundation.org/focus-areas/improving-use-research-evidence
* Frankenberg and Jacobsen (2011) Erica Frankenberg and Rebecca Jacobsen. 2011. The Polls–Trends: School Integration Polls. _Public Opinion Quarterly_ 75, 4 (2011), 788–811.
* Frankenberg et al. (2003) Erica Frankenberg, Chungmei Lee, and Gary Orfield. 2003\. A Multiracial Society with Segregated Schools: Are We Losing the Dream? The Civil Rights Project, Harvard University. https://civilrightsproject.ucla.edu/research/k-12-education/integration-and-diversity/a-multiracial-society-with-segregated-schools-are-we-losing-the-dream/frankenberg-multiracial-society-losing-the-dream.pdf
* Gerber and Green (2012) Alan S. Gerber and Donald Green. 2012. _Field Experiments: Design, Analysis, and Interpretation_. New York: W. W. Norton & Company.
* Gillani et al. (2023) Nabeel Gillani, Doug Beeferman, Christine Vega-Pourheydarian, Cassandra Overney, Pascal Van Hentenryck, and Deb Roy. 2023. Redrawing attendance boundaries to promote racial and ethnic diversity in elementary schools. _Forthcoming in Educational Researcher_ (2023). https://papers.ssrn.com/sol3/papers.cfm?abstract_id=4387899
* Gillani and Eynon (2014) Nabeel Gillani and Rebecca Eynon. 2014. Communication patterns in massively open online courses. _The Internet and Higher Education_ 23 (2014), 18–26.
* Gurnee and Shmoys (2021) Wes Gurnee and David B. Shmoys. 2021. Fairmandering: A column generation heuristic for fairness-optimized political districting. _arXiv: 2103.11469_ (2021).
* Hailey (2021) Chantal A. Hailey. 2021\. Racial Preferences for Schools: Evidence from an Experiment with White, Black, Latinx, and Asian Parents and Students. _Sociology of Education_ (2021).
* Hall and Hibel (2017) Matthew Hall and Jacob Hibel. 2017. Latino Students and White Migration from School Districts, 1980-2010. _Social Problems_ 64, 4 (2017), 457–475.
* Harris (2022) Adam Harris. 2022\. School Boards are No Match for America’s Political Dysfunction. The Atlantic. https://www.theatlantic.com/politics/archive/2022/04/parents-partisan-school-board-meeting-covid/629669/
* Hasan and Kumar (2019) S. Hasan and A. Kumar. 2019\. Digitization and Divergence: Online School Ratings and Segregation in America. SSRN. https://ssrn.com/abstract=3265316
* Johnson (2011) Rucker C. Johnson. 2011\. Long-run Impacts of School Desegregation & School Quality on Adult Attainments. _NBER Working Paper No. 16664_ (2011).
* Kane et al. (2005) Thomas J. Kane, Douglas O. Staiger, and Stephanie K. Riegg. 2005\. School Quality, Neighborhoods and Housing Prices: The Impacts of school Desegregation. _NBER Working Paper No. 11347_ (2005).
* King et al. (2017) Gary King, Benjamin Schneer, and Ariel White. 2017\. How the news media activate public expression and influence national agendas. _Science_ 358, 6364 (2017), 776–780.
* Kizilcec et al. (2013) Rene F. Kizilcec, Chris Piech, and Emily Schneider. 2013\. Deconstructing disengagement: analyzing learner subpopulations in massive open online courses. In _Third International Conference on Learning Analytics and Knowledge_. 170–179.
* Lockhart et al. (2023) Jeffrey W. Lockhart, Molly M. King, and Chris Munsch. 2023\. Name-based demographic inference and the unequal distribution of misrecognition. _Nature Human Behaviour_ (2023).
* Macartney and Singleton (2018) Hugh Macartney and John D. Singleton. 2018. School boards and student segregation. _Journal of Public Economics_ 164 (2018), 165–182.
* Massey and Denton (1988) D. S. Massey and N. A. Denton. 1988. The Dimensions of Residential Segregation. _Social Forces_ 67, 2 (1988), 281–315.
* Mayer (2014) Susan Mayer. 2014\. How Economic Segregation Affects Children’s Educational Attainment. _Annual Review of Sociology_ 40 (2014), 199–218.
* McFarland et al. (2021) Daniel A. McFarland, Saurabh Khanna, Benjamin W. Domingue, and Zachary A. Pardos. 2021. Education Data Science: Past, Present, Future. _AERA Open_ (2021).
* McMillan (2018) Susan M. McMillan. 2018\. Common Practices in Changing School Attendance Zone Boundaries. Educational Data Systems. https://eddata.com/wp-content/uploads/2018/10/Common-Practices-in-Changing-School-Attendance-Zone-Boundaries.pdf
* Mobarak (2022) Ahmed Mushfiq Mobarak. 2022\. Assessing social aid: the scale-up process needs evidence, too. _Nature Comment_ (2022). https://www.nature.com/articles/d41586-022-03039-2
* Moir (2018) Taryn Moir. 2018\. Why Is Implementation Science Important for Intervention Design and Evaluation Within Educational Settings? _Frontiers in Education_ (2018).
* Monarrez and Chien (2021) Tomas Monarrez and Carina Chien. 2021. Dividing Lines: Racially Unequal School Boundaries in US Public School Systems. The Urban Institute. https://www.urban.org/sites/default/files/publication/104736/dividing-lines-racially-unequal-school-boundaries-in-us-public-school-systems.pdf
* Monarrez et al. (2022) Tomas Monarrez, Brian Kisida, and Matthew Chingos. 2022\. The Effect of Charter Schools on School Segregation. _American Economic Journal: Economic Policy_ 14, 1 (2022), 1–42.
* Nakajima (2021) Nozomi Nakajima. 2021\. Evidence-Based Decisions and Education Policymakers. _Working Paper_ (2021). https://nozominakajima.github.io/files/nakajima_policymaker.pdf
* Natanson (2021) Hannah Natanson. 2021\. How and why Loudoun County became the face of the nation’s culture wars. The Washington Post. https://www.washingtonpost.com/local/education/loudoun-critical-race-theory-transgender-rights/2021/07/05/3dab01b8-d4eb-11eb-ae54-515e2f63d37d_story.html
* Nosek et al. (2018) Brian A. Nosek, Charles R. Ebersole, Alexander C. DeHaven, and David T. Mellor. 2018. The preregistration revolution. _Proceedings of the National Academy of Sciences_ 115, 11 (2018), 2600–2606.
* of Education (2021) US Department of Education. 2021\. Fast Facts: Public school choice programs. Digest of Education Statistics, 2019 (NCES 2021-009). https://nces.ed.gov/fastfacts/display.asp?id=6
* Office (2022) U.S. Government Accountability Office. 2022\. K-12 Education: Student Population Has Significantly Diversified, but Many Schools Remain Divided Along Racial, Ethnic, and Economic Lines. U.S. Government Accountability Office. https://www.gao.gov/assets/gao-22-104737.pdf
* OR-Tools (2022) Google OR-Tools. 2022\. CP-SAT Solver. https://developers.google.com/optimization/cp/cp_solver
* Orfield and Lee (2005) Gary Orfield and Chungmei Lee. 2005. Why Segregation Matters: Poverty and Educational Inequality. The Civil Rights Project, Harvard University. https://civilrightsproject.ucla.edu/research/k-12-education/integration-and-diversity/why-segregation-matters-poverty-and-educational-inequality/orfield-why-segregation-matters-2005.pdf
* Owens (2018) Ann Owens. 2018\. Income Segregation between School Districts and Inequality in Students’ Achievement. _Sociology of Education_ 91, 1 (2018).
* Perron and Didier (2020) Laurent Perron and Frederic Didier. 2020. CPAIOR 2020 Master Class: Constraint Programming. Youtube. https://youtu.be/lmy1ddn4cyw
* Piech et al. (2015) Chris Piech, Jonathan Bassen, Jonathan Huang, Surya Ganguli, Mehran Sahami, Leonidas Guibas, and Jascha Sohl-Dickstein. 2015. Deep Knowledge Tracing. _Advances in Neural Information Processing Systems_ 28 (2015).
* Quick (2016) Kimberly Quick. 2016\. Chicago Public Schools: Ensuring Diversity in Selective Enrollment and Magnet Schools. The Century Foundation. https://tcf.org/content/report/chicago-public-schools/
* Reardon and Owens (2014) Sean Reardon and Ann Owens. 2014. 60 Years After Brown: Trends and Consequences of School Segregation. _Annual Review of Sociology_ 40 (2014), 199–218.
* Reardon and Bischoff (2011) Sean F. Reardon and K. Bischoff. 2011. Income Inequality and Income Segregation. _American Journal of Sociology_ 116, 4 (2011), 1092–1153.
* Reardon et al. (2018) Sean F. Reardon, D. Kalogrides, and K. Shores. 2018\. The Geography of Racial/Ethnic Test Score Gaps. _Stanford Center for Education Policy Analysis, Working Paper No. 16-10_ (2018).
* Reber (2005) Sarah J. Reber. 2005\. Court-Ordered Desegregation: Successes and Failures Integrating American Schools since Brown versus Board of Education. _Journal of Human Resources_ 40, 3 (2005), 559–590.
* Ritter et al. (2007) Steven Ritter, Jonna Kulikowich, Pui-Wa Lei, and Christy Mcguire. 2007. What Evidence Matters? A Randomized Field Trial of Cognitive Tutor Algebra I. _Frontiers in Artificial Intelligence and Applications_ 162, 13 (2007).
* Robertson et al. (2021) Samantha Robertson, Tonya Nguyen, and Niloufar Salehi. 2021\. Modeling Assumptions Clash with the Real World: Transparency, Equity, and Community Challenges for Student Assignment Algorithms. In _CHI ’21_.
* Samuels (2020) Christina A. Samuels. 2020\. Why School Board Diversity Matters. EdWeek. https://www.edweek.org/leadership/why-school-board-diversity-matters/2020/11
* Sawchuck (2021) Stephen Sawchuck. 2021\. Why School Boards Are Now Hot Spots for Nasty Politics. EdWeek. https://www.edweek.org/leadership/why-school-boards-are-now-hot-spots-for-nasty-politics/2021/07
* Small (2009) Mario L. Small. 2009\. _Unanticipated Gains: The Origins of Network Inequality in Everyday Life_. Oxford University Press.
* Staff (2019) Baltimore Sun Staff. 2019\. In Howard County, a ‘courageous’ plan to redraw school boundaries tests community’s commitment to diversity. The Baltimore Sun. https://www.baltimoresun.com/education/bs-md-howard-school-redistricting-20190906-xhzkmkf2zvgcxdkbd3vqdanblm-story.html
* Thille (2016) Candace M. Thille. 2016\. Bridging Learning Research and Teaching Practice for the Public Good: The Learning Engineer. TIAA Institute. https://origin-www.tiaainstitute.org/sites/default/files/presentations/2017-02/bridging_learning_research_and_teaching_practice.pdf
* Van Hentenryck (1989) Pascal Van Hentenryck. 1989\. _Constraint Satisfaction in Logic Programming_. MIT Press.
* Whitehurt (2017) Grover J. Whitehurt. 2017\. New evidence on school choice and racially segregated schools. https://www.brookings.edu/research/new-evidence-on-school-choice-and-racially-segregated-schools/
* Xie (2022) Fangzhou Xie. 2022\. rethnicity: An R package for predicting ethnicity from names. _SoftwareX_ 17 (2022), 317–372.
* Yeung (2022) Peter Yeung. 2022\. What Happened When France Sent Low-Income Kids to Wealthy Schools. The 74 Million. https://www.the74million.org/article/what-happened-when-france-sent-low-income-kids-to-wealthy-schools/
## Appendix A Email subject lines and corresponding hypotheses
Table 2 shows the different subject lines, motivating hypotheses, and waves
these emails were sent during.
Email subject line | Hypothesis | Wave(s)
---|---|---
Diversify Learning | Concise subject line that highlights project topic | Test, 1
Diversify Learning and get $20 for a school | A small incentive that board members can allocate to a teacher of their choice may increase engagement | 1
Data science to diversify attendance boundaries | “Data science” might pique interest, especially in the context of boundary planning | 2
$20 for your views on diversifying attendance boundaries | A small incentive that board members can allocate to a teacher of their choice may increase engagement | 2
Data science to improve learning experiences | “Data science”, focusing on learning instead of diversity may increase engagement | 2
Data science to diversify learning experiences | Same as above, but mentioning diversity and learning together | 2, 5
More diversity with shorter commutes? | Stating the main finding of (Gillani et al., 2023) up front may increase engagement | 3, 5
More diversity with shorter commutes? $20 for your thoughts! | Stating the main finding of (Gillani et al., 2023) up front and adding the incentive may increase engagement | 3
More diversity, less driving? | Pithy description of findings from (Gillani et al., 2023) | 4
2,000 pennies for your thoughts | Pithy version of incentive email | 4
Improve discussions about school diversity with families | Heard from districts that they find it difficult to talk with families about diversifying schools | 5
Data science to diversify learning and reduce commuting times | Combing learning, diversity, and shorter commute messages from earlier | 6
Data science to decide which schools to open or close | Framing in terms of a problem school districts often face that triggers boundary planning may increase engagement | 6
Data science to diversify learning and reduce commute times | Has elements of promising subject lines throughout runs | Final
Table 2. Email subject lines and corresponding hypotheses motivating their
inclusion across different email waves.
## Appendix B Dashboard survey questions
The optional survey linked off of the dashboard included the following
questions:
* •
How happy would you be with this rezoning? [Very happy / Somewhat happy /
Neutral / Somewhat unhappy / Very unhappy / Other:]
* •
How happy do you believe other families would be with this rezoning? [Very
happy / Somewhat happy / Neutral / Somewhat unhappy / Very unhappy / Other:]
* •
How much do you trust the computer program that created this rezoning? [I
trust it completely / I sort of trust it / I’m not sure if I trust it or not /
I don’t trust it / Other:]
* •
After learning about this project, how do you feel about the idea of redrawing
attendance boundaries to increase diversity in your district’s schools? [More
interested in the idea than before / Less interested in the idea than before /
About the same level of interest as before / Other:]
* •
How much do you agree with the following statement: ”I believe changing
attendance boundaries is a valuable strategy for promoting more diverse
schools in my district” [Strongly agree / Agree / Neutral / Disagree /
Strongly disagree / Other:]
* •
What’s something you found interesting, surprising, or novel about the
proposed rezoning, if anything? [Open-ended response]
* •
What’s something you found troubling or concerning about the proposed
rezoning, if anything? [Open-ended response]
* •
Do you have any other thoughts or reactions that you’d like to share? [Open-
ended response]
* •
Please share your contact information if you would like for a member of our
team to reach out to you to discuss this project further, including how it
might be applied to your school district.
## Appendix C Dashboard landing page IRB text
Email recipients who clicked through saw the following text on the dashboard
landing page.
> Hello! We are scientists at the Massachusetts Institute of Technology, and
> we’ve found that it may be possible to increase diversity across many
> districts’ schools without creating large disruptions for families. We
> invite school district leaders like yourself to explore these results and
> share your feedback through this website, which is part of a research study
> we are conducting in order to better understand how education leaders
> respond to different diversity policies. Exploring the dashboard is
> completely voluntary, and you can stop anytime. We don’t expect any
> discomforts from doing so, and hope you’ll find our results interesting and
> informative for your work. Additionally, there will be no public record or
> knowledge of your participation, so you can feel free to engage (or not
> engage!) in a way that’s best for you. If you have any questions or
> comments, feel free to email us at<EMAIL_ADDRESS>or contact
> MIT’s research office at<EMAIL_ADDRESS>Please click “Continue” to get
> started!
|
observer and source, respectively. The Einstein radius $\theta_{E}$ cannot be
obtained directly from the study of the magnification light curve of the
microlensing event181818In the rare case of perfect alignment
observer/lens/source it is possible to infer the angular Einstein radius
$\theta_{E}$ by using photometric information alone., hence other quantities
need to be defined. The only quantity that contains some information about the
physical properties of the event, especially about the lens itself, and can be
directly inferred from the light curve is instead the timescale $t_{E}$ of the
event, i.e. the time it takes the source to traverse an angular distance of
$\theta_{E}$ and can be written as follows:
$t_{E}=\frac{\theta_{E}}{\mu_{LS}}$ (XVI.3)
where $\mu_{LS}$ is the relative proper motion between lens and source. In a
microlensing event, $t_{E}$ can vary from a few days up to several hundreds
days. In long-duration microlensing events, the annual parallax tends to lead
to prominent departures in the photometric signature [1043, 1044], so that we
can also infer the microlensing parallax parameter $\pi_{E}$:
$\pi_{E}=\frac{\pi_{LS}}{\theta_{E}}$ (XVI.4)
From Eq. XVI.1 we know that the more massive the lens, the longer the duration
of the event itself. In this context, microlensing events involving massive
and compact objects such as BHs as lenses are the most likely to have
sufficient duration to detect the microlensing parallax effect using
photometry.
To obtain the mass of the lens, however, the photometric information is
unfortunately not sufficient, even when the parallax $\pi_{E}$ is known, as
$M_{L}$ and $\pi_{LS}$ are degenerate. The only way to disentangle these two
contributions and get an accurate estimate of the mass of the lens is by
monitoring and measuring the small astrometric shift of the position of the
source induced by the presence of the lens. This can be done using high-
precision astrometry from space, e.g. with HST (see an example later in the
text). Indeed, the astrometric shift ${\bf\delta(u)}$ is a two-dimensional
position angle which is proportional to the angular Einstein radius
$\theta_{E}$ according to the expression:
$\delta({\bf u})=\frac{{\bf u}}{u^{2}+2}\theta_{E}$ (XVI.5)
where the vector ${\bf u}$ defines the angular separation between lens and
source and it changes with time. By substituting Eq. XVI.1 in Eq. XVI.4, we
can derive an expression for the lens mass:
$M_{L}=\frac{\theta_{E}}{k\pi_{E}}$ (XVI.6)
where $\theta_{E}$ comes from precise, multi-epoch astrometry and $\pi_{E}$
comes from photometry. If the distance of the source is known, for example
because the microlensing event is detected in a specific region of the Galaxy
(e.g. the Galactic bulge, whose distance is known), then we can use this
information to constrain the distance of the lens as well, from Eq. XVI.2. If
this is not the case, we can still overcome the limitation if some
spectroscopic data of the source is available.
Searching for isolated stellar-mass BHs via gravitational microlensing is a
promising method as massive objects such as BHs are capable of producing
greater space-time distortions than any other medium-mass star, so they can be
distinguished. In addition, detailed Milky Way microlensing simulation codes
have recently demonstrated that when plotting different sources (stars, white
dwarfs, NS or BHs) in the $\pi_{E}$ \- $t_{E}$ plane, BHs tend to segregate in
a specific region of this parameter space, corresponding to long Einstein
crossing times and relatively low microlensing parallaxes (see Fig. 13 in
[1045]). This means that selecting microlensing events with such properties is
an ideal way to maximize the chances of detecting events involving BH lenses.
Figure 16: from Pietrzyński et al. [1046]: it illustrates the gravitational
microlensing produced by a BH. The deflection of the light of a background
star when it passes in front of a BH produces two images of the same star,
temporarily amplifying its brightness. These two images are too close to being
spatially resolved but the difference in brightness between them produces a
shift in the source’s position that can be measured using astrometry and can
provide strong constraints on the nature of the lens.
Over the past twenty years there has been a growing interest in microlensing
events and several microlensing survey programs have been designed for this
purpose, some of which are currently still in operation. The Optical
Gravitational Lensing Experiment (OGLE-IV; [1047, 1048]), the Expérience pour
la Recherche d’Objets Sombres (EROS; [1049]), the Microlensing Observations in
Astrophysics collaboration (MOA-II; [1050, 1051]), and the Korea Microlensing
Telescope Network (KMTNet; [1052]) are the best examples. These surveys carry
out photometric monitoring of several fields, both in the Milky Way (mostly
the Galactic bulge) and in Galactic satellites (for example the Magellanic
Clouds), allowing to construct light curves for an unlimited amount of
objects. Thanks to their precious work over many years, these surveys have
provided the scientific community not only with more than 30,000 microlensing
events, but also with a detailed and long list of classified variables, from
pulsators (e.g., RR-Lyrae or Cepheids) to eclipsing or ellipsoidal variables.
Unfortunately, it is more difficult to have accurate astrometric measurements
and it is only recently that people have started to have access to this type
of data, thanks to important international collaborations (e.g., HSTPROMO
[1053] which exploits high-resolution HST observations of globular clusters
across multiple epochs) or space missions (e.g. GAIA [1054], which instead
scans the sky multiple times to measure stars’ positions and proper
motions191919We note here that the astrometric power of GAIA is very limited
in crowded fields like in the direction of the Galactic bulge.), opening to
the concrete possibility of carrying out a search for isolated BHs through
astrometric microlensing in the Universe.
To date, only a few isolated BHs have claimed to be detected via this
technique, and interestingly, all of them have been found in the direction of
the Galactic bulge, where there is a higher rate of microlensing events due to
the high density of stars. It is interesting to note that, historically, all
the searches for isolated BHs have focused primarily on microlensing events
with $\pi_{E}$ $\sim$ 0.1 or higher, essentially excluding all those events we
now know are most likely produced by BHs. This could be one of the reasons for
the lack of isolated BH detections, despite the enormous observational effort
of the last few decades.
MACHO-96-BLG-5 and MACHO-98-BLG-6 are the first two microlensing events that
scientists have proposed to be linked to the presence of isolated BHs, with a
mass of the order of 6 $M_{\odot}$ [1055]. These events were first detected
and observed using long-term photometric monitoring [1056] as part of the
MACHO project, which used microlensing events to specifically search for
MAssive Compact Halo Objects [1057]. However, since these estimates were
derived by fitting only the light curves (an approach that may lead to
degeneracies), they were strongly influenced by the imposed Galactic model
prior. Abdurrahman et al. [1058] recently re-analyzed the data of these two
sources, complementing the light curves with new high-resolution near infrared
data. Their work strengthens the case for an isolated BH to be the lens in
both microlensing events, for relative lens-source proper motions above 0.81
mas/yr for MACHO-96-BLG-5, and 2.48 mas/yr for MACHO-98-BLG-6. A future
astrometric/spectroscopic examination of the sources will be crucial: 1) to
obtain an independent measurement of the main parameters of the sources (i.e.
lens-source parallax, lens mass); and 2) to search for every possible
contribution from a second object such as a non-BH compact object or a brown
dwarf lens. This will provide a definitive answer on the nature of these
objects.
The first and only unambiguous detection of an isolated stellar-mass BH in the
Galactic bulge, but more generally thus far in the Universe, has been recently
reported by two different groups, Lam et al. [1059] and Sahu et al. [1060],
almost at the same time. The microlensing event has been detected by MOA and
OGLE, has a long duration ($t_{E}$ $\approx$ 270 d) and is labeled as
MOA-2011-BLG-191 or OGLE-2011-BLG-0462. The magnification of the source caused
by the passage of the lens is shown in Figure 17 in two different ways: as a
sample of HST observations acquired from 2011 to 2017 where both the apparent
change in luminosity and the astrometric deflection can be visualized (top
panel) and as the photometric light curve of the source drawn over time by
multiple facilities (bottom panel). The two teams have used almost the same
set of multi-epoch observations from Hubble to obtain precise astrometry of
this magnified source and derived both the mass of the lens and its distance.
They both agree that the lens must be a compact source rather than a luminous
object, but they disagree on its current mass, thus on the interpretation of
its nature. Lam and colleagues [1059] estimated the source to be relatively
nearby, at a distance of 0.70-1.92 kpc and derived a mass of 1.6-4.4
$M_{\odot}$, consistent with either a NS or a BH, while Sahu and collaborators
[1060] measured a distance of 1.58 $\pm$ 0.18 kpc, implying an inferred lens
mass of 7.1 $\pm$ 1.3 $M_{\odot}$, only compatible with it being an isolated
BH.
Both groups noted an inconsistency between the best fit solution preferred by
the photometric data and the one from the astrometric data but they adopted
slightly different approaches to try and investigate this aspect further. Lam
et al. [1059] in particular used two different weights to analyze the results
of the microlensing models: a default weight (DW) which gave the same weight
to each measurement and uncertainty, and an equal weight (EW) where the two
data sets were weighted in the same way. Of the two, the DW solution, that
better agrees with the photometric data, is the one closer to the solution
found in the study by Sahu et al. [1060], although some differences still
remain. The EW solution, instead, is more in agreement with the astrometric
data but predicts a slightly lower distance for the lens, as well as a lower
mass, falling in the NS mass regime.
Shortly after this detection, Mereghetti et al. [1061] have started an
observational campaign with the Chandra, XMM-Newton and INTEGRAL satellites to
search for any X-ray emission coming from this object, with the aim of
independently binding its nature. They found that the inferred X-ray
luminosity of this object is consistent with the small radiative efficiency
expected for a BH more than for a NS. Upon reanalysis of the available
photometric and astrometric data, Mróz et al. [1062] were able to say the last
word on this story: according to them, the lens is indeed an isolated BH (with
mass M = 7.88 $\pm$ 0.82 $M_{\odot}$ at a distance of $D_{L}$= 1.49 $\pm$ 0.12
kpc). They investigated the presence of systematic errors both in the
photometric and/or astrometric data but they also evaluated the impact of the
presence of a bright star located $\approx$ 0.4” away from the microlensing
source (see Fig. 17) in the estimate of the astrometric shift of the source.
They came to the conclusion that, although in general the presence of a
blending source might produce erroneous results and need to be properly taken
into account, in this case systematic errors in the astrometric solution were
the primary cause of the low mass measured by the Lam et al. team.
The blind search for isolated BHs is still in an embryonic state, so any new
detections coming in the future will represent an important step forward in
the field. The third data release (DR3) of the GAIA satellite (more details
about the mission can be found in Section XVII.2.2), for example, already
contains 363 new microlensing events [1063] that no other surveys have
detected before and the prospect of combining both the light curves and the
astrometric information that GAIA will continue to provide until the end of
the mission looks extremely promising. The development of new dedicated tools
for searching for transient astrometric lensing events caused by dark compact
objects in GAIA DR4 will also greatly help in this regard [1064].
The detection of isolated BHs through gravitational microlensing will receive
a decisive boost in the future thanks to the advent of a new and advanced
facility, the Nancy Grace Roman Space Telescope and its Galactic Bulge Time
Domain Survey (GBTDS). With its wide field of view (0.28 square degrees),
super resolution (0.11 arcsec/pixel) and ability to simultaneously obtain
precise photometry and astrometry, Roman is expected to detect and allow the
characterization of several thousand microlensing events [1065, 1066]. It will
observe the Galactic bulge and other fields multiple times over the years,
allowing us to collect a rich sample of photometric and astrometric
measurements of known and unknown microlensing events. For such a reason,
Roman will enable the GBTDS to expand the catalog of isolated BHs by order(s)
of magnitude, yielding exquisite determinations of their masses, distances,
and proper motions, as well as opening up the opportunity to finally perform
BH population studies.
A recent work by Saradjan & Sahu [1067] have simulated a numerous sample of
microlensing events as seen by Roman, in order to effectively quantify the
role that this instrument will have in the future research for isolated BHs.
Since there is not yet an accurate mass function for isolated BHs based on
observations, they performed the simulations by considering several mass
functions (e.g. $dN/dM\propto M^{-0.5},\propto M^{-1}\leavevmode\nobreak\
\text{and}\propto M^{-2}$). The conclusion of the study is that, considering
the entire lifetime of the mission, the Roman telescope will be able to detect
3-4, 15-17 and 22-24 isolated BHs through astrometric microlensing (depending
of the mass function adopted) and will estimate their physical properties with
reasonable accuracy. In particular, they estimate that the relative errors on
physical parameters such as mass, lens-observer distance and proper motion
will be respectively less than 1, 5, 10%. Advances in this field will provide
important and independent insights into our understanding of the formation and
evolution of BHs.
Figure 17: combined from Sahu et al. [1060]: Top panel: The source of the
microlensing event MOA-11-191/OGLE-11-462 as observed in eight epochs of HST
observations (from 2011 to 2017). The source is highlighted with a white arrow
in the leftmost panel. All stars in the field have similar luminosity over the
epochs while the magnification of MOA-11-191/OGLE-11-462 in Epochs 1 and 2
(produced by the lens passage) is evident compared to the remaining epochs.
Epochs 2 and 3 are instead those where a maximum astrometric shift is
measured. Bottom panel: The photometric light curve of the microlensing event
MOA-11-191/OGLE-11-462 built by combining ground-based observational data
collected from different facilities is shown along with the best-fitting model
light curve. The instruments are labeled directly in the figure with different
colors. The shape of the magnification curve is characteristic of a
microlensing event.
### XVII Black Holes in binary systems with luminous companions
The second family of stellar-mass BHs consists of all those objects that orbit
a star in a binary system. The physics governing binary systems is rather
complicated and still not fully understood but for the purpose of
understanding this Section, it is important to mention some basic concepts. In
a binary system, each of the two components is surrounded by a teardrop-shaped
region, called Roche Lobe, where self-gravity dominates over the gravitational
pull of the companion. If the star is inside of this region, the material is
considered bound. Usually approximated as a sphere of the same volume, the
Roche Lobe have size which depends on the mass difference between the two
stars and on the orbital separation of the system, therefore on how far apart
the two components are from each other. According to [412], the size of a
Roche Lobe can be approximated to:
$\frac{r_{1}}{A}=\frac{0.49q^{2/3}}{0.6q^{2/3}+ln(1+q^{1/3})}$ (XVII.1)
where $r_{1}$ is the radius of the Roche Lobe of the primary component (the
more massive by convention), A is the angular separation of the system, while
$q$ is the mass ratio defined here as $M_{1}/M_{2}$. To obtain the Roche Lobe
size for the secondary component, it is sufficient to replace label 1 with
label 2 in the previous formula.
During the evolution of a binary system, both the radius of the stars and the
size of the Roche Lobes can vary, for several reasons. The first happens as
the stars evolve to different stages of their evolution (for example expanding
when evolving from the main sequence to the red giant branch), while the
latter occurs when the orbit of the binary shrinks or widens due to dynamical
interactions and close passages of third stars (especially in crowded fields)
or the mutual influence of the two stars. As soon as the size of the most
massive object (the one that evolves faster as predicted by stellar evolution
theory) exceeds that of its Roche Lobe, the material “located outside” the
Roche Lobe becomes unbound, triggering three possible scenarios. This material
can indeed: 1) escape the system completely, reducing the total mass of the
system; 2) orbit both stars; 3) fall onto the companion’s Roche Lobe, with
consequent mass transfer and accretion. Each of these events significantly
modifies the configuration of the binary. For those interested in the details,
a comprehensive review on binary systems can be found in Chapter 1.
Depending on the status of the system, it can be classified as either non-
interacting or interacting binary, with pretty significant differences in how
they can be observed. The simplest and most secure way to detect a BH (or a
NS) in a binary system with a star is if the two objects are actually
interacting, as the accretion of material from the star onto the compact
object can trigger the emission of energy at high-frequency (for example
X-rays or gamma-rays) but also at low-frequency (in radio) which can be
detected through tailored observations in these bands. Unfortunately this is
not the case if the BH is in a non-interacting system, also called a detached
configuration. As the word itself explains, this means that the companion has
not yet filled its Roche Lobe and has not yet begun to transfer material to
the BH, therefore they are in a quiescent state and their presence can only be
revealed using alternative techniques. These two configurations will be
discussed in more detail in two dedicated sections later.
#### XVII.1 Black Holes in interacting systems
In this Section we will discuss interacting black hole binary systems, their
observational history and their connection to gravitational wave physics.
X-ray binaries (XRBs) consist of a compact object (black hole or neutron star)
accreting matter from a gravitationally bound companion (or donor star). These
sources are multiwavelength objects and are most luminous in the X-ray regime,
with $\rm{L_{X}}>10^{36}$ erg/s. The relativistic accretion of material from
the companion star to the compact object is what powers this enormous energy
output. XRBs are the only class of binary system that can be
electromagnetically detected outside the Local Group, a small ($\sim$1 Mpc)
group of 50 galaxies that includes the Milky Way [see e.g. 1068, 1069], making
these sources very useful in the study of binary stellar systems.
X-ray binaries were some of the first X-ray sources to be detected and have
been actively studied for several decades [see e.g. 1070]. There are over 400
known XRBs in the Galaxy and the Magellanic Clouds [1071, 1072].
The study of X-ray binaries allow us to probe physics in extreme conditions;
from regions of high magnetic field strength to relativistic accretion and
strong gravity [see e.g. 1070]. They also help us to gain insight into other
astrophysical systems.
For instance, black hole X-ray binaries (BHBs) can allow us to explore the
physics of black holes on much shorter time scales than exhibited by
supermassive black holes. A study of BHBs with low accretion rates uncovered a
power law relation between radio ($L_{R}$) and X-ray luminosity ($L_{X}$),
with $L_{R}\sim L_{X}^{0.7}$ [1073, 1074]. Extending this work to include
supermassive black holes in active galactic nuclei, [1075, 1076] revealed a
relation between X-ray luminosity ($L_{X}$), radio luminosity ($L_{R}$) and
black hole mass $M$, often called the Fundamental Plane of black hole activity
(see Fig. 18), with
$\rm{log}\,\rm{L_{R}}\sim 0.6\,log\,\rm{L_{X}}+log\,M$ (XVII.2)
This relation suggests that for black holes with low accretion rates, the
physical processes governing how accreted material is converted into into
radiative energy could be universal over the entire black hole mass range,
from stellar mass black holes to supermassive black holes [see 1077, and
refernces therein].
Figure 18: Taken from [1075]. This is an edge-on view of the Fundamental Plane
of black hole activity, across a black hole mass range spanning 9 orders of
magnitude.
XRBs are predicted to be possible progenitors of some of the type of
gravitational wave sources that are already being detected [1078, 1079, 718,
and references therein]. They are also predicted to be sources of detectable
gravitational waves themselves [see e.g. 1080, and references therein]. In
light of this, there has been a renewed interest in XRBs from many branches of
astronomy.
In this work, we will be exclusively focusing on XRBs with a black hole
accretor (black hole X-ray binaries; BHBs). In the following sections, we will
go over the past, present and future of X-ray detectors and observatories. We
will discuss the types of X-ray binaries and their links to gravitational wave
physics, and give an overview of the observational signatures of black hole
binaries.
##### XVII.1.1 Development of electromagnetic detection methods for X-ray
binaries
The Earth’s atmosphere is opaque to X-rays from space. Thus, in order to
detect this high energy emission from astronomical objects, X-ray telescopes
must be placed above the atmosphere using balloons, rockets or satellites. The
first X-ray detection outside of the Earth’s atmosphere was from the our Sun’s
corona, in 1949, using Geiger counters launched on a sounding rocket [1081].
To date, over 60 X-ray instruments and telescopes have been used by
astronomers202020A list of all satellite and rocketborne X-ray missions can be
found at https://heasarc.gsfc.nasa.gov/docs/heasarc/missions/.
The first X-ray binary, and indeed the first extrasolar X-ray source, to be
discovered was Scorpius X-1 [1082]. In a mission lasting 350 s, an Aerobee
rocket equipped with proportional counter detectors observed what is still the
brightest apparent source of X-rays in the sky, after the Sun. Figure 19 shows
some early observational X-ray data of Scorpius X-1 taken in 1967 with
detectors on a sounding rocket. After considering the X-ray and optical data
for Scorpius X-1 it was concluded that it must be a neutron star accreting
from a binary companion [1083].
In 1964, another rocket equipped with Geiger counters observed a strong
Galactic X-ray source in the Cygnus constellation [1084]. This source, called
Cygnus X-1, would later be found to be a XRB containing the first compact
object widely accepted to be black hole.
Figure 19: Taken from [1085]. This data shows the X-ray photon count rate from
a three minute long observation taken with a rocket-borne detector in 1967 in
the direction of Scorpius X-1 and including the Galactic centre. We see
clearly that Scorpius X-1 is a very bright X-ray source.
The first dedicated X-ray satellite, _Uhuru_ was launched in 1970 [1086]. The
mission lasted for just over two years and a total of 339 X-ray sources were
detected [1087], including galaxy clusters, supernova remnants and binary star
systems (see Fig. 20). _Uhuru_ detections allowed us to understand that most
the X-ray stars that had been observed were in fact black holes or neutron
stars accreting from a binary companion. Indeed it was observations from
_Uhuru_ that helped to determine that Cygnus X-1 contained a black hole [1088,
1089].
Figure 20: Taken from [1087]. The X-ray sources from the fourth and final
_Uhuru_ source catalogue displayed in Galactic coordinates. The size of the
symbols is proportional to the logarithm of the peak source intensity.
The systematic study of XRBs became possible with the launch of the _Einstein_
X-ray Observatory [_Einstein_ ; 1090]. A space-based instrument launched in
November 1978, it was the first fully imaging X-ray satellite telescope.
_Einstein_ boasted an angular resolution of tens of arcseconds (compared to
the few arcminute resolution of the rocket borne telescopes at the time [see
e.g. 1091]), a field of view of a few arcminutes and a sensitivity nearly 1000
times better than previous instruments and operated in the 0.3–4 keV range.
Prior to _Einstein_ ’s launch, only six extragalactic X-ray binaries had been
detected (excluding the $\sim$60 XRBs in the Milky Way); five in the Large
Magellanic Cloud and one in the Small Magellanic Cloud [1092, and references
therein]. With _Einstein_ , over 100 XRBs or XRB candidates were found in 10
Local Group galaxies, making it possible to compare XRB populations in
different environments. For instance, it was found that high mass X-ray
binaries (see section XVII.1.2) were found preferentially in later type
galaxies whereas low mass X-ray binaries (see section XVII.1.2) were more
likely to be found in earlier types. This finding was not thought to be only
due to the increasing number of massive stars in later type galaxies.
_Einstein_ also revealed that a significant number of XRBs were associated
with globular clusters and that XRB host clusters were brighter than average [
e.g. 1093, see also sections XVII.1.2 and XVII.1.2].
The _Einstein Observatory_ (also called HEAO-2)) ushered in a new and exciting
age in X-ray astronomy. The three large X-ray satellite telescopes that
followed, the _Röntgen_ Satellite [_ROSAT_ ; launched in June 1990 1094] , the
_Advanced Satellite for Cosmology and Astrophysics_ [_ASCA_ ; launched in
Febraury 1993 1095] and the _Rossi X-ray Timing Explorer_ [_RXTE_ ; launched
in December 1995 1096] had instruments that provided the next big improvements
on Einstein. Each of these three telescopes were optimized for one aspect of
X-ray observations that complimented the other two. _ASCA_ , with it’s focus
on X-ray spectroscopy, had a large energy range and good energy resolution. It
was also the first X-ray telescope to make use of large CCD detectors. _ROSAT_
was designed for X-ray imaging, therefore it had a large field of view (2°)
and very good spatial resolution. _RXTE_ was a timing instrument and offered
microsecond time resolution. Figure 21 shows a selection of sources detected
by _ROSAT_ during the all-sky survey (see Table 1 for a comparison of various
telescope characteristics.)
Figure 21: Taken from https://www.mpe.mpg.de/971847/RH_all_sky_survey. This
image shows the 18 811 bright point sources detected by _ROSAT_ in the all-sky
survey. A total of 150 000 sources were detected by _ROSAT_ in this survey, 25
times more than in all previous X-ray satellites combined.
The next significant advancement in X-ray astronomy came with the launching of
the _Chandra_ X-ray Observatory and the _XMM-Newton_ Observatory. The
_Chandra_ X-ray Observatory was launched in 1999 [1097]. This telescope has
subarcsecond spatial resolution and an energy range of 0.1–10 keV. The _XMM-
Newton_ Observatory, launched later that same year, operates in a similar
energy range (0.1–15 keV). _XMM_ has a much coarser angular resolution ($\sim$
$15"$) than _Chandra_ , but has an effective area many times larger (4500 cm2
at 1 keV) [1098]. Figure 22 shows over 40,000 _XMM_ sources detected until the
end of 2013. Using these telescopes, it is now possible to carry detailed
studies of X-ray binaries, and other X-ray sources, in galaxies well outside
the Local Group [see e.g. 1099, 1100].
Figure 22: Taken from [1101]. This image shows the nearly 400 000 X-ray
sources detected by _XMM_ using public observations up to 2013 December 31.
With their complimentary specifications and their greatly improved
capabilities compared to predecessors, _Chandra_ and _XMM_ have been the main
work horse instruments for the study of X-ray binaries for over two decades.
Table 1 shows the comparison between the technical characteristics of
_Chandra_ , _XMM_ and previous X-ray satellites.
Table 1: A comparison of X-ray telescope characteristics. This table has been
adapted from Table 28 of the _XMM-Newton_ Users Handbook (https://xmm-
tools.cosmos.esa.int/external/xmm_user_support/documentation/uhb/xmmcomp.html).
Data are taken from this table unless otherwise indicated. See also
https://heasarc.gsfc.nasa.gov/docs/heasarc/missions/comparison.html for
comparisons of _Chandra_ , _XMM_ and other X-ray telescopes.
Telescope | Energy | Effective | Angular | Energy | Time | Orbital
---|---|---|---|---|---|---
| range | area | resolution | resolution | resolution | target
| (keV) | at 1 keV (cm2) | (arcsec) | at 1 keV (eV) | (s) | visibility (hour)
_Chandra_ | 0.1$-$10 | 555 (ACIS-S)212121For more information on _Chandra_ ’s ACIS instrument see https://cxc.harvard.edu/proposer/POG/html/chap6.html#tth_chAp6. | 0.5 | 1 (HETG)222222For more information on _Chandra_ ’s ACIS instrument see https://cxc.harvard.edu/proposer/POG/html/chap8.html. | 3.2232323taken from https://cxc.harvard.edu/cal/Acis/index.html. (ACIS) | 44.4
_XMM_ | $0.15-12$ | 4650 | 15 | 4 (RGS)242424Fore more infomation on _XMM_ ’s RGS instrument see https://www.cosmos.esa.int/web/xmm-newton/technical-details-rgs. | 0.07252525Taken from https://www.cosmos.esa.int/web/xmm-newton/technical-details-epic. (pn)262626For more information on _XMM_ ’s pn instrument see https://www.cosmos.esa.int/web/xmm-newton/technical-details-epic#5.1.2. | 36.7
_ASCA_ | $0.5-10$ | 350 | 174 | 100 | $6\times 10^{-5}$ (GIS)272727Taken from [1102] and see same source for more information of _ASCA_ ’s GIS instrument. | 0.9
_ROSAT_ | $0.1-2.4$ | 400 | 3.5 | 500 | $6\times 10^{-5}$ (HRI)282828Taken from https://heasarc.gsfc.nasa.gov/docs/rosat/hri.html and see same link for more information on _ROSAT_ ’s HRI instrument. | 1.3
_RXTE_ | $2-250$ | _n.a._ | _n.a._ | _n.a._ | $10^{-6}$ | 1
_Einstein_ 292929Data taken from Table 1 in[1090], see same source for more information on _Einstein_ instruments | $0.3-4$ | 300 | 2 | $10-100$ (FPCS) | 8$\times 10^{-6}$ (HRI) | $0.3-0.6$303030Taken from https://heasarc.gsfc.nasa.gov/docs/einstein/HEAO-B_Guest_Observers_Guide.pdf
There are currently about a dozen X-ray satellite observatories operating,
including _Chandra_ and _XMM_ 313131see
https://heasarc.gsfc.nasa.gov/docs/heasarc/missions/active.html for a list of
active X-ray observatories. This is a global effort with China, Europe, India,
Japan, Russia and the US all involved in launching and running missions, many
of them jointly [1103, 1104, 1105, 1106, 1107, 1108, 1109, 1110, 1111, 1112,
1113]. A list of planned upcoming and future X-ray missions can be found at
https://heasarc.gsfc.nasa.gov/docs/heasarc/missions/upcoming.html.
##### XVII.1.2 Types of X-ray binaries
X-ray binaries are generally classified by the mass of the companion star, as
the nature of this companion determines the time scales and accretion mode of
the binary. Thus we have high mass X-ray binaries, low mass X-ray binaries
(see Fig. 23), intermediate mass X-ray binaries, depending on the mass of the
companion. XRBs can also be classified by their X-ray emission profiles, such
as ultraluminous X-ray sources and super soft X-ray sources (the latter are
sources with a white dwarf accretor and so will not be discussed further in
this).
###### High Mass X-ray Binaries
High mass X-ray binaries (HMXBs) are XRBs with a giant or subgiant donor star
with mass $>10$ M⊙, typically O or B main sequence stars [see e.g. 371] and
the optical luminosity is greater than the X-ray. HMXBs are associated with
young stellar populations and they are a good tracer of recent star formation
[see e.g. 1114, 1115, 1116].
There are two categories of HMXBs: standard HMXBs and Be star XRBs [371, and
references therein]. The standard HMXBs have supergiant O or B type companion
stars, are persistent X-ray sources, have low eccentricity ($e<0.1$) and short
orbital periods ($<10$ days). The mass transfer in these sources proceeds via
one of two processes:
* •
Roche lobe overflow. Roche lobe overflow (RLOF) occurs in a binary when the
companion star expands due to its evolution, filling its Roche lobe, the tear-
shaped surface of gravitational equipotential. Matter from the companion then
flows through the inner Lagrangian point (the point between the binary
components where their respective Roche lobes intersect, i.e. their respective
gravitational potentials are equal; see Fig. 23) onto the compact object [see
e.g. 1117].
* •
Wind-fed accretion. The O and B companion stars in HMXBs have dense, strong,
high-velocity stellar winds. The orbit of the compact object is inside this
stellar wind and the compact object accretes material directly from the
companion’s wind [1118, 1119].
The second class of HMXBs are Be X-ray binaries (BeXRBs), so called because
they have B-emission star (Be-star) companions, rapidly rotating stars with a
variable disk of circumstellar materal around its equator. These systems have
higher eccentricity ($\lesssim 0.2-0.5$) and wider orbits (P$\sim 10-35$ days)
than the standard HMXBs [1120]. In these systems, accretion occurs when the
compact object’s orbit intersects with the circumstellar disk of the Be-star;
rare instances of transient RLOF have also been observed. Due to the high
eccentricity of the compact object orbit and the variable nature of the
circumstellar disk, BeXRBs are transient and highly variable X-ray sources
[1118, 371].
Only 10 BH HXMBs or BH HMXB candidates are currently known, with 7 of these
being extragalactic sources [1121, 1122, 1123, 1124, 1125, 1126, 1127, 1128,
1129, 1130, 1131, 1132, 1133]. These sources are of particular interest since
HMXBs are thought to be progenitors of some of the gravitational wave sources
we observe with instruments such as the advanced Laser Interferometer
Gravitational-Wave Observatory (a _LIGO_ 323232See https://www.ligo.org/ for
details on this instrument.), such as BBHs and BH-NS binaries [see e.g. 645,
291, 98, 1079, 465, and references therein. See also section XVIII].
However, the current models for the evolution of HMXBs to double degenerate
binaries that merge within a Hubble timescale depends on several poorly
constrained aspects of stellar and binary evolution. These uncertainties
include metallicity, the common envelop phase of binary evolution, star-
formation rates and massive star evolution [e.g. 1134, 718].
###### Low Mass X-ray Binaries
Low mass X-ray binaries (LMXBs) consist of an accreting compact object with a
low mass ($\lesssim$1 M⊙) companion [371]. The companion stars are faint and
rarely observed, with $\rm{L_{opt}}/\rm{L_{X}}\ll 0.1$ [1135]. Mass transfer
in LMXBs occurs via RLOF.
These systems have orbital periods of between 11 minutes and about 20 days.
LMXBs are associated with old stellar populations like those found in the
Galactic bulge, elliptical galaxies and globular clusters [371].
LMXB population size scales with the stellar mass and age of the host galaxy
[e.g. 1136, and references therein]. It was also found that younger stellar
populations have fewer faint LMXBs and more bright ones per unit mass than
older populations. LMXB populations can also be used to probe the nature and
distribution of dark matter in galaxies [see e.g. 1137, 1138].
_LMXBs in dense environments_
Globular clusters (GCs) are old ($\gtrsim 1$ Gyr), gravitationally bound
stellar clusters that orbit galaxies. These clusters are compact systems with
half light radii of a few parsecs, masses in the range $\sim 10^{4}-10^{6}$
M⊙, optical luminosities of MV = -5 to -10 and increasing stellar density
towards the cluster centre [e.g. 1139, 1140].
Globular clusters have a high rate of dynamical stellar interactions, making
these clusters very efficient at forming LMXBs. Binaries are formed in GCs via
such processes as tidal capture, exchange interactions and direct collisions
[1093, 1141, 1142, 1143, 1144, and references therein].
Ten percent of the LMXBs associated with the Milky Way reside in its GCs
[1072, and references therein]. Milky Way LMXBs are therefore more than two
orders of magnitude more likely to be found in a GC than in the field of the
Galaxy per unit stellar mass [see e.g. 1145]. For elliptical galaxies,
$20\%-70\%$ of LMXBs are found in GCs [see e.g. 1146, 1147, 1148, 1149].
LMXBs with hydrogen-poor donors and orbital periods of less than an hour are
called ultracompact X-ray binaries (UCXBs). Currently, there are about 30
UCXBs and UCXB candidates known in the Milky Way [1150]. UCXBs are fives time
more common in globular clusters than in the field of the Milky Way. Only one
black hole UCXB in a GC has been discovered to date [1151].
Future gravitational wave observatories, such as the Laser Interferometer
Space Antenna (_LISA_) will be able to detect binary black holes and black
hole UCXBs in Milky Way globular clusters and perhaps even in GCs as far out
at the Virgo Cluster [see e.g. 1152, 1153, 1154]. These gravitational wave
detections will provide robust mass estimates for the binary components,
thereby constraining GC black hole formation and evolutionary models.
Figure 23: Taken from [371]. The typical configurations for a high mass (left)
and low mass (right) X-ray binary (this sketch depicts neutron star binaries,
but black hole binaries have the same composition). The Roche lobe overflow
accretion process is depicted here for the LMXB and windfed accretion is shown
for the HMXB, although Roche lobe overflow can take place in HMXBs as well.
Note the difference in size and mass for two types of binary.
###### IMXBs
For completeness, we note that X-ray binaries with intermediate mass (IMXB)
companion stars (1–10 M⊙) do exist [see e.g. 1155]. Accretion onto the compact
object in an IMXB proceeds via Roche lobe overflow. Due to the relatively high
mass ratio between the compact object and donor star, this accretion phase
proceeds very rapidly, lasting only a few thousand years [see 371, and
references therein]. These short time scales make these systems difficult to
observe and only a small number of these systems are known. Additionally, the
evolution of IMXBs is thought to be a formation channel for low mass X-ray
binaries [see e.g. 1156, and references therein]. Thus IMXBs are not discussed
further in this work.
###### ULXs
Ultraluminous X-ray sources (ULXs) are extragalactic point-like X-ray sources
located outside of the centre of a galaxy, with an apparent peak X-ray
luminosity $>10^{39}$ erg/s, the Eddington luminosity of an 8 M⊙ black hole
[see 595, for a review] (see also XVII.1.3). With their high X-ray
luminosities, ULXs are important objects in the study of relativistic
accretion as they challenge our understanding of black hole formation and
relativistic accretion [1157, 595, 1158, 1159, 1160, and references therein].
Figure 24: Figure 2 from [1161], showing the X-ray contour map of the spiral
galaxy IC 342 overlaid on the Palomar plate image of the galaxy. The image
shows two bright point sources to the NE and SE of the central source with
estimated X-ray luminosities of $\gtrsim 2\times 10^{39}$ erg/s.
ULXs were first detected in the 1970s by the _Einstein_ Observatory [1162,
1163] (see Fig. 24). These sources were initially thought to be intermediate
mass black holes (IMBHs) with mass range $100-10^{5}$ M⊙, since the Eddington
luminosity, $L_{\rm X}$ scales with accretor mass, $M$, as
$\rm{L_{X}}\approx 1.3\times 10^{38}(M/M_{\odot})\,erg/s$ (XVII.3)
[1164, and references therein] (see also XVII.1.3). IMBHs are important
objects from a cosmological standpoint, as they are thought to be the seed
black holes from which the observed supermassive black holes in the universe
grow [1165, 1166, 1167]. Subsequent studies showed that the predicted
formation rates of accreting IMBHs cannot explain the entirety of the observed
population of ULXs [1168, 1169].
_ULXs in low-density density environments_
ULXs are more likely to be found in star-forming galaxies [1170] and younger
ellipticals with recent star formation [1171, 1172], suggesting that ULXs are
an extension of the high luminosity X-ray binary population, which scales with
both star-formation rate and stellar mass of the host galaxy [1173, 1174,
1116]. Using observational data from the _Chandra_ and _XMM-Newton_ X-ray
telescopes and cross-matching with optical catalogues, [1175] compiled a
catalogue of over 700 ULX sources and candidates. They found that ULXs are
preferentially located in spiral galaxies and areas with active star
formation, consistent with previous work (see Fig. 25).
Figure 25: Figure 9 from [1175], showing the ULX population size by host
galaxy Hubble type ($T_{H}$). The vast majority of ULXs in the catalogue are
found in spiral galaxies ($0<T_{H}<10$) with a spike in elliptical galaxies
($T_{H}<-3$). The blue histogram represents the distance and luminosity
complete sample and the red represents the bright ($\rm{L_{X}}$ $>5\times
10^{40}$ erg/s) subsample.
Today, ULXs are mostly thought to be the result of super-Eddington accretion
onto stellar mass black holes and neutron stars in X-ray binaries [1176, 1177,
1178, 1179, 1180]. There is a subset of extremely high luminosity sources, so
called hyperluminous X-ray sources (HLXs; $\rm{L_{X}}>10^{41}$ erg/s), that
are better explained by accretion onto higher mass black holes [1181, 1182,
1183].
[1184] compiled a catalogue of over 100 HLX candidates out to a redshift of
0.9 by combining X-ray and optical observations. This sample of HLXs is a
useful starting point to search for IMBHs electromagnetically and they
identified 22 sources as IMBH candidates.
###### ULXs in high-density environments
There is a small population of about 20 ULXs that resides in extragalactic
globular clusters (GCs) [1185, and references therein]. Unlike ULXs in the
field of galaxies, GC ULXS form via dynamical interactions, due to the high
stellar density of GCs. The donor stars for GC ULXs are also old, low mass and
hydrogen poor, compared to the high mass stars we find in star forming regions
in the field of galaxies. The presence of ULXs in GCs means these X-ray
sources can be easily separated from background AGN. An analysis of this
population of X-ray sources revealed that ULXs are preferentially found in
brighter GCs, but that the presence of a ULX in a GC is not correlated with
the optical colour of the cluster [1186, 1187].
The first globular cluster ULX source was discovered by [1188, see Fig. 26]
using _XMM_ data. The source flux decreased by a factor of seven from an
estimated peak flux of $\sim 4\times 10^{39}$ erg/s over the course of a few
hours. This rapid variability does not support the scenario that the X-ray
flux is from a superposition of neutrons stars in the GC. Indeed, rapid X-ray
flux variability is one of the ways we can distinguish a black hole X-ray
binary (BHB) from other X-ray sources in GCs [1189].
Figure 26: Taken from [1188], showing the X-ray light curve of XMMU
J122939.7+075333, the first ULX detected in a globular cluster (GC). The
source is located with in the GC RZ2019 of the giant elliptical galaxy NGC
44772. The light curve shows an estimated peak X-ray luminosity of $\sim
4\times 10^{39}$ erg/s.
BHBs are the most likely sources of GC ULXs. Thus the study of ULXs in GCs can
help shed light on the formation, evolution and merger history of black holes
in GCs [see e.g 1078]. However, it is important to note given the
extragalactic nature of these sources, we are unable to determine mass
estimates for the components of these X-ray binaries electromagnetically.
Fortunately, we can turn to gravitational wave detections to help solve this
problem.
Initially, it was thought that any black holes formed in GCs would either
cause the cluster to be destroyed [1190] or the BHs would be ejected via
dynamical interactions fairly early on in the cluster lifetime. [1191, 1192].
Globular clusters are now thought to be very efficient at forming binary black
hole systems (BBHs) and other binaries due to the high density of stars,
especially in the cluster core [see e.g. 1193, 1194]. Indeed, GCs are now
thought to be likely hosts of some of the gravitational waves sources observed
by a _LIGO_[see e.g. 1195]. As with other LMXBs in GCs, the study of GC ULXs
will be greatly enhanced with future gravitational wave observations [see e.g.
1196].
###### Finding black hole X-ray binaries
Figure 27: Taken from [1197], last updated in December 2021. The cumulative
histogram of stellar mass black hole candidates in transient XRBs. The red
histogram shows the BH candidates and the blue bars represent dynamically
confirmed BHs. The horizontal black lines indicate the lifetimes of the X-ray
observatories that have discovered SXTs during an X-ray outburst.
As of the end of 2022, there are over 60 BHs and BH candidates known in our
Galaxy [1197]. The vast majority are LMXBs detected during an outburst, where
the X-ray luminosty increases by at least two orders of magnitude for weeks to
months before returning to a quiescent state for between one and 50 years,
with L${}_{X}\sim 10^{32}$ erg/s [371]. These types of LMXBs are called soft
X-ray transients (SXTs). The first black hole SXT, A0620-00, was identified by
[1198] in the 1980s. The source was initially discovered in August 1975 when
it went into outburst, becoming the brightest extra-solar X-ray source ever
detected at the time [1199]. A0620-00 remained bright for several weeks before
returning to quiescence.
With most LMXBs, the optical flux from the low mass donor star is not detected
due to the very bright optical emission produced from the X-ray heated gas
around the compact object. In the case of soft X-ray transients, when the
X-ray source is in a quiescent X-ray emission state ($<10^{33}$ erg/s; see
next section), the X-ray heating and subsequent optical emission are
diminished to the point where the optical emission from the donor star becomes
detectable. The donor star optical spectrum can then be used to determine the
orbital parameters of the SXT, including the masses of the binary components
[see e.g. 1198, 371, and references therein], (see also XVII.2.1 later in this
chapter).
In the next sections, we discuss the accretion disk physics that leads to
observed X-ray outbursts as well as the X-ray emission states of black hole
binaries.
##### XVII.1.3 Accretion physics in X-ray binaries
###### The accretion disk
As mentioned before, the vast energy output of XRBs is the result of accretion
of material from the donor star onto the compact object. In the case of RLOF
matter moving from the donor to towards the black hole object will have too
much angular momentum to fall directly onto the compact object [1200]. Instead
the material flows into circular orbit around the black hole, since this is
the lowest energy orbit for a given amount of angular momentum [e.g. 1201].
Viscous processes within the accreting material then cause the angular
momentum and energy to be transferred outwards. The material spreads outwards
into a series of concentric annuli moving in circular, Keplarian orbits,
called an accretion disk.
The viscosity in accretion disks is not the typical molecular viscosity due to
interactions between particles. It widely thought that the viscosity is due to
the interactions between the weak magnetic fields within the accretion disk
and the accretion disk material; this is called the magneto-rotational
instability model [1202, 1203, 1204].
As the accretion material loses angular momentum and kinetic energy, it moves
further down the gravitational potential well of the black hole, releasing
gravitational potential energy. The amount of rest mass energy released as
radiation depends on the type of compact object. For black holes, up to 40% of
of the accreted material rest mass energy can be released as radiation [1205]
.
The conditions to form an accretion disk are not necessarily present for wind-
fed HMXBs. The stellar wind of O and B star companions do not carry as much
angular momentum as the material coming from a RLOF scenario. Only one wind-
fed HMXB has shown evidence of having formed an accretion disk [1206, 1207,
and references therein].
###### Properties of the accretion disk
The structure and radiation emission of the accretion disk is largely
determined by the rate at which material flows from the companion star to the
accretion disk and towards the black hole [1200]. This accretion rate
($\dot{M}$) is related to the disk luminosity by
$L_{disk}=\eta\dot{M}c^{2}\,erg/s,$ (XVII.4)
where $\eta$ is the efficiency with which gravitational potential energy
released as radiation and $c$ is the speed of light.
At a critical value of the accretion rate $\dot{M}_{crit}$, the resultant disk
luminosity produces enough of radiation pressure on the disk electrons to
balance the compact object’s gravitational force on the disk protons. For a
compact object of mass $M$,
$\dot{M}_{crit}=3\times 10^{8}\eta M/year.$ (XVII.5)
The resultant disk luminosity, called the Eddington luminosity ($L_{Edd}$) is
then,
$L_{Edd}=\frac{4\pi GMm_{P}c}{\sigma_{T}}\,erg/s,$ (XVII.6)
where $G$ is the gravitational constant, $m_{P}$ is the mass of a proton, and
$\sigma_{T}$ is the Thompson scattering cross section of an electron [see e.g.
1200].
We note that the Eddington luminosity is calculated under the assumptions that
the accretion disk is composed entirely of pure hydrogen that is fully ionised
and that the radiation is isotropic. Despite this, $L_{Edd}$ is used widely to
parameterise the luminosity of XRBs.
When $\dot{M}<\dot{M}_{crit}$, radiative processes cool the disk enough for it
to be geometrically thin, but optically thick, such that the disk scale
height, $H\ll R$, where $R$ is the disk radius [see e.g. 1200, 1205]. The thin
accretion disk temperature profile is then
$T(R)=\biggl{\\{}\frac{3GM\dot{M}}{8\pi\sigma
R^{3}}\left[1-\beta\left(\frac{R_{in}}{R}\right)\right]\biggr{\\}}^{1/4}\,K,$
(XVII.7)
where $R_{in}$ is the inner radius if the accretion disk, $\beta$ is a
dimensionless parameter determined by the boundary condition at $R_{in}$ and
$\sigma$ is the Stefan-Boltzmann constant.
The smallest, stable, Keplerian circular orbit that the accretion disk can
extend to depends on the type of compact object [1200]. For a BH, this is the
innermost stable circular orbit (ISCO), equal to 3 times the Schwarzschild
radius ($R_{Schwarzschild}=2\frac{GM}{c^{2}}$). The general relativistic
effects in the strong gravitational field of the BH prevent stable orbits
smaller than the ISCO.
Because the thin accretion disk rotates at roughly Keplerian velocities, half
the disk energy is released as radiation and the other half is changed to
kinetic energy. Black holes have no solid surface, instead they are surrounded
by an intangible boundary called an event horizon that encompasses the region
of spacetime that can no longer communicate with the universe outside it. Thus
the kinetic energy within the accretion disk is lost once it crosses the event
horizon [1208].
###### The multicolour disk model
The geometrically thin, optically thick accretion disk described above is most
commonly modeled as a superposition of black body spectra, with a maximum
temperature at the innermost radius ($T_{in}$), the so-called ”multi-colour
disk” (MCD) model [1209]. This a non-relativistic model that does not take
into account the effects of strong gravity near the inner disk radius
($R_{in}$). Yet this MCD approximation is widely used to parameterise XRB
spectra and is able to fit the spectral data well. The disk luminosity
($L_{disk}$) is then obtained by integrating over the radial distance of the
disk and we have
$L_{disk}=4\pi\sigma R_{in}^{2}T^{4}_{in}\,erg/s.$ (XVII.8)
For black holes, we have
$\rm{M}\propto T^{-2}_{in}L^{1/2}_{disk},$ (XVII.9)
therefore more massive black holes have a lower characteristic disk
temperature [e.g. 1210].
Merloni et al. (2000) [1211] do caution that MCD model is an
oversimplification of the accretion disk than can have significant impact on
disk parameters obtained when the model is applied. Taking into account such
processes as Doppler blurring and gravitational redshift, they showed that the
MCD model systematically underestimates the value of $R_{in}$.
###### Deviation from the standard accretion disk model
When $\dot{M}>\dot{M}_{crit}$, the accretion disk structure changes; it
becomes both optically and geometrically thick, a so-called ’slim’ disk. The
radial temperature profile then goes from $T(R)\propto R^{-3/4}$ in the thin
disk regime, to $T(R)\propto R^{-1/2}$ [and references therein 1212].
When the accretion rate is very low ($<0.01M_{crit}$), the energy released
from viscous processes is not radiated away as efficiently as in the standard
thin disk scenario and the energy is instead stored in the disk material. This
causes the inner disk to swell and become geometrically thick, optically thin
and become hot, with the electron temperatures of about 100 keV [1213, 1212].
As a result, the inner accretion disk moves away from the compact object and
the disk becomes truncated. The hot, optically thin accretion material and the
energy inside it are then advected towards the black hole. Only a small
fraction of the energy is released before the accretion material reaches the
event horizon and is lost. This regime is called the advection dominated
accretion flow [ADAF; 1214] and the radiative efficiency is less than 1%.
###### The disk instability model
For an accretion disk to have a stable flow of matter through it, the
accretion material must be fully ionised throughout the disk. This stable flow
will result in a persistently bright X-ray source. As mentioned earlier, XRBs
can display strong flux variability in the form out outbursts on timescales of
weeks to months. The most widely accepted explanation for this variability is
the disk instability model, whereby instabilities in the accretion disk are
caused by changing ionisation states of the accretion disk material [1215,
1216].
Over a range of accretion rates, the accretion disk is thought to be in either
a cool, low ionisation, low viscosity state or a hot, high ionisation, high
viscosity state. The accretion disk opacity is depends very sensitively on the
ionisation level of the accretion disk material and therefore on the disk
temperature. Thus small changes in disk temperature cause rapid changes
between the hot and cool states.
To ensure that the hydrogen in the accretion disk is fully ionised, the disk
temperature must be greater than 6500 K, the ionisation temperature for
hydrogen [see e.g. 1217]. The disk temperature declines with increasing disk
radius, therefore for a stable accretion disk, the outer disk temperature must
be greater than 6500 K.
If the outer accretion disk temperature is lower than 6500 K, the disk will
contain areas of neutral hydrogen and these will quickly spread throughout the
disk. Cool, neutral accretion material does not flow as as easily as ionised
matter, thus the accretion flow towards the black hole decreases. This
reduction in accretion rate then leads to a reduction in luminosity and also
changes the structure of the accretion disk.
The changes in $\dot{M}$ described by the disk instability model, and the
subsequent changes in the nature of the accretion disk, lead to very distinct
radiation emission states. In the next section, we discuss the salient XRB
properties that arise due to the variations in the accretion rate.
##### XVII.1.4 Electromagnetic observational signatures of black hole
binaries
###### Black hole binary X-ray emission states
Black hole binaries are known to transition between quasi-stable emission
states, where the X-ray spectrum is dominated by either thermal or non-thermal
emission. These emission states are a result of the disk instability model
described above [see 1218, for a full review of black hole emission states].
In the following sections I will describe the canonical BHB X-ray emission
states as well as the accretion flow and physical accretion disk features
related to them.
_The quiescent state_
As described in previous sections, BH LMXBs spend most of their lifetime in a
quiescent X-ray state, where $10^{30.5}<L_{X}<10^{33}$ erg/s. The accretion
rate is very low in this state, with $\dot{M}<0.01M_{crit}$ [see e.g. 1214,
1208]. In the quiescent state, the accretion disk does not extend to the ISCO.
Instead, observations of BHB show that the disk is truncated at a large inner
disk radius [1213, 1214, 1219, 1220]. The ADAF model is invoked to explain the
low luminosity of the quiescent state as discussed in the previous section.
The leading model for the origin of the X-ray emission in the quiescent state
is as follows: A corona of hot ($100\textendash 300$ keV) particles surrounds
the accretion disk and the ADAF [e.g. 1214, 371, and references therein]. The
X-ray emission in the quiescent state comes from soft (low energy) photons
originating in the accretion disk being Compton up-scattered by the corona,
giving rise to to non-thermal, hard emission with energies up to 100 keV. The
energy spectrum of this emission is well fit by a power law (PL) spectral
model, with
$F(E)\propto E^{-\Gamma},$ (XVII.10)
where F is the flux, E in the energy and $\Gamma$ is the photon specrtal
index. For the quiescent state $\Gamma$ is in the range $1.7-2.1$ [e.g. 1208].
_The thermal dominant state_
This emission state is characterised by a large ($>75\%$) thermal emission
contribution component and the accretion rate is estimated be $0.1-0.5$
$\dot{M}/\dot{M}_{crit}$. The spectrum can be modelled by a MCD, with inner
disk temperature of $0.7-1.5$ keV [1214, 1208]. In this state, there is also
sometimes a hard, non-thermal component to the spectrum, modelled by steep
power law, $2.1<\Gamma<4.8$. The flux variability in the thermal dominant
state is weak and the accretion disk is thought to be in the classic thin disk
configuration of [1200].
_The hard state_
The hard emission state is defined by hard, non-thermal emission, where
$>80\%$ of the emission in the $2-20$ keV range can be well modelled by a
power law with $1.7<\Gamma<2.1$ [1208]. A soft X-ray contribution is also
present, which can be modelled as a cooler disk emission component of
$T_{in}=0.1-0.2$ keV. For some BHBs, there is also an excess of hard emission
($20-100$ keV), thought to arise from the reflection of the PL component
emission off the inner accretion disk.
The hard state is sometimes regarded as a less extreme version of the
quiescent state and the accretion rate is thought lie between those of the
quiescent and thermal dominant state. As with the quiescent state, the inner
accretion disk in the hard state is truncated to a larger radius than the ISCO
[1214, 1208].
The nature of the accretion flow within the inner disk radius is still a
matter of debate. Several versions of the ADAF model have been put forward,
but no one set of models can entirely predict or replicate the observed X-ray
behaviour [see 1221, 1222, for reviews].
The hard emission state in BHBs also is associated with synchrotron radio
emission from the a quasi-steady, compact jet [1223, and references therein].
The jets emission disappears when the BHB transitions from the hard state to
the thermal dominant state and there is a strong correlation between the jet
launching mechanisms and the nature of the accretion disk.
_The steep power law state_
The steep power law state (SPL; formerly called the very high state) is
characterised by non-thermal emission which is well modelled by a power law
model with a spectral index, $\Gamma>2.4$. At least 20% of the overall flux is
from the SPL component. In this state, $L_{X}$ is typically $>0.2L_{Edd}$ and
the accretion rate is $\gtrsim 0.5\dot{M}_{crit}$ [see 1208, and references
therein].
The exact physical processes that give rise to the SPL are still being
debated. [1224] and [1225] present a model where the steep power law emission
is due to Compton upscattering of the lower energy accretion disk photons by
the high energy corona electrons. Later, [1226] put forward the following
scenario: high energy ($>$ MeV energies) photons produced via synchrotron
emission from relativistic electrons in highly magnetised areas near the ISCO.
These high energy seed photons are then down-scattered by electrons in the
corona to produce the SPL. Another model uses the expected nature of the inner
accretion disk at high luminosities to produce the steep power law emission
[1227]. At high luminosites, the inner accretion disk is radiation dominated,
resulting in optically thin material producing significant flux. As a
consequence, the gas temperature increases rapidly inwards, leading to locally
saturated Compton spectra with rapidly increasing peak energies, the sum of
which results in a steep power law spectral component.
###### Black hole binary timing signatures
X-ray binaries display a wide range of flux variability across the
electromagnetic spectrum and on various timescales. We have already mentioned
flux variations on timescales of years to decades (quiescent state), weeks to
months (X-ray outbursts) and days to minutes (orbital periods). In the X-ray
regime, flux variability can also occur on millisecond timescales. This
aperiodic, fast variability is an important tool in the study of the strong
gravity near compact objects [see 1228].
The spectral emission states and flux variability in XRBs are correlated, with
both arising due to the physical processes occurring in the inner accretion
disk. [1229] first postulated that accretion onto stellar mass compact objects
will result in millisecond timescale variability. This fast, quasi-periodic
variability was thought to be the result of clumps of accretion material
orbiting in the inner disk near the compact object [1230]. Observational
evidence of this short timescale variability was provided by the _RXTE_
mission, which had millisecond time resolution.
In BHBs, this variability is made up of longer (noise) and shorter timescale
features. The short timescale features are called quasi-periodic oscillations
(QPOs) with frequencies in the range 0.01 to 450 Hz. QPOs are modelled as
Lorentzian functions and have coherence parameters, $Q=\nu/FWHM\gtrsim 2$,
where $\nu$ is the centroid frequency [e.g. 1231, 1228]. QPOs are associated
with X-ray spectral state transitions as well as non-thermal emission states.
Quasi-periodic oscillations can be divided into two subgroups: low frequency
QPOs (LFGQPOs; $\sim 0.1\textendash 30$ Hz) and high frequency QPOs (HFQPOs;
$40\textendash 450$ Hz).
LFQPOs have been observed in BHBs in the SPL, hard and SPL to hard transition
states. The frequencies and amplitudes of LFQPOs have been found to be
correlated to both the PL and thermal components of these spectral states [see
e.g. 1232, 1233, 1234, 1235]. Thus studying LFQPOs can help us understand the
PL spectral component and LFQPOs data can be used to constrain models of non-
thermal emission in BHBs [see 1218].
HFQPOs are transient and have lower amplitudes than LFQPOs and they have been
observed to occur when the BHB is in the steep power law state. Their
frequencies correspond to that expected for accretion material orbiting near
the ISCO of a roughly 10 M⊙ BH. It is thought that HFQPOs are related to the
mass and spin of the BH. Combining HFQPO data with BH mass estimates are a
good way to constrain the spin of the BH [see §8.2.4 in 1218].
#### XVII.2 Black Holes in non-interacting systems
In this Section we will describe what are the main observational strategies
that can be used to unveil stellar-mass BHs which are part of non-interacting
(or quiescent333333In this Chapter we give the words ”non-interacting” and
”quiescent” the same meaning. In the literature you can find these words
referring to slightly different configurations or evolutionary phases.) binary
systems.
Not being able to rely on X-ray, gamma-ray or radio emission from these
sources, the only way we can unveil a BH in a quiescent binary is by measuring
the dynamics of the companion luminous star. There are two main indirect
techniques that can be used to unveil a BH-star binary: 1) spectroscopically
by measuring period changes on the spectral lines of the luminous companion or
2) astrometrically (for example with GAIA), by tracking the orbital motion of
a luminous star as a NS or BH pulls it around. We will explore these two
approaches by highlighting the observational studies that have been conducted
so far in the field.
##### XVII.2.1 Spectroscopic detection of quiescent Black Holes
In a binary system made of two stars, the reciprocal motion of the two
companions with respect to their mutual center of mass causes the line-of-
sight components of their velocities to vary in antiphase with each other
(also referred to as reflex motion) as a function of time, and this produces a
periodic signal, whose periodicity depends on the orbital period of the system
under investigation, directly linked to the distance between the two stars in
the system. The amplitude of the signal is instead related to the masses
involved in the binary. By measuring the change in wavelengths of a few lines
of interest and/or the full spectrum (i.e., Doppler shift) for the two stars
over a period of time, it is possible to construct a radial velocity curve.
Figure 28 illustrates the case of a stellar binary. Depending on their
positions relative to the binary’s center of mass, star 1 and star 2 (as
labeled in the figure) appear to move towards and away from the observer. This
motion makes the light from the stars appear slightly bluer when they are
moving towards the observer, and slightly redder when moving away. This is
what is called Doppler effect.
Figure 28: Schematic representation of the methodology used to detect
spectroscopic binaries. This is the case of a stellar binary. Depending on the
position of a star (star 1 or 2) relative to the binary’s center of mass it
may appear to move towards or away from the observer. In the first case the
light emitted by this star appears blueshifted, however in the second it
appears redshifted. By monitoring these shifts over time it is possible to
construct the radial velocity curve of both star 1 and star 2, independently.
Binaries detected using this method (also called radial velocity method) are
named spectroscopic binaries and can be divided in two groups: 1) Binaries
where the two stars are almost equally bright, called double-lined
spectroscopic binaries (or SB2) ; 2) Binaries where one component is
significantly brighter than the other, called single-lined spectroscopic
binaries (or SB1). While for SB2 binaries we can determine the radial
velocities of the two components ($v_{1}$ and $v_{2}$), for SB1 binaries we
can instead only measure the radial velocities of the brightest component
(usually called $v_{1}$).
The radial velocity or Doppler shift method has an important limitation. It
does not allow to determine the position of the binaries’ orbits in space,
thus the observed velocities $v$ are not the true velocities $v_{0}$ of the
stars but their velocities projected into the line of sight. These two
quantities are linked to each other via the following relationship:
$v=v_{0}\leavevmode\nobreak\ sin\leavevmode\nobreak\ i$ (XVII.11)
where the inclination $i$ is the angle between the line of sight and the
normal of the orbital plane. This directly affects the measurement of the
radial velocity semi-amplitude of the binary $K$, which is also defined as
$K=K_{0}sin\leavevmode\nobreak\ i$, where $K_{0}$ and $K$ represent the true
and observed amplitudes, respectively. By using a simple mathematical
formulation we can understand what are the main properties of a binary
obtainable with the radial velocity method, and what differentiates SB1 from
SB2 binaries. If we consider a binary system where the two stars move in a
circular orbit around their common center of mass, and we assume that the
radii of the orbits of the two components are $a_{1}$ and $a_{2}$,
respectively, by using the equation of the center of mass $m_{1}a_{1}$ =
$m_{2}a_{2}$ and considering $a=a_{1}+a_{2}$, we can rewrite the orbital
radius of the first component as a function of the masses involved in the
system as follows:
$a_{1}=\frac{a\leavevmode\nobreak\ m_{2}}{m_{1}+m_{2}}$ (XVII.12)
The true semi-amplitude velocity $K_{0}$ can instead be rewritten as:
$K_{0,1}=\frac{2\pi a_{1}}{P},$ (XVII.13)
where $P$ is the orbital period of the binary. By substituting $K_{0}$ from
the above equation, we obtain:
$K_{1}=\frac{2\pi a_{1}\leavevmode\nobreak\ sin\leavevmode\nobreak\ i}{P}$
(XVII.14)
Substituting Eq. XVII.12 in Eq. XVII.14, we then obtain:
$K_{1}=\frac{2\pi a}{P}\frac{m_{2}sin\leavevmode\nobreak\ i}{m_{1}+m_{2}}$
(XVII.15)
By solving the equation for the semi-major axis $a$ of the binary and by
applying the Third Kepler’s law, we derive the mass function of the binary
system:
$f(m)=\frac{m_{2}^{3}\leavevmode\nobreak\ sin\leavevmode\nobreak\
i^{3}}{(m_{1}+m_{2})^{2}}=\frac{K_{1}^{3}P}{2\pi G}$ (XVII.16)
This formula, which only applies to binaries in circular orbits, can be
generalized to take into account the effect of eccentricity $e$ as follows:
$f(m)=\frac{m_{2}^{3}\leavevmode\nobreak\ sin\leavevmode\nobreak\
i^{3}}{(m_{1}+m_{2})^{2}}=\frac{K_{1}^{3}P}{2\pi G}(1-e^{2})^{3/2}$ (XVII.17)
The right hand side of Eq. XVII.16 (or Eq. XVII.17) can be fully inferred from
radial velocity data in a spectroscopic binary, as $K_{1}$, $P$ and $e$ are
all measurable quantities. In an SB2 binary, where the spectral lines of both
components can be observed, we measure not only $K_{1}$ but also $K_{2}$ and
an expression similar to Eq. XVII.16 (or Eq. XVII.17 in the general case) can
be written, with the labels reversed. By using Eq. XVII.15 and the definition
of the center of mass, we obtain the following:
$\frac{K_{1}}{K_{2}}=\frac{a_{1}}{a_{2}}=\frac{m_{2}}{m_{1}}$ (XVII.18)
This means that in an SB2 binary we can derive the mass ratio of the two
components. However, since we do not have any direct information about
$sin\leavevmode\nobreak\ i$, we cannot obtain the true masses $m_{1}$ and
$m_{2}$. Only if the true mass of one of the two stars is determined via
alternative methods (e.g. using photometric data to derive the spectral energy
distribution of the star, hence its mass), then the mass of the companion is
immediately deduced.
The situation becomes significantly more complicated in SB1 binaries, where
only the spectral lines of the brightest component can be observed. In this
case the mass ratio of the visible vs unseen object in the binary is unknown
and the only information that can be retrieved is the system mass function in
Eq. XVII.17. Even if we know the mass of the bright component (that we call
here primary), we are not able to determine the true mass $m_{0,2}$ of the
secondary unseen star, but only $m_{2}=m_{0,2}\leavevmode\nobreak\
sin^{3}\leavevmode\nobreak\ i$. This is the so-called
$m\;sin\leavevmode\nobreak\ i$ degeneracy, i.e. the same value of $m_{2}$ can
be produced either by a low inclination $i$ (close to a face-on orbit,
$i=0^{\circ}$) and a high mass or by a high $i$ (close to an edge-on orbit,
$i=90^{\circ}$) and a low mass for the star. There are a few cases in which
this degeneracy between mass and inclination can eventually be broken and we
will discuss them briefly in the following. In all other cases scientists
determine a minimum mass for the secondary unseen star, by assuming that the
binary orbit is perfectly edge-on, thus by assuming that it has an orbital
inclination of $90^{\circ}$. This way $sin^{3}\leavevmode\nobreak\
(i=90^{\circ})$=1 and this term can be removed. Every other inclination value
in the range between $0^{\circ}$ and $90^{\circ}$ provides a larger value for
the true mass of the secondary unseen companion.
As anticipated, there is a way to break the degeneracy between mass and
inclination in spectroscopic binaries, and that happens when the light
modulation of the system as a function of time (light curve) is observed
together with its radial velocity curve. More specifically:
* •
In eclipsing binaries, i.e. when the system inclination is such that one star
passes in front of the other and partially obscures the light coming from it.
The shape and depth of the eclipses can be uniquely solved to give the
inclination and hence the true masses of the individual stars.
* •
In non-eclipsing close binary systems, or when one component is not seen, then
the ellipsoidal modulation of the seen component depends on the mass ratio and
the inclination. Together with the radial velocity curve, this can then give
unique masses for the components.
Promising BH-stellar binary candidates are those that meet two criteria: i)
the periodic signal due to the motion of the star can be clearly detected but
the reflex motion due to a visible companion is not observed; and ii) the
analysis of the radial velocity curve suggests the unseen component to be more
massive than the observed one. However, before misinterpreting the data, some
caution is required as there may be several reasons why a visible companion,
although present, does not appear in the composite spectra (see the discussion
in the next two Sections for details).
Spectroscopic surveys are one of the main strategy for dynamically detecting
non-interacting binaries. Any spectrograph can be used to perform these
studies, with the most useful instruments for this purpose being either multi-
object spectrographs (e.g. GIRAFFE at the Very Large Telescope - VLT) or
spectrographs equipped with an integral field unit (IFU) such as the VLT/MUSE.
Both can be used to obtain simultaneous spectra of many objects, hence
performing blind but efficient searches. By observing the same field for a
number of times, and extracting a spectrum for a significant number of sources
in the field of view, it is possible to identify those objects showing radial
velocity variations and among them, those orbiting massive (and likely) dark
companions. Unfortunately, the only surveys that have multi-epoch spectroscopy
that is rich enough for a time-series analysis are APOGEE (Apache Point
Observatory Galactic Evolution Experiment [1236]), LAMOST (Large Sky Area
Multi- Object Fiber Spectroscopic Telescope [1237]), RAVE (RAdial Velocity
Experiment [1238]) and Gaia-ESO [1239], which came into play only in recent
years. This limitation has severely hampered the detection of BH-stellar
binaries in the past via this method. More surveys (WEAVE [1240] and 4MOST
[1241]) will be in operation in the coming years, surely providing a boost in
the field, thanks to a significant increase in both the number and depth of
the observations.
As mentioned above, an important caveat to keep in mind in this type of study
is that, when only radial velocity measurements are available, the binary
system cannot be fully characterized and only a minimum companion mass can be
inferred for the unseen source. This is due to the fact that the inclination
of the system is unconstrained. When radial velocities are instead
complemented with additional pieces of information, for example, data showing
the photometric variability of the system as a function of time (i.e. light
curves), then these degeneracies can be removed and the current mass of the
two components can be finally estimated. Unlike spectroscopy, there are many
photometric surveys, some even quite dated, which scan the sky both in the
direction of the Milky Way and of Galactic satellites (e.g. the Large and
Small Magellanic Clouds). Most of them have been designed to study variable
stars in different regions of the Universe and for different purposes. Among
the most popular we find VVV [1242], VMC [1243], MACHO [1244], EROS-2 [1245],
ASAS-SN [1246] and TESS [1247]. As already mentioned in Section XVI, OGLE also
has to be added to the list, which over the last twenty years has been an
extraordinary resource of interesting objects.
When talking about dynamically detecting quiescent BHs in binaries, it is
important to make a distinction between two different types of environment
where they can be found. They can be located in high density fields, like the
central regions of massive star clusters (dense collections of thousands to
million of gravitationally bound stars) or in low density fields, which are
the most abundant in the Universe. Various mechanisms are at play in these two
environments, so the BH population they host can be very different. Below we
will describe the low and high density fields separately, highlighting the
main results on the detection of quiescent BHs in both.
###### Quiescent Black Holes in low-density fields
Our Universe is largely very low-density, populated by collisionless systems,
such as galaxies, where stars are mainly moving in the collective
gravitational field, and interactions or collisions between individual stars
are extremely rare if at all absent. When a stellar-mass BH is detected in
this type of environments, it is very likely that it might be the remnant of
the death of a massive star that formed either isolated or in a ”primordial”
binary that exploded as a supernova, rather than the product of a more complex
formation mechanism (e.g., hierarchical assembly of low-mass BHs). The
population of BHs that can be sampled by randomly observing the field is as
close as possible to an “undisturbed” distribution but does not necessarily
resemble what we can find in other types of environments.
Many studies in the literature have postulated that millions of BHs should
exist in our Galaxy [1248, 1249, 1035, 1036, 1037, 1250, 1038], and detecting
a good fraction of them would be extremely important not only to understand
the physical processes that trigger their formation but also to characterize
their properties (mass distribution, spin), to date completely unknown.
Unfortunately, until very recently, no quiescent non-interacting BH-stellar
binaries have been found in radial velocity searches. As previously
anticipated, one of the reasons is the lack of dedicated spectroscopic surveys
capable of scanning the sky a sufficient number of times to capture a
significant number of sources to analyze. Another, even more critical, is the
difficulty of interpreting the results coming from the few available.
Casares et al. [1251] reported the first detection of an X-ray quiescent
stellar-mass BH of 3-6 $M_{\odot}$ in the Milky Way, in a binary system with a
classical Be star (i.e., fast rotating B star that displays strong emission
lines) in a mass range between 10 and 16 $M_{\odot}$. The system, MWC 656, has
been identified as the candidate counterpart of a potential $\gamma$-ray
flare, hence it does not represent the best example of a fully dynamically
detected system. Subsequent radio and X-ray emissions have been found to be
associated with the system, potentially strengthening the case for this to be
a BH. A re-analysis of the spectral variability properties of the source,
however, has recently questioned this interpretation, suggesting a hot
subdwarf as the most likely companion [1252]. Another interesting object, AS
386, has been studied by Khokhlov et al. [1253], showing H$\alpha$ emission
and near-infrared excess. Thanks to low and high-resolution spectroscopy and
multi-band optical and near-infrared photometry the authors have suggested
that the system is as follows: a binary of a B-type star surrounded by
circumstellar gas and dust, exhibiting the B[e] phenomenon and a high mass
($\sim$7 $M_{\odot}$) invisible companion which could not be anything other
than a BH. This system is one of the very few to date that has not been
questioned.
The first detection of a Galactic non-interacting stellar-mass BH in a binary
system with a sensibly smaller luminous companion was reported by Thompson et
al. [1254]. Indeed, the invisible companion has an estimated mass of $\sim$3.3
$M_{\odot}$ and it is in a binary system with a giant star (2MASS
J05215658+4359220). The system has an orbital period of $\sim$83 days, a semi-
amplitude velocity K$\sim$44.6 km/s, near-zero eccentricity and an inclination
close to 90 degrees. The detection has been possible thanks to the analysis of
the radial velocity curve of the system provided by APOGEE and TRES,
complemented by multi-band ASAS-SN and TESS light curves. In Figure 29, left
panel, both radial velocity and light curves of the binary system are shown.
Given the uncertainties in the estimated mass of the invisible companion, it
is compatible with both a BH or a NS, falling in the so-called mass gap
between these two types of objects [1255]. Right panel of Figure 29 shows the
constraints on the mass of the compact object based on the properties of the
visible star (L, Teff) and on the predicted inclination of the system itself
based on the observed photometric variability. Van den Heuvel & Tauris [1256]
analyzed the same set of data available for the system but came to a rather
different conclusion. Apparently, if the estimated mass of the giant star is
lower by a factor of 3 from what suggested by [1254] ($\sim$ 1 $M_{\odot}$),
the system would likely be a tertiary instead of a binary system. In this
case, the unseen component would be compatible with a close binary system,
made of two low mass main sequence stars, which would produce no detectable
signals in the spectral energy distribution, as well as providing a simple
explanation why no X-ray emission is detected. To date, none of the two
scenarios have been investigated further, hence it is still unclear whether
this system contains or not a BH. If confirmed, it would be an extremely
important discovery as it would help building statistics on these systems.
Figure 29: from Thompson et al. [1254]: Left panel: The ASAS-SN light curves
of the binary in 4 different filters, and the TRES radial velocity curve used
to constrain the orbital properties of the system. Right panel: The mass of
the unseen companion (the compact object) as a function of the mass of the
giant. The blue vertical shaded areas define the allowable mass range for the
giant given its Teff and L, while the solid black curves correspond to
different inclinations of the orbits. The black and white dots identify the
more likely masses for both components in the binary.
Starting in 2019, a number of non-interacting BH candidates have been reported
in the literature, most of which with associated massive companions (i.e.
O-type and B-type stars). For most of those systems, alternative explanations
have been proposed that do not require the presence of a BH in them. While the
alternative interpretation has been confirmed by further analysis/data for a
number of these objects, additional investigations need to be performed for
others, in order to provide a firm conclusion on the nature of these systems.
We briefly discuss some of these individual systems here as they represent the
most recent claims of BHs found in the field, and even when it was later
proved that a BH did not exist in the systems, they are still very useful
examples that help us understand 1) how challenging these types of studies
are; 2) what are the main caveats that scientists need to be aware of when
dealing with these systems; and 3) what are the main tests to be performed to
be sure that a stellar-mass BH actually resides in the system.
\- LB-1 is a Galactic binary system. According to Liu et al. [1257], it is
made of a massive BH of $\sim$68 $M_{\odot}$ orbiting around a B-type star
with period P = 78.9 $\pm$ 0.3 d, semi-amplitude velocity K = 52.8 $\pm$ 0.7
km/s, and eccentricity e = 0.03 $\pm$ 0.01. The radial velocity curve of the
system was measured with a sample of 26 LAMOST, 21 GTC and 7 Keck observations
obtained over two years. By studying the motion of the B-type star and the
accompanying H$\alpha$ emission line, that they interpreted linked to an
accretion disk orbiting around the invisible companion, the authors concluded
that the invisible companion should be relatively massive, hence it could only
be a BH. When published, this finding drew much attention from the scientific
community as it was the most massive stellar BH ever reported (even more
massive than theoretical predictions), with strong implications for our
interpretations of gravitational wave (GW) signals. A number of groups
independently analyzed the system from different perspectives, using
alternative techniques, and the final answer came shortly after: LB-1 does not
contain a BH. Instead, it is a rare binary system of a stripped star (the
former mass donor) and a Be star rotating nearly at its critical velocity (the
former mass accretor) [1258, 777, 1259, 1260]. The high rotation of the B-type
star creates a rotationally supported equatorial disk responsible for the
H$\alpha$ emission that Liu and collaborators wrongly attributed to the
presence of a BH in the system. Moreover, when a star rotates very fast, its
spectral lines widen due to this effect, so that the lines of a presumed
second bright companion can be easily overlooked, making it impossible to
detect and separate the two contributions. The use of the shift-and-add
spectral disentangling technique ([1261, 1262]) was fundamental to separate
the flux contribution of the two sources from the composite spectrum and guide
to the correct interpretation of the data.
\- HR 6819 is a naked-eye Galactic binary system. Using FEROS echelle spectra
and TESS light curves, Rivinius et al. [1263] suggested that HR 6819 is a
hierarchical triple made of a classical Be star in a wide orbit with an
unconstrained period around an inner 40 d binary consisting of a B3 III star
and a BH in a circular orbit. They measured a radial-velocity semi-amplitude K
of 61.3 km/s for the inner star and by assuming a minimum mass of $\sim$5
$M_{\odot}$ for it, they deduced a mass for the unseen object $>$ 4.2
$M_{\odot}$, indeed compatible with a BH. Similarly to what happened for LB-1,
also HR 6819 triggered lots of investigations within the community which ended
up with different teams independently suggesting that the system is not a
triple, but instead a binary system consisting of a stripped B-type primary
and a rapidly-rotating Be star formed from a previous mass transfer ([1264,
1265]). This is basically the same kind of system proposed for LB-1, although
the interpretation of the two systems was initially very different, and the
presence of the companion in the spectra has been overlooked for exactly the
same reason. The final word on whether or not HR 6819 contains a BH came very
recently, when new high-angular resolution data from the VLTI/GRAVITY revealed
the presence of a stellar companion with an angular resolution of $\sim$1.2
mas from the primary, that moves on the plane of the sky over a timescale
compatible with the known spectroscopic period of 40 d [1266]. Thanks to
further data and more sophisticated tools, it has now been established that
neither LB-1 nor HR 6819 contain a BH.
In the past couple of years, two more claims of quiescent BH detections in the
Milky Way have been reported in the literature: V723 Mon (“the Unicorn”,
[1267]) and 2M04123153+6738486 (“the Giraffe”, [1268]). Both of them have been
proposed to be binary systems of giant stars with low-mass/faint companions,
which they interpreted as BHs of a few solar masses, around 2-3 $M_{\odot}$.
Masuda and collaborators [1269] also presented the results of further
investigations in support of the dark nature of the companion in “the
Unicorn”. El-Badry and colleagues [1270], on the other hand, have recently
proposed an alternative explanation for both systems, namely that they are
stripped giants with subgiant companions. The authors used a combination of
spectral disentangling and fitting of binary spectral models to investigate
the presence of any luminous companion in both systems. They applied the
spectral disentangling technique to the Keck/HIRES spectra available for “the
Unicorn” and “the Giraffe” and were able to actually identify the presence of
an additional stellar source in both systems. In “the Giraffe”, the companion
was also detected in the near-infrared APOGEE spectra. They found the
secondaries to contribute about half the light in the optical, while they
dominate at the bluest observed wavelengths.
HD 96670 in the Carina OB2 association is one of the last binary systems
claimed to contain a quiescent stellar-mass BH [1271]. It is a binary system
of a $\sim$22 $M_{\odot}$ O-type star orbiting around a $\sim$7 $M_{\odot}$ BH
with an orbital period of $\sim$5 d and possibly a third component. Similarly
to the case of MWC 656, an emission at high-frequency (in X-rays) was found to
be associated with the system triggering further photometric and spectroscopic
analyses, hence the detection, if later confirmed from further analyses, would
not come from a blind dynamical search.
Very recently, a new X-ray quiet BH (VFTS 243) has been discovered in the
field of the Large Magellanic Cloud (LMC), in the direction of the Tarantula
nebula, to be the companion of a massive (M$\sim$25 $M_{\odot}$) O-type star
[566]. The authors have used both radial velocity curves from the Fibre Large
Array Multi Element Spectrograph (FLAMES) at the VLT and OGLE light curves to
constrain the nature of the binary system. In particular, it has an orbital
period P$\sim$10.05 d, an almost circular orbit (e$\sim$0.017) and a semi-
amplitude velocity K$\sim$81 km/s. Shenar and collaborators have applied a
spectral disentangling technique to set an upper limit to the flux
contribution of any putative luminous companion and found that any luminous
companion would exceed this limit significantly. From the analysis of the
light curves and the very small ellipsoidal variations they estimated the
possible inclination of the system to be larger than 40 degrees and the mass
of the unseen companion to be of at least 9 $M_{\odot}$, which can only be a
BH. Under the same observational campaign targeting the Tarantula nebula in
the LMC, Shenar et al. [1272] have also looked at other SB1 binaries and they
have identified two other binaries hosting as many putative BH candidates.
Their nature, however, is much more uncertain and a proper study of these
systems is required.
Given the number of binaries with quiescent BHs that have been questioned in
the recent years, Mahy et al. [1273] have suggested a method to overcome this
problem in the future, which is based on the combination of 1) high-resolution
spectroscopy, 2) high-precision space-based photometry, and 3) state-of-the-
art spectral disentangling. These three ingredients together can allow to
actually constrain the nature of unseen companions in systems classified as
SB1 in the literature. To show the power of this approach, they selected a
sample of 32 Galactic binaries hosting an O-type or an early B-type star. This
sample contained also two systems that, from X-rays observations, are already
known to host a BH (Cyg X-1 [1274, 1275]) and a NS (HD 74194 [1276]),
respectively. Their analysis demonstrates that, when good data are available,
even very faint companions can be detected. They found a secondary luminous
companion in 17 out of 32 binaries in their sample, while in the remaining
systems they were not able to detect any secondary star, either because it was
too faint, or because it was dark. The two most promising candidates hosting a
BH were actually Cyg X-1 and HD 130298, where the estimated mass of the unseen
companion was more than 7 $M_{\odot}$. We know that Cyg X-1 hosts a BH since
many years now, while HD 130298 was presented as another system likely hosting
a quiescent BH candidate. Further multi-wavelength observations will confirm
the nature of the latter but thus far it represents one of the few promising
candidates in the literature. Two more binaries have been recently identified
by Fu et al. [1277] as potential hosts of as many compact objects, by
inspecting the GAIA DR3 catalog and modelling TESS light curves of the
targets, but even in this case there is no certainty about the nature of the
companions. Follow-up observations will establish this.
To conclude, currently no quiescent BHs in stellar binaries have been
unambiguously unveiled in the Milky Way field through a purely dynamical
approach based on radial velocity searches, other than that reported by
Khokhlov et al. [1253] and by Mahy et al. [1273] (high-mass companions). In HD
96670, the only other system not questioned so far, a counterpart at high
frequency has triggered the dynamical search, so technically we cannot
classify it as a binary with a dormant BH. The only extra-galactic quiescent
BH detected and confirmed to date is that recently reported in the LMC field
by Shenar et al. [566]. The near lack of BHs detected so far is interesting
but not unexpected, given that they are all individual detections based on
relatively small samples. It is even more justified if we consider how
challenging these studies are.
It is interesting to note, however, that most of the binary systems so far
proposed to host a dormant BH (confirmed or later disputed) have relatively
massive O- and B-type star companions. Due to the large masses involved, these
systems are by far the most difficult to model, thus being the most prone to
interpretation errors. On the other side, they are the ones with the highest
expected fraction of BH companions. Indeed, recent theoretical calculations
predict that about 3% of massive O or early-B stars in binary systems should
have a dormant BH as a companion [1278, 1279]. If this theory is correct,
large numbers of these binaries are actually hiding out in the Universe and it
may be only a matter of time before they are detected. However, it is
important to underline that O and B stars are very massive and short-lived
objects. The reason they are largely undetected may simply be an observational
limitation: their lifetime are short enough to make their observations
unlikely. An intriguing aspect to mention is also that many of these systems,
at some point in their evolution, may effectively become double degenerate
binaries (i.e., binaries with two compact objects such as BH-NS or BH-BH
binaries), thus the possible progenitors of GW sources.
In light of these theoretical studies [1278, 1279], the small fraction of
quiescent BHs detected so far in the field, the increasing number of
interacting BHs in binaries detected through X-ray or radio observations, and
the large number of GW signals produced by BH-NS or BH-BH binaries revealed by
the LIGO-VIRGO-KAGRA consortium since 2015 [1280] can probably be reconciled.
The main reasons for the apparent discrepancy we see in their numbers may
indeed be the following: 1) the latter detection methods are much more
efficient in identifying stellar-mass BHs as their signature is clear and
cannot be misinterpreted; 2) in comparison, radio and X-ray detectors tend to
scan a much larger area of the sky and for GWs facilities it is even more the
case. The study of these complementary detections (see the related sections of
this Chapter) is very instructive, since it is expected that the local density
of BHs in the field is made up of BHs randomly but almost equally distributed
in these three different groups: 1) non-interacting BHs; 2) interacting BHs;
3) BH-BH binaries, hence there is real hope that future investigations may
finally reveal the conspicuous number of these quiescent BHs that are now
missing.
###### Quiescent Black Holes in high-density fields
In a Universe predominantly dominated by low-density structures, high-density
fields such as star clusters are very special environments, where specific
mechanisms can occur, triggering interesting phenomena or the formation of
exotic objects (such as cataclysmic variables, blue straggler stars or
millisecond pulsars [1281, 1282, 1283]) that are not observed elsewhere. Star
clusters are a class of cosmic objects made up of $10^{3}-10^{6}$
gravitationally bound stars, which share nearly the same metallicity and age
among them, since they are born during the same burst of star formation (at
least as a first approximation, but see the review by [1284] for a recent
discussion about the multiple population phenomenon in star clusters).
Due to the very high number of stars located in a very limited volume, star
clusters are collisional systems, meaning that all stars during their
lifetimes experience several interactions with other cluster members, both
stellar or compact objects. These close interactions are the main engines of
their dynamical evolution and may lead to changes in the properties of the
clusters themselves with time but also trigger the formation and destruction
of binary systems, which are present in significant numbers within them. In
addition, there is observational evidence that more massive stars (e.g., main-
sequence OB stars) are found more preferentially in binaries or higher order
multiples than lighter ones (see [363, 95, 365]). Initially made up of many
massive stars, star clusters are the ideal places to look at for finding
stellar-mass BHs, both interacting and dormant. In this regard, massive star
clusters have recently been placed under the magnifying glass, after the
GW190521 event was detected [51, 50]. This was the first intermediate mass BH
discovered ($\sim$150 $M_{\odot}$), which formed from the merger of two
massive BHs (60 $M_{\odot}$ and 85 $M_{\odot}$, respectively). Interestingly,
these progenitor BHs might be themselves the product of a previous binary BH
merger or stellar collision (as single star evolution struggles to produce
such massive BHs; refer to Chapter 1, Section 1a, for an exhaustive discussion
of the topic). Two more GW events of this type (GW190426 and GW200220) have
been detected in recent years that involved very massive BH candidates, which
merged to produce final BHs of more than 100 $M_{\odot}$, or even close to 200
$M_{\odot}$. BHs in star clusters have the real chance to pair up dynamically,
thus high-density environments can actually be considered potential nurseries
of many GW sources.
Until very recently, the innermost regions of star clusters, both in the Milky
Way and in external galaxies, were completely inaccessible to perform
spectroscopic studies, due to observational limitations, so until 2018 no
dynamical detections of dormant stellar-mass BHs in clusters were claimed.
Thanks to spectrographs equipped with IFUs, such as MUSE at the VLT (the best
currently available in terms of field coverage and both spatial and spectral
resolution), scientists have started to observe the innermost regions of
clusters, trying to resolve and extract a spectra for every single source
detected. The central regions are the most interesting for this type of
science, as this is where massive and compact objects are most likely to be
found, due to a mechanism at play in star clusters called mass segregation,
i.e. more massive objects sink into the center while lighter sources tend to
lie on the outskirts.
Giesers et al. [1285] used MUSE multi-epoch spectroscopic data of the old
($\sim$ 12 Gyr) Galactic globular cluster NGC 3201 spanning a time baseline of
a few years to start a blind systematic survey of binaries and, among them,
those with likely dark companions. Performing a blind search is very powerful
for two main reasons: 1) we know that the highest chance to detect stellar-
mass BHs is in the center compared to the outskirts, but we do not know
exactly where they are in the cluster (at which direction / what distance from
the center); 2) we are not biased toward detecting only a specific sample of
objects, which always happens when handling targeted observations.
Using this novel technique, they were able to detect a stellar-mass BH in the
cluster, in an orbit rather eccentric (e$\sim$0.6) with a period of $\sim$167
d and a semi-amplitude velocity K$\sim$70km/s with a subgiant star companion.
Since the luminous star was part of the cluster, they were able to exploit the
advantage of knowing exactly the age, the metallicity, the distance and the
extinction in the direction of the cluster, to infer the mass of the luminous
star through a comparison with stellar evolutionary models (MIST, [152]). They
found a mass of $\sim$0.8 $M_{\odot}$ for the star, and thanks to the orbital
properties of the binary system, they derived a minimum mass of 4.36
$M_{\odot}$ for the unseen companion, which can be confidently claimed to be a
BH (see Figure 30). After analyzing the entire sample of binaries in NGC 3201,
they identified another BH candidate and a binary whose orbital solutions were
compatible with a BH or NS companion. These solutions have been published
along with the properties of several other stellar binaries in [1286].
This was a very important result for two reasons: 1) it represented the first
detection of non-interacting stellar-mass BHs in star clusters; 2) given the
old age of NGC 3201, these detections were a further confirmation that BHs can
actually reside in star clusters for a Hubble time or more. The latter is in
contrast with the general understanding of star clusters in the early 1990s.
In fact, it was believed that, due to interactions and dynamical processes,
any BH formed within them would be expelled, leaving star clusters essentially
devoid of these objects in a very short timescale (t $<$ 1 Gyr, [1191]). On
the other side, the results by Giesers and colleagues [1285, 1286] are in good
agreement with what more advanced and sophisticated cluster models have
actually predicted, i.e. star clusters are able to retain their BHs for longer
times, even longer than a Hubble time [1287, 1288, 1289, 1290, 1291].
Figure 30: from Giesers et al. [1285]: The radial velocity curve of the first
BH candidate detected in NGC 3201, from MUSE observations. The red solid line
represents the best fit solution for the system, thanks to a comparison with a
large library of possible orbits. The radial velocities of a nearby cluster
member showing no signs of variability (i.e. a single star) are presented in
blue for comparison.
Unlike in the field, uncovering the BH population of star clusters can
actually help to answer some crucial long-standing questions about stellar
astrophysics, such as: 1) how strong are natal kicks in clusters? 2) does a
merger event always produce a natal kick? 3) how likely BHs are to be ejected
from globular clusters through dynamical interactions? Answering these
questions would be important to understand, for example, if globular clusters
are the main factories of GWs, but also to investigate how likely it is that
BHs with masses larger than 100 $M_{\odot}$ (like GW190521) are formed (via
hierarchical merging of lighter stellar-mass BHs) and are held within globular
clusters, or whether denser environments with deeper gravitational potential
are required for this, such as nuclear star clusters [1292].
Until very recently, no other attempts have been made to reveal further
dormant BHs in star clusters using the radial velocity method. Saracino et al.
[1293] exploited the approach developed by Giesers and collaborators to look
for any evidence of stellar-mass BHs in a massive ($\sim 10^{5}$ $M_{\odot}$)
cluster of $\sim$100 Myr located in the LMC. They chose a young cluster, where
dynamical processes have not yet occurred, to study the initial distribution
of BHs in clusters, which is still completely unknown. In this work, the
authors claimed to have detected a binary system consisting of a BH of
$\sim$11 $M_{\odot}$ orbiting almost circularly a main-sequence B-type star
with a period of 5.04 d and a semi-amplitude of K $\sim$ 140.5$\pm 3$ km/s,
which they named NGC1850 BH1. This finding was the result of a combined
analysis of the radial velocity curve constructed from 17 multi-epoch
observations with MUSE and the I and V light curves provided by OGLE-IV. Since
the binary is a member of the cluster and its position in the CMD is
consistent with a main sequence turn-off star, the authors derived a mass for
the luminous object of $\sim$4.9 $M_{\odot}$ from the comparison with stellar
isochrones of age, metallicity, extinction and distance of NGC 1850.
Right after the detection, an alternative explanation has been proposed for
the binary system NGC1850 BH1 [1294, 1295], which does not require the
presence of a BH. Both groups suggested that this system is a post-mass
transfer binary in a rare state of its evolution and it consists of a bloated-
stripped star and a more massive (possibly rapidly rotating) B-type star. They
argued that the stripped star is about 5 times less massive than what Saracino
and collaborators assumed, but that its luminosity is still consistent with
that of the observed star. Furthermore, they predicted that the massive B-type
star would be expected to contribute 10 to 100% to the total flux in the MUSE
spectra and would be overlooked due to the rapid rotation of the star itself.
After a re-analysis of the MUSE spectra, the original authors [1296] confirmed
that the visible star has a lower mass compared to what initially thought but
they also made the following findings: 1) a larger radial velocity semi-
amplitude ($K_{\rm 2,revised}=176\pm 3$ km/s) than in the discovery paper,
which translates in a mass function $f_{\rm revised}$=2.83 $M_{\odot}$ for the
system; 2) the unseen source contributes less than 10% to the total light in
the visual; 3) an accretion disk seems to be present in NGC1850 BH1. These new
observational constraints indicate that, regardless of the mass of the visible
star, the unseen companion must be rather massive. Its nature, however, is
still unconstrained. This could either be a BH with an accretion disk or a
massive luminous star enshrouded in an optically think disk that almost
completely blocks its light. Further data and modelling are needed to provide
a definitive answer as to whether or not NGC1850 BH1 contains a BH.
A recent addition to the list of dormant BHs detected in star clusters is NGC
2004-115 [1297]. It lies on the periphery of the young cluster NGC 2004, on
the northern edge of the LMC. The authors used VLT/FLAMES and SALT-HRS
spectroscopic data covering a baseline of about 20 years to shed light on the
nature of the system: a triple system consisting of an inner binary with
period P=2.92 d, semi-amplitude velocity K$\sim$62.4 km/s and mass function
f(m)=0.07 $M_{\odot}$. By assuming the visible star is a B-type main sequence
of 8.6 $M_{\odot}$, they inferred a mass for the invisible companion greater
than 25 $M_{\odot}$. An upper limit on the X-ray luminosity coming from the
binary has been set by XMM-Newton and it is consistent with having a quiescent
BH in the system. The presence of a massive BH in NGC 2004-115 was later
questioned by [1298] as such a high mass results from an inferred orbital
inclination of only 9 degrees for the system, which is not consistent with the
modelling of both the OGLE and the MACHO light curves of the system. A triple
system in which the 25 $M_{\odot}$ BH is replaced by a 2-3 $M_{\odot}$
luminous companion, and with an outer Be star appears to be the only scenario
capable of reproducing the observed light curves and radial velocity curves of
the system simultaneously.
As part of a multi-epoch VLT/FLAMES spectroscopic campaign aimed at
constraining the binary fraction of OB stars in high density environments,
Banyard et al. [1299, 1300] have recently provided an important contribution
to the search for quiescent BHs in clusters. Indeed, they studied the
population of B stars in the young (7 Myrs) open cluster NGC 6231 in the Milky
Way, measuring an observed binary fraction of 33 $\pm$ 5% (which increases to
52 $\pm$ 8% if corrected for the observational biases) and characterised the
multiplicity properties of the identified binary systems. By integrating
medium-resolution optical spectroscopy, ground-based photometry and high-
cadence space-based photometry, the authors found that 7 of the 15 SB1 systems
in the cluster are indeed SB2 systems with mass ratios as low as q = 0.1. Of
the 8 remaining systems, they ruled out 4 ambiguous systems and identified a
sample of 4 binaries with a high probability of harboring a compact object.
They calculated the minimum mass for the unseen companions, finding that two
have masses between 1 and 3.5 $M_{\odot}$ and fall within the NS mass range,
while the other two have mass ranges between 2.5 to 8 $M_{\odot}$ and 1.6 to
26 $M_{\odot}$, respectively, and so are possible BH candidates. The most
massive unseen source is part of the CD-41 11038 system. However, as the
authors themselves mention in the paper, the detection limits of their
methodology do not exclude that some or all of these candidates have lower-
mass main sequence companions, stripped star companions or are part of triple
systems. Follow up photometric and/or interferometric observations are
therefore needed to answer this question.
To recap, only the two (possibly three) BH candidates in NGC 3201 are
currently recognized as robust non-interacting BH detections in star clusters
within the scientific community, but NGC 6231 appears to be another promising
source of compact objects, again underscoring the fact that crowded fields
such as clusters are ideal places to look for these degenerate sources. The
methods that Giesers and Banyard have developed together with their
collaborators are extremely promising and have the potential to lead to the
discovery of many more of these objects in other clusters if such a systematic
search is performed. The work by Giesers et al. is especially important in
light of the new instruments that will be available to the community in the
coming years, starting from HARMONI at the ELT in 2027 [1301]. This IFU
spectrograph will have a superb spatial resolution (up to 0.004 arcsec,
compared to 0.2 arcsec of MUSE-WFM) and thanks to the extremely large
collecting area will be able to observe and resolve the central regions of
star clusters more distant than the Magellanic Clouds into individual stars, a
science impossible to accomplish with current facilities. By overcoming the
crowding issue in nearby star clusters, scientists will be able to look for
non-interacting stellar-mass BHs in binaries where the luminous stars are much
fainter on the main sequence than has been possible hitherto.
To conclude and improve the readability of Section XVII.2.1 (a, b), we present
in Table 2 a summary of all the systems discussed, where those hosting
quiescent BHs are included, together with those subsequently questioned and
proven to be BH imposters.
Table 2: A summary of all systems claimed to date to host a quiescent BH is
presented below, including only those systems detected via spectroscopic
investigation. Both low-density and high-density environments are included,
and the systems are classified and catalogued based on the estimated mass of
the luminous companions. For completeness, confirmed BHs and BH imposters are
listed.
Name | Z | Envr. | Proposed | Alternative(s) | References
---|---|---|---|---|---
High - mass | | | | |
companions | | | | |
LB-1 | MW | Field | B + BH | Be + stripped (str) | Liu+[1257]; Shenar+[1259]; El-Badry[1260]
MWC 656 | MW | Field | Be + BH | Be + sdO ? | Casares+[1251]; Rivinius+[1252]
NGC 1850 BH1 | LMC | Cluster | B + BH | B + str ? | Saracino+[1293, 1296]; El-Badry+[1294];
| | | | | Stevance+[1295]
HR 6819 | MW | Field | (B+BH)+Be | Be + str | Rivinius+[1263]; Bodensteiner+[1264];
| | | | | El-Badry+[1265]; Frost+[1266]
NGC 2004 # 115 | LMC | Cluster | (B+BH)+Be | (B+A) + Be ? | Lennon+[1297]; El-Badry+[1298]
HD 96670 | MW | Field | (O+BH) + ? | \- - - | Gomez+[1271]
AS 386 | MW | Field | B[e] + BH | \- - - | Khokhlov+[1253]
VFTS 243 | LMC | Field | O + BH | \- - - | Shenar+[566]
HD 130298 | MW | Field | O + BH | \- - - | Mahy+[1273]
Low - mass | | | | |
companions | | | | |
NGC3201 #12560 | MW | Cluster | MS + BH | \- - - | Giesers+[1285]
NGC3201 #21859 | MW | Cluster | MS + BH | \- - - | Giesers+[1286]
NGC3201 #5132 | MW | Cluster | MS + BH/NS | \- - - | Giesers+[1286]
Unicorn (V723 Mon) | MW | Field | RG + BH | RG + str subgiant ? | Jayasinghe+[1267]; Masuda+[1269];
| | | | | El-Badry+[1270]
Giraffe (2M0412) | MW | Field | RG + BH | RG + str subgiant ? | Jayasinghe+[1268]; El-Badry+[1270]
2MASS | MW | Field | RG + BH/NS | RG + (MS+MS) ? | Thompson+[1254]; Van den Heuvel+
J05215658+4359220 | | | | | [1256]
##### XVII.2.2 Astrometric detection of quiescent Black Holes
As already explained in the previous Sections, the dynamical detection of a
stellar-BH binary totally relies on the possibility to obtain multi-epoch
spectroscopic data for the source of interest or for a very large sample of
objects in order to conduct a blind search. When they are not available,
alternative methods can instead be used to detect a quiescent BH orbiting
around a luminous companion. Indeed, binary systems can also be detected
“astrometrically”, by measuring the perturbations in the motion of the stars
by their unseen companions. This technique has been used previously to
identify “ordinary” binary systems and exoplanets, but can also be extended to
discover any kind of binaries and it is particularly useful in identifying
non-interacting compact binaries. Briefly, astrometric binaries are systems
made of two sources visible in astronomical observations as single sources.
Three are the main reasons why the companion source is not observed: it is
either too faint to be seen or too close to the primary to be resolved as two
stars. Alternatively, it is a compact remnant (a BH or a NS) and cannot be
observed due to its dark nature. What is observed in these cases is that the
primary has a telltale wavy motion across the sky inconsistent with it being a
single star. It instead wobbles about an unseen centre of mass, clearly
indicating that another object exists in the system and gravitationally
affects the motion of the bright, observed star. By analysing the motion of
this star over time it is possible to place important constraints on the
orbital properties of the system and on the masses involved.
As discussed in Section XVII.2.1, the dynamical detection of binaries through
radial velocity measurements allows for a robust estimate of the mass function
of the system, but unfortunately does not provide any constraints on its
inclination. This limitation is effectively exceeded by astrometry, as it also
allows measuring the binary’s orientation in space, including its inclination
angle. This means that if the mass of the luminous component can be derived
from independent analyses/data, this is sufficient to break the degeneracy
between the different parameters and have a complete characterization of the
system [1302]. In order to detect astrometric binaries, it is necessary to
measure the position on the sky of many objects and to trace their motion over
time. Moreover, this has to be done with extremely high precision in order to
pick up even very small displacements. In this respect, Figure 31 shows the
astrometric motion of a 1 $M_{\odot}$ visible star orbiting around an unseen
companion of 1.4 $M_{\odot}$ for different orbital periods (increasing from
left to right) and proper motion (increasing from top to bottom), from Andrews
et al. [1302]. The astrometric effects are amplified as the binary is placed
at a distance of only 35 pc.
Figure 31: from Andrews et al. [1302]: The astrometric motion of a luminous
star of 1 $M_{\odot}$ orbiting around an unseen companion of 1.4 $M_{\odot}$,
for different orbital periods (from 10 d to 10000 d) and two different proper
motions (10 mas/yr and 50 mas/yr). The binary is placed at a relatively close
distance of 35 pc to magnify any astrometric effects and the black dots show
the positions of the star over many GAIA epochs. For orbital periods much
shorter than a year, the orbital motion and parallax can be disentangled.
However, for P = 100-1000 d, the positions become extremely complex.
The Hipparcos space astrometry mission led by the European Space Agency (ESA)
was a pioneering project in this respect and it was able to pinpoint the
positions of more than 100,000 stars with high precision, opening up the
concrete possibility of detecting new binaries using this method. Several
astrometric binaries have been detected thanks to this mission in the
literature [1303, 1304, 1305] but there have been no claims of binaries
containing a non-interacting BH companion. Now that Hipparcos has been
replaced by the new ESA space astrometry mission GAIA, the ability to detect
quiescent BHs by analyzing the “wobble” in their proper motions over time is
more real than ever (see [1306] and [1307]). The GAIA satellite was launched
at the end of 2013 with the aim of providing 3D motion, luminosity,
temperature and composition of almost a billion stars in our Galaxy and
beyond, and so far it has been a revolution in many astronomical fields (e.g.
galactic archaeology).
Although the first GAIA astrometric orbital solutions for binary sources have
been published only in its third release in June 2022, many scientists have
started to analyze the data searching for a way to uncover astrometric BH
binaries well before this date. As an example, [1308] focused on the number of
OB + BH systems (i.e. BHs in binary systems with massive companions, $>$ 8
$M_{\odot}$) that GAIA would have been detected as binaries in the DR3 and
DR4. They took as a reference the 2nd Alma Luminous Star Catalogue (ALS II)
containing a sample of massive OB-type stars in GAIA and estimated that 77% of
the OB + BH binaries in ALS II would be detected as binaries in DR3, of which
89% would be unambiguously identifiable as OB + BH binaries. These systems
would cover an uncharted parameter space of long-period binaries (10 d $<$ P
$<$ 1000 d). They predicted that these numbers would increase to 85% and 82%,
respectively, by the end of the mission, as they would also detect systems
with both shorter and longer periods than those predicted in DR3. It is
interesting to note that, after an accurate cleaning procedure performed by
the GAIA team, most of these sources have been removed from the DR3 sample,
hence we do not have any information on these sources yet. However, GAIA DR3
still represents the game changer in this field, as scientists from around the
world have finally access to orbital solutions (periods, eccentricities, etc.)
for a sample of over $10^{5}$ astrometric binaries [1309, 1310, 1311]. The
analysis is still ongoing but this huge data set will grow and refine year by
year, with the concrete possibility of starting to identify which of these
astrometric binaries may actually contain massive invisible companions, i.e.
compact objects such as NSs or BHs.
Andrews et al. [1312] performed a first attempt in this direction: they
analyzed the whole sample of GAIA astrometric binaries under the conservative
assumption that the companions were dark and therefore the photocenters of the
systems followed the observed stars. They assumed the luminous stars to be of
1 $M_{\odot}$ and selected only those systems having a companion more massive
than 1.4 $M_{\odot}$, this to exclude any contamination from white dwarf-like
objects. After removing binaries with suspiciously long orbital periods
relative to the duration of the mission to date and applying further cuts,
they ended up with a sample of 24 binaries with companion masses ranging from
1.35 to 2.7 $M_{\odot}$, compatible with both NS or low-mass BH companions.
These systems show interesting characteristics, like periods of a few years
and orbital velocities of the order of $\sim$20 km/s. For at least 8 of these
objects a subsequent spectroscopic follow-up was performed and the inspection
of their spectra has confirmed that there is no evidence for a second luminous
component.
On the same line but using a different approach, Shahaf et al. [1313] also
looked at the sample of GAIA astrometric binaries searching for possible
compact object companions. Their triage technique, already proposed in [1314],
relies on a good knowledge of the mass-luminosity relation (for main-sequence
stars only), in order to discard all the single stars that for one reason or
another are contaminating the sample. They compiled a sample of 177 systems
with highly-probable non-luminous massive companions, only 20% of the total
sample provided by GAIA. Among them, they identified 8 BH candidates with
masses larger than 2.4 $M_{\odot}$. The most promising candidate among them
was soon after the subject of two independent spectroscopic follow-ups which
recently confirmed its dark and compact nature, estimating a mass for the BH
of approximately 10 $M_{\odot}$ [1315, 1316]. As in [1315], we refer to the
source as GAIA BH1.
GAIA BH1 represents the first BH identified via its astrometric orbital
solution from GAIA as well as the nearest BH ever detected (in the solar
neighborhood, d = 474 pc). Although a slight discrepancy in the predicted mass
of the BH between the two groups, they both confirm that it is in a binary
system with a slowly-rotating G dwarf (Teff = 5850 K, log g = 4.5, and M =
0.93 $M_{\odot}$) and the system has an orbital period of 185.6 d. The
predicted motion of GAIA BH1’s photocentre on the sky over a 6-yr window and
its predicted orbit are shown in Figure 32, from [1315]. This detection is
pretty robust, as consistent results have been obtained by using 1) GAIA
astrometric data alone; 2) spectroscopic data alone (acquired as a follow-up
program to the astrometric detection); or 3) a joint analysis of both
measurements. The latter obviously provided the most stringent constraints on
the properties of the system.
A very similar claim has been made almost at the same time from another group
[1317]. They analyzed a sample of $\sim$65,000 binaries with both time-
resolved spectroscopic and astrometric data in Gaia DR3 and identified a
peculiar system made of a red giant star (Gaia DR3 5870569352746779008) and an
unseen companion of a minimum mass $>$ 5.25 $M_{\odot}$. Although the mass of
the visible star was rather uncertain, they argued that the unseen source is a
BH candidate at a 99% confidence level, based on the orbital parameters of the
system. Their conclusions were entirely dependent on the reliability of the
Gaia solution.
El-Badry and collaborators [1318] obtained additional spectra for this
specific target, in the course of a broader program to spectroscopically
follow-up Gaia DR3 binaries suspected to contain compact objects. These data
confirmed the compact and dark nature of the invisible companion, which they
called Gaia BH2. They were also able to better constrain the properties of the
system itself. A bright (G = 12.3), nearby (d = 1.16 kpc) star on the lower
red giant branch with mass M = 1.07 $\pm$ 0.19 $M_{\odot}$ orbiting a BH of
mass $M_{BH}$=8.9 $\pm$ 0.3 $M_{\odot}$ with a moderate eccentricity e=0.52
and a relative low inclination angle i = 35°. Interestingly, this system has
the longest period (1277 $\pm$ 1 days) among those discovered so far, slightly
lower than what predicted by the Gaia solution (1352.25 $\pm$ 45.50 days) and
reported by Tanikawa and colleagues [1317].
Figure 32: from El-Badry et al. [1315]: Left: The predicted astrometric orbit
of the photocentre of GAIA BH1 on the sky over a period of 6 years. The black
line indicates the total motion due to proper motion, parallax, and orbital
motion while the dotted blue line shows the contribution due to parallax and
proper motion alone. The predicted motion of the source, with parallax and
proper motion removed is instead shown in the middle panel based on the joint
astrometry and RV solution and in the right panel by using the pure astrometry
solution. The joint analysis provides the most stringent constraints on the
system.
Despite extensive research, to date GAIA BH1 and BH2 appear to be the only two
quiescent BHs enclosed in the Gaia DR3 sample of binaries that can be
detected. Figure 33 shows where the population of known accreting BHs detected
via X-ray observations are located as a function of period and distance from
us. When Gaia BH1 and Gaia BH2 are included (large black points) they occupy a
very different parameter space based on their properties. They have
significantly longer periods and closer to Earth than the rest. We do not have
enough statistics to make any claims, but it appears that Gaia BH1 and BH2 may
be the tip of the iceberg of the hitherto totally unknown binary population of
wide-orbit black holes.
Figure 33: from El-Badry et al. [1318]: Gaia BH1 and BH2 (black points) are
compared to Galactic BHs detected via accretion (in X-rays) in the reference
plane of distance and orbital period. As indicated in the legend, the red and
blue dots represent LMXBs and HMXBs respectively, while the cyan and purple
dots identify detached BHs detected so far either in the field or in star
clusters. As can be seen, the systems discovered by GAIA are in a part of this
parameter space unexplored by other known BHs, i.e. with longer periods and
closer to Earth than the rest of BHs.
Gaia BH1 and Gaia BH2 unfortunately will not be progenitors of any GW signals
as the luminous companion of the compact object is a low-mass star and there
is no chance that the system may become a BH-NS binary at any point in time.
However, the study of these systems remains extremely important as it allows
us to build a more complete view of the actual population(s) of NSs and BHs
existing in the Universe and if (at all) their properties change as a function
of the environment.
The release of Gaia DR4 data in late 2025 as well as further investigations of
GAIA DR3 will probably reveal many other interesting systems (e.g. massive
white dwarfs and NSs). For what concerns BHs, based on recent theoretical
predictions, Chawla et al. [1319]) argued that the extended Gaia mission can
astrometrically resolve $\approx$ 30-300 detached BH-stellar binaries in the
Milky Way, with a high probability of finding even some compatible with being
|
# Navigation Turing Test (NTT): Learning to Evaluate Human-Like Navigation
Sam Devlin Raluca Georgescu Ida Momennejad Jaroslaw Rzepecki Evelyn Zuniga
Gavin Costello Guy Leroy Ali Shaw Katja Hofmann
###### Abstract
A key challenge on the path to developing agents that learn complex human-like
behavior is the need to quickly and accurately quantify human-likeness. While
human assessments of such behavior can be highly accurate, speed and
scalability are limited. We address these limitations through a novel
automated Navigation Turing Test (ANTT) that learns to predict human judgments
of human-likeness. We demonstrate the effectiveness of our automated NTT on a
navigation task in a complex 3D environment. We investigate six classification
models to shed light on the types of architectures best suited to this task,
and validate them against data collected through a human NTT. Our best models
achieve high accuracy when distinguishing true human and agent behavior. At
the same time, we show that predicting finer-grained human assessment of
agents’ progress towards human-like behavior remains unsolved. Our work takes
an important step towards agents that more effectively learn complex human-
like behavior.
Machine Learning, ICML
## 1 Introduction
Developing agents capable of learning complex human-like behaviors is a key
goal of artificial intelligence research (Winston, 1984). Progress in this
area would lead to new applications, from engaging video games (Soni &
Hingston, 2008), to intuitive virtual assistants (Cowan et al., 2017) and
robots (Scheutz et al., 2007). Yet, despite striking successes in training
_highly skilled_ agents (Vinyals et al., 2019), quantifying and making
progress towards learning _human-like_ behavior remains an open challenge.
We propose an automated Navigation Turing Test (automated NTT or ANTT), a
novel approach to learn to quantify the degree to which behavior is perceived
as human-like by humans. While human judges can be highly accurate when
judging human-like behavior, their scalability and speed are limited. As a
result, it is prohibitive to use human assessments in, e.g., iterations of
training machine learning-based agents. We aim to remove this bottleneck,
providing a scalable and accurate proxy of human assessments.
We demonstrate our NTT approach on the complex task of goal-directed
navigation in 3D space. This task is highly relevant and practical because:
(1) it has been studied from a wide range of perspectives in both biological
(Banino et al., 2018; de Cothi, 2020) and artificial (Alonso et al., 2020;
Chaplot et al., 2020) agents, so is relatively well understood; and (2)
artificial approaches achieve a high level of task success (Alonso et al.,
2020), so proficient agents can be obtained enabling us to focus on human-
likeness. While the majority of research on learning 3D navigation has focused
on task success, we show that a high success is not sufficient for human-like
behavior.
Our key contributions compared to prior work (Sec. 2) are: (1) we introduce an
automated NTT for assessing human-like behavior and investigate six
representative architectures that capture principal design choices of this
task (Sec. 3); (2) we devise methodology for validating our automated NTT
against a human Navigation Turing Test (HNTT) and empirically demonstrate its
effectiveness (Sec. 4); (3) our results show high agreement of ANTT with human
judges when comparing human behavior to state-of-the-art reinforcement
learning agents (Sec. 5). At the same time, we demonstrate that capturing
finer gradations of human-likeness in artificial agents remains an open
challenge. In addition to the potential to drive learning of more human-like
behavior, our work opens the door to a deeper understanding of human decision
making when assessing human-like behavior.
## 2 Related Work
The question of how to systematically assess whether an automated agent
behaves in a human-like manner was compellingly addressed by Alan Turing
(Turing, 1950). Initially called the “imitation game”, and now adapted and
widely known as “Turing Test”, it asks a human judge to identify which of two
respondents is human, and which is machine. Turing’s initial thought
experiment considered a setup where the judge could interact with each
respondent through an appropriate natural language interface. Since its
conception, the idea has been widely adapted to assess the degree of human-
likeness in a range of scenarios, including to assess human-like play in video
games (Yannakakis & Togelius, 2018). Closest to our focus, a Turing Test was
used to assess bot behavior in the video game Unreal Tournament as part of the
2K BotPrize Contest (Hingston, 2009) and human-like play in the Mario AI
Championship (Shaker et al., 2013).
While a Turing Test, if well designed, can provide strong ground truth
assessments of whether humans judge behavior as human-like, several drawbacks
exist. Most important from a machine learning perspective is the need to
collect human judgments for every iteration of an agent, making the approach
extremely costly. This makes it prohibitive for use in typical machine
learning settings. Consequently, previous work has proposed automated proxies
of human-likeness (Kirby et al., 2009; Karpov et al., 2013; de Cothi, 2020).
Domain specific approaches to detecting (non) human-like behavior
automatically often rely on heuristics or rules. In navigation scenarios this
can include methods for detecting mistakes such as “getting stuck” (Kirby et
al., 2009; Karpov et al., 2013). Kirby et al. (2009) focus on socially
acceptable robot navigation with experiments performed in simulated navigation
tasks (e.g., passing people in a hallway). Human preferences were explicitly
encoded as constraints (e.g., a preference for moving to the right to let
someone pass), and learned behaviors were evaluated in simulation, by
assessing how often a given preference was violated. Karpov et al. (2013)
tackle the problem of “believable” bot navigation in the context of the
BotPrize competition (Hingston, 2009). Believable behavior is assessed using
rules for detecting mistakes such as getting stuck in cyclic behavior,
collisions with other entities, failing to move, or falling off a map. A key
drawback of rule-based automation is that they require high manual effort by
domain experts to construct. In addition, due to this high effort, they are
necessarily coarse and unable to capture nuanced aspects of human-likeness.
Addressing both the limitations of scalability of human Turing Tests and rule-
based approaches, learned models can serve as a proxy measure of human-
likeness. If such a model can learn accurate predictions of human-likeness
given ground truth judgments, these approaches could be deployed rapidly and
at scale to drive the development and training of more human-like agents.
Closely aligned with this goal of our work is the recent development of
imitation learning approaches that rely on a discriminator to learn to
distinguish between learned imitation and demonstrated expert behavior (Ho &
Ermon, 2016; Peng et al., 2018). However, while closely aligned with the goals
of our work in principle, the current state of the art in this field has not
yet reached the level required for empirically strong performance on complex
tasks and human-like behavior, with many evaluated on relatively less complex
behaviors and imitating other automated agents. Our work has the potential to
pave the way towards learning complex human-like behavior, and provides
important insights into the key design choices required for capturing notions
of human-likeness. We include representative examples closest to this approach
as baselines (model SYM-FF and VIS-FF, Tbl. 1).
Closely related to our work is the approach taken by de Cothi et al. (2020) to
compare artificial agent behavior in a maze navigation task to human and rat
behavior. They used a support vector machine classifier trained on a small set
of expert designed features to distinguish between model-free, model-based,
and successor-representation-based trained agents. The trained classifier was
then applied to behavior of humans and rats to estimate which agent behavior
was closest to that of biological agents (de Cothi et al., 2020; de Cothi,
2020). We base our approach on this setup, but extend it to take as input
lower-level features (e.g., trajectories, images) to avoid the need for expert
designed feature extraction and better align with the line of work on
imitation learning that forms the second inspiration to our work.
While our approach makes no specific assumption about the type of task for
which we aim to detect human-like behavior, we focus on the task of navigation
in 3D space. This task is well suited as it has been widely studied from a
range of perspectives, including neuroscience (Iaria et al., 2003; Banino et
al., 2018; de Cothi, 2020) and machine learning (Alonso et al., 2020; Chaplot
et al., 2020). In addition, recent years have seen rapid progress on machine
learned agents that achieve high task success, allowing us to focus our
attention on whether the learned successful behavior is indeed human-like. A
driver of progress on learning to navigate has been the availability of
simulated environments and evaluation frameworks. Anderson et al. (2018)
survey recent progress and evaluation frameworks for 3D navigation. In line
with the majority of work in this area, the focus is on evaluating task
success. Recent work on learning to navigate with a machine learning focus
includes improvements of sample efficiency, e.g., through unsupervised
pretraining (Li et al., 2020) and object semantics (Chaplot et al., 2020).
Here we leverage the work of Alonso et al. (2020), as it most closely aligns
with our requirement for achieving high task success in a 3D game environment.
At the intersection of machine learning and neuroscience, machine learned
models have been proposed as models for studying biological behavior (Banino
et al., 2018; de Cothi, 2020). As before, the primary evaluation of these
focused on task success (Banino et al., 2018), with the notable exception of
de Cothi et al. (2020) which forms the starting point of our work.
## 3 Method
In this work we approach the task of predicting human judgments of whether an
observed behavior appears human-like as a classification task, as detailed in
Section 3.1. Our key focus is the question of which type of input space (Sec.
3.2) and model (Sec. 3.3) can best capture human judgments.111See Appendix A.1
for training details and hyperparameters.
### 3.1 Problem setup
We formalize the task of assessing human-likeness as binary classification. We
assume input data in the form of time-indexed trajectories $x_{0:T}$ of length
$T$, where $x_{t}\in\mathbb{R}^{d}$ denotes the $d$-dimensional observation at
time step $t$.
Our goal is to obtain a classifier that achieves high alignment with ground
truth human judgments (detailed in Sec. 4.4). Ground truth takes the form
$y^{*}:x^{(1)}\succ x^{(2)}$ to indicate that a single human judged the
trajectory represented by $x^{(1)}$ to be more human-like than $x^{(2)}$. We
obtain repeat measurements from several human judges for each pair. We focus
on two evaluation measures: (1) accuracy compared to human ground truth
(aggregated by majority vote), and (2) Spearman correlation (a non-parametric
rank correlation measure robust to outliers (Croux & Dehon, 2010)) compared to
a ranking of agent trajectories by the proportion of judges that assessed them
as human-like. Together, these capture coarse and fine-grained agreement
between our classification models and human judgments of human-likeness.
In this work, we assume that human ground truth judgments are only available
at test time. This is a realistic assumption given the resource intensive
nature of this type of data. To address this challenge, we propose training
our models with _proxy labels_ , which are derived from known agent identity
$\hat{y}$. Thus, our classifiers are trained to maximize accuracy when
predicting whether a trajectory was truly generated by a human player or an
artificial agent, using a binary cross entropy loss:
$L=-[\hat{y}_{H=1}\log(p_{H=1})+(1-\hat{y}_{H=1})\log(1-p_{H=1})]$, where we
denote by $\hat{y}_{H=1}$ the pseudo label indicating that trajectory was
truly generated by a human player, and by $p_{H=1}$ the probability that the
current model assigns to the trajectory having been generated by a human
player.
A key question is whether the resulting models can generalize to align well
with the degree to which human judges actually _perceive_ a trajectory as more
or less human. A key design choice that we expect to influence this ability to
capture human judgments is the input space the classifier operates on. We
discuss these choices in the next section.
### 3.2 Choice of input space
Classifier input can be represented in a variety of ways. For example,
presenting input in the form of video frames derived directly from the game
footage, is likely to align well with the information available to human
judges when they decide on whether a given behavior appears human-like. We
hypothesize that observation spaces that are well aligned with the information
available to human judges will result in higher accuracy on the test task of
capturing human judgments of human-like behavior. However, more complex
observations, such as video, may suffer from low amounts of training data,
leading to a potential trade-off between model fidelity and sample efficiency.
(a) Bar-code input.
(b) Sample of a top-down trajectory ($320\times 200$ pixels).
(c) Video frames ($320\times 200$)
Figure 1: Examples of the inputs used by our classifiers: (1) bar-code of a
video (used by BC-CNN), (1) top-down trajectory (TD-CNN), (1) video (frame
sequence, VIS-FF and VIS-GRU).
We consider the following choices (samples in Fig. 1):
SYM. _Symbolic_ observations provide a low-dimensional representation of each
step along the agent trajectory. Here, we use absolute agent location in 3D
space as it provides a minimal representation of agent state.
VIS. _Visual_ observations provide images of size 320x200 of the actual game
footage, rendered without overlays such as game score. This observation type
is expected to be well aligned with the information available to a human judge
when assessing whether a behavior appears human-like.
In our architectures, we explore feed-forward and recurrent models for
encoding observations, as detailed in Section 3.3. In addition, we consider
two compressed representations that abstract the respective observations in a
single image:
TD. _Top-down_ trajectory projection (a single 2D image per trajectory),
obtained by projecting the _symbolic_ representation (agent position) along
the ”up” direction (z-coordinate).
Name | Input (Sec. 3.2) | Encoder | Pre-Training | Notes
---|---|---|---|---
SYM-FF | _Symbolic_ | Feedforward Network | [None] | Equivalent to typical discriminator architectures in adversarial imitation learning (Ho & Ermon, 2016)
SYM-GRU | _Symbolic_ | GRU (Cho et al., 2014) | [None] |
VIS-FF | _Visual_ | VGG-16 (Simonyan & Zisserman, 2014) | Imagenet (CNN) | Equivalent to discriminators for adversarial imitation learning from visual observations (Li et al., 2017)
VIS-GRU | _Visual_ | CNN (VGG-16) + GRU | Imagenet (CNN) |
TD-CNN | _Top-down_ | CNN (VGG-16) | Imagenet (CNN) |
BC-CNN | _Bar-code_ | CNN (VGG-16) | Imagenet (CNN) |
Table 1: Overview of the ANTT model architectures investigated in this work.
See Sections 3.2 and 3.3 for details.
BC. _Bar-code_ encodes _visual_ observations (a single 2D image per trajectory
video). It is obtained by collapsing every video frame into a single line by
averaging along the y-dimension of the frame, and then stacking those lines
into a single image. If the input video had 600 frames of size 320x200, the
resulting bar-code has size 320x600.
We hypothesize that the compressed TD and BC representations capture key
patterns of human-likeness while providing higher information density than
symbolic or visual observations, leading to more effective classification.
### 3.3 Models
Following from our problem definition above (Section 3.1), we now detail our
proposed classification models. Each model tests specific hypotheses about how
to best learn to align with human judgments. A summary of all models is
provided in Table 1, with details as follows:
FF. Feedforward models extract features from individual states, without taking
into account the sequential structure along a trajectory. We construct this
type of model as a representative baseline in line with discriminators typical
of adversarial imitation learning (Ho & Ermon, 2016).
GRU. Recurrent models explicitly model sequence information and are expected
to more directly align with human assessments (human assessors view
observations in the form of videos, c.f., Section 4). We encode sequences
using Gated Recurrent Unit (GRU) networks (Cho et al., 2014).
CNN. Convolutional models are applied to image input (visual, top-down and
bar-code observations). We use a VGG-16 (Simonyan & Zisserman, 2014) pre-
trained on Imagenet (Deng et al., 2009) to extract visual features.
For all model training, input batches of individual observations (feedforward)
and sequences (GRU) are sampled uniformly from trajectories. At test time,
predictions are aggregated per trajectory using the majority vote. Learning
aggregation at the trajectory level is an important direction of research,
however improving simple and robust aggregation baselines with learned
approaches is challenging (Radford et al., 2018; Lan et al., 2020).
Figure 2: Validation accuracy over training epochs for our ANTT models (5-fold
cross validation). Symbolic models achieve relatively lower validation
accuracy, however, our later results show they are less prone to overfitting
(See Section. 5.2).
Before moving forward, we validate that the proposed models are effective on
the task they are trained to perform - distinguishing between behavior
generated by human vs artificial agents. The validation curves in Figure 2
show that all proposed models learn to distinguish between these on our
validation set. This confirms that we have a strong basis for evaluating which
of the proposed approaches matches ground truth provided by human assessors.
Our main results on this question will be presented in Section 5, after we
detail our approach to validating the classifiers in Section 4.
## 4 Validating ANTT by Human NTT
This section details our approach to validating the methods proposed for
automatically evaluating human-like behavior (Sec. 3) with a human NTT (HNTT).
The following components are required for this validation, and are discussed
in turn below. First, we detail the 3D navigation task on which human-like
behavior is assessed (Sec. 4.1). Second, we need to collect behavior data from
human players (Sec. 4.2) and artificial agents (Sec. 4.3). Third, we obtain
ground truth human judgments of human-likeness (Sec. 4.4).222See Appendices
for further details: A.2 (agent training), A.3 (human data collection) and A.4
(evaluation).
### 4.1 Definition of the 3D navigation task
Our experiments are conducted in a modern AAA video game.333We thank
[anonymized] for providing access to this game. A screenshot of the navigation
task is shown in Figure 3, together with a top-down view of the map. The task
is set up on a map that is representative in size and complexity for a modern
3D video game, and comparable to recent navigation experiments, such as the
large game map explored by Alonso et al. (2020). The players’ spawning
position is on an island outside the main map, connected by a set of 3
available jump areas. Agents must learn to navigate to a target location on
the main map, represented by the blue containers in the left screenshot in
Figure 3. On each navigation episode, the target spawns uniformly at random in
one of 16 possible locations. In preliminary tests we confirmed that human
players can successfully navigate to all goal locations within $26$ seconds on
average.
Figure 3: Screenshot of the 3D navigation task as perceived by human players
(left). Mini map of the level (right), indicating as a blue circle the
player’s spawning location (bottom) and in green the 16 possible goal
locations. The main map is approximately 500x320 meters in size and includes
complex 3D elements. The screenshot is not representative of the final game
play and visuals.
### 4.2 Collecting human navigation data
We now describe how we collected human navigation data.
Requirements. Data from a total of 7 human players was collected using the
replay capture engine in an experimental game build implementing the
navigation task (Section 4.1). Human players’ expertise with video games
varied, ranging from casual to advanced players. All players had high
familiarity with the game map and task, and recorded replays on machines that
met the system requirements of the experimental game build, including GPU
rendering support.
Parameters and Constraints. The navigation target was indicated on-screen in a
mini-map that was shown throughout the navigation task (Fig. 3). There was a
limited action space: agents and human players could only navigate (e.g.,
jump, attack and special abilities were disabled). The human players used
keyboard and mouse navigation only.
Video Generation. The episode replays were saved as MP4. A total of 140 human
recordings were collected, which we split into 100 videos (from 4 players) for
classifier training and validation, and 40 (3 remaining players) for testing.
Post-Processing. We applied post-processing edits to the videos for final
inclusion into the HNTT ground truth assessment (Section 4.4). These edits are
defined by a set of rules which were uniformly applied to all the videos. Rule
1: Mask identifying information (computer name tag). Rule 2: Cut all videos to
start the moment the character lands on the main map.444This is to focus our
current validation on ground-based navigation. Assessing human-like flying is
left for future work. Rule 3: Cut the last 1-3 seconds of the human player
videos to just before they stopped moving to correct for an effect of the
human data collection process, where human players had to manually end the
recording and thus appeared briefly “stopped” at the end of each episode.
### 4.3 Collecting agent navigation data
This section details training of the artificial agents used to generate agent
navigation data, including agent architectures, training process, and agent
navigation data collection.
Figure 4 shows our two reinforcement learning agent architectures: symbolic
and hybrid. Both are representative examples of current state-of-the-art
approaches to learning 3D navigation in video games, closely following very
recent work by Alonso et al. (2020). We chose this starting point as a
representative approach that learns effective navigation in a game similar to
ours, and select two observation spaces and architectures for which we expect
high task performance based on the ablation studies of Alonso et al. (2020).
Symbolic. The _symbolic_ agent (Fig. 4) observes a low-dimensional
representation of the goal (relative angle and distance) and game state (agent
position, average frame depth), processed using a feed forward network. We
hypothesize that this compact observation space enables sample-efficient
learning but relatively less human-like behavior.
Hybrid. The _hybrid_ agent (Fig. 4) receives the symbolic observation, as
above, and a 32x32 depth buffer of the game footage. Image features are
extracted using a CNN before concatenating with the output of the feed forward
layers that process symbolic features. Given access to (coarse) visual and
depth information, we hypothesize the agent to be better aligned with the type
of information used by human players and to consequently learn more human-like
behavior.
Both agents use a discrete action space: moving forwards, left and right to a
set of given degrees (30, 45, 90), as well as turning in place. Both agents
receive the same reward signal during training. This consists of a dense
reward for minimizing the distance between the goal location and the agent’s
position, a +1 reward for reaching the target, and a -1 penalty for dying, as
agents are able to jump off the edges of the map. In addition, a small per-
step penalty of $-0.01$ encourages efficient task completion. Episodes end
when agents reach the goal radius or after $3,000$ game ticks ($\approx 50$
seconds, $210$ agent steps).
The agents were trained using PPO (Schulman et al., 2017), implemented in
Tensorflow (Abadi et al., 2015). Behavior policy rollouts during training were
collected through a scalable distributed framework on $60$ parallel game
instances. In each training episode, the agent spawned in one of the 26
predefined locations, both on the main map and in the spawn area, with a
$34\%$ preference for the latter. This uniform domain randomization (Tobin et
al., 2017) was found sufficient for sample efficient learning in preliminary
experiments.
(a) Symbolic agent model architecture
(b) Hybrid agent model architecture Figure 4: Overview of the agent
architectures used in this study.
Before proceeding to the next stage, we verify that the selected agent models
successfully learn the navigation task. Figure 5 verifies that the agent
models used here are representative of the state of the art of learning to
navigate in complex 3D games. As shown, agents learn to reliably reach the
goal within the first 5M training steps. On average, the hybrid agent achieves
task completion 10% faster than the symbolic agent. This demonstrates that our
trained agents achieve the required level of task success, allowing us to
focus on whether the learned behavior is also human-like.
Figure 5: Time to goal over training steps (5 seeds each). Both models achieve
high task success. The high variance observed in the hybrid agent is due to
one seed that had a delay in convergence.
To capture agent navigation data we rolled out the model checkpoints after 20M
training steps. The rolled out policies are non-deterministic. The starting
position for all rollouts was the spawn area (as for human players). The
episodes were saved as videos and then sampled to match the navigation goals
in the selected human navigation samples. The agent navigation was post-
processed in line with the human navigation data (Sec. 4.2).
### 4.4 Collecting ground truth human judgments
We designed two behavioral human NTT (HNTT) studies to establish ground truth
based on collecting human judgments of agent vs. human navigation in the game.
Each study was administered in the form of an online survey.
IRB Approval. The survey was approved by our institution’s Institutional
Review Board (IRB), authorizing us to collect data from internal employees. An
IRB approved digital consent form was obtained before participation.
Participants. A total of 60 human assessors participated in the HNTT. We
recruited participants from internal mailing lists, none of the participants
were familiar with our study. Participants were required to be 18 years or
older. No additional demographic information was collected. Participants were
asked to indicate their familiarity with third-person action games and the
game title used in this study.
Study Design. Each HNTT study consisted of 10 Turing Test trials, in each of
which the stimuli consisted of two side-by-side videos, A and B, of an avatar
navigating a game environment (Figure 6). The task of the participant was to
watch each video and answer the question “Which video is more likely to be
human?”, explain their rationale, and indicate uncertainty of their decision
on a 5-point Likert scale varying from “extremely certain” to “extremely
uncertain”.
In each study all participants viewed the same ten HNTT trials with the same
stimuli, presented in randomized order per participant (within-subject
design). Videos A and B were selected from a set of short videos in which the
symbolic or hybrid agent (Section 4.3), or a human player (Section 4.2)
completed the navigation task (Section 4.1).
Stimuli and conditions. The presented video stimuli and trial conditions
differed between Study 1 and Study 2 according to Table 2, administered as
follows.
vs. | Human | Hybrid | Symbolic
---|---|---|---
Human | - | Study 1 | Study 2
Hybrid agent | Study 1 | - | Study 1,2
Symbolic agent | Study 2 | Study 1,2 | -
Table 2: Table of HNTT comparison conditions in Study 1 and Study 2. Each
study consisted of 6 human vs. agent comparison trials, and 4 trials comparing
the two agents.
Study 1. Human participants were asked to judge a human player vs. a hybrid
agent in 6 trials with side-by-side video pairs, and a symbolic agent vs. a
hybrid agent in 4 trials with side-by-side video pairs (Table 2).
Study 2. Human participants were asked to judge a human player vs. a symbolic
agent in 6 trials with side-by-side video pairs, and a symbolic agent vs. a
hybrid agent in 4 trials with side-by-side video pairs (Table 2).
The video stimuli used in the study were sampled uniformly at random from our
available recordings, matching the video pairs on task goal location. See
Figure 6 for a screenshot.
Figure 6: Example of a Human Navigation Turing Test (HNTT) trial. Videos are
not representative of actual game play and visuals.
## 5 Results
In this section we detail our main empirical results of two Navigation Turing
Test studies, obtained according to the validation approach detailed above
(Sec. 4).
### 5.1 HNTT: Human assessments of human-likeness
We analyzed human behavioral data from two human NTT (HNTT) studies, in each
of which participants observed 10 pairs of video stimuli, judged which is more
human-like, and marked their uncertainty. These results were used to establish
ground truth for automated NTT studies (Sec. 4.4). We first assessed the
extent to which human participants accurately judged videos of real human
players as more human-like than videos of RL agents (Fig. 7). Participants
accurately detected human players above chance in both Study 1 (mean=$0.84$,
std=$0.16$, n=$30$ throughout for both studies) and 2 (mean=$0.77$,
std=$0.16$). On average, participants were slightly more accurate in judging
humans as more human-like in Study 1 compared to Study 2 (Mann-Whitney
$U=341$, $p=0.046$). We then analyzed the reported uncertainty of
participants, reported on a scale of 1 (extremely certain) to 5 (extremely
uncertain), Fig. 8. In Study 1 participants indicated a mean certainty of
$2.1$ (std=$0.47$), corresponding to the level “somewhat certain”. In Study 2,
participants indicated an average certainty of $2.6$ (std=$0.84$), between
“somewhat certain” and “neutral”. The distributions in the two groups differed
significantly (Mann–Whitney $U=257.5$, $p=0.002$), indicating that
participants were less certain of their decisions in Study 2.
Figure 7: HNTT Results. Left: violin plots display proportion of videos with
true human behavior (ground truth) that were judged more human-like. Each
participant’s agreement with ground truth is a dot in the swarm plot. Right:
Reported uncertainty for the same judgment (1: extremely certain to 5:
extremely uncertain).
Next we compared how participants judged symbolic and hybrid agents (Sec.
4.3). In both studies, and in line with our prediction, human participants
judge the hybrid agent to be more human-like when directly compared with
symbolic agents (Fig. 8). At the same time, we observed substantial
variability among human judges. A comparison across studies reveals an
interesting pattern. In hybrid-symbolic comparisons of Study 1, on average
$0.78$ of all participants judged the hybrid agent to be more human-like
(std=$0.25$). This proportion is substantially higher than that of Study 2
(mean=$0.62$, std=$0.30$), with a statistically significant difference (Mann-
Whitney $U=328.0$, p=$0.013$).
Figure 8: Responses in direct comparisons of symbolic and hybrid agents in
Study 1 (left) and 2 (right). Y-axis: number of times a participant judged an
agent more human-like (out of $4$). X-axis: participants, ordered by how often
they chose the hybrid agent.
Classifier | Identity Accuracy | Human-Agent Accuracy | Human-Agent Rank | Hybrid-Symbolic Accuracy | Hybrid-Symbolic Rank
---|---|---|---|---|---
SYM-FF | $0.850$ $(0.062)$ | $0.850$ $(0.062)$ | $0.364$ $(0.043)$ | $0.475$ $(0.166)$ | $-0.244$ $(0.252)$
SYM-GRU | $0.850$ $(0.082)$ | $0.850$ $(0.082)$ | $0.173$ $(0.049)$ | $0.400$ $(0.200)$ | $-0.249$ $(0.210)$
VIS-FF | $0.633$ $(0.041)$ | $0.633$ $(0.041)$ | $-0.041$ $(0.160)$ | $0.225$ $(0.050)$ | $-0.165$ $(0.286)$
VIS-GRU | $0.767$ $(0.097)$ | $0.767$ $(0.097)$ | $0.220$ $(0.267)$ | $0.425$ $(0.127)$ | $-0.056$ $(0.331)$
TD-CNN | $0.583$ $(0.075)$ | $0.583$ $(0.075)$ | $0.222$ $(0.059)$ | $0.525$ $(0.094)$ | $-0.093$ $(0.149)$
BC-CNN | $0.717$ $(0.145)$ | $0.717$ $(0.145)$ | $-0.009$ $(0.131)$ | $0.475$ $(0.050)$ | $-0.095$ $(0.412)$
Table 3: Classifier accuracy and rank compared to human judgments on held-out
test data. All results are the mean (and standard deviation) from 5 repeats of
training the classifier with hyperparameter settings chosen by their average
validation accuracy in 5-fold cross-validation.
One interpretation of this finding is that participants’ judgments are non-
transitive: they judge humans more human-like when directly compared to hybrid
agents (Study 1), and less human-like when compared to the symbolic agent
(Study 2), while at the same time judging the symbolic agent less human-like
in a direct comparison to the hybrid agent. Another interpretation is that
this may be due to the specific videos used in Study 2. Discerning between
these hypotheses is an interesting direction for further investigation.
Overall, our analysis of human judgments show that, in line with expectations
and on average, videos of human navigation are judged more human-like than
those of artificial agents. In addition, as hypothesized in Sec. 4.3, the
hybrid agent is judged more human-like when directly compared with the
symbolic agent (Figure 8). These results validate our HNTT methodology,
enabling us to validate our classifiers (artificial judges) against the human
judgment ground truth reported in this section. Future behavioral studies can
explore individual differences in human judgments of both human and artificial
navigation.
### 5.2 Evaluating the ANTT
We now turn to our results of evaluating our proposed ANTT. We aim to identify
classification models (Sec. 3.3) that accurately capture human judgments of
human-likeness. All models were trained to distinguish human from agent
trajectories. Training and hyperparameter tuning was performed using 5-fold
cross validation on trajectories generated by agent checkpoints and human
players that were fully separate from those that generated test data. We
collected human judgments (Sec. 5.1) for test trajectories only.
Table 3 shows our results for all considered classifier models (see Table 1
for model details). First, we verify that all models generalize to the held
out training data. We assess this by comparing classifier predictions
(aggregated on the trajectory level by majority vote) against the true
identity of the agent (i.e., human or agent) termed identity accuracy. We find
that all models generalize to the test data, with results for the symbolic
models particular robust to overfitting. Results range from an accuracy of
$0.583$ for TD-CNN to $0.850$ for both symbolic models. These results also
validate our trajectory level aggregation by majority voting approach,
demonstrating that is achieves high accuracy when comparing to trajectory
level human judgments.
Our main results compare classifier models to human judgments, split into
human-agent accuracy and hybrid-symbolic accuracy (Table 3). As the majority
vote aggregation of human responses was equal to the true identity for all
questions that compared human videos with agent videos, we again observe the
same high accuracy on human-agent judgments. However, accuracy is low for
hybrid-symbolic pairs across all classifiers.
Our results are further confirmed when assessing human-agent and hybrid-
symbolic rank correlation between human judgments and model predictions. We
compute these using Spearman’s rank correlation (Croux & Dehon, 2010), as
follows. We aggregate human ground truth judgments for human-agent
(separately: symbolic-hybrid) pairs based on the percentage of human judges
that labeled a human (or hybrid agent in hybrid-symbolic) trajectory as more
human-like. This was compared to a ranking of model output logits. We observe
medium rank agreement and low variance for the symbolic and topdown CNN models
on human-agent rankings. However, rankings of all models on hybrid-symbolic
agent rankings are low (negative values indicate anti-correlated rankings) and
have high variance.
Our classification results demonstrate that classifier models do generalize
well to unseen test trajectories. Agreement with human judgments is high when
assessing the human-likeness of agents compared to human players. Highest
agreement is observed for symbolic models. This relative performance of
classifiers is unexpected, given that the symbolic classifiers observe a
representation that is very different from that used by human judges for
decision making. Based on our results, it is likely that the visual models
overfit given the richer representations and higher model capacity. In
addition, we find that current classifiers perform poorly when assessing the
degree of human-likeness of pairs of agents. Our results motivate important
follow up work to close this gap between matching human judgments on human-
agent vs agent-agent pairs.
## 6 Conclusion
We introduce the automated Navigation Turing Test (ANTT), a new approach to
automatically evaluating the human-likeness of navigation behavior. Our
results, comparing to a human NTT, show that a high level of agreement between
automated and human judgments can be achieved, opening the door to using
classification approaches as scalable proxies for driving the development of
more human-like agents. Best performance is achieved by classification models
that operate on compact representations of agent trajectories, in particular
_bar-code_ as well as symbolic representations. At the same time, we show that
detecting finer-grained human judgments that capture agents’ improvements
towards human-likeness remains an open challenge for future work. Our work
takes an important step on the dual goals of understanding human assessment of
human-like behavior and towards agents capable of learning complex human-like
behaviors.
## Acknowledgements
We thank Batu Aytemiz, Dave Bignell, Mikhail Jacob, Mingfei Sun, Robert
Loftin, Sebastian Nowozin, and Stephanie Reikine for their feedback and
valuable discussions. We are grateful to Ninja Theory and Microsoft for their
support. We thank the anonymous participants of our human NTT studies for
their time and effort.
## References
* Abadi et al. (2015) Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G. S., Davis, A., Dean, J., Devin, M., Ghemawat, S., Goodfellow, I., Harp, A., Irving, G., Isard, M., Jia, Y., Jozefowicz, R., Kaiser, L., Kudlur, M., Levenberg, J., Mané, D., Monga, R., Moore, S., Murray, D., Olah, C., Schuster, M., Shlens, J., Steiner, B., Sutskever, I., Talwar, K., Tucker, P., Vanhoucke, V., Vasudevan, V., Viégas, F., Vinyals, O., Warden, P., Wattenberg, M., Wicke, M., Yu, Y., and Zheng, X. TensorFlow: Large-scale machine learning on heterogeneous systems, 2015\.
* Alonso et al. (2020) Alonso, E., Peter, M., Goumard, D., and Romoff, J. Deep reinforcement learning for navigation in aaa video games. _arXiv preprint:2011.04764_ , 2020.
* Anderson et al. (2018) Anderson, P., Chang, A., Chaplot, D. S., Dosovitskiy, A., Gupta, S., Koltun, V., Kosecka, J., Malik, J., Mottaghi, R., Savva, M., et al. On evaluation of embodied navigation agents. _arXiv preprint:1807.06757_ , 2018.
* Banino et al. (2018) Banino, A., Barry, C., Uria, B., Blundell, C., Lillicrap, T., Mirowski, P., Pritzel, A., Chadwick, M. J., Degris, T., Modayil, J., et al. Vector-based navigation using grid-like representations in artificial agents. _Nature_ , 557(7705):429–433, 2018.
* Chaplot et al. (2020) Chaplot, D. S., Gandhi, D. P., Gupta, A., and Salakhutdinov, R. R. Object goal navigation using goal-oriented semantic exploration. _Advances in Neural Information Processing Systems_ , 33, 2020.
* Cho et al. (2014) Cho, K., Van Merriënboer, B., Bahdanau, D., and Bengio, Y. On the properties of neural machine translation: Encoder-decoder approaches. _arXiv preprint:1409.1259_ , 2014.
* Cowan et al. (2017) Cowan, B. R., Pantidi, N., Coyle, D., Morrissey, K., Clarke, P., Al-Shehri, S., Earley, D., and Bandeira, N. ” what can i help you with?” infrequent users’ experiences of intelligent personal assistants. In _Proceedings of the 19th International Conference on Human-Computer Interaction with Mobile Devices and Services_ , pp. 1–12, 2017\.
* Croux & Dehon (2010) Croux, C. and Dehon, C. Influence functions of the spearman and kendall correlation measures. _Statistical methods & applications_, 19(4):497–515, 2010.
* de Cothi et al. (2020) de Cothi, W., Nyberg, N., Griesbauer, E.-M., Ghanamé, C., Zisch, F., Lefort, J., Fletcher, L., Newton, C., Renaudineau, S., Bendor, D., Grieves, R., Duvelle, É., Barry, C., and Spiers, H. J. Predictive maps in rats and humans for spatial navigation. _bioRxiv preprint: 2020.09.26.314815_ , 2020.
* de Cothi (2020) de Cothi, W. J. _Predictive maps in rats and humans for spatial navigation_. PhD thesis, UCL (University College London), 2020.
* Deng et al. (2009) Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., and Fei-Fei, L. Imagenet: A large-scale hierarchical image database. In _2009 IEEE conference on computer vision and pattern recognition_ , pp. 248–255. Ieee, 2009.
* Hingston (2009) Hingston, P. A turing test for computer game bots. _IEEE Transactions on Computational Intelligence and AI in Games_ , 1(3):169–186, 2009.
* Ho & Ermon (2016) Ho, J. and Ermon, S. Generative adversarial imitation learning. In _Proceedings of the 30th International Conference on Neural Information Processing Systems_ , pp. 4572–4580, 2016.
* Iaria et al. (2003) Iaria, G., Petrides, M., Dagher, A., Pike, B., and Bohbot, V. D. Cognitive strategies dependent on the hippocampus and caudate nucleus in human navigation: variability and change with practice. _Journal of Neuroscience_ , 23(13):5945–5952, 2003.
* Karpov et al. (2013) Karpov, I. V., Schrum, J., and Miikkulainen, R. Believable bot navigation via playback of human traces. In _Believable Bots_ , pp. 151–170. Springer, 2013.
* Kirby et al. (2009) Kirby, R., Simmons, R., and Forlizzi, J. Companion: A constraint-optimizing method for person-acceptable navigation. In _RO-MAN 2009-The 18th IEEE International Symposium on Robot and Human Interactive Communication_ , pp. 607–612. IEEE, 2009.
* Lan et al. (2020) Lan, O., Huang, X., Lin, B. Y., Jiang, H., Liu, L., and Ren, X. Learning to contextually aggregate multi-source supervision for sequence labeling. In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pp. 2134–2146, 2020.
* Li et al. (2020) Li, J., Wang, X., Tang, S., Shi, H., Wu, F., Zhuang, Y., and Wang, W. Y. Unsupervised reinforcement learning of transferable meta-skills for embodied navigation. In _Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition_ , pp. 12123–12132, 2020.
* Li et al. (2017) Li, Y., Song, J., and Ermon, S. Inferring the latent structure of human decision-making from raw visual inputs. _arXiv preprint:1703.08840_ , 2, 2017.
* Peng et al. (2018) Peng, X. B., Kanazawa, A., Toyer, S., Abbeel, P., and Levine, S. Variational discriminator bottleneck: Improving imitation learning, inverse rl, and gans by constraining information flow. In _International Conference on Learning Representations_ , 2018.
* Radford et al. (2018) Radford, B. J., Richardson, B. D., and Davis, S. E. Sequence aggregation rules for anomaly detection in computer network traffic. _arXiv preprint:1805.03735_ , 2018.
* Scheutz et al. (2007) Scheutz, M., Schermerhorn, P., Kramer, J., and Anderson, D. First steps toward natural human-like hri. _Autonomous Robots_ , 22(4):411–423, 2007.
* Schulman et al. (2017) Schulman, J., Wolski, F., Dhariwal, P., Radford, A., and Klimov, O. Proximal policy optimization algorithms. _arXiv preprint:1707.06347_ , 2017.
* Shaker et al. (2013) Shaker, N., Togelius, J., Yannakakis, G. N., Poovanna, L., Ethiraj, V. S., Johansson, S. J., Reynolds, R. G., Heether, L. K., Schumann, T., and Gallagher, M. The turing test track of the 2012 mario ai championship: entries and evaluation. In _2013 IEEE Conference on Computational Inteligence in Games (CIG)_ , pp. 1–8. IEEE, 2013.
* Simonyan & Zisserman (2014) Simonyan, K. and Zisserman, A. Very deep convolutional networks for large-scale image recognition. In _International Conference on Learning Representations (ICLR)_ , 2014.
* Soni & Hingston (2008) Soni, B. and Hingston, P. Bots trained to play like a human are more fun. In _2008 IEEE International Joint Conference on Neural Networks (IEEE World Congress on Computational Intelligence)_ , pp. 363–369. IEEE, 2008\.
* Tobin et al. (2017) Tobin, J., Fong, R., Ray, A., Schneider, J., Zaremba, W., and Abbeel, P. Domain randomization for transferring deep neural networks from simulation to the real world. In _2017 IEEE/RSJ international conference on intelligent robots and systems (IROS)_ , pp. 23–30. IEEE, 2017.
* Turing (1950) Turing, A. Computing machinery and intelligence. _Mind_ , 59(236):433–460, 1950.
* Vinyals et al. (2019) Vinyals, O., Babuschkin, I., Czarnecki, W. M., Mathieu, M., Dudzik, A., Chung, J., Choi, D. H., Powell, R., Ewalds, T., Georgiev, P., et al. Grandmaster level in starcraft ii using multi-agent reinforcement learning. _Nature_ , 575(7782):350–354, 2019.
* Winston (1984) Winston, P. H. _Artificial intelligence, an MIT perspective_. MIT Press, 1984.
* Yannakakis & Togelius (2018) Yannakakis, G. N. and Togelius, J. _Artificial intelligence and games_ , volume 2. Springer, 2018.
|
# cGraph: Graph Based Extensible Predictive Domain Threat Intelligence
Platform
Wathsara Daluwatta†, Ravindu De Silva†, Sanduni Kariyawasam†, Mohamed Nabeel‡,
Charith Elvitigala†, Kasun De Zoysa†, Chamath Keppitiyagama† †University of
Colombo School of Computing‡Qatar Computing Research InstituteSri Lanka, Qatar
(2022)
###### Abstract.
Ability to effectively investigate indicators of compromise and associated
network resources involved in cyber attacks is paramount not only to identify
affected network resources but also to detect related malicious resources.
Today, most of the cyber threat intelligence platforms are reactive in that
they can identify attack resources only after the attack is carried out.
Further, these systems have limited functionality to investigate associated
network resources. In this work, we propose an extensible predictive cyber
threat intelligence platform called cGraph that addresses the above
limitations. cGraph is built as a graph-first system where investigators can
explore network resources utilizing a graph based API. Further, cGraph
provides real-time predictive capabilities based on state-of-the-art inference
algorithms to predict malicious domains from network graphs with a few known
malicious and benign seeds. To the best of our knowledge, cGraph is the only
threat intelligence platform to do so. cGraph is extensible in that additional
network resources can be added to the system transparently.
malicious domains, belief propagation, graph inference, kubernetes, apache
spark
††copyright: acmcopyright††journalyear: 2022††conference: Online, hosted by
Lyon, France; April 25-29, 2022; The Web Conference 2022
## 1\. Introduction
Internet domains are the launch pad for many cyber attacks we observe
nowadays. One effective way to reduce the damage caused by such attacks is to
identify the domains involved early and take actions such as blocking, taking
down or sinkholing. Further, these malicious domains have underlying
associations with other Internet resources that help to uncover malicious
infrastructures. We observe that existing threat intelligence systems ,such as
Cisco Umbrella Investigate and Anomali, fail to take advantage of such
associations to detect stealthy malicious domains nor do they capture such
associations in a user-friendly graphical form that allows users to easily
explore related Internet resources. All these solutions are reactive in nature
and by the time security operation teams observe them the damage is already
done. A predictive solution would immensely assist in mitigating such therats.
Motivated by these gaps in the industry, we build a proactive threat
intelligence platform that can uncover malicious domains early and visualize
associated Internet resources in a graph format for easy analysis.
Building an investigation platform in this domain involves a unique set of
challenges. First, the amount of data one needs to process on a daily basis is
huge. On average, we observe around 850 million network records per day.
Second, the network data is highly dynamic. For example, domains change their
hosting IPs periodically, the ownership of IPs change over time, new domains
are created and existing domains cease to exist. Third, intelligence sources
such as Phishtank, VirusTotal and GSB, contain limited information about the
status of domain and they are slow to detect malicious domains (Nabeel et al.,
2020). This is due to the fact that most of such systems rely on either
website content or user accessing of these websites to ascertain whether they
are malicious or not. Providing timely and up-to-date intelligence is thus a
challenge on its own. Fourth, network data, though mostly structured, is
available in different data formats and these formats over time. Fifth,
collecting additional network data and intelligence seeds is often rate
limited and encounter frequent network failures. Sixth, storage and retrieval
of huge amounts of graph data, especially in community databases, are not
demonstrated to work reliably in practice.
In this work, we build a system called cGraph to address the above mentioned
limitations. We design to implement a scalable big data processing pipeline
that can ingest billions of disparate reports daily. In order to track the
changes over time, we build succinct snapshots of the whole Internet on a
daily basis. cGraph provides predictive intelligence by incorporating state of
the art graph inference based malicious domain prediction techniques (Nabeel
et al., 2020). In order to make the system oblivious to data format changes,
cGraph provides one level of indirection to ingest data as common schema into
the system. This allows one to incorporate data format changes without
requiring to change the data processing pipeline allowing to extend with
additional data sources transparently. Further, unlike prior work (Nabeel et
al., 2020) which utilizes static graphs to infer malicious domains, cGraph is
able to construct a graph around the domain in question on the fly and
generate real-time intelligence. This gives cGraph to predict the
maliciousness of any domain provided there are seed domains in the
neighborhood.
Demonstration Scenarios: cGraph assists in domain threat investigation as well
as identification of associated malicious domains 111Demonstrations of a
combosquatting domain and a benign domain investigation are available at
https://youtu.be/yu47CZkp_eY and https://youtu.be/waxczBk7vA8 respectively..
We show three use cases: (1) searching for impersonating domains and
discovering related malicious domains, (2) real-time reputation score for a
domain via a browser based plugin, and (3) investigation of the evaluation of
network resources over time around a domain under consideration.
## 2\. Heterogeneous Network Graph
### 2.1. Data Sources
The system ingests four types of data/intelligence sources to build the graph:
(1) Farsight Security PDNS (Farsight Security, Inc., 2022) \- Passive DNS data
observed from all around the world which covers the majority of the DNS
resolutions in the Internet., (2) VirusTotal (VT) (VirusTotal, Subsidiary of
Google, 2022) \- Reputation service that provides aggregated intelligence on
any URL, which is the most popular aggregated threat intelligence platform
that provides threat reports using more than 80 antivirus engines, blacklists,
and sandbox systems., (3) Maxmind (MaxMind, 2022) \- A comprehensive database
consisting of geolocations and ASN information of IP addresses (4) Alexa top 1
million domains (ale, 2022) \- Alexa is a widely used resource for search
engine optimization and capturing the status of domains.
In summary, we use Farsight passive DNS records to generate comprehensive
knowledge graphs and use Alexa Top 1 million domains, VirusTotal, and Maxmind
information to inject the graph with benign and malicious seed domains for
running belief propagation and predicting new malicious domains (Nabeel et
al., 2020).
### 2.2. Graph Modeling
To create the knowledge graph (KG), we employ the DNS record types A, NS, MX,
SOA, and CNAME, which include the most important information on domains.
Specifically, Type A records contain domain to IPv4 address resolutions, Type
NS records contain domain to nameserver resolutions, Type MX records contain
domain to mail server resolutions, and CNAME records provide domain to alias
domains resolutions. All records are created by having a domain name as their
pivot. We create a heterogeneous KG utilizing the above DNS records with
different edge types representing different record types and nodes
representing domains or IP addresses. Our intuition of building such a KG is
that attackers are likely to host their domains in the same infrastructure
where some of their blacklisted domains are hosted. This allows one to utilize
label propagation algorithms to predict malicious domains from a small seed.
We build a heterogeneous KG based on the following different types of DNS
records available in PDNS.
* •
Vertices: Apex, FQDNs, IPs, Name Servers, Mail Servers, CNAME, SOA
* •
Edges: Apex-IP, Apex-FQDNs, FQDNs-IP, Apex-NS, FQDNs-NS, Apex-MX, FQDNs-MX,
Apex-CNAME, FQDNs-CNAME, Apex-SOA, FQDNs-SOA
It is important to store the graph optimized for lookup as cGraph should be
able to retrieve the KG for inference as well as replaying of network
infrastructure over time quickly. Thus, the graph information is stored along
with timestamps at the granularity of a day.
## 3\. System Architecture
### 3.1. Overall System
The system consists of four main layers: data processing, database and caching
layer, middleware and API layer, and consumer applications. The data
processing layer is the component where all the data is cleaned and processed
at the first phase. The infrastructure layer orchestrates the resources to
insert/update the cleaned and reshaped data into the database cluster. Caching
layer handles the data caching and the middleware layer handles the real-time
inference process and intermediate data processing tasks. API layer connects
the backend to the consumer products, which are web application, browser
extension, and public developer API 222More information is available at
https://qcri.github.io/cgraph/..
Figure 1. Overall system from infrastructure point of view
All resources have been provisioned on top of VMware vSphere private cloud.
Kubernetes cluster has been installed on top of the provisioned VMs and an
abstracted computation resources layer has been created in order to manage all
computation resources together.
### 3.2. Graph Inference
Inference. Belief propagation (BP) is an iterative algorithm that passes the
belief iteratively. In each iteration t, belief propagation algorithm updates
the belief value on the node and passes the beliefs (messages) to its
neighbors based on the belief values it received from the previous iteration
t-1. As mentioned, this interactive process continues until each message value
passed by a node to a neighbor node converges to a stable value. The final
beliefs are extracted after convergence.
Execution. The main idea is to construct a KG on the fly and inject ground
truth labels to the KG from credible sources where VT, Alexa, and Maxmind and
then run the BP algorithm on top of the knowledge graph by using the labeled
values as initial beliefs. It propagates a probabilistic values for each
unlabeled domain indicating the maliciousness. For bravity, we explain the
execution for a selected example graph. The ground truth extraction process is
conducted in two stages; benign ground truth extraction and malicious ground
truth extraction.
For the benign ground truth extraction, Alexa Rank data is used. Alexa Rank
data feed for the latest seven consecutive days is taken and the domains with
Alexa Rank $\leq 10000$ is taken for each day. From this, the unique set of
domains is taken as the benign ground truth. Similar to prior research (De
Silva et al., 2021), we observe that domains with Alexa Rank $\leq 10000$
which appear consistently throughout a week are likely to be benign as
malicious domains do not live longer than a day or two in Alexa 10K.
For the malicious ground truth, the data from the daily feed of VirusTotal is
used. VirusTotal reports for latest seven consecutive days are taken and and
the reports with a VT positive count of at least 3 is extracted as malicious
ground truth as VT reports with 2 or less positive counts are known to be a
less reliable indicator of maliciousness (Nabeel et al., 2020). We manually
verify random samples of benign and malicious ground truth and ascertain that
they are indeed correctly labeled.
After seeding the KG with the ground truth domains, we run a multi-threaded
version of the BP algorithm which executes within a few seconds. Our
experiments show that our inference based classification achieves a high
precision of 92.17% and high recall of 99.41%. We observe that a domain’s
maliciousness score does not affect significantly by considering only its two
hop neighbors compared to the complete graph. Hence, for the real-time
inference on an unknown domain, we build the two-hop neighborhood KG and run
the inference algorithm to further improve the computational performance.
## 4\. Current Threat intelligent Solutions vs cGraph
We thoroughly evaluate the key threat intelligence solutions in the market
against cGraph, our threat intelligence platform: AT&T Cybersecurity,
VirusTotal Graph, Cisco Umbrella Investigate, and Anomali Threat Intelligence
Platform. Below, we provide a summary of key differentiation in our system:
(1) Dynamic graph-based visualization and search of Internet artifacts for
threat intelligence (all the above solutions lack an intuitive graph based
navigation), (2) The ability to integrate various data sources and expand the
network coverage and the ability to integrate any intelligence sources
seamlessly, (3) Predictive domain threat intelligence based on an efficient
graph inference algorithm (all above solutions provide only reactive
intelligence), and (4) The ability to traverse the network infrastructure
based on time and observe the changes in the physical layout as well as the
maliciousness of the domains in the network over time, which greatly assist
security operation teams in identifying root causes of attacks and the extent
of the damage.
## 5\. Demonstration
In this section, we show three common use cases of our system.
Searching for impersonating domains and related other malicious domains:
Attackers increasingly impersonate popular brands such as Apple, Paypal and
Microsoft. cGraph allows to search for any popular brands and easily identify
likely combosquatting domains.We demonstrate the use case of hunting for
combosuqatting domains of paypal.com.
Figure 2. Search results for the keyword paypal
Figure 2 shows the search results for the keyword paypal ranked based on VT
positive counts. An investigator may pick any of these domains to further
investigate. Assume that they select paypal-assist.com, which takes them to
the graph based interface shown in Figure 3. This view includes the domain
summary information, network graph, Alexa ranking and VT statistics over time,
IP geolocation information and historical hosting information.
Figure 3. Dashboard for the domain paypal-assit.com
The ability expand the network graph based on the investigator’s findings is
quite useful in practice. cGraph not only allows to expand any network node
when it has children. but also provides real-time predict intelligence on the
unknown domains in the network. Figure 4 shows the expansion of the IP address
47.254.170.24 that adds 4 additional domains to the graph which are hosted on
this IP address.
Figure 4. Expansion of the IP address 47.254.170.24
The inference graph view shows the maliciousness score of each domain in the
network graph calculated on the fly using the BP algorithm based on known seed
domains, paypal-assist.com in this case. Figure 5 shows that cGraph inference
identifies the domains google-standard.com, google-money.com, paypal-debit.com
and google-sale.com to be highly malicious.
Figure 5. Malicious domains identified by inference
Evaluation of maliciousness of any domain via the browser plugin: In this use
case, we show how cGraph API can be consumed to obtain reputation scores for
websites on the fly. While blacklists such as GSB that powers the Chrome
browser identify many malicious websites, they are either slow to react or
unable to make a judgement if the website content is cloaked from the
blacklisting service. cGraph’s reputation engine takes a different approach
and infers the maliciousness of domains based on the maliciousness of the
neighboring network resources such as sibling domains, name servers, CNAME
domains, IP addresses and certificates. As mentioned earlier, the reputation
score is computed dynamically based on the status of the network
infrastructure at the present moment.
Figure 6. Browser plugin: real time reputation score
Figure 6 shows the cGraph inference API in action when a user types the
website soaponline.org. The maliciousness score indicates how likely the
domain is malicious or benign. In this example, the maliciousness score of
0.87 indicates that soaponline.org is highly likely to be malicious.
Investigating the network over time: Network resources associated with domains
change over time due to both malicious as well as benign reasons. For example,
cloud IP rotations, CDN hostings and IP load balancing change IPs associated
with domains frequently. Attackers also move from one IP to another as a
technique to either resist take down or evade detection. cGraph timeline
feature allows to investigate the network graph over a time period. Coupled
with the real-time inference feature, one can observe how the reputation of
any of the domains change over time. Figure 7 shows the network graph on day 1
and the evolved network graph on day 7 respectively. This allows the
investigator to learn the fluxing behavior of domains and take necessary
actions.
Figure 7. Network timeline for day 1 (left) and day 7 (right)
## 6\. Conclusions
We propose, cGraph, which takes domain threat investigation a step forward.
cGraph can efficiently store and process large amount of disparate network
data, build network graph for any given time period and perform real-time
graph inference to predict domains as malicious or benign, compared to
reactive threat intelligence platforms available. We demonstrate three
frequent use cases of cGraph: domain search, real-time reputation computation
and network graph evolution along with the maliciousness scores of domains in
the network over time.
## References
* (1)
* ale (2022) 2022\. Alexa: The Web Information Company. https://www.alexa.com/topsites. Accessed January 2022.
* ano (2022) 2022\. Anomali Threat Intelligence Platform. https://www.anomali.com/. [Online].
* cis (2022) 2022\. Cisco Umbrella Investigate. https://umbrella.cisco.com/products/umbrella-investigate. [Online].
* De Silva et al. (2021) R. De Silva, M. Nabeel, C. Elvitigala, I. Khalil, T. Yu, and C. Keppitiyagama. 2021\. Compromised or Attacker-Owned: A Large Scale Classification and Study of Hosting Domains of Malicious URLs. Usenix Security.
* Farsight Security, Inc. (2022) Farsight Security, Inc. 2022\. DNS Database. https://www.dnsdb.info/.
* MaxMind (2022) MaxMind. 2022\. GeoLite2 Databases.
* Nabeel et al. (2020) M. Nabeel, I. M. Khalil, B. Guan, and T. Yu. 2020\. Following Passive DNS Traces to Detect Stealthy Malicious Domains Via Graph Inference. _ACM Trans. Priv. Secur._ 23, 4, Article 17 (July 2020), 36 pages. https://doi.org/10.1145/3401897
* VirusTotal, Subsidiary of Google (2022) VirusTotal, Subsidiary of Google. 2022\. VirusTotal. https://www.virustotal.com/.
|
# Black-Box Detection of Language Model Watermarks
Thibaud Gloaguen
Department of Mathematics
ETH Zurich
<EMAIL_ADDRESS>&Nikola Jovanović
Department of Computer Science
ETH Zurich
<EMAIL_ADDRESS>&Robin Staab
Department of Computer Science
ETH Zurich
&Martin Vechev
Department of Computer Science
ETH Zurich
###### Abstract
Watermarking has emerged as a promising way to detect LLM-generated text. To
apply a watermark an LLM provider, given a secret key, augments generations
with a signal that is later detectable by any party with the same key. Recent
work has proposed three main families of watermarking schemes, two of which
focus on the property of preserving the LLM distribution. This is motivated by
it being a tractable proxy for maintaining LLM capabilities, but also by the
idea that concealing a watermark deployment makes it harder for malicious
actors to hide misuse by avoiding a certain LLM or attacking its watermark.
Yet, despite much discourse around detectability, no prior work has
investigated if any of these scheme families are detectable in a realistic
black-box setting. We tackle this for the first time, developing rigorous
statistical tests to detect the presence of all three most popular
watermarking scheme families using only a limited number of black-box queries.
We experimentally confirm the effectiveness of our methods on a range of
schemes and a diverse set of open-source models. Our findings indicate that
current watermarking schemes are more detectable than previously believed, and
that obscuring the fact that a watermark was deployed may not be a viable way
for providers to protect against adversaries. We further apply our methods to
test for watermark presence behind the most popular public APIs: GPT4, Claude
3, Gemini 1.0 Pro, finding no strong evidence of a watermark at this point in
time.
## 1 Introduction
With the rapid increase in large language model (LLM) capabilities and their
widespread adoption, researchers and regulators alike are raising concerns
about their potential misuse for generating harmful content [1, 3]. To tackle
this issue, the idea of watermarking, a process of embedding a signal
invisible to humans into generated texts, is gaining significant traction.
#### Language model watermarking
More formally, in LLM watermarking [8, 9, 10, 6, 22, 21, 23, 19] we consider
the setting of a _model provider_ that offers black-box access to their
proprietary model $LM$ while ensuring that each generation $y$ in response to
a prompt $q$ can be reliably attributed to the model. To enable this, the
provider modifies the generations using a secret watermark key $\xi$, which a
corresponding watermark detector can later use to detect whether a given text
was generated by $LM$.
The prominent family of _Red-Green_ watermarks [8, 9] achieves this by, at
each step of generation, selecting a context-dependent pseudorandom set of
logits to be boosted, modifying the model’s next-token distribution. In
contrast to these, the recently proposed families of _Fixed-Sampling_ [10] and
_Cache-Augmented_ [22, 6] schemes have focused on developing watermarks that
aim to preserve the output distribution of the LLM, achieving such guarantees
in ideal theoretical settings. This goal can be motivated by two main
benefits: (i) it is a tractable proxy for preservation of capabilities of the
original LLM, and (ii) intuitively, the deployment of the watermark should
become inherently harder to detect for malicious actors that would otherwise
be able to avoid or attack the watermark [7, 15, 19].
#### Practical detectability
Yet, despite watermark detection being a key concern in current discourse,
there have so far been no rigorous investigations into the _practical
detectability_ of any of the three watermark scheme families. It thus remains
unclear if the distribution preservation guarantees of Fixed-Sampling and
Cache-Augmented schemes, that hold in ideal settings, offer any detectability
benefits in real-world deployments. Recent work has even argued that Red-Green
schemes, whose detectability has also never been thoroughly studied in
realistic settings, avoid some undesirable tradeoffs of distribution-
preserving watermarks, and are ultimately more practical [16].
#### This work: practical black-box watermark detection
In this work, for the first time, we comprehensively investigate the question
of watermark detection in practical real-world settings. Faithful to how LLMs
are exposed in current public deployments, we assume a minimal setting in
which the adversary cannot control any input parameters (e.g., temperature,
sampling strategy) apart from the prompt $q$, and does not receive any
additional information (e.g., log-probabilities) apart from the instruction-
tuned model’s textual response $y$. In this setting, the goal of the adversary
is to identify if the model is watermarked and determine which scheme was
used.
To this end, we propose rigorous and principled statistical tests for the
black-box detection of seven scheme variants of the three most prominent
watermark families. Our tests, illustrated in Fig. 1, are based on fundamental
properties of the three respective scheme families: Red-Green (Section 2),
Fixed-Sampling (Section 3), and Cache-Augmented (Section 4) watermarks. In our
extensive experimental evaluation, we find that in practice, both
distribution-modifying as well as distribution-preserving schemes are _easily
detectable_ by our tests, even in the most restricted black-box setting.
Highlighting their practicality, we further test several real-world LLM
deployments (GPT4, Claude 3, Gemini 1.0 Pro) and detect no watermarks,
suggesting that watermark deployment at this scale is an ongoing challenge.
#### Implications
While the impact of watermarking schemes on model capabilities (point (i)
above) remains an interesting avenue for future work, our findings question
whether non-detectability (point (ii) above) should be a key focus in the
development of LLM watermarks. Guided by our results, we suspect that avoiding
practical detectability may be an inherently hard problem given the
fundamental nature of watermarks. In light of this, we advocate for a more
diverse and practically oriented development and evaluation of LLM watermarks,
allowing for broader consideration of other relevant properties such as attack
robustness, text quality, and efficiency.
#### Main contributions
We make the following key contributions:
* •
We present the first principled and statistically rigorous tests for practical
black-box detection of popular LLM watermarks across the three prominent
scheme families: Red-Green (Section 2), Fixed-Sampling (Section 3), and Cache-
Augmented (Section 4).
* •
We confirm the effectiveness of all our tests in an extensive experimental
evaluation across seven schemes and five open-source models, and present
additional results that verify the applicability of our tests in real-world
settings (Sections 5.1 and 5.2).
* •
We further highlight the practicality of our tests by executing them on
several real-world black-box LLM deployments—GPT, Claude, and Gemini (Section
5.3).
## 2 Detecting Red-Green Watermarks
Figure 1: Overview of our key contribution. Given black-box textual access to
a language model, a client can query the model and run statistical tests to
rigorously test for presence of a watermark. In this example, both the test
for _Cache-Augmented_ watermarks (Section 4) and the test for _Fixed-Sampling_
watermarks (Section 3) fail, while the test for _Red-Green_ watermarks
(Section 2) successfully detects the watermark.
In this section, we introduce our statistical test for detecting watermarks
from the _Red-Green_ family, illustrated in Fig. 1 (bottom). To this end, we
exploit their core property: applying the watermark introduces a _context-
dependent logit bias_ , i.e., the model’s output distribution is biased in a
way that can greatly vary depending on the last few tokens that were generated
(_context_), yet is unaffected by the tokens preceding the context (_prefix_).
We first recall the background related to these schemes. We then introduce our
modeling assumptions, describe the querying strategy we use to obtain the data
for the test, and finally describe the test itself. In Section 5, we
experimentally confirm the effectiveness of the test in realistic settings and
study its query cost, and in App. C explore methods to estimate scheme
parameters once a watermark is confirmed.
#### Background
Assume a watermark key $\xi\in\mathbb{N}$, a pseudorandom function (PRF) $f$,
and a hashing function $H\colon\mathbb{N}\to\mathbb{N}$. At each generation
step $t$, a Red-Green watermark modifies the logits $l_{1:|V|}$ of tokens from
the vocabulary $V$ to promote a subset of tokens (_green tokens_) before
applying standard sampling. We consider two popular variants, LeftHash [8] and
SelfHash [9], both parametrized by $\delta,\gamma\in\mathbb{R}^{+}$, and using
$h=1$ and $h=3$ previous tokens as context, respectively. LeftHash seeds $f$
with $H(y_{t-1})\cdot\xi$, and uses it to split $V$ into $\gamma|V|$ green
tokens and remaining _red_ tokens. For each green $i$, it then increases
$l_{i}$ by $\delta$. SelfHash differs by seeding $f$ with
$\min(H(y_{t-3}),\ldots,H(y_{t}))\cdot H(y_{t})\cdot\xi$, effectively using a
context size of $4$ by including the token $y_{t}$ yet to be generated. For
both schemes, the watermark detector is based on observing a significant
number of green tokens in the input text of length $m$, above the expectation
of $\gamma m$. Other schemes from this family include, among else, varying the
aggregation function or the context size.
#### Modeling Red-Green watermarks
Assume a setting where the model chooses a token from some restricted set
$\Sigma\subset V$, following some _context_ $t_{2}$ (longer than the watermark
context $h$), which is preceded by a _prefix_ $t_{1}$. We discuss how to
construct such a setting shortly. To model the watermark, we assume the
following distribution for $p_{t_{1},t_{2}}(x)$, the probability the model
assigns to some $x\in\Sigma$:
$p_{t_{1},t_{2}}(x)=\frac{\exp\left((x^{0}+\delta_{t_{2}}(x)+\varepsilon_{t_{1},t_{2}}(x))/T\right)}{\sum_{w\in\Sigma}\exp\left((w^{0}+\delta_{t_{2}}(w)+\varepsilon_{t_{1},t_{2}}(w))/T\right)}.$
(1)
Here, we assume the existence of the _true logit_ $x^{0}$, modeling the model
bias towards $x$. The true logit is shifted by $\delta_{t_{2}}(x)$, a
$\delta$-Bernoulli random variable, where $\delta\in\mathbb{R}^{+}$ is the
watermark parameter introduced above. Finally, $\varepsilon_{t_{1},t_{2}}(x)$
for different $t_{1},t_{2}$ are iid symmetric error terms with mean $0$ and
variance $\sigma^{2}$. Applying the _logit_ function $p\to\log(p/(1-p))$ to
Eq. 1, we obtain:
$l_{t_{1},t_{2}}(x)=\frac{x^{0}}{T}+\frac{\delta_{t_{2}}(x)}{T}+\frac{\varepsilon_{t_{1},t_{2}}(x)}{T}-\log\left(\sum_{w\in\Sigma\setminus\\{x\\}}\exp\left(\frac{w^{0}}{T}+\frac{\delta_{t_{2}}(j)}{T}+\frac{\varepsilon_{t_{1},t_{2}}}{T}\right)\right).$
(2)
Approximating the log-sum-exp with a maximum, and WLOG setting $w^{0}=0$ for
$w$ which is the maximum in the log-sum-exp (as logits are shift-invariant),
the above simplifies to
$l_{t_{1},t_{2}}(x)=x^{0}/T+\delta^{\prime}_{t_{2}}(x)+\varepsilon^{\prime}_{t_{1},t_{2}}(x),$
(3)
where $\varepsilon^{\prime}_{t_{1},t_{2}}(x)$ absorbs the previous error
terms. The resulting $\delta^{\prime}_{t_{2}}(x)$ is a random variable that is
$0$ for unwatermarked models, and has $3$ possible values for watermarked
models: $\delta/T$ (where $\delta$ is the watermark parameter), if $x$ is the
only token from $\Sigma$ that is in the green partition of the vocabulary,
$-\delta/T$, if $x$ is in the red partition and some other tokens from
$\Sigma$ are in the green partition, and $0$ otherwise. Our test is based on
detecting cases of $\delta^{\prime}_{t_{2}}(x)\neq 0$ and checking that their
occurrence is indeed independent of $t_{1}$, distinguishing model variance
from the watermark bias.
#### Querying strategy
Recall that our goal is to steer a model into a setting (via an instruction,
as we have no access to the completion model) where it makes a choice from
some restricted set $\Sigma$. Importantly, we want to ensure enough
variability in the model’s choices to be able to observe the behavior specific
to a Red-Green watermark and perform efficient estimation (in terms of query
cost).
Assuming an upper bound $H$ on the context size $h$ of the watermark, we use
the following prompt template, parametrized by $t_{1}$ (an arbitrary length
string), $d$ (a single token), and a word list $\Sigma$.
⬇
f"Complete the sentence \"{$t1$} {$\underbrace{d\cdot H}_{t_{2}}$}\" using a
random word from: [{$\Sigma$}]."
Here, $t_{1}$ serves as the prefix, and $t_{2}=d\cdot H$ as the context. For a
watermarked model, changing $t_{2}$ is the only way to change the red-green
vocabulary split, which can greatly affect the model’s choice. For
unwatermarked models, while we often see bias towards some choices from
$\Sigma$, this bias will not strongly depend on $t_{2}$. This holds for all
context sizes $\leq H$ and most aggregation functions, making our setting and
the corresponding test directly applicable to different variants of Red-Green
schemes.
In our instantiation (illustrated in Fig. 1), we use $N$ different $t_{1}$ of
the form “I bought”, varying the verb, $M$ different values of $d$ from the
set of digits, a word list $\Sigma$ of four different plural fruit names, and
an example that uses different words to not bias the model towards a specific
choice. We empirically observe that this setup minimizes the chance that the
model collapses to a single choice or fails to follow the instruction, which
often occurred for similar settings based on repetition of words or numbers
outside of natural context.
#### Estimating the logits
To collect the data for the test, we query the model in two phases. First, we
choose different values of $\Sigma$ until we find one where the model does not
always make the same choice, inducing $Q_{1}$ total queries. We set $x$ to be
the most commonly observed word from $\Sigma$. Next, for each $(t_{1},t_{2})$
pair we query the model until we obtain $K$ valid responses (filtering out
failures to follow the instruction), inducing $Q_{2}$ additional queries.
We use these samples to estimate the model _logits_ corresponding to $x$ as
$\hat{l}_{t_{1},t_{2}}(x)=\log\frac{\hat{p}_{t_{1},t_{2}}(x)}{1-\hat{p}_{t_{1},t_{2}}(x)}$,
where $\hat{p}_{t_{1},t_{2}}(x)$ is the empirically estimated probability of
$x$ in the responses. The result of the adversary’s querying procedure is a
matrix $L_{N\times M}$ (visualized in Fig. 1) of such logit estimates.
#### Testing watermark presence
Finally, we describe the statistical test based on the logit estimates
$L_{N\times M}$. We first estimate $\sigma$, the standard deviation of
$\varepsilon^{\prime}_{t_{1},t_{2}}(x)$, as follows:
$\hat{\sigma}^{2}=\text{median}[\text{Var}_{t_{2}}(L)],$ (4)
using the empirical median to improve robustness to unpredictable behavior
caused by different $t_{1}$. Then, we calculate the following two binary
functions, which flag cases where we believe the model’s probability was
affected by a watermark:
$R_{x}(t_{1},t_{2})=\mathbbm{1}\\{\hat{l}_{t_{1},t_{2}}(x)-\text{median}[L]<-r\hat{\sigma}\\},$
(5)
and
$G_{x}(t_{1},t_{2})=\mathbbm{1}\\{\hat{l}_{t_{1},t_{2}}(x)-\text{median}[L]>r\hat{\sigma}\\},$
(6)
with $r\in\mathbb{R}^{+}$ a parameter of the test. In practice, to account for
model unpredictability, we use the empirical median conditioned on $t_{1}$ in
Eqs. 5 and 6. For simplicity, let us denote $t_{1}\in\\{1,\ldots,N\\}$ and
$t_{2}\in\\{1,\ldots,M\\}$. Let
$cnt_{x}(t_{2})=\max\left(\sum_{t_{1}=1}^{N}R_{x}(t_{1},t_{2}),\sum_{t_{1}=1}^{N}G_{x}(t_{1},t_{2})\right)$
count the number of consistently flagged values for fixed $t_{2}$. We define
the following test statistic:
$S_{x}(L)=\max_{t_{2}\in[M]}cnt_{x}(t_{2})-\min_{t_{2}\in[M]}cnt_{x}(t_{2}).$
(7)
The null hypothesis of our test is $\forall
t_{2}\colon~{}\delta^{\prime}_{t_{2}}(x)=0$, i.e., the model is not
watermarked. To obtain a $p$-value, we apply a Monte Carlo permutation test to
$S_{x}$, checking if the flagged values are correlated with $t_{2}$ in a way
that indicates a Red-Green watermark. Namely, we sample a set of permutations
$\sigma$ of the matrix $L$ uniformly at random, and calculate a 99% confidence
interval of $\Pr[S_{x}(\sigma(L))\geq S_{x}(L)]$, whose upper bound we take as
our $p$-value. When this value is small, we interpret that as evidence of a
watermark. Because Eq. 3 is permutation invariant when
$\delta^{\prime}_{t_{2}}(x)=0$, this ensures that the test does not reject
under the null. This completes our method for detection of Red-Green
watermarks.
## 3 Detecting Fixed-Sampling Watermarks
Unlike Red-Green watermarks, the recently proposed _Fixed-Sampling_ watermarks
do not modify the logit vectors during generation, so estimating the
probabilities of model outputs as above is not informative. Instead, the
sampling is fully determined by the rotation of the watermark key, making the
natural vector to exploit when detecting this watermark its _lack of
diversity_. Given a prompt for which an unwatermarked model is expected to
produce highly diverse outputs, we can use this observation to distinguish
between the two, as illustrated in Fig. 1 (middle).
As in Section 2, we start by introducing the needed background. We then
formally model the diversity of model outputs, discuss our querying strategy
that ensures our assumptions are met, and describe the resulting statistical
test. The effectiveness of our method is evaluated in Section 5.
#### Background
For Fixed-Sampling watermarks, the secret watermark key sequence $\xi$ of
length $n_{key}$ is cyclically shifted uniformly at random for each generation
to obtain $\bar{\xi}$, and the entry $\bar{\xi}_{t}$ is used to sample from
$l$. In the ITS variant, $\bar{\xi}_{t}$ is a pair $(u,\pi)\in[0,1]\times\Pi$,
where $\Pi$ defines the space of permutations of the vocabulary $V$. Given the
probability distribution $p$ over $V$, obtained by applying the softmax
function to $l$, ITS samples the token with the smallest index in the
permutation $\pi$ such that the CDF of $p$ with respect to $\pi$ is at least
$u$. In the EXP variant, $\bar{\xi}_{t}$ is a value $u\in[0,1]$, and we sample
the token $\operatorname*{arg\,min}_{i\in V}-\log(u)/p_{i}$. The detection,
for both variants, is based on testing the correlation between the text and
$\xi$ using a permutation test. As noted in Section 1, the key design goal of
ITS and EXP is that, in expectation w.r.t. $\xi$, they do not distort the
distribution of the responses. How close to this ideal is achieved in
practical implementations is the question we aim to answer.
#### Modeling the diversity
Let $U_{n}(q,t)$ denote a random variable that counts the number of unique
outputs to a fixed prompt $q$ in $n$ queries, each of length exactly $t$ in
tokens. We introduce the _rarefaction curve_ (visualized in Fig. 1) as
$R(n)=\mathbb{E}[U_{n}(q,t)].$ (8)
For suitable $q$ that enables diversity and large enough $t$, the
unwatermarked model exhibits $R(n)=n$. For a Fixed-Sampling watermark (both
ITS and EXP variants), the watermark key segment used for sampling is
determined by choosing a rotation of the key $\xi$ uniformly at random. As
choosing the same rotation for the same prompt and sampling settings will
always yield the same output, the number of unique outputs is at most equal to
the key size $n_{key}$. The probability that an output $i$ was not produced is
given by $(1-1/n_{key})^{n}$. By linearity of expectation, we have the
rarefaction curve
$R(n)=n_{\text{key}}\left(1-(1-\frac{1}{n_{\text{key}}})^{n}\right).$ (9)
#### Querying strategy
To estimate $R(n)$ of $LM$, we query it with a fixed prompt $q$, using
rejection sampling to discard short responses until we obtain a set of $N$
responses of length $t$ tokens (inducing $Q$ total queries). We then
repeatedly sample $n$ responses from this set to get a Monte-Carlo estimation
of $R(n)$. There are two key considerations. First, we need to ensure that we
are in a setting where an unwatermarked model would have $R(n)=n$. To do this,
we use the prompt ‘‘This is the story of’’ that reliably causes diverse
outputs, and set $t$ high enough to minimize the chance of duplicates. In
Section 5 we experimentally confirm that the number of unique outputs scales
exponentially with $t$, and investigate the effect of small temperatures.
Second, as larger $n_{key}$ make $R(n)$ closer to linear, we must ensure that
$n$ is large enough for our test to be reliable. To do this, we can set an
upper bound $\bar{n}_{key}$ on key size, and estimate the number of samples
needed for a given power by simulating the test—our experiments show that our
test succeeds even on values of $\bar{n}_{key}$ far above practical ones.
#### Testing watermark presence
Finally, to test for presence of a Fixed-Sampling watermark, we use a Mann-
Whitney U test to compare the rarefaction curve $R(n)=n$ with the observed
rarefaction data obtained as above. If the test rejects the null hypothesis,
we interpret this as evidence that the model is watermarked with a Fixed-
Sampling scheme. We confirm the effectiveness of our test in Section 5. In
App. C we discuss estimation of the watermark key size $n_{key}$ once the
presence of a Fixed-Sampling watermark is confirmed.
## 4 Detecting Cache-Augmented Watermarks
We proceed to the detection of Cache-Augmented watermarks. While the
underlying techniques used in these schemes are often similar to those of the
Red-Green and Fixed-Sampling families, we focus on a general approach that
exploits the presence of a cache on a fundamental level, and can be generally
applied to any Cache-Augmented scheme. Namely, we note that the cache
sometimes _reveals the true distribution_ of the model, which was also noted
in recent work [15] as an undesirable property for a watermarking scheme. This
implies that the distribution of choices is _cache-conditioned_ (as seen in
Fig. 1, top), which our adversary will exploit to detect the presence of a
watermark.
We first recall relevant background, and then discuss our method: we query the
model in two phases to probe the two cache-conditioned distributions, and
apply a statistical test to detect when they differ, which would indicate the
presence of a Cache-Augmented watermark.
#### Background
The watermarks that we consider in this section use previous $h$ tokens to
reweigh the distribution or apply deterministic sampling at each step. As a
key feature motivated by practical distribution preservation, these schemes
introduce a _cache_ of previously seen contexts. Namely, whenever
$y_{t-h:t-1}$ is already present in the cache, they ignore the watermarking
procedure and instead fall back to standard generation. The cache can be
either global or local (per user) and clears after a certain number $G$ of
generations. We consider three variants: $\delta$-reweight [6],
$\gamma$-reweight [6], and DiPmark [22] with parameter $\alpha$, the details
of which we defer to App. B.
#### Probing the true distribution
In the first phase of querying, our goal is to find a setting where the
distribution of the model under the watermark will differ from its true
distribution, and estimate the true distribution. For schemes we focus on,
this corresponds to a setting with two choices, where the model is not
significantly biased towards any of them. In particular, we use the following
prompt:
⬇
Pick a fruit between: {$f_{1}$} and {$f_{2}$}. Use the following format:
{$uc$}{$f_{example}$},
and modify $f_{1},f_{2},$ and $f_{example}\neq f_{1},f_{2}$ until we find a
setting where the model outputs the two choices roughly uniformly. Crucially,
we prefix the prompt with a randomly sampled sufficiently long string of
tokens $uc$. As $LM$ will repeat $uc$ before providing the answer, this
ensures that if a cache is present, after our first query (the result of which
we discard) the choice of the model will be made according to the true
distribution, as the relevant part of $uc$ was cached. Assuming WLOG that
$f_{1}$ is the more likely choice for the model, we query it $Q_{1}$ times
with the same input to obtain $\hat{p_{1}}$, the estimate of the true
probability of the model to pick $f_{1}$.
#### Probing the watermarked distribution
In the second phase, we query $LM$ with the same input, while ensuring that
the cache is reset between each query, i.e., the model will respond according
to the watermarked distribution. In case of a global cache, it is sufficient
to wait for a set amount of time—resetting the cache too infrequently is not a
realistic setting for a deployment, as it would on average lead to a weak
watermark. The uncommon prefix $uc$ ensures that no other user will
accidentally insert the same context into the cache. In case of a per-user
cache, we can either saturate the cache by asking diverse queries, or use
multiple user accounts. We query $LM$ $Q_{2}$ times and obtain $\hat{p_{2}}$,
the estimate of the probability of $f_{1}$ under the watermarked distribution.
#### Testing watermark presence
For unwatermarked models or those watermarked with a scheme from another
family, both of the previous steps were sampling from the same distribution,
thus for high enough $Q_{1}$ and $Q_{2}$ we expect $\hat{p_{1}}=\hat{p_{2}}$.
However, for all Cache-Augmented watermarking schemes, these probabilities
will differ, indicating that the cache has revealed the true distribution of
the model. To test this, we apply a Fischer’s exact test with the null
hypothesis $\hat{p_{1}}=\hat{p_{2}}$. If we observe a low $p$-value, we
interpret this as evidence that the model is watermarked with a Cache-
Augmented scheme. Our experiments in Section 5 demonstrate that this test is
robust to different scheme variants, and does not lead to false positives when
$LM$ is unwatermarked or uses a scheme from another family. In App. C we
discuss how to distinguish different variants of Cache-Augmented schemes.
## 5 Experimental Evaluation
Table 1: Main results of our watermark detection tests across different models
and watermarking schemes. We report median p-values across 100 repetitions of
the experiment, and for Red-Green schemes additionally over 5 watermarking
keys. p-values below $0.05$ (test passing) are highlighted in bold. $\delta$R
and $\gamma$R denote $\delta$-reweight and $\gamma$-reweight schemes,
respectively.
| | | | | Red-Green | | Fixed-Sampling | | Cache-Augmented
---|---|---|---|---|---|---|---|---|---
| | Unwatermarked | | LeftHash | SelfHash | | ITS/EXP | | DiPmark/$\gamma$R | $\delta$R
Model | Test | $T=$ $1.0$ | $T=$ $0.7$ | | $\delta,\gamma=$ $2,0.25$ | $\delta,\gamma=$ $4,0.5$ | $\delta,\gamma=$ $2,0.5$ | $\delta,\gamma=$ $4,0.25$ | | $n_{key}=$ $256$ | $n_{key}=$ $2048$ | | $\alpha=$ $0.3$ | $\alpha=$ $0.5$ |
Mistral 7B | R-G (Section 2) | 1.000 | 1.000 | | 0.000 | 0.000 | 0.000 | 0.000 | | 1.000 | 1.000 | | 1.000 | 1.000 | 1.000
Fixed (Section 3) | 0.938 | 0.938 | | 0.938 | 0.938 | 0.938 | 0.938 | | 1.3e-125 | 1.1e-9 | | 0.938 | 0.938 | 0.938
Cache (Section 4) | 0.570 | 0.667 | | 0.607 | 0.608 | 1.00 | 0.742 | | 0.638 | 0.687 | | 2.4e-4 | 2.1e-3 | 5.6e-27
Llama2 13B | R-G (Section 2) | 0.149 | 0.663 | | 0.000 | 0.000 | 0.000 | 0.000 | | 0.121 | 0.128 | | 0.149 | 0.149 | 0.149
Fixed (Section 3) | 0.968 | 0.868 | | 0.938 | 0.938 | 0.938 | 0.938 | | 5.5e-124 | 7.5e-8 | | 0.938 | 0.938 | 0.968
Cache (Section 4) | 0.708 | 0.573 | | 0.511 | 0.807 | 0.619 | 0.710 | | 0.518 | 0.692 | | 1.8e-2 | 5.3e-3 | 6.7e-32
Llama2 70B | R-G (Section 2) | 1.000 | 1.000 | | 0.000 | 0.020 | 0.000 | 0.000 | | 1.000 | 1.000 | | 1.000 | 1.000 | 1.000
Fixed (Section 3) | 0.938 | 0.526 | | 0.966 | 0.965 | 0.984 | 0.975 | | 4.2e-125 | 1.7e-8 | | 0.938 | 0.966 | 0.938
Cache (Section 4) | 0.596 | 0.620 | | 0.657 | 0.639 | 0.651 | 0.608 | | 0.463 | 0.818 | | 1.5e-3 | 4.4e-3 | 5.8e-28
In this section, we apply the tests introduced in Sections 2, 3 and 4 to a
wide range of models and watermarking schemes, and confirm their effectiveness
in detecting watermarks. In Section 5.1 we show that the adversary can
reliably detect the watermarking scheme used (if any) at a low cost, across a
wide range of practical settings. In Section 5.2, we provide further
experimental insights into the tests’ robustness. Finally, in Section 5.3, we
demonstrate that our tests can be directly applied to real-world LLM
deployments, revealing no significant evidence of any watermarking schemes at
this point in time.
### 5.1 Main Experiments: Detecting Watermarking Schemes
We perform our main set of experiments to verify the effectiveness of our
tests introduced in Sections 2, 3 and 4 in detecting watermarking schemes in
realistic scenarios.
#### Experimental setup
We run all our tests on five different models (Mistral-7B, Llama2-7B, -13B,
and -70B, and Llama3-8B), in eleven different scenarios. We here present
results of a subset of those, and defer the rest to App. A. In each scenario,
each model is either unwatermarked (where we vary the temperature) or
watermarked with a certain scheme from the three main families (where we vary
the particular scheme and its parameters). If our tests are reliable, we
expect to see low p-values only when the model is watermarked exactly with the
scheme family that we are testing for.
For Red-Green tests, we set $N=10,M=9,r=1.96$, a different $\Sigma$ per model
based on the first $Q1$ samples, use $100$ samples to estimate the
probabilities, and use $10000$ permutations in the test. For Fixed-Sampling
tests, we use $n=1000$ queries and set $t=50$. For Cache-Augmented tests, we
use $Q1=Q2=75$ and assume the cache is cleared between queries in the second
phase.
#### Results: reliable watermark detection
Our main results are shown in Table 1, where we report the median p-values for
each (model, test, scenario) tuple across 100 repetitions of each experiment.
Across all experiments, all three tests reject the null hypothesis (_the
specific watermark is not present_) at a 95% confidence level only when the
scheme from the target family is indeed applied to the model. This confirms
that our tests are reliable in detecting watermarks, and robust with respect
to the model and the specific parameters of the scheme, emphasizing our tests’
generalization to all schemes that are based on the same foundational
principles. In particular, our Red-Green tests are robust to the seeding
scheme, the logit bias $\delta$, and the green token fraction $\gamma$; our
Fixed-Sampling tests maintain high confidence even for unusually high values
of $n_{key}$ and those higher than the number of queries; finally, our Cache-
Augmented tests are robust to all details of the underlying scheme. Our
results also provide evidence that unrelated model modifications do not cause
false positives, as no test passes when the model is unwatermarked or
watermarked with a different scheme family.
Our tests do not incur significant costs for the adversary, making them easily
applicable in practice. While the cost of a particular run varies across
models and scenarios, we can estimate the average cost of the tests to be
below $20 for Red-Green, $1 for Fixed-Sampling, and $0.1 for Cache-Augmented
tests, assuming latest OpenAI GPT4o pricing.
### 5.2 Additional Experiments: Validating the Assumptions
Figure 2: _Left_ : distribution of bootstrapped p-values of the Red-Green test
on Llama2-13B with $(\delta,\gamma)=(2,0.25)$, for different sample sizes. We
see reliable results for $100$ or more samples. _Right_ : the diversity gap
$n-R(n)$ on log scale in different settings. Linear behavior means that
diversity scales exponentially with $t$, and we see that the assumption of
$R(n)=n$ can be easily met in practice.
We present two additional experiments to validate the assumptions made in our
tests and provide further insight into their behavior in practical settings.
#### Sampling in Red-Green tests
The Red-Green test (Section 2) relies on the sampling of model outputs to
estimate the probabilities of the model. As the resulting p-value is computed
assuming knowledge of true probabilities, this raises the question of the
impact of the sampling error on our results. To heuristically mitigate this,
we propose a bootstrapping procedure, where for fixed $(t_{1},t_{2})$, we
sample with replacement from a single set of model outputs, and report the
median p-value $p_{med}$ across such samples. In Fig. 2 (Left) we report the
resulting distribution of $p_{med}$, where one point corresponds to one
independent run. We see that already for $100$ samples per $(t_{1},t_{2})$ (as
used in Table 1), the p-value distribution is narrow and the false positive
rate due to sampling error is well controlled. This confirms that our test is
robust against sampling error and can be used in realistic settings, without
additional access to model logprobs. For computational reasons, we did not
apply this correction in Table 1, where we still observe reliable results in
the median case across experiment repetitions.
#### Diversity assumption in Fixed-Sampling tests
As detailed in Section 3, the Fixed-Sampling test relies on the unwatermarked
model being sufficiently diverse, i.e., for the number of unique outputs
$R(n)=\mathbb{E}[U_{n}(q,t)]$ after $n$ queries with prompt $q$ and response
length $t$, it should hold that $R(n)=n$. Our goal is to show that we can
easily choose $t$ such that this property holds across different settings.
To this end, we hypothesize that the number of unique outputs converges to $n$
exponentially fast as the response length $t$ increases. In particular, we
assume
$R(n)=n-\lfloor n\cdot\exp(-\alpha(T)t)\rfloor,$ (10)
where $\alpha(T)$ is a monotonic function of the temperature $T$. To verify
this hypothesis, we measure $n-R(n)$ on several models and temperatures, and
show the result on log scale in Fig. 2 (Right). If Eq. 10 holds, we expect the
resulting relationship to be linear, which is indeed confirmed by our results.
While $\alpha$ (the slope of the line) varies across settings, we see that a
bit over $200$ tokens would be sufficient for the line to drop to $0$ (not
visible on the log scale plot). This holds even in cases impractical for
deployment of Fixed-Sampling watermarks such as $T=0.4$ [10, 16], indicating
that $R(n)=n$ and our assumption is met, validating our p-values.
### 5.3 Detecting watermarks in deployed models
Table 2: The results of our watermark detection tests on popular black-box LLM
deployments.
| | GPT4 | Claude 3 | Gemini 1.0 Pro
---|---|---|---|---
R-G (Section 2) | | 0.998 | 0.638 | 0.683
Fixed (Section 3) | | 0.938 | 0.844 | 0.938
Cache (Section 4) | | 0.51 | 0.135 | 0.478
Finally, we demonstrate the applicability of the statistical tests introduced
in Sections 2, 3 and 4, by applying them to popular black-box LLM deployments:
GPT4, Claude 3, and Gemini 1.0 Pro. We use the same experimental setup as in
Section 5.1, and use the API access for efficiency reasons—we do not rely on
any additional capabilities, and our tests could be as easily run in the web
interface. For the Cache-Augmented tests, we assume a global cache, that if
present clears after $1000$ seconds. For the Fixed-Sampling test on Claude 3,
due to its lack of diversity, we used $t=75$ tokens per query to ensure the
test’s hypotheses are met.
#### Results
Our results in Table 2 show that the null hypothesis is not rejected for any
of the models and any of the watermark tests. This suggests that none of the
deployed models tested are watermarked at this point in time. The demonstrated
applicability of our tests makes it simple to monitor these deployments and
detect any watermarking schemes that may be introduced in the future.
## 6 Related Work
#### Language model watermarking
Besides the approaches by [6, 8, 10] introduced above, there are various
methods building on similar ideas. Hou et al., [5], Liu et al., 2024b [13],
Ren et al., [18] all apply variations of [8] on semantic information, while Gu
et al., [4] aims to distill a new model from the output of a watermarked
model. Similarly, Liu et al., 2024a [12] apply a Red-Green scheme using a
learned classifier instead of hash functions. A range of works on multi-bit
watermarking [21, 23] aim to not only watermark generated texts but encode
additional information in the watermark itself.
#### Attacks on language model watermarking
Attacks on LLM watermarks have so far been mainly investigated in terms of
scrubbing [7, 8, 19] (i.e., removal of a watermark) and spoofing [7, 19, 4]
(i.e., applying a watermark without knowing $\xi$). Notably, Jovanović et al.,
[7] showed that observing watermarked texts can facilitate both attacks on
various distribution-modifying schemes, disproving common robustness
assumptions [9]. However, using this and similar attacks as means of practical
watermark detection is infeasible, as they generally offer no way to quantify
the attack’s success—in contrast, we aim to provide rigorous statements about
scheme presence. Further, such attacks incur significantly higher query costs
than necessary for detection (as our work demonstrates), and in some cases
assume certain knowledge of the watermark parameters, a setting fundamentally
at odds with our threat model of black-box watermark detection.
The closest related work to ours is Tang et al., [20], that tackles the
problem of watermark detection in strictly simpler settings where the
adversary either has access to an unwatermarked counterpart of the target
model, or can access full model logits. Such knowledge is commonly not
available in practice, limiting the applicability of this approach. To the
best of our knowledge, no work has developed methods for detecting the
presence of a watermark in a realistic black-box setting.
#### Extracting data from black-box models
With many of the most potent LLMs being deployed behind restricted APIs, the
extraction of model details has been an active area of research. This
includes, e.g., the reconstruction of a black-box model tokenizer [17] as well
as the extraction of the hidden-dimensionality or the weights of the embedding
projection layer [2]. Naseh et al., [14] have shown practical attacks to
recover the decoding mechanism of non-watermarked black-box models. Given
access to output logits, Li and Fung, [11] have further demonstrated that it
is possible to train an inversion model that aims to recover the input prompt.
## 7 Conclusion, Impact and Limitations
In this paper, we have focused on the problem of detecting watermarks in large
language models (LLMs) given only black-box access. We developed rigorous
statistical tests for the three most prominent scheme families, and validated
their effectiveness on a wide range of schemes and real-world models. Our
results imply that protecting from detectability should not be the key focus
of LLM watermarks, and that other properties such as robustness to attacks,
text quality, and efficiency should be prioritized.
#### Broader Impact
While our work primarily enables malicious parties to more easily circumvent
attempts at tracing LLM-generated text, we believe the benefits of our work
outweigh those risks, as our conclusions help model providers calibrate their
expectations in terms of watermark detectability, and avoid a false sense of
security.
#### Limitations
One limitation of our tests is that they are restricted to the three scheme
families discussed in Sections 2, 3 and 4. These are indeed the most prominent
in the literature, and as our tests are based on fundamental properties of
these scheme families, they should generalize to more variants and
combinations of the underlying ideas. However, it is possible that a model
provider deploys a scheme based on an entirely novel idea, which our tests
would not be able to detect. Further, our p-values are based on several model
assumptions, such as symmetric error terms and perfect sampling. While we
validate that these assumptions are sufficiently met on several open-source
models, we cannot guarantee that all models adhere to them. Finally, another
conceptual limitation of our Red-Green test is that it does not take into
account the possibility that the red-green vocabulary split is the same (on
the observed domain) for all contexts. The probability of this event decreases
exponentially with the number of different contexts, and is thus unlikely to
affect the test’s performance in practice.
## References
* Bommasani et al., [2021] Bommasani, R., Hudson, D. A., Adeli, E., Altman, R. B., Arora, S., von Arx, S., Bernstein, M. S., Bohg, J., Bosselut, A., Brunskill, E., Brynjolfsson, E., Buch, S., Card, D., Castellon, R., Chatterji, N. S., Chen, A. S., Creel, K., Davis, J. Q., Demszky, D., Donahue, C., Doumbouya, M., Durmus, E., Ermon, S., Etchemendy, J., Ethayarajh, K., Fei-Fei, L., Finn, C., Gale, T., Gillespie, L., Goel, K., Goodman, N. D., Grossman, S., Guha, N., Hashimoto, T., Henderson, P., Hewitt, J., Ho, D. E., Hong, J., Hsu, K., Huang, J., Icard, T., Jain, S., Jurafsky, D., Kalluri, P., Karamcheti, S., Keeling, G., Khani, F., Khattab, O., Koh, P. W., Krass, M. S., Krishna, R., Kuditipudi, R., and et al. (2021). On the opportunities and risks of foundation models. CoRR.
* Carlini et al., [2024] Carlini, N., Paleka, D., Dvijotham, K. D., Steinke, T., Hayase, J., Cooper, A. F., Lee, K., Jagielski, M., Nasr, M., Conmy, A., Wallace, E., Rolnick, D., and Tramèr, F. (2024). Stealing part of a production language model. CoRR.
* Council of the European Union, [2024] Council of the European Union (2024). Proposal for a regulation of the european parliament and of the council laying down harmonised rules on artificial intelligence (artificial intelligence act) and amending certain union legislative acts - analysis of the final compromise text with a view to agreement.
* Gu et al., [2024] Gu, C., Li, X. L., Liang, P., and Hashimoto, T. (2024). On the learnability of watermarks for language models. ICLR.
* Hou et al., [2023] Hou, A. B., Zhang, J., He, T., Wang, Y., Chuang, Y., Wang, H., Shen, L., Durme, B. V., Khashabi, D., and Tsvetkov, Y. (2023). Semstamp: A semantic watermark with paraphrastic robustness for text generation. arXiv.
* Hu et al., [2024] Hu, Z., Chen, L., Wu, X., Wu, Y., Zhang, H., and Huang, H. (2024). Unbiased watermark for large language models. ICLR.
* Jovanović et al., [2024] Jovanović, N., Staab, R., and Vechev, M. (2024). Watermark stealing in large language models. arXiv.
* Kirchenbauer et al., [2023] Kirchenbauer, J., Geiping, J., Wen, Y., Katz, J., Miers, I., and Goldstein, T. (2023). A watermark for large language models. In ICML.
* Kirchenbauer et al., [2024] Kirchenbauer, J., Geiping, J., Wen, Y., Shu, M., Saifullah, K., Kong, K., Fernando, K., Saha, A., Goldblum, M., and Goldstein, T. (2024). On the reliability of watermarks for large language models. ICLR.
* Kuditipudi et al., [2024] Kuditipudi, R., Thickstun, J., Hashimoto, T., and Liang, P. (2024). Robust distortion-free watermarks for language models. TMLR.
* Li and Fung, [2013] Li, Y. and Fung, P. (2013). Improved mixed language speech recognition using asymmetric acoustic model and language model with code-switch inversion constraints. In ICASSP.
* [12] Liu, A., Pan, L., Hu, X., Li, S., Wen, L., King, I., and Yu, P. S. (2024a). A private watermark for large language models. ICLR.
* [13] Liu, A., Pan, L., Hu, X., Meng, S., and Wen, L. (2024b). A semantic invariant robust watermark for large language models. ICLR.
* Naseh et al., [2023] Naseh, A., Krishna, K., Iyyer, M., and Houmansadr, A. (2023). On the risks of stealing the decoding algorithms of language models. CoRR.
* Pang et al., [2024] Pang, Q., Hu, S., Zheng, W., and Smith, V. (2024). Attacking LLM watermarks by exploiting their strengths. arXiv.
* Piet et al., [2023] Piet, J., Sitawarin, C., Fang, V., Mu, N., and Wagner, D. A. (2023). Mark my words: Analyzing and evaluating language model watermarks. arXiv.
* Rando and Tramèr, [2024] Rando, J. and Tramèr, F. (2024). The worst (but only) claude 3 tokenizer. https://github.com/javirandor/anthropic-tokenizer.
* Ren et al., [2023] Ren, J., Xu, H., Liu, Y., Cui, Y., Wang, S., Yin, D., and Tang, J. (2023). A robust semantics-based watermark for large language model against paraphrasing. arXiv.
* Sadasivan et al., [2023] Sadasivan, V. S., Kumar, A., Balasubramanian, S., Wang, W., and Feizi, S. (2023). Can ai-generated text be reliably detected? arXiv.
* Tang et al., [2023] Tang, L., Uberti, G., and Shlomi, T. (2023). Baselines for identifying watermarked large language models. CoRR.
* Wang et al., [2024] Wang, L., Yang, W., Chen, D., Zhou, H., Lin, Y., Meng, F., Zhou, J., and Sun, X. (2024). Towards codable text watermarking for large language models. ICLR.
* Wu et al., [2023] Wu, Y., Hu, Z., Zhang, H., and Huang, H. (2023). Dipmark: A stealthy, efficient and resilient watermark for large language models. arXiv.
* [23] Yoo, K., Ahn, W., and Kwak, N. (2024s). Advancing beyond identification: Multi-bit watermark for language models. NAACL.
## Appendix A Extending Experiments across Diverse Settings
Table 3: Additional results of our watermark detection tests across different
models and watermarking schemes. We report median p-values across 100
repetitions of the experiment, and for Red-Green schemes additionally over 5
watermarking keys. p-values below $0.05$ (test passing) are highlighted in
bold. $\delta$R and $\gamma$R denote $\delta$-reweight and $\gamma$-reweight
schemes, respectively.
| | | | | Red-Green | | Fixed-Sampling | | Cache-Augmented
---|---|---|---|---|---|---|---|---|---
| | Unwatermarked | | LeftHash | SelfHash | | ITS/EXP | | DiPmark/$\gamma$R | $\delta$R
Model | Test | $T=$ $1.0$ | $T=$ $0.7$ | | $\delta,\gamma=$ $2,0.25$ | $\delta,\gamma=$ $2,0.5$ | $\delta,\gamma=$ $4,0.25$ | $\delta,\gamma=$ $4,0.5$ | $\delta,\gamma=$ $2,0.25$ | $\delta,\gamma=$ $2,0.5$ | $\delta,\gamma=$ $4,0.25$ | $\delta,\gamma=$ $4,0.5$ | | $n=$ $256$ | $n=$ $2048$ | | $\alpha=$ $0.3$ | $\alpha=$ $0.5$ |
Mistral 7B | R-G (Section 2) | 1.000 | 1.000 | | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | | 1.000 | 1.000 | | 1.000 | 1.000 | 1.000
Fixed (Section 3) | 0.938 | 0.938 | | 0.938 | 0.938 | 0.938 | 0.938 | 0.938 | 0.938 | 0.938 | 0.938 | | 1.3e-125 | 1.1e-9 | | 0.938 | 0.938 | 0.938
Cache (Section 4) | 0.570 | 0.667 | | 0.607 | 0.639 | 0.632 | 0.608 | 1.000 | 1.000 | 0.742 | 0.824 | | 0.638 | 0.687 | | 2.4e-4 | 2.1e-3 | 5.6e-27
Llama2 13B | R-G (Section 2) | 0.149 | 0.663 | | 0.000 | 0.014 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | | 0.121 | 0.128 | | 0.149 | 0.149 | 0.149
Fixed (Section 3) | 0.968 | 0.868 | | 0.938 | 0.966 | 0.938 | 0.938 | 0.938 | 0.938 | 0.938 | 0.967 | | 5.5e-124 | 7.5e-8 | | 0.938 | 0.938 | 0.968
Cache (Section 4) | 0.708 | 0.573 | | 0.511 | 0.596 | 0.623 | 0.807 | 0.657 | 0.619 | 0.710 | 0.583 | | 0.518 | 0.692 | | 1.8e-2 | 5.3e-3 | 6.7e-32
Llama2 70B | R-G (Section 2) | 1.000 | 1.000 | | 0.000 | 0.000 | 0.000 | 0.020 | 0.000 | 0.020 | 0.000 | 0.000 | | 1.000 | 1.000 | | 1.000 | 1.000 | 1.000
Fixed (Section 3) | 0.938 | 0.526 | | 0.966 | 0.964 | 0.889 | 0.965 | 0.963 | 0.984 | 0.975 | 0.990 | | 4.2e-125 | 1.7e-8 | | 0.938 | 0.966 | 0.938
Cache (Section 4) | 0.596 | 0.620 | | 0.657 | 0.797 | 0.824 | 0.639 | 0.535 | 0.651 | 0.608 | 0.593 | | 0.463 | 0.818 | | 1.5e-3 | 4.4e-3 | 5.8e-28
Llama2 7B | R-G (Section 2) | 1.000 | 1.000 | | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | | 1.000 | 1.000 | | 1.000 | 1.000 | 1.000
Fixed (Section 3) | 0.993 | 0.988 | | 0.938 | 0.938 | 0.938 | 0.938 | 0.968 | 0.938 | 0.938 | 0.938 | | 7.6e-124 | 1.0e-8 | | 0.938 | 0.938 | 0.993
Cache (Section 4) | 0.604 | 0.602 | | 0.623 | 0.705 | 0.728 | 0.593 | 0.620 | 0.718 | 0.610 | 0.593 | | 0.476 | 0.588 | | 4.2e-6 | 4.5e-7 | 1.3e-21
Llama3 8B | R-G (Section 2) | 1.000 | 1.000 | | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | | 1.000 | 1.000 | | 1.000 | 1.000 | 1.000
Fixed (Section 3) | 0.938 | 0.938 | | 0.938 | 0.938 | 0.938 | 0.938 | 0.938 | 0.938 | 0.986 | 0.938 | | 1.2e-124 | 3.8e-8 | | 0.938 | 0.938 | 0.938
Cache (Section 4) | 0.734 | 0.504 | | 0.605 | 0.514 | 0.712 | 0.605 | 0.600 | 0.731 | 0.729 | 0.714 | | 0.618 | 0.605 | | 5.2e-5 | 3.2e-8 | 3.5e-18
#### Watermark detection with additional models and schemes
We extend the experiments from Section 5.1 using two additional models,
Llama2-7B and Llama3-8B, as well as more variations of the watermarking
schemes’ parameters to further assess the robustness of the tests. The
experimental setup for the additional results is consistent with the one
described in Section 5.1.
Our exhaustive results are presented in Table 3. The same conclusion applies
to the two additional models: the null hypothesis (_the specific watermark is
not present_) is rejected at a 95% confidence level only when the
corresponding watermarking scheme is applied to the model. These results
confirm that the modeling assumptions for each test are satisfied across a
wide range of models, indicating the tests’ relevance in practical scenarios.
#### Multiple keys in Red-Green watermarks
To demonstrate that the Red-Green test is robust to variations in the
watermarking scheme within the same watermark family, we consider the case of
Red-Green watermarks with multiple keys, where the key $\xi$ is uniformly
selected from a predefined pool of keys at each generation. Using $n$ keys
turns Eq. 3 into
$l_{t_{1},t_{2}}(x)=x^{0}/T+\delta^{\prime\prime}_{t_{2}}(x)+\varepsilon^{\prime}_{t_{1},t_{2}}(x),$
(11)
with $\delta^{\prime\prime}_{t_{2}}(x)$ in $\\{k\delta/(nT)\,|\,\forall
k\in\\{-n,...,n\\}\\}$ is obtained by averaging the variables
$\delta^{\prime}_{t_{2}}(x)$ over the set of keys. Despite modeling changes,
the core assumption of logit bias being conditioned on $t_{2}$ remains
unchanged. Therefore, we can apply the same test as in Section 2 to detect the
watermark. Consequently, we conducted the Red-Green test on both the LeftHash
and SelfHash variants using $n=3$ keys on three models (Llama2-13B, Llama2-70B
and Mistral-7B). Recent work [15] shows that using too many keys can lead to
other vulnerabilities.
Across all three models and scheme parameters, the null hypothesis (_the Red-
Green watermark is not present_) is rejected at a 95% confidence level, with
median p-values lower than $1e\text{-}4$ for each combination of model and
setting. It shows that the Red-Green test is robust even in settings that
slightly deviate from the original modeling considered in Section 2. It
emphasizes the test’s reliance on the foundational principles behind Red-Green
schemes rather than on specific implementation details.
## Appendix B Details of Cache-Augmented Schemes
We provide the details of the three Cache-Augmented watermarking schemes
considered in this work: $\delta$-reweight [6], $\gamma$-reweight [6], and
DiPmark [22], that were omitted from Section 4.
All three variants, at each generation step $t$, jointly hash the watermark
key $\xi\in\mathbb{Z}_{2}^{K}$ and the preceding context $y_{t-h:t-1}$
(commonly setting $h=5$) using SHA256, and use the result as a seed to sample
a _code_ $E_{t}\in P_{E}$ uniformly at random. Let $p$ denote the probability
distribution over $V$, obtained by applying the softmax function to the
logits. For the $\delta$-reweight variant, $P_{E}=[0,1]$, and the code $E_{t}$
is used to sample the token in $V$ with the smallest index, such that the CDF
of $p$ is at least $E_{t}$. For the $\gamma$-reweight variant and DiPmark,
$P_{E}$ is the space of permutations of $V$. For $\gamma$-reweight, we
transform $p$ to a new distribution $p^{\prime}$ by, for each token $i\in V$,
setting $p^{\prime}(i)=f_{2}(f_{1}(E_{t}(i)))-f_{2}(f_{1}(E_{t}(i)-1))$, where
we have $f_{1}(i^{\prime})=\sum_{j\in V}{\mathbbm{1}}(E_{t}(j)\leq
i^{\prime})p(j)$ and $f_{2}(v)=\max(2v-1,0)$, effectively dropping the first
half of the permuted CDF. For DiPmark, given parameter $\alpha\in[0,0.5]$,
this is generalized by using $f_{2}(v)=\max(v-\alpha,0)+\max(v-(1-\alpha),0)$,
recovering $\gamma$-reweight for $\alpha=0.5$. The former two variants perform
detection using a log-likelihood ratio test (requiring access to $LM$), while
DiPmark uses a model-independent test.
## Appendix C Estimating Scheme Parameters
In this section, we describe how, mostly leveraging the queries already
performed during the detection tests, we can estimate the main parameters of
the detected watermarking scheme. This demonstrates the soundness of our
modeling assumptions in Sections 2, 3 and 4 from which we derive all the
estimators.
### C.1 Estimation of Red-Green Watermarking Scheme Parameters
If the null hypothesis (_the model not being Red-Green watermarked_) is
rejected, we can then estimate the scheme-specific watermark parameters
$\delta$ and the context size $h$ using mostly the same data as used in the
test. First, we describe the estimators for both parameters, and then discuss
their practicality by analyzing their performance on multiple models.
#### Description of estimators
To estimate $\delta$, we establish a parametrized model based on Eq. 2 that
relies on our flagging function from Eq. 6, and additional estimates
$\hat{l}_{t_{1},t_{2}}(w)$ for every $w\in\Sigma$, computed on the same data
as above, requiring no additional queries. For each $w\in\Sigma$, we set:
$\hat{l}_{t_{1},t_{2}}(w)=\bar{w}_{t_{1}}+G_{w}(t_{1},t_{2})\bar{\delta}-\log\left(\sum_{w^{\prime}\in\Sigma\setminus\\{w\\}}\exp\left(\bar{w}^{\prime}_{t_{1}}+G_{w^{\prime}}(t_{1},t_{2})\bar{\delta}\right)\right),$
(12)
and set $\forall t_{1}$, $\bar{w}_{t_{1}}=0$ for a single $w\in\Sigma$, as
logits are shift-invariant. Fitting the parameters $\bar{\delta}$ and all
$\bar{w}_{t_{1}}$ by minimizing the mean squared error with gradient descent
allows us to recover $\delta/T$ as $\bar{\delta}$. If $T$ is known or
estimated separately, this term can be removed.
Let $h$ denote the context size, i.e., the number of _previous_ tokens
considered by the watermarking scheme. To estimate $h$, we use the same prompt
template as in Section 2, with a fixed prefix $t_{1}$ and digit $d$, but with
a varying $H$ and perturbation digit $d^{\prime}$ prepended to $d$.
⬇
f"Complete the sentence \"{$t1$} {$d^{\prime}$}{$d\cdot H$}\" using a random
word from: [{$\Sigma$}]."
The probability distribution of the model output will abruptly change when $H$
exceeds the context size $h$, as the change in $d^{\prime}$ will not alter the
red/green split of the vocabulary. By performing a series of pairwise Mood’s
tests on the estimated log-probabilities of $x$ we can find the largest $H$
for which $t_{1}$ is part of the context. This corresponds to the first value
of $H$ for which the test is rejected.
Estimating $\gamma$ is more challenging, as in contrast to $\delta$, this
parameter is not directly reflected in the logits but rather defines a more
global behavior of the scheme. This is particularly difficult for schemes with
self-seeding, as the rejection sampling interferes with the behavior of
$\gamma$. We leave further exploration of this problem to future work.
Figure 3: Estimation of $\delta$ for different models using LeftHash with
$\gamma=0.25$. The number of samples used increases from left to right, with
the leftmost plot assuming direct access to the log-probabilities. The
estimation is done on the same data as the test. Error bars are given by the
95% bootstrapped confidence interval with respect to the sampling of the model
outputs.
#### Experimental results
We computed the estimator for $\delta$ on the LeftHash variant with
$\gamma=0.25$ and varying $\delta$ from 0 to 4. The results are shown in Fig.
3, with the 95% confidence intervals reflecting the sampling error. The
estimator successfully estimates $\delta$ for all models with a sufficient
number of samples, using only the estimated output probabilities of the model.
It is also consistent across all models tested, suggesting that the model
assumptions in Eq. 1 are met in practice.
Estimating the context size for Red-Green schemes requires performing a new
attack once the model is flagged as watermarked. We estimate the context size
for three different models (Mistral-7B, Llama2-13B and Llama2-70B) using
LeftHash with $\delta=2$ and $\gamma=0.25$. The estimation process requires an
additional 5,000 queries, and the estimator successfully determines the
context size for all models. However, the estimator is less robust on the
SelfHash variant due to the self-seeding algorithm, which leads to higher
probabilities for tokens in $\Sigma$ being in the green vocabulary, and thus
diminishes the perturbation’s significance and resulting in false negatives in
Mood’s test. Therefore, the procedure stated above produces a lower bound for
the context size. To mitigate this issue, we use the estimator across 10
different $t_{2}$ and then consider the median of the $10$ estimators as our
final estimator. This estimator applied on the SelfHash variant with
$\delta=2$ and $\gamma=0.25$ is successful on all three models. It also does
not change the results on LeftHash and can be used as a more robust estimator
for $h$ in all cases, when the additional cost of $50,000$ queries is not
prohibitive.
### C.2 Estimation of Fixed-Sampling Watermarking Scheme Parameters
Our approach does not distinguish between the variant of the Fixed-Sampling
watermark used (ITS or EXP), as the diversity property that we exploit is
common to both. The only other relevant parameter of Fixed-Sampling schemes is
$n_{\text{key}}$. To estimate it, we use non-linear regression on the
rarefaction curve using (9) and the same data that we used for the presence
test, and compute the confidence intervals using bootstrapping.
Our results are given in Table 4. We see that the estimator is consistent
across different models and remains relatively precise even for values of
$n_{\text{key}}$ higher than the number of queries.
Table 4: Key length estimation for Fixed-Sampling watermarks using non-linear
regression on the rarefaction curve.
Key length | | Llama2-7B | Llama2-13B | Llama2-70B | Llama3-8B | Mistral-7B
---|---|---|---|---|---|---
256 | | $259\pm 0.6$ | $259\pm 0.5$ | $256\pm 0.5$ | $257\pm 0.5$ | $256\pm 0.6$
2048 | | $1978\pm 10$ | $2107\pm 12$ | $2006\pm 13$ | $2070\pm 14$ | $1831\pm 10$
### C.3 Estimation of Cache-Augmented Watermarking Scheme Parameters
For Cache-Augmented watermarks, we can estimate which scheme variant is
present, and if the variant is DiPmark, attempt to learn the value of $\alpha$
(recall that $\alpha=0.5$ corresponds to $\gamma$-reweight). To do this, we
use the same approach as in Section 4 to obtain $\hat{p_{1}}$ and
$\hat{p_{2}}$, where WLOG we assumed $p_{1}>0.5$. If we observe
$\hat{p_{2}}=0$ this directly implies a $\delta$-reweight watermark. If we
observe $\hat{p_{2}}\in(0,1)$, we learn the following: if
$\hat{p_{2}}=2\hat{p_{1}}-1$ then $\alpha>1-\hat{p_{1}}$, otherwise
$\alpha=|\hat{p_{1}}-\hat{p_{2}}|$. The bound in the first case can be further
improved with additional queries with different $p_{1}$. Finally, if we
observe $\hat{p_{2}}=1$ we repeat the whole procedure in total $K$ times,
following the same case distinction—if $\hat{p_{2}}=1$ repeats in all $K$
runs, we conclude that the model is watermarked with $\delta$-reweight.
Using the same parameters as the one for the test, we distinguish with 100%
accuracy between a DiPmark and a $\delta$-reweight watermark. However, the
estimation of $\alpha$ becomes unreliable for higher values of $\alpha$,
especially for smaller models. One of the reasons for this are the failures of
the model to follow the instruction, that are more common in the presence of
the uncommon prefix $uc$. While the detection test in Section 4 was robust to
such behavior, this does not hold for the estimation of $\alpha$.
|
# The Surprising Lack of Effect from Stellar Feedback on the Gas Stripping
Rate from Massive Jellyfish Galaxies
Nina Akerman,1,2 Stephanie Tonnesen,3 Bianca Maria Poggianti,1 Rory Smith,4
Antonino Marasco,1 Andrea Kulier,1 Ancla Müller,5 and Benedetta Vulcani,1
1INAF - Astronomical Observatory of Padova, vicolo dell’Osservatorio 5,
IT-35122 Padova, Italy
2Dipartimento di Fisica e Astronomia ‘Galileo Galilei’, Università di Padova,
vicolo dell’Osservatorio 3, IT-35122, Padova, Italy
3Flatiron Institute, CCA, 162 5th Avenue, New York, NY 10010, USA
4Departamento de Física, Universidad Técnica Federico Santa María, Vicuña
Mackenna 3939, San Joaquín, Santiago de Chile
5Ruhr University Bochum (RUB), Faculty of Physics and Astronomy, Astronomical
Institute, Universitätsstr. 150, 44801 Bochum, Germany E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
We study the role of star formation and stellar feedback in a galaxy being ram
pressure stripped on its infall into a cluster. We use hydrodynamical wind-
tunnel simulations of a massive galaxy ($M_{\text{star}}=10^{11}M_{\odot}$)
moving into a massive cluster ($M_{\text{cluster}}=10^{15}M_{\odot}$). We have
two types of simulations: with and without star formation and stellar
feedback, SF and RC respectively. For each type we simulate four realisations
of the same galaxy: a face-on wind, edge-on wind, $45^{\circ}$ angled wind,
and a control galaxy not subject to ram pressure. We directly compare the
stripping evolution of galaxies with and without star formation. We find that
stellar feedback has no direct effect on the stripping process, i.e. there is
no enhancement in stripping via a velocity kick to the interstellar medium
gas. The main difference between RC and SF galaxies is due to the indirect
effect of stellar feedback, which produces a smoother and more homogeneous
interstellar medium. Hence, while the average gas surface density is
comparable in both simulation types, the scatter is broader in the RC
galaxies. As a result, at the galaxy outskirts overdense clumps survive in RC
simulation, and the stripping proceeds more slowly. At the same time, in the
inner disc, underdense gas in the RC holes is removed faster than the smoothly
distributed gas in the SF simulation. For our massive galaxy, we therefore
find that the effect of feedback on the stripping rate is almost negligible,
independent of wind angle.
###### keywords:
methods: numerical – galaxies: evolution – galaxies: clusters: intracluster
medium – galaxies: ISM
††pubyear: 2023††pagerange: The Surprising Lack of Effect from Stellar
Feedback on the Gas Stripping Rate from Massive Jellyfish Galaxies–C
## 1 Introduction
A cluster galaxy may be subject to processes specific to high-density
environments. Among them is ram pressure stripping (RPS; Gunn & Gott, 1972) in
which the intracluster medium (ICM) interacts with the interstellar medium
(ISM) of the galaxy, removing it as the galaxy falls into the cluster
potential well. As a result, over time the galaxy quenches its star formation,
while the stripped gas forms tails trailing behind the galaxy disc (Gavazzi et
al., 2001; Sun et al., 2010, 2021; Fumagalli et al., 2014; Kenney et al.,
2014; Fossati et al., 2016; Jáchym et al., 2017, 2019; George et al., 2018;
Moretti et al., 2020a). Observational evidence (corroborated by simple
theoretical predictions, e.g. Gunn & Gott, 1972) suggests that as the gas is
stripped outside-in (Cortese et al., 2012; Merluzzi et al., 2016; Gullieuszik
et al., 2017; Fossati et al., 2018; Cramer et al., 2019; Vulcani et al., 2020)
and the disc is left truncated with a disturbed morphology (Koopmann & Kenney,
2004; Crowl & Kenney, 2008; Boselli et al., 2014; Fritz et al., 2017; Kenney
et al., 2015). The effects outlined above have long been observed in HI gas
(Cayatte et al., 1990; Kenney et al., 2004; Chung et al., 2007), which can be
ionised by in-situ star formation or other processes, giving rise to H$\alpha$
emitting-tails (Fumagalli et al., 2014; Merluzzi et al., 2016; Poggianti et
al., 2017; Bellhouse et al., 2017; Gullieuszik et al., 2017). Recent
observations showed that such tails also contain large amounts of molecular
gas, partly formed in-situ and partly stripped in the vicinity of the disc
(Lee et al., 2017; Jáchym et al., 2017, 2019; Lee & Chung, 2018; Moretti et
al., 2018, 2020a, 2020b).
Galaxy evolution under RPS has also been studied in cosmological and idealised
simulations. Some of the first simulations used a constant ICM wind (Schulz &
Struck, 2001; Roediger & Brüggen, 2006; Tonnesen & Bryan, 2009), though
Tonnesen (2019) showed that modelling ram pressure (RP) whose magnitude
increases with time (as experienced by a galaxy falling into a cluster) is
crucial for obtaining realistic stripping profiles of galaxies that can be
compared to the observed ones. Different angles at which the wind hits the
galaxy have also been modelled (Roediger & Brüggen, 2006, 2007; Jáchym et al.,
2007; Bekki, 2014; Steinhauser et al., 2016; Akerman et al., 2023), with a
conclusion that there is a weak dependence of the stripping process on the
inclination angle unless it is close to edge-on stripping. There are also some
simulations that include a magnetised ISM (Dursi & Pfrommer, 2008; Ruszkowski
et al., 2014; Tonnesen & Stone, 2014; Ramos-Martínez et al., 2018) and cosmic
rays (Farber et al., 2022), which generally find small changes to the global
stripping rate but more significant differences in the gas phase distribution.
Whether RP immediately quenches a galaxy or is able to cause some star
formation enhancement via gas compression has long been debated. The latest
observations show that some enhancement can indeed take place (Vulcani et al.,
2018; Roberts & Parker, 2020; Roberts et al., 2021, 2022a, 2022b). Simulations
agree as well, noting that the star formation rate (SFR) would be increased
only temporarily and that whether or not the SFR enhancement would happen
depends on many factors, such as galaxy mass, wind inclination angle and RP
strength (Tonnesen & Bryan, 2012; Bekki, 2014; Roediger et al., 2014;
Steinhauser et al., 2016; Ruggiero & Lima Neto, 2017; Lee et al., 2020).
Göller et al. (2023) find similar results in a large cosmological simulation
(TNG50), where SFR is enhanced in individual stripped galaxies at some point
of their evolution, although as a population RPS galaxies have low SFRs.
Although simulations have extensively studied the effect of RPS on the SFR, it
is still unclear how, in turn, star formation and stellar feedback (hereafter
just ‘feedback’ since this is the only type of feedback that we model)
influence gas stripping. Bahé & McCarthy (2015) show that feedback assists RP
in removing gas from a galaxy, with the effect being stronger for low-mass
galaxies with $M_{\text{star}}=(2-5)\times 10^{9}M_{\odot}$. Kulier et al.
(2023) also find that in the EAGLE simulation (Schaye et al., 2015) stripping
proceeds quicker due to the stellar feedback, and they note that the feedback
might be overestimated in EAGLE (see also Bahé et al., 2016) due to the
subgrid implementation and cosmological-scale resolution.
In this work, we study how star formation and, importantly, stellar feedback,
affect the process of RPS by comparing gas in galaxies from two sets of
simulations, a first set that includes radiative cooling, and a second set
that adds star formation and feedback prescriptions to the radiative cooling
simulations. We can naively imagine two scenarios. In the first one, feedback
might impede gas stripping by adding more thermal and nonthermal pressure
support to and above the ISM that will block the RP from making direct contact
with the disc (this will be most effective when the wind has a significant
component in the face-on direction). On the other hand, feedback might enhance
gas removal by moving ISM gas higher above and below the disc where it is less
tightly gravitationally bound. In this work we find that the impact of
feedback is more nuanced than either of the above simple sketches.
The paper is organised as follows. First, we outline our simulations in
Section 2. In Section 3 we calculate and compare the stripping rates of star-
forming versus radiative-cooling-only galaxies, and in Section 4 we explore
what drives the differences in RPS in these sets of simulations. We compare
our results to other works in Section 5 and draw conclusions in Section 6.
## 2 Methodology
We describe our simulations in detail in our previous work, Akerman et al.
(2023), and here will briefly highlight their main features. Using the
adaptive mesh refinement code Enzo (Bryan et al., 2014), we make a simulation
box with 160 kpc on a side, with 5 levels of refinement allowing for a maximum
resolution of 39 pc. The refinement criteria are Jeans length and baryon mass
of $\approx 7500M_{\odot}$. In Appendix A we perform a resolution test and
show how our refinement criteria resolve the galaxy disc.
The simulations include radiative cooling using the grackle library (Smith et
al., 2017) with metal cooling and the UV background by Haardt & Madau (Haardt
& Madau, 2012). The disc starts with metallicity $Z=1.0\;Z_{\odot}$ and the
ICM has $Z=0.3\;Z_{\odot}$.
We use star formation and stellar feedback recipes by Goldbaum et al. (2015,
2016). If the mass of a cell exceeds the Jeans mass and the minimum threshold
number density of $n_{\rm{thresh}}=10\,\rm{cm}^{-3}$, a stellar particle will
form with a minimum mass of $10^{3}M_{\odot}$, assuming a star formation
efficiency of 1 per cent. We also simulated galaxies from the same initial
conditions (described below) with $n_{\rm{thresh}}=3\,\rm{cm}^{-3}$ and
$30\,\rm{cm}^{-3}$ and found that galaxies with all three $n_{\rm{thresh}}$
evolve similarly in their star formation history. We study the effect that the
choice of the threshold number density might have on our results in Appendix
B, where we also outline the role of star formation itself in shaping the ISM
distribution. Thus, the parameter choice for the $n_{\rm{thresh}}$ does not
notably affect the results. The feedback includes momentum and energy input
from supernovae (SNe) (that also increase cell metallicity), ionising
radiation from young stars (heating up to $10^{4}$ K), and winds from evolved
massive stars. SNe have combined energy budget of $10^{51}$ erg; first the
terminal momentum input is added from the number of SNe expected at a given
time step, then any additional energy is added as thermal energy. The total
momentum of $3\times 10^{5}\text{ M}_{\odot}\text{ km }\text{s}^{-1}$ is
distributed equally among the 26 nearest neighbour cells. The maximum change
in velocity of any given cell is limited to 1000 km s-1.
We follow the setup by Roediger & Brüggen (2006) and Tonnesen & Bryan (2009),
modelling static potentials for the stellar disc and the spherical dark matter
halo, while calculating the self-gravity of the gas component at each time
step. For the Plummer–Kuzmin stellar disc (Miyamoto & Nagai, 1975) we use a
mass of $M_{\text{star}}=10^{11}M_{\odot}$, a scale length of
$r_{\text{star}}=5.94$ kpc and a scale height of $z_{\text{star}}=0.58$ kpc.
We model a Burkert profile for the dark matter halo (Burkert, 1995; Mori &
Burkert, 2000) with a core radius of $r_{\text{DM}}=17.36$ kpc111While an
Navarro–Frenk–White dark matter potential has been found by many to be more
appropriate for massive galaxies (e.g. Di Cintio et al., 2014), a Burkert
profile has been found to be a better match to some observational samples of
massive galaxies (e.g. Rodrigues et al., 2017; Li et al., 2020). Importantly
for this work, the Burkert potential we use leads to a good match between the
rotation curve of our simulated galaxy and JO201. To orient the reader, the
R200 and M200 of our halo are 340 kpc and 1.15$\times$1012 M⊙, respectively..
Initial parameters of the gaseous disc are the following: mass
$M_{\text{gas}}=10^{10}M_{\odot}$, scale length $r_{\text{gas}}=10.1$ kpc and
scale height $z_{\text{gas}}=0.97$ kpc. The values were chosen to match a
well-studied RPS galaxy, JO201 (Bellhouse et al., 2017, 2019). As with the
previous work, our intention here is to merely select realistic initial
conditions based on an observed stripped galaxy, not to model it exactly. A
more detailed comparison between our simulated galaxies and JO201 can be found
in Akerman et al. (2023), Section 7.1.
We fix the galaxy in the centre of the simulation box and add an ICM wind via
inflow boundary conditions (and outflow on the opposite side) to simulate RPS.
We include a time-varying (in density and velocity) ICM wind (Tonnesen, 2019),
and to find its parameters, we model a galaxy falling into a massive cluster
($M_{\text{cluster}}=10^{15}M_{\odot}$) following the procedure described in
Bellhouse et al. (2019) from a clustercentric radius of 1.9 Mpc and with an
initial velocity of $1785\;\mathrm{km\,s}^{-1}$. We assume hydrostatic
equilibrium of an isothermal ICM with constant temperature of $T=7.55\times
10^{7}$ K and a beta profile. From Rankine–Hugoniot jump conditions for Mach
number of 3 we find the pre-wind ICM conditions using the ICM wind parameters
at the initial radius.
Since our galaxy is fixed in space, we define a wind angle as the angle
between the wind direction and the galaxy rotation axis, and model three wind
angles: $0^{\circ}$ (a face-on wind that flows along the $z$-direction in our
simulated box, W0), $45^{\circ}$ (W45, in which the wind has equal components
along the $y$\- and $z$-directions), and $90^{\circ}$ (edge-on wind that flows
along the $y$-direction in our simulated box, W90). The wind angle is constant
throughout a simulation. As a control, we simulate a galaxy that does not
undergo RPS (no wind, NW).
We simulate galaxies that include only radiative cooling (denoted as RC) and
those that also include star formation and feedback (denoted as SF), thus
modelling in total eight galaxies: RCNW, RCW0, RCW45, RCW90 and SFNW, SFW0,
SFW45, SFW90.
We evolve our RC galaxies in isolation for 130 Myr, during which the gas cools
down and collapses into a thin disc. The wind starts inflowing through the
lower boundary of the simulation box, and we restart our simulation as four
separate runs: RCNW, RCW0, RCW45, RCW90. The wind reaches the galaxies after
70 Myr, taking 200 Myr in total for the galaxies to evolve before the onset of
RP. This amount of time allowed the majority of the gas in the disc
(particularly within the stripping radius of 15 kpc, discussed below) to
fragment into clouds. We repeat the procedure for the SF galaxies, allowing
them to evolve for 230 Myr in isolation in order to stabilise the SFR, such
that the variation of the SFR on a 5 Myr time-scale decreases to 5 per cent.
In total, the four galaxies (SFNW, SFW0, SFW45, SFW90) evolve for 300 Myr
before the beginning of RPS.
The RC simulations run for a total of 900 Myr and the SF simulations for a
full Gyr. We used 3.0M node-hours to simulate the four RC galaxies and 3.7M
node-hours for the four SF galaxies.
## 3 Stripping rate
Figure 1: Stripping rate of gas with metallicity $Z>0.35Z_{\odot}$ in a galaxy
disc (defined with a radius 30 kpc and a height of $\pm 2$ kpc from the disc
plane) as a function of time. From top to bottom, rows show data for different
wind angles, where W0 is a face-on stripped galaxy and W90 is stripped edge-
on. Each panel shows $\dot{M}$ for SF (solid) and RC (dashed) simulations.
Overplotted in black is the SFR for each galaxy normalised by the SFR in SFNW
at the same time step (right $y$-axis), with a solid (dotted) line plotting
the ratio above (below) one. Notice that for W0 and W45 (top and middle
panels) the left $y$-axis range is the same, while for W90 (bottom panel) it
is smaller. The right $y$-axis range is the same in all the panels.
We start by analysing differences in the gas distribution in the galactic
disc, a region defined as having a radius of $R=30$ kpc and $h=\pm 2$ kpc
height from the plane of the disc (we note that we also used a disc with
$h=\pm 5$ kpc height and found similar results). To quantify the effect that
the feedback has on the process of stripping we calculate the stripping rate:
$\dot{M}=\frac{\Delta M}{\Delta t}-\text{SFR},$ (1)
where $\Delta t=t_{i}-t_{i-1}=5$ Myr time step, $\Delta M=M_{i}-M_{i-1}$ is
mass difference of gas with metallicity $Z>0.35Z_{\odot}$. The metallicity cut
allows us to exclude the pure ICM, and the choice of this metallicity
threshold does not have a significant impact on our results, as we tested up
to a much more strict metallicity criterion of $0.7Z_{\odot}$. While equation
1 could be positive in the case of accretion from galaxy fountain flows or the
surrounding ICM, once the ICM wind begins removing gas from the galaxy
(shortly after RPS begins) $\dot{M}$ is always negative. Because we are
examining our simulations while stripping is occurring, we call $\dot{M}$ the
stripping rate. SFR is found at the same time step:
$\text{SFR}=\frac{M_{\text{star}}^{i}}{\Delta t},$ (2)
where $M_{\text{star}}^{i}$ is the total mass of stars born within the last
time step $\Delta t$, which is always 5 Myr. For RC galaxies, SFR is defined
to be zero. By subtracting SFR from the stripping rate we account for the gas
lost to star formation, thus the differences between stripping rates of RC and
SF galaxies should be due to the effect the feedback plays in RPS.
Figure 2: Top: SF to RC stripping rate (from Fig. 1) ratio as a function of
time, colour-coded by wind angle. The ratio is calculated for $\dot{M}<0$
only, to avoid confusion when gas is accreted at the beginning of the
simulation. The grey line of equal ratio serves to guide the eye. The W90
ratio starts at $\sim 150$ Myr because before that time the gas is not
stripped but accreted onto both of the galaxies. Bottom: Cumulative mass of
stripped gas (accounting for star formation) with metallicity
$Z>0.35Z_{\odot}$ as a function of time, for SF (solid) and RC (dashed)
simulations.
Fig. 1 shows the stripping rate, with each row for a different wind angle (W0
is a face-on stripped galaxy and W90 is stripped edge-on), solid lines
plotting SF galaxies and dashed lines plotting the RC ones. Note that here and
in other plots the $x$-axis (and later any time measurements) is ‘time since
RPS’, where $t=0$ denotes the start of RPS and excludes the 200 and 300 Myr
during which RC and SF galaxies, respectively, were evolving in isolation. We
also confirm that by the start of RPS the SF galaxy has only 5 per cent less
gas mass than the RC one, due to star formation, meaning that any variation in
the stripping process is not a result of a simple mass difference. We overplot
in black the SFR for each galaxy divided by the SFR in SFNW at the same
simulation time, with a solid (dotted) line indicating that the ratio is above
(below) one.
Once we have accounted for the gas loss due to star formation, the stripping
rates of RC and SF galaxies are very similar for any given wind angle. This
might be expected in SFW0 and SFW45 galaxies, since they slowly quench their
star formation after the onset of RPS due to the removal of gas in the
outskirts, following a brief slight enhancement of SFR during the initial disc
compression by the wind. Hence, the global role of feedback in these galaxies
will be decreasing with time, and they would become more and more similar to
their RC counterparts. Moreover, SFW0 and SFW45 (as well as RCW0 and RCW45)
are similar to each other. In Akerman et al. (2023) we show that under RP,
SFW0 and SFW45 evolve rather closely on the galaxy scale in terms of their gas
mass and quenching histories, so their comparable stripping rates (and the
striking difference from SFW90) are no surprise. Note that the small delay in
the peak stripping rates between W0 and W45 galaxies is simply due to the fact
that it takes the angled wind more time to reach the galaxy since it has a
larger distance to travel to the centre of the simulation box. At the peak
stripping rate of W0 and W45 (at $\sim$30–50 Myr) the RC galaxies are stripped
more rapidly than the SF galaxies, indicating that the small SFRs in our W0
and W45 simulations do not enhance stripping. We also note that the W0 maxima
are larger than those in W45, suggesting that more angled winds may strip
galaxies slower (in agreement with previous works such as Roediger & Brüggen,
2006). It is more surprising to see that even in SFW90, where the SFR is
increased by a factor of 2–3 due to RP, the feedback has almost no apparent
effect on the stripping rate. On the contrary, from 350–500 Myr after
stripping has begun RCW90 is stripped faster. We will go into the details of
where RPS removes mass from the galaxies in the next section.
To emphasise the variations in the stripping rates in the top panel of Fig. 2
we plot their ratio, $\dot{M}_{\text{SF}}$ over $\dot{M}_{\text{RC}}$. The
ratio is calculated for $\dot{M}<0$ only (i.e. outflow), to avoid confusion
when gas is accreted at the beginning of the simulation, as the ICM wind
starts mixing with the ISM prior to the gas removal. For W0 and W45 the ratio
is quite unstable at first, but between 100 to 200 Myr the RC galaxies are
stripped systematically faster. The apparent delay in the ratios of W0 and W45
is, again, due to the delay in stripping of the W45 galaxies. Finally, at 200
Myr the W0 galaxies reach the point of ‘stripping equilibrium’
($\dot{M}_{\text{SF}}/\dot{M}_{\text{RC}}\approx 1$) that lasts until 400 Myr
when $\geq 80$ per cent of these galaxies’ original gas has been stripped or
used for star formation. The W45 galaxies during most of this period have a
higher ratio than that of the W0 at identical times, and also higher than one.
However, the stripping rate itself is quite low in all the four galaxies, with
$\dot{M}\approx 7M_{\odot}\text{yr}^{-1}$ on average, which means that even
small differences in the stripping rate will be reflected in the ratio.
The differences in the stripping rates of SFW90 and RCW90 show a different
pattern, with a more clear trend in the ratio. We note that at the earliest
times we do not plot the ratio in the top panel because at the leading edge of
the galaxy, where the disc is hit by the wind, the ICM begins mixing with the
ISM at a rate higher than that of the gas removal on the opposite side of the
galaxy. Because of this, for the first 120 Myr the gas is not stripped but
accreted onto both of the galaxies. In RCW90 the total mass of the gas
accreted over this period equates to $6.5\times 10^{7}M_{\odot}$ or 0.67 per
cent of the galaxy mass at the start of RP, while in SFW90 it is $2\times
10^{8}M_{\odot}$ or 2 per cent of the galaxy mass. This accretion does not
affect the course of stripping. When the gas starts being removed at a steady
rate in both SFW90 and RCW90, the stripping rate is higher in the SF galaxy.
The ratio then starts gradually but steadily decreasing until 400 Myr, after
which it begins rising again.
The bottom panel in Fig. 2 shows the cumulative amount of stripped gas (this
is the integral of Fig. 1), i.e. the gas that was removed from a galaxy by
RPS, not the total gas mass lost by a galaxy (which would also include gas
used in star formation). This distinction is important to keep in mind, since
in the figure SFW0 and SFW45 have seemingly lost 15 per cent less gas than the
respective RC galaxies by the end of the simulation. This is due to the fact
that less gas was available to be stripped from these galaxies, because it was
used for star formation, and due to the difference in the stripping rates
during 30–50 Myr (to be discussed in Section 4). In combination, the two
panels in the figure illustrate that even though the SF-to-RC stripping rate
ratio in the W90 galaxies varies more in the first 500 Myr of RPS, while the
ratio in the W0 and W45 stays, in comparison, more constant, the actual mass
of stripped gas in the W90 galaxies is much lower. As also shown in Fig. 1,
before 200 Myr, the stripping rate in the W0 and W45 galaxies is much higher
than in the W90 galaxy, and even small disagreements between the RC and SF
stripping rates, when accumulated, lead to noticeable differences in the
amount of stripped gas. Conversely, RCW90 and SFW90 lose much less gas,
especially in the initial 100 Myr.
In none of the three galaxies can the behaviour of the stripping rate ratio be
easily tied to the presence of feedback. One could imagine that it would
either assist or impede the RPS. By expelling the gas high above the plane of
the disc (up to 15 kpc, see below), the feedback makes the gas less tightly
gravitationally bound and thus more easily stripped. On the other hand, the
feedback adds thermal and nonthermal pressure to the surrounding gas that in
the cases of face-on stripped W0 and W45 galaxies acts against the ICM wind.
In our simulations, even as the SFR in SFW0 and SFW45 quickly decreases, we
might expect the feedback to have a systemic effect on the stripping rate
prior to complete quenching (quenching only happens after 500 Myr). The role
of feedback is even less clear in the SFW90 galaxy, where the SFR is always
2–3 times higher than in SFNW galaxy, and hence, there is no obvious star-
formation-related reason for the long-term fluctuations of the stripping rate
ratio. To explain the behaviour we see, we now take a closer look at the gas
in their discs.
Figure 3: Mean cell-mass-weighted outward $z$-velocity, $v_{\text{out}}$,
(outward velocities $v_{\text{out}}>0$ and inward velocities
$v_{\text{out}}<0$) of ISM ($Z>0.35Z_{\odot}$) in a thin ($x=\pm 2$ kpc) slice
of the disc for RCNW (top) and SFNW (bottom), taken at 0 Myr. In the bottom
panel the black dashed lines indicate the galaxy boundaries, as we define them
with $z=\pm 2$ kpc.
## 4 What drives the differences between RC and SF galaxies
Here we directly compare the RC and SF simulations in the NW, W0 and W45, and
W90 cases separately. We begin by examining the impact of star formation and
feedback on an isolated galaxy, build a physical interpretation of how it will
affect stripping rates, and then use our understanding to examine the
differences in the stripping rates in detail.
Figure 4: Density projections for ISM ($Z>0.35Z_{\odot}$) in the central 5 kpc
of the disc, within $\pm 2$ kpc of the disc plane for RCNW (top) and SFNW
(bottom) at 0 Myr. While star formation and stellar feedback locally smooth
out the gas distribution in the disc, in RCNW the gas distribution remains
patchy, with holes between the dense clumps.
### 4.1 General impact of stellar feedback
We will start by analysing the impact that star formation and stellar feedback
have on gas in general, using our isolated simulations. To illustrate the
effect of SN feedback on gas flows around our simulated galaxies, in Fig. 3 we
plot the mean cell-mass weighted outward $z$-velocity, $v_{\text{out}}\equiv
v_{z}\text{sgn}(z)$ (outward velocities from the galaxy $v_{\text{out}}>0$ are
red and inward velocities to the galaxy $v_{\text{out}}<0$ are blue), of ISM
($Z>0.35Z_{\odot}$) in a thin ($x=\pm 2$ kpc) edge-on slice of the RCNW (top)
and SFNW (bottom) discs, taken at 0 Myr of RPS (200 Myr total evolution of
RCNW and 300 Myr for SFNW). As we begin the simulation, the initially
homogeneous ISM starts collapsing into a thin disc, as shown in the top panel
of Fig. 3, where the disc is about 400 pc thick. While there is no mechanism
to stop the collapse into dense clumps in RCNW (illustrated also with the blue
inward flows), in SFNW the feedback expels the SN gas from the galaxy via
galactic fountains. The bottom panel in Fig. 3 illustrates these fountains, as
they eject the gas (red) and it later falls back on to the galaxy (blue).
Along with ionising radiation from young stars and winds from evolved massive
stars the fountains act against collapse, and the SFNW galaxy disc becomes
much thicker compared to its RCNW counterpart. Although the outflows can eject
some material beyond 10 kpc from the disc plane, we highlight that there is
very little gas mass in these flows, with only about 0.1 per cent of the total
gaseous disc mass extending above 5 kpc from the disc plane.
In addition to this, star formation and stellar feedback also smooth out the
gas distribution in the disc, while in the RC galaxies there is no mechanism
to inject energy into cold clumps. As a result, the RC gas distribution is
patchy, with low-density holes between the dense clumps that would otherwise
turn into stars. To illustrate this, in Fig. 4 we plot the density projection
of ISM ($Z>0.35Z_{\odot}$) in the central 5 kpc of the disc within $\pm 2$ kpc
of the disc plane for the RCNW (top) and SFNW (bottom) at 0 Myr. We stress,
however, that the feedback is only able to smooth out the ISM locally and does
not cause radial mass redistribution across the whole disc.
Another feature present in RC galaxies is a ‘ring’, which manifests itself at
$\sim$200 Myr as a non-fragmented density distribution (therefore without
high-density clumps) located at $14\text{kpc}<r<20\text{kpc}$. It can be seen
in the top panel of Fig. 3 as a white ‘gap’ where the gas is not moving, and
is a result of a non-uniform collapse of the initial disc. The ring is a
numerical effect present in many galaxy-scale idealised simulations (e.g.
Goldbaum et al., 2015; Behrendt et al., 2015) that disappears with time (by
300 Myr of total simulation time or 0 Myr of RPS) in SF simulations as the gas
distribution becomes more homogeneous and less clumpy due to SN feedback. In
RCNW the ring disappears by 350 Myr of total simulation time, when the whole
galaxy disc collapses into dense clumps. Both the added clumpiness in the
central disc and the more uniform density distribution in the unfragmented
ring play a role in the comparative stripping rates of the RC and SF galaxies.
To summarise, star formation and stellar feedback homogenised the gas
distribution in the galaxy disc and prevented it from collapsing into the
highly dense clouds found in the inner regions of RCNW, with SN fountains
launching the gas as far as 15 kpc above the plane of the galaxy (Fig. 3).
### 4.2 The Importance of the Disc Gas Distribution
Figure 5: Surface density of ISM ($Z>0.35Z_{\odot}$) as a function of
galactocentric radius for RCW0 (top) and SFW0 (bottom) at 10 Myr. Surface
density is found within our galaxy disc boundaries $z\pm 2$ kpc (see the text
for details). The points are coloured by the mean cell-mass-weighted $v_{z}$.
The orange line plots average gas surface density as a function of radius. The
red line represents threshold surface density from the Gunn & Gott (1972)
model below which (hatched area) the gas could be stripped at the current RP.
This line illustrates that at the galaxy outskirts even the highest density
gas will be removed by the ICM wind, while at the centre the gas is protected.
To gain a physical understanding of how the gas distribution could affect a
galaxy’s gas stripping rate, we consider the gas distribution in the disc and
the outside-in nature of RPS, which can be quantified by calculating the
stripping radius at any time. Gunn & Gott (1972) define the stripping radius
as the radius at which the gravitational restoring force per unit area equals
the RP: $P_{\text{ram}}=2\pi G\Sigma_{\text{star}}\Sigma_{\text{gas}}$, where
$\Sigma_{\text{star}}$ is the stellar surface density and
$\Sigma_{\text{gas}}$ is the gas surface density. While other equations can be
used to calculate whether gas will be removed (e.g. McCarthy et al., 2008),
this simple criterion has been shown to match reasonably well with simulations
(Roediger & Brüggen, 2006; Steinhauser et al., 2016; Lee et al., 2020) and is
enough to guide our understanding.
Here, to find the restoring force we measure $\Sigma_{\text{gas}}$ as a
function of radius along the wind direction ($z$-axis), within $z=\pm 2$ kpc
(the galaxy boundaries). We have repeated the calculations for the disc height
of $z\pm 5$ kpc and found qualitatively similar results. We measure the
$\Sigma_{\text{stars}}$ only from the static stellar disc, since compared to
it the surface density of our star particles is negligible. Assuming
instantaneous stripping, this is the radius to which a galaxy would be
stripped at any given RP. In our galaxies, immediately after the onset of RP,
the stripping radius is $\sim 15$ kpc, and due to the increasing RP it drops
down to 10 kpc in the next 100 Myr. Note that in this paper we define the
stripping radius solely by comparing the gas surface density in our simulated
discs to the surface density at which the Gunn & Gott (1972) criterion would
predict stripping. It is not a measure of where the gas is actually stripped
from the disc in our simulation, and indeed the goal of this section is to
predict when and understand why simulated galaxies do not follow the simple
Gunn & Gott (1972) prescription.
The use of the stripping radius to calculate where the ISM will be removed
assumes that gas only varies in surface density as a function of radius, which
is clearly an oversimplification. Fig. 4 shows that in both SF and RC galaxies
gas fragments into dense clumps embedded in a lower density ISM. To understand
the impact of the level of homogeneity in the gas density distribution on gas
removal by RPS, in Fig. 5 we plot surface density of the ISM
($Z>0.35Z_{\odot}$) as a function of galactocentric radius for RCW0 (top) and
SFW0 (bottom) at 10 Myr. In order to use a more spatially-refined density
distribution in the disc, we select the area of $60\times 60\text{ kpc}^{2}$
around the galaxy centre and divide it into $1024\times 1024$ squares (we
confirm that the following results are independent of the resolution). We then
integrate the density in each square over the galaxy height of $z\pm 2$ kpc.
We find qualitatively similar results for the disc height of $z\pm 5$ kpc. In
contrast with finding an average surface density at a given radius, this
method allows us to capture high-density clumps and low-density holes in RCW0
and SFW0, while the choice of surface density instead of the volume/number
density informs us about the stripping using the Gunn & Gott (1972) criterion.
We colour the points by the mean cell-mass-weighted $v_{z}$. The orange line
plots average gas surface density as a function of radius. The red line
represents threshold surface density from the Gunn & Gott (1972) model. This
threshold surface density is found by equating the restoring force to the RP
and calculating for each radius the surface density below which (hatched area)
the gas could be stripped at the current RP. This line illustrates that at the
galaxy outskirts even gas with surface densities well above the average will
be removed by the ICM wind, while at the centre nearly all of the gas is
protected.
In this figure we first notice that the surface density in both the RC and SF
galaxies varies by orders of magnitude at every radius. Also, although the
average value of the surface density is quite similar in both galaxies, the RC
galaxy has a larger range of $\Sigma_{\text{gas}}$ at most radii, except for
in the ring region that we introduced in Section 4.1. This figure also shows
that SFW0 disc is more radially extended ($r>25$ kpc) than the RCW0 disc with
low $\Sigma_{\text{gas}}$ from stellar feedback. We again emphasise that this
is a local effect that does not cause mass redistribution over large radial
scales, nor does feedback cause the disappearance of the ring region that
happens in RCNW at a similar rate compared to SFNW – this naturally happens
over time due to fragmentation from radiative cooling. We also point out that
lower $\Sigma_{\text{gas}}$ tends to have a higher vz.
We can now discuss how the gas inhomogeneity in the discs will influence gas
stripping, starting at the galaxy outskirts. Beyond about 15 kpc, gas must
have a higher $\Sigma_{\text{gas}}$ than the average in order to remain
unstripped according to our criterion. In Fig. 5 we see that at the outermost
regions of the galaxy ($\sim$23 kpc), RCW0 has more gas at
$\Sigma_{\text{gas}}$ above the red line. Therefore, at early times, when the
outermost gas is removed, we would expect SFW0 to be stripped more quickly.
In the ring region of RCW0, at 14–20 kpc, the gas has not yet fragmented and
therefore largely lies below the red line. In the same region in SFW0, there
is gas at higher $\Sigma_{\text{gas}}$ that is likely to survive longer.
Therefore, in the outer region, we predict that gas homogeneity increases gas
stripping as more gas falls below the red line.
In the inner region we might expect the gas homogeneity to assist RPS as it
does in the disc outskirts, but Fig. 5 reveals a different picture. Within the
inner 10 kpc gas inhomogeneity leads to some regions of the disc having very
low $\Sigma_{\text{gas}}$. While the mass in these regions is low, the
extremely low $\Sigma_{\text{gas}}$ ISM in RCW0 has high velocity in the wind
direction ($v_{z}$). In fact, in the inner disc, the gas below the red line in
RCW0 has a higher $v_{z}$ than the same $\Sigma_{\text{gas}}$ ISM in SFW0.
Although there are similar amounts of mass below the gas surface density
stripping threshold, the higher velocities in the RC ISM result in higher mass
fluxes.
We highlight that this is a snapshot of the ISM $\Sigma_{\text{gas}}$
distribution, and this distribution will vary with time over the simulation.
The most obvious differences will be a lack of low $\Sigma_{\text{gas}}$ ISM
at large radii as that gas is removed, and that as RP increases, the red line
will shift to higher $\Sigma_{\text{gas}}$ values. However, all of the general
trends we have outlined here persist over time: stellar feedback continues to
allow for low $\Sigma_{\text{gas}}$ gas just beyond the bulk of the disc,
dense clumps survive at large radius, and low-density regions are accelerated
to high velocities. In the next subsections we will see how this plays out in
our simulations.
Figure 6: For the W0 galaxies at 10 Myr of RPS, from top to bottom: difference
between the azimuthally-averaged distributions of cell-mass-weighed velocity
$v_{z}$, vertical mass flux, and total cell mass in RCW0 and SFW0 galaxies, as
a function of cylindrical radius $r$ and $z$. We calculate the data only for
the stripped gas with $v_{z}>0$ to avoid confusion when plotting the
difference. The solid cyan line separates downwind and upwind halves of the
disc (wind moves upward in the $z$-direction), while the dashed cyan line
denotes the the $z=2$ kpc height at which we define the galactic disc
boundaries. The green line outlines the gas distribution in RCW0 to illustrate
that it is more compact compared to SFW0 at the same time step.
### 4.3 Face-on stripped galaxies
Now that we have built intuition for how we expect stripping to be influenced
by the ISM gas distribution, we will begin looking at RPS-galaxies, starting
with W0, as it is stripped the most rapidly. We measure the mass flux of gas
($Z>0.35Z_{\odot}$) in the galaxy disc ($R=30$ kpc and $h=\pm 2$ kpc) in RCW0
and SFW0 galaxies. Then, to understand the evolution of the SF-to-RC stripping
rate ratio we will map the difference in the mass flux distribution between
the two galaxies. Since in this case RPS is axially-symmetric, we can make use
of cylindrical coordinates, focusing only on variations as a function of
radius $r$ and $z$-height. Here, we select the data only for the cells with
upward outflows $v_{z}>0$ to avoid confusion when plotting the difference
between the two galaxies (see below), but for completeness, in Appendix C we
show maps for cells with downward inflows $v_{z}<0$. The mass flux is:
$\dot{M}=\sum_{i}\frac{m_{i}v_{z,i}}{dL},$ (3)
where $m_{i}$ is mass of the $i$-th cell, $v_{z,i}$ is the $z$ velocity
component and $dL=200$ pc is the height of a bin in Fig. 6, and the sum is
extended to all cells in a given $(r,z)$ bin.
We can now spatially compare the mass flux between the RC and SF galaxies. In
Fig. 6 we plot for the two W0 galaxies at 10 Myr from top to bottom:
difference between the $(r,z)$ maps of cell-mass weighted azimuthally-averaged
velocity $v_{z}$, vertical mass flux, and total cell mass in RCW0 and SFW0
galaxies, as a function of cylindrical radius $r$ and cylindrical $z$. The bin
size in each of the panels is $200\times 200\text{ pc}^{2}$. The goal of this
plot is to map which of the two galaxies, SFW0 or RCW0, has more mass flux in
the wind direction (central panel), and which of the mass flux components
contributes more to it. Each panel shows RC $-$ SF, so orange indicates higher
RC values, and purple indicates higher SF values. With the solid cyan line
($z=0$) we separate downwind and upwind halves of the disc, while the dashed
cyan line denotes the $z=2$ kpc height at which we define the galactic disc
boundaries. We also overplot a green contour that outlines the gas
distribution in RCW0 as it would appear if in this figure we plotted real RCW0
values222We note that these green contours are only denoting the edge of the
distribution of disc gas, defined as gas with a metallicity above 0.35. There
is gas in the cells beyond this contour, but it is ICM gas. instead of the
difference. This illustrates that RCW0 is more compact compared to SFW0 at the
same time step, since all gas outside the contour is only from the SFW0 disc.
As mentioned earlier, the SF-to-RC stripping rate ratio for W0 galaxies
changes periodically in the first 200 Myr. Here we look in detail at these
different periods and what causes the change, always referring to the top
panel of Fig. 2. At the beginning, up to 25 Myr, SFW0 is stripped faster,
illustrated in the middle panel of Fig. 6 with purple outflows beyond $r>10$
kpc. At $r<10$ kpc, however, the difference between the RC and SF mass fluxes
is quite small but tends towards higher mass flux in the RC simulation.
Interestingly, the inner region of the SF galaxy has stronger outflows from
feedback (Fig. 3), so those feedback-driven outflows do not seem to be
dominating the mass flux. Close to the disc plane we see enhanced mass flux in
the RC galaxy, which will result in larger RC stripping rates at later times.
Since stellar feedback expels gas in all directions, the gas disc of SFW0 is
much thicker and more radially extended (by $\sim 2$ kpc) than in RCW0, as
shown in Fig. 3 and in the bottom panel of Fig. 6. This added thickness does
not change the gas surface density, and therefore does not affect the
stripping rate. The feedback does not directly remove gas or assist RP in
doing so, as it would be with the gas getting an additional ‘kick’ to its
vertical velocity by galactic fountains, but, rather, enhanced stripping is a
result of star formation and stellar feedback homogenising the ISM (as
illustrated in Fig. 3 and 4), and radially extending the gas disc (as seen in
Fig. 5). We note that the homogeneity and the added radial extent persist in
the SF runs because star formation and feedback continually affect the ISM.
Although most of the gas is lost from the outskirts and the gas disc generally
shrinks as the RP continuously increases, these figures can also be used to
explain the difference in stripping rates in the inner disc. In the inner 10
kpc the SFW0 gas seems to be protected from RPS, as the mass-flux distribution
is dominated by the high-velocity gas from RCW0 (orange area in the top panel
of Fig. 6). The ICM wind can easily flow through the underdense regions
created by continuous cooling and gas fragmentation (Fig. 4 and 5). As it does
so, it rapidly accelerates the low density ISM, and it is this gas that can be
stripped even from the galaxy centre that we see in orange in the middle panel
of Fig. 6.
In addition to the enhanced mass flux from the SFW0 galaxy crossing $z=2$ kpc,
at 10 Myr we can see enhanced mass flux formed in the RCW0 galaxy in a narrow
strip directly above the $z=0$ line (Fig. 6, middle and bottom panels, in
orange). By 35 Myr, this stripped slice evolves into a whole arch as shown in
Fig. 7. This arch is the reason why the stripping rates are higher in RCW0
during 30–50 Myr and why at that time there is a peak in the top panel of Fig.
1. This peak is also present in SFW0 but is due to the gas removal at large
radii. The enhanced stripping of the RCW0 ring once again illustrates that, as
in the outskirts of SFW0, homogeneity of gas leads to more gas below the
stripping surface density (as there is an absence of dense clumps that are
present in this region in SFW0), which makes it easier to be stripped.
In the middle panel of Fig. 7 at $12\text{ kpc}<r<20\text{ kpc}$ we can see
SFW0 gas that begins to move (the dark purple area under the orange arch).
Later, during 55–85 Myr, as SFW0 is stripped outside-in, the same gas is
leaving the galaxy. Since by this time there is no gas left in RCW0 at these
radii (the gas arch seen here has passed beyond $z=2$ kpc), in this period the
stripping rate is stronger in the SF galaxy.
Figure 7: Same as Fig. 6 but for 35 Myr of RPS. At this time, the stripping
rate is dominated by gas removed from a more uniform, unfragmented low-density
region between $14\text{kpc}<r<20\text{kpc}$ in RCW0 galaxy.
The final change to the SF-to-RC stripping rate ratio happens at 90 Myr. With
a smooth ISM distribution, a galaxy subject to face-on RPS will be gradually
reduced in size up to its stripping radius. This largely occurs in SFW0, which
has a constant peak gas surface density at all radii as shown in Fig. 5.
Therefore, increasing RP smoothly strips it from the outside-in. In RCW0,
however, while the loosely-bound gas was easily stripped from the former ring
area, the dense clumps of gas in the outskirts at $r>20$ kpc remain until 90
Myr. We can see these dense clouds surviving at 35 Myr in the bottom panel of
Fig. 7 by the orange region of excess mass and in Fig. 5 as the high surface
density gas at $\sim$23 kpc. Hence, from 90 to 170 Myr while SFW0 has already
been reduced to 12 kpc radius, RCW0 once again dominates the stripping rate as
the remaining gas clumps in the outskirts are being quickly stripped in
addition to the gas at $\sim$12 kpc that is being stripped from both galaxies.
During 170–400 Myr we see similar stripping rates in the RC and SF galaxies.
By this point both the W0 galaxies have been reduced to the same stripping
radius (using the Gunn & Gott (1972) gas surface density definition), around
12 kpc, with the gas distribution in SFW0 still being 1–2 kpc more extended
than in RCW0. Because feedback continues to push some of the ISM to larger
radii, the SFW0 galaxy is stripped faster in the outskirts. On the other hand,
in the galaxy centre RCW0 is stripped faster due to the low surface density
regions in the gas distribution, making the overall picture of RPS similar to
what we saw during the first 30 Myr. This also agrees well with our intuition
derived from the gas surface density distributions seen in Fig. 5. Therefore,
feedback both enhances outer stripping by extending the ISM and depresses
inner stripping by homogenising the gas surface density. At this time in our
simulations these two effects balance each other out, and globally the
galaxies have almost identical stripping rates (Fig. 2, top panel).
From 400 Myr onward the SFW0 galaxy starts losing gas more and more slowly.
During this time its calculated stripping radius and measured gas disc radius
are both smaller than that of RCW0. As shown for earlier times by Fig. 6 and 7
the mass flux is determined mostly by the mass distribution: where there is
more mass, the mass flux is stronger, even if that gas is moving slower. By
400 Myr, SFW0 has lost gas not only to RPS, but also to star formation, which
is especially true in the inner 5 kpc, where most of the stars are formed.
This means that in the later part of the simulation there is simply very
little gas left in the centre of SFW0, and therefore RCW0 starts being
stripped faster.
As evident from the stripping rates in Fig. 1 and, consequently, from the mass
of the stripped gas in the bottom panel of Fig. 2, the W45 galaxies follow a
very similar evolution to the W0; something that was also shown by Roediger &
Brüggen (2006) and Akerman et al. (2023). It is not surprising then that the
SF-to-RC stripping rate ratio of W45 galaxies also follows the one for the W0
(Fig. 2, top panel). The main difference between the two ratios is due to the
time delay that, as explained above, stems from the fact that it takes an
inclined wind more time to reach a galaxy and begin the stripping process.
Another difference shown in the top panel of Fig. 2 is that during two
relatively short periods of time, 225–250 Myr and 305–375 Myr, SFW45 is
stripped faster than RCW45. This is another effect of the continuous collapse
of gas, as a result of which dense clumps form. These clumps are then pushed
in the shadow of the disc, where they are protected from the ICM wind. The
stripping rate in RCW45 drops and the SF-to-RC stripping rate ratio goes up.
In conclusion, the added inhomogeneities in RC galaxies result in more gas
both above and below the stripping surface density (Fig. 5). Dense gas is
harder to strip, as evidenced by the lower $v_{z}$ at high surface density. In
the galaxy outskirts, ISM inhomogeneities are able to enhance gas surface
density to the point where it cannot be stripped, slowing down the global
stripping rate. We see this in the delayed stripping of dense gas beyond 20
kpc in the RC galaxies relative to SF galaxies, as well as in the earlier
stripping of the homogeneous gas ring at 14–20 kpc in the RC galaxies. In the
galaxy centre, ISM inhomogeneities can result in regions with very low gas
surface density. These regions are stripped very quickly, which leads to
faster stripping rates in the central regions of RC galaxies.
Figure 8: For the W90 galaxies, $v_{y}$ mass flux of ISM ($Z>0.35Z_{\odot}$)
as a function of $z$ measured in $29.5\text{kpc}<r<30.5\text{kpc}$ at (from
left to right) 150, 400 and 550 Myr of RPS for SF (solid) and RC (dashed)
simulations. We select the data for the downwind part, $y>0$, and for the
stripped gas with $v_{y}>0$.
In this section we have focused on the early times when the RC and SF
stripping rates differ in order to understand the physical reason for the
distinct rates. However, generally, the lower stripping rates at the outskirts
of the RC galaxies balance the higher stripping rates in their centres,
leading to similar stripping rates in the SF and RC simulations.
### 4.4 Edge-on stripped galaxies
After having discussed the detailed evolution of the face-on stripped
galaxies, we examine the edge-on ones, RCW90 and SFW90. Since in this case the
feedback does not act parallel to the direction of the wind, its effect is
harder to predict. Still, we could expect RP from the edge-on ICM wind to
easily remove the gas expelled above (and below) the disc by the galaxy
fountains. Surprisingly, while this effect indeed takes place, it does not
dominate the stripping rates in SFW90.
To illustrate this, in Fig. 8 we show the $v_{y}$ mass flux (similar to eq. 3)
measured in $29.5\text{kpc}<r<30.5\text{kpc}$ as a function of $z$. We remind
the reader that in the edge-on case (W90) the wind moves along the
$y$-direction. Solid (dashed) lines plot SFW90 (RCW90) at three different
times (from left to right): 150 Myr, 400 Myr, and 550 Myr. We chose these time
steps to have three different SF-to-RC stripping rate ratios in the top panel
of Fig. 2, where at 150 Myr SFW90 is stripped faster, at 400 Myr RCW90
dominates and at 550 Myr the galaxies lose their gas at a similar rate. We
also select the data only for the downwind part, $y>0$, and for $v_{y}>0$
only. Thus, shown here is that while (for the most part) outside of the disc
plane there is more mass flux in SFW90 than in RCW90, the bulk of the
stripping at all times is in the galaxy plane.
Moreover, shown in the bottom panel of Fig. 1 is SFR of SFW90, enhanced
compared to SFNW on the whole duration of the simulation. This means that the
scenario outlined above of SN outflows being swept away by the ICM wind plays
out during all 700 Myr. Globally, evolution of the SF-to-RC stripping rate
ratio (Fig. 2, top panel) can be divided into two periods: from 120 to 400 Myr
the ratio goes down, indicating a slow but steady decrease in the relative
outflow rates of SFW90 compared to RCW90; from 400 to 700 Myr it goes up
again, meaning that now the SF galaxy gets stripped systematically faster.
This means that despite the ongoing stripping of the expelled feedback gas,
RCW90 manages to overcome the effect, and is even stripped faster than SFW90
from 330 to 530 Myr (as also evident in the middle panel of Fig. 8).
To understand the change in the stripping rate, we look at the gas
($Z>0.35Z_{\odot}$) distribution in the disc plane ($z\pm 2$ kpc), shown in
Fig. 9 by an $xy$ density projection for 250 Myr. Again, we calculate the data
only for the gas with $v_{y}>0$. This leads to a lack of gas at positive
$x$-values in the figure, where galaxy rotation leads to negative $v_{y}$. In
both panels, the dashed pink line denotes the outer boundary of $r=30$ kpc
with which we define the galactic disc. The yellow line outlines the gas
distribution in SFW90 to illustrate that it is more compact compared to RCW90
at the same time step. Also note that unlike in the W0 runs, where the gas
needs to travel up to $z=2$ kpc to be considered stripped, in W90 this path is
increased to $r=30$ kpc. Any change in the gas distribution in the galaxy disc
will affect the stripping rate only about 100 Myr later, since the gas needs
to travel 30 kpc on average to the downwind edge. This needs to be kept in
mind when analysing the plots, since differences in the immediate gas
distribution of RCW90 and SFW90 may not reflect the current SF-to-RC stripping
rate ratio.
Figure 9: Density (cell-volume weighted) distribution of gas ($z\pm 2$ kpc) in
RCW90 (left) and SFW90 (right) at 250 Myr. We calculate the data only for the
stripped gas with $v_{y}>0$. The dashed pink line denotes the upper boundary
of $r=30$ kpc with which we define the galactic disc. The yellow line outlines
the gas distribution in SFW90 to illustrate that it is more compact compared
to RCW90 at the same time step.
The gas distribution in RCW90 and SFW90 reveals that, just like in the face-on
stripped galaxies, the main factor contributing to the differences between SF
and RC galaxies is not direct removal of gas from the disc plane due to
feedback (such as galaxy fountains) but the homogenised gas distribution in
the disc. When hit by an edge-on ICM wind, the dense clouds are hard to
remove; at the same time, they shield the easily-stripped low-density gas.
Compared to this, the more homogeneous gas in SFW90 is being easily pushed as
evident in the top panel of Fig. 2. This is reflected in the more extended
distribution of gas in RCW90 compared to SFW90 – the higher density clouds in
RCW90 have been less affected by the ICM wind and therefore less gas has been
removed.
However, while the lower density ISM in SFW90 has been stripped more quickly
than the higher density clouds in RCW90, the ISM is not simply directly
stripped from the galaxy. As described above, the gas has a long way to travel
before being considered ‘lost’ by the galaxy (the longest path from the upwind
to the downwind side is $\sim 55$ kpc). On this course, the combination of
wind and galaxy rotation makes gas from different galactic radii collide with
each other to create a dense ‘arch’ (right panel of Fig. 9, at the leading
edge of the disc in the bottom left quadrant). At this point, the SFW90 galaxy
is also substantially smaller than RCW90, as outlined by the yellow line. As
the arch forms, its dense gas becomes harder and harder to strip, at which
point the SF-to-RC stripping rate ratio crosses the equality line. In the
bottom panel of Fig. 1 the decreasing $\dot{M}$ after 330 Myr indicates that
SFW90 runs out of easily stripped gas at this point. The remaining gas from
the arch is pushed into a big dense knot on the wind-facing side of the disc
about 10 kpc from the galaxy centre. At this point the stripping rate in SFW90
drops, while the RC galaxy maintains its stripping rate by removing the low-
density gas.
The steady pushing of gas to the galaxy centre also proceeds in RCW90, but
more slowly since the dense clumps can only be pushed by higher RP (which
occurs later in the simulations). By 400 Myr RCW90 repeats the fate of SFW90:
gas forms an even denser (compared to the SF simulation) knot in the galaxy
centre, and after that the stripping rate starts to drop (Fig. 1, bottom
panel).
In the next $\sim 150$ Myr, SFW90 slowly overtakes RCW90 in the stripping rate
(Fig. 2, top panel). The dense central knot will inevitably be destroyed by
the increasing RP (as it starts to at $\sim$550 Myr), and its gas stripped.
From $\sim 550$ Myr onward the two galaxies are stripped at about the same
rate.
In conclusion, the main differences between the stripping process in the edge-
on stripped galaxies, just like in the face-on ones, stem from the fact that
in the SF galaxies the gas is homogenised and thus more easy to push at first,
while in the RC galaxies it is patchy with dense clumps that are comparatively
harder to move. This however does not simply mean that SFW90 is always
stripped faster, because the edge-on RP can compress gas to higher densities
as well as directly remove gas, and once the high density region in the SF
galaxy matches that in the RC galaxy, stripping occurs at the same rate.
## 5 Discussion
Our simulations show that the stripping rates of star-forming and radiative-
cooling-only galaxies do not differ significantly, and that on galactic scales
the stellar feedback does not directly assist or impede RPS by driving
significant gas mass outside the disc plane (Fig. 1). While there are not many
papers that study the effect of stellar feedback on the process of stripping,
the ones that do, Bahé & McCarthy (2015) and Kulier et al. (2023), show
results different from ours. Both of these works explore cosmological smoothed
particle hydrodynamic simulations, the former with Galaxies-Intergalactic
Medium Interaction Calculations (GIMIC) and the latter with EAGLE data.
Bahé & McCarthy (2015) reveal two things. Firstly, the fraction of gas that
can be stripped from a galaxy declines with galaxy mass, while the fraction
that can be removed (but still remain bound to the galaxy halo) via galaxy
fountains, on the contrary, increases. If we consider their galaxies that are
the most similar to ours, with stellar masses $M_{\text{star}}\approx
10^{10.75}M_{\odot}$ in massive clusters with
$M_{\text{cluster}}=10^{15.2}M_{\odot}$, then it appears that RPS accounts for
60 per cent of all removed gas, while the rest is due to galactic fountains.
Secondly, they also show that the stellar feedback implemented in GIMIC (and
EAGLE) gave a prior ‘kick’ to a significant part of the stripped gas that
later allowed RPS to quickly remove it. Still, for the most massive galaxies
in the most massive clusters the amount of stripped gas when accounting for
the ‘kick’ is within the boundaries predicted by Gunn & Gott (1972) model.
Kulier et al. (2023) come to a similar conclusion, where they measure the
‘excess’ gas that is stripped due to the feedback and reveal that the amount
increases with RP. This indicates a synergy between RP and stellar feedback in
removing galactic gas.
These findings are, perhaps, similar to our own, but rather than the direct
effect of SN feedback, SFW0 is stripped faster at the galaxy outskirts due to
the feedback homogenising and radially extending gas in the galaxy disc, thus
lowering its maximum surface density. However, we also show that in a direct
comparison with a non-star-forming galaxy this effect proves to be
insignificant. It is worth emphasising again that both Bahé & McCarthy (2015)
and Kulier et al. (2023) use cosmological smoothed-particle hydrodynamics
simulations. By their nature such simulations cannot have high resolution and
sophisticated feedback physics of galaxy-scale adaptive-mesh-refinement
simulations. Due to these necessary technical limitations the effect of
stellar feedback might be overestimated. For example, in EAGLE the feedback is
ejective, with galactic fountains launching gas to large distances (Mitchell
et al., 2020); it is also known to create excessively large ‘holes’ in the ISM
(Bahé et al., 2016). In this light, the role of the homogenised ISM density
distribution in RPS is likely underestimated in these simulations.
Bahé & McCarthy (2015) may give us insight into the possible galaxy mass
dependence of our result. They find that while the highest-mass galaxies seem
to be the least affected by the stellar feedback when it comes to stripping,
the lower-mass galaxies $M_{\text{star}}<10^{10}M_{\odot}$ lose much more gas
than estimated by Gunn & Gott (1972). Muratov et al. (2015) in a cosmological
zoom-in simulation FIRE measured the mass loading factor (mass carried out by
stellar feedback per mass of newly formed stars) and found a decreasing trend
with galaxy stellar mass, although their galaxies do not reach the stellar
mass of our simulated galaxy. This means that low-mass galaxies lose more gas
via galaxy fountains, in a seeming agreement with Bahé & McCarthy (2015). In
this case, it is possible that stellar feedback would assist the gas removal
via RPS of low-mass galaxies. We stress that our specific result that SN
feedback has little effect on the stripping rate of galaxies may be quite mass
dependent. However, our result that the homogeneity of the ISM will have
important consequences on the radial distribution of stripping is general.
Another important note about this work is that we did not test a range of
feedback models, in which we would expect feedback implementations that
strongly eject mass to large distances from the disc to have a bigger effect
on the stripping rate of galaxies. However, we can compare our results with
other works studying SN feedback. Kim et al. (2020) conducted local
simulations of several regions in a Milky-Way-like galaxy with a resolution of
2 to 8 pc. Similar to Li & Bryan (2020) they find that the hot ($T>10^{6}$ K)
galactic outflows are much more energetic than the cool ($T<10^{4}$ K) ones,
which, conversely, are responsible for driving the mass out of a galaxy. We
can compare mass flux from different regions in SFNW that correspond to the
regions in Kim et al. (2020). We find that at the radius where our
$\Sigma_{\text{gas}}$ and $\Sigma_{\text{star}}$ are similar to those in Kim
et al. (2020) (R8), our mass fluxes are also comparable to theirs. This
therefore gives us confidence that the scale of our mass fluxes is reasonable,
and our galaxy should not have dramatically different mass fluxes that would
change our main results.
We also reiterate that these are purely hydrodynamical simulations, so we do
not include magnetic fields or cosmic rays. Magnetic fields have been shown in
simulations to broaden the density PDF (Burkhart et al., 2015), especially for
low-Mach number turbulence (Burkhart et al., 2009), and may decrease the SFR
(Federrath & Klessen, 2012; Federrath, 2015; Wibking & Krumholz, 2023). There
is also some observational evidence that enhanced magnetic fields may reduce
gas fragmentation (e.g. Añez-López et al., 2020). Simulations also indicate
that cosmic rays smooth the distribution of gas density in the ISM and the
circumgalactic medium (Butsky & Quinn, 2018; Butsky et al., 2020; Farber et
al., 2022). The impact of these sources of non-thermal pressure on RPS could
be an interesting avenue for future work.
## 6 Conclusions
In this paper we present a suite of wind-tunnel galaxy-scale simulations to
determine the role of star formation and stellar feedback in stripping a
galaxy of its gas via RP. The initial conditions are the same in each
simulation, and model a galaxy slightly more massive than the Milky Way
($M_{\text{star}}=10^{11}M_{\odot}$) falling into a massive cluster
($M_{\text{cluster}}=10^{15}M_{\odot}$). We have two sets of simulations:
radiative-cooling-only (denoted as RC) and with star formation and stellar
feedback (SF). In each of the two sets there are four galaxies, of which one
is not undergoing RPS (no wind, NW). The other three galaxies are on their
first infall into a cluster and have different galaxy-wind impact angle: face-
on (W0), edge-on (W90) and $45^{\circ}$ (W45). We compare the stripping
histories of a RC and SF galaxy at the same impact angle and find that:
1. 1.
Star formation and stellar feedback vertically puff up the galaxy disc,
expelling galactic fountains up to 15 kpc, and homogenise the ISM density
distribution. In RC galaxies, the gas continuously cools down and fragments
into dense clumps (Fig. 3 and 4), resulting in a broader distribution of gas
densities in the ISM. In addition, since the fountains eject gas in all
directions, including radially, at 0 Myr (the time when the stripping begins)
the radius of a SF galaxy is $\sim 2$ kpc bigger than that of a RC one.
2. 2.
Globally, the stripping rates are quite similar for RC and SF galaxies of the
same impact angle (Fig. 1), although since part of the gas is used up for star
formation, the SF galaxies lose less total gas through RPS (Fig. 2, bottom
panel). Comparing how the stripping rates change relative to each other (Fig.
2, top panel) reveals that the relative stripping rates of the SF and RC
galaxies invert multiple times over the course of the simulations.
3. 3.
The differences in the stripping rates of the RC and SF galaxies are related
to the differences in the density distributions in the galaxy discs. This is
because a large distribution in gas density results in a large distribution in
gas surface density (Fig. 5). While the average surface density is not
affected, more inhomogeniety in the ISM of RC galaxies results in more gas
both above and below the stripping surface density at which the restoring
force equals the RP. Gas that falls below the stripping surface density may be
more easily removed, while the gas above it may survive unstripped.
4. 4.
In the outskirts of W0 and W45 where the average surface density is below the
stripping surface density, inhomogeneity makes stripping slower by allowing
dense clouds to survive (Fig. 5 and 6).
5. 5.
At inner radii where the average surface density is higher than the stripping
surface density, inhomogeneity leads to more gas at lower surface densities,
allowing more stripping and faster acceleration of low density gas. We see
this in the enhanced stripping rates in RC in the central regions (Fig. 5, 6
and 7).
6. 6.
In the edge-on stripped SFW90, low-density galaxy fountain gas is stripped
faster by the ICM wind (Fig. 8). However, this has a very small effect on the
overall stripping rate, since the bulk of the gas is stripped in the galaxy
plane. Instead, we find that the homogenised gas in SFW90 is easier to push at
the beginning of RPS, thus allowing for the galaxy to be stripped faster (Fig.
9). Once this lower density gas has been removed or shifted towards the centre
of the galaxy (550 Myr), both RCW90 and SFW90 have almost identical stripping
rates.
We find that regardless of the wind impact angle stellar feedback has no
direct effect on the stripping process, such as giving an additional velocity
kick to the stripped gas. Instead, the indirect effect of homogenising the gas
is what makes the difference between star-forming and radiative-cooling only
galaxies.
An important insight we have gained from this comparison is that correctly
modelling the small-scale distribution of gas in a disc undergoing RPS is
important for understanding how quickly it will be stripped. We argue that
cosmological simulations, which may find that feedback can enhance gas removal
by RPS due to the strong ejection of gas particles to large distances from the
galaxy plane (e.g. Bahé & McCarthy, 2015; Kulier et al., 2023), cannot be used
for detailed comparisons with RPS galaxy populations because they do not
resolve the density range or distribution in the discs of satellite galaxies.
We find that a more smooth gas distribution will result in more (compared to
clumpy and underdense gas in RC simulation) stripping in regions with low
average restoring force versus RP, and less stripping in regions with high
average restoring force versus RP.
## Acknowledgements
This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(grant agreement No. 833824). The simulations were performed on the Frontera
supercomputer operated by the Texas Advanced Computing Center (TACC) with LRAC
allocation AST20007 and on the Stampede2 supercomputer operated by the Texas
Advanced Computing Center (TACC) through allocation number TG-MCA06N030. B. V.
acknowledges the Italian PRIN-Miur 2017 (PI A. Cimatti). We use the yt python
package (Turk et al., 2011) for data analysis and visualisation.
## Data Availability
The data underlying this article will be shared on reasonable request to the
corresponding author.
## References
* Añez-López et al. (2020) Añez-López N., et al., 2020, A&A, 644, A52
* Akerman et al. (2023) Akerman N., Tonnesen S., Poggianti B. M., Smith R., Marasco A., 2023, ApJ, 948, 18
* Bahé & McCarthy (2015) Bahé Y. M., McCarthy I. G., 2015, MNRAS, 447, 969
* Bahé et al. (2016) Bahé Y. M., et al., 2016, MNRAS, 456, 1115
* Behrendt et al. (2015) Behrendt M., Burkert A., Schartmann M., 2015, MNRAS, 448, 1007
* Bekki (2014) Bekki K., 2014, MNRAS, 438, 444
* Bellhouse et al. (2017) Bellhouse C., et al., 2017, ApJ, 844, 49
* Bellhouse et al. (2019) Bellhouse C., et al., 2019, MNRAS, 485, 1157
* Boselli et al. (2014) Boselli A., Cortese L., Boquien M., Boissier S., Catinella B., Gavazzi G., Lagos C., Saintonge A., 2014, A&A, 564, A67
* Bryan et al. (2014) Bryan G. L., et al., 2014, ApJS, 211, 19
* Burkert (1995) Burkert A., 1995, ApJ, 447, L25
* Burkhart et al. (2009) Burkhart B., Falceta-Gonçalves D., Kowal G., Lazarian A., 2009, ApJ, 693, 250
* Burkhart et al. (2015) Burkhart B., Collins D. C., Lazarian A., 2015, ApJ, 808, 48
* Butsky & Quinn (2018) Butsky I. S., Quinn T. R., 2018, ApJ, 868, 108
* Butsky et al. (2020) Butsky I. S., Fielding D. B., Hayward C. C., Hummels C. B., Quinn T. R., Werk J. K., 2020, ApJ, 903, 77
* Cayatte et al. (1990) Cayatte V., van Gorkom J. H., Balkowski C., Kotanyi C., 1990, AJ, 100, 604
* Chung et al. (2007) Chung A., van Gorkom J. H., Kenney J. D. P., Vollmer B., 2007, ApJ, 659, L115
* Cortese et al. (2012) Cortese L., et al., 2012, A&A, 544, A101
* Cramer et al. (2019) Cramer W. J., Kenney J. D. P., Sun M., Crowl H., Yagi M., Jáchym P., Roediger E., Waldron W., 2019, ApJ, 870, 63
* Crowl & Kenney (2008) Crowl H. H., Kenney J. D. P., 2008, AJ, 136, 1623
* Di Cintio et al. (2014) Di Cintio A., Brook C. B., Dutton A. A., Macciò A. V., Stinson G. S., Knebe A., 2014, MNRAS, 441, 2986
* Dursi & Pfrommer (2008) Dursi L. J., Pfrommer C., 2008, ApJ, 677, 993
* Farber et al. (2022) Farber R. J., Ruszkowski M., Tonnesen S., Holguin F., 2022, MNRAS, 512, 5927
* Federrath (2015) Federrath C., 2015, MNRAS, 450, 4035
* Federrath & Klessen (2012) Federrath C., Klessen R. S., 2012, ApJ, 761, 156
* Fossati et al. (2016) Fossati M., Fumagalli M., Boselli A., Gavazzi G., Sun M., Wilman D. J., 2016, MNRAS, 455, 2028
* Fossati et al. (2018) Fossati M., et al., 2018, A&A, 614, A57
* Fritz et al. (2017) Fritz J., et al., 2017, ApJ, 848, 132
* Fumagalli et al. (2014) Fumagalli M., Fossati M., Hau G. K. T., Gavazzi G., Bower R., Sun M., Boselli A., 2014, MNRAS, 445, 4335
* Gavazzi et al. (2001) Gavazzi G., Boselli A., Mayer L., Iglesias-Paramo J., Vílchez J. M., Carrasco L., 2001, ApJ, 563, L23
* George et al. (2018) George K., et al., 2018, MNRAS, 479, 4126
* Goldbaum et al. (2015) Goldbaum N. J., Krumholz M. R., Forbes J. C., 2015, ApJ, 814, 131
* Goldbaum et al. (2016) Goldbaum N. J., Krumholz M. R., Forbes J. C., 2016, ApJ, 827, 28
* Göller et al. (2023) Göller J., Joshi G., Rohr E., Zinger E., Pillepich A., 2023, arXiv e-prints, p. arXiv:2304.09199
* Gullieuszik et al. (2017) Gullieuszik M., et al., 2017, ApJ, 846, 27
* Gunn & Gott (1972) Gunn J. E., Gott J. R. I., 1972, ApJ, 176, 1
* Haardt & Madau (2012) Haardt F., Madau P., 2012, ApJ, 746, 125
* Jáchym et al. (2007) Jáchym P., Palouš J., Köppen J., Combes F., 2007, A&A, 472, 5
* Jáchym et al. (2017) Jáchym P., et al., 2017, ApJ, 839, 114
* Jáchym et al. (2019) Jáchym P., et al., 2019, ApJ, 883, 145
* Kenney et al. (2004) Kenney J. D. P., van Gorkom J. H., Vollmer B., 2004, AJ, 127, 3361
* Kenney et al. (2014) Kenney J. D. P., Geha M., Jáchym P., Crowl H. H., Dague W., Chung A., van Gorkom J., Vollmer B., 2014, ApJ, 780, 119
* Kenney et al. (2015) Kenney J. D. P., Abramson A., Bravo-Alfaro H., 2015, AJ, 150, 59
* Kim et al. (2020) Kim C.-G., et al., 2020, ApJ, 900, 61
* Koopmann & Kenney (2004) Koopmann R. A., Kenney J. D. P., 2004, ApJ, 613, 851
* Kulier et al. (2023) Kulier A., et al., 2023, arXiv e-prints, p. arXiv:2305.03758
* Lee & Chung (2018) Lee B., Chung A., 2018, ApJ, 866, L10
* Lee et al. (2017) Lee B., et al., 2017, MNRAS, 466, 1382
* Lee et al. (2020) Lee J., Kimm T., Katz H., Rosdahl J., Devriendt J., Slyz A., 2020, ApJ, 905, 31
* Li & Bryan (2020) Li M., Bryan G. L., 2020, ApJ, 890, L30
* Li et al. (2020) Li P., Lelli F., McGaugh S., Schombert J., 2020, ApJS, 247, 31
* McCarthy et al. (2008) McCarthy I. G., Frenk C. S., Font A. S., Lacey C. G., Bower R. G., Mitchell N. L., Balogh M. L., Theuns T., 2008, MNRAS, 383, 593
* Merluzzi et al. (2016) Merluzzi P., Busarello G., Dopita M. A., Haines C. P., Steinhauser D., Bourdin H., Mazzotta P., 2016, MNRAS, 460, 3345
* Mitchell et al. (2020) Mitchell P. D., Schaye J., Bower R. G., Crain R. A., 2020, MNRAS, 494, 3971
* Miyamoto & Nagai (1975) Miyamoto M., Nagai R., 1975, PASJ, 27, 533
* Moretti et al. (2018) Moretti A., et al., 2018, MNRAS, 480, 2508
* Moretti et al. (2020a) Moretti A., et al., 2020a, ApJ, 889, 9
* Moretti et al. (2020b) Moretti A., et al., 2020b, ApJ, 897, L30
* Mori & Burkert (2000) Mori M., Burkert A., 2000, ApJ, 538, 559
* Muratov et al. (2015) Muratov A. L., Kereš D., Faucher-Giguère C.-A., Hopkins P. F., Quataert E., Murray N., 2015, MNRAS, 454, 2691
* Poggianti et al. (2017) Poggianti B. M., et al., 2017, ApJ, 844, 48
* Ramos-Martínez et al. (2018) Ramos-Martínez M., Gómez G. C., Pérez-Villegas Á., 2018, MNRAS, 476, 3781
* Roberts & Parker (2020) Roberts I. D., Parker L. C., 2020, MNRAS, 495, 554
* Roberts et al. (2021) Roberts I. D., et al., 2021, A&A, 650, A111
* Roberts et al. (2022a) Roberts I. D., van Weeren R. J., Timmerman R., Botteon A., Gendron-Marsolais M., Ignesti A., Rottgering H. J. A., 2022a, A&A, 658, A44
* Roberts et al. (2022b) Roberts I. D., et al., 2022b, ApJ, 941, 77
* Rodrigues et al. (2017) Rodrigues D. C., del Popolo A., Marra V., de Oliveira P. L. C., 2017, MNRAS, 470, 2410
* Roediger & Brüggen (2006) Roediger E., Brüggen M., 2006, MNRAS, 369, 567
* Roediger & Brüggen (2007) Roediger E., Brüggen M., 2007, MNRAS, 380, 1399
* Roediger et al. (2014) Roediger E., Bruggen M., Owers M. S., Ebeling H., Sun M., 2014, MNRAS, 443, L114
* Ruggiero & Lima Neto (2017) Ruggiero R., Lima Neto G. B., 2017, MNRAS, 468, 4107
* Ruszkowski et al. (2014) Ruszkowski M., Brüggen M., Lee D., Shin M. S., 2014, ApJ, 784, 75
* Schaye et al. (2015) Schaye J., et al., 2015, MNRAS, 446, 521
* Schulz & Struck (2001) Schulz S., Struck C., 2001, MNRAS, 328, 185
* Smith et al. (2017) Smith B. D., et al., 2017, MNRAS, 466, 2217
* Steinhauser et al. (2016) Steinhauser D., Schindler S., Springel V., 2016, A&A, 591, A51
* Sun et al. (2010) Sun M., Donahue M., Roediger E., Nulsen P. E. J., Voit G. M., Sarazin C., Forman W., Jones C., 2010, ApJ, 708, 946
* Sun et al. (2021) Sun M., et al., 2021, NatAs, 6, 270
* Tasker & Bryan (2006) Tasker E. J., Bryan G. L., 2006, ApJ, 641, 878
* Tonnesen (2019) Tonnesen S., 2019, ApJ, 874, 161
* Tonnesen & Bryan (2009) Tonnesen S., Bryan G. L., 2009, ApJ, 694, 789
* Tonnesen & Bryan (2012) Tonnesen S., Bryan G. L., 2012, MNRAS, 422, 1609
* Tonnesen & Stone (2014) Tonnesen S., Stone J., 2014, ApJ, 795, 148
* Turk et al. (2011) Turk M. J., Smith B. D., Oishi J. S., Skory S., Skillman S. W., Abel T., Norman M. L., 2011, ApJS, 192, 9
* Vulcani et al. (2018) Vulcani B., et al., 2018, ApJ, 866, L25
* Vulcani et al. (2020) Vulcani B., et al., 2020, ApJ, 892, 146
* Wibking & Krumholz (2023) Wibking B. D., Krumholz M. R., 2023, MNRAS, 521, 5972
## Appendix A Refinement criteria and resolution test
Our simulation box has 160 kpc on a side, divided into $128\times 128\times
128$ cells. We use the adaptive approach to include additional 5 levels of
refinement, with each level having twice the resolution of the previous one,
and get a maximum resolution of 39 pc. Our refinement criteria are Jeans
length and baryon mass of $\approx 7500M_{\odot}$. In Fig. 10 we show how
using these criteria we are able to refine the galaxy discs ($R=30$ kpc and
$z\pm 2$ kpc) in RCNW (left) and SFNW (centre) and low-resolution SFNW (right,
see the text below). The cell sizes are colour-coded by cell mass. We select
the data at 10 Myr before RPS begins. Firstly, this plot shows that the chosen
refinement criteria are able to highly resolve the disc, since in all of the
galaxies the bulk of the mass is concentrated at the finest level. Secondly,
compared to RCNW, SFNW has more mass on levels 4 and 5 (156 and 78 pc,
respectively) since in this case the ISM disc within the $z\pm 2$ kpc
boundaries is much thicker (Fig. 3).
Figure 10: Cell sizes $dx$ (we have cubical cells, $dx=dy=dz$) as a function
of cylindrical radius $r$ colour-coded by cell mass in a galaxy disc ($R=30$
kpc and $z\pm 2$ kpc) in RCNW (left) and SFNW (centre) and low-resolution SFNW
(right) at 10 Myr before RPS begins.
Our galaxy, which is the best-resolved region in the simulation box, has a
maximum of $3.33\times 10^{7}$ of cells (of any resolution) in the SFNW run,
while the median number of cells over the 1 Gyr of total simulation time is
$2.86\times 10^{7}$. In the SF wind runs, during the first 100 Myr after the
ICM has just hit the galaxies, the peak cell number value for different wind
angles is $[3.36,3.39]\times 10^{7}$. After that, the number of cells only
continues to decrease with time, as the gas is being stripped and less and
less regions are left to be resolved. In RCNW, as the lack of stellar feedback
allows for the gas to collapse into a thin disc, the number of cells in the
galaxy disc is somewhat smaller with the maximum and median values being
$2.17\times 10^{7}$ and $1.76\times 10^{7}$, respectively.
Here we discuss how resolution may impact our results. We run a short
simulation that includes only 4 levels of refinement, with the finest cell
being 78 pc. In the central and right panels of Fig. 10 we can compare the
resolution distribution in the fiducial and low-resolution runs, respectively.
The bulk of the mass is located at the highest-resolution level, while the
mass distribution on levels 1 to 4 (1248 to 156 pc) is similar in both
simulations.
The main result of this paper, pictured in Fig. 5, is the role of the surface
density distribution in gas stripping. The resolution used, therefore, must
capture both the low- and high-density gas. In Fig. 11 we repeat Fig. 5 for
the fiducial SFNW (top) and low-resolution (bottom) runs 10 Myr before RPS
begins. Note that the colour bar range is different in this figure from Fig.
5, since before the RP, there is no high-velocity gas in the disc (in other
words, we are missing gas that is being rapidly stripped). Overall, the
surface density distribution in both galaxies looks quite similar, with some
extension to lower surface densities in the SFNW low-res run. Meanwhile, the
average surface densities are almost identical, as illustrated in the bottom
panel with the white dashed line that overplots the average in the fiducial
SFNW over the average in the low-resolution run. The similar average values
mean that at any radius the bulk of the mass is distributed at comparable
surface densities. We also note that this means that the calculated truncation
radius (for which the average surface density is used) is resolution-
independent.
Figure 11: Same as Fig. 5, but for the fiducial SFNW (top) and low-resolution
(bottom) runs 10 Myr before RPS begins. Note that the colour bar range is
different in this figure, since before RP there is no high-velocity gas in the
disc. In the bottom panel, the white dashed line overplots average surface
density in the fiducial SFNW over the average in the low-resolution run to
illustrate that they are almost identical.
## Appendix B The effect of star formation on density distribution
In this work we compared radiative-cooling-only simulations to the ones that
include star formation and stellar feedback. Here, we will consider the effect
that star formation alone has on the gas distribution in the galaxy. This will
help us separate the role of star formation from that of stellar feedback in
shaping the ISM.
We will start by outlining the findings of previous works that used, like us,
the Enzo code and considered the impact of star formation and feedback on the
ISM. For example, Tasker & Bryan (2006) run a suite of hydrodynamical
simulations of a Milky-Way-like galaxy. They study the effect that star
formation threshold density and star formation efficiency play in shaping the
Kennicutt-Schmidt law, and run radiative-cooling-only, star-formation-only and
simulations that include both star formation and stellar-feedback. Comparison
of the density-temperature gas distribution between their star-formation-only
and feedback-including simulations (their Figs 11 and 12) reveals that the
low-density-high-temperature ISM is produced (for the most part) and is most
strongly affected by the stellar feedback. This is evident from both the mass
and the volume distributions. Conversely, the high-density-low-temperature ISM
(‘molecular’ or star-forming gas) is affected mostly by star formation,
although some density redistribution by stellar feedback is also evident.
Goldbaum et al. (2016) also makes a comparison between star-formation-only and
feedback-including simulations. Although this is not the focus of their work,
they also show that it is the feedback that is responsible for smoothing the
density distribution of the multi-phase ISM by preventing much of the mass
from being bound into dense clouds. Therefore, star formation alone should not
cause the lack of the lowest surface density gas found when comparing SFW0 to
RCW0 in Fig. 5.
To confirm these previous works, we run a short simulation similar to SFW0 but
without stellar feedback. In order to save computational resources, we restart
RCW0 40 Myr prior to RPS (160 Myr from the beginning of the simulation) and
turn on only the star formation with the same parameters as in our fiducial
model. In Fig. 12 we repeat the Fig. 5 but for RCW0 (top) SFW0-no-stellar-
feedback (middle) and SFW0 (bottom) galaxies (50 Myr later, 10 Myr after the
ICM wind has hit the galaxy). As expected from previous studies, the star
formation alone is not enough to affect the ISM, especially its low-density
phase, and, in fact, the only noticeable difference from RCW0 is in slightly
lower density peaks as the dense gas can now form stars. The run with star
formation and feedback (bottom panel), however, has the lowest density peaks
and, particularly in the central regions, has a narrower range in gas surface
density. We conclude that it is the stellar feedback that creates the more
homogeneous multi-phase ISM in SFW0.
Figure 12: Same as Fig. 5, but for RCWO (top) SFW0-no-stellar-feedback
(middle) and SFW0 (bottom) runs at 10 Myr.
Another factor that could potentially influence our results is the value of
the star formation threshold density. As we explained in Section 2, according
to the chosen star formation recipe, if the mass of a cell exceeds the Jeans
mass and the minimum threshold number density, a stellar particle will form
with a minimum mass of $10^{3}M_{\odot}$, with a chosen star formation
efficiency of 1 per cent. In the fiducial SFNW (in this Appendix referred to
as SF10) we choose a threshold number density of
$n_{\rm{thresh}}=10\,\rm{cm}^{-3}$. We also simulate galaxies from the same
initial conditions with $n_{\rm{thresh}}=3\,\rm{cm}^{-3}$ (SF03) and
$30\,\rm{cm}^{-3}$ (SF30) for 300 Myr (the time at which the ICM wind hits the
galaxies in the fiducial star-forming runs).
We find that the differences in the ISM brought by the change of threshold
number density are overall minor. We illustrate this by once again repeating
Fig. 5. In Fig. 13 we plot surface density distribution, this time for the
SF03 (top), SF10 (middle) and SF30 (bottom) runs. The central regions, where
the bulk of star formation occurs, are quite similar in SFR and gas surface
density across all the density thresholds. The gas surface density average and
distributions are very similar in all cases until at the very edge of the SF03
disc where there is little ISM gas as it has been used in star formation. The
differences in the ISM brought by the change of threshold number density are
overall unsubstantial. We further prove this by plotting in Fig. 14 the volume
density histograms of all the cells in the ISM of the galaxy disc ($R=30$ kpc
and $z\pm 2$ kpc) for SF03 (blue), SF10 (yellow) and SF30 (red) runs, which
are very close to each other. Thus, we conclude that the choice of the number
density threshold for star formation would not have a significant influence
over our results.
Figure 13: Same as Fig. 5, but for the SF03
($n_{\rm{thresh}}=3\,\rm{cm}^{-3}$, top), SF10 (or SFNW,
$n_{\rm{thresh}}=10\,\rm{cm}^{-3}$, middle) and SF30
($n_{\rm{thresh}}=30\,\rm{cm}^{-3}$, bottom) runs at 300 Myr of total
evolution, at which point the wind would hit the galaxies in the fiducial
runs. Note that the colour bar range is different in this figure, since
without RP there is no high-velocity gas in the disc. Figure 14: For SF03
($n_{\rm{thresh}}=3\,\rm{cm}^{-3}$, blue), SF10 (or SFNW,
$n_{\rm{thresh}}=10\,\rm{cm}^{-3}$, yellow) and SF30
($n_{\rm{thresh}}=30\,\rm{cm}^{-3}$, red) runs, the volume density histogram
of gas ($Z>0.35Z_{\odot}$) in a galaxy disc ($R=30$ kpc and $z\pm 2$ kpc)
taken at 300 Myr of total evolution.
## Appendix C Gas fallback in face-on stripped galaxies
Here for completeness we show the gas fallback in the face-on stripped
galaxies. We repeat the plots from Section 4.3 but for $v_{z}<0$ so that Fig.
15 is for 10 Myr (same as Fig. 6) and Fig. 16 is for 35 Myr (same as Fig. 7).
To avoid confusion, we show the difference of absolute values of velocity (top
panels) and mass flux (middle panels), so that, like in the previous plots,
orange colour means faster gas, more mass flux in RCW0 and purple colour means
that SFW0 dominates.
In Akerman et al. (2023) we show that fallback is negligible compared to the
radial flows withing the galaxy when it is subject to a varying, ever-
increasing ICM wind. It is no surprise, then, to see that on the maps at both
10 and 35 Myr the mass flux is almost 0 (notice that the colour bar scale is
the same as in Fig. with $v_{z}>0$). Since in green we outline the gas
distribution in RCW0, we can easily compare it to Fig. 6 and 7 and find that
the gas with negative $v_{z}$ is several kpc less radially extended than the
gas with positive $v_{z}$. This means that there is no fallback in the
outskirts or in the regions with the most violent stripping.
We can only see some fallback in the inner 10 kpc where it is dominated by
SFW0. As explained in Section 4.3, in SFW0 this part of the galaxy is
protected from RPS, since it is located within the stripping radius and the
outflows driven by feedback are able to fall back to the galaxy in fountain
flows. On the other hand, through the big low-density holes in the RCW0 gas
distribution the ICM wind can move freely, dragging some of the ICM with it.
As we can see here, this gas is truly stripped and does not fall back on the
galaxy.
Figure 15: Same as Fig. 6 but for $v_{z}<0.$ Figure 16: Same as Fig. 7 but for
$v_{z}<0.$
|
\\[1.3cm]
\quad
\quad
\quad
\quad
\quad
\quad
\\[0.7cm]
\hline
\quad
\tikz{0.6}{
\draw[lgray,line width=1.5pt,->] (-1,0) -- (1,0);
\draw[lgray,line width=1.5pt,->] (0,-1) -- (0,1);
\node[left] at (-1,0) {\tiny $1$};\node[right] at (1,0) {\tiny $1$};
\node[below] at (0,-1) {\tiny $1$};\node[above] at (0,1) {\tiny $1$};
\quad
\quad
\tikz{0.6}{
\draw[lgray,line width=1.5pt,->] (-1,0) -- (1,0);
\draw[lgray,line width=1.5pt,->] (0,-1) -- (0,1);
\node[left] at (-1,0) {\tiny $1$};\node[right] at (1,0) {\tiny $1$};
\node[below] at (0,-1) {\tiny $0$};\node[above] at (0,1) {\tiny $0$};
\quad
\quad
\tikz{0.6}{
\draw[lgray,line width=1.5pt,->] (-1,0) -- (1,0);
\draw[lgray,line width=1.5pt,->] (0,-1) -- (0,1);
\node[left] at (-1,0) {\tiny $1$};\node[right] at (1,0) {\tiny $0$};
\node[below] at (0,-1) {\tiny $0$};\node[above] at (0,1) {\tiny $1$};
\quad
\\[1.3cm]
\quad
\quad
\quad
\quad
\quad
\quad
\\[0.7cm]
\hline
\end{tabular}
\end{align}
Observe that these weights are stochastic, that is,
∑_i_2, j_2 ℛ_y / x (i_1, j_1; i_2, j_2) = 1.
§.§ Partition Function $Z(x_1,\dots,x_N;y_1,\dots,y_N;k)$
Fix two alphabets $(x_1,\dots,x_N) \in \mathbb{C}^N$ and $(y_1,\dots,y_N) \in \mathbb{C}^N$, and an integer $k \in \mathbb{Z}_{\geq 0}$. Define the following partition function in the quadrant:
\begin{align}
\label{hybrid-pf}
\tikz{0.75}{
\foreach\y in {1,...,5}{
\draw[lgray,line width=1.5pt,->] (0,\y) -- (7,\y);
\foreach\x in {1,...,5}{
\draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,7);
\draw[lgray,line width=4pt,->] (6,0) -- (6,6) -- (0,6);
%top labels
\node[above] at (5,7) {$1$};
\node[above] at (4,7) {$1$};
\node[above] at (3,7) {$1$};
\node[above] at (2,7) {$1$};
\node[above] at (1,7) {$1$};
%bottom labels
\node[below] at (6,0) {$k$};
\node[below] at (5,0) {$0$};
\node[below] at (4,0) {$0$};
\node[below] at (3,0) {$0$};
\node[below] at (2,0) {$0$};
\node[below] at (1,0) {$0$};
\node[below,circle,scale=0.75,draw] at (5,-0.75) {$y_N$};
\node[below] at (3.5,-0.75) {$\cdots$};
\node[below] at (2.5,-0.75) {$\cdots$};
\node[below,circle,scale=0.75,draw] at (1,-0.75) {$y_1$};
%left labels
\node[left] at (0,1) {$1$};
\node[left] at (0,2) {$1$};
\node[left] at (0,3) {$1$};
\node[left] at (0,4) {$1$};
\node[left] at (0,5) {$1$};
\node[left] at (0,6) {$k$};
\node[left,circle,scale=0.75,draw] at (-0.75,1) {$\overline{x}_1$};
\node[left] at (-0.75,2.5) {$\vdots$};
\node[left] at (-0.75,3.5) {$\vdots$};
\node[left,circle,scale=0.75,draw] at (-0.75,5) {$\overline{x}_N$};
%right labels
\node[right] at (7,1) {$0$};
\node[right] at (7,2) {$0$};
\node[right] at (7,3) {$0$};
\node[right] at (7,4) {$0$};
\node[right] at (7,5) {$0$};
\end{align}
Vertices in the topmost row and the rightmost column are of the form (<ref>); all other vertices are given by (<ref>). The variables associated to horizontal lines are reciprocated; namely, we set $\overline{x}_a = 1/x_a$, for all $1 \leq a \leq N$. Partition functions of the form (<ref>) were originally studied in [52], where they appeared in connection with Cauchy identities and as a one-parameter generalization of domain wall partition functions [33, 37].
We now show that (<ref>) admits a nice probabilistic interpretation. Fix parameters $t$, $(x_1,\dots,x_N)$ and $(y_1,\dots,y_N)$ such that each vertex (<ref>) within (<ref>) has real weight in the interval $[0,1]$. We may then associate to any configuration $\mathfrak{C}$ in the (finite) quadrant $[1,N] \times [1,N]$ (that does not include the thick arrow along the northeast boundary in (<ref>)) a probability weight $\mathbb{P}_{6{\rm v}}(\mathfrak{C})$, defined as the product of weights of all vertices within $\mathfrak{C}$.
Let $\mathfrak{h}(N,N)$ denote the height function assigned to the vertex $(N,N)$ within the stochastic six-vertex model. We have that
\begin{align}
\label{6v-E}
\prod_{a=1}^N
\cdot
\mathbb{E}_{6{\rm v}}
\left[
\frac{(t^{k+1};t)_{\infty}}{(t^{k+1+\mathfrak{h}(N,N)};t)_{\infty}}
\right],
\end{align}
where the expectation is taken with respect to the measure $\mathbb{P}_{6{\rm v}}$ defined above.
We begin by decomposing the partition function (<ref>) along the edges where vertices of the types (<ref>) and (<ref>) meet. This produces the equation
\begin{multline}
\label{Z-expand}
\\
= \sum_{\{i_1,\dots,i_N\} \in \{0,1\}^N}
\sum_{\{j_1,\dots,j_N\} \in \{0,1\}^N}
\mathbb{P}_{6{\rm v}}(i_1,\dots,i_N;j_1,\dots,j_N)
\end{multline}
where we have defined two new partition functions. The first is given by
\begin{align}
\label{square}
\mathbb{P}_{6{\rm v}}(i_1,\dots,i_N;j_1,\dots,j_N)
\tikz{0.8}{
\foreach\y in {1,...,5}{
\draw[lgray,line width=1.5pt,->] (0,\y) -- (6,\y);
\foreach\x in {1,...,5}{
\draw[lgray,line width=1.5pt,->] (\x,0) -- (\x,6);
%top labels
\node[above] at (5,6) {$i_N$};
\node[above] at (4,6) {$\cdots$};
\node[above] at (3,6) {$\cdots$};
\node[above] at (2,6) {$\cdots$};
\node[above] at (1,6) {$i_1$};
%bottom labels
\node[below] at (5,0) {$0$};
\node[below] at (4,0) {$0$};
\node[below] at (3,0) {$0$};
\node[below] at (2,0) {$0$};
\node[below] at (1,0) {$0$};
\node[below,circle,scale=0.75,draw] at (5,-0.75) {$y_N$};
\node[below] at (3.5,-0.75) {$\cdots$};
\node[below] at (2.5,-0.75) {$\cdots$};
\node[below,circle,scale=0.75,draw] at (1,-0.75) {$y_1$};
%left labels
\node[left] at (0,1) {$1$};
\node[left] at (0,2) {$1$};
\node[left] at (0,3) {$1$};
\node[left] at (0,4) {$1$};
\node[left] at (0,5) {$1$};
\node[left,circle,scale=0.75,draw] at (-0.75,1) {$\overline{x}_1$};
\node[left] at (-0.75,2.5) {$\vdots$};
\node[left] at (-0.75,3.5) {$\vdots$};
\node[left,circle,scale=0.75,draw] at (-0.75,5) {$\overline{x}_N$};
%right labels
\node[right] at (6,1) {$j_1$};
\node[right] at (6,2) {$\vdots$};
\node[right] at (6,3) {$\vdots$};
\node[right] at (6,4) {$\vdots$};
\node[right] at (6,5) {$j_N$};
\end{align}
where all vertices are of the type (<ref>); this quantity is the probability that a random configuration $\mathfrak{C}$ in the quadrant $[1,N] \times [1,N]$ has state $i_a \in \{0,1\}$ exiting vertically from vertex $(a,N)$ and state $j_a \in \{0,1\}$ exiting horizontally from vertex $(N,a)$, for all $1 \leq a \leq N$. The second is a tower of vertices of the type (<ref>) (which one may also view as a straightened version of the thick arrow along the northeast boundary in (<ref>)):
\begin{align}
\label{tower}
\tikz{0.85}{
\foreach\y in {1,...,6}{
\draw[lgray,line width=1.5pt,->] (0,\y) -- (2,\y);
\draw[lgray,line width=4pt,->] (1,0) -- (1,7);
%top labels
\node[above] at (1,7) {$k$};
%bottom labels
\node[below] at (1,0) {$k$};
%left labels
\node[left] at (0,1) {$j_1$};
\node[left] at (0,2) {$\vdots$};
\node[left] at (0,3) {$j_N$};
\node[left] at (0,4) {$i_N$};
\node[left] at (0,5) {$\vdots$};
\node[left] at (0,6) {$i_1$};
\node[left,circle,scale=0.75,draw] at (-0.75,1) {$\overline{x}_1$};
\node[left] at (-0.75,2) {$\vdots$};
\node[left,circle,scale=0.75,draw] at (-0.75,3) {$\overline{x}_N$};
\node[left,circle,scale=0.75,draw] at (-0.75,4) {$y_N$};
\node[left] at (-0.75,5) {$\vdots$};
\node[left,circle,scale=0.75,draw] at (-0.75,6) {$y_1$};
%right labels
\node[right] at (2,1) {$0$};
\node[right] at (2,2) {$0$};
\node[right] at (2,3) {$0$};
\node[right] at (2,4) {$1$};
\node[right] at (2,5) {$1$};
\node[right] at (2,6) {$1$};
\end{align}
In view of the arrow conservation property of the vertices (<ref>), it is easy to see that each internal vertical edge within (<ref>) admits a unique state such that the tower has non-vanishing weight. This allows us to compute $H_k(j_1,\dots,j_N;i_1,\dots,i_N)$ explicitly:
\begin{align}
\label{H-prod}
\prod_{a=1}^{N}
\mathcal{L}_{\overline{x}_a}(k+J_{a-1},j_a;k+J_a,0)
\mathcal{L}_{y_a}(k+I_a,i_a;k+I_{a-1},1),
\end{align}
where we have defined the partial sums $J_0 = I_0 = 0$, $J_a = \sum_{b=1}^{a} j_b$, $I_a = a-\sum_{b=1}^{a} i_b$ for $1 \leq a \leq N$, and where we note that $J_N = I_N$. From the table (<ref>), the weights
$\mathcal{L}_{\overline{x}_a}(k+J_{a-1},j_a;k+J_a,0)$ are all equal to $1$; the remaining terms in the product (<ref>) are given by
\begin{align*}
\mathcal{L}_{y_a}(k+I_a,i_a;k+I_{a-1},1)
y_a \cdot
\left\{
\begin{array}{ll}
1-t^{k+I_a}, & \qquad i_a=0,
\\
\\
1, & \qquad i_a =1.
\end{array}
\right.
\end{align*}
The product (<ref>) then simplifies to
\begin{align*}
\prod_{a=1}^{N} y_a
\cdot
\prod_{b=1}^{I_N} (1-t^{k+b})
\prod_{a=1}^{N} y_a
\cdot
\prod_{b=1}^{J_N} (1-t^{k+b}).
\end{align*}
In particular, $H_k(j_1,\dots,j_N;i_1,\dots,i_N)$ depends on $(i_1,\dots,i_N)$ and $(j_1,\dots,j_N)$ only via $I_N = J_N = \sum_{b=1}^{N} j_b$. In fact, returning to the quadrant (<ref>), we see that
$J_N = \mathfrak{h}(N,N)$; accordingly, one has
\begin{align*}
\prod_{a=1}^{N}
\cdot
\prod_{b=1}^{\mathfrak{h}(N,N)}
\\
\prod_{a=1}^{N}
\cdot
\prod_{b=1}^{\infty}
\frac{1-t^{k+b}}{1-t^{k+b+\mathfrak{h}(N,N)}}
\prod_{a=1}^{N}
\cdot
\frac{(t^{k+1};t)_{\infty}}{(t^{k+1+\mathfrak{h}(N,N)};t)_{\infty}}.
\end{align*}
Coming back to (<ref>), we then find that
\begin{multline*}
\\
\prod_{a=1}^{N}
\cdot
\sum_{\{i_1,\dots,i_N\} \in \{0,1\}^N}
\sum_{\{j_1,\dots,j_N\} \in \{0,1\}^N}
\mathbb{P}_{6{\rm v}}(i_1,\dots,i_N;j_1,\dots,j_N)
\frac{(t^{k+1};t)_{\infty}}{(t^{k+1+\mathfrak{h}(N,N)};t)_{\infty}}.
\end{multline*}
Conditioning on the possible values of $\mathfrak{h}(N,N)$, this may be written as
\begin{align*}
%\sum_{\{i_1,\dots,i_N\} \in \{0,1\}^N}
%\sum_{\{j_1,\dots,j_N\} \in \{0,1\}^N}
%\mathbb{P}_{6{\rm v}}(i_1,\dots,i_N;j_1,\dots,j_N)
\prod_{a=1}^{N}
\cdot
\sum_{m=0}^{N}
\frac{(t^{k+1};t)_{\infty}}{(t^{k+1+m};t)_{\infty}}
\mathbb{P}_{6{\rm v}}(\mathfrak{h}(N,N)=m),
\end{align*}
which proves the claim (<ref>).
§.§ Determinant Evaluation
Following Section 4.2 and Appendix B of [52], the partition function (<ref>) may be computed in closed form:
For any $N \geq 1$ and $k \geq 0$, one has
\begin{align}
\label{Z-det}
\frac{\prod_{i=1}^{N}y_i \prod_{1 \leq i,j \leq N} (1-x_i y_j)}
{\prod_{1 \leq i<j \leq N} (x_i-x_j)(y_i-y_j)}
\det_{1 \leq i,j \leq N}
\left[
\frac{1-t^{k+1}-t(1-t^k) x_i y_j}{(1-x_i y_j)(1-t x_i y_j)}
\right].
\end{align}
The proof relies on finding a list of properties of $Z(x_1,\dots,x_N;y_1,\dots,y_N;k)$ that determine it uniquely; one then shows that the right hand side of (<ref>) obeys the same properties. We list these properties below without derivation; for more information, we refer the reader to <cit.>.
* $Z(x_1,\dots,x_N;y_1,\dots,y_N;k)$ is symmetric in the alphabet $(x_1,\dots,x_N)$ and separately in the alphabet $(y_1,\dots,y_N)$;
* $\prod_{1 \leq i,j \leq N} (1-t x_i y_j) \cdot Z(x_1,\dots,x_N;y_1,\dots,y_N;k)$ is a polynomial in $x_N$ of degree $N$;
* Setting $x_N = 1/y_N$ we have
\begin{align*}
\Big|_{x_N = 1/y_N}
\cdot
\end{align*}
* Setting $x_i = 0$ for all $1 \leq i \leq N$, we have
\begin{align*}
\prod_{b=1}^{N}
\end{align*}
* For $N=1$, there holds
\begin{align*}
y_1 \cdot \left( \frac{1-t^{k+1}-t(1-t^k)x_1 y_1}{1-t x_1 y_1} \right).
\end{align*}
§.§ Schur Expectation
Having computed (<ref>) in determinant form, it is now easy to pass to its Schur expansion.
Define the following Schur measure with respect to two alphabets $(x_1,\dots,x_N)$ and $(y_1,\dots,y_N)$:
\begin{align}
\label{schur-meas}
\mathbb{P}_{\SM}(\lambda)
\prod_{1 \leq i,j \leq N}
(1-x_i y_j)
\cdot
\end{align}
For fixed $N \geq 1$ and $k \geq 0$ we then have the identity
\begin{align}
\label{Z-schur}
%\prod_{i=1}^{N}y_i \prod_{1 \leq i,j \leq N} (1-x_i y_j)
\prod_{i=1}^{N} y_i
\cdot
\mathbb{E}_{\SM}
\left[
\prod_{i=1}^{N}
\right],
\end{align}
where the expectation is taken with respect to the measure (<ref>).
We begin by manipulating the determinant present in (<ref>). One has
\begin{align*}
\det_{1 \leq i,j \leq N}
\left[
\frac{1-t^{k+1}-t(1-t^k) x_i y_j}{(1-x_i y_j)(1-t x_i y_j)}
\right]
\det_{1 \leq i,j \leq N}
\left[
\frac{1}{1-x_i y_j}
\frac{t^{k+1}}{1-t x_i y_j}.
\right],
\end{align*}
Replacing the two fractions on the right hand side by their corresponding geometric series, we obtain the identity
\begin{align*}
\det_{1 \leq i,j \leq N}
\left[
\frac{1-t^{k+1}-t(1-t^k) x_i y_j}{(1-x_i y_j)(1-t x_i y_j)}
\right]
%\det_{1 \leq i,j \leq N}
%(x_i y_j)^a
%t^a (x_i y_j)^a
\det_{1 \leq i,j \leq N}
\left[
\sum_{a=0}^{\infty}
(x_i y_j)^a
\right].
\end{align*}
To the latter we apply the Cauchy–Binet identity, which yields
\begin{align*}
\det_{1 \leq i,j \leq N}
\left[
\frac{1-t^{k+1}-t(1-t^k) x_i y_j}{(1-x_i y_j)(1-t x_i y_j)}
\right]
\sum_{0 \leq a_1 < \cdots < a_N}
\prod_{i=1}^{N}
\det_{1 \leq i,j \leq N}
\left[ x_i^{a_j} \right]
\det_{1 \leq i,j \leq N}
\left[ y_j^{a_i} \right];
\end{align*}
including Vandermonde factors on both sides of the equation, and making the change of summation indices $a_i = \lambda_{N-i+1}+i-1$, we then find that
\begin{multline*}
\prod_{1 \leq i<j \leq N}
\frac{1}{(x_i-x_j)(y_i-y_j)}
\cdot
\det_{1 \leq i,j \leq N}
\left[
\frac{1-t^{k+1}-t(1-t^k) x_i y_j}{(1-x_i y_j)(1-t x_i y_j)}
\right]
\\
\sum_{\lambda_1 \geq \cdots \geq \lambda_N \geq 0}
\
\prod_{i=1}^{N}
\end{multline*}
This recovers the claim (<ref>), after multiplying through by $\prod_{i=1}^{N} y_i \cdot \prod_{1 \leq i,j \leq N} (1-x_i y_j)$.
§.§ Final Match
Comparing (<ref>) and (<ref>), we have proved that
\begin{align}
\label{final-match}
\mathbb{E}_{6{\rm v}}
\left[
\frac{(t^{k+1};t)_{\infty}}{(t^{k+1+\mathfrak{h}(N,N)};t)_{\infty}}
\right]
\mathbb{E}_{\SM}
\left[
\prod_{i=1}^{N}
\right].
\end{align}
Both sides of (<ref>) are polynomial in $t^{k}$; since (<ref>) holds for all integer values $k \in \mathbb{Z}_{\geq 0}$ and the set $\{t^k\}_{k \geq 0}$ has a point of accumulation for $|t|<1$, we may extend the equality to all complex values by the analytic continuation $t^{k+1} = -\zeta \in \mathbb{C}$. This yields
\begin{align*}
\mathbb{E}_{6{\rm v}}
\left[
\frac{(-\zeta;t)_{\infty}}{(-\zeta t^{\mathfrak{h}(N,N)};t)_{\infty}}
\right]
\mathbb{E}_{\SM}
\left[
\prod_{i=1}^{N}
(1+\zeta t^{\lambda_i-i+N})
\right],
\end{align*}
which is precisely the result (<ref>) in the case $q=t$, $r_i=s_i=t^{-1/2}$ and $M=N$.
Extending this result to $M<N$ may be achieved by taking $y_{M+1} = \cdots = y_N = 0$ in (<ref>). On the side of the six-vertex model, this choice trivializes the contribution of the final $N-M$ thin vertical lines in the picture (<ref>) (as no arrows can enter them through the left), leading to a rectangular domain; on the side of the Schur expectation, this choice does not damage the measure (<ref>) in view of the stability property of Schur polynomials. A similar reduction is possible in the case $M>N$.
[1]
A. Aggarwal.
Current fluctuations of the stationary ASEP and six-vertex model.
Duke Math. J., 167(2):269–384, 2018.
[2]
A. Aggarwal and A. Borodin.
Phase transitions in the ASEP and stochastic six-vertex model.
Ann. Probab., 47(2):613–689, 2019.
[3]
A. Aggarwal, A. Borodin, and A. Bufetov.
Stochasticization of solutions to the Yang-Baxter equation.
Ann. Henri Poincaré, 20(8):2495–2554, 2019.
[4]
A. Aggarwal, A. Borodin, and M. Wheeler.
Colored fermionic vertex models and symmetric functions.
Preprint, arXiv:2101.01605.
[5]
D. Aldous and P. Diaconis.
Longest increasing subsequences: from patience sorting to the
Baik-Deift-Johansson theorem.
Bull. Amer. Math. Soc. (N.S.), 36(4):413–432, 1999.
[6]
G. Amir, I. Corwin, and J. Quastel.
Probability distribution of the free energy of the continuum directed
random polymer in $1+1$ dimensions.
Comm. Pure Appl. Math., 64(4):466–537, 2011.
[7]
J. Baik, G. Ben Arous, and S. Péché.
Phase transition of the largest eigenvalue for nonnull complex sample
covariance matrices.
Ann. Probab., 33(5):1643–1697, 2005.
[8]
J. Baik, P. Deift, and K. Johansson.
On the distribution of the length of the longest increasing
subsequence of random permutations.
J. Amer. Math. Soc., 12(4):1119–1178, 1999.
[9]
J. Baik and E. M. Rains.
Limiting distributions for a polynuclear growth model with external
J. Statist. Phys., 100:523–541, 2000.
[10]
G. Barraquand.
A phase transition for $q$-TASEP with a few slower particles.
Stoch. Proc. Appl., 125(7):2674–2699, 2015.
[11]
A. Borodin.
On a family of symmetric rational functions.
Adv. Math., 306:973–1018, 2017.
[12]
A. Borodin.
Stochastic higher spin six vertex model and Macdonald measures.
J. Math. Phys., 59(2):023301, 17, 2018.
[13]
A. Borodin, A. Bufetov, and M. Wheeler.
Between the stochastic six vertex model and Hall–Littlewood
Preprint, arXiv:1611.09486.
[14]
A. Borodin and I. Corwin.
Macdonald processes.
Probab. Theory Related Fields, 158(1-2):225–400, 2014.
[15]
A. Borodin, I. Corwin, and P. Ferrari.
Free energy fluctuations for directed polymers in random media in
$1+1$ dimension.
Comm. Pure Appl. Math., 67(7):1129–1214, 2014.
[16]
A. Borodin and V. Gorin.
Lectures on integrable probability.
In Probability and statistical physics in St. Petersburg,
volume 91 of Proc. Sympos. Pure Math., pages 155–214. Amer. Math.
Soc., Providence, RI, 2016.
[17]
A. Borodin and V. Gorin.
Moments match between the KPZ equation and the Airy point
SIGMA Symmetry Integrability Geom. Methods Appl., 12:Paper No.
102, 7, 2016.
[18]
A. Borodin, A. Okounkov, and G. Olshanski.
Asymptotics of Plancherel measures for symmetric groups.
J. Amer. Math. Soc., 13(3):481–515, 2000.
[19]
A. Borodin and G. Olshanski.
The ASEP and determinantal point processes.
Comm. Math. Phys., 353(2):853–903, 2017.
[20]
A. Borodin and S. Péché.
Airy kernel with two sets of parameters in directed percolation and
random matrix theory.
J. Stat. Phys., 132(2):275–290, 2008.
[21]
A. Borodin and L. Petrov.
Nearest neighbor Markov dynamics on Macdonald processes.
Adv. Math., 300:71–155, 2016.
[22]
A. Borodin and L. Petrov.
Higher spin six vertex model and symmetric rational functions.
Selecta Math. (N.S.), 24(2):751–874, 2018.
[23]
A. Borodin and M. Wheeler.
Coloured stochastic vertex models and their spectral theory.
Preprint, arXiv:1808.01866.
[24]
A. Bufetov and K. Matveev.
Hall-Littlewood RSK field.
Selecta Math. (N.S.), 24(5):4839–4884, 2018.
[25]
P. Calabrese, P. Le Doussal, and A. Rosso.
Free-energy distribution of the directed polymer at high temperature.
EPL (Europhysics Letters), 90(2):20002, apr 2010.
[26]
I. Corwin.
The Kardar-Parisi-Zhang equation and universality class.
Random Matrices Theory Appl., 1(1):1130001, 76, 2012.
[27]
I. Corwin, P. Ghosal, H. Shen, and L.-C. Tsai.
Stochastic PDE limit of the six vertex model.
Comm. Math. Phys., 375(3):1945–2038, 2020.
[28]
I. Corwin and L. Petrov.
The $q$-PushASEP: a new integrable model for traffic in $1+1$
J. Stat. Phys., 160(4):1005–1026, 2015.
[29]
I. Corwin and L. Petrov.
Stochastic higher spin vertex models on the line.
Comm. Math. Phys., 343(2):651–700, 2016.
[30]
V. Dotsenko.
Bethe ansatz derivation of the tracy-widom distribution for
one-dimensional directed polymers.
EPL (Europhysics Letters), 90(2):20003, apr 2010.
[31]
P. L. Ferrari and H. Spohn.
Random growth models.
In The Oxford handbook of random matrix theory, pages
782–801. Oxford Univ. Press, Oxford, 2011.
[32]
T. Imamura and T. Sasamoto.
Fluctuations of the one-dimensional polynuclear growth model with
external sources.
Nuclear Phys. B, 699(3):503–544, 2004.
[33]
A. G. Izergin.
Partition function of a six-vertex model in a finite volume.
In Doklady Akademii Nauk, volume 297, pages 331–333. Russian
Academy of Sciences, 1987.
[34]
K. Johansson.
Shape fluctuations and random matrices.
Comm. Math. Phys., 209(2):437–476, 2000.
[35]
K. Johansson.
Discrete orthogonal polynomial ensembles and the Plancherel
Ann. of Math. (2), 153(1):259–296, 2001.
[36]
M. Kardar, G. Parisi, and Y.-C. Zhang.
Dynamic scaling of growing interfaces.
Phys. Rev. Lett., 56:889–892, Mar 1986.
[37]
V. E. Korepin.
Calculation of norms of Bethe wave functions.
Communications in Mathematical Physics, 86(3):391–418, 1982.
[38]
P. P. Kulish, N. Y. Reshetikhin, and E. K. Sklyanin.
Yang-Baxter equations and representation theory. I.
Lett. Math. Phys., 5(5):393–403, 1981.
[39]
C. T. MacDonald, J. H. Gibbs, and A. C. Pipkin.
Kinetics of biopolymerization on nucleic acid templates.
Biopolymers, 6(1):1–25, 1968.
[40]
I. G. Macdonald.
Symmetric functions and Hall polynomials.
Oxford Classic Texts in the Physical Sciences. Oxford University
Press, New York, second edition, 2015.
[41]
V. V. Mangazeev.
On the Yang-Baxter equation for the six-vertex model.
Nuclear Phys. B, 882:70–96, 2014.
[42]
P. Meakin.
Fractals, scaling and growth far from equilibrium, volume 5 of
Cambridge Nonlinear Science Series.
Cambridge University Press, Cambridge, 1998.
[43]
A. Okounkov.
Infinite wedge and random partitions.
Selecta Math. (N.S.), 7(1):57–81, 2001.
[44]
Y. Pei.
A $q$-Robinson-Schensted-Knuth algorithm and a $q$-polymer.
Electron. J. Combin., 24(4):Paper No. 4.6, 38, 2017.
[45]
M. Prähofer and H. Spohn.
Universal distributions for growth processes in $1+1$ dimensions and
random matrices.
Phys. Rev. Lett., 84:4882–4885, May 2000.
[46]
M. Prähofer and H. Spohn.
Scale invariance of the PNG droplet and the Airy process.
J. Statist. Phys., 108(5-6):1071–1106, 2002.
[47]
J. Quastel.
Introduction to KPZ.
In Current developments in mathematics, 2011, pages 125–194.
Int. Press, Somerville, MA, 2012.
[48]
T. Sasamoto and H. Spohn.
Exact height distributions for the KPZ equation with narrow wedge
initial condition.
Nucl. Phys. B, 834(3):523–542, 2010.
[49]
F. Spitzer.
Interaction of Markov processes.
Adv. Math., 5:246–290 (1970), 1970.
[50]
C. A. Tracy and H. Widom.
Asymptotics in ASEP with step initial condition.
Comm. Math. Phys., 290(1):129–154, 2009.
[51]
B. Virag.
The heat and the landscape I.
Preprint, arXiv:2008.07241.
[52]
M. Wheeler and P. Zinn-Justin.
Refined Cauchy/Littlewood identities and six-vertex model
partition functions: III. Deformed bosons.
Advances in Mathematics, 299:543–600, 2016.
|
11institutetext: François Lamothe 22institutetext: Alain Haït 33institutetext:
Emmanuel Rachelson 44institutetext: ISAE-SUPAERO, Université de Toulouse
44email<EMAIL_ADDRESS>55institutetext: Claudio Contardo
66institutetext: Concordia University, Canada 77institutetext: Bernard Gendron
88institutetext: Université de Montréal, Canada
# On the integration of Dantzig-Wolfe and Fenchel decompositions via
directional normalizations
François Lamothe Alain Haït Emmanuel Rachelson Claudio Contardo Bernard
Gendron
###### Abstract
The strengthening of linear relaxations and bounds of mixed integer linear
programs has been an active research topic for decades. Enumeration-based
methods for integer programming like linear programming-based branch-and-bound
exploit strong dual bounds to fathom unpromising regions of the feasible
space. In this paper, we consider the strengthening of linear programs via a
composite of Dantzig-Wolfe and Fenchel decompositions. We provide geometric
interpretations of these two classical methods. Motivated by these geometric
interpretations, we introduce a novel approach for solving Fenchel sub-
problems and introduce a novel decomposition combining Dantzig-Wolfe and
Fenchel decompositions in an original manner. We carry out an extensive
computational campaign assessing the performance of the novel decomposition on
the unsplittable flow problem. Very promising results are obtained when the
new approach is compared to classical decomposition methods.
###### Keywords:
Mixed Integer Linear Programming Decomposition methods Unsplittable flows
Dantzig-Wolfe decomposition Fenchel decomposition
## Acknowledgments
This document is the result of a research project funded by the Centre
national d’études spatiales (CNES) and Thales Alenia Space. C. Contardo and B.
Gendron thank the Natural Sciences and Engineering Research Council of Canada
(NSERC) for its financial support under Discovery Grants no. 2020-06311 and
2017-06054.
## 1 Introduction
Enumeration-based algorithms are arguably the main algorithmic frameworks to
solve mixed-integer linear programs among which the branch-and-bound (B&B)
method land1960automatic (33) is perhaps the most efficient and versatile one.
It relies on the ability to compute primal and dual bounds on the value of the
solutions to the problem at hand. The method is most efficient when quick
computing of tight primal and dual bounds is available, resulting in short
enumerations. Traditionally, primal bounds are found via heuristics and dual
bounds via solving relaxations of the problem. The most classical such
relaxation is the so-called linear relaxation in which the integrality
constraints are ignored. The resulting problem, an ordinary linear program,
can be solved efficiently either in polynomial time via interior point methods
KHACHIYAN198053 (30, 38), or via a greedy method — namely the simplex method
dantzig1951maximization (19) — of exponential worst-case complexity, but very
efficient in practice borgwardt2012simplex (10).
A common practice to improve the linear relaxation of mixed integer linear
programs is to apply a decomposition method to a subset of the problem’s
constraints whose associated polyhedron does not have the integrality property
(some vertices of the polyhedron have non-integer components). The
decomposition tightens the polyhedron which in turn strengthens the linear
relaxation of the problem. One of the most classical decomposition methods is
the one by dantzig1960decomposition (20), known as the Dantzig-Wolfe
decomposition. It has proven successful in various applications which explains
the attention it has received over the years desaulniers2006column (22). The
Dantzig-Wolfe decomposition is closely related to the Lagrangian decomposition
which has also been successful in many practical applications. The main
difference between the Lagrangian and Dantzig-Wolfe decomposition is that in
the former only dual information is extracted and exploited in a sub-gradient
algorithm whereas the latter also extracts primal information which can be
embedded within an enumeration scheme, usually referred to as branch-and-price
barnhart1998branch (3). Because in this work we will consider column
generation and cutting plane procedures, we will focus on the Dantzig-Wolfe
decomposition.
Our analysis restricts to decompositions that are able to exploit integer
subproblems —as opposed to pure linear problems—, and for that reason, we do
not consider Benders decomposition benders1962partitioning (7). Fenchel
decomposition, on the other hand, is a cutting plane method similar to Benders
decomposition that can be applied when the sub-problem is integer. It is thus
able to improve the relaxation quality of mixed integer linear programs. This
technique has had success in several applications such as knapsack problems
boyd1993generating (11, 29), generalized assignment problems
avella2010computational (1), network design problems with unsplittable flow
chen2021exact (15) or even stochastic optimization problems ntaimo2013fenchel
(40, 4).
Although Fenchel and Dantzig-Wolfe decompositions have been largely studied in
the context of linear relaxation strengthening, several limitations of both
decomposition methods have been identified in the literature which may affect
their convergence. In particular, the Dantzig-Wolfe decomposition is known to
suffer from the degeneracy of its master problem (which happens when the
master problem admits several optimal dual solutions). This explains why a
considerable effort has been made by the scientific community to improve these
decomposition methods and overcome their weaknesses. This work is in this line
of contribution. We show that Dantzig-Wolfe and Fenchel decompositions can be
seen in a similar light as constructing inner and outer approximations of the
polyhedron being decomposed and that the synergy between the two
approximations may be beneficial for the entire approach when applied to some
large-scale mixed-integer linear programs.
We outline the main contributions of our work as follows:
* •
We provide geometric interpretations of Dantzig-Wolfe and Fenchel
decompositions which may fuel complementary insights on these two approaches
compared to purely analytical interpretations.
* •
We provide a critical overview of normalizations at the core of the separation
problems arising in Fenchel decomposition. The normalization impacts the type
of cut, its properties, and ultimately the convergence speed of the
decomposition method. We provide geometric interpretations of several types of
normalization and their properties.
* •
We introduce a novel approach to the Fenchel sub-problem when a directional
normalization is used. The proposed method possesses reduces the numerical
instabilities of a direct resolution approach commonly used. We show that the
new approach solves the Fenchel sub-problem in finitely many iterations.
* •
We introduce a new decomposition method inspired by both the Dantzig-Wolfe and
the Fenchel decompositions. The proposed method uses a Dantzig-Wolfe master
problem and a Fenchel master problem. A Fenchel sub-problem guided with a
directional normalization is used to coordinate the two master problems. The
resulting method is shown to perform especially well on instances presenting
high degrees of degeneracy. We provide a possible explanation of this
phenomenon based on our findings.
In the following, we start by giving in Section 2 the context in which we
apply the various decomposition methods together with some notation. Then, we
present in Section 3 a geometric interpretation of Dantzig-Wolfe and Fenchel
decomposition methods. Section 4 is dedicated to an overview of the resolution
of the Fenchel subproblem and in particular, the impact of the normalization
used. This concept will be used in the next section for the new methods
introduced in the paper. We then present in Section 5 a new approach to
solving the Fenchel subproblem. In Section 6, we introduce the new
decomposition method which integrates both Dantzig-Wolfe and Fenchel master
problems as well as a Fenchel subproblem. In Section 7, we illustrate our
method by applying it to the unsplittable flow problem. An experimental
evaluation of all proposed methods is then made in Section 8 on small and
medium-sized instances of the problem.
## 2 Context
In the remainder of this paper, we will apply decomposition methods to the
following general mixed integer linear program:
$\displaystyle(P)\quad$ $\displaystyle\max_{x}$ $\displaystyle cx$ (1a)
subject to $\displaystyle A_{1}x\leq b_{1}$ (1b) $\displaystyle A_{2}x\leq
b_{2}$ (1c) $\displaystyle x\in X$ (1d)
where $X$ is a product set $\mathbb{R}^{p}\times\mathbb{Z}^{q}$ of appropriate
dimension whose linear relaxation $\mathbb{R}^{p+q}$ will be noted $\bar{X}$.
Moreover, to explain the methods used, we will consider the following
polyhedra which are assumed to be bounded to simplify the explanations:
$\displaystyle LR_{1}=\\{x\in\bar{X}|A_{1}x\leq b_{1}\\}$ $\displaystyle
LR_{2}=\\{x\in\bar{X}|A_{2}x\leq b_{2}\\}$ $\displaystyle Q_{2}=conv(\\{x\in
X|A_{2}x\leq b_{2}\\})$
In order to bound from above the value of the optimal solution of a general
mixed integer linear program as ($P$) one can optimize its objective function
over a relaxation containing the set of valid solutions. Usually, the linear
relaxation of the set of solutions $LR_{1}\cap LR_{2}$ is used. However, one
might want a tighter relaxation to obtain a better upper bound. This can be
obtained by using the relaxation $LR_{1}\cap Q_{2}$ which still contains the
set of valid solutions while being included in $LR_{1}\cap LR_{2}$. As
illustrated in Figure 1, this relaxation usually returns strictly better
bounds than the linear relaxation when the polyhedron $Q_{2}$ is strictly
included in $LR_{2}$ which happens when the polyhedron $LR_{2}$ does not have
the integrality property (some of the vertices of the polyhedron have non-
integer coordinates). However, the drawback of $Q_{2}$ compared to $LR_{2}$ is
that one usually only has a representation of $Q_{2}$ with an exponential
number of variables or constraints which is not manageable directly by a
linear programming solver. In compensation, we assume to have at our disposal
an efficient algorithm to optimize a linear function on the polyhedron
$Q_{2}$. This algorithm is called the optimization oracle and solves the
following mixed integer linear program:
$\displaystyle(O)\quad$ $\displaystyle\max_{x}$ $\displaystyle\pi x$ subject
to $\displaystyle A_{2}x\leq b_{2}$ $\displaystyle x\in X$
where $x\mapsto\pi x$ is any linear function. Thus, to be able to optimize
over $LR_{1}\cap Q_{2}$, the goal of a decomposition method is to use this
oracle to compute an approximation of $Q_{2}$ of manageable size. In
particular, the Dantzig-Wolfe decomposition iteratively grows an inner
approximation of $Q_{2}$ while the Fenchel decomposition iteratively refines
an outer approximation of $Q_{2}$.
$LR_{1}$$LR_{2}$ (a) Polyhedra associated with the linear relaxation
$LR_{1}$$Q_{2}$ (b) Polyhedra obtained after reformulation by the
decomposition methods
Figure 1: Representation of solution spaces of the linear relaxation and the
models obtained from the decomposition methods
Decomposition methods are known to work very well when the matrix $A_{2}$ is
block diagonal because it enables decomposing the problem of optimizing over
$Q_{2}$ into several smaller problems (one per block of the matrix). For
instance, in the case of the application considered in this study, the
unsplittable flow problem, the capacity constraints lead to a block diagonal
structure. In this context, the constraints $A_{1}x\leq b_{1}$ would
correspond to the flow conservation constraints while the constraints
$A_{2}x\leq b_{2}$ would correspond to the capacity constraints. However, our
work makes use of a different rationale. Let us assume that the matrix $A_{2}$
is sparse with non-zero elements for only a few variables; this may happen for
instance when $A_{2}$ corresponds to one of the blocks of a block diagonal
matrix. With this choice, it is possible to apply decomposition methods even
in contexts where the main problem ($P$) does not have any block diagonal
structure. For clarity purposes, we will present the methods as if we were
decomposing only one polyhedron at a time. However, in practice, one would
decompose several polyhedra at the same time; for example, all the blocks of a
block diagonal matrix.
## 3 Geometric interpretation of Dantzig-Wolfe and Fenchel decompositions
In this section, we provide a geometric interpretation of both decomposition
methods with the potential to fuel new intuitions regarding their strengths
and weaknesses.
### 3.1 Dantzig-Wolfe decomposition
Instead of optimizing the objective function of mixed integer linear program
(P) over the solution set $LR_{1}\cap LR_{2}$ of its linear relaxation,
Dantzig-Wolfe decomposition allows for the optimization over the smaller set
$LR_{1}\cap Q_{2}$. Because one does not usually have a manageable description
of the set $Q_{2}$, it is necessary to compute an approximation of $Q_{2}$ in
order to optimize over the intersection $LR_{1}\cap Q_{2}$. To compute this
approximation, we assume the availability of an optimization oracle over
$Q_{2}$. The main idea in the Dantzig-Wolfe decomposition is to iteratively
grow an inner approximation of $Q_{2}$. One can create such an approximation
by obtaining a set of points $x_{i}$ belonging to $Q_{2}$ (usually extreme
points of $Q_{2}$) and setting the approximation as the convex envelope of
these points. We will denote this inner approximation with $\widehat{Q}_{2}$.
The Dantzig-Wolfe decomposition illustrated in Figure 2 proceeds as follows:
1. 1.
Initialize the approximation $\widehat{Q}_{2}$ with points of $Q_{2}$
2. 2.
Find the optimal solution $\widehat{x}$ over $LR_{1}\cap\widehat{Q}_{2}$
3. 3.
Search for a point of $Q_{2}$ whose addition to $\widehat{Q}_{2}$ may improve
the value of the optimal solution $\widehat{x}$
4. 4.
If such a point exist, add it to $\widehat{Q}_{2}$ and go to Step 2.
5. 5.
Else: $\widehat{x}$ is the optimal solution over $LR_{1}\cap Q_{2}$. Stop.
#### 3.1.1 Optimizing over $LR_{1}\cap\widehat{Q}_{2}$
To optimize over $LR_{1}\cap\widehat{Q}_{2}$, a master linear program is
created in which the condition $x\in\widehat{Q}_{2}$ must be enforced. This
can be done by rewriting $x$ as a convex combination of its extreme points
$x_{i}$. To that end, a variable $\lambda_{i}$ is introduced for each point
$x_{i}$ used to create $\widehat{Q}_{2}$. This variable represents the weight
of the vertex $x_{i}$ in the convex combination. The Dantzig-Wolfe master
linear program can then be written as follows:
$\displaystyle(DW)\quad$ $\displaystyle\max_{x,\lambda_{i}}$ $\displaystyle
c^{T}x$ subject to $\displaystyle A_{1}x\leq b_{1}$ $\displaystyle
x=\sum_{i\in I}\lambda_{i}x_{i}$ $\displaystyle\sum_{i\in I}\lambda_{i}=1$
$\displaystyle x\in\bar{X},~{}\lambda_{i}\in\mathbb{R}^{+},\quad\forall i\in
I$
(a) Iteration 1. Initialization of $\widehat{Q}_{2}$ with points of $Q_{2}$
and optimization over $LR_{1}\cap\widehat{Q}_{2}$. Black arrow: objective
function of the master problem
(b) A direction (red arrow) in which to grow the polyhedron $\widehat{Q}_{2}$
is created with the dual variables of the master problem. The farthest point
of $Q_{2}$ in this direction is found and added to $\widehat{Q}_{2}$.
(c) Iteration 2. Re-optimization over $LR_{1}\cap\widehat{Q}_{2}$. The
farthest point of $Q_{2}$ in the dual direction is found and added to
$\widehat{Q}_{2}$.
(d) Iteration 3. Even though the optimal solution is found, due to degeneracy,
a new point of $Q_{2}$ is found in the dual direction to strengthen the dual
bound.
(e) Iteration 4. Once the current solution is the farthest point of $Q_{2}$ in
the dual direction then it is optimal and the method stops.
Figure 2: Geometric illustration of the Dantzig-Wolfe decomposition
#### 3.1.2 Finding an improving point of $Q_{2}$
In the Dantzig-Wolfe decomposition, the master problem returns the farthest
point $x^{*}$ of $LR_{1}\cap\widehat{Q}_{2}$ in the direction $c$ and one
would like to know whether this point is also the optimal for $LR_{1}\cap
Q_{2}$ or if the inner approximation $\widehat{Q}_{2}$ needs to be improved.
The dual point of view of this statement, on which is based the Dantzig-Wolfe
subproblem, is that the bound $cx\leq cx^{*}$ is valid for
$LR_{1}\cap\widehat{Q}_{2}$ and we would like to know if it also holds true
for $LR_{1}\cap Q_{2}$.
Ideas from linear programming duality theory: The following concepts are
illustrated in Figure 3. Linear programming duality informs us that because
$LR_{1}\cap\widehat{Q}_{2}$ is the intersection of two polyhedra, the bound
$cx\leq cx^{*}$ can always be decomposed as the sum of two inequalities, one
valid for $LR_{1}$ and the other valid for $\widehat{Q}_{2}$. Furthermore, the
dual variables of the master problem yield such a decomposition of the optimal
bound for $LR_{1}\cap\widehat{Q}_{2}$. Indeed, let us denote $u$, $\pi$ and
$\pi_{0}$ the optimal dual variables of the constraints $A_{1}x\leq b_{1}$,
$x=\sum_{i}\lambda_{i}x_{i}$ and $\sum_{i}\lambda_{i}=1$, respectively. By
construction of the dual of the master problem (whose derivation we let to the
reader), $uA_{1}x\leq ub_{1}$ is valid for $LR_{1}$, $\pi x\leq\pi_{0}$ is
valid for $\widehat{Q}_{2}$ and these two inequalities sum to the optimal
bound $cx\leq cx^{*}$ over $LR_{1}\cap\widehat{Q}_{2}$ (note: the equality
$ub+pi_{0}=cx^{*}$ follows thanks to the strong duality theorem of linear
programming matouvsek2007understanding (37)).
Figure 3: The optimal bound (black) on the optimization direction (black
arrows) over the intersection (red fill) of $LR_{1}$ (grey) and
$\widehat{Q}_{2}$ (red) can be decomposed as the sum of a valid inequality
(blue) for $LR_{1}$ and a valid inequality (red) for $\widehat{Q}_{2}$.
The Dantzig-Wolfe subproblem: In order to show that the solution $x^{*}$ of
the master problem is not the farthest point of $LR_{1}\cap Q_{2}$ in the
direction $c$, one must at least prove that the bound $cx\leq cx^{*}$ implied
by the dual variables of the master problem is not valid for $LR_{1}\cap
Q_{2}$. However, if the inequality $\pi x\leq\pi_{0}$ were valid for $Q_{2}$,
because we know that $uA_{1}x\leq ub_{1}$ is valid for $LR_{1}$ and that these
two inequalities sum to the bound $cx\leq cx^{*}$ then this bound would be
valid for $LR_{1}\cap Q_{2}$. Thus, the goal of the sub-problem is to show
that the inequality $\pi x\leq\pi_{0}$ is not valid for $Q_{2}$. To that end,
the subproblem is tasked to find the farthest point of $Q_{2}$ in the
direction $\pi$. If this point violates $\pi x\leq\pi_{0}$ then we have found
a point of $Q_{2}$ violating the inequality. The point can then be added to
the master problem to grow the inner approximation $\widehat{Q}_{2}$ and at
least prevent the master problem from yielding the same dual variables again.
Otherwise, the current solution is optimal because the bound implied by the
dual variables is valid for $LR_{1}\cap Q_{2}$.
#### 3.1.3 A note on degeneracy
We have seen above that the dual variables of the master problem imply a bound
on the value of its objective function. This bound can be considered a
certificate that the current value of the master problem is optimal.
Meanwhile, a linear program such as a Dantzig-Wolfe master problem is said to
be degenerate when it has several dual optimal solutions. Each of these dual
solutions is a certificate of optimality for the current value of the master
problem. Thus, to improve the value of the master problem, one must invalidate
each of these certificates. However, the subproblem of the Dantzig-Wolfe
procedure only invalidates one of these certificates without any guarantee
about the one implied by the other dual solutions. Thus the Dantzig-Wolfe
decomposition is susceptible to having many iterations without improvement of
the objective function when the master problem is highly degenerate. This can
slow down the convergence of the method considerably.
### 3.2 Fenchel decomposition
In Fenchel decomposition, instead of growing an inner approximation of
$Q_{2}$, an outer approximation of the polyhedron $Q_{2}$ is refined to enable
the optimization over $LR_{1}\cap Q_{2}$. Such an outer approximation can use
any collection of inequalities valid for $Q_{2}$. The decomposition is
illustrated in Figure 4 and proceeds as follows:
1. 1.
Initialize an outer approximation $\widecheck{Q}_{2}$ with inequalities valid
for $Q_{2}$; typically one can take $\widecheck{Q}_{2}=LR_{2}$.
2. 2.
Optimize over $LR_{1}\cap\widecheck{Q}_{2}$ and recover a solution
$\widecheck{x}$.
3. 3.
Search for a cut separating $\widecheck{x}$ from $Q_{2}$.
4. 4.
If such a cut exists, add the cut to $\widecheck{Q}_{2}$ and go to Step 2.
5. 5.
Else: $\widecheck{x}$ is the optimal solution over $LR_{1}\cap Q_{2}$. Stop.
(a) Initialization of $\widecheck{Q}_{2}$ with $LR_{2}$ and optimization over
$LR_{1}\cap\widecheck{Q}_{2}$. Black arrow: objective function of the master
problem
(b) Creation of a cut separating $Q_{2}$ from the solution of the optimization
over $LR_{1}\cap\widecheck{Q}_{2}$
(c) After addition of the cut to $\widecheck{Q}_{2}$, re-optimization over
$LR_{1}\cap\widecheck{Q}_{2}$.
(d) Creation of a cut separating $Q_{2}$ from the solution of the optimization
over $LR_{1}\cap\widecheck{Q}_{2}$
(e) Once enough cuts have been added to the approximation $\widecheck{Q}_{2}$,
the solution found cannot be separated from $Q_{2}$ and is thus optimal.
Figure 4: Geometric illustration of the Fenchel decomposition
In the second step, in order to optimize over $LR_{1}\cap\widecheck{Q}_{2}$,
the following Fenchel master problem is used:
$\displaystyle(F)\quad$ $\displaystyle\max_{x}$ $\displaystyle c^{T}x$ subject
to $\displaystyle A_{1}x\leq b_{1}$ $\displaystyle\pi
x\leq\pi_{0}\quad\forall(\pi,\pi_{0})\in\mathcal{C}$ $\displaystyle
x\in\bar{X}$
where $\mathcal{C}$ is the set of cuts describing $\widecheck{Q}_{2}$.
The main challenge in the Fenchel decomposition is to generate a cut $\pi
x\leq\pi_{0}$ separating the solution of the Fenchel master problem
$\widecheck{x}$ from the polyhedron $Q_{2}$. The classical approach to
generating such cuts is based on a linear program as described in Section 4.
## 4 The Fenchel separation subproblem and its normalizations
In the Fenchel decomposition, a cut $\pi x\leq\pi_{0}$ separating the solution
of the Fenchel master problem $\widecheck{x}$ from the polyhedron $Q_{2}$ must
be found. Such a cut can be created by finding a solution of non-negative
value of the following separation linear program:
$\displaystyle(S)\quad$ $\displaystyle\max_{\pi,\pi_{0}}$
$\displaystyle\pi\widecheck{x}-\pi_{0}$ subject to $\displaystyle\pi
x_{i}\leq\pi_{0},\quad\forall x_{i}\in Q_{2}$
$\displaystyle\pi,\pi_{0}\in\mathbb{R}$
where the objective maximizes the violation of the generated cut by
$\widecheck{x}$ while the constraints ensure that the cut is valid for every
point of the polyhedron $Q_{2}$. However, this separation problem has too many
constraints to consider them all explicitly. It is thus initially solved with
a subset of its constraints ensuring that the generated cut is valid for a few
points of $Q_{2}$. New points will progressively be taken into account in the
constraints until the validity of the cut for the whole polyhedron can be
ensured. To check if a cut $\bar{\pi}x\leq\bar{\pi_{0}}$ is valid for $Q_{2}$,
we search for the point of $Q_{2}$ that most violates the cut. This can be
done with a call to the optimization oracle ($O$) for the linear function
associated with $\bar{\pi}$. If the point returned by ($O$) violates the cut,
it is added as a constraint to ($S$). Otherwise, the cut
$\bar{\pi}x\leq\bar{\pi_{0}}$ is the optimal solution of ($S$) and maximizes
the separation of the solution of the Fenchel master problem from $Q_{2}$.
Another issue preventing the direct resolution of the problem ($S$) is that
its solution space is a cone. Indeed, if a cut $\pi x\leq\pi_{0}$ is valid for
all the points of $Q_{2}$ then so is the cut $\alpha\pi x\leq\alpha\pi_{0}$
for any non-negative constant $\alpha$. Thus, the problem ($S$) is often
unbounded. To prevent this, a normalization process must be performed which
classically consists in adding a constraint to the separation problem ($S$)
that will make the set of possible coefficients of the generated cut bounded.
The choice of this normalization greatly impacts the generated cut as well as
the convergence speed of the Fenchel decomposition. In addition, the
normalization may have an effect on the quality of the cut obtained, for
instance by favoring (or not) the obtention of facets of the polyhedron
$Q_{2}$. As we will see in the following sections, the normalization also
impacts the dual problem of ($S$) which becomes “Find the point of $Q_{2}$
minimizing some criterion of proximity to $\widecheck{x}$". It is often more
intuitive to interpret the impact of a normalization on the dual and therefore
we use this approach for our geometric interpretations.
### 4.1 Normalization $\|\pi\|\leq 1$
boyd1995convergence (12) analyzes the normalization consisting in adding to
the separation problem the constraint $\|\pi\|\leq 1$ for any norm (i.e. a
function that satisfies the triangular inequality and for which for every
$x\in\mathbb{R}^{n}$ and $\lambda\in\mathbb{R}$ satisfies $\|\lambda
x\|=|\lambda|\|x\|$ and $\|x\|=0\Rightarrow x=0$). With this choice, we can
consider that the optimized quantity is
$\frac{\pi\widecheck{x}-\pi_{0}}{\|\pi\|}$, i.e. the distance from
$\widecheck{x}$ to the hyperplane $\pi x=\pi_{0}$. Note that this distance is
not measured in the norm $\|.\|$ but in its dual norm $\|.\|^{*}$:
$\|\lambda\|^{*}=\ max_{\|x\|\leq 1}\lambda x$. In particular, note that the
norms $\|.\|_{1}$ and $\|.\|_{\infty}$ are dual and that the norm $\|.\|_{2}$
is self-dual. When using this normalization, the dual problem ($D$) becomes
"Find the point of $Q_{2}$ closest to $\widecheck{x}$ in the sense of the dual
norm $\|.\|^{*}$". This point is located on the border of $Q_{2}$ and
corresponds to the point of contact between $Q_{2}$ and the smallest sphere
centered in $\widecheck{x}$ touching $Q_{2}$. The generated cut is then a
tangent cut to $Q_{2}$ and to the sphere at the point of contact. One
important result of the work of boyd1995convergence (12) is that the
polyhedron $Q_{2}$ can be computed in finite time with this normalization. On
the other hand, the generated cuts are not always facets of the polyhedron
$Q_{2}$. All these geometric interpretations are illustrated in the
$\|.\|_{2}$ norm in Figure 5.
$\widecheck{x}$ (a) Cut associated to a $\|.\|_{2}$ normalization: a facet of
$Q_{2}$ is generated
$\widecheck{x}$ (b) Cut associated to a $\|.\|_{2}$ normalization: the
generated cut is not a facet of $Q_{2}$
Figure 5: Example of cuts generated with a $\|.\|_{2}$ normalization
### 4.2 Normalizations guaranteeing the generation of facets
We are now interested in normalizations guaranteeing the generation of facets
of $Q_{2}$. To do this, we start by presenting a theorem linking the facets of
$Q_{2}$ and the extreme rays of the cone of the solutions of the separation
problem ($S$). This theorem can be found in conforti2019facet (17)
(Proposition 1). In this theorem, $cone(X)$ (resp. $vect(X)$) is the set of
all conical (resp. linear) combinations of elements of a set $X$.
###### Theorem 4.1
Let P be a nonempty polyhedron. Let $(\pi^{j}x=\pi_{0}^{j})_{j\in J^{=}}$ be a
non-redundant representation of the affine envelope of P and let
$(\pi^{j}x\leq\pi_{0}^{j})_{j\in J^{\leq}}$ be the set of facets of P. Then
the set of valid cuts for P is:
$\mathcal{C}(P)=cone\left(\begin{pmatrix}0\\\
1\end{pmatrix},\begin{pmatrix}\pi^{j}\\\ \pi_{0}^{j}\end{pmatrix}_{j\in
J^{\leq}}\right)+vect\left(\begin{pmatrix}\pi^{j}\\\
\pi_{0}^{j}\end{pmatrix}_{j\in J^{=}}\right)$
In addition, all these vectors are necessary for the previous representation.
This theorem states that apart from the trivial ray $(0,1)$, the extreme rays
of the cone of the solutions of ($S$) are associated with the facets of the
polyhedron $Q_{2}$. When the polyhedron is not fully dimensional (_i.e._
$J^{=}\neq\emptyset$), the cuts corresponding to $(\pi^{j},\pi_{0}^{j})_{j\in
J^{=}}$ are the improper faces of $Q_{2}$. Let us suppose for the following
that $Q_{2}$ is full-dimensional. This happens once all the improper faces
have been generated.
A sufficient condition ensuring that the cuts generated are facets of $Q_{2}$
is to use a normalization condition which can be applied by adding a unique
linear constraint to the separation problem. If only one linear constraint is
added to ($S$), then all the extreme points of the new solution space
correspond to old extreme rays and therefore to facets of $Q_{2}$. If the
problem ($S$) is indeed made bounded by the addition of the linear constraint,
then it suffices to find an optimal vertex to guarantee the construction of a
facet. More generally, it is possible to add several linear constraints, as
long as they do not intersect inside the cone of solution of ($S$). For
example, adding a constraint $-1\leq f(\pi,\pi_{0})\leq 1$, where $f$ is a
linear function, corresponds to adding two linear constraints that do not
intersect. The cuts generated will then induce facets of $Q_{2}$. When, for
reasons specific to the problem, we know that valid cuts satisfy $\pi\geq 0$
(for example if $Q_{2}$ is a knapsack polyhedron), then the condition
$\|\pi\|_{1}\leq 1$ reduces to $\sum_{i}\pi_{i}\leq 1$ and can be used to
generate facet inducing inequalities.
Normalization of $\pi_{0}$: A normalization guaranteeing the generation of
facets is $|\pi_{0}|\leq$ 1\. This normalization is studied in detail in
conforti2019facet (17). It makes the problem ($S$) bounded if and only if
$\widecheck{x}$ can be written as a combination of elements in $Q_{2}$ using
only non-negative coefficients. Indeed, let us look at the impact of
normalization on the dual problem ($D$) which becomes:
$\displaystyle(D)\quad$ $\displaystyle\min_{\lambda_{i},~{}z}$
$\displaystyle|z|$ subject to $\displaystyle\sum_{i\in
I}\lambda_{i}x_{i}=\widecheck{x}$ $\displaystyle\sum_{i\in I}\lambda_{i}=1+z$
$\displaystyle z\in\mathbb{R},~{}\lambda_{i}\in\mathbb{R}^{+},\quad\forall
i\in I$
In simple words, this problem can be interpreted as follows: find a
combination with non-negative coefficients of elements of $Q_{2}$ equal to
$\widecheck{x}$ whose sum of the coefficients is as close as possible to 1.
The separation problem ($S$) is bounded if and only if its dual ($D$) is
feasible. However, the problem ($D$) is feasible if and only if
$\widecheck{x}$ can be written as a combination with non-negative coefficients
of elements of $Q_{2}$. This condition is naturally reached when the origin is
in the interior of $Q_{2}$ since the set of combinations with non-negative
coefficients of elements of $Q_{2}$ is then the entire space. If we know a
point in the interior of $Q_{2}$, it is possible to ensure this condition by
translating the problem to place the origin on this interior point. Once the
origin is inside $Q_{2}$, the generated cut is a facet of $Q_{2}$ intersecting
the segment connecting $\widecheck{x}$ to the origin.
Directional normalization of $\pi$: A second normalization, studied by
bonami2003etude (9) and guaranteeing the generation of facets, consists in
bounding the coefficients of $\pi$ in a given direction using the constraint
$|(\widehat{x}-\widecheck{x})\pi|\leq 1$ where $\widehat{x}$ is an arbitrary
point in $\mathbb{R}^{n}$. In order to determine when this normalization makes
the problem ($S$) bounded, let us look at its impact on the dual problem ($D$)
which becomes:
$\displaystyle(D)\quad$ $\displaystyle\min_{\lambda_{i},~{}z}$
$\displaystyle|z|$ subject to $\displaystyle\sum_{i\in
I}\lambda_{i}x_{i}=z\widehat{x}+(1-z)\widecheck{x}$ $\displaystyle\sum_{i\in
I}\lambda_{i}=1$ $\displaystyle
z\in\mathbb{R},~{}\lambda_{i}\in\mathbb{R}^{+},\quad\forall i\in I$
This problem can be interpreted as follows: find the point of $Q_{2}$ closest
to $\widecheck{x}$ on the line containing $\widehat{x}$ and $\widecheck{x}$.
The problem ($S$) is bounded when its dual ($D$) is feasible which happens
when there is a point belonging to both $Q_{2}$ and to the previous line. This
condition can be fulfilled for instance by choosing $\widehat{x}$ equal to a
known point of $Q_{2}$. In this case, the generated cut is a facet of $Q_{2}$
intersecting the segment connecting $\widecheck{x}$ and $\widehat{x}$. This is
illustrated in Figure 6(a). Unlike the previous normalization, the known point
$\widehat{x}$ does not need to be in the interior of $Q_{2}$. However, note
that when it is on the border of $Q_{2}$, the optimal value of the problem
($S$) is bounded but the set of optimal solutions may be unbounded. Indeed,
let us consider the following example illustrated in Figure 6(b).
Example: Suppose that $\widehat{x}$ is located on a vertex of $Q_{2}$ and that
$\widecheck{x}$ is located such that the line $(\widecheck{x},\widehat{x})$
intersects $Q_{2}$ only in $\widehat{x}$. Let $\pi x\leq\pi\widehat{x}$ be an
optimal cut separating $\widecheck{x}$ from $Q_{2}$ thus passing through
$\widehat{x}$. Let $\pi^{\bot}$ be the projection of $\pi$ on the orthogonal
of the vector space induced by the vector $\widecheck{x}-\widehat{x}$. Finally
let us assume that the cut $\pi^{\bot}x\leq\pi^{\bot}\widehat{x}$ is valid for
$Q_{2}$. This is for instance the case in the 2D example illustrated in Figure
6(b) as the hyperplane $\pi^{\bot}x=\pi^{\bot}\widehat{x}$ is, in this case,
the line $(\widecheck{x},\widehat{x})$. Then for all $\alpha\geq 0$, the cut
$(\pi+\alpha\pi^{\bot})x\leq(\pi+\alpha\pi^{\bot})\widehat{x}$ is also an
optimal cut for the separation problem. Indeed, it is valid for $Q_{2}$ as a
non-negative combination of valid cuts for $Q_{2}$. Moreover, the violation of
this cuts by $\widecheck{x}$ is the same as the violation of $\pi
x\leq\pi\widehat{x}$ because by construction of $\pi^{\bot}$ the product
$\pi^{\bot}(\widecheck{x}-\widehat{x})$ is equal to zero. Thus, in this
example, the separation problem is bounded because its dual is feasible but
the set of optimal cuts is unbounded because the norm of
$\pi+\alpha\pi^{\bot}$ approaches infinity when $\alpha$ does the same.
Despite this unbounded set of solutions, only the facets of the polyhedron
$Q_{2}$ are vertices of the solution space of ($S$). Thus, if the algorithm
solving ($S$) always returns a vertex, the generated cut will always be a
facet of $Q_{2}$.
$\widecheck{x}$$\widehat{x}$ (a) Cut associated with directional normalization
$\pi$$\pi^{\bot}$$\widecheck{x}$$\widehat{x}$ (b) Different normals of optimal
cuts for the directional normalization: the norm of optimal normals can tend
to infinity
Figure 6: Examples of results obtained when using directional normalization.
The above normalizations have been presented several times in the literature
and are applicable in the general case. However, certain problems can admit
normalizations particularly adapted to their structure. One can find such
normalizations for example in the separation problems of disjunctive
programming balas2002lift (2). In addition, the unsplittable flow problem,
which will be used in our experimental study, admits a normalization that
seems natural and which be presented in Section 7.1.
## 5 A new approach for the Fenchel sub-problem
In this section, we present a new approach to solve the separation problem of
the Fenchel decomposition when the directional normalization presented in
Section 4 is used.
### 5.1 Presentation of the method
The proposed method is described in Algorithm 1 and illustrated in Figure 7.
Note that the cut associated with a directional normalization toward
$\widehat{x}$ is the same for $\widecheck{x}$ than for any other point of the
segment $(\widecheck{x},\widehat{x})$ not belonging to $Q_{2}$. The underlying
idea is to compute intermediate cuts using an alternative normalization. By
projecting a point $x^{\prime}$ (initially equal to $\widecheck{x}$) onto
these intermediate cuts, the algorithm gradually shifts the point $x^{\prime}$
along the segment $(\widecheck{x},\widehat{x})$ in the direction of
$\widehat{x}$. Once $x^{\prime}$ reaches the frontier of $Q_{2}$ the procedure
stops. Indeed, the last cut generated is the one associated with the
directional normalization toward $\widehat{x}$ since it contains the
intersection between the segment $(\widecheck{x},\widehat{x})$ and the
frontier of $Q_{2}$.
Algorithm 1 New approach for solving the Fenchel sub-problem
1:a polyhedron $Q_{2}$, a point $\widecheck{x}$ to separate from $Q_{2}$, a
point $\widehat{x}$ belonging to $Q_{2}$
2:a cut $C$ separating $\widecheck{x}$ from $Q_{2}$ and containing the
intersection point the frontier of $Q_{2}$ and the segment
$(\widecheck{x},\widehat{x})$, a list $LS$ of vertices of $Q_{2}$ satisfying
the cut $C$ to equality
3:Set $x^{\prime}$ equal to $\widecheck{x}$
4:while $x^{\prime}\notin Q_{2}$ do
5: $C$, $LS$ = Secondary_separation($Q_{2}$, $x^{\prime}$)
6: Set $x^{\prime}$ equal to the intersection point between the segment
$(\widecheck{x}$, $\widehat{x})$ and the cut $C$
7:return $C$, $LS$
$\widecheck{x}$$\widehat{x}$ (a) The goal is to separate the blue point using
a directional normalization with the red point as an interior point
$\widecheck{x}$$\widehat{x}$ (b) The separation problem is solved with a
secondary normalization (here the norm $\|.\|_{2}$)
$\widecheck{x}$$\widehat{x}$ (c) The point to be separated is projected on the
generated section
$\widecheck{x}$$\widehat{x}$ (d) The separation problem is solved again with
the secondary normalization (here the norm $\|.\|_{2}$)
$\widecheck{x}$$\widehat{x}$ (e) Once the point to be separated is projected
on the border of the polyhedron, the procedure stops.
Figure 7: Example of solving the Fenchel sub-problem for directional
normalization using a secondary normalization
The advantage of this iterative approach is that the linear separation program
($S$) is never directly solved with the directional normalization, which often
presents numerical instabilities. Indeed, for this normalization, the
separation problem ($S$) seeks a face of the polyhedron $Q_{2}$ intersecting
the segment $(\widecheck{x},\widehat{x})$. If, for example, this segment
intersected a facet of $Q_{2}$ by forming an angle close to $0$ with it then a
small error on the parameters of the segment or of the facet can induce a
large error on the position of the point of intersection. For this reason, the
problem ($S$) is sometimes too numerically unstable to be solved directly. The
new method presented above yields an alternative way of computing the cut
associated with the directional normalization without ever solving the
numerically unstable linear program of the direct method. Unfortunately, note
that this iterative algorithm requires the computation of the intersection
between the segment $(\widecheck{x},\widehat{x})$ and the cuts returned by the
secondary separation problem. If the segment is almost parallel to one of
these cuts, the computation of this point of intersection may still be
numerically unstable. However, this is only an intermediate point not returned
by the separation oracle. The results of the method are a cut and vertices
which are computed with a secondary normalization. In practice, this seems
sufficient as no numerical instabilities were found during our thorough
numerical campaign.
Our experimental section yields insights into the practical performance of
this new approach for the separation oracle. From a theoretical standpoint, we
provide proofs in A that the method converges in a finite number of iterations
in two cases: 1) when the secondary normalization always generates facets of
$Q_{2}$, and 2) when the secondary normalization is $\|\pi\|\leq 1$ for any
norm. These two cases cover most normalizations used in the literature
including all those mentioned in Section 4.
## 6 Coupling the Fenchel and Dantzig-Wolfe decompositions
In this section, we present a decomposition method that integrates Dantzig-
Wolfe and Fenchel decompositions. In our experimental campaign, we show that
the new method presents a much superior performance than Fenchel decomposition
alone, and is competitive against Dantzig-Wolfe on non-degenerate problems.
For degenerate problems, again the proposed method shows a superior perormance
to a classical Dantzig-Wolfe decomposition.
### 6.1 Presentation of the method
When generating a Fenchel cut, the primal variables of the separation sub-
problem ($S$) are the coefficients of the generated cut and the active
constraints correspond to the vertices of the separated polyhedron $Q_{2}$
verifying this cut at equality. Thus, the Fenchel sub-problem generates both
valid cuts for the polyhedron $Q_{2}$ and vertices of this polyhedron.
The main idea of the new decomposition method is to use two master problems
operating in tandem. The first corresponds to the Fenchel formulation ($F$) in
which the generated Fenchel cuts are added to improve an outer approximation
of $Q_{2}$. The second one is the Dantzig-Wolfe formulation ($DW$) in which
the vertices generated in the separation sub-problem are added to grow an
inner approximation of $Q_{2}$. One of the key points of the method is that
points $\widecheck{x}$ provided by the Fenchel master problem are separated
using directional normalization. This normalization requires the knowledge of
a point in the polyhedron $Q_{2}$, for which we use the solution $\widehat{x}$
computed by the Dantzig-Wolfe master problem.
The steps of the algorithm, illustrated in Figure 8, are as follows:
1. 1.
Initialize an inner approximation $\widehat{Q}_{2}$ of the polyhedron $Q_{2}$
as in the Dantzig-Wolfe decomposition and an outer approximation
$\widecheck{Q}_{2}$ as in the Fenchel decomposition.
2. 2.
Optimize over $LR_{1}\cap\widecheck{Q}_{2}$ using a Fenchel master problem: a
solution $\widecheck{x}$ is obtained whose value is an upper bound of the
problem.
3. 3.
Optimize over $LR_{1}\cap\widehat{Q}_{2}$ using a Dantzig-Wolfe master
problem: a solution $\widehat{x}$ is obtained whose value is a lower bound of
the problem.
4. 4.
If the two bounds are equal: end of the algorithm
5. 5.
With the Fenchel separation problem ($S$), separate the point $\widecheck{x}$
from $Q_{2}$ using directional normalization with $\widehat{x}$ as the
interior point: a cut is obtained as well as vertices of $Q_{2}$.
6. 6.
Add the cut to the outer approximation $\widecheck{Q}_{2}$ and the vertices to
the inner approximation $\widehat{Q}_{2}$. Then, go to Step 2.
(a) The outer approximation $\widecheck{Q}_{2}$ and the inner approximation
$\widehat{Q}_{2}$ are computed. The solutions $\widecheck{x}$ and
$\widehat{x}$ are computed by solving over the intersection of $LR_{1}$ and
the corresponding approximation.
(b) The solution $\widecheck{x}$ is separated from $Q_{2}$ using the
directional normalization toward $\widehat{x}$. This separation problem
returns a cut and vertices of the polyhedron $Q_{2}$.
(c) The cut is added to the outer approximation $\widecheck{Q}_{2}$ and the
vertices to the inner approximation $\widehat{Q}_{2}$.
(d) After enough iterations, the value of the solutions given by the two
approximations coincide and the method stops.
Figure 8: Illustration of the Dantzig-Wolfe-Fenchel decomposition
Although this new decomposition method uses ideas taken from both the Dantzig-
Wolfe and the Fenchel decompositions, the way in which they operate yields a
few remarkable observations and interpretations that are worth describing:
* •
Intuitively, the inner and outer approximations constructed disagree on the
location of the boundary of $Q_{2}$ on the segment
$(\widecheck{x},\widehat{x})$. The separation problem finds the exact position
of the frontier and gives information to the two approximations so that they
can approximate exactly this part of the frontier.
* •
Another way of looking at the method is to say that it gradually improves the
point $\widehat{x}$. At each iteration, the algorithm tests a direction of
potential improvement $\widecheck{x}-\widehat{x}$. The method then finds
either new vertices of $Q_{2}$ allowing to improve $\widehat{x}$ or a facet of
$Q_{2}$ passing through $\widehat{x}$ proving that it is not possible to
improve $\widehat{x}$ in this direction. This interaction between an exterior
point and an interior point of $Q_{2}$ is reminiscent of the in-out separation
proposed for the Benders decomposition ben2007acceleration (6).
* •
Compared to a classic Fenchel decomposition, this method concentrates its cut
generation around the point $\widehat{x}$. It thus refines the knowledge of
the polyhedron $Q_{2}$ around this point. On the other hand, a classical
Fenchel decomposition might spend multiple iterations searching for cuts in
regions of the solution space that end up being far from the optimum.
* •
Compared to a classic Dantzig-Wolfe method, this method devotes more time to
the resolution of its sub-problem which allows it to generate a greater number
of vertices to add to the master problem.
### 6.2 Degeneracy
The Dantzig-Wolfe decomposition is known to present convergence issues when
its master problem is highly degenerate. Our experiments reveal that the
method presented in this section does not suffer from the same issues. In the
following, we give a partial theoretical explanation of the absence of
degeneracy issues in the new method.
A linear program is said to be degenerate when it has multiple dual optima. We
have seen in Section 3.1 that each dual solution of the Dantzig-Wolfe master
problem implies a bound on the value of its objective function $cx\leq c_{0}$
that certifies the optimality of its current primal solution on the solution
set $LR_{1}\cap\widehat{Q}_{2}$. Thus, in order to improve the solution of the
master problem, one needs to be able to separate each of these dual solutions
from the polyhedron $LR_{1}\cap Q_{2}$. However, the sub-problem of the
Dantzig-Wolfe decomposition may only separate one such dual vector which
explains why the Dantzig-Wolfe decomposition is so affected by degeneracy.
The new proposed Dantzig-Wolfe-Fenchel decomposition seems to be unaffected by
the degeneracy of its Dantzig-Wolfe master problem. One possible explanation
of this behavior is supported in the following observation and proposition.
First, let us remark that the new decomposition neither computes nor uses any
dual information, therefore showing no sensitivity to the quality of the
duals. Second:
###### Proposition 1
At each iteration, the sub-problem invalidates either all the dual solutions
of the Dantzig-Wolfe master program or none of them.
###### Proof
The sub-problem finds the farthest point in $Q_{2}$ along the segment
$(\widehat{x},\widecheck{x})$. On the one hand, if this farthest point
coincides with $\widehat{x}$ then it does not invalidate any of the bounds
implied by the dual solutions because $\widehat{x}$ belongs to
$LR_{1}\cap\widehat{Q}_{2}$. On the other hand, suppose the farthest point
$x^{*}$ does not coincide with $\widehat{x}$. First, by construction, $x^{*}$
belongs to $Q_{2}$ but also to $LR_{1}$ because it is a convex combination of
$\widehat{x}$ and $\widecheck{x}$, both belonging to $LR_{1}$. Second,
assuming that $\widecheck{x}$ has a strictly better objective value than
$\widehat{x}$ (which is always the case except when the method is about to
terminate) then $x^{*}$ also has a strictly better objective value than
$\widehat{x}$. Thus, it must invalidate all the bounds implied by the dual
solutions of the Dantzig-Wolfe master problem (remember that these bounds
certify that no better solution than $\widehat{x}$ exists). ∎
Therefore, whether the Dantzig-Wolfe master problem admits multiple dual
optima or not does not influence the algorithm’s capacity to find a strictly
improving point.
Although we presented all the decomposition methods as if only one polyhedron
was decomposed at once (e.g. a block of a block diagonal matrix), in practice
several polyhedra are decomposed at the same time (e.g. all the blocks of a
block diagonal matrix). In this case, Proposition 1 does not hold. However,
the ideas discussed in its proof may still impact positively the practical
computations. In any case, the decomposition method still does not use any
dual information which renders it oblivious to the number of dual solutions of
its Dantzig-Wolfe master problem.
## 7 Application to the unsplittable flow problem
In the unsplittable flow problem (UFP), one is given a weighted directed graph
$\mathcal{G}=(V,A,c)$ where $c_{a}$ is the capacity of the arc $a$ for every
$a\in A$. We are also given a family $K$ of commodities, each composed of an
origin $o(k)$, a destination $d(k)$, and a demand $D_{k}$, for every $k\in K$.
Each commodity has to be routed from its origin to its destination through an
unique path. We consider the problem where the capacity constraints are soft,
meaning that they can be violated at a certain unit penalty. The objective is
to design routing paths for every commodity on the network so as to minimize
the sum of the violations of the arcs’ capacities.
The UFP is an extensively studied NP-hard variant of the classic maximum-flow
problem. It has multiple applications, as for instance in telecommunication
networks (_e.g._ optical networks, telecommunication satellites
rivano2002lightpath (18, 31)), and logistics farvolden1993primal (23). The
early work of belaidouni2007minimum (5) studied the UFP from a polyhedral
perspective, proposing cutting planes to strengthen the linear relaxation of a
three-index model that uses variables $x_{ijk}$ for every arc $(i,j)$ in the
network and every commodity $k$ to be transported. park2003integer (41) also
strengthen the linear relaxation by applying the Dantzig-Wolfe decomposition
to the capacity constraints of the UFP. The resulting relaxation is as strong
as if all the inequalities valid for the capacity constraints of the problem
were added. Thus, faster decomposition methods able to compute the Dantzig-
Wolfe linear relaxation could yield improvements in the resolution of the UFP.
We will consider an arc-path formulation where the meaning of the variables is
the following:
* •
$x_{pk}$ indicates whether commodity $k$ uses path $p$ to push its flow,
* •
$\Delta_{a}$ represents the overflow on arc $a$.
In addition to the decision variables, we also denote, for a given commodity
$k\in K$, $P_{k}$ the set of all $o(k)$-$d(k)$-paths in $G$. For every $p\in
P_{k}$ and arc $a\in A$, we define a constant $\alpha_{ap}$ that takes the
value 1 iff $p$ uses the arc $a$. The path formulation of the UFP is the
following:
$\displaystyle\min_{x_{pk},\Delta_{a}}$ $\displaystyle\sum_{a\in A}\Delta_{a}$
(9a) subject to $\displaystyle\sum_{p\in P_{k}}x_{pk}=1$ $\displaystyle\forall
k\in K,$ (9b) $\displaystyle\sum_{k\in K}\sum_{p\in P_{k},a\in
p}x_{pk}D_{k}\leq c_{a}+\Delta_{a}$ $\displaystyle\forall a\in A,$ (9c)
$\displaystyle x_{pk}\in\\{0,1\\},~{}\Delta_{a}\in\mathbb{R}^{+}$
$\displaystyle\forall p\in P_{k},~{}\forall k\in K,~{}\forall a\in A$ (9d)
The objective function minimizes the sum of the overflows on the arcs.
Equation (9b) ensures that exactly one path is chosen for each commodity.
Equation (9c) corresponds to the soft capacity constraints. It ensures that
any overflow on an arc $a$ is recorded on the corresponding variable
$\Delta_{a}$. The fact that $x_{pk}\in\\{0,1\\}$ ensures that the flow is
unsplittable.
The polyhedron associated with the capacity constraints does not have the
integrality property and its relaxation can thus be tightened with any of the
previously discussed decomposition methods. If we denote $f_{a}^{k}=\sum_{p\in
P_{k},a\in p}x_{pk}$, this polyhedron can be written as follows:
$\left\\{f_{a}^{1},...,f_{a}^{k}\in[0,1],o_{a}\in\mathbb{R}^{+}\bigg{|}\sum_{k\in
K}f_{a}^{k}D^{k}\leq c_{a}+o_{a}\right\\}.$
Studies have been carried out on the structure, the cut selection, and the
strengthening of the linear relaxation of this type of polyhedron by
Marchand19990 (36) as well as in the more general framework of linear programs
in mixed variables dash2011mixed (21, 24, 16). Moreover, optimization methods
on this polyhedron have been studied by buther2012reducing (14, 34, 46, 28,
35).
In the following, we present a specialized normalization for this polyhedron.
This normalization guarantees the generation of facets and will be used in our
implementation of the Fenchel decomposition and as secondary normalization in
the new procedure to solve the Fenchel sub-problem presented in Section 4.
Then, we describe our implementation of the oracle that optimizes a linear
function on the polyhedron associated with the capacity constraints of the
UFP.
### 7.1 Natural normalization for unsplittable flows
Let $\widecheck{f}=(\widecheck{f}^{k})_{k\in K}\in[0,1]^{|K|}$ be a flow
distribution for each commodity which induces an overflow $\widecheck{o}$ on a
given arc. In this section, we assume that the arc is fixed and will therefore
drop the arc index for the sake of simplicity. In the context of unsplittable
flows, vertices of the polyhedron $Q_{2}$ correspond to commodity patterns
which will be indexed by a superscript $g$. A naturally occurring question is:
how does this distribution break down into a combination of commodity patterns
inducing a minimum capacity overflow? This question can be solved using the
following linear program:
$\displaystyle(D)\quad$ $\displaystyle\min_{\lambda^{g},z}$ $\displaystyle z$
subject to $\displaystyle\sum_{g\in G}\lambda^{g}f^{g}=\widecheck{f}$
$\displaystyle\sum_{g\in G}\lambda^{g}o^{g}=\widecheck{o}+z$
$\displaystyle\sum_{g\in G}\lambda^{g}=1$
$\displaystyle\lambda^{g}\in\mathbb{R}^{+},~{}z\in\mathbb{R}^{+}\quad\forall
g\in G$
where $\lambda^{g}$ is the coefficient in the decomposition associated with a
commodity pattern $f^{g}$ inducing an overflow $o^{g}$.
Now the dual of this decomposition program is the following program:
$\displaystyle(S)\quad$ $\displaystyle\max_{\pi,\pi_{o}\pi_{0}}$
$\displaystyle\pi\widecheck{f}+\pi_{o}\widecheck{o}-\pi_{0}$ subject to
$\displaystyle\pi f^{g}+\pi_{o}o^{g}\leq\pi_{0},\quad\forall(f^{g},o^{g})\in
Q_{2}$ $\displaystyle\pi_{o}\leq-1$
$\displaystyle\pi,\pi_{0},\pi_{o}\in\mathbb{R}$
This program corresponds exactly to the problem ($S$) of separating the point
$\widecheck{x}=(\widecheck{f_{1}},...,\widecheck{f}_{|K|},\widecheck{o})$ with
a constraint of normalization imposing that the coefficient $\pi_{o}$
associated with the overflow variable satisfies $\pi_{o}\leq-1$. This
normalization constraint is what we will call the natural normalization for
the unsplittable flow problem. This normalization is very close to a
particular case of directional normalization for the direction
$\widehat{x}-\widecheck{x}=(0,...,0,1)$. Just like directional normalization,
the natural normalization guarantees the generation facets because it is
imposed using a single linear constraint.
### 7.2 Knapsack oracle resolution
All the decomposition methods presented in this work assume that there exists
an efficient algorithm capable of optimizing a linear function on the
polyhedron $Q_{2}$. In this section, we detail the problem solved by the
oracle in the context of unsplittable flows.
In the version of the unsplittable flow problem that we are studying, the
capacity constraints do not require that the flow of commodities respect the
capacities $c_{a}$ of the arcs. However, the overflow must be stored in a
variable $o_{a}$. Thus, the polyhedron of variables satisfying the soft
capacity constraint associated with the arc $a$ is written:
$\left\\{(f_{a}^{k}\in\\{0,1\\})_{a\in A,k\in
k},o_{a}\in\mathbb{R}^{+}\Bigg{|}\sum_{k\in K}f_{a}^{k}D^{k}\leq
c_{a}+o_{a}\right\\}.$
The optimization of a linear function whose coefficients are $(\pi^{k})_{k\in
K}$ and $-\pi_{o}$ on this polyhedron can be written as follows:
$\displaystyle(O_{a})\quad$ $\displaystyle\max_{f_{a}^{k},o_{a}}$
$\displaystyle\sum_{k\in K}\pi^{k}f_{a}^{k}-\pi_{o}o_{a}$ subject to
$\displaystyle\sum_{k\in K}f_{a}^{k}D^{k}\leq c_{a}+o_{a}$ $\displaystyle
f_{a}^{k}\in\\{0,1\\},~{}o_{a}\in\mathbb{R}^{+}$
This problem can be solved as a sequence of two 0-1 knapsack problems using a
case disjunction. This method was presented by buther2012reducing (14) and is
recalled in B. In our experiments, we use the MINKNAP algorithm proposed by
pisinger1997minimal (43) to solve the two associated knapsack problems.
## 8 Experimental study
In this section, we present an experimental comparison of different
decomposition methods. The datasets and code used in this section can be
accessed at https://github.com/TwistedNerves/decomposition_paper_code. The
code was written in Python 3 and the experiments carried on an Intel Core
i9-9900K 3.60 GHz $\times 16$ cores CPU, 60 Gbit of RAM, running Ubuntu 20.10.
### 8.1 Datasets
An instance of the unsplittable flow problem is composed of a graph and a list
of commodities. The method used to create instances in our experiments is the
one presented in lamothe2021randomized (32). All the graphs used are strongly
connected random graphs. To create demands for the commodities,
lamothe2021randomized (32) used two formulas. In this work, we use the formula
that creates mainly commodities with large demands because it tends to create
instances that are harder to solve. Moreover, in each instance, all the
commodities can be unsplittably routed without exceeding the arc capacities.
Therefore, the lower bound given by the linear relaxation is optimal. In order
to create an optimality gap in the instances, we slightly modify the
capacities of some arcs as follows a number of times equal to 100 times the
number of nodes:
* •
Randomly select the origin of a commodity.
* •
Randomly select two arcs coming out of this origin.
* •
Add 1 to the capacity of one arc and subtract 1 from the ability of the other
arc.
Because of the way instances are created, all outgoing arcs from each origin
node are saturated in the solutions without overflow while the other arcs are
often non-saturated. Therefore, in most cases, transferring some capacity
between outgoing arcs of origins does not change the value of the linear
relaxation. On the other hand, this transfer of capacity can have an impact on
the value of the best unsplittable solution. Indeed, there is no longer
necessarily a combination of commodities whose sum of demands is exactly equal
to the capacity of each arc. In this case, the best unsplittable solution has
a non-zero overflow.
Another change made to the instances is that the commodities have only access
to a restricted set of paths to push their flow. The restricted set of paths
of a commodity is chosen to be the k-shortest paths from the origin to the
destination of the commodity with k = 4. Because this study explores the
strengthening of the linear relaxation of the unsplittable flow problem
through its capacity constraints, this modification should not change the
relative behavior of the tested algorithms but does make the instances much
simpler to solve which enables the testing of the different algorithms on
larger instances.
##### The datasets
Three different datasets are used during the experiments in which ten
instances are generated for each value of the varying parameter.
* •
Low maximum demand dataset: this dataset considers strongly connected random
graphs from 50 nodes to 145 nodes. The maximum commodity demand is set at
$\widecheck{d}_{max}=$ 100 and the arc capacity at 1000. This choice of
maximum demand implies that a large number of commodities can pass through
each arch. However, in our tests, the optimal solution for these instances
often does not contain overflow. Therefore, these instances do not contain an
optimality gap. We hypothesize that the large number of commodities allows
them to rearrange themselves to exactly fill the capacity of each arc.
* •
High maximum demand dataset: this dataset considers strongly connected random
graphs from 145 nodes to 1000 nodes. The maximum commodity demand is set at
$\widecheck{d}_{max}=$ 1000 and the arc capacity at 1000. Because of this
maximum demand choice, these instances contain only a small number of
commodities. However, they generally have an optimality gap which allows us to
study the evolution of the lower bounds given by the algorithms.
* •
Size of capacities dataset: this dataset considers strongly connected random
graphs of 70 nodes. The maximum demand of the $\widecheck{d}_{max}$
commodities is fixed at $1/10$ of the common capacity of the arcs which varies
from 100 to 100,000. The knapsack problem is known to have algorithms that are
pseudo-polynomial in the capacity of the knapsack. One such algorithm is the
MINKNAP algorithm we use. In the case of unsplittable flows, the capacity of
the knapsack corresponds to the capacity of the arcs. The instances of this
dataset all have the same structure (same graph size, same size of commodities
relative to the capacity of the arcs) but varying arc capacities. This impacts
the resolution time of the MINKNAP algorithm.
### 8.2 Comparison of Dantzig-Wolfe variants
A large number of works have proposed variations and improvements to Dantzig-
Wolfe decomposition. Indeed, the method presented in Section 3.1 sometimes
suffers from a slow convergence which has been associated with the following
observations pessoa2013out (42):
* •
Dual oscillations: the dual variables $\pi$ used to generate the vertices of
$Q_{2}$ perform large oscillations and do not converge monotonically toward
their optimal value.
* •
The tailing-off effect: during the last iterations, the space of the dual
solutions is only marginally reduced and the dual bound progresses very
slowly.
* •
Degenerate primal and equivalent dual solutions: The master problem ($DW$) is
regularly degenerate because it has several dual optimal solutions. The method
iterates between equivalent dual solutions without making progress on the
value of the objective function.
In order to overcome these difficulties, stabilization methods for the dual
variables have been considered. These methods can be classified into three
categories pessoa2013out (42):
* •
Penalization: the penalization methods are best interpreted by considering the
dual of the problem ($DW$). In order to stabilize the dual variables $\pi$, a
penalty $f(\|\pi-\bar{\pi}\|)$ is added to the objective function of the dual.
In this penalty, $f()$ is an increasing function, which pushes $\pi$ to stay
close to a value $\bar{\pi}$ which evolves slowly during the algorithm.
Typically, $\bar{\pi}$ is one of the values taken by the dual variables during
the previous iterations. A widely studied special case is to penalize
proportionally to $\|\pi-\bar{\pi}\|_{2}^{2}$, which is done in the Bundle
methods briant2008comparison (13).
* •
Smoothing: in smoothing methods, the dual variables $\pi$ of the problem
($DW$) are not used directly in the sub-problem in order to generate new
vertices of $Q_{2}$. We note, for the iteration $j$ of column generation,
$\pi_{j}$ the values of the variables resulting from the dual of the problem
($DW$) and $\bar{\pi}_{j}$ the values used in the sub-problem. A smoothing
method proposed by neame2000nonsmooth (39) uses the following formula:
$\bar{\pi}_{j}=\alpha\bar{\pi}_{j-1}+(1-\alpha)\pi_{j}$. This method amounts
to adding an momentum effect to the dual variables. Another method, proposed
by wentges1997weighted (44), performs a convex combination with a fixed dual
value $\bar{\pi}$, i.e. $\bar{\pi}_{j}=\alpha\bar{\pi}+(1-\alpha)\pi_{j}$.
* •
Centralization: The idea of centralization methods is that it is more
efficient to use in the sub-problem dual values located inside the dual
polyhedron rather than on an extreme vertex of the dual polyhedron. On the
other hand, such interior points are more expensive to compute than extreme
points. The interior point used in the Primal-Dual Column Generation
gondzio2013new (26) is obtained by approximately solving the problem ($DW$) by
an interior point method. Another classic point is the analytical center used
in the analytical center cutting plane method goffin2002convex (25).
In order to have a suitable comparison for the decomposition methods we
proposed in this work, we experimentally compared the following three
variations of Dantzig-Wolfe decomposition:
DW: Dantzig-Wolfe decomposition method. No stabilization of the dual variables
is used. The lower bounds are computed using the dual variables and the value
of the solution of the knapsack sub-problems.
DW-momentum: similar to the DW method except that the dual variables are
stabilized by smoothing using the formula of neame2000nonsmooth (39),
$\bar{\pi}_{t}=\alpha\bar{\pi}_{t-1}+(1-\alpha)\pi_{t}$ with the coefficient
$\alpha$ set to $0.8$.
DW-interior-point: similar to the DW method except that the Dantzig-Wolfe
master problem is solved with an interior point method in order to return a
non-optimal but centered solution. To that end, we ask the gurobi (27) solver
to solve the linear program using an interior point method with a precision of
$10^{-3}$ and without using its crossover method. However, the first time the
sub-problem fails to generate a new negative reduced cost variable, the solver
Gurobi is reset to its default settings to ensure an exact computation of the
last reduced costs. With the default parameters, the generation of columns is
no longer stabilized.
These methods are compared in Figure 9. For the rest of the experiments, we
will use the variation based on the interior point solver as it always returns
the best results in our tests.
Figure 9: Compararison of Dantzig-Wolfe decomposition variations, Low maximum
demand dataset, 70 nodes
### 8.3 The decomposition methods studied
In the following, we experimentally compare the following decomposition
methods:
Fenchel: Fenchel decomposition method. The cuts generated are added to the
linear relaxation while the generated vertices are added to a Dantzig-Wolfe
formulation. The Fenchel sub-problem is solved with the natural normalization
presented in Section 7.1. Therefore, the two master problems do not act in
tandem.
DW-Fenchel: method combining the Fenchel and Dantzig-Wolfe decompositions
presented in Section 6, the cuts generated are added to the linear relaxation
while the generated vertices are added to a Dantzig-Wolfe formulation. The
Fenchel sub-problem is solved with directional normalization with the optimal
point of the Dantzig-Wolfe formulation as the interior point. The use of this
normalization couples the two formulations.
DW-Fenchel-iterative: similar to the DW-Fenchel method except that the Fenchel
sub-problem is solved using the iterative method presented in Section 5.
DW-interior-point: Dantzig-Wolfe decomposition introduced in Section 8.2 where
the master problem is solved with an interior point method in order to return
a non-optimal but centered solution.
### 8.4 Algorithms’ parameters
Authorized paths Because the focus of this work is on the capacity constraints
and not on how to generate the paths for each commodity, each commodity is
restrained to a small set of allowed paths. This set is made up of the four
shortest paths between the origin and destination of the commodity as well as
the path used to create the commodity in the method of lamothe2021randomized
(32).
Algorithm termination condition The decomposition methods considered are
stopped when the absolute difference between their bounds is $10^{-3}$.
Pre/post-processings for the sub-problem of Fenchel: Solving directly a
Fenchel subproblem is sometimes too computationally expensive to be integrated
into a decomposition method. However, the resolution time of this subproblem
can be greatly reduced with pre/post-processing steps. Indeed, boccia2008cut
(8) showed that it is possible to solve the Fenchel separation problem by
focusing on a sub-polyhedron of $Q_{2}$ of far lesser dimension which
decreases the computing time. However, the generated cut is not directly valid
for $Q_{2}$ and one must use a lifting procedure to create a cut valid for
$Q_{2}$. These concepts are explained in C. Together, the techniques of
Dimensionality reduction and Lifting induce a drastic decrease in the
resolution time of the Fenchel subproblem. This fact was confirmed by our
experiments and the results we present in our experimental study reflect this
algorithmic choice.
### 8.5 Experimental results
We now present the results of our experimental campaign for the different
decomposition methods presented in this work. In each figure, we display the
evolution of the lower and upper bounds achieved by the algorithms as a
function of the computational time (in seconds). Note that all the displayed
values are not directly the bounds but their deviation from the value of an
optimal solution of the Dantzig-Wolfe reformulation. The plotted curves
represent the average results of the algorithms aggregated on instances using
the same parameters while the confidence intervals at $95\%$ for the mean are
plotted in semi-transparency around the main curve. A problem encountered when
generating these curves is that the algorithms only return bounds at the end
of each of their iterations but these iterations take a variable time for the
same algorithm depending on the instance. It is therefore not possible to
directly aggregate the curves using the points given at the end of each
iteration because they do not correspond to the same computing time. To obtain
points on which we can appropriately average the values, the points defining
the curves are replaced with points sampled every ten seconds by considering
that the bounds evolve linearly between two iterations. The confidence
intervals are created using the statistical method called Bootstrapping with a
number of resamplings equal to 1000. Because the Bootstrapping method is
applied independently for every ten seconds of the curves the resulting
confidence intervals have jitters. These jitters can be interpreted as the
uncertainty on the bound of the confidence intervals due to the Bootstrapping
method.
Figure 10: Legend of the Figures 11 to 17
Figure 11: Low maximum demand dataset, 70 nodes
Figure 12: Low maximum demand dataset, 90 nodes
Figure 13: Low maximum demand dataset, 145 nodes
Figure 14: High maximum demand dataset, 250 nodes
Figure 15: High maximum demand dataset, 400 nodes
Figure 16: Size of commodities dataset, capacity 100
Figure 17: Size of commodities dataset, capacity 10000
Solving the Fenchel sub-problem with a secondary normalization. The new method
of solving the Fenchel sub-problem presented in Section 5 is used in the DW-
Fenchel-iterative method. This method appears to be slightly slower than the
DW-Fenchel method which uses a direct approach for the sub-problem. On the
other hand, the direct approach sometimes fails to solve the sub-problem
because of numerical instabilities which prevent the decomposition method from
converging. For instances with 145 nodes of the Low maximum demand dataset,
this happens every 10 to 20 instances. The 10 instances presented in Figure 13
did not suffer from instability in this set of experiments thus it does not
appear in the figure. However, we were able to identify seeds for which the
instability appears. Unfortunately, these seeds appear to be hardware-
dependent thus researchers trying to reproduce the results will have to find
their own seeds.
Impact of coupling the two master problems using directional normalization.
The DW-Fenchel and DW-Fenchel-iterative methods couple the Dantzig-Wolfe and
Fenchel master problems using a directional normalization in their Fenchel
sub-problem. The impact of this coupling can be studied by comparing these
methods to the Fenchel method whose only difference is to use the natural
normalization of the unsplittable flow problem in its sub-problem. As
illustrated in Figure 13, one notes that all the methods are similar during
the first iterations. However, the Fenchel method stalls rapidly. On the other
hand, this is not the case with the methods using directional normalization
which yield much better results. Our interpretation of this phenomenon is as
follows. There are many equivalent optimal solutions of the linear relaxation
of the unsplittable flow problem. When a cut is generated with the natural
normalization, it only cuts a subset of these solutions, and the new solution
to the Fenchel master problem is in a completely different place in the
solution space. The method then fails to cut all the solutions because of this
large number of symmetries. In contrast, the cuts generated using directional
normalization focus on the optimal solution of the Dantzig-Wolfe master
problem and try to prove its optimality. By focusing on a sub-part of the
solution space, this method avoids the problem of symmetries which improves
its convergence. This hypothesis is supported by the results of a preliminary
study on a variation of the unsplittable flow problem where a path is favored.
Indeed, because of the presence of a privileged path for each commodity, this
variant does not have as many symmetries. In this context, there is a smaller
difference between the methods based on the two normalizations.
Comparison between DW-interior-point and DW-Fenchel-iterative. In the case of
the low maximum demand dataset where the number of commodities is high, the
DW-Fenchel-iterative method behaves a lot better than the Dantzig-Wolfe
methods. We assume that this is because the large number of commodities
implies a greater degeneracy of the master problem which should less bother
the DW-Fenchel methods. Indeed, this degeneracy seems to be the cause of the
rather slow start of the Dantzig-Wolfe methods on these instances. In
contrast, for the high maximum demand dataset, the DW-interior-point and DW-
Fenchel-iterative methods show more similar results. On these instances, the
DW-Fenchel-iterative method shows a faster start of convergence, but slows
down at the end of convergence, in particular for the lower bound.
Impact of capacity size. The knapsack problem is known to have pseudo-
polynomial resolution algorithms in the knapsack capacity such as the MINKNAP
algorithm that we use. In the case of unsplittable flows, this capacity of the
knapsack corresponds to the capacity of the arcs. In Figure 17 and 17, we vary
the capacities of the arcs. Note that the results for capacities of 1000 are
given in Figure 11. This variation in capacities impacts the computation time
of the two methods making them slower. However, the DW-Fenchel-iterative
method is much more impacted because the methods having a Fenchel sub-problem
spend more time in their sub-problem than the Dantzig-Wolfe methods. This
emphasizes the fact that having a fast optimization oracle for the polyhedron
$Q_{2}$ is much more important for the methods based on a Fenchel sub-problem
than those based on a Dantzig-Wolfe sub-problem.
General comments. The new methods presented that couples the Dantzig-Wolfe and
Fenchel decompositions shows very promising results. In particular, they seems
to be far less affected by degeneracy than the Dantzig-Wolfe decomposition and
possess better convergence than the Fenchel decomposition. On the other hand,
they can end up converging slightly less rapidly than the Dantzig-Wolfe
decomposition on instances where degeneracy is not an issue. The new methods
are particularly effective when the optimization oracle ($O$) can be
implemented by a fast algorithm.
## 9 Conclusions
In this work we have revisited Dantzig-Wolfe and Fenchel decompositions for
some hard combinatorial problems with block structures. We have provided
geometrical and intuitive interpretations of several types of normalizations
used in the literature to stabilize the sub-problems associated. This
intuition has fueled the development of a novel methodology capable of
coupling both decomposition approaches acting in tandem via a directional
normalization. We have conducted a thorough computational campaign to
demonstrate the effectiveness of the newly proposed approach for the
unsplittable flow problem. We have observed that on problems suffering from
high degrees of degeneracy, the new approach is superior to its competitors.
Meanwhile, it is also competitive with the classical approaches on the less
degenerate cases. We also proposed a new approach to solve the Fenchel
subproblem with directional normalization by using an alternative
normalization. We provide theoretical guarantees for the finiteness of this
new approach for several classes of alternative normalizations and our
experimental campaign revealed that it presents far less numerical
instabilities.
A likely lead for future research will therefore be to investigate the
performance of this new method in different contexts than the unsplittable
flow problems. Moreover, one of the central points of this new method is the
use of directional normalization in the Fenchel sub-problem. It would be
interesting to use this normalization inside other decomposition methods.
## References
* (1) Pasquale Avella, Maurizio Boccia and Igor Vasilyev “A computational study of exact knapsack separation for the generalized assignment problem” In _Computational Optimization and Applications_ 45.3 Springer, 2010, pp. 543–555 DOI: 10.1007/s10589-008-9183-8
* (2) Egon Balas and Michael Perregaard “Lift-and-project for mixed 0–1 programming: recent progress” In _Discrete Applied Mathematics_ 123.1-3 Elsevier, 2002, pp. 129–154 DOI: 10.1016/S0166-218X(01)00340-7
* (3) Cynthia Barnhart et al. “Branch-and-Price: Column Generation for Solving Huge Integer Programs” In _Operations Research_ 46.3, 1998, pp. 316–329 DOI: 10.1287/opre.46.3.316
* (4) Eric Beier, Saravanan Venkatachalam, Luca Corolli and Lewis Ntaimo “Stage-and scenario-wise fenchel decomposition for stochastic mixed 0-1 programs with special structure” In _Computers & Operations Research_ 59 Elsevier, 2015, pp. 94–103 DOI: https://doi.org/10.1016/j.cor.2014.12.011
* (5) Meriema Belaidouni and Walid Ben-Ameur “On the minimum cost multiple-source unsplittable flow problem” In _RAIRO-Operations Research_ 41.3 EDP Sciences, 2007, pp. 253–273 DOI: 10.1051/ro:2007023
* (6) Walid Ben-Ameur and José Neto “Acceleration of cutting-plane and column generation algorithms: Applications to network design” In _Networks: An International Journal_ 49.1 Wiley Online Library, 2007, pp. 3–17 DOI: 10.1002/net.20137
* (7) JF Benders “Partitioning procedures for solving mixed-variables programming problems.” In _Numerische Mathematik_ 4, 1962, pp. 238–252 DOI: 10.1007/BF01386316
* (8) Maurizio Boccia, Antonio Sforza, Claudio Sterle and Igor Vasilyev “A cut and branch approach for the capacitated p-median problem based on Fenchel cutting planes” In _Journal of mathematical modelling and algorithms_ 7.1 Springer, 2008, pp. 43–58 DOI: 10.1007/s10852-007-9074-5
* (9) Pierre Bonami “Etude et mise en œuvre d’approches polyédriques pour la résolution de programmes en nombres entiers ou mixtes généraux”, 2003
* (10) Karl Heinz Borgwardt “The Simplex Method: a probabilistic analysis” Springer Science & Business Media, 2012 DOI: /10.1007/978-3-642-61578-8
* (11) E Andrew Boyd “Generating Fenchel cutting planes for knapsack polyhedra” In _SIAM Journal on Optimization_ 3.4 SIAM, 1993, pp. 734–750 DOI: 10.1137/0803038
* (12) E Andrew Boyd “On the convergence of Fenchel cutting planes in mixed-integer programming” In _SIAM Journal on Optimization_ 5.2 SIAM, 1995, pp. 421–435 DOI: 10.1137/0805021
* (13) Olivier Briant et al. “Comparison of bundle and classical column generation” In _Mathematical programming_ 113.2 Springer, 2008, pp. 299–344 DOI: 10.1007/s10107-006-0079-z
* (14) Marcel Büther and Dirk Briskorn “Reducing the 0-1 knapsack problem with a single continuous variable to the standard 0-1 knapsack problem” In _International Journal of Operations Research and Information Systems (IJORIS)_ 3.1 IGI Global, 2012, pp. 1–12 DOI: 10.4018/joris.2012010101
* (15) Liang Chen, Wei-Kun Chen, Mu-Ming Yang and Yu-Hong Dai “An exact separation algorithm for unsplittable flow capacitated network design arc-set polyhedron” In _Journal of Global Optimization_ Springer, 2021, pp. 1–31 DOI: 10.1007/s10898-020-00967-z
* (16) Vašek Chvátal, William Cook and Daniel Espinoza “Local cuts for mixed-integer programming” In _Mathematical Programming Computation_ 5.2 Springer, 2013, pp. 171–200 DOI: 10.1007/s12532-013-0052-9
* (17) Michele Conforti and Laurence A Wolsey ““Facet” separation with one linear program” In _Mathematical Programming_ 178.1 Springer, 2019, pp. 361–380 DOI: 10.1007/s10107-018-1299-8
* (18) D. Coudert and H. Rivano “Lightpath assignment for multifibers WDM networks with wavelength translators” In _Global Telecommunications Conference, 2002. GLOBECOM ’02. IEEE_ 3, 2002, pp. 2686–2690 vol.3 DOI: 10.1109/GLOCOM.2002.1189117
* (19) George B Dantzig “Maximization of a linear function of variables subject to linear inequalities” In _Activity analysis of production and allocation_ 13, 1951, pp. 339–347
* (20) George B Dantzig and Philip Wolfe “Decomposition principle for linear programs” In _Operations research_ 8.1 INFORMS, 1960, pp. 101–111 DOI: 10.1287/opre.8.1.101
* (21) Sanjeeb Dash “Mixed integer rounding cuts and master group polyhedra” In _Combinatorial Optimization_ IOS Press, 2011, pp. 1–32 DOI: 10.3233/978-1-60750-718-5-1
* (22) Guy Desaulniers, Jacques Desrosiers and Marius M Solomon “Column generation” Springer Science & Business Media, 2006 DOI: 10.1007/b135457
* (23) Judith M Farvolden, Warren B Powell and Irvin J Lustig “A primal partitioning solution for the arc-chain formulation of a multicommodity network flow problem” In _Operations Research_ 41.4 INFORMS, 1993, pp. 669–693 DOI: 10.1287/opre.41.4.669
* (24) Ricardo Fukasawa and Marcos Goycoolea “On the exact separation of mixed integer knapsack cuts” In _Mathematical programming_ 128.1 Springer, 2011, pp. 19–41 DOI: 10.1007/s10107-009-0284-7
* (25) Jean-Louis Goffin and Jean-Philippe Vial “Convex nondifferentiable optimization: A survey focused on the analytic center cutting plane method” In _Optimization methods and software_ 17.5 Taylor & Francis, 2002, pp. 805–867 DOI: 10.1080/1055678021000060829a
* (26) Jacek Gondzio, Pablo González-Brevis and Pedro Munari “New developments in the primal–dual column generation technique” In _European Journal of Operational Research_ 224.1 Elsevier, 2013, pp. 41–51 DOI: 10.1016/j.ejor.2012.07.024
* (27) LLC Gurobi Optimization “Gurobi Optimizer Reference Manual”, 2020 URL: http://www.gurobi.com
* (28) Yichao He et al. “Encoding transformation-based differential evolution algorithm for solving knapsack problem with single continuous variable” In _Swarm and Evolutionary Computation_ 50 Elsevier, 2019, pp. 100507 DOI: 10.1016/j.swevo.2019.03.002
* (29) Konstantinos Kaparis and Adam N Letchford “Separation algorithms for 0-1 knapsack polytopes” In _Mathematical programming_ 124.1 Springer, 2010, pp. 69–91 DOI: 10.1007/s10107-010-0359-5
* (30) L.G. Khachiyan “Polynomial algorithms in linear programming” In _USSR Computational Mathematics and Mathematical Physics_ 20.1, 1980, pp. 53–72 DOI: 10.1016/0041-5553(80)90061-0
* (31) François Lamothe et al. “Dynamic unsplittable flows with path-change penalties: New formulations and solution schemes for large instances” In _Computers & Operations Research_ 152, 2023, pp. 106–154 DOI: 10.1016/j.cor.2023.106154
* (32) François Lamothe et al. “Randomized rounding algorithms for large scale unsplittable flow problems” In _Journal of Heuristics_ , 2021 DOI: 10.1007/s10732-021-09478-w
* (33) AH Land and AG Doig “An Automatic Method of Solving Discrete Programming Problems” In _Econometrica_ 28.3, 1960, pp. 497–520 DOI: 10.1007/978-3-540-68279-0_5
* (34) Geng Lin, Wenxing Zhu and M Montaz Ali “An exact algorithm for the 0–1 linear knapsack problem with a single continuous variable” In _Journal of global optimization_ 50.4 Springer, 2011, pp. 657–673 DOI: 10.1007/s10898-010-9642-5
* (35) Hongtao Liu “An exact algorithm for the biobjective 0-1 linear knapsack problem with a single continuous variable” In _2017 18th International Conference on Parallel and Distributed Computing, Applications and Technologies (PDCAT)_ , 2017, pp. 81–85 IEEE DOI: 10.1109/PDCAT.2017.00022
* (36) Hugues Marchand and Laurence A Wolsey “The 0-1 knapsack problem with a single continuous variable” In _Mathematical Programming_ 85.1 Springer, 1999, pp. 15–33 DOI: 10.1007/s101070050044
* (37) Jiří Matoušek and Bernd Gärtner “Understanding and using linear programming” Springer, 2007 DOI: 10.1007/978-3-540-30717-4
* (38) Sanjay Mehrotra “On the implementation of a primal-dual interior point method” In _SIAM Journal on optimization_ 2.4 SIAM, 1992, pp. 575–601 DOI: 10.1137/0802028
* (39) Philip James Neame “Nonsmooth dual methods in integer programming” University of Melbourne, Department of MathematicsStatistics, 2000
* (40) Lewis Ntaimo “Fenchel decomposition for stochastic mixed-integer programming” In _Journal of Global Optimization_ 55.1 Springer, 2013, pp. 141–163 DOI: 10.1007/s10898-011-9817-8
* (41) Sungsoo Park, Deokseong Kim and Kyungsik Lee “An integer programming approach to the path selection problems” In _Proceedings of the International Network Optimization Conference INOC, Evry-Paris, France_ , 2003, pp. 448–453
* (42) Artur Pessoa, Ruslan Sadykov, Eduardo Uchoa and Francois Vanderbeck “In-out separation and column generation stabilization by dual price smoothing” In _International Symposium on Experimental Algorithms_ , 2013, pp. 354–365 Springer DOI: 10.1007/978-3-642-38527-8_31
* (43) David Pisinger “A minimal algorithm for the 0-1 knapsack problem” In _Operations Research_ 45.5 INFORMS, 1997, pp. 758–767 DOI: 10.1287/opre.45.5.758
* (44) Paul Wentges “Weighted Dantzig-Wolfe decomposition for linear mixed-integer programming” In _International Transactions in Operational Research_ 4.2 Elsevier, 1997, pp. 151–162 DOI: 10.1016/S0969-6016(97)00001-4
* (45) Laurence A Wolsey and George L Nemhauser “Integer and combinatorial optimization” John Wiley & Sons, 1999 DOI: 10.1002/9781118627372
* (46) Chenxia Zhao and Xianyue Li “Approximation algorithms on 0–1 linear knapsack problem with a single continuous variable” In _Journal of Combinatorial optimization_ 28.4 Springer, 2014, pp. 910–916 DOI: 10.1007/s10878-012-9579-3
## Appendix A Proof of the finite convergence for the iterative resolution of
the Fenchel sub-problem
In this appendix, we give proofs of the finite convergence for the method
presented in Section 5 for two types of secondary normalizations. First, when
the secondary normalization guarantees that a facet of $Q_{2}$ will be
generated by the linear program. Second, when the secondary normalization is
$\|\pi\|\leq 1$ where $\|.\|$ is any norm.
### A.1 Secondary normalization generating facets
In this section, we are interested in the termination of the proposed method
to solve the Fenchel sub-problem when the secondary normalization guarantees
that a facet of $Q_{2}$ will be generated by the separation linear program.
###### Theorem A.1
If the secondary normalization guarantees the generation of facets then the
method generates a cut associated with a directional normalization in finite
time.
###### Proof
We will show that in the worst case the method ends after the secondary
separation has generated all the facets of $Q_{2}$. For this, we show that
each facet of $Q_{2}$ is generated at most once. As the algorithm progresses,
the point $x^{\prime}$ separated during secondary separations advances along
the segment $(\widecheck{x},\widehat{x})$ in the direction of $\widehat{x}$.
Once a facet of $Q_{2}$ is generated, the point $x^{\prime}$ is projected onto
that facet along the segment $(\widecheck{x},\widehat{x})$. The future points
$x^{\prime}$ will therefore all satisfy the inequality associated with this
facet which can thus no longer be generated by the secondary separation
problem. Since a polyhedron has a finite number of facets, a facet
intersecting the segment $(\widecheck{x},\widehat{x})$ is generated in a
finite number of steps. Once this happens, the method ends after a single call
to the alternate separation problem. Indeed, the procedure places the point
$x^{\prime}$ on the point of intersection between this facet and the segment
$(\widecheck{x},\widehat{x})$. On the next iteration, the secondary separation
problem indicates that the point $x^{\prime}$ belongs to $Q_{2}$ and the
method stops. ∎
### A.2 Secondary normalization using any norm
We now present a proof of convergence when the secondary normalization is
$\|\pi\|\leq 1$ where $\|.\|$ is any norm. This proof uses the Lemmas 1 and 2
presented in boyd1995convergence (12). Previously, we recall the following
properties and notations. In the case of a normalization $\|\pi\|\leq 1$, the
dual problem of the separation problem is: find the point of $Q_{2}$
minimizing the distance $\|x-\widecheck{x}\|^{*}$ where $\|.\|^{*}$ is the
dual norm of $\|.\|$: $\|\lambda\|^{*}=\max_{\|x\|\leq 1}\lambda x$. The
solution point of the dual problem is denoted $x^{*}$ and always satisfies at
equality the cut generated by the separation problem. We now present the
lemmas used in the proof.
###### Lemma 1 (boyd1995convergence (12))
Let $\lambda^{*}x\leq\lambda^{*}x^{*}$ be the cut generated during the
separation of a point $\widehat{x}$ from a polyhedron $Q_{2}$ with the
normalization $\|\pi\|\leq 1$ where $x^{*}$ is the optimal solution of the
dual of the separation problem. Then $-\lambda^{*}$ is a sub-gradient of
$x\mapsto\|x-\widecheck{x}\|^{*}$ in $x^{*}$.
In the second lemma $\angle(\lambda,x)$ will denote the angle between the
vectors $\lambda$ and $x$ (the one lower than $\pi$ radiant).
###### Lemma 2 (boyd1995convergence (12))
For each norm, there exists an angle $\theta_{min}>0$ such that at any point
$x$ and for any sub-gradient $\lambda$ of this norm in $x$:
$\angle(\lambda,x)\geq\frac{\pi}{2}-\theta_{min}$
We now present the main theorem of this section.
###### Theorem A.2
If the secondary normalization is $\|\pi\|\leq 1$ for any norm then the method
generates a cut associated with a directional normalization in finite time.
###### Proof
Scheme of the proof: First we will show that once a face intersecting the
segment $(\widecheck{x},\widehat{x})$ has been generated the method ends after
a single call to the secondary separation problem. Secondly, we will show that
if another face of $Q_{2}$ is generated, the point $x^{\prime}$ advances more
than $\epsilon$ along the segment $(\widecheck{x},\widehat{x})$ toward
$\widehat{x}$ where $\epsilon$ is a strictly non-negative distance independent
of the iteration. Thus, this second case cannot happen more than
$\frac{\|\widehat{x}-\widecheck{x}\|_{2}}{\epsilon}$ times so the procedure
ends in a finite number of steps.
1) Suppose that at one iteration, a face intersecting the segment
$(\widecheck{x},\widehat{x})$ is generated. After generating the face, the
procedure places the point $x^{\prime}$ on the intersection point. At the next
iteration, the secondary separation problem indicates that the point
$x^{\prime}$ belongs to $Q_{2}$ and the method stops.
2) For the rest of this proof, we will denote by $x^{(i)}$ the point
$x^{\prime}$ separated during iteration $i$ of the algorithm. We will show
that if a face $F$ of $Q_{2}$ generated does not intersect the segment
$(\widecheck{x},\widehat{x})$, then the point $x^{\prime}$ advances along the
segment $(\widecheck{x},\widehat{x})$ a strictly non-negative distance
$d_{min}\sin(\theta_{min})$ independent of the face $F$, i.e.
$\|x^{(i+1)}-x^{(i)}\|_{2}\geq d_{min}\sin(\theta_{min})$. To that end, we
will use trigonometry on the triangle formed by the points associated with
$x^{(i)}$, $x^{*}$, and $x^{(i+1)}$ where $x^{*}$ is the dual optimal solution
of the secondary separation problem. This triangle will be denoted $\triangle
x^{(i)}x^{*}x^{(i+1)}$. We will use $\theta_{min}$ as lower bound for the
angle $\angle(x^{(i+1)}-x^{*},x^{(i)}-x^{*})$ and $d_{min}$ as lower bound for
the distance $\|x^{(i)}-x^{*}\|_{2}$.
2.1) Suppose that at iteration $i$, the secondary separation problem of
$x^{(i)}$ returns a primal-dual solution pair $(\lambda^{*},x^{*})$ which thus
correspond to the cut $\lambda^{*}x\leq\lambda^{*}x^{*}$. Recall that $x^{*}$
is a point of $Q_{2}$ satisfying the generated cut to equality. It is thus on
the generated face $F$. From Lemma 1, since the secondary normalization is
$\|\pi\|\leq 1$, the vector $-\lambda^{*}$ is a sub-gradient of
$\|x-x^{(i)}\|^{*}$ in $x^{*}$. Thus, according to Lemma 2, we have
$\angle(-\lambda^{*},x^{*}-x^{(i)})\leq\frac{\pi}{2}-\theta_{min}$ for a
$\theta_{min}>0$ depending on the used norm but neither on the face $F$ nor on
the iteration. This is equivalent to
$\frac{\pi}{2}-\angle(\lambda^{*},x^{(i)}-x^{*})\geq\theta_{min}$. After
generating the cut $\lambda^{*}x\leq\lambda^{*}x^{*}$, the procedure projects
the point $x^{(i)}$ on the hyperplane $\lambda^{*}x=\lambda^{*}x^{*}$ along
the segment $(\widecheck{x},\widehat{x})$. The result of this projection is
the point $x^{(i+1)}$. Since both $x^{(i+1)}$ and $x^{*}$ are point of the
hyperplane $\lambda^{*}x=\lambda^{*}x^{*}$, the vector $x^{(i+1)}-x^{*}$ is on
the hyperplane $\lambda^{*}x=0$. Let us consider the smallest angle between
$x^{(i)}-x^{*}$ and a point of the hyperplane $\lambda^{*}x=0$. Firstly, it is
smaller than the angle $\angle(x^{(i+1)}-x^{*},x^{(i)}-x^{*})$. Secondly, this
smallest angle can be expressed in terms of the normal $\lambda^{*}$ as
$\frac{\pi}{2}-\angle(\lambda^{*},x^{(i)}-x^{*})$ which is shown above to be
greater than $\theta_{min}$. Thus, we have shown that the angle
$\bm{\angle(x^{(i+1)}-x^{*},x^{(i)}-x^{*})}$ is greater than
$\bm{\theta_{min}}$.
2.2) Let $d_{min}$ be the distance in norm $\|.\|_{2}$ between the segment
$(\widecheck{x},\widehat{x})$ and the union of the faces of $Q_{2}$ that do
not intersect the segment $(\widecheck{x},\widehat{x})$. Since the generated
face $F$ does not intersect the segment $(\widecheck{x},\widehat{x})$, the
distance between the segment $(\widecheck{x},\widehat{x})$ and the face $F$ is
greater than $d_{min}$. However, $x^{(i)}$ belongs to the segment
$(\widecheck{x},\widehat{x})$ and $x^{*}$ to the face $F$ so the distance
$\bm{\|x^{(i)}-x^{*}\|_{2}}$ is greater than $\bm{d_{min}}$.
2.3) We will now show with simple trigonometry on the triangle $\triangle
x^{(i)}x^{*}x^{(i+1)}$ that $\|x^{(i+1)}-x^{(i)}\|_{2}\geq
d_{min}\sin(\theta_{min})$. First, the distance $\|x^{(i+1)}-x^{(i)}\|_{2}$ is
larger than the length of the altitude of the triangle $\triangle
x^{(i)}x^{*}x^{(i+1)}$ associated with $x^{(i)}$. However, the length of this
altitude is $\|x^{(i)}-x^{*}\|_{2}\sin(\angle(x^{(i)}-x^{*},x^{(i+1)}-x^{*}))$
which according to paragraph 2.1) and 2.2) is greater than
$d_{min}\sin(\theta_{min})$. Thus finally, we have:
$\bm{\|x^{(i+1)}-x^{(i)}\|_{2}\geq d_{min}\sin(\theta_{min})}$.
The distance $d_{min}\sin(\theta_{min})$ is strictly non-negative and
independent of the iteration of the algorithm which completes the proof. ∎
## Appendix B Solution of the knapsack oracle
All the decomposition methods presented in this work assume that there exists
an efficient algorithm capable of optimizing a linear function on the
polyhedron $Q_{2}$ which in our version of the unsplittable flow problem can
be written as:
$\\{(f_{a}^{k}\in\\{0,1\\})_{a\in A,k\in k}o_{a}\in\mathbb{R}^{+}|\sum_{k\in
K}f_{a}^{k}D_{k}\leq c_{a}+o_{a}\\}.$
Optimizing a linear function whose coefficients are $(\pi^{k})_{k\in K}$ and
$-\pi_{o}$ on this polyhedron can thus be done by following mixed integer
linear program:
$\displaystyle(O_{a})\quad$ $\displaystyle\max_{f_{a}^{k},o_{a}}$
$\displaystyle\sum_{k\in K}\pi^{k}f_{a}^{k}-\pi_{o}o_{a}$ subject to
$\displaystyle\sum_{k\in K}f_{a}^{k}D_{k}\leq c_{a}+o_{a}$ $\displaystyle
f_{a}^{k}\in\\{0,1\\},~{}o_{a}\in\mathbb{R}^{+}$
This problem can be addressed by solving two classic 0-1 knapsack problems
using a case disjunction. This method was presented by buther2012reducing (14)
but we recall it here. Consider the following disjunction: either the flow of
the commodities respects the capacity of the arc $a$, or the flow of the
commodities exceeds the capacity of the arc $a$. Finding the best solution in
the first case amounts to solving the following problem:
$\displaystyle\max_{f_{a}^{k}}$ $\displaystyle\sum_{k\in K}\pi^{k}f_{a}^{k}$
subject to $\displaystyle\sum_{k\in K}f_{a}^{k}D_{k}\leq c_{a}$ $\displaystyle
f_{a}^{k}\in\\{0,1\\}$
Indeed, we suppose that the coefficient $\pi_{o}$ is non-negative because
otherwise the problem $O_{a}$ would be unbounded. Since the flow of
commodities respects the capacity of the arc, the variable $o_{a}$ always
takes the value zero and can be removed from the problem. Note that in this
first case of the disjunction, the problem to be solved is a classic knapsack
problem. In the second case of disjunction, finding the best solution amounts
to solving the following problem:
$\displaystyle\max_{f_{a}^{k},o_{a}}$ $\displaystyle\sum_{k\in
K}\pi^{k}f_{a}^{k}-\pi_{o}o_{a}$ subject to $\displaystyle\sum_{k\in
K}f_{e}^{k}D_{k}\leq c_{a}+o_{a}$ $\displaystyle\sum_{k\in
K}f_{a}^{k}d^{k}\geq c_{a}$ $\displaystyle
f_{a}^{k}\in\\{0,1\\},~{}o_{a}\in\mathbb{R}^{+}$
Since we assume that $\pi_{o}$ is non-negative and that the flow of
commodities does not respect the capacity of the arcs, the variable $o_{e}$ is
always equal to $\sum_{k\in K}f_{a}^{k}D_{k}-c_{a}$. By performing the
replacement in the objective function, by replacing the variables $f_{a}^{k}$
by their complement $\bar{f}_{a}^{k}=1-f_{a}^{k}$ and by multiplying the
constraint $\sum_{k\in K}f_{a}^{k}D_{k}\geq c_{a}$ by -1 we get the following
reformulation:
$\displaystyle\max_{\bar{f}_{a}^{k}}$ $\displaystyle\sum_{k\in
K}(\pi_{o}D_{k}-\pi^{k})\bar{f}_{a}^{k}+C$ subject to $\displaystyle\sum_{k\in
K}\bar{f}_{a}^{k}D_{k}\leq\sum_{k\in K}D_{k}-c_{a}$
$\displaystyle\bar{f}_{a}^{k}\in\\{0,1\\}$
where the constant $C$ is equal to $\sum_{k\in
K}(\pi^{k}-\pi_{o}d^{k})-\pi_{o}c_{a}$. This reformulation shows that the
problem to be solved in the second case of the disjunction is also a classic
knapsack problem.
Note: in the experimental study of Section 8, in order to solve these two
knapsack problems, we use the MINKNAP algorithm proposed by
pisinger1997minimal (43). Since MINKNAP only accepts integer-valued weights
and profits, we multiply the dual vector $(\pi,\pi_{0},\pi_{o})$ by $10^{7}$
and truncate before invoking MINKNAP.
## Appendix C Pre/post-processing for Fenchel subproblems
Solving directly a Fenchel subproblem is sometimes too computationally
expensive to be integrated into a decomposition method. However, the
resolution time of this subproblem can be greatly reduced with pre/post-
processing steps. Indeed, boccia2008cut (8) showed that it is possible to
solve the Fenchel separation problem by focusing on a sub-polyhedron of
$Q_{2}$ of far lower dimension. However, the cut generated may not be valid
for $Q_{2}$ and one must use a lifting procedure to create a cut valid for
$Q_{2}$. These concepts are explained in the following.
Dimensionality reduction: This technique typically applies when the variables
$x$ of the initial problem ($P$) are binary. Instead of separating a point
$\widecheck{x}$ from $Q_{2}$, this point is separated from the sub-polyhedron
$Q_{2}^{f}(\widecheck{x})$ induced by the variables taking a value different
from their bounds in $\widecheck{x}$ (in the binary case, induced by variables
taking a fractional value in $\widecheck{x}$). More precisely,
$Q_{2}^{f}(\widecheck{x})=\\{x\in Q_{2}|x_{i}=\widecheck{x}_{i}\text{ if
}\widecheck{x}_{i}\in\\{0,1\\}\\}$. Indeed, $\widecheck{x}$ belongs to $Q_{2}$
if and only if it belongs to $Q_{2}^{f}(\widecheck{x})$ which implies that
$\widecheck{x}$ is separable from $Q_{2}$ if and only if it is separable from
$Q_{2}^{f}(\widecheck{x})$. Replacing $Q_{2}$ with $Q_{2}^{f}(\widecheck{x})$
in the separation problem has two advantages. First, the problem only
considers variables taking a fractional value in $\widecheck{x}$ which reduces
the number of dimensions of the problem and therefore speeds up its
resolution. Forcing certain variables to take a binary value in the
optimization oracle on $Q_{2}$ is also often easy to do. Second, if the
separation problem ($S$) is solved directly through constraint generation,
most of the computational time is used to generate constraints associated with
vertices of $Q_{2}$ not belonging to $Q_{2}^{f}(\widecheck{x})$ and which are
therefore mostly unnecessary. Reducing the search space to
$Q_{2}^{f}(\widecheck{x})$ allows the computation to concentrate on the part
of the space containing the most important vertices.
Lifting: Upon applying the dimensionality reduction described before, the cut
generated will be valid for $Q_{2}^{f}(\widecheck{x})$ but not necessarily for
$Q_{2}$. A procedure able to create a valid cut for $Q_{2}$ from one valid for
$Q_{2}^{f}(\widecheck{x})$ is the sequential lifting procedure
wolsey1999integer (45) that we will now describe. In the following, we will
name fixed variables, the variables eliminated from the separation problem
when replacing the polyhedron $Q_{2}$ by the polyhedron
$Q_{2}^{f}(\widecheck{x})$. For the fixed variables, the coefficients of the
generated cut are all zero. At each step of the sequential lifting procedure,
a new value for the coefficient of one of the fixed variables is computed so
that the cut becomes valid for the polyhedron where this variable is no longer
fixed. These new coefficients are optimal in the sense that if the initial cut
was a facet of the polyhedron $Q_{2}^{f}(\widecheck{x})$ then the cut
resulting from the sequential lifting is a facet of $Q_{2}$. We now present an
iteration of the sequential lifting procedure. Suppose therefore that
$Q_{2}^{f}(\widecheck{x})=\\{x\in Q_{2}|x_{1}=1\\}$; the procedure will then
contain only one iteration because only the variable $x_{1}$ is fixed. The cut
generated for $Q_{2}^{f}(\widecheck{x})$ is $\pi x\leq\pi_{0}$ where the
coefficient $\pi_{1}$ associated with the variable $x_{1}$ is zero. The
lifting procedure consists in creating a new cut $\gamma x\leq\gamma_{0}$ such
that the two cuts are identical when $x_{1}=1$ and such that the new cut is
valid for $Q_{2}$. Since $\pi x\leq\pi_{0}$ is already valid for
$Q_{2}^{f}(\widecheck{x})$, the two previous conditions can be written:
$\forall x\in\\{x|x_{1}=1\\},~{}\gamma x-\gamma_{0}=\pi x-\pi_{0}$ $\max_{x\in
Q_{2}~{}|~{}x_{1}=0}\gamma x\leq\gamma_{0}$
Let $e_{i}$ be the $i^{\text{th}}$ vector of the canonical basis. Applying the
first condition for $x=e_{1}$, we get $\gamma_{1}-\gamma_{0}=\pi_{1}-\pi_{0}$.
Moreover, by applying it in $x=e_{1}+e_{i}$ for all $i>1$, we obtain
$\gamma_{1}+\gamma_{i}-\gamma_{0}=\pi_{1}+\pi_{i}-\pi_{0}$ which becomes
$\gamma_{i}=\pi_{i}$ by subtracting the previous equality. We therefore know:
$\displaystyle\forall i>1,\gamma_{i}=\pi_{i}$
$\displaystyle\gamma_{1}=\gamma_{0}+\pi_{1}-\pi_{0}$
Thus, the value of $\gamma_{0}$ is the only element missing to know the value
of all the $\gamma_{i}$. By replacing $\gamma_{i}$ by $\pi_{i}$ for all $i>1$
in the second condition we get $\gamma_{0}\geq\max_{x\in\\{x\in
Q_{2}|x_{1}=0\\}}\pi x$. The smallest value of $\gamma_{0}$ satisfying the
second condition is, therefore, $\gamma_{0}=\max_{x\in\\{x\in
Q_{2}|x_{1}=0\\}}\pi x$ which can be computed with a call to the optimization
oracle on $Q_{2}$ where the variable $x_{1}$ set to $0$. It is therefore
possible to compute each lifted coefficient using a single call to the oracle
($O$).
|
AND
# Bounded-error constrained state estimation in presence of sporadic
measurements
Yasmina BECIS-AUBRY *The author is with Université d’Orléans, Laboratoire
PRISME EA 4229 (Univ. Orléans - INSA CVL). 63 av. de Lattre de Tassigny, 18020
Bourges Cedex, FRANCE. Tel. +33 2 48 23 84 78<EMAIL_ADDRESS>
###### Abstract
This contribution proposes a recursive set-membership method for the
ellipsoidal state characterization for linear discrete-time models with
additive unknown disturbances vectors, bounded by possibly degenerate
zonotopes and polytopes, impacting respectively, the state evolution equation
and the sporadic measurement vectors, which are expressed as linear inequality
and equality constraints on the state vector. New algorithms are designed
considering the unprecedented fact that, due to equality constraints, the
shape matrix of the ellipsoid characterizing all possible values of the state
vector is non invertible. The two main size minimizing criterions (volume and
sum of squared axes lengths) are examined in the time update step and also in
the observation updating, in addition to a third one, minimizing some error
norm and ensuring the input-to-state stability of the estimation error.
## 1 INTRODUCTION
There is no more need to praise the interests of the set-membership state
estimation techniques neither is there a necessity to recall how interesting
alternative they offer to conventional state estimation methods where the
statistical assumptions on the disturbances can not be satisfied in certain
practical situations, nor how increasing attention they are currently
receiving since the noises of their models are assumed only to be bounded.
Nevertheless, the stability question is rarely addressed in this kind of
estimation approach.
On the other hand, the constrained state filtering has been widely studied in
stochastic context [1], [2], [3]. In [4], constrained Kalman filter variations
were reexamined and an alternative derivation of the optimal constrained
Kalman filter for time variant systems was proposed. The literature is less
abundant on this subject when it comes to bounded error framework. LMI
techniques were employed for ellipsoidal set-membership constrained state
filtering with linear [5] and linearized [6] equalities. In [7], the authors
used a combined stochastic and set-membership uncertainty representation by
integrating, into the Kalman filter structure, ellipsoidal constraints on the
state vector as a relaxation of equality constraints. All these works faced a
same challenge, not arising here, in inverting the estimation error covariance
matrix, becoming inevitably singular, when dealing with equality constraints.
Most algorithms of set-membership techniques, when applied to linear discrete-
time systems, are composed of two stages, in the manner of Kalman filter: 1.
the time prediction consisting of the linear transformation of the set
containing the state vectors at previous time step, on one hand and the
Minkowski sum (cf. $\S$ 1.1.9.) of the set containing the processes errors, on
the other; followed by 2. the measurement correction phase, where an
intersection of the predicted set just obtained is performed with the one
provided by the measurements with bounded noises.
Four main approaches are sharing the research interests in this area: those
using intervals or boxes (i.e.balls for the $\infty-$norm), parallelotopes
(affine images of boxes, obtained by injective, usually bijective, maps) or
more general zonotopes (usually surjective, affine maps of boxes) and
ellipsoids (affine maps of balls for the $2-$norm).
The asset of the interval methods is the simplicity of its conceptualization
and the availability of libraries allowing the approximation of a large range
of linear and nonlinear functions. On the other hand, the recourse to interval
computing softwares including time costly operations such as subpavings and
contractors are inevitable to overcome the conservatism of such aligned with
the coordinates’ axes boxes [8], [9].
The zonotope has the attractive structure of a centrally symmetric convex
polytope where every facet is a parallelotope having a parallel congruent
facet. This class has the interesting quality to be closed under the Minkowski
sum (operated at the time prediction stage) resulting in the concatenation of
the generator vectors of the operands, thus inducing their inevitable growth.
The zonotopic approximation has the advantage of the accuracy. However, this
accuracy is again obtained at the expense of memory and time complexity.
As intervals, ellipsoids benefit from the simplicity of their formulation and
the constant minimal space storage required ($n(\frac{n}{2}+2)$ at each
iteration, where $n$ is the state dimension). Also, this class has, in common
with zonotopes, the useful property to be closed under the linear
transformation. The only drawback of choosing ellipsoids for the
characterization of the disturbances lies at the necessity of solving two
polynomial equations of order $n$ (the size of the state vector), at each time
step: one arising at the prediction and the other at the correction step,
while optimizing the ellipsoids overbounding, respectively, the Minkowski sum
and the intersection of ellipsoids111This equation can be avoided at the
prediction step, if minimizing the squared axes’ sum of the resulting
ellipsoid rather than its volume. [10, 11, 12, 13, 14]. It is nonetheless
important to note that the parallelotope is a particular zonotope and that the
ellipsoidal and parallelotopic overbounding methods are tightly related: the
ellipsoid of center $\bm{c}$ and of shape matrix $P$ is the unique minimum
volume ellipsoid (called Löwner-John ellipsoid) inscribed in the parallelotope
of exactly the same center and the same shape matrix. This ellipsoid also
circumscribes the parallelotope of the same center and of shape matrix
$\frac{1}{\sqrt{n}}P$. It is therefore easy to deduce the parallelotopic
characterization from the ellipsoidal one and vice versa.
This is the reason we opted for the ellipsoid to characterize only the set of
all possible values of the state vector. The disadvantages of using the
ellipsoid as bounding set for both process and disturbances vectors are
overcome by choosing zonotopes to bound the process disturbances and, since we
are interested here in the state estimation of linear discrete-time systems
subject not only to bounded disturbances but also to all kinds of linear
constraints on the state vector, the output disturbances vector is modeled by
a more general class of polyhedrons (intersection of halfspaces, each modeling
an inequality constraint on the state vector), which can be unbounded and/or
flattened, due to some possible hyperplane intersections (introduced by
equality constraints). In fact, among all the classes considered here, the
class of ellipsoids is the only one that is closed under the intersection with
a hyperplane. Neither intervals nor zonotopes have this advantageous feature
[15], which will be favorably exploited while processing the equality
constraints. Moreover, the choice of the ellipsoidal set for the state
characterization is the only one that guarantees an optimal (_w.r.t._ chosen
criterions) set, without any conservatism.
This work is the extension of the author’s recent one, [16], where only one
criterion could be chosen at each phase in the presence of equality
constraints: the SSAL (sum of squared axes lengths) at the prediction step and
the scaling parameter $\varsigma_{k}$ representing a weighted estimation error
norm in the worst noise case was minimized at the correction step. Since the
volume of the state bounding ellipsoid becomes zero in presence of equality
constraints, it cannot be minimized. To address this issue, a new notion of
pseudo-volume is introduced. Our concern here is to propose a complete, easily
applicable solution. For this purpose, the trace minimization approach during
the measurement correction stage, was also thoroughly examined. In fact, even
if the trace and determinant minimizations were widely used in literature, for
ellipsoidal set-membership approaches, [10, 12], especially in the parameter
identification framework, [17, 18], at our knowledge, no analytical ready-to-
use solution (without $n$${}^{\textrm{th}}$ order equations or LMI’s to solve)
was developed for the state estimation problem.
The paper is organized as follows. After this introduction, which is completed
by some notations and definitions, the second section recaptures the
constrained set-membership state estimation problem with sporadic
measurements, already formulated in [16]. The third section concerns the time
prediction stage where two size minimization criterions are considered:
pseudo-volume and SSAL (sum of squared axes lengths) the measurement
correction stage of the estimation algorithm is detailed in forth section
using three different optimization criterions. The algorithm’s convergence is
studied in the fifth section. Numerical simulations are presented in the sixth
and finally, a brief conclusion terminates the paper.
### 1.1 Notations and definitions
1. 1.
The symbol ${\,:=\,}$ (_resp_. ${\,=:\,}$) means that the Left Hand Side
(_resp_. RHS) _is defined to be equal to_ the Right Hand Side (_resp_. LHS).
Normal lowercase letters are used for scalars, capital letters for matrices,
bold lowercase letters for vectors and calligraphic capital letters for sets.
${\mathop{\rm I\mkern-3.5muR}}$, ${\mathop{\rm I\mkern-3.5muR}}^{*}$, $R_{+}$,
$R_{+}^{*}$ denote the sets of real, non-zero, nonnegative and positive
numbers _resp_. ${\mathop{\rm I\mkern-3.5muN}}$ and ${\mathop{\rm
I\mkern-3.5muN}}^{*}$ are the sets of nonnegative and positive integers
_resp_. $l,m,n,p,q\in{\mathop{\rm I\mkern-3.5muN}}$ designate vectors and
matrices dimensions. The subscript $k\in{\mathop{\rm I\mkern-3.5muN}}$ is the
discrete time step and $i,j\in{\mathop{\rm I\mkern-3.5muN}}^{*}$ are vector
and matrix component indices.
2. 2.
$x_{i}$ is the $i$${}^{\textrm{th}}$ component of the vector $x$. $a_{ij}$ is
the $i$${}^{\textrm{th}}$ row and $j$${}^{\textrm{th}}$ column element of
$A{\,:=\,}\big{[}a_{ij}\big{]}=[\bm{a}_{j}]\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n\times m}}$ and $\bm{a}_{j}\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}$ is its $j$${}^{\textrm{th}}$ column vector (if $n=0$
or $m=0$, $A$ is an empty matrix).
3. 3.
$\mbox{\boldmath$0_{n}$}\in{\mathop{\rm I\mkern-3.5muR}}^{n}$ and
$0_{n,m}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n\times m}}$ are vector and
matrix of zeros; $I_{n}{\,:=\,}[\bm{i}_{n_{1}}\ldots\bm{i}_{n_{n}}]$ is the
$n\times n$ identity matrix.
4. 4.
$A^{T}$, $A^{\dagger}$, $\operatorname*{rank}(A)$, $\mathpzc{Ker}(A)$ and
$\mathpzc{R}(A)$ stand _resp_. for the transpose, Moore-Penrose inverse, rank,
kernel and range of the matrix $A$. If A is square, $\operatorname*{tr}(A)$,
${\left|A\right|}$ and $A^{-1}$, are its trace, determinant and inverse (if
any) _resp_.
5. 5.
$\operatorname*{Diag}(x_{i})_{i=1}^{k}$ is a diagonal matrix where
$x_{1},\ldots,x_{k}$ are its diagonal elements.
6. 6.
A Symmetric matrix $M$ is Positive Definite, denoted by SPD or $M>0$ (_resp_.
Positive Semi-Definite or non-negative definite, denoted by SPSD or $M\geq 0$)
if and only if $\forall\mbox{\boldmath$x$}\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}\text{--}\\{{\mathbf{0}}\\}$,
$\mbox{\boldmath$x$}^{T}Mx>0$ (_resp_. $\mbox{\boldmath$x$}^{T}Mx\geq~{}0$).
This condition is met if and only if all its eigenvalues are real (because of
its symmetry) and positive (_resp_. non-negative). The matrix inequality $M>N$
(_resp_. $M\geq N$) means that $M-N>0$ (_resp_. $M-N\geq 0$).
7. 7.
$\left\|\mbox{\boldmath$x$}\right\|{\,:=\,}\left\|\mbox{\boldmath$x$}\right\|_{2}{\,:=\,}\sqrt{\mbox{\boldmath$x$}^{T}\mbox{\boldmath$x$}}$
is the 2-norm of the vector $x$ and
$\left\|A\right\|{\,:=\,}\left\|A\right\|_{2}{\,:=\,}\sup_{\bm{x}\neq{\mathbf{0}}}\frac{\left\|A\bm{x}\right\|}{\left\|\bm{x}\right\|}=$
$\sigma_{\max}(A)$, where $\sigma_{\max}(A)=\max_{i}\sigma_{i}(A)$ is the
largest singular value of $A$.
8. 8.
${\mathcal{B}}^{n}_{p}{\,:=\,}\\{\mbox{\boldmath$z$}\in{\mathop{\rm
I\mkern-3.5muR}}^{n}|\left\|\mbox{\boldmath$z$}\right\|_{p}\leq 1\\}$ is a
unit ball in ${\mathop{\rm I\mkern-3.5muR}}^{n}$ for the $p-$norm.
${\mathcal{B}}^{n}_{2}$ and ${\mathcal{B}}^{n}_{\infty}{\,:=\,}[-1,1]^{n}$ are
the centred unit hypersphere and hypercube/box _resp_.
9. 9.
${\mathcal{S}}_{1}\oplus{\mathcal{S}}_{2}{\,:=\,}\\{\mbox{\boldmath$x$}\in{\mathop{\rm
I\mkern-3.5muR}}^{n}|\mbox{\boldmath$x$}=\mbox{\boldmath$x$}_{1}+\mbox{\boldmath$x$}_{2},\mbox{\boldmath$x$}_{1}\in{\mathcal{S}}_{1},\mbox{\boldmath$x$}_{2}\in{\mathcal{S}}_{2}\\}$
is the Minkowski sum of the sets
${\mathcal{S}}_{1},{\mathcal{S}}_{2}\subset{\mathop{\rm I\mkern-3.5muR}}^{n}$
and
$\operatorname*{\oplus}_{i=1}^{m}{\mathcal{S}}_{i}{\,:=\,}{\mathcal{S}}_{1}\oplus\cdots\oplus{\mathcal{S}}_{m}$.
10. 10.
${\mathcal{E}}(\bm{c},P){\,:=\,}\\{x\in{\mathop{\rm I\mkern-3.5muR}}^{n}|\
(x-\bm{c})^{T}P^{-1}(x-\bm{c})\leq 1\\}$ is an ellipsoid in ${\mathop{\rm
I\mkern-3.5muR}}^{n}$, where $\bm{c}\in{\mathop{\rm I\mkern-3.5muR}}^{n}$ is
its center and $P\in{\mathop{\rm I\mkern-3.5muR}}^{n\times n}$ is a SPD matrix
that defines its shape, size and orientation in the ${\mathop{\rm
I\mkern-3.5muR}}^{n}$ space. When $P$ is SPSD not invertible, then the
ellipsoid is degenerate. It is as an affine transformation of matrix $M$
(where $M^{T}M=P$) of the unit Euclidean ball
${\mathcal{B}}_{2}^{n}$:${\mathcal{E}}(\bm{c},M^{T}M)=~{}\\{\mbox{\boldmath$x$}\in{\mathop{\rm
I\mkern-3.5muR}}^{n}|\
\mbox{\boldmath$x$}=\bm{c}+M\mbox{\boldmath$z$},\mbox{\boldmath$z$}\in{\mathcal{B}}_{2}^{p}\\}$.
11. 11.
${\mathcal{H}}(\bm{d},a){\,:=\,}\\{\mbox{\boldmath$x$}\in{\mathop{\rm
I\mkern-3.5muR}}^{n}|\mbox{\boldmath$x$}^{T}\bm{d}=a\\}$ is a hyperplane in
${{{\mathop{\rm I\mkern-3.5muR}}}^{n}}$ of normal vector
$\bm{d}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n}}$ and whose signed distance
from the origin is $\frac{{a}}{\left\|\bm{d}\right\|}$. Let also
${{\mathcal{G}}}(\bm{d},a){\,:=\,}\\{\mbox{\boldmath$x$}:\mbox{\boldmath$x$}^{T}\bm{d}\leq
a\\}$ be one of the two halfspaces into which the hyperplane divides the
${\mathop{\rm I\mkern-3.5muR}}^{n}$ space and ${{\mathcal{G}}}(-\bm{d},-a)$ is
the other one. Now let
${\mathcal{D}}(\bm{d},a){\,:=\,}{{\mathcal{G}}}(\bm{d},1+a)\cap{{\mathcal{G}}}(-\bm{d},1-a)$,
i.e., ${\mathcal{D}}(\bm{d},a){\,:=\,}\\{\mbox{\boldmath$x$}\in{\mathop{\rm
I\mkern-3.5muR}}^{n}:{\left|\mbox{\boldmath$x$}^{T}\bm{d}-a\right|}\leq 1\\}$
which is the strip of ${\mathop{\rm I\mkern-3.5muR}}^{n}$, of width
${2}{\left\|\bm{d}\right\|}^{-1}$, that can also be seen as a degenerate
unbounded ellipsoid or zonotope centered at ${\mathcal{H}}(\bm{d},a)$.
${\mathcal{P}}(C,\bm{d})=\bigcap_{i=1}^{m}{\mathcal{G}}(\bm{c}_{i},d_{i})$ is
a polyhedron.
12. 12.
${\mathcal{Z}}(\bm{c},L){\,:=\,}\\{\mbox{\boldmath$x$}\in{\mathop{\rm
I\mkern-3.5muR}}^{n}|\
\mbox{\boldmath$x$}=\bm{c}+L\mbox{\boldmath$z$},\mbox{\boldmath$z$}\in{\mathcal{B}}_{\infty}^{m}\\}$$=\operatorname*{\oplus}_{j=1}^{q}\\{t_{j}\bm{l}_{j},{\left|t_{j}\right|}\leq
1\\}\oplus\\{\bm{c}\\}$ is a zonotope of center $\bm{c}$, obtained by affine
transformation, of shape matrix $L\in{\mathop{\rm I\mkern-3.5muR}}^{n\times
m}$, of the unit box ${\mathcal{B}}_{\infty}^{m}$, where $m$ can be smaller,
equal to or greater than $n$. A zonotope is also a convex polyhedron with
centrally symmetric faces in all dimensions.
13. 13.
The support function of a set ${\mathcal{S}}\subset{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}$ is $\rho_{\mathcal{S}}~{}:{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}\rightarrow{\mathop{\rm I\mkern-3.5muR}}$,
$\displaystyle\mbox{\boldmath$x$}\mapsto\rho_{\mathcal{S}}(\mbox{\boldmath$x$}){\,:=\,}\sup_{\bm{u}\in{\mathcal{S}}}\mbox{\boldmath$u$}^{T}\mbox{\boldmath$x$}$.
${\mathcal{H}}\big{(}\mbox{\boldmath$x$},\rho_{\mathcal{S}}(\mbox{\boldmath$x$})\big{)}$
is the supporting hyperplane of ${\mathcal{S}}$ and
${\mathcal{S}}\subset{\mathcal{G}}\big{(}\mbox{\boldmath$x$},\rho_{\mathcal{S}}(\mbox{\boldmath$x$})\big{)}$.
$\displaystyle\rho_{{\mathcal{E}}(\bm{c},P)}(\mbox{\boldmath$x$})=\bm{c}^{T}\mbox{\boldmath$x$}+\sqrt{\mbox{\boldmath$x$}^{T}P\mbox{\boldmath$x$}}$
cf. [19].
## 2 PROBLEM FORMULATION
Consider the following linear discrete time system 0pt
$\displaystyle\mbox{\boldmath$x$}_{k}$
$\displaystyle=A_{k-1}\mbox{\boldmath$x$}_{k-1}+B_{k-1}\mbox{\boldmath$\tau$}_{k-1}+R_{k-1}\bm{w}_{k-1},\quad
k\in{\mathop{\rm I\mkern-3.5muN}}^{*}$ (1a) where
$\displaystyle\mbox{\boldmath$x$}_{0}$
$\displaystyle\in{\mathcal{E}}(\mbox{\boldmath$\hat{x}_{0}$},\varsigma_{0}P_{0}){\,=:\,}{\mathcal{E}}_{0}\subset{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}\text{ and }\bm{w}_{k}\in{\mathcal{B}}_{\infty}^{m},$
(1b)
where $\mbox{\boldmath$x$}_{k}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n}}$,
$\mbox{\boldmath$\tau$}_{k}\in{\mathop{\rm I\mkern-3.5muR}}^{l}$ and
$\bm{w}_{k}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{m}}$ are the unknown state
vector to be estimated, a known and bounded control vector and an unobservable
bounded process noise vector with unknown statistical characteristics,
_resp_.,
${\mathcal{E}}(\mbox{\boldmath$\hat{x}_{0}$},\varsigma_{0}P_{0}){\,=:\,}{\mathcal{E}}_{0}$
is a known ellipsoid (cf. $\S$ 1.1.10.), where
$\mbox{\boldmath$\hat{x}_{0}$}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n}}$ is the
initial estimate of $\mbox{\boldmath$x$}_{k}$ at $k=0$,
$P_{0}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n\times n}}$ is a SPD matrix,
$\varsigma_{0}\in{\mathop{\rm I\mkern-3.5muR}}_{+}^{*}$ is a scaling positive
scalar, the product $\varsigma_{0}P_{0}$ is chosen as large as the confidence
in $\hat{x}_{0}$ is poor; ${\mathcal{B}}_{\infty}^{m}$ is the unit ball for
the $\infty-$norm in ${{{\mathop{\rm I\mkern-3.5muR}}}^{m}}$ (cf. $\S$
1.1.8.); $A_{k}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n\times n}}$ and
$B_{k}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n\times l}}$ are known state and
input matrices, _resp_. and $R_{k}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n\times
m}}$ is the generator matrix defining the shape of the zonotope bounding the
unknown input vector
$\bar{\bm{w}}_{k}{\,:=\,}R_{k}\bm{w}_{k}\in{\mathcal{Z}}({\mathbf{0}}_{n},R_{k})$.
Now consider the output equation for the system (1):
$\displaystyle
F_{k}^{T}\mbox{\boldmath$x$}_{k}=\mbox{\boldmath$y$}_{k},\quad\mbox{\boldmath$y$}_{k}\in{\mathop{\rm
I\mkern-3.5muR}}^{p_{k}}$ (2a)
$\displaystyle\underaccent{\bar}{y}_{k_{i}}\leq{y_{k_{i}}}\leq\bar{y}_{k_{i}},\
i\in\mathscr{P}_{k}{\,:=\,}\\{1,\ldots,{p_{k}}\\}=\bar{\mathscr{D}}_{k}\cup\mathscr{H}_{k}\subset{\mathop{\rm
I\mkern-3.5muN}},$ (2b)
where, the output matrix
$F_{k}{\,:=\,}[\bm{f}_{k_{1}}\ldots,\bm{f}_{k_{p_{k}}}]\in{\mathop{\rm
I\mkern-3.5muR}}^{n\times{p_{k}}}$ is time varying and so is the number of its
columns, ${p}{\,:=\,}p_{k}\in{\mathop{\rm I\mkern-3.5muN}}$, which can be zero
sometimes (in the absence of measurements)222The output equation (2) can be
roughly derived from the one with measurements vector:
$\mbox{\boldmath$z$}_{k}=G_{k}^{T}\mbox{\boldmath$x$}_{k}+\mbox{\boldmath$v$}_{k}$,
subject to a noise vector belonging to a polyhedron:
$\mbox{\boldmath$v$}_{k}\in{\mathcal{P}}([C_{k}\
-C_{k}],[\bar{\bm{d}}_{k}^{T}\
-\underaccent{\bar}{\db}_{k}^{T}]^{T})\subset{\mathop{\rm
I\mkern-3.5muR}}^{2{p}}$, where
$\bar{\bm{y}}_{k}{\,:=\,}[\bar{{y}}_{k_{1}}\cdots\bar{{y}}_{k_{p}}]^{T}=~{}\\!\bar{\bm{d}}_{k}-C_{k}^{T}\mbox{\boldmath$z$}_{k}$,
$\underaccent{\bar}{\y}_{k}{\,:=\,}~{}\\![\underaccent{\bar}{{y}}_{k_{1}}\cdots\underaccent{\bar}{{y}}_{k_{p}}]^{T}=\underaccent{\bar}{\db}_{k}-C_{k}^{T}\mbox{\boldmath$z$}_{k}$
and $F_{k}=-G_{k}C_{k}$. . Indeed, the measurements are available in varying
amounts, at not all but only some sporadic, not a priori known, time steps
$k$. Three cases can be exhaustively enumerated: 1) for some
$i\in\mathscr{D}_{k}\subset\mathscr{P}_{k}$, both (finite and distinct) bounds
are available: $\underaccent{\bar}{y}_{k_{i}}<\bar{y}_{k_{i}}$; 2) for some
$i\in\mathscr{G}_{k}{\,:=\,}(\underaccent{\bar}{\mathscr{G}}_{k}\cup\bar{\mathscr{G}}_{k})\subset\mathscr{P}_{k}$,
only one bound, either $\bar{y}_{k_{i}}$ (if $i\in\bar{\mathscr{G}}_{k}$) or
$\underaccent{\bar}{y}_{k_{i}}$ (if $i\in\underaccent{\bar}{\mathscr{G}}_{k}$)
is available, in this case, the other (unavailable) bound is considered as
$\mp\infty$. 3) for some other $i\in\mathscr{H}_{k}\subset\mathscr{P}_{k}$,
the bounds are equal: $\underaccent{\bar}{y}_{k_{i}}=\bar{y}_{k_{i}}$. The
sets $\mathscr{D}_{k}$, $\bar{\mathscr{G}}_{k}$,
$\underaccent{\bar}{\mathscr{G}}_{k}$ and $\mathscr{H}_{k}$ form a partition
for $\mathscr{P}_{k}$:
$\mathscr{P}_{k}=\mathscr{D}_{k}\cup\bar{\mathscr{G}}_{k}\cup\underaccent{\bar}{\mathscr{G}}_{k}\cup\mathscr{H}_{k}$.
The measurement inequalities (2) can be rewritten as
0pt
$\displaystyle\bm{f}_{k_{i}}^{T}\mbox{\boldmath$x$}_{k}\leq\bar{y}_{k_{i}}\text{
and
}\underaccent{\bar}{y}_{k_{i}}\rightarrow-\infty\Leftrightarrow\mbox{\boldmath$x$}_{k}\in\bar{{\mathcal{G}}}_{k_{i}}{\,:=\,}{\mathcal{G}}(\bm{f}_{k_{i}},\bar{y}_{k_{i}}),$
$\displaystyle\text{ if }i\in\bar{\mathscr{G}}_{k},$ (3a)
$\displaystyle\bm{f}_{k_{i}}^{T}\mbox{\boldmath$x$}_{k}\geq\underaccent{\bar}{y}_{k_{i}}\text{
and
}\bar{y}_{k_{i}}\rightarrow+\infty\Leftrightarrow\mbox{\boldmath$x$}_{k}\in\underaccent{\bar}{\GC}_{k_{i}}{\,:=\,}{\mathcal{G}}(-\bm{f}_{k_{i}},-\underaccent{\bar}{y}_{k_{i}}),$
$\displaystyle\text{ if }i\in\underaccent{\bar}{\mathscr{G}}_{k},$ (3b)
$\displaystyle\bm{f}_{k_{i}}^{T}\mbox{\boldmath$x$}_{k}=\bar{y}_{k_{i}},\text{
and }\underaccent{\bar}{y}_{k_{i}}=\bar{y}_{k_{i}}\
\Leftrightarrow\mbox{\boldmath$x$}_{k}\in{\mathcal{H}}_{k_{i}}{\,:=\,}{\mathcal{H}}(\bm{f}_{k_{i}},\bar{y}_{k_{i}}),$
$\displaystyle\text{ if }i\in\mathscr{H}_{k},$ (3c)
$\displaystyle{\left|\tfrac{1}{\gamma_{k_{i}}}\bm{f}_{k_{i}}^{T}\mbox{\boldmath$x$}_{k}-{y_{k_{i}}}\right|}\leq
1\Leftrightarrow\mbox{\boldmath$x$}_{k}\in{\mathcal{D}}_{k_{i}}{\,:=\,}{\mathcal{D}}\Big{(}\tfrac{1}{\gamma_{k_{i}}}\bm{f}_{k_{i}},{y_{k_{i}}}\Big{)},\text{
otherwise }$ $\displaystyle(i\in\mathscr{D}_{k})$ (3d)
$\displaystyle\text{where
}\gamma_{k_{i}}{\,:=\,}\tfrac{\bar{y}_{k_{i}}-\underaccent{\bar}{y}_{k_{i}}}{2}\text{
and
}{y_{k_{i}}}{\,:=\,}\tfrac{\bar{y}_{k_{i}}+\underaccent{\bar}{y}_{k_{i}}}{2\gamma_{k_{i}}};$
(3e)
where ${\mathcal{G}}$, ${\mathcal{H}}$ and ${\mathcal{D}}$ are a halfspace, a
hyperplane and a strip resp. (cf. $\S$ 1.1.11.). The linear inequality
constraint on the state vector of the form (3a) (resp. (3b)) stands for a
measurement corrupted by an error whose only upper (_resp_. lower) bound is
known; the linear equality constraint on the state vector (3c) represent a
noiseless output; as for (3e), it acts as a measurement that is affected by a
bounded noise. All these three categories of outputs can obviously coexist,
i.e., occur at the same time-step $k$ and it is possible to add a forth one,
in which $\bar{y}_{k_{i}}\rightarrow+\infty$ and
$\underaccent{\bar}{y}_{k_{i}}\rightarrow+\infty$ meaning that no measurement
is available.
###### Assumptions 2.1
From now on, we assume that
1. 1.
all known matrices and vectors intervening in (1) and (3), as well as the SPD
$P_{0}$ and $\varsigma_{0}\in{\mathop{\rm I\mkern-3.5muR}}_{+}^{*}$ are
bounded;
2. 2.
all the columns of all the matrices intervening in (1) and those of $F_{k}$,
if any, are nonzero;
3. 3.
the matrix
$F_{k}({\mathscr{H}_{k}}){\,:=\,}[\bm{f}_{k_{i}}]_{i\in{\mathscr{H}_{k}}}$,
intervening in (3c), has full column rank (thus avoiding contradictory
constraints leading to an empty set);
###### Aims 2.2
We are intending here to design an estimator $\hat{\mbox{\boldmath$x$}}_{k}$
for the state vector $\mbox{\boldmath$x$}_{k}$ of the system (1)-(2), such
that,
1. 1.
a set (ellipsoid ${\mathcal{E}}_{k}$ of center
$\hat{\mbox{\boldmath$x$}}_{k}$) containing all possible values of the true
state vector $\mbox{\boldmath$x$}_{k}$ is quantified, at each time step
$k\in{\mathop{\rm I\mkern-3.5muN}}^{*}$ (standard requirement for a set-
membership approach);
2. 2.
the state estimate vector $\hat{\mbox{\boldmath$x$}}_{k}$ is _acceptable_ ,
i.e., it belongs to all the sets defined in (3).
3. 3.
under some conditions, the estimator $\hat{\mbox{\boldmath$x$}}_{k}$ is ISS,
(Input-to-State Stable, cf. Theorem 5.2). This is one of the distinguishing
features of the algorithm designed here.
The other distinguishing feature is that, unlike the other set-membership
techniques, such as those using exclusively intervals, zonotopes or polytopes,
the one detailed here delivers an optimal (_w.r.t._ some chosen criterions)
set, without any conservatism. Since the only measured information about the
true state vector $\mbox{\boldmath$x$}_{k}$ consists in its belonging to the
sets defined in (3), there is no better estimate than the one that belongs to
these sets. But such an estimator is not unique and is not necessarily stable
so the most suitable one will be chosen among the set of all possible
estimators by optimizing a given criterion.
Let
${\mathcal{E}}_{k}{\,:=\,}{\mathcal{E}}(\hat{\mbox{\boldmath$x$}}_{k},\varsigma_{k}P_{k})$
be the ellipsoid containing all possible values of the true state vector
$\mbox{\boldmath$x$}_{k}$. Note that the singular values of the shape matrix
$\varsigma_{k}P_{k}$ correspond to the semi-lengths of its axes, whose
directions are defined by the associated–orthogonal since $P_{k}$ is
symmetric–eigenvectors. In what follows, we have to determine the progression
law for the ellipsoid ${\mathcal{E}}_{k}$ (and thence for the state estimate
vector $\hat{\mbox{\boldmath$x$}}_{k}$) such that the aims $i$.–$iii$. are
fulfilled.
## 3 Time update (prediction stage)
In the two first paragraphs of this subsection, useful tools are established
in view of the development of the prediction algorithm in $\S$ 3.3.
### 3.1 Minkowski sum of an ellipsoid and a line segment
The lemma below gives the parametrized family of ellipsoids,
$\bar{{\mathcal{E}}}(\mu)$, that contains the Minkowski sum of the ellipsoid
$A{\mathcal{E}}$ (resulting from the endomorphism of matrix $A$ applied to the
ellipsoid ${\mathcal{E}}$) on one hand and the segment
${\mathcal{Z}}({\mathbf{0}}_{n},\bm{r}\bm{r}^{T})$, on the other.
###### Lemma 3.1
Let $\bm{c},\bm{u},\bm{r}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n}}$, with
$\bm{r}\neq{\mathbf{0}}_{n}$, $A\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n\times
n}}$ and $P\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n\times n}}$ SPSD.
For any $\bm{x}\in{\mathcal{E}}{\,:=\,}{\mathcal{E}}(\bm{c},\varsigma P)$,
$\bm{w}\in{\mathcal{Z}}(\bm{u},\bm{r})$ and for any $\mu\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}$,
0pt $\displaystyle A\bm{x}+\bm{w}$
$\displaystyle\subset\bar{{\mathcal{E}}}(\mu){\,:=\,}{\mathcal{E}}\big{(}\bar{\bm{c}},\varsigma\bar{P}(\mu)\big{)},$
(4a) $\displaystyle\text{where }\qquad\qquad\qquad\qquad\bar{\bm{c}}$
$\displaystyle{\,:=\,}A\bm{c}+\bm{u},$ (4b) $\displaystyle\bar{P}(\mu)$
$\displaystyle{\,:=\,}(1+\mu){Q}+\tfrac{1+\mu}{\varsigma\mu}\bm{r}\bm{r}^{T}\text{,
where }{Q}{\,:=\,}APA^{T}.$ (4c)
Proof. cf. Appendix A.1. ❑
$\mu$ is a positive scalar parameter, chosen in such a way as to minimize the
size of $\bar{{\mathcal{E}}}(\mu)$, as detailed in the next paragraph.
### 3.2 Optimal values for the parameter $\bm{\mu}$
Now, the most telling two measures of the size of an ellipsoid, i.e.the volume
and the SSAL (sum of the squared axes lengths) will be minimized. Since the
eigenvalues of $\varsigma\bar{P}(\mu)$ are the squared semi-axes lengths of
$\bar{{\mathcal{E}}}(\mu)$, the former is proportional to their product, i.e.,
to ${\left|\varsigma\bar{P}(\mu)\right|}$ and the latter is equal to
$\varsigma\operatorname*{tr}\big{(}\bar{P}(\mu)\big{)}$.
#### 3.2.1 Pseudo-volume minimization
The equality constraints on the state vector, introduced by the measurements
$i~{}\\!\in\mathscr{H}_{k}$ and resulting in the intersection of the state
ellipsoid ${\mathcal{E}}_{k}$ with hyperplanes (studied in $\S$ 4.4), causes
the ellipsoid’s shape matrix $P_{k}$ to loose rank during the correction
stage, ensuing in its dimension reduction by zeroing some axes lengths and,
therefore, bestowing this ellipsoid a zero volume. Thereupon, we shall
introduce a generalized volume, the _pseudo-volume_ of an ellipsoid, when its
usual volume can be zero. Let us first recall that if
$P\in~{}\\!{{{\mathop{\rm I\mkern-3.5muR}}}^{n\times n}}$ is SPD, then the
usual volume of an ellipsoid is
$\operatorname*{vol}\Big{(}{\mathcal{E}}(\bm{c},P)\Big{)}{\,=:\,}\dfrac{2\pi^{\frac{n}{2}}}{n\Gamma(\frac{n}{2})}{\left|P\right|}$,
where ${\left|P\right|}$ is the usual determinant of $P$ and $\Gamma$ denotes
the $\Gamma-$function [20].
###### Definition 3.1
For any SPSD matrix $P$ and any $\bm{c}\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}$, the pseudo-volume of the ellipsoid
${\mathcal{E}}\big{(}\bm{c},P)$ is proportional to the determinant of $P$:
$\displaystyle\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}\big{(}{\mathcal{E}}(\bm{c},P)\big{)}$
$\displaystyle{\,=:\,}\operatorname*{vol}({\mathcal{B}}^{q}_{2}){\left|P\right|}_{{\dagger}},\text{
where
}\operatorname*{vol}({\mathcal{B}}^{q}_{2})=\tfrac{\pi^{\frac{q}{2}}}{\Gamma(\frac{q}{2}+1)},$
(5)
where $q{\,:=\,}\operatorname*{rank}(P)$ and
${\left|P\right|}_{{\dagger}}{\,=:\,}\displaystyle\lim_{t\rightarrow
0}\dfrac{{\left|P+tI_{n}\right|}}{t^{n-q}}$ is the pseudo-determinant of the
matrix $P$, i.e., the product of all its nonzero singular values. The pseudo-
volume of ${\mathcal{E}}(\bm{c},P)$ is nothing else than the volume of the
projection of ${\mathcal{E}}(\bm{c},P)$ onto $\mathpzc{R}(P)$.
###### Proposition 3.2
Let ${Q}_{+}{\,:=\,}b\big{(}Q+a\bm{r}\bm{r}^{T})$, $a,\>b\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}$; then
0pt
$\displaystyle\textit{i.}\qquad\qquad{q_{+}{\,:=\,}\operatorname*{rank}\big{(}{Q_{+}}\big{)}}$
$\displaystyle=\begin{cases}q{\,:=\,}\operatorname*{rank}(Q),&\text{ if
}\mbox{\boldmath$v$}={\mathbf{0}}_{n},\\\ q+1,&\text{ otherwise.}\end{cases}$
(6a) $\displaystyle\textit{ii.}\qquad\qquad{\left|{Q}_{+}\right|}_{{\dagger}}$
$\displaystyle=\begin{cases}b^{q}{\left|{Q}\right|}_{{\dagger}}\big{(}1+a\bm{r}^{T}\bm{u}\big{)},&\text{if
}\mbox{\boldmath$v$}={\mathbf{0}}_{n};\\\
b^{q+1}a\mbox{\boldmath$v$}^{T}\mbox{\boldmath$v$},&\text{otherwise;}\end{cases}$
(6b) 0pt $\displaystyle\text{where
}\qquad\bm{u}{\,:=\,}Q^{\dagger}\bm{r}\qquad\text{ and
}\qquad\mbox{\boldmath$v$}{\,:=\,}(I_{n}-QQ^{\dagger})\bm{r}.$ (6c)
Proof. cf. Appendix A.2. ❑
###### Theorem 3.3
$\bar{{\mathcal{E}}}(\mu)$ defined in (4) has the minimum pseudo-volume if
$\mu=~{}\\!\mu_{\text{v}}{\,:=\,}\arg\min_{\mu\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}}\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}(\bar{{\mathcal{E}}}(\mu)):$
0pt
$\displaystyle\mu_{\text{v}}{\,:=\,}\begin{cases}\tfrac{1}{2{q}}\sqrt{({q}-1)^{2}{h}^{2}+4{q}{h}}-\tfrac{{q}-1}{2{q}}{h},&\text{
if }\mbox{\boldmath$v$}={\mathbf{0}}_{n},\\\ \tfrac{1}{q},\text{ otherwise;
}\end{cases}$ (7a) $\displaystyle\text{where $\bm{u}$ and
$\mbox{\boldmath$v$}$ are defined in \eqref{eq_u_v_def},
}{q}{\,:=\,}\operatorname*{rank}(Q)\text{ and
}{h}{\,:=\,}\varsigma^{-1}\bm{r}^{T}\bm{u}.$ (7b)
Proof. cf. Appendix A.3. ❑
It is worth noting that he minimization of the volume of
$\bar{{\mathcal{E}}}(\mu)$ requires to determine the pseudo-inverse of the
matrix $P$. The result hereafter will serve to express
$\bar{P}(\mu)^{\dagger}$ by means of $Q^{\dagger}$, allowing to deduce it
recursively, without the need to compute it anew at each step. 0pt
###### Proposition 3.4
Let ${Q}_{+}{\,:=\,}b\big{(}Q+a\bm{r}\bm{r}^{T})$, $a\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}$; then
$\displaystyle{Q_{+}^{\dagger}}$
$\displaystyle=\tfrac{1}{b}\big{(}Q^{\dagger}+{\Delta}\big{)},\text{ with
}{\Delta}{\,:=\,}\begin{cases}\tfrac{1}{\left\|\bm{r}\right\|^{2}}\Big{(}\\!\tfrac{c}{\left\|\bm{r}\right\|^{2}}\mbox{\boldmath$v$}\mbox{\boldmath$v$}^{T}-\bm{u}\mbox{\boldmath$v$}^{T}-\mbox{\boldmath$v$}\bm{u}^{T}\\!\Big{)},&\\!\\!\\!\text{if
}\mbox{\boldmath$v$}\neq{\mathbf{0}}_{n};\\\
-\tfrac{1}{c}\bm{u}\bm{u}^{T},&\\!\\!\\!\text{otherwise.}\end{cases}$ (8a)
where $c{\,:=\,}\tfrac{1+a\bm{r}^{T}Q^{\dagger}\bm{r}}{a}$ and $\bm{u}$ and
$v$ are defined in (6c).
Proof. (6) can be obtained by applying Thm 1 and Thm 3 of [21]. ❑
#### 3.2.2 SSAL minimization
As for the minimization of the sum of the squared axes lengths of the
ellipsoid, _a.k.a._ the trace criterion, it is given directly by the following
theorem ensuing from the literature, where
$\operatorname*{ssal}\big{(}{\mathcal{E}}(\bm{c},\varsigma
P)\big{)}{\,:=\,}\varsigma\operatorname*{tr}(P)$.
###### Theorem 3.5 ([10])
$\bar{{\mathcal{E}}}(\mu)$ defined in (4) has the minimum SSAL if
$\mu=\mu_{\text{s}}$:
$\displaystyle\mu_{\text{s}}$
$\displaystyle{\,:=\,}\arg\min_{\mu\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}}\operatorname*{ssal}\big{(}\bar{{\mathcal{E}}}(\mu)\big{)}=\arg\min_{\mu\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}}\operatorname*{tr}\big{(}\bar{P}(\mu)\big{)}=\sqrt{\dfrac{\bm{r}^{T}\bm{r}}{\varsigma\operatorname*{tr}(APA^{T})}}.$
(9)
### 3.3 The time update algorithm
Let
${\mathcal{E}}_{k+1/k}{\,:=\,}{\mathcal{E}}(\hat{\bm{x}}_{k+1/k},\varsigma_{k}P_{k+1/k})$
be the ellipsoid including the “reachable set” of every possible value of
$\mbox{\boldmath$x$}_{k}\in{\mathcal{E}}_{k}{\,:=\,}{\mathcal{E}}(\hat{\mbox{\boldmath$x$}}_{k},\varsigma_{k}P_{k})$
that evolves according to the plant dynamics eq. (1a), subject to (1b). The
following theorem gives the parametrized family of ellipsoids
${\mathcal{E}}_{k+1/k}$ (of parameter $\bm{\mu}$) that contains
$A_{k}{\mathcal{E}}_{k}\oplus{\mathcal{Z}}({\mathbf{0}}_{n},{R}_{k})$.
###### Theorem 3.6 (Prediction stage)
If
$\mbox{\boldmath$x$}_{k}\in{\mathcal{E}}_{k}{\,:=\,}{\mathcal{E}}(\hat{\mbox{\boldmath$x$}}_{k},\varsigma_{k}P_{k})$
and $\mbox{\boldmath$x$}_{k}$ obeys to (1), then
$\forall\bm{\mu}{\,:=\,}(\mu_{1},\ldots,\mu_{m})^{T}\in]0,+\infty[^{m}$,
0pt
$\displaystyle\bm{x}_{k+1}\in{\mathcal{E}}(\hat{\bm{x}}_{k+1/k},\varsigma_{k}P_{k+1/k}){\,=:\,}{\mathcal{E}}_{k+1/k}{\,:=\,}{\mathcal{E}}_{{k+1/k}_{m}}$
0pt $\displaystyle\text{where }\quad{\mathcal{E}}_{{k+1/k}_{i}}$
$\displaystyle{\,:=\,}{\mathcal{E}}(\hat{\bm{x}}_{k+1/k},\varsigma_{k}{Q}_{k_{i}})\supseteq{\mathcal{E}}_{{k+1/k}_{i-1}}\oplus{\mathcal{Z}}({\mathbf{0}}_{n},{\bm{r}}_{k_{i}}{\bm{r}}_{k_{i}}^{T}),$
(10a) $\displaystyle\text{where }\quad\hat{\bm{x}}_{k+1/k}$
$\displaystyle{\,:=\,}A_{k}\hat{\mbox{\boldmath$x$}}_{k}+B_{k}\mbox{\boldmath$\tau$}_{k},$
(10b) $\displaystyle P_{k+1/k}$ $\displaystyle{\,:=\,}{Q}_{k_{m}},$ (10c)
$\displaystyle{Q}_{k_{0}}$ $\displaystyle{\,:=\,}A_{k}P_{k}A_{k}^{T};$ (10d)
$\displaystyle{Q}_{k_{i}}$
$\displaystyle{\,:=\,}(1+\mu_{i})\left({Q}_{k_{i-1}}+\tfrac{1}{\mu_{i}\varsigma_{k}}{\bm{r}}_{k_{i}}{\bm{r}}_{k_{i}}^{T}\right),\forall
i\in\\{1,\ldots,m\\};$ (10e)
${\bm{r}}_{k_{i}}$ (the $i$${}^{\textrm{th}}$ column of $R_{k}$) being the
generator vector of the zonotope containing all possible values of the process
noise $\bar{\bm{w}}_{k}$.
Proof. cf. Appendix A.4. ❑
Now, the results of $\S$ 3.2 are employed in order to express the optimal
predicted ellipsoid ${\mathcal{E}}_{k/k-1}$ according to the volume and trace
criterion respectively.
#### 3.3.1 Pseudo-volume minimization
###### Theorem 3.7
${\mathcal{E}}_{{k+1/k}_{i}}$ (cf. Thm 3.6) has the minimum pseudo-volume
if$\displaystyle\mu_{i}{\,:=\,}\mu_{k_{i}}{\,:=\,}\arg\min_{\mu_{i}\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}}{\left|Q_{k_{i}}\right|}_{{\dagger}}=\mu_{\text{v}}$,
given by (7a), where 0pt
0pt $\displaystyle q$ $\displaystyle{\,:=\,}{q}_{{k+1/k}_{i-1}},\
h{\,:=\,}\varsigma_{k}^{-1}{\bm{r}}_{k_{i}}^{T}\bm{u},$ (11a)
$\displaystyle\bm{u}$
$\displaystyle{\,:=\,}\bar{Q}_{k_{i-1}}{\bm{r}}_{k_{i}},\
\mbox{\boldmath$v$}{\,:=\,}{\bm{r}}_{k_{i}}-Q_{k_{i-1}}\bm{u},$ (11b)
$\displaystyle{q}_{{k+1/k}_{i}}$
$\displaystyle{\,:=\,}\begin{cases}{q}_{{k+1/k}_{i-1}},&\text{ if
}\mbox{\boldmath$v$}={\mathbf{0}}_{n},\\\ {q}_{{k+1/k}_{i-1}}+1&\text{
otherwise;}\end{cases}$ (11c) $\displaystyle{{q}_{{k+1/k}_{0}}}$
$\displaystyle{\,:=\,}\begin{cases}{q}_{k}{\,:=\,}{q}_{k_{p}},&\text{ if
}\operatorname*{rank}(A_{k})=n;\\\ \operatorname*{rank}({Q}_{k_{0}}),&\text{
otherwise;}\end{cases}$ (11d) where
${q}_{k_{p}}{\,:=\,}\operatorname*{rank}(P_{k})$ (cf. (26q));
${q}_{{k+1/k}_{i}}\\!{\,:=\,}\operatorname*{rank}({Q}_{k_{i-1}})$,
$q_{k+1/k}{\,:=\,}{q}_{{k+1/k}_{m}}$ and
$\bar{Q}_{k_{i}}{\,:=\,}Q_{k_{i}}^{\dagger}$: $\displaystyle\bar{Q}_{k_{i}}$
$\displaystyle{\,:=\,}\tfrac{1}{1+\mu_{i}}\big{(}\bar{Q}_{k_{i-1}}+{\Delta_{k_{i}}}\big{)},\
i\in\\{1,\ldots,m\\},$ (11e) $\displaystyle\bar{Q}_{k_{0}}$
$\displaystyle{\,:=\,}Q_{k_{0}}^{\dagger}=\big{(}A_{k}P_{k}A_{k}^{T}\big{)}^{\dagger},$
(11f) $\displaystyle\text{with }\Delta_{k_{i}}$ $\displaystyle=\Delta\text{
given in \eqref{Delta_pseudo-inv-rec}, where
}\bm{r}{\,:=\,}{\bm{r}}_{k_{i}}\text{ and
}c{\,:=\,}{\varsigma_{k}\mu_{i}+{\bm{r}}_{k_{i}}^{T}\bm{u}}.$ (11g)
Proof. Direct consequence of Thm 3.3 and Proposition 3.4. ❑
Algorithm 1 resumes the last two theorems and computes
${\mathcal{E}}_{k+1/k}$, where
$\hat{\mbox{\boldmath$x$}}_{k}\rightarrow~{}\\!\mbox{\boldmath$x$}$,
$P_{k}\rightarrow~{}\\!P$, $\varsigma_{k}\rightarrow~{}\\!\varsigma$,
$q_{k}\rightarrow~{}\\!q$, $A_{k}\rightarrow~{}\\!A$,
$B_{k}\rightarrow~{}\\!B$,
$R_{k}=[{\bm{r}}_{k_{1}}\cdots{\bm{r}}_{k_{m}}]\rightarrow~{}\\!R$,
$\mbox{\boldmath$\tau$}_{k}\rightarrow~{}\\!\mbox{\boldmath$\tau$}$ and
$\bar{\mbox{\boldmath$x$}}\rightarrow~{}\\!\hat{\bm{x}}_{k+1/k}$,
$\bar{P}\rightarrow~{}\\!P_{k+1/k}$,
$\bar{\varsigma}\rightarrow~{}\\!\varsigma_{k+1/k}$,
$\bar{q}\rightarrow~{}\\!q_{k+1/k}{\,:=\,}{q}_{{k+1/k}_{m}}$. Let Predic_$\mu$
be a boolean variable set to “True” when choosing the pseudo-volume
minimization,
$\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}({\mathcal{E}}_{k/k-1_{i}})$,
and to “False” when minimizing the SSAL of ${\mathcal{E}}_{k/k-1_{i}}$.
Algorithm 1 Computation of the minimal pseudo-volume predicted ellipsoid
0: $x$, $P$, $\varsigma$, $A$, $B$, $R=[\bm{r}_{1}\cdots\bm{r}_{m}]$, $\tau$,
$q$
0: $\bar{\mbox{\boldmath$x$}}$, $\bar{P}$, $\bar{\varsigma}$, Predic_VolMin,
$\bar{q}$
1: $n\leftarrow$ size of $x$
2: $m\leftarrow$ number of columns of $R$
3: ${Q}{\,:=\,}APA^{T}$; {cf. (10d)}
4: $\bar{Q}{\,:=\,}{Q}^{\dagger}$; {cf. (11f)}
5: $q{\,:=\,}\operatorname*{rank}({Q})$; {this line can be skipped if
$\operatorname*{rank}({A_{k}})=n$; cf. (11d)}
6: for $i=1,\cdots,m$ do
7: $\bm{u}{\,:=\,}\bar{Q}\bm{r}_{i}$;
$\mbox{\boldmath$v$}{\,:=\,}\bm{r}_{i}-Q\bm{u}$; {cf. (11b)}
8: if $\mbox{\boldmath$v$}={\mathbf{0}}_{n}$ then
9: $h{\,:=\,}\varsigma\bm{r}_{i}^{T}\bm{u}$;
$\mu=\tfrac{1}{2{q}}\sqrt{({q}-1)^{2}{h}^{2}+4{q}{h}}-\tfrac{{q}-1}{2{q}}{h}$;
{cf. (11a), (7a) _resp_.}
10:
${\Delta}{\,:=\,}-\tfrac{1}{\varsigma\mu+\bm{r}_{i}^{T}\bm{u}}\bm{u}\bm{u}^{T}$;
{cf. (11g) and (8a)}
11: else
12: $\mu=\tfrac{1}{{q}}$; {cf. (7a)}
13: $q\leftarrow q+1$; {cf. (11d)}
14:
${\Delta}{\,:=\,}\tfrac{1}{\left\|\bm{r}_{i}\right\|^{2}}\bigg{(}\tfrac{{\varsigma\mu+\bm{r}_{i}^{T}\bm{u}}}{\left\|\bm{r}_{i}\right\|^{2}}\mbox{\boldmath$v$}\mbox{\boldmath$v$}^{T}-\bm{u}\mbox{\boldmath$v$}^{T}-\mbox{\boldmath$v$}\bm{u}^{T}\bigg{)}$;
{cf. (11g) and (8a)}
15: end if{cf. (7a)}
16:
${Q}{\,:=\,}(1+\mu)\big{(}{Q}+\tfrac{1}{\mu\varsigma}\bm{r}_{i}\bm{r}_{i}^{T}\big{)}$;
{cf. (10e)}
17: $\bar{Q}{\,:=\,}(1+\mu)^{-1}\big{(}\bar{Q}+{\Delta}\big{)}$; {cf. (11e)}
18: end for
19: $\bar{\mbox{\boldmath$x$}}\leftarrow
A\mbox{\boldmath$x$}+B\mbox{\boldmath$\tau$}$ {cf. (10b)};
$\bar{P}\leftarrow{Q}$ {cf. (10c)}; $\bar{\varsigma}\leftarrow\varsigma$;
$\bar{q}\leftarrow q$;Predic_VolMin $\leftarrow$ True;
###### Remark 3.1
It is worth noting that the volume minimization problem
$\displaystyle\arg\min_{\mu_{i}}{\left|{Q}_{k_{i}}\right|}_{{\dagger}}$ has an
explicit solution here. If the unknown input vector was bounded by an
ellipsoid, as was the case in [10, 12, 14, 22], rather than by an interval-
like set, such as a zonotope, $\displaystyle\mu_{{\text{v}}_{i}}$ would be the
unique positive root of an $n-$order polynomial. Nevertheless, considering
that the pseudo-inverse of a $n\times n$ matrix is needed at each time step
$k$, in line with (11f), the trace minimization is more appealing, at least
from the computational point of view.
#### 3.3.2 Sum of the squared axes’ lengths minimization
###### Theorem 3.8
${\mathcal{E}}_{k+1/k}$ defined in Thm 3.6 has the minimum SSAL, if
$\bm{\mu}{\,:=\,}\bm{\mu}_{k}{\,:=\,}\arg\min_{\mu_{i}\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}}\operatorname*{ssal}({\mathcal{E}}_{k+1/k})=\bm{\mu}_{\text{s}}$,
0pt
$\displaystyle\text{where }\mu_{{\text{s}}_{i}}$
$\displaystyle{\,:=\,}\arg\min_{\mu_{i}\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}}\operatorname*{tr}(Q_{k_{i}})=\sqrt{\dfrac{{\bm{r}}_{k_{i}}^{T}{\bm{r}}_{k_{i}}}{\varsigma_{k}\operatorname*{tr}(Q_{k_{i-1}})}},i\in\\{1,\cdots,m\\},$
(12)
and the recursive formula (10c)-(10e) becomes:
0pt $\displaystyle P_{k+1/k}$
$\displaystyle=\big{(}1+\tfrac{\bar{\mu}}{\bar{\mu}_{0}}\big{)}\big{(}Q_{k_{0}}+\tfrac{\bar{\mu}_{0}}{\varsigma_{k}}\tilde{R}_{k}\big{)},$
(13a) $\displaystyle{\text{where }}\qquad\bar{\mu}_{0}$
$\displaystyle{\,:=\,}\sqrt{\varsigma_{k}\operatorname*{tr}(Q_{k_{0}})}\text{
and $Q_{k_{0}}$ given in \eqref{P_pred0},}$ (13b)
$\displaystyle\bar{\mu}_{k}{\,:=\,}\sum_{i=1}^{m}{\bar{\mu}_{i}},\
\bar{\mu}_{i}{\,:=\,}{\left\|{\bm{r}}_{k_{i}}\right\|}\text{ and
}\tilde{R}_{k}{\,:=\,}\sum_{i=1}^{m}\bar{\mu}_{i}^{-1}{\bm{r}}_{k_{i}}{\bm{r}}_{k_{i}}^{T}=R_{k}{\operatorname*{Diag}\big{(}\tfrac{1}{\bar{\mu}}\big{)}_{i=1}^{m}}R_{k}^{T}.$
(13c)
Proof. cf. Appendix A.5. ❑
Algorithm 2 computes the value of $\bm{\mu}_{\text{s}}$ given in Thm 3.8 .
Algorithm 2 Computation of the minimal SSAL predicted ellipsoid
0: $x$, $P$, $\varsigma$, $A$, $B$, $R=[\bm{r}_{1}\cdots\bm{r}_{m}]$, $\tau$
0: $\bar{}\mbox{\boldmath$x$}$, $\bar{P}$, $\bar{\varsigma}$, Predic_$\mu$
1: $n\leftarrow$ size of $x$;
2: $m\leftarrow$ number of columns of $R_{k}$;
3: ${Q}{\,:=\,}AP_{k}A^{T}$; {cf. (10d)}
4: $\bar{\mu}_{0}{\,:=\,}\sqrt{\varsigma\operatorname*{tr}(Q)}$; {cf. (13b)}
5: $\bar{\mu}=0$;
6: $\tilde{R}=0_{n\times n}$;
7: for $i=1,\cdots,m$ do
8: $\epsilon{\,:=\,}{\left\|\bm{r}_{i}\right\|}$;
9: $\bar{\mu}\leftarrow\bar{\mu}+\epsilon$;
10: $\tilde{R}\leftarrow\tilde{R}+\frac{1}{\epsilon}\bm{r}_{i}\bm{r}_{i}^{T}$;
11: end for
12: $\bar{}\mbox{\boldmath$x$}\leftarrow
A\mbox{\boldmath$x$}+B\mbox{\boldmath$\tau$}$ {(10b)};
$\bar{P}\leftarrow\big{(}1+\tfrac{\bar{\mu}}{\bar{\mu}_{0}}\big{)}\big{(}Q+\tfrac{\bar{\mu}_{0}}{\varsigma}\tilde{R}\big{)}${(13a)};
$\bar{\varsigma}\leftarrow\varsigma$; Predic_$\mu$ $\leftarrow$ False;
###### Remark 3.2
When minimizing the SSAL of ${\mathcal{E}}_{k/k-1}$, it is not necessary to
compute the $m$ intermediate values of ${Q}_{k_{i}}$, given by the recursive
formula (10c)-(10e). $P_{k/k-1}$ can be computed directly using (13) instead.
Note also that all ${\bar{\mu}_{k_{i}}}$ are nonzero thanks to the assumption
2.
###### Remark 3.3
It is possible to minimize the weighted sum of the squared axes lengths of
${\mathcal{E}}_{k+1/k}$: $\operatorname*{tr}(CP_{k+1/k}C^{T})$, for any
$C\in{\mathop{\rm I\mkern-3.5muR}}^{n_{C}\times n}$, $n_{C}\in{\mathop{\rm
I\mkern-3.5muN}}^{*}$. In this case, instead of (12), the optimal value for
$\bm{\mu}$ would be
$\bm{\mu}_{k_{\text{s}}}{\,:=\,}\bm{\mu}_{\text{s}}{\,:=\,}[\mu_{{\text{s}}_{1}},\ldots,\mu_{{\text{s}}_{m}}]$,
where
$\displaystyle\mu_{{\text{s}}_{i}}$
$\displaystyle{\,:=\,}\sqrt{\frac{\mbox{\boldmath$r$}_{{k-1}_{i}}^{T}C^{T}C\mbox{\boldmath$r$}_{{k-1}_{i}}}{\varsigma_{k-1}\operatorname*{tr}(C{Q}_{k_{i-1}}C^{T})}},\
i\in\\{1,\cdots,m\\},$ (14a) and the equations (13c) would be replaced by
$\displaystyle\bar{\mu}_{k}$
$\displaystyle{\,:=\,}\sum_{i=1}^{m}{\bar{\mu}_{k_{i}}},\quad\bar{\mu}_{k_{i}}{\,:=\,}{\left\|C{\bm{r}}_{k_{i}}\right\|}\text{
and
}\tilde{R}_{k}{\,:=\,}\sum_{i=1}^{m}\bar{\mu}_{k_{i}}^{-1}C{\bm{r}}_{k_{i}}{\bm{r}}_{k_{i}}^{T}C^{T}.$
(14b)
Given the ellipsoid ${\mathcal{E}}_{k}$ at the time step $k$, Thm 3.6 provides
the predicted ellipsoid ${\mathcal{E}}_{k+1/k}$ whose center is given by (10b)
and whose shape matrix is given, up to the factor $\varsigma_{k}$, by the
recursive formula (10c)-(10e), which depends on $\bm{\mu}$; Thms 3.7 and 3.8
offer the optimal values for this parameter according to two criterions, whose
choice is left to the reader. The author will prefer the trace minimization,
for computational reason explained in Remark 3.1.
Discussions on how to choose the parameters $\mu_{i}$ to achieve more
optimality conditions for the ellipsoid bounding the reachable set for
continuous-time systems are proposed in [13] and [23]. See also survey books
[19], [11] for a more complete overview.
## 4 Measurement update (correction stage)
The dynamic state evolution equation (1) allowed to compute the predicted
ellipsoid ${\mathcal{E}}_{k/k-1}$ which contains all possible values of the
state vector $\mbox{\boldmath$x$}_{k}$ taking into account all the
measurements up to time step $k-1$ if any. Now, let us recall the other sets
containing $\mbox{\boldmath$x$}_{k}$, obtained from the measurements:
$\displaystyle\eqref{bounds}$
$\displaystyle\Leftrightarrow\mbox{\boldmath$x$}_{k}\in\bigcap_{i\in\mathscr{G}_{k}}{\mathcal{G}}_{k_{i}}\cap\bigcap_{i\in\mathscr{D}_{k}}{\mathcal{D}}_{k_{i}}\cap\bigcap_{i\in\mathscr{H}_{k}}{\mathcal{H}}_{k_{i}},\text{
if }{p}\neq 0.$ (15)
${p}{\,:=\,}{p_{k}}$ is the number of measurements at time-step $k$. It is
interesting to note that the intersection of half-spaces can be considered as
a possibly unbounded polyhedron and that the intersection of strips is a
zonotope:
$\displaystyle\bigcap_{i\in\mathscr{G}_{k}}{\mathcal{G}}_{k_{i}}$
$\displaystyle{\,=:\,}{\mathcal{P}}_{k}{\,:=\,}{\mathcal{P}}\big{(}[\bm{f}_{k_{i}}]_{i\in\mathscr{G}_{k}},[{y_{k_{i}}}]_{i\in\mathscr{G}_{k}}\big{)},$
(16) $\displaystyle\bigcap_{i\in\mathscr{D}_{k}}{\mathcal{D}}_{k_{i}}$
$\displaystyle{\,=:\,}{\mathcal{Z}}_{k}{\,:=\,}{\mathcal{Z}}^{{\mathcal{H}}}\big{(}[\bm{f}_{k_{i}}]_{i\in\mathscr{D}_{k}},[{y_{k_{i}}}]_{i\in\mathscr{G}_{k}}\big{)}.$
(17)
The correction stage consists in performing the intersection between
${\mathcal{E}}_{k/k-1}$ and the set (15), allowing to find
${\mathcal{E}}_{k}\supset{\mathcal{S}}_{k}$ in light of the current
measurements, where
$\displaystyle{\mathcal{S}}_{k}$
$\displaystyle{\,:=\,}\Big{(}\big{(}{\mathcal{E}}_{k/k-1}\cap\bigcap_{i\in\mathscr{G}_{k}}{\mathcal{G}}_{k_{i}}\big{)}\cap\bigcap_{i\in\mathscr{D}_{k}}{\mathcal{D}}_{k_{i}}\Big{)}\cap\bigcap_{i\in\mathscr{H}_{k}}{\mathcal{H}}_{k_{i}}$
$\displaystyle=\Big{(}\big{(}{\mathcal{E}}_{k/k-1}\cap{\mathcal{P}}_{k}\big{)}\cap{\mathcal{Z}}_{k}\Big{)}\cap\bigcap_{i\in\mathscr{H}_{k}}{\mathcal{H}}_{k_{i}}.$
(18)
It will be shown that this intersection is the one between
${\mathcal{E}}_{k/k-1}$ and the possibly degenerate (if
$\mathscr{H}_{k}\neq\emptyset$) zonotope. It does not result in an ellipsoid
in general and has to be circumscribed by such a set, which is the subject of
the upcoming paragraphs. We shall begin by working on the intersection
$\displaystyle{\mathcal{E}}_{k/k-1}\cap{\mathcal{G}}_{k_{i}}$ in $\S$ 4.1.
Secondly, we’ll be dealing with the intersection between an ellipsoid and a
strip in order to carry out the set obtained in $\S$ 4.1 and intersecting it
with $\displaystyle\bigcap_{i\in\mathscr{D}_{k}}{\mathcal{D}}_{k_{i}}$; $\S$
4.2 provides the optimal ellipsoid overbounding this intersection. Thirdly,
the intersection of an ellipsoid with a hyperplane will be presented in $\S$
4.4, in order to handle the intersection of the previously obtained ellipsoid
with $\displaystyle\bigcap_{i\in\mathscr{H}_{k}}{\mathcal{H}}_{k_{i}}$.
Finally, all these results will be compiled in a unique state estimation
algorithm in $\S$ 5.1.
### 4.1 Intersection of an ellipsoid with a halfspace
The intersection between the ellipsoid ${\mathcal{E}}_{k/k-1}$ obtained in
$\S$ 3 and the polyhedron ${\mathcal{P}}_{k}$ can be reformulated as the
intersection of ${\mathcal{E}}_{k/k-1}$ and a series of strips
${\mathcal{D}}_{k_{i}}$. To grasp this idea, take any closed convex set
${\mathcal{S}}$ and a hyperplane ${\mathcal{H}}$ intersecting it. The
intersection of ${\mathcal{S}}$ with a halfspace ${\mathcal{G}}$ delimited by
${\mathcal{H}}$ is nothing else that its intersection with the strip formed
between ${\mathcal{H}}$ and a support hyperplane of ${\mathcal{S}}$, parallel
to ${\mathcal{H}}$ and contained in ${\mathcal{G}}$. Now, if ${\mathcal{H}}$
doesn’t intersect ${\mathcal{S}}$, the latter is either a subset of
${\mathcal{G}}$ or lies outside of it, and if ${\mathcal{H}}$ is tangent to
${\mathcal{S}}$ (being its support hyperplane), then ${\mathcal{S}}$ is either
again a subset of ${\mathcal{G}}$ or it has only one point in common with it.
In the case where ${\mathcal{S}}$ is an ellipsoid and the intersecting
halfspace corresponds to the constraint (3a), the theorem below provides the
parameters of the intersecting strip. To obtain the intersection of an
ellipsoid with the halfspace given by the constraint (3b), it suffices to
replace $\bm{f}$ by $-\bm{f}$ and $\bar{y}$ by $\underaccent{\bar}{y}$:
###### Theorem 4.1 (ellipsoid-halfspace intersec.)
Let $\bm{c}\in~{}{{{\mathop{\rm I\mkern-3.5muR}}}^{n}}$,
$\bm{f}\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}\text{--}\\{{\mathbf{0}}_{n}\\}$, $P\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n\times n}}$ SPSD, $\varsigma\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}$ and $\bar{y}\in{\mathop{\rm I\mkern-3.5muR}}$.
If $\bar{y}<-\underaccent{\bar}{\rho}$, (case 1)
$\displaystyle{\mathcal{E}}(\bm{c},\varsigma
P)\cap{\mathcal{G}}(\bm{f},\bar{y})$ $\displaystyle=\emptyset;$ (19b) else if
$\bar{y}\geq\bar{\rho}$, (case 2) $\displaystyle{\mathcal{E}}(\bm{c},\varsigma
P)\cap{\mathcal{G}}(\bm{f},\bar{y})$
$\displaystyle={\mathcal{E}}(\bm{c},\varsigma P);$ (19c) else if
$\bar{y}=-\underaccent{\bar}{\rho}$, (case 3)
$\displaystyle{\mathcal{E}}(\bm{c},\varsigma
P)\cap{\mathcal{G}}(\bm{f},\bar{y})$
$\displaystyle={\mathcal{E}}(\bm{c},\varsigma
P)\cap{\mathcal{H}}(\bm{f},-\underaccent{\bar}{\rho})=\\{\bm{c}-\varsigma^{\frac{1}{2}}(\bm{f}^{T}P\bm{f})^{-\frac{1}{2}}P\bm{f}\\};$
(19d) else ($-\underaccent{\bar}{\rho}<\bar{y}<\bar{\rho}$), (case 4)
$\displaystyle{\mathcal{E}}(\bm{c},\varsigma
P)\cap{\mathcal{G}}(\bm{f},\bar{y})$
$\displaystyle={\mathcal{E}}(\bm{c},\varsigma
P)\cap{\mathcal{D}}(\tfrac{1}{\gamma}\bm{f},{y}),$ (19e) where
$\displaystyle\gamma$
$\displaystyle{\,:=\,}\tfrac{1}{2}(\bar{y}+\underaccent{\bar}{\rho})\text{ and
}{y}{\,:=\,}\tfrac{1}{2\gamma}(\bar{y}-\underaccent{\bar}{\rho})$ (19f)
$\displaystyle\underaccent{\bar}{\rho}$
$\displaystyle{\,:=\,}\rho_{{\mathcal{E}}(\bm{c},\varsigma
P)}(-\bm{f})=-\bm{c}^{T}\bm{f}+\sqrt{\varsigma\bm{f}^{T}P\bm{f}}\text{ ({\it
cf.\ }$\S$~{}\ref{subsec_notations} \ref{Support function})}$ (19g)
$\displaystyle\bar{\rho}$
$\displaystyle{\,:=\,}\rho_{{\mathcal{E}}(\bm{c},\varsigma
P)}(\bm{f})=\bm{c}^{T}\bm{f}+\sqrt{\varsigma\bm{f}^{T}P\bm{f}}.$ (19h)
Proof. cf. Appendix B.1. ❑
The figure 1 illustrates the above theorem in the case where $n=3$
($\mbox{\boldmath$x$}_{k}\in{\mathop{\rm I\mkern-3.5muR}}^{3}$). The halfspace
is depicted by the shaded region and its upper boundary is the colored
hyperplane.
Case 1:
when $\bar{y}-\bm{c}^{T}\bm{f}<-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$, the
ellipsoid ${\mathcal{E}}(\bm{c},\varsigma P)$ is outside the halfspace
${\mathcal{G}}(\bm{f},\bar{y})$ and their intersection is thus empty.
Case 2:
when $\bar{y}-\bm{c}^{T}\bm{f}>\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$, the
ellipsoid is entirely contained in the halfspace and their intersection is
nothing else than the ellipsoid ${\mathcal{E}}(\bm{c},\varsigma P)$ itself.
Case 3:
when $\bar{y}-\bm{c}^{T}\bm{f}=-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$, the
ellipsoid is tangent to the halfspace and the intersection is reduced to one
single point given by (19d) represented by a red “o” on the figure 1\- (Case
3).
Case 4:
when
${\left|\bar{y}-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$,
the intersection between the ellipsoid, ${\mathcal{E}}(\bm{c},\varsigma P)$
and the halfspace ${\mathcal{G}}(\bm{f},\bar{y})$ is the same as the
intersection of this ellipsoid with a strip
${\mathcal{D}}(\frac{1}{\gamma}\bm{f},y)$ (dark shaded area), bounded on one
hand by the hyperplane ${\mathcal{H}}(\bm{f},\bar{y})$ (cyan), the boundary of
${\mathcal{G}}(\bm{f},\bar{y})$, and on the other hand, by the support
hyperplane, ${\mathcal{H}}(-\bm{f},-\underaccent{\bar}{\rho})$ (violet) of the
ellipsoid.
(a) Case 1: $\bar{y}-\bm{c}^{T}\bm{f}<-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
(b) Case 2: $\bar{y}-\bm{c}^{T}\bm{f}>\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
(c) Case 3: $\bar{y}-\bm{c}^{T}\bm{f}=-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
(d) Case 4:
${\left|\bar{y}-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
Figure 1: Illustration of the four cases of Theorem 4.1 for $n=3$
### 4.2 Ellipsoid bounding the intersection of an ellipsoid with a strip
In the previous paragraph, we showed that the incorporation of the
measurements $i\in\mathscr{G}_{k}$ result, as for those $i\in\mathscr{D}_{k}$,
from the intersection of the predicted ellipsoid with a zonotope, formulated
as an intersection of several strips. We need now to overbound this
intersection by an ellipsoid. To begin with, the theorem below presents a
family of parametrized ellipsoids (of parameter $\beta$) that contain an
ellipsoidal layer, coming out of the intersection of
${\mathcal{E}}(\bm{c},\varsigma P)$ with the strip ${\mathcal{D}}(\bm{f},y)$,
which can be considered–interestingly enough–as an ellipsoid unbounded in the
direction orthogonal to $\bm{f}$.
###### Theorem 4.2 (ellips./strip inters.)
Let $\bm{c}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n}}$,
$\varsigma\in{\mathop{\rm I\mkern-3.5muR}}_{+}^{*}$, $y\in{\mathop{\rm
I\mkern-3.5muR}}$, $P\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n\times n}}$ SPSD
and $\bm{f}\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}\text{--}\\{{\mathbf{0}}_{n}\\}$,
if $y-1>\bar{\rho}\vee y+1<-\underaccent{\bar}{\rho}$, (case 1)
$\displaystyle{\mathcal{D}}(\bm{f},y)\cap{\mathcal{E}}(\bm{c},\varsigma P)$
$\displaystyle=\emptyset;$ (20b) else if $y+1\geq\bar{\rho}\wedge
y-1\leq-\underaccent{\bar}{\rho}$, (case 2)
$\displaystyle{\mathcal{D}}(\bm{f},y)\cap{\mathcal{E}}(\bm{c},\varsigma P)$
$\displaystyle={\mathcal{E}}(\bm{c},\varsigma P);$ (20c) else if
$y=-\underaccent{\bar}{\rho}-1$, (case 3.a)
$\displaystyle{\mathcal{D}}(\bm{f},y)\cap{\mathcal{E}}(\bm{c},\varsigma P)$
$\displaystyle={\mathcal{H}}(\bm{f},-\underaccent{\bar}{\rho})\cap{\mathcal{E}}(\bm{c},\varsigma
P)=\\{\bm{c}-\varsigma^{\frac{1}{2}}(\bm{f}^{T}P\bm{f})^{-\frac{1}{2}}P\bm{f}\\};\\!\\!\\!\\!\\!$
(20d) else if $y=\bar{\rho}+1$, (case 3.b)
$\displaystyle{\mathcal{D}}(\bm{f},y)\cap{\mathcal{E}}(\bm{c},\varsigma P)$
$\displaystyle={\mathcal{H}}(\bm{f},\bar{\rho})\cap{\mathcal{E}}(\bm{c},\varsigma
P)=\\{\bm{c}+\varsigma^{\frac{1}{2}}(\bm{f}^{T}P\bm{f})^{-\frac{1}{2}}P\bm{f}\\};$
(20e) else
$(-\underaccent{\bar}{\rho}<y+1<\bar{\rho}\vee-\underaccent{\bar}{\rho}<y-1<\bar{\rho}),\
\forall\beta\in]0,1[$, (case 4)
$\displaystyle{\mathcal{D}}(\bm{f},y)\cap{\mathcal{E}}(\bm{c},\varsigma
P)={\mathcal{D}}(\breve{\bm{f}},\breve{y})\cap{\mathcal{E}}(\bm{c},\varsigma
P)\subset{\mathcal{E}}\big{(}\breve{\bm{c}}(\beta),\breve{\varsigma}(\beta)\breve{P}(\beta)\big{)}{\,=:\,}\breve{{\mathcal{E}}}(\beta),$
(21b) 0pt $\displaystyle\text{where
}\breve{\bm{f}}{\,:=\,}\tfrac{1}{\gamma}\bm{f}\text{ and
}\breve{y}{\,:=\,}\tfrac{1}{2\gamma}(\bar{y}+\underaccent{\bar}{y})=\tfrac{1}{\gamma}(\bm{f}^{T}\bm{c}+\delta),$
(21c) $\displaystyle\breve{P}(\beta)$
$\displaystyle{\,:=\,}P-{\alpha\beta}P\bm{f}\bm{f}^{T}P=P-{\alpha\beta}\bm{\varphi}\bm{\varphi}^{T},\quad$
(21d) $\displaystyle\breve{\bm{c}}(\beta)$
$\displaystyle{\,:=\,}\bm{c}+{\alpha\beta}\delta
P\bm{f}=\bm{c}+{\alpha\beta}\delta\bm{\varphi},$ (21e)
$\displaystyle\breve{\varsigma}(\beta)$
$\displaystyle{\,:=\,}\varsigma+{\alpha\beta}\left(\tfrac{\gamma^{2}}{1-\beta}-\delta^{2}\right),$
(21f) $\displaystyle\alpha$
$\displaystyle{\,:=\,}\big{(}\bm{f}^{T}P\bm{f}\big{)}^{-1}=\big{(}\bm{f}^{T}\bm{\varphi}\big{)}^{-1}\text{
and }\bm{\varphi}{\,:=\,}P\bm{f},$ (21g) $\displaystyle\delta$
$\displaystyle{\,:=\,}\tfrac{1}{2}(\bar{y}+\underaccent{\bar}{y})-\bm{f}^{T}\bm{c}=\tfrac{1}{2}(\bar{y}+\underaccent{\bar}{y}-\bar{\rho}+\underaccent{\bar}{\rho}),$
(21h) $\displaystyle\gamma$
$\displaystyle{\,:=\,}\tfrac{1}{2}(\bar{y}-\underaccent{\bar}{y}),$ (21i)
$\displaystyle\bar{y}$ $\displaystyle{\,:=\,}\min(y+1,\bar{\rho})\text{ and
}\underaccent{\bar}{y}{\,:=\,}\max(y-1,-\underaccent{\bar}{\rho}),$ (21j)
and $\underaccent{\bar}{\rho}$ and $\bar{\rho}$ are defined in (19g)–(19h).
Proof. cf. Appendix B.2. ❑
(a) (1a): ${y-\bm{c}^{T}\bm{f}}<-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}-1$
(b) (1b): ${y-\bm{c}^{T}\bm{f}}>\sqrt{\varsigma\bm{f}^{T}P\bm{f}}+1$
(c) (2):
${\left|y-\bm{c}^{T}\bm{f}\right|}<1-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
(d) (3a): ${y-\bm{c}^{T}\bm{f}}=-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}-1$
(e) (3b): ${y-\bm{c}^{T}\bm{f}}=\sqrt{\varsigma\bm{f}^{T}P\bm{f}}+1$
(f) (4a):
${\left|y+1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
and ${\left|y-1-\bm{c}^{T}\bm{f}\right|}>\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
(g) (4b):
${\left|y+1-\bm{c}^{T}\bm{f}\right|}>\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
and ${\left|y-1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
(h) (4c):
${\left|y+1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
and ${\left|y-1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
Figure 2: Illustration of Theorem 4.2 for $n=3$
The figure 2 illustrates the above theorem in the case where $n=3$. The strip
is depicted by the shaded region and its boundaries are the colored
hyperplanes.
Case 1:
when ${\left|y-\bm{c}^{T}\bm{f}\right|}>\sqrt{\varsigma\bm{f}^{T}P\bm{f}}+1$,
the strip is either below (cf. fig. 2.1a) or above (cf. fig. 2.1b) with empty
intersection.
Case 2:
when ${\left|y-\bm{c}^{T}\bm{f}\right|}<1-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$,
the ellipsoid is entirely contained in the strip (cf. fig. 2.2) and the
intersection is the ellipsoid ${\mathcal{E}}(\bm{c},\varsigma P)$ itself.
Case 3:
when ${\left|y-\bm{c}^{T}\bm{f}\right|}=\sqrt{\varsigma\bm{f}^{T}P\bm{f}}+1$,
the ellipsoid is tangent to the strip and the intersection is reduced to a
single point given by (20d) (represented by a red “o” on fig. 2.3a) or (20e)
(white “o” on fig. 2.3.b).
Case 4:
when ${\left|y+1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
or ${\left|y-1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$.
This case can be decomposed in three sub-cases (not appearing in the theorem):
4.a:
if ${\left|y+1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$ and
${\left|y-1-\bm{c}^{T}\bm{f}\right|}>\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$,
meaning that the hyperplane ${\mathcal{H}}(\bm{f},y+1)$ representing the upper
boundary (cyan) of the strip (the strip is depicted by all the shaded region
in fig. 2.4a) intersects the ellipsoid while the lower one
${\mathcal{H}}(\bm{f},y-1)$ (magenta) does not; the strip is then reduced in
width (to the new dark shaded strip) replacing the latter hyperplane by a
support hyperplane (violet) of the ellipsoid,
${\mathcal{H}}(\bm{f},\underaccent{\bar}{y})$, where
$\underaccent{\bar}{y}{\,:=\,}\bm{c}^{T}\bm{f}-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$.
4.b:
if ${\left|y+1-\bm{c}^{T}\bm{f}\right|}>\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$ and
${\left|y-1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$,
meaning that the hyperplane ${\mathcal{H}}(\bm{f},y-1)$ (green) representing
the lower boundary of the strip is intersecting the ellipsoid while the upper
one ${\mathcal{H}}(\bm{f},y+1)$ (magenta) is not; the strip is then reduced in
width replacing the latter by ${\mathcal{H}}(\bm{f},\bar{y})$ (yellow) where
$\bar{y}{\,:=\,}\bm{c}^{T}\bm{f}+\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$.
4.c:
if ${\left|y+1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$ and
${\left|y-1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$,
meaning that both the hyperplanes ${\mathcal{H}}(\bm{f},y-1)$ and
${\mathcal{H}}(\bm{f},y+1)$ representing _resp_. the lower and upper
boundaries of the strip are intersecting the ellipsoid.
(a) (4a):
${\left|y+1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
and ${\left|y-1-\bm{c}^{T}\bm{f}\right|}>\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
(b) (4b):
${\left|y+1-\bm{c}^{T}\bm{f}\right|}>\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
and ${\left|y-1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
(c) (4c):
${\left|y+1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
and ${\left|y-1-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
Figure 3: Illustration of the case 4 Theorem 4.2 for $n=2$
After the strip is reduced, if necessary (case 4.a and 4.b), the intersection
between the new strip and the ellipsoid is overbounded by a family of
parametrized, function of $\beta$, ellipsoids. The red ellipsoids depicted in
3.4a, 3.4b and 3.4c are only one possible occurrence of this family of
ellipsoids for a particular value of $\beta$ given in (22).
### 4.3 Optimal values for the parameter $\beta$
In this section, the optimal value of the weighting parameter $\beta$ with
respect to a judiciously chosen criterion will be derived.
Three optimization criteria have been used in literature for the parameter
identification. The main two involve set measures on the ellipsoid and were
first proposed by Fogel and Huang in their well-known pioneer paper [17], in
the parameter identification framework. They gave two optimal values for the
weighting parameter $\omega{\,:=\,}\tfrac{\alpha\beta}{1-\beta}$: the first
minimizing the determinant of $\tilde{\varsigma}(\omega)\tilde{P}(\omega)$
(where $\varsigma=1$) in Lemma B.2 and the second, its trace, thus optimizing
the volume and the sum, _resp_., of the squared semi-axes lengths of the
ellipsoid ${\mathcal{E}}\big{(}\tfrac{\omega}{\omega+\alpha}\big{)}$, defined
in (21). Five years later, Dasgupta and Huang, in [24], designed a modified
least-squares parameter identification algorithm with a forgetting factor,
where they used a weighting parameter $0\leq\nu\leq 1$ (the forgetting factor
being $1-\nu$), which could be roughly related to the one used in [17] by
$\nu=\tfrac{\omega}{\omega+1}$ and they introduced a new optimization
criterion for $\nu$ based on the minimization of $\tilde{\varsigma}$. Nayeri,
Deller, Liu et al. , actively studied all these aspects during the nineties
[25, 26, 27, 28, 29], proposing a set-membership stochastic approximation
identification algorithm and, in [18], a unified framework of the general
class of optimal bounding ellipsoid (OBE) for all the methods previously
cited, on the basis of the weighted least squares identification method. They
focused on the volume minimization criterion, for evident reasons.
Even if the convergence of such algorithms in the particular case of
unconstrained set-membership parameter identification framework333The
parameter identification can be seen, in a reductive manner, as a particular
case of the state estimation where the estimated vector is supposed to be
constant, i.e., $A_{k}=I_{n}$, $B_{k}=0_{n,l}$ and $R_{k}=0_{n,m}$ in (1a).
was proven for the three addressed criteria, each having its own interesting
properties, when it comes to the set-membership state estimation, it is
definitely an open issue.
#### 4.3.1 Minimization of the worst case weighted estimation error
In contrast with all the algorithms in the literature, [10, 11, 12, 13], that
minimize the size of the ellipsoid $\breve{{\mathcal{E}}}(\beta)$, the optimal
value of $\beta$ developed in this paragraph is obtained by minimizing some
quadratic measure of the estimation error vector444represented by its
candidate Lyapunov function $\breve{\mathscr{V}}_{\beta}(\mbox{\boldmath$x$})$
in the worst noise case, embodied by $\breve{\varsigma}$, in the manner of
[14, 22], inspired by some identification algorithms [24, 30, 31].
###### Theorem 4.3.1
Consider the case 4 of Thm 4.2 where $\breve{{\mathcal{E}}}(\beta)$,
$\beta\in]0,1[$, is given by (21), then $\breve{\varsigma}(\beta)$ defined in
(21f) satisfies
0pt $\displaystyle{\varsigma}(\beta)$
$\displaystyle=\\!\max_{\bm{x}\in{\mathcal{D}}(\breve{\bm{f}},\breve{y})\cap{\mathcal{E}}(\bm{c},\varsigma
P)}\\!\\!{\mathscr{V}}_{\breve{P}(\beta)}\big{(}\mbox{\boldmath$x$}-\breve{\bm{c}}(\beta)\big{)},\text{
where }{\mathscr{V}}_{{P}}(\bm{x}){\,:=\,}\bm{x}^{T}{P}^{\dagger}\bm{x},$
(22a)
$\displaystyle{\mathscr{V}}_{\breve{P}(\beta)}\big{(}\mbox{\boldmath$x$}-\breve{\bm{c}}(\beta)\big{)}-{\mathscr{V}}_{{P}}(\mbox{\boldmath$x$}-\bm{c})$
$\displaystyle\leq\breve{\varsigma}(\beta)-\varsigma=\tfrac{\alpha\beta\gamma^{2}}{1-\beta}-\alpha\beta{\delta^{2}}.$
(22b) and its minimum is achieved at $\displaystyle\beta_{\varsigma}$
$\displaystyle{\,:=\,}\arg\min_{\beta\in]0,1[}\breve{\varsigma}(\beta)=\begin{cases}1-\gamma{\left|\delta\right|}^{-1},&\text{if
}{\left|\delta\right|}>\gamma,\\\ 0,&\text{otherwise};\end{cases}$ (22c) where
$\breve{\bm{f}}$, $\breve{y}$, $\alpha$ and $\delta$ are defined in (21c),
(21g) and (21h). 0pt $\displaystyle\text{Furthermore, if
}{\left|\delta\right|}>\gamma,\text{ then
}\breve{\varsigma}(\beta_{\varsigma})<\varsigma\text{ and
}\breve{\varsigma}(\beta_{\varsigma})-\varsigma=-\alpha\beta_{\varsigma}^{2}{\delta}^{2}<0,$
(22d)
$\displaystyle\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}\big{(}\breve{{\mathcal{E}}}(\beta_{\varsigma})\big{)}<\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}\big{(}{\mathcal{E}}(\bm{c},\varsigma
P)\big{)}\text{ and
}\operatorname*{ssal}\big{(}\breve{{\mathcal{E}}}(\beta_{\varsigma})\big{)}<\operatorname*{ssal}\big{(}{\mathcal{E}}(\bm{c},\varsigma
P)\big{)}.$ (22e)
Proof. cf. Appendix B.3. ❑
The value of $\beta_{\varsigma}$ is resumed in Algorithm 3.
Algorithm 3 Optimal value of parameter $\beta$ minimizing
$\breve{\varsigma}(\beta)$
0: $\gamma$, $\delta$ {defined in (21)}
0: $\beta_{\varsigma}$
1: if ${\left|\delta\right|}>\gamma$ then
2: $\beta_{\varsigma}{\,:=\,}1-\gamma{\left|\delta\right|}^{-1}$; {cf. (22c)};
3: else
4: $\beta_{\varsigma}{\,:=\,}0$; {cf. (22c)};
5: end if
###### Remark 4.1
The representation of the output noise vector’s bounding set as an
intersection of strips, rather than as an ellipsoid, enables this optimization
problem to have an analytical solution.
As for fig. 2 (case 4), the figure 4 on the right shows the (red/small)
ellipsoid $\breve{{\mathcal{E}}}(\beta_{\varsigma})$ containing the
intersection of the (blue/big) ellipsoid ${\mathcal{E}}(\bm{c},\varsigma P)$
with the (colored) strip ${\mathcal{D}}(\bm{f},y)$, for $\beta_{\varsigma}$
given by (22c) and $n=2$. It is worth noting that with this value of
$\beta_{\varsigma}$, the center $\breve{\bm{c}}(\beta_{\varsigma})$, of
$\breve{{\mathcal{E}}}(\beta_{\varsigma})$ is the projection of $\bm{c}$ (the
center of ${\mathcal{E}}(\bm{c},\varsigma P)$), on the hyperplan representing
the nearest strip boundary, in the direction $P\bm{f}$.
Figure 4:
$\breve{{\mathcal{E}}}(\beta_{\varsigma})\supset{\mathcal{E}}(\bm{c},\varsigma
P)\cap{\mathcal{D}}(\bm{f},y)$
#### 4.3.2 Minimization of the ellipsoid’s volume
As already mentioned in Section 3, the ellipsoid can possibility have zero
axes lengths and, therefore, a zero volume and a noninvertible shape matrix.
To our knowledge, this issue was not addressed in set-membership state
estimation algorithms of the literature reviewed here.
The optimal value for the weighting parameter $\beta$ adopted in this paper
and intervening in (21), minimizing the pseudo-volume (cf. Definition 3.1) of
the ellipsoid $\breve{{\mathcal{E}}}(\beta)$, when its usual volume can be
zero, is derived in what follows.
###### Theorem 4.3.2
If teh ellipsoid $\breve{{\mathcal{E}}}(\beta)$ given by (21) of Thm 4.2, for
which ${q}{\,:=\,}\operatorname*{rank}(P)>1$, then
$\beta_{\text{v}}{\,:=\,}\arg\min_{\beta\in]0,1[}\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}\big{(}\breve{{\mathcal{E}}}(\beta)\big{)}$
is the unique solution of the quadratic equation
$a_{2}\beta^{2}+a_{1}\beta+a_{0}=0$, where 0pt $\displaystyle a_{0}$
$\displaystyle{\,:=\,}{q}\alpha(\gamma^{2}-\delta^{2})-\varsigma,$ (23a)
$\displaystyle a_{1}$
$\displaystyle{\,:=\,}(2{q}+1)\alpha\delta^{2}+\varsigma-\gamma^{2}\alpha,$
(23b) $\displaystyle a_{2}$ $\displaystyle{\,:=\,}-({q}+1)\alpha\delta^{2},$
(23c) if $a_{0}<0$; otherwise $\beta_{\text{v}}=0$. 0pt
$\displaystyle\text{Furthermore, if $a_{0}<0$, then
}\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}\big{(}\breve{{\mathcal{E}}}(\beta_{\text{v}})\big{)}<\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}\big{(}{\mathcal{E}}(\bm{c},\varsigma
P)\big{)}.$ (23d)
Proof. cf. Appendix B.4. ❑
Algorithm 4 Optimal value of parameter $\beta$ minimizing the volume of
$\breve{\mathcal{E}}(\beta)$
0: $q$, $\alpha$, $\gamma$, $\delta$, $\varsigma$ {defined in (21)}
0: $\beta_{\textit{v}}$
1: $a_{0}{\,:=\,}{q}\alpha(\gamma^{2}-\delta^{2})-\varsigma$; {cf. (23a)};
2: if $a_{0}\geq 0$ then
3: $\beta_{\textit{v}}{\,:=\,}0$;
4: else
5: $a_{1}{\,:=\,}(2{q}+1)\alpha\delta^{2}+\varsigma-\gamma^{2}\alpha$; {cf.
(23b)};
6: $a_{2}{\,:=\,}-({q}+1)\alpha\delta^{2}$; {cf. (23c)};
7:
$\beta_{\textit{v}}{\,:=\,}\dfrac{-a_{1}-\sqrt{a_{1}^{2}-4a_{0}a_{2}}}{2a_{2}}$;
8: end if
#### 4.3.3 Minimization of the sum of the squared axes’ lengths
###### Theorem 4.3.3
If the ellipsoid $\breve{{\mathcal{E}}}(\beta)$ is given by (21) of Thm 4.2,
then
$\beta_{\text{s}}{\,:=\,}\arg\min_{\beta\in]0,1[}\operatorname*{tr}\big{(}\breve{\varsigma}(\beta)\breve{P}(\beta)\big{)}$
is the unique positive solution to the cubic equation $\displaystyle
b_{3}\beta^{3}+b_{2}\beta^{2}+b_{1}\beta+b_{0}{\,:=\,}0,$ (24a) if $b_{0}<0$;
$\beta_{\text{s}}=0$ otherwise; where $\displaystyle b_{0}$
$\displaystyle{\,:=\,}\nu\varsigma-\tau(\delta^{2}-\gamma^{2}),\
\tau{\,:=\,}\operatorname*{tr}(P)\text{ and
}\nu{\,:=\,}\bm{\varphi}^{T}\bm{\varphi}=\bm{f}^{T}P^{2}\bm{f},$ (24b)
$\displaystyle b_{1}$
$\displaystyle{\,:=\,}2\big{(}\tau\delta^{2}-\nu\varsigma+\alpha\nu(\delta^{2}-\gamma^{2})\big{)},$
(24c) $\displaystyle b_{2}$
$\displaystyle{\,:=\,}\nu\varsigma-\tau\delta^{2}+\alpha\nu(\gamma^{2}-4\delta^{2}),$
(24d) $\displaystyle b_{3}$ $\displaystyle{\,:=\,}2\alpha\nu\delta^{2}.\quad$
(24e) 0pt $\displaystyle\text{Furthermore, if $b_{0}<0$, then
}\operatorname*{ssal}\big{(}\breve{{\mathcal{E}}}(\beta_{\text{s}})\big{)}<\operatorname*{ssal}\big{(}{\mathcal{E}}(\bm{c},\varsigma
P)\big{)}.$ (24f)
Proof. cf. Appendix B.5. ❑
The computation of $\beta_{\text{s}}$ is elaborated in Algorithm 5.
Algorithm 5 Optimal value of the parameter $\beta$ minimizing the SSAL of
$\breve{\mathcal{E}}(\beta)$
0: $P$, $\alpha$, $\gamma$, $\delta$, $\varsigma$, $\bm{\varphi}$ {defined in
(21)}
0: $\beta_{\text{s}}$
1: $\tau{\,:=\,}\operatorname*{tr}(P)$;
$\nu{\,:=\,}\bm{\varphi}^{T}\bm{\varphi}$;
2: $b_{0}{\,:=\,}\nu\varsigma-\tau(\delta^{2}-\gamma^{2})$;
3: if $b_{0}>=0$, then
4: $\beta_{\text{s}}{\,:=\,}0$;
5: else
6:
$b_{1}{\,:=\,}2\big{(}\tau\delta^{2}-\nu\varsigma+\alpha\nu(\delta^{2}-\gamma^{2})\big{)}$;$b_{2}{\,:=\,}\nu\varsigma-\tau\delta^{2}+\alpha\nu(\gamma^{2}-4\delta^{2})$;
$b_{3}{\,:=\,}2\alpha\nu\delta^{2}$;
7: ${s}{\,:=\,}\tfrac{b_{1}}{b_{3}}-\tfrac{b_{2}^{2}}{3b_{3}^{2}}$;
${t}{\,:=\,}\tfrac{b_{0}}{b_{3}}-\tfrac{b_{1}b_{2}}{3b_{3}^{2}}+\tfrac{2b_{2}^{3}}{27b_{3}^{3}}$;
8:
${u}{\,:=\,}\left(\tfrac{{s}}{3}\right)^{3}+\big{(}\tfrac{{t}}{2}\big{)}^{2}$;
${v}{\,:=\,}\sqrt[3]{-\tfrac{{t}}{2}+\sqrt{{u}}}$;
${w}{\,:=\,}\sqrt[3]{-\tfrac{{t}}{2}-\sqrt{{u}}}$;
9: $\omega{\,:=\,}-\tfrac{1}{2}+\tfrac{1}{2}\sqrt{3}i$;
$\omega_{s}{\,:=\,}-\tfrac{1}{2}-\tfrac{1}{2}\sqrt{3}i$; { $\omega$ is the
cubic root of 1 and $\omega_{s}{\,:=\,}\omega^{2}$}
10: $\bar{\beta}_{1}{\,:=\,}{v}+{w}$;
$\bar{\beta}_{2}{\,:=\,}\omega{v}+\omega_{s}{w}$;
$\bar{\beta}_{3}{\,:=\,}\omega_{s}{v}+\omega{w}$;
11: $\beta_{1}{\,:=\,}\bar{\beta}_{1}-\tfrac{b_{2}}{3b_{3}}$;
$\beta_{2}{\,:=\,}\bar{\beta}_{2}-\tfrac{b_{2}}{3b_{3}}$;
$\beta_{3}{\,:=\,}\bar{\beta}_{3}-\tfrac{b_{2}}{3b_{3}}$;
12: $\beta_{\text{s}}{\,:=\,}\beta_{i}$, $i\in\\{1,2,3\\}$, _s.t._
$\beta_{i}\in{\mathop{\rm I\mkern-3.5muR}}^{*}_{+}$.
13: end if
### 4.4 Intersection of an ellipsoid with a hyperplane
In this paragraph, the equality-type constraint (3c) is examined. This
constraint on the state vector can be also viewed as a noiseless measurement,
_a.k.a._ pseudo-measurement. Let us examine the intersection of an ellipsoid
with a hyperplane. This intersection leads to a degenerate ellipsoid of lesser
dimension, whose shape matrix loses one rank with each intersecting (not
parallel) hyperplane.
The theorem below gives the expression of thusly obtained ellipsoid.
###### Theorem 4.3 (ellips./hyperplane inters.)
Let $\bm{c}\in~{}{{{\mathop{\rm I\mkern-3.5muR}}}^{n}}$, $P\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n\times n}}$ SPSD, $\varsigma\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}$, $\bm{f}\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}\text{--}\\{{\mathbf{0}}_{n}\\}$ and $y\in{\mathop{\rm
I\mkern-3.5muR}}$,
if $y>\bar{\rho}\vee y<-\underaccent{\bar}{\rho}$, (case 1)
$\displaystyle{\mathcal{E}}(\bm{c},\varsigma P)\cap{\mathcal{H}}(\bm{f},y)$
$\displaystyle=\emptyset;$ (25b) else if
$y=\bar{\rho}=-\underaccent{\bar}{\rho}$, (case 2)
$\displaystyle{\mathcal{E}}(\bm{c},\varsigma P)\cap{\mathcal{H}}(\bm{f},y)$
$\displaystyle={\mathcal{E}}(\bm{c},\varsigma P);$ (25c) else if
$y=-\underaccent{\bar}{\rho}$, (case 3a)
$\displaystyle{\mathcal{E}}(\bm{c},\varsigma P)\cap{\mathcal{H}}(\bm{f},y)$
$\displaystyle=\\{\bm{c}-\varsigma^{\frac{1}{2}}(\bm{f}^{T}P\bm{f})^{-\frac{1}{2}}P\bm{f}\\};$
(25d) else if $y=\bar{\rho}$, (case 3b)
$\displaystyle{\mathcal{E}}(\bm{c},\varsigma P)\cap{\mathcal{H}}(\bm{f},y)$
$\displaystyle=\\{\bm{c}+\varsigma^{\frac{1}{2}}(\bm{f}^{T}P\bm{f})^{-\frac{1}{2}}P\bm{f}\\};$
(25e) otherwise $(\text{if }-\underaccent{\bar}{\rho}<y<\bar{\rho})$ (case 4)
$\displaystyle{\mathcal{E}}(\bm{c},\varsigma P)\cap{\mathcal{H}}(\bm{f},y)$
$\displaystyle={\mathcal{E}}(\check{\bm{c}},\check{\varsigma}\check{P}),$
(25f) where $\displaystyle\check{\bm{c}}$
$\displaystyle{\,:=\,}\bm{c}+\alpha\delta{P\bm{f}},$ (25g)
$\displaystyle\check{P}$
$\displaystyle{\,:=\,}P-{\alpha}{P\bm{f}\bm{f}^{T}P},$ (25h)
$\displaystyle\check{\varsigma}$
$\displaystyle{\,:=\,}\varsigma-{\alpha}{\delta^{2}},$ (25i)
$\displaystyle\delta$ $\displaystyle{\,:=\,}y-\bm{f}^{T}\bm{c}$ (25j) and
where $\alpha$, $\underaccent{\bar}{\rho}$ and $\bar{\rho}$ are defined in
(21g), (19g) and (19h) _resp_.
Proof. cf. Appendix B.6. ❑
The figure 5 is self-explanatory. It is plain to see that the intersection of
a three-dimensional ellipsoid with an intersecting non parallel hyperplane is
a degenerate three-dimensional, corresponding to a two-dimensional, ellipsoid
and the good news is that there is no need to circumscribe it by an other one,
as it is done in case of intersection with strip or halfspace. Please note
that the case 2 happens only when the ellipsoid
${\mathcal{E}}(\bm{c},\varsigma P)$ is already degenerate and contained in the
hyperplane ${\mathcal{H}}(\bm{f},y)$, meaning that $y=\bm{c}^{T}\bm{f}$ and
that $\bm{f}\in\mathpzc{Ker}(P)$.
(a) (1a): $y-\bm{c}^{T}\bm{f}>\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
(b) (1b): $y-\bm{c}^{T}\bm{f}<-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
(c) (2):
${y-\bm{c}^{T}\bm{f}}\\!=\\!{\bm{f}^{T}P\bm{f}}\\!=\\!-{\bm{f}^{T}P\bm{f}}=0$
(d) (3a): $y-\bm{c}^{T}\bm{f}=\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
(e) (3b): $y-\bm{c}^{T}\bm{f}=-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
(f) (4): ${\left|y-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
Figure 5: Illustration of the four cases of Theorem 4.3 for $n=3$
### 4.5 The output update algorithm
Hereafter, the measurement update part of the state estimation algorithm is
summarized. All the variables are depending on the time step $k$ even if the
subscript $k$ was skipped for some of them.
###### Theorem 4.4 (Correction stage)
Let
0pt
$\displaystyle\bar{y}_{k_{i}}{\,:=\,}\min(\bar{y}_{k_{i}},\bar{{\rho}}_{k_{i}})$
$\displaystyle\text{ and
}\underaccent{\bar}{y}_{k_{i}}{\,:=\,}\max(\underaccent{\bar}{y}_{k_{i}},-\underaccent{\bar}{{\rho}}_{k_{i}}),\
i\in\\{1,\ldots,{p}\\}.$ (26a) If $\mbox{\boldmath$x$}_{k}$ satisfies (1)
meeting (3), then $\displaystyle\mbox{\boldmath$x$}_{k}\in{\mathcal{S}}_{k}$
$\displaystyle\subseteq{\mathcal{E}}_{k}{\,:=\,}{\mathcal{E}}(\hat{\mbox{\boldmath$x$}}_{k},\varsigma_{k}P_{k}),$
(26b) 0pt ${\mathcal{S}}_{k}$ is defined in (18) and, $\forall
k\in{\mathop{\rm I\mkern-3.5muN}}^{*}$, $\displaystyle\varsigma_{k}$
$\displaystyle{\,:=\,}\varsigma_{k_{p}},P_{k}{\,:=\,}P_{k_{p}}\text{ and
}\hat{\mbox{\boldmath$x$}}_{k}{\,:=\,}\mbox{\boldmath$\hat{x}$}_{k_{p}};\quad\text{and
for $i\in\\{1,\ldots,{p}\\}$:}$ (26d) $\displaystyle P_{k_{i}}$
$\displaystyle{\,:=\,}P_{k_{i-1}}-{\alpha_{k_{i}}\beta_{k_{i}}}{{\bm{\varphi}}_{k_{i}}{\bm{\varphi}}_{k_{i}}^{T}},$
(26e) $\displaystyle\hat{\mbox{\boldmath$x$}}_{k_{i}}$
$\displaystyle{\,:=\,}\hat{\mbox{\boldmath$x$}}_{k_{i-1}}+{\alpha_{k_{i}}\beta_{k_{i}}}{\delta_{k_{i}}}{\bm{\varphi}}_{k_{i}},$
(26f) $\displaystyle\varsigma_{k_{i}}$
$\displaystyle{\,:=\,}\begin{cases}\varsigma_{k_{i-1}}-{\alpha_{k_{i}}\beta_{k_{i}}^{2}{\delta_{k_{i}}^{2}}},&\text{if
}\big{(}\underaccent{\bar}{y}_{k_{i}}=\bar{y}_{k_{i}}\wedge-\underaccent{\bar}{{\rho}}_{k_{i}}\neq\bar{{\rho}}_{k_{i}}\big{)}\vee\beta^{*}=\beta_{\varsigma},\\\
\varsigma_{k_{i-1}}+{\alpha_{k_{i}}\beta_{k_{i}}\bigg{(}\tfrac{\gamma_{k_{i}}^{2}}{1-\beta_{k_{i}}}-{\delta_{k_{i}}^{2}}\bigg{)}},&\text{otherwise;}\end{cases}$
(26g) $\displaystyle P_{k_{0}}$
$\displaystyle{\,:=\,}P_{k/k-1},\hat{\mbox{\boldmath$x$}}_{k_{0}}{\,:=\,}\hat{\mbox{\boldmath$x$}}_{k/k-1}\text{
and }\varsigma_{k_{0}}{\,:=\,}\varsigma_{k-1},$ (26h) where
$\hat{\mbox{\boldmath$x$}}_{k/k-1}\text{ and }P_{k/k-1}$ are computed
according to Thms 3.6 and 3.7 or 3.8, with
${q}_{k_{0}}{\,:=\,}q_{k/k-1}{\,:=\,}{q}_{{k/k-1}_{m}}$ computed at the
prediction stage, in (11d) (line 13 of Algorithm 1) and
${q}_{k}{\,:=\,}\operatorname*{rank}(P_{k}){\,:=\,}{q}_{k_{m}}$.
$\displaystyle\alpha_{k_{i}}$
$\displaystyle{\,:=\,}\begin{cases}\theta_{k_{i}}^{-1},&\text{if }{p_{k}}\neq
0\text{ and }\lambda_{k_{i}}\neq 0,\\\ 0,&\text{otherwise};\end{cases}$ (26i)
$\displaystyle\beta_{k_{i}}$ $\displaystyle{\,:=\,}\begin{cases}1,&\text{if
}\underaccent{\bar}{y}_{k_{i}}=\bar{y}_{k_{i}}\wedge-\underaccent{\bar}{{\rho}}_{k_{i}}\neq\bar{{\rho}}_{k_{i}},(i\in\mathscr{H}_{k}),\\\
\beta^{*},&\text{else if
}-\underaccent{\bar}{{\rho}}_{k_{i}}<\underaccent{\bar}{y}_{k_{i}}\vee\bar{y}_{k_{i}}<\bar{{\rho}}_{k_{i}},(i\in\mathscr{D}_{k}\cup\mathscr{G}_{k})\\\
0,&\text{otherwise;}\end{cases}$ (26j) where, either
$\beta^{*}=\beta_{\varsigma}$, cf. (22c) - Algo. 3,
$\beta^{*}=\beta_{\text{v}}$, given by Thm 4.3.2 \- Algo. 4 or
$\beta^{*}=\beta_{\text{s}}$, cf. Thm 4.3.3 \- Algo. 5; where
$\alpha{\,:=\,}\alpha_{k_{i}}$, $\gamma{\,:=\,}\gamma_{k_{i}}$,
$\delta{\,:=\,}{\delta_{k_{i}}}$, $\varsigma{\,:=\,}\varsigma_{k_{i-1}}$,
$\bm{\varphi}{\,:=\,}{\bm{\varphi}}_{k_{i}}$, $P{\,:=\,}P_{k_{i-1}}$ and
${q}{\,:=\,}{q}_{k_{i-1}}$and $\displaystyle{\delta_{k_{i}}}$
$\displaystyle{\,:=\,}\gamma_{k_{i}}{y_{k_{i}}}-\tfrac{1}{2}(\bar{{\rho}}_{k_{i}}-\underaccent{\bar}{{\rho}}_{k_{i}}),\quad{y_{k_{i}}}{\,:=\,}\tfrac{1}{2\gamma_{k_{i}}}({\bar{y}_{k_{i}}+\underaccent{\bar}{y}_{k_{i}}}),$
(26k) $\displaystyle\gamma_{k_{i}}$
$\displaystyle{\,:=\,}\tfrac{1}{2}(\bar{y}_{k_{i}}-\underaccent{\bar}{y}_{k_{i}}),$
(26l) $\displaystyle\theta_{k_{i}}$
$\displaystyle{\,:=\,}\bm{f}_{k_{i}}^{T}{\bm{\varphi}}_{k_{i}},$ (26m)
$\displaystyle{\bm{\varphi}}_{k_{i}}$
$\displaystyle{\,:=\,}P_{k_{i-1}}\bm{f}_{k_{i}},$ (26n)
$\displaystyle\bar{{\rho}}_{k_{i}}$
$\displaystyle{\,:=\,}\lambda_{k_{i}}+\bm{f}_{k_{i}}^{T}\hat{\mbox{\boldmath$x$}}_{k_{i-1}}\text{
and
}\underaccent{\bar}{{\rho}}_{k_{i}}{\,:=\,}2\lambda_{k_{i}}-\bar{{\rho}}_{k_{i}},$
(26o) $\displaystyle\lambda_{k_{i}}$
$\displaystyle{\,:=\,}(\varsigma_{k_{i-1}}\theta_{k_{i}})^{\frac{1}{2}};$
(26p) and, if ${\mathcal{E}}_{k/k-1}$ is computed according to Thms 3.6 and
3.7 by (10)-(11), then $\displaystyle{q}_{k_{i}}$
$\displaystyle\big{(}{\,:=\,}\operatorname*{rank}(P_{k_{i}})\big{)}{\,:=\,}\begin{cases}{q}_{k_{i-1}}-1,&\text{if
}\underaccent{\bar}{y}_{k_{i}}=\bar{y}_{k_{i}}\wedge-\underaccent{\bar}{{\rho}}_{k_{i}}\neq\bar{{\rho}}_{k_{i}}\wedge\alpha_{k_{i}}\neq
0,\\\ {q}_{k_{i-1}},&\text{otherwise.}\end{cases}$ (26q)
Proof. cf. Appendix B.7. ❑
The correction stage described in Thm 4.4 is resumed in Algorithm 6 and
${\mathcal{E}}_{k}$ is computed where
$\hat{\mbox{\boldmath$x$}}_{k/k-1}\rightarrow~{}\\!\bar{}\mbox{\boldmath$x$}$,
$P_{k/k-1}\rightarrow~{}\\!\bar{P}$,
$\varsigma_{k-1}\rightarrow~{}\\!\bar{\varsigma}$,
$q_{k/k-1}\leftarrow~{}\\!\bar{q}$ (from the prediction stage Algorithm 1 or
2), ${F}_{k}\rightarrow~{}\\!F=[\bm{f}_{i}]$,
$\bar{\bm{y}}_{k}\rightarrow~{}\\!\bar{\mbox{\boldmath$y$}}=~{}\\![\bar{y}_{i}]$,
${i\in\\{1,\dots,p\\}}$,
$\underaccent{\bar}{\y}_{k}\rightarrow~{}\\!\underaccent{\bar}{\yb}$ (as in
(3)) and $\mbox{\boldmath$x$}\rightarrow~{}\\!\hat{\mbox{\boldmath$x$}}_{k}$,
$P\rightarrow~{}\\!P_{k}$, $\varsigma\rightarrow~{}\\!\varsigma_{k}$. Note
that the updating of the rank of the shape matrix $P_{k_{i}}$, by (26q), is
done only when the volume minimization criterion is chosen at the prediction
step, by setting Predic_$\mu$ to “True”. Let Correc_$\beta$ $\in\\{0,1,2\\}$
set to 0, 1 or 2 when choosing the minimization of
$\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}({\mathcal{E}}_{k_{i}})$,
the SSAL of ${\mathcal{E}}_{k_{i}}$ or $\varsigma_{k_{i}}$, _resp_., during
the correction stage.
Algorithm 6 Computation of the corrected ellipsoid
0: $\bar{}\mbox{\boldmath$x$}$, $\bar{P}$, $\bar{\varsigma}$ ${F}$,
$\bar{\mbox{\boldmath$y$}}$,
$\underaccent{\bar}{\yb}=[\underaccent{\bar}{y}_{i}]_{i\in\\{1,\dots,p\\}}$,
Predic_$\mu$, Correc_$\beta$ , $\bar{q}$ (if Predic_$\mu$)
0: $x$, $P$, $\varsigma$, $q$ (if Predic_$\mu$)
1: $p\leftarrow$ number of columns of $F$;
2: $\mbox{\boldmath$x$}\leftarrow\bar{}\mbox{\boldmath$x$}$;
$P\leftarrow\bar{P}$; $\varsigma\leftarrow\bar{\varsigma}$;
$q\leftarrow\bar{q}$ (if Predic_$\mu$);
3: if $p\neq 0$ then
4: for $i=1,\ldots p$ do
5: $\bm{\varphi}{\,:=\,}P\bm{f}_{i}$;
$\theta{\,:=\,}\bm{f}_{i}^{T}\bm{\varphi}$; {cf. (26n)};
6: if $\theta\neq 0$ then
7: $\alpha{\,:=\,}\theta^{-1}$;
$\lambda{\,:=\,}(\varsigma\theta)^{\frac{1}{2}}$; {cf. (26i), (26n)};
8: $\bar{\rho}{\,:=\,}\lambda+\bm{f}_{i}^{T}\mbox{\boldmath$x$}$;
$\underaccent{\bar}{\rho}{\,:=\,}2\lambda-\bar{\rho}$; {cf. (26o)};
9: $\bar{y}{\,:=\,}\min(\bar{y}_{i},\bar{\rho})$;
$\underaccent{\bar}{y}{\,:=\,}\max(\underaccent{\bar}{y}_{i},-\underaccent{\bar}{\rho})$;
{cf. (26a)};
10:
$\delta{\,:=\,}\tfrac{1}{2}({\bar{y}+\underaccent{\bar}{y}}-\bar{\rho}+\underaccent{\bar}{\rho})$;
{cf. (26k)};
11: $\gamma{\,:=\,}\frac{1}{2}(\bar{y}-\underaccent{\bar}{y})$; {cf. (26l)};
12: if $\underaccent{\bar}{y}=\bar{y}\text{ and
}-\underaccent{\bar}{\rho}\neq\bar{\rho}$ then
13: $\beta=1$;
14: if Predic_$\mu$ $=$ True then
15: $q\leftarrow q-1$
16: end if
17: else if $-\underaccent{\bar}{\rho}<\underaccent{\bar}{y}$ or
$\bar{y}<\bar{\rho}$ then
18: if Correc_$\beta$ $=0$ then
19: Algorithm 4 (Input: $q$, $\alpha$, $\gamma$, $\delta$, $\varsigma$;
Output: $\beta_{\text{v}}$)
20: $\beta{\,:=\,}\beta_{\text{v}}$;
21: else if Correc_$\beta$ $=1$ then
22: Algorithm 5 (Input: $P$, $\alpha$, $\gamma$, $\delta$, $\varsigma$,
$\bm{\varphi}$; Output: $\beta_{\text{s}}$)
23: $\beta{\,:=\,}\beta_{\text{s}}$;
24: else
25: Algorithm 3 (Input: $\delta$, $\gamma$; Output: $\beta_{\varsigma}$ )
26: $\beta{\,:=\,}\beta_{\varsigma}$;
27: end if
28: else
29: $\beta=0$;
30: end if
31: $P\leftarrow P-{\alpha\beta}{\bm{\varphi}\bm{\varphi}^{T}}$;
$\mbox{\boldmath$x$}\leftarrow\mbox{\boldmath$x$}+{\alpha\beta}\delta\bm{\varphi}$;
$\varsigma\leftarrow\varsigma-{\alpha\beta^{2}\delta^{2}}$; {cf. (26e)–(26g)};
32: end if
33: end for
34: end if
###### Remark 4.2
This algorithm is of low computational complexity : $O(n^{2})$. Indeed, all
the operations are simple sums and products: they were optimized in this
regard and are thence suitable for systems with high dimensional state vector
(big $n$) and with many measurements (big ${p_{k}}$). The intermediate
variables $\alpha_{k_{i}}$, $\theta_{k_{i}}$, $\lambda_{k_{i}}$,
${\bm{\varphi}}_{k_{i}}$ were added on to perform redundant vector and matrix
operations only once. Thereby noticing that
$\bm{f}_{k_{i}}^{T}\hat{\mbox{\boldmath$x$}}_{k_{i-1}}=\tfrac{1}{2}(\bar{{\rho}}_{k_{i}}-\underaccent{\bar}{{\rho}}_{k_{i}})$
allows to determine
${\delta_{k_{i}}}{\,:=\,}\tfrac{1}{2}({\bar{y}_{k_{i}}+\underaccent{\bar}{y}_{k_{i}}})-\bm{f}_{k_{i}}^{T}\hat{\mbox{\boldmath$x$}}_{k_{i-1}}$
and
$\underaccent{\bar}{{\rho}}_{k_{i}}{\,:=\,}\lambda_{k_{i}}-\bm{f}_{k_{i}}^{T}\hat{\mbox{\boldmath$x$}}_{k_{i-1}}$
using addition of scalars, in (26k) and (26o) _resp_., rather than
multiplication of possibly high dimensional vectors.
###### Remark 4.3
In the case where $\mathscr{H}_{k}\neq\emptyset$, the matrix $P_{k_{i}}$ loses
rank with each intersecting hyperplane ${\mathcal{H}}_{k_{i}}$,
$i\in\mathscr{H}_{k}$, thusly entailing the progressive flattening of the
ellipsoid ${\mathcal{E}}_{k_{i}}$. Depending on the rank of the matrix $R_{k}$
(on which no assumption is made), the rank of $P_{k+1/k}$ can be recovered at
the time-update phase. The value of $\operatorname*{rank}(P_{k_{i}})$ is
needed at each $i=1\ldots,m,\ldots,m+{p_{k}}$ for each time step $k$, whenever
the volume of the ellipsoid ${\mathcal{E}}_{k}$ is minimized. So keeping track
of this parameter through simple relations, as (11d) during the time update
and (26q) during the observation update, spares the need to calculate it each
time, provided that $\operatorname*{rank}(A_{0}P_{0}A_{0}^{T})$ is given.
###### Remark 4.4
Setting either $\alpha_{k_{i}}=0$ or $\beta_{k_{i}}=0$ results in freezing
${\mathcal{E}}_{k_{i-1}}$, meaning that the corresponding measurements
$\bm{f}_{k_{i}},\underaccent{\bar}{y}_{k_{i}},\bar{y}_{k_{i}}$ do not bring
any useful information.
###### Remark 4.5
The cases 1 of Thms 4.1, 4.2 and 4.3 are not explicitly treated in this
theorem assuming that they can not occur since the intervening measurements
should be consistent with the system model; yet the case where the measurement
$\\{\bm{f}_{k_{i}},\underaccent{\bar}{y}_{k_{i}},\bar{y}_{k_{i}}\\}$ is
aberrant is implicitly considered, setting again either $\alpha_{k_{i}}=0$ or
$\beta_{k_{i}}=0$, preventing so the updating of the ellipsoid
${\mathcal{E}}_{k_{i-1}}$.
## 5 Algorithm properties and stability analysis
### 5.1 The overall state estimation algorithm
The time prediction stage given by Thm 3.6 with either Thm 3.7 or Thm 3.8 on
one hand, and on the other, the measurement correction phase, given by Thm 4.4
are concatenated to form the hole state estimation algorithm presented in
Algorithm 7, where $N$ is the number of samples.
Algorithm 7 Computation of the ellipsoid
${\mathcal{E}}(\hat{\mbox{\boldmath$x$}}_{k},\varsigma_{k}P_{k})$
0: $\hat{x}_{0}$, $\varsigma_{0}$, $P_{0}$, $N$, Predic_$\mu$, Correc_$\beta$
0: $\hat{\mbox{\boldmath$x$}}_{k}$, $\varsigma_{k}$, $P_{k}$,
$k\in\\{1,\ldots,N\\}$
1: $n\leftarrow$ size of $\hat{x}_{0}$
2: for $k=0,1,\ldots,N-1$ do
{// Prediction //}
3: if Predic_$\mu$ then
4: Algorithm 1 (Input: $\hat{\mbox{\boldmath$x$}}_{k}$, $P_{k}$,
$\varsigma_{k}$, $A_{k}$, $B_{k}$, $R_{k}$, $\mbox{\boldmath$\tau$}_{k}$,
Predic_$\mu$, $q_{k}$; Output $\hat{\bm{x}}_{k+1/k}$, $P_{k+1/k}$,
$\varsigma_{k+1/k}$, Predic_$\mu$, $q_{k+1/k}$)
5: else
6: Algorithm 2 (Input: $\hat{\mbox{\boldmath$x$}}_{k}$, $P_{k}$,
$\varsigma_{k}$, $A_{k}$, $B_{k}$, $R_{k}$,
$\mbox{\boldmath$\tau$}_{k}$;Output: $\hat{\bm{x}}_{k+1/k}$, $P_{k+1/k}$,
$\varsigma_{k+1/k}$, Predic_$\mu$)
7: end if
{// Correction //}
8: $k\leftarrow k+1$
9: Algorithm 6 (Input: $\hat{\mbox{\boldmath$x$}}_{k/k-1}$, $P_{k/k-1}$,
$\varsigma_{k/k-1}$, ${F}_{k}$, $\bar{\bm{y}}_{k}$,
$\underaccent{\bar}{\y}_{k}$, Predic_$\mu$, Correc_$\beta$ , $q_{k/k-1}$;
Output: $\hat{\mbox{\boldmath$x$}}_{k}$, $P_{k}$, $\varsigma_{k}$, $q_{k}$)
10: $P_{k}\leftarrow\frac{\varsigma}{\varsigma_{0}}P_{k}$;
$\varsigma_{k}\leftarrow\varsigma_{0}$;
11: end for
###### Remark 5.1
For more numerical stability and in order to avoid the explosion of the matrix
$P_{k}$, caused by the set summations at the prediction step, the assignments
(26h) corresponding to the line 10 of the Algorithm 7 can be replaced by
$P_{k}\leftarrow\frac{\varsigma_{k_{p}}}{\varsigma_{0}}P_{k_{p}}$,
$\bar{\varsigma}_{k}\leftarrow\frac{\varsigma_{0}\varsigma_{k_{p}}}{\bar{\varsigma}_{k-1}}$
and $\varsigma_{k}\leftarrow\varsigma_{0}$. Then $P_{k}$ would, by itself,
represent the shape of the ellipsoid ${\mathcal{E}}_{k}$ up to a constant
factor $\varsigma_{0}^{-1}$ and the new variable $\bar{\varsigma}_{k}$ is
introduced to keep track of the decreasing parameter $\varsigma_{k}$, with
$\bar{\varsigma}_{0}=\varsigma_{0}$.
The proposed algorithm is designed in such a way as to fulfill the
requirements 1. \- 3., expressed in the $\mathsection$2 and this is what will
be shown in this section.
### 5.2 Algorithm properties
In this paragraph, some algorithm’s properties are shown, while the stability
will be examined in the next.
###### Theorem 5.1
Consider the system (1) subject to (3) and its state estimation algorithm
given by Thms 3.6 and 3.7 or 3.8 on one hand and Thm 4.4, on the other.
1. 1.
If
$\mbox{\boldmath$x_{0}$}\in{\mathcal{E}}(\mbox{\boldmath$\hat{x}_{0}$},\varsigma_{0}P_{0})$,
then $\forall\bm{\beta}\in[0,1[^{p_{k}}$, $\forall\bm{\mu}\in]0,+\infty[^{m}$,
$\mbox{\boldmath$x$}_{k}\in{\mathcal{E}}(\hat{\mbox{\boldmath$x$}}_{k},\varsigma_{k}P_{k})$,
$\forall k\in{\mathop{\rm I\mkern-3.5muN}}^{*}$;
2. 2.
The vector $\hat{\mbox{\boldmath$x$}}_{k}$ is _acceptable_ , i.e., it
satisfies (3a)–(3e):
$\displaystyle\forall k\in{\mathcal{K}},\
\hat{\mbox{\boldmath$x$}}_{k}\in\bigcap_{i\in\mathscr{G}_{k}}{\mathcal{G}}_{k_{i}}\cap\bigcap_{i\in\mathscr{D}_{k}}{\mathcal{D}}_{k_{i}}\cap\bigcap_{i\in\mathscr{H}_{k}}{\mathcal{H}}_{k_{i}},$
(27a)
where ${\mathcal{K}}{\,:=\,}\\{k\in{\mathop{\rm I\mkern-3.5muN}}|{p_{k}}\neq
0\\}$.
3. 3.
$\varsigma_{k}$, defined in (26g), satisfies,
$\displaystyle\varsigma_{k}$
$\displaystyle=\max_{\bm{x}_{k}\in{\mathcal{S}}_{k}}\mathscr{V}_{{P}_{k}}(\mbox{\boldmath$x$}_{k}-\hat{\mbox{\boldmath$x$}}_{k}),\
\forall\beta^{*}\in]0,1[,$ (27b)
where ${\mathcal{S}}_{k}$ and $\mathscr{V}_{{P}}$ are given by (18) and (22a)
_resp_.
Moreover, if in (26j), $\beta^{*}=\beta_{\varsigma}$, given by (22c) - Algo.
3, then the sequence $\left\\{\varsigma_{k}\right\\}_{k\in{\mathop{\rm
I\mkern-3.5muN}}^{*}}$ is decreasing and convergent on ${\mathop{\rm
I\mkern-3.5muR}}_{+}$.
Proof. The proof of this lemma is detailed in the Appendix C.1. ❑
### 5.3 Stability analysis
The stability requirement 3. exploits the Input-to-State stability concept:
roughly speaking, for an ISS system, inputs that are bounded, “eventually
small”, or convergent, should lead to the state vector with the respective
property; and the ${\mathbf{0}}$-input system should be globally stable. We
shall now recall more formal definitions and results about the ISS concept.
Before doing so, let us recall some comparison functions, widely used in
stability analysis. A continuous function $\psi_{1}:{\mathop{\rm
I\mkern-3.5muR}}_{+}\rightarrow{\mathop{\rm I\mkern-3.5muR}}_{+}$ is called
positive definite if it satisfies $\psi_{1}(0)=0$ and $\psi_{1}(t)>0$,
$\forall t>0$. A positive definite function is of class $\mathscr{K}$ if it is
strictly increasing and of class $\mathscr{K}_{\infty}$ if it is of class
$\mathscr{K}$ and unbounded. A continuous function $\psi_{2}:{\mathop{\rm
I\mkern-3.5muR}}_{+}\rightarrow{\mathop{\rm I\mkern-3.5muR}}_{+}$ is of class
$\mathscr{L}$ if $\psi_{2}(t)$ is strictly decreasing to 0 as
$t\rightarrow\infty$ and a continuous function $\psi_{3}:{\mathop{\rm
I\mkern-3.5muR}}_{+}\times{\mathop{\rm
I\mkern-3.5muR}}_{+}\rightarrow{\mathop{\rm I\mkern-3.5muR}}_{+}$ is of class
$\mathscr{KL}$ if it is of class $\mathscr{K}$ in the first argument and of
class $\mathscr{L}$ in the second argument.
###### Definition 5.1 (based on [32])
The system 0pt
$\displaystyle\mbox{\boldmath$z$}_{k+1}$
$\displaystyle=f_{k}\big{(}\mbox{\boldmath$z$}_{k},\bm{u}_{k}\big{)},$ (28a)
$\displaystyle\text{where }\qquad f({\mathbf{0}},{\mathbf{0}})$
$\displaystyle={\mathbf{0}},$ (28b)
is globally input-to-state stable (ISS), if there exists a
$\mathscr{KL}$-function $\phi$ and a $\mathscr{K}$-function $\psi$ such that,
for each bounded input sequence
$\bm{u}_{[0,k]}{\,:=\,}\left\\{\bm{u}_{0},\ldots,\bm{u}_{k}\right\\}$ and each
$\mbox{\boldmath$z$}_{0}\in~{}{{{\mathop{\rm I\mkern-3.5muR}}}^{n}}$,
$\displaystyle\left\|\mbox{\boldmath$z$}(k,\mbox{\boldmath$z$}_{0},\bm{u}_{[0,k-1]})\right\|\leq\phi(\left\|\mbox{\boldmath$z$}_{0}\right\|,k)+\psi({\\!\sup_{i\in\\{1,\ldots,k-1\\}}\\!\left\|\bm{u}_{i}\right\|}),$
where $\mbox{\boldmath$z$}(k,\mbox{\boldmath$z$}_{0},\bm{u}_{[0,k-1]})$ is the
trajectory of the system (28), for the initial state
$\mbox{\boldmath$z$}_{0}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n}}$ and the
input sequence $\bm{u}_{[0,k-1]}$.
###### Definition 5.2 ([32])
_A continuous function $\mathscr{V}:{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}\rightarrow{\mathop{\rm I\mkern-3.5muR}}_{+}$ is an ISS-
Lyapunov function for the system (28), if both conditions 1 et 2 are met:_
1. 1.
there exists $\mathscr{K}_{\infty}$-functions $\psi_{1}$ and $\psi_{2}$ such
that for all $\mbox{\boldmath$z$}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n}}$,
$\displaystyle\psi_{1}(\left\|\mbox{\boldmath$z$}\right\|)\leq\mathscr{V}(\mbox{\boldmath$z$})\leq\psi_{2}(\left\|\mbox{\boldmath$z$}\right\|);$
(29)
2. 2.
there exists a $\mathscr{K}_{\infty}$\- function $\psi_{3}$ and a
$\mathscr{K}$-function $\chi$ such that for all $k\in{\mathop{\rm
I\mkern-3.5muN}}^{*}$, $\mbox{\boldmath$z$}\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}$ and all $\bm{u}\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{m}}$,
$\displaystyle\mathscr{V}\big{(}f_{k}(\mbox{\boldmath$z$},\bm{u})\big{)}-\mathscr{V}(\mbox{\boldmath$z$})\leq-\psi_{3}(\left\|\mbox{\boldmath$z$}\right\|)+\chi(\left\|\bm{u}\right\|).$
(30)
###### Lemma 5.2 ([32])
The system (28) is ISS, if it admits a continuous ISS- Lyapunov function.
To prove the ISS stability, we will be using the candidate Lyapunov function
defined in (27b). For this purpose, the shape matrix $P_{k}$ should be bounded
above and below. Before studying the boundedness of $P_{k}$, we need to recall
the uniform controllability and observability notions:
###### Definition 5.3 (uniform observability and controllability)
Consider time-varying matrices $\bar{F}_{k}\in{\mathop{\rm
I\mkern-3.5muR}}^{n\times p}$, ${R}_{k}$ and invertible
${A}_{k}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n\times n}}$. The pair
$\\{{A}_{k},\bar{F}_{k}^{T}\\}$ is _uniformly observable_ , if there exist
positive constants ${\varrho}_{1}$ and ${\varrho}_{2}$ and a positive integer
$h$, such that, for all $k\geq h$,
$\displaystyle{\varrho}_{1}I_{n}\leq{{\mathcal{O}}}_{k,k-h}\leq{\varrho}_{2}I_{n},$
(31)
where ${{\mathcal{O}}}_{k+l,k}$ is an observability gramian of length $l$: 0pt
$\displaystyle{{\mathcal{O}}}_{k+l,k}$
$\displaystyle{\,:=\,}\sum_{i=k}^{k+l}{\Phi}_{i,k}^{T}\bar{F}_{i}\bar{F}_{i}^{T}{\Phi}_{i,k}.$
(32) $\displaystyle\text{where \quad}{\Phi}_{k+l,k}$
$\displaystyle{\,:=\,}{A}_{k+l-1}\ldots{A}_{k},\text{ with
}{\Phi}_{k,k+l}={\Phi}_{k+l,k}^{-1}.$ (33)
The pair $\\{{A}_{k},{R}_{k}\\}$ is _uniformly controllable_ , if there exist
positive constants ${\varrho}_{3}$ and ${\varrho}_{4}$ and a positive integer
${h}$, such that, for all $k\geq h$,
$\displaystyle{\varrho}_{3}I_{n}\leq{{\mathcal{C}}}_{k,k-{h}}\leq{\varrho}_{4}I_{n},$
(34)
where ${{\mathcal{C}}}_{k+l,k}$ is a controllability gramian of length $h$:
$\displaystyle{{\mathcal{C}}}_{k+l,k}$
$\displaystyle{\,:=\,}\sum_{i=k}^{k+l-1}{\Phi}_{k,i+1}{R}_{i}{R}_{i}^{T}{\Phi}_{k,i+1}^{T}.$
(35)
It is needless to say that it is difficult to ensure the full rank for the
matrix sum ${{\mathcal{O}}}_{k,k-{h}}$ on a time window of constant length $h$
when dealing with sporadic measurements. The system (1)-(3) can therefore not
be uniformly observable. For this purpose, let us introduce the new
observability criterion for systems with sporadic measurements, by allowing
the observability gramian, used in _uniform observability_ to have a variable
length $\kappa_{k}(h)$ instead of the fixed one $h$:
###### Definition 5.4 (sporadic observability)
The pair $\\{{A}_{k},\bar{F}_{k}^{T}\\}$ is said _sporadically observable_ ,
if there exist positive constants $\bar{\varrho}_{1}$ and $\bar{\varrho}_{2}$
and a positive integer $h$, such that, for all $k\geq\kappa_{k}(h)$,
$\displaystyle\bar{\varrho}_{1}I_{n}\leq{{\mathcal{O}}}_{k,k-\kappa_{k}(h)}\leq\bar{\varrho}_{2}I_{n},$
where ${{\mathcal{O}}}_{k+l,k}$ is the observability gramian given in (32) and
${\Phi}_{k+l,k}$, defined in (33), is the state transition matrix associated
to ${A}_{k}\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n\times n}}$, which is assumed
to be _invertible_ ; $\kappa_{k}(h)\in{\mathop{\rm I\mkern-3.5muN}}$ is _s.t._
$\displaystyle\operatorname*{Card}\big{(}\\{i\in\bar{{\mathcal{K}}}|k-\kappa_{k}(h)\leq
i\leq k\\}\big{)}=h$ (36)
where $\operatorname*{Card}({\mathcal{S}})$ stands for the cardinality (number
of elements) of the set ${\mathcal{S}}$.
###### Lemma 5.3
Consider the system (1) subject to (3) and the matrix $P_{k}$ computed in line
with either ((10c)-(10e) and (11)) or (13) on one hand and (26), on the other.
Let $H_{k_{1}}\in~{}{\mathop{\rm I\mkern-3.5muR}}^{n\times{q}_{k}}$ whose
columns form an orthonormal basis for $\mathpzc{R}\big{(}P_{k}\big{)}$, where
${q}_{k}{\,:=\,}\operatorname*{rank}(P_{k})$ and let
$\bar{P}_{k}{\,:=\,}H_{k_{1}}P_{k}H_{k_{1}}^{T}$. If the pair
$\\{A_{k},\bar{F}_{k}^{T}\\}$ is _sporadically observable_ and
$\\{A_{k},R_{k}\\}$ is uniformly controllable, then there exist positive
finite numbers $\varrho_{k_{1}}$ and $\varrho_{k_{2}}$, _s.t._ for all
$k\geq\kappa_{k}(l)$,
$\displaystyle\varrho_{k_{1}}I_{{q}_{k}}$
$\displaystyle\leq\bar{P}_{k}\leq\varrho_{k_{2}}I_{{q}_{k}},$ (37)
$\kappa_{k}$ being given in (36) of Definition 5.4.
Proof. The proof of this lemma is detailed in Appendix C.2. ❑
The following theorem shows the stability of the estimation algorithm
according to the choice of the optimisation criterion made for $\beta_{k_{i}}$
in (26j).
###### Theorem 5.4
Consider the system (1) subject to (3) and its state estimation algorithm
given by either Thms 3.6 and 3.7 or 3.8, on one hand and 4.4, on the other.
Let
0pt $\displaystyle\bar{\mathscr{D}}_{k}$
$\displaystyle{\,:=\,}\bar{\mathscr{G}}_{k}\cup\underaccent{\bar}{\mathscr{G}}_{k}\cup\mathscr{D}_{k}=\mathscr{P}_{k}-\mathscr{H}_{k},\text{
{\it cf.\ }\eqref{bounds},}$ (38a) $\displaystyle\bar{F}_{k}$
$\displaystyle{\,:=\,}{F}_{k}(\bar{\mathscr{D}}_{k}){\,:=\,}[\bm{f}_{k_{i}}]_{i\in\bar{\mathscr{D}}_{k}}\in{\mathop{\rm
I\mkern-3.5muR}}^{n\times\bar{p}},$ (38b) $\displaystyle\bar{p}$
$\displaystyle{\,:=\,}\bar{p}_{k}{\,:=\,}\operatorname*{Card}(\bar{\mathscr{D}}_{k})\text{
and }\bar{{\mathcal{K}}}=\\{k\in{\mathop{\rm
I\mkern-3.5muN}}^{*}|\bar{p}_{k}\neq 0\\},$ (38c) $\displaystyle\text{setting
\quad}\bar{F}_{k}\bar{F}_{k}^{T}$ $\displaystyle{\,:=\,}0_{n,n},\quad\forall
k\notin\bar{{\mathcal{K}}}.$ (38d) For each time step $k\in{\mathop{\rm
I\mkern-3.5muN}}^{*}$ and each measurement $i\in\bar{\mathscr{D}}_{k}$,
consider $\beta^{*}$ the value of $\beta_{k_{i}}$ defined in (26j). If $A_{k}$
is invertible and the pairs $\\{A_{k},\bar{F}_{k}^{T}\\}$ and
$\\{A_{k},R_{k}\\}$ are _sporadically observable_ 555cf. Definition 5.4. and
uniformly controllable666cf. Definition 5.3. _resp_., then
1. 1.
the volume of ${\mathcal{E}}_{k}$ and all its axes lengths are bounded,
$\forall\beta^{*}\in]0,1[$, in (26j);
2. 2.
If $\beta^{*}=\beta_{\varsigma}$ given in (22c) of Thm 4.3.1 or Algo. 3, then
$\mathscr{V}_{{P}_{k}}:{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}\rightarrow{\mathop{\rm I\mkern-3.5muR}}_{+}$,
$\bm{x}\mapsto~{}\\!\mathscr{V}_{{P}_{k}}(\bm{x}){\,:=\,}\bm{x}^{T}P_{k}^{\dagger}\bm{x}$
is an ISS-Lyapunov function777cf. Definition 5.2 for the estimation error
$\tilde{\bm{x}}_{k}{\,:=\,}\mbox{\boldmath$x$}_{k}-\hat{\mbox{\boldmath$x$}}_{k}$,
which is ISS888cf. Lemma 5.2.
3. 3.
If $\beta^{*}=\beta_{\text{v}}$ given by Thm 4.3.2 and Algo 4, then
$\tfrac{1}{{v}_{k}}\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}\big{(}{\mathcal{E}}_{k}\big{)}$
is nonincreasing and
$\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}\big{(}{\mathcal{E}}_{k}\big{)}\leq{v}_{k}\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}\big{(}{\mathcal{E}}_{0}\big{)}$;
4. 4.
If $\beta^{*}=\beta_{\text{s}}$ given by Thm 4.3.3 and Algo. 5, then
$\tfrac{1}{{s}_{k}}\operatorname*{ssal}\big{(}{\mathcal{E}}_{k}\big{)}$ is
nonincreasing and
$\operatorname*{ssal}\big{(}{\mathcal{E}}_{k}\big{)}\leq{{s}_{k}}\operatorname*{ssal}\big{(}{\mathcal{E}}_{0}\big{)}$;
where
$\displaystyle{v}_{k}$
$\displaystyle{\,:=\,}{\prod_{j=0}^{k-1}{\left|\bar{A}_{j}P_{j}P_{j}^{\dagger}\right|}_{{\dagger}}^{2}{\left|\bar{Q}_{j}^{\dagger}\bar{Q}_{j}+\big{(}\bar{Q}_{j}^{\dagger}+I_{n}-\bar{Q}_{j}^{\dagger}\big{)}W_{j}\right|}_{{\dagger}}},\
{v}_{0}=1;$ (38e) $\displaystyle{s}_{k}$
$\displaystyle{\,:=\,}{\prod_{j=0}^{k-1}\Big{(}\operatorname*{tr}\big{(}A_{j}A_{j}^{T}\big{)}+\tfrac{\operatorname*{tr}(W_{j})}{\operatorname*{tr}(P_{j})}\Big{)}},\
{s}_{0}=1;$ (38f) $\displaystyle\bar{Q}_{j}$
$\displaystyle{\,:=\,}\bar{A}_{j}P_{j}\bar{A}_{j}^{T};$ (38g)
$\displaystyle\bar{A}_{j}$ $\displaystyle{\,:=\,}\lambda_{j}A_{j},\text{ where
}\lambda_{j}{\,:=\,}\prod_{i=1}^{m}\sqrt{1+\mu_{i}}=\sqrt{\chi_{1}},\
\forall\mu_{i}\in]0,1[;$ (38h) $\displaystyle W_{j}$
$\displaystyle{\,:=\,}\tfrac{1}{\varsigma_{j}}R_{j}\operatorname*{Diag}\big{(}\tfrac{\chi_{i}}{{\mu_{i}}}\big{)}_{i=1}^{m}R_{j}^{T},\text{
where }\chi_{i}=\prod_{j=i}^{m}(1+\mu_{j}).$ (38i)
5. 5.
Now, let $R_{k}=0_{n,m}$, $\forall k\in{\mathop{\rm I\mkern-3.5muN}}$. If
$\beta^{*}=\beta_{\varsigma}$ $\mathscr{V}_{{P}_{k}}$ is a nonincreasing
Lyapunov function for the estimation error which is Lyapunov stable.
6. 6.
Furthermore, if $\left\|A_{k}\right\|\leq 1$, $\forall k\in{\mathop{\rm
I\mkern-3.5muN}}$, then999There is an error in the formulation of this point
in the paper submitted to L-CSS.
1. $i$.
all the axes lengths of ${\mathcal{E}}_{k}$ are nonincreasing, if
$\beta^{*}=\beta_{\varsigma}$;
2. $ii$.
$\operatorname*{\textstyle\operatorname*{vol}_{\dagger}\\!}\big{(}{\mathcal{E}}_{k}\big{)}$
is nonincreasing, if $\beta^{*}=\beta_{\text{v}}$;
3. $iii$.
$\operatorname*{ssal}\big{(}{\mathcal{E}}_{k}\big{)}$ is nonincreasing, if
$\beta^{*}=\beta_{\text{s}}$.
Proof. The proof is detailed in the Appendix C.3. ❑
## 6 NUMERICAL SIMULATIONS
First, for the sake of graphic illustration, the presented algorithm is
applied to a second order randomly generated system with coil-shaped input and
one (either strip or halfspace type) measurement, also randomly generated at
each time-step for $k=0,\cdots,100$. The figures 6(a) and 6(b) show the
evolution of the ellipsoid ${\mathcal{E}}_{k}$ for a stable model (the
eigenvalues of state matrix $A{\,:=\,}A_{k}$ are less than 1) and a model at
the stability limit (the eigenvalues $A{\,:=\,}A_{k}$ are 1), _resp_.
(a) Stable state model
(b) Unstable state model
Secondly, in order to evaluate the algorithm performances, the matrices of the
system model (1) and (3) are generated randomly for two values of the state
dimension: $n=10$ and $n=~{}100$, with $\varrho=q=r=s=\frac{n}{2}$, $m=n$,
$\pi=\frac{n}{5}$ and $\bm{\mu}=\bm{\mu}^{\operatorname*{tr}}_{k}$; the input
vector $B_{k-1}\mbox{\boldmath$\tau$}_{k-1}$ contains sine entries of random
magnitude and frequency. $P_{0}=100I_{n}$, $\varsigma_{0}=1$ and
$\hat{\mbox{\boldmath$x$}}_{k_{0}}$ randomly chosen on the boundary of
${\mathcal{E}}_{0}$. The measurements are available at all time steps
${\mathcal{K}}=\\{1,\ldots,N\\}$ in the case 1; at some randomly chosen time
steps, in case 2 and ${\mathcal{K}}=\emptyset$ in the case 3 (where only
prediction stage is performed without any measurement correction). For each
case, the simulations are run 25 times under MATLAB R2018b on Intel Core i7
(2.3GHz, 8G RAM), each one for a different system model and containing
$N=~{}100$ time steps. The results are summarized in Table 1. Let
$\overline{\varsigma\operatorname*{tr}(P)}{\,:=\,}\underset{k\in\\{1,\ldots,N\\}}{\operatorname*{mean}\varsigma_{k}\operatorname*{tr}(P_{k})}$:
the average sum of ${\mathcal{E}}_{k}$’s squared axes lengths,
$\overline{\left\|\tilde{\bm{x}}\right\|}{\,:=\,}\underset{k\in\\{1,\ldots,N\\}}{\operatorname*{mean}\left\|\tilde{\bm{x}}_{k}\right\|}$:
the mean estimation error vector norm and $T$: the average computational time
for the simulation horizon of $N$ time steps.
$n$ | ${\mathcal{K}}$ | $\dfrac{\varsigma_{0}\operatorname*{tr}(P_{N})}{\varsigma_{N}\operatorname*{tr}(P_{0})}$ | $\overline{\varsigma\operatorname*{tr}(P)}$ | $\dfrac{\left\|\tilde{\bm{x}}_{N}\right\|}{\left\|\tilde{\bm{x}}_{0}\right\|}$ | $\overline{\left\|\tilde{\bm{x}}\right\|}$ | $T(ms)$
---|---|---|---|---|---|---
$10$ | case 1 | 0.010 | 1 141 | 0.027 | 3.02 | 18
case 2 | 0.026 | 2 206 | 0.053 | 4.60 | 15
case 3 | 0.058 | 7 540 | 0.066 | 7.25 | 10
$100$ | case 1 | 0.712 | 7$\cdot 10^{5}$ | 0.264 | 26.68 | 1870
case 2 | 0.707 | 2$\cdot 10^{6}$ | 0.269 | 46.19 | 1321
case 3 | 5.774 | 5$\cdot 10^{6}$ | 0.702 | 71.16 | 700
Table 1: Simulation results
It is plain to see that the algorithm considered here exhibits better
performances for systems of rather smaller dimension but it can still be
fairly efficiently used with very high dimensional systems provided enough
measurements are available. Moreover, given its low running time, it can be
implemented online with such systems.
## 7 CONCLUSION
An ellipsoidal state bounding approach was proposed for discrete-time linear
dynamic models with sporadic measurements, corrupted by additive unknown
process and measurement disturbances and subject to equality constraints on
the state vector. The following aspects were of special concern in the design
of the state estimation algorithm: 1\. the optimization of its inherent
parameters ($\mu$ and $\beta$), while dealing with equality constraints that
induce rank loss of the ellipsoid’s shape matrix, thus making a difference
with the existing algorithms. 2\. the numerical stability and the
computational undemandingness, in terms of time and memory, what makes this
algorithm suitable for high dimensional systems. 3\. the formulation of the
conditions of convergence in spite of the sporadic character of the
measurements and the state constraints manifesting themselves episodically,
now and then, not necessarily together and not at all time steps. 4\. the easy
applicability of the proposed technique by presenting detailed and turnkey
pseudocodes.
## Appendix A Proofs of the results of section 3
### A.1 Proof of Lemma 3.1
The set containing every possible value of
$\bar{\bm{x}}{\,:=\,}A\bm{x}+\bm{\eta}$ can be schematized by $\bar{\bm{x}}\in
A{\mathcal{E}}\oplus{\mathcal{Z}}\subset\bar{{\mathcal{E}}}(\mu)$; where
$A{\mathcal{E}}{\,:=\,}{\mathcal{E}}(A\bm{c},\varsigma APA^{T})$ is the image
of the ellipsoid ${\mathcal{E}}{\,:=\,}{\mathcal{E}}(\bm{c},\varsigma P)$ by
the endomorphism of matrix $A$; and
$\bar{{\mathcal{E}}}(\mu){\,:=\,}{\mathcal{E}}\big{(}\bar{\bm{c}},\varsigma\bar{P}(\mu)\big{)}$
is the outer-bounding ellipsoid of the Minkowski sum (cf. $\mathsection$.1.9.)
of $A{\mathcal{E}}$ and the one-dimensional zonotope, i.e., interval segment,
that is also a one-dimensional ellipsoid:
${\mathcal{Z}}{\,:=\,}{\mathcal{Z}}(\bm{u},\bm{r})={\mathcal{E}}(\bm{u},\bm{r}\bm{r}^{T})$.
Now, the Minkowski sum of two ellipsoids ${\mathcal{E}}(\bm{c}_{1},P_{1})$ and
${\mathcal{E}}(\bm{c}_{2},P_{2})$ is not an ellipsoid, in general, yet can be
bounded by a parametrized ellipsoid [19]:
$\displaystyle{\mathcal{E}}(\bm{c},P(\mu))$
$\displaystyle\supset{\mathcal{E}}(\bm{c}_{1},P_{1})\oplus{\mathcal{E}}(\bm{c}_{2},P_{2}),\
\forall\mu\in{\mathop{\rm I\mkern-3.5muR}}_{+}^{*},$ (39a)
$\displaystyle\text{ where }\bm{c}$ $\displaystyle=\bm{c}_{1}+\bm{c}_{2}\text{
and }P(\mu)=(1+\mu)P_{1}+(1+\leavevmode\kern 1.00006pt\raise
2.15277pt\hbox{$1$}\kern-1.00006pt/\kern-1.49994pt\lower
1.07639pt\hbox{$\mu$})P_{2}.$ (39b)
Setting $P_{1}{\,:=\,}\varsigma APA^{T}$, $\bm{c}_{1}=A\bm{c}$,
$P_{2}{\,:=\,}\bm{r}\bm{r}^{T}$ and $\bm{c}_{2}=\bm{u}$ completes the proof.
### A.2 Proof of Proposition 3.2
$\operatorname*{rank}\big{(}{Q_{+}}\big{)}=\operatorname*{rank}(Q+a\bm{r}\bm{r}^{T})$.
0pt
1. 1.
Let $\bar{q}{\,:=\,}n-{q}$. There exists a unitary matrix $U$, _s.t._
$U^{T}QU=\Big{[}{{\Sigma}^{2}\atop 0_{\bar{q},{q}}}{0_{{q},\bar{q}}\atop
0_{\bar{q},\bar{q}}}\Big{]}$, where $\Sigma\in~{}\\!{{{\mathop{\rm
I\mkern-3.5muR}}}^{q\times q}}$ is a diagonal matrix with nonzero singular
values of ${Q}$ on its diagonal. Let
$\bar{\bm{r}}{\,:=\,}\sqrt{a}{\Sigma}^{-1}\big{[}I_{q}\
{\mathbf{0}}_{q,\bar{q}}\big{]}U^{T}\bm{r}\in~{}\\!{{{\mathop{\rm
I\mkern-3.5muR}}}^{q}}$ and
$\underaccent{\bar}{\rb}{\,:=\,}\sqrt{a}\big{[}{\mathbf{0}}_{\bar{q},q}\
I_{\bar{q}}\big{]}U^{T}\bm{r}\in{\mathop{\rm I\mkern-3.5muR}}^{\bar{q}}$.
$\displaystyle Q+a\bm{r}\bm{r}^{T}$ $\displaystyle=U\left[{\Sigma\atop
0_{\bar{q},{q}}}{0\atop
0}{\Sigma\bar{\bm{r}}\atop\underaccent{\bar}{\rb}}\right]\left[{\Sigma\atop
0_{\bar{q},{q}}}{0\atop
0}{\Sigma\bar{\bm{r}}\atop\underaccent{\bar}{\rb}}\right]^{T}U^{T}=U\left[{\Sigma\atop
0_{\bar{q},{q}}}{\Sigma\bar{\bm{r}}\atop\underaccent{\bar}{\rb}}\right]\left[{\Sigma\atop
0_{\bar{q},{q}}}{\Sigma\bar{\bm{r}}\atop\underaccent{\bar}{\rb}}\right]^{T}\\!U^{T},\\!\\!\\!$
(40)
$\displaystyle\operatorname*{rank}(Q+\bm{r}\bm{r}^{T})=\operatorname*{rank}\left(\left[{\Sigma\atop
0_{\bar{q},{q}}}{0_{{q},\bar{q}}\atop I_{\bar{q}}}\right]\left[{I_{q}\atop
0_{\bar{q},q}}{\bar{\bm{r}}\atop\underaccent{\bar}{\rb}}\right]\right)=\operatorname*{rank}\left(\left[{I_{q}\atop
0_{\bar{q},q}}{\bar{\bm{r}}\atop\underaccent{\bar}{\rb}}\right]\right)$. Then
if
$\mbox{\boldmath$v$}{\,:=\,}(I_{n}-QQ^{\dagger})\bm{r}\neq{\mathbf{0}}_{n}$,
which is the projection of $\bm{r}$ onto $\mathpzc{Ker}({Q})$, meaning that
$\underaccent{\bar}{\rb}\neq{\mathbf{0}}_{\bar{q}}$, implying that
$\operatorname*{rank}\big{(}{Q_{+}}\big{)}=\operatorname*{rank}(Q+a\bm{r}\bm{r}^{T})=~{}\\!q+1$;
otherwise $\operatorname*{rank}\big{(}{Q_{+}}\big{)}=~{}\\!q$.
2. 2.
From (40),
$\displaystyle Q+a\bm{r}\bm{r}^{T}$ $\displaystyle=U\left[{\Sigma\atop
0_{\bar{q},{q}}}{0_{{q},\bar{q}}\atop
I_{\bar{q}}}\right]\left[{I_{q}+\bar{\bm{r}}\bar{\bm{r}}^{T}\atop\underaccent{\bar}{\rb}\bar{\bm{r}}^{T}}{\bar{\bm{r}}\underaccent{\bar}{\rb}^{T}\atop\underaccent{\bar}{\rb}\underaccent{\bar}{\rb}^{T}}\right]\left[{\Sigma\atop
0_{\bar{q},{q}}}{0_{{q},\bar{q}}\atop I_{\bar{q}}}\right]^{T}U^{T}$
$\displaystyle{\left|Q+a\bm{r}\bm{r}^{T}\right|}_{{\dagger}}$
$\displaystyle={\left|\Sigma\right|}^{2}{\left|\left[\begin{smallmatrix}I_{q}&0_{q,\bar{q}}&{\mathbf{0}}_{q}\\\
0_{\bar{q},q}&I_{\bar{q}}&{\mathbf{0}}_{\bar{q}}\\\
\bar{\bm{r}}^{T}&\underaccent{\bar}{\rb}^{T}&1\end{smallmatrix}\right]\left[\begin{smallmatrix}I_{q}+\bar{\bm{r}}\bar{\bm{r}}^{T}&\bar{\bm{r}}\underaccent{\bar}{\rb}^{T}&\bar{\bm{r}}\\\
\underaccent{\bar}{\rb}\bar{\bm{r}}^{T}&\underaccent{\bar}{\rb}\underaccent{\bar}{\rb}^{T}&\underaccent{\bar}{\rb}\\\
0_{1,q}&0_{1,\bar{q}}&1\end{smallmatrix}\right]\left[\begin{smallmatrix}I_{q}&0_{q,\bar{q}}&{\mathbf{0}}_{q}\\\
0_{\bar{q},q}&I_{\bar{q}}&{\mathbf{0}}_{\bar{q}}\\\
-\bar{\bm{r}}^{T}&-\underaccent{\bar}{\rb}^{T}&1\end{smallmatrix}\right]\right|}_{{\dagger}}$
$\displaystyle={\left|\Sigma\right|}^{2}{\left|\left[\begin{smallmatrix}I_{q}&0_{q,\bar{q}}&\bar{\bm{r}}\\\
0_{\bar{q},q}&0_{\bar{q}}&\underaccent{\bar}{\rb}\\\
{\mathbf{0}}_{q}^{T}&-\underaccent{\bar}{\rb}^{T}&\bar{\bm{r}}^{T}\bar{\bm{r}}+\underaccent{\bar}{\rb}^{T}\underaccent{\bar}{\rb}+1\end{smallmatrix}\right]\right|}_{{\dagger}}=\begin{cases}{\left|\Sigma\right|}^{2}\big{(}\bar{\bm{r}}^{T}\bar{\bm{r}}+1\big{)},&\text{
if }\underaccent{\bar}{\rb}\neq{\mathbf{0}}_{\bar{q}}\\\
{\left|\Sigma\right|}^{2}\underaccent{\bar}{\rb}^{T}\underaccent{\bar}{\rb},&\text{
otherwise.}\end{cases}$ (41)
Now, noticing that
${\left|bQ_{+}\right|}_{{\dagger}}=\prod_{\sigma_{i}(Q_{+})\neq
0}b\sigma_{i}(Q_{+})=b^{q_{+}}{\left|Q_{+}\right|}_{{\dagger}}$, that
${\left|\Sigma\right|}^{2}={\left|Q\right|}_{{\dagger}}$ and since
$\underaccent{\bar}{\rb}^{T}\underaccent{\bar}{\rb}=a\mbox{\boldmath$v$}^{T}\mbox{\boldmath$v$}$
and $\bar{\bm{r}}^{T}\bar{\bm{r}}=a\bm{r}^{T}Q^{\dagger}\bm{r}$, the proof of
(6b) is completed.
### A.3 Proof of Theorem 3.3
The pseudo-volume of an ellipsoid being proportional to the pseudo-determinant
of its shape matrix and $\varsigma$ being considered as constant _w.r.t._
$\mu$, 0pt
$\displaystyle\mu_{\text{v}}{\,:=\,}\arg\min_{\mu\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}}{\left|\bar{P}(\mu)\right|}_{{\dagger}}=\arg\min_{\mu\in{\mathop{\rm
I\mkern-3.5muR}}_{+}^{*}}\log{\left|Q\right|}_{{\dagger}}.$
According to Proposition 3.2, where $b{\,:=\,}(1+\mu)$ and
$a{\,:=\,}\tfrac{1}{\varsigma\mu}$, two cases are distinguished.
Case 1:
the projection of the vector $\bm{r}$ onto the nullspace of ${Q}$, embodied by
$v$ is a zero vector, in this case
$\operatorname*{rank}\big{(}\bar{P}(\mu)\big{)}=\operatorname*{rank}\big{(}Q_{+})=\operatorname*{rank}({Q})=q$
and
$\displaystyle\bar{P}(\mu)$
$\displaystyle{\,:=\,}(1+\mu)\left({Q}+\tfrac{1}{\varsigma\mu}\bm{r}\bm{r}^{T}\right)\Rightarrow{\left|\bar{P}(\mu)\right|}_{{\dagger}}={\left|Q\right|}_{{\dagger}}(1+\mu)^{q}\big{(}1+\tfrac{h}{\mu}\big{)}$
$\displaystyle\tfrac{\partial}{\partial\mu}\log{\left|\bar{P}(\mu)\right|}_{{\dagger}}=0$
$\displaystyle\Leftrightarrow\tfrac{{q}}{1+\mu}-\tfrac{h}{\mu(\mu+h)}=0\Leftrightarrow\
{q}\mu^{2}+h({q}-1)\mu-h=0.$ (42)
$\mu_{\text{v}}$ in (7a), is the only positive solution to the quadratic
equation (42).
Case 2:
the projection of $\bm{r}$ onto the nullspace of ${Q}$ is nonzero, thus
implying the rank incrementation of the shape matrix:
$\operatorname*{rank}\big{(}\bar{P}(\mu)\big{)}=q+1$ and
$\displaystyle{\left|\bar{P}(\mu)\right|}_{{\dagger}}={\left|Q\right|}_{{\dagger}}\left\|\mbox{\boldmath$v$}\right\|^{2}\tfrac{(1+\mu)^{q+1}}{\varsigma\mu}\Leftrightarrow\tfrac{\partial}{\partial\mu}\log{\left|\bar{P}(\mu)\right|}_{{\dagger}}=0$
$\displaystyle\Leftrightarrow\tfrac{{q}+1}{1+\mu}-\tfrac{1}{\mu}=0,$
this results in $\mu=\tfrac{1}{q}$ of (7a).
### A.4 Proof of Theorem 3.6
Conforming to (1a), the set containing every possible value of
$\mbox{\boldmath$x$}_{k}$ can be schematized by
$\displaystyle
A_{k}{\mathcal{E}}_{k}\oplus{\mathcal{Z}}(B_{k}\mbox{\boldmath$\tau$}_{k},R_{k})={\mathcal{E}}_{{k+1/k}_{0}}\oplus{\mathcal{Z}}({\mathbf{0}}_{n},R_{k}).$
(43)
Now, the zonotope ${\mathcal{Z}}({\mathbf{0}}_{n},R_{k})$, where
$R_{k}=\big{[}\mbox{\boldmath$r$}_{k_{1}}\cdots\mbox{\boldmath$r$}_{k_{m}}\big{]}$,
can be viewed as the sum of $m$ degenerate ellipsoids [33]:
$\displaystyle{\mathcal{Z}}({\mathbf{0}}_{n},R_{k})=\operatorname*{\oplus}_{i=1}^{m}{\mathcal{Z}}({\mathbf{0}}_{n},{\bm{r}}_{k_{i}})=\operatorname*{\oplus}_{i=1}^{m}{\mathcal{E}}({\mathbf{0}}_{n},{\bm{r}}_{k_{i}}{\bm{r}}_{k_{i}}^{T}).$
Applying Lemma 3.1 sequentially to
$\displaystyle\displaystyle{\mathcal{E}}_{k+1/k}\supset{\mathcal{E}}_{{k+1/k}_{0}}\oplus\left(\operatorname*{\oplus}_{i=1}^{m}{\mathcal{E}}({\mathbf{0}}_{n},{\bm{r}}_{k_{i}}{\bm{r}}_{k_{i}}^{T})\right),$
(44)
eventuates in (10).
### A.5 Proof of Theorem 3.8
Since the SSAL of an ellipsoid is the trace of its shape matrix,
$\displaystyle\mu_{{\text{s}}_{i}}=~{}\\!\arg\min_{\mu_{i}}\operatorname*{tr}(P_{{k+1/k}_{i}})$;
(12) is the direct corollary of Theorem 3.8. As for (13), it is a direct
consequence of the result [12] saying that the minimum trace ellipsoid
containing the Minkowski sum of $m+1$ ellipsoids is :
$\displaystyle{\mathcal{E}}(\bm{c},P)$
$\displaystyle{\,:=\,}\oplus_{i=0}^{m}{\mathcal{E}}(\bm{c}_{i},P_{i}),$ (45a)
where $\displaystyle\bm{c}{\,:=\,}\sum_{i=0}^{m}\bm{c}_{i}\text{ and }P$
$\displaystyle{\,:=\,}\Big{(}\sum_{i=0}^{m}\sqrt{\operatorname*{tr}(P_{i})}\Big{)}\Big{(}\sum_{i=0}^{m}\big{(}\sqrt{\operatorname*{tr}(P_{i})}\big{)}^{-1}{P_{i}}\Big{)}.$
(45b)
Then, after noticing that
$\displaystyle\operatorname*{tr}(\mbox{\boldmath$r$}_{{k}_{i}}\mbox{\boldmath$r$}_{{k}_{i}}^{T})=\mbox{\boldmath$r$}_{{k}_{i}}^{T}\mbox{\boldmath$r$}_{{k}_{i}}=\left\|\mbox{\boldmath$r$}_{{k}_{i}}\right\|^{2}$
(46) and that
$\displaystyle\sum_{i=1}^{m}\left({\sqrt{\operatorname*{tr}(\mbox{\boldmath$r$}_{{k}_{i}}\mbox{\boldmath$r$}_{{k}_{i}}^{T})}}\right)^{-1}{{\bm{r}}_{k_{i}}{\bm{r}}_{k_{i}}^{T}}=\tilde{R}_{k}$
(47)
(45) applied to
$\displaystyle{\mathcal{E}}_{{k+1/k}_{0}}\oplus\big{(}\oplus_{i=1}^{m}{\mathcal{E}}({\mathbf{0}}_{n},\mbox{\boldmath$r$}_{{k}_{i}}\mbox{\boldmath$r$}_{{k}_{i}}^{T}\big{)}$,
leads clearly to (13). It is also stated in [12] that such an ellipsoid is the
same that the one obtained sequentially in (9).
## Appendix B Proofs of the results of section 4
### B.1 Proof of Theorem 4.1
###### Definition B.1
The signed distance from a set ${\mathcal{S}}\subset{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}$ to a vector $\mbox{\boldmath$x$}\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}$ is
$\displaystyle\zeta({\mathcal{S}},\mbox{\boldmath$x$}){\,:=\,}\max_{\left\|\bm{u}\right\|=1}\bm{u}^{T}\mbox{\boldmath$x$}-\rho_{\mathcal{S}}(\bm{u})$.
###### Proposition B.1 ([34])
The signed distance from an ellipsoid to a hyperplane is given by:
$\displaystyle\zeta\big{(}{\mathcal{E}}(\bm{c},P),{\mathcal{H}}(\bm{d},a)\big{)}{\,:=\,}\left\|\bm{d}\right\|^{-1}\Big{(}{\left|a-\bm{c}^{T}\bm{d}\right|}-\sqrt{\bm{d}^{T}P\bm{d}}\Big{)}.$
(48)
Let ${\mathcal{E}}{\,:=\,}{\mathcal{E}}(\bm{c},\varsigma P)$. The signed
distance from ${\mathcal{E}}$ to ${\mathcal{H}}(\bm{f},\bar{y})$ is
$\displaystyle\zeta{\,:=\,}\left\|\bm{f}\right\|^{-1}\left({\left|\bar{y}-\bm{c}^{T}\bm{f}\right|}-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}\right).$
(49)
* •
$\zeta\geq 0$ means that ${\mathcal{H}}(\bm{f},\bar{y})$ does not intersect
${\mathcal{E}}$ in more than one point:
1. 1.
if $\bm{c}^{T}\bm{f}>\bar{y}$, then
${\mathcal{E}}\subset~{}{\mathcal{G}}(-\bm{f},-\bar{y})$ and
${\mathcal{E}}\cap{\mathcal{G}}(\bm{f},\bar{y})=\emptyset\Leftrightarrow$
(19b);
2. 2.
if $\bm{c}^{T}\bm{f}\leq\bar{y}$, then
${\mathcal{E}}\subset{\mathcal{G}}(\bm{f},\bar{y})\Rightarrow{\mathcal{E}}\cap{\mathcal{G}}(\bm{f},\bar{y})={\mathcal{E}}\Leftrightarrow$
(19c);
3. 3.
if $\bm{c}^{T}\bm{f}-\bar{y}=\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$, then
${\mathcal{H}}(\bm{f},\bar{y})$ is tangent to ${\mathcal{E}}$ and
${\mathcal{E}}\cap{\mathcal{G}}(\bm{f},\bar{y})={\mathcal{E}}\cap{\mathcal{H}}(\bm{f},\bar{y})=\\{\check{\bm{c}}\\}$,
where $\check{\bm{c}}$ is given in (25e) and can be calculated using (25g),
letting $\delta\leftarrow\bar{y}-\bm{c}^{T}\bm{f}$, with
$\delta^{2}={\varsigma\bm{f}^{T}P\bm{f}}$, $\bm{f}\leftarrow\bm{f}$,
$P\leftarrow\varsigma^{-1}P$.
* •
If $\zeta\leq 0$, then ${\mathcal{H}}(\bm{f},\bar{y})$ intersects
${\mathcal{E}}$ and ${\mathcal{H}}(-\bm{f},\rho(-\bm{f}))$ is the ellipsoid’s
supporting hyperplane of normal vector $-\bm{f}$ which is contained in
${\mathcal{G}}(\bm{f},\bar{y})$. Indeed,
$\displaystyle\mbox{\boldmath$x$}\in{\mathcal{H}}(-\bm{f},\rho(-\bm{f}))$
$\displaystyle\Leftrightarrow\mbox{\boldmath$x$}^{T}\bm{f}-\bm{c}^{T}\bm{f}=-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}\leq\bar{y}-\bm{c}^{T}\bm{f}$
$\displaystyle\Rightarrow\mbox{\boldmath$x$}^{T}\bm{f}\leq\bar{y}\Leftrightarrow\mbox{\boldmath$x$}\in{\mathcal{G}}(\bm{f},\bar{y}).$
(50) Thence, ${\mathcal{G}}\big{(}{-\bm{f}},\rho(-\bm{f})\big{)}$ is its
supporting halfspace and
${\mathcal{E}}\subset{\mathcal{G}}\big{(}-\bm{f},\rho(-\bm{f})\big{)}$.
Therefore, $\displaystyle{\mathcal{E}}\cap{\mathcal{G}}(\bm{f},\bar{y})$
$\displaystyle=\Big{(}{\mathcal{G}}\big{(}{-\bm{f}},\rho(-\bm{f})\big{)}\cap{\mathcal{G}}(\bm{f},\bar{y})\Big{)}\cap{\mathcal{E}}$
(51)
${\left|\bar{y}-\bm{c}^{T}\bm{f}\right|}<\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$
means that $0<\bar{y}+\rho(-\bm{f})<2\sqrt{\varsigma\bm{f}^{T}P\bm{f}}$,
entailing, on one hand, $\displaystyle{\mathcal{G}}(\bm{f},\bar{y})$
$\displaystyle=\big{\\{}\mbox{\boldmath$x$}\big{|}\mbox{\boldmath$x$}^{T}\bm{f}\leq\bar{y}\big{\\}}=\bigg{\\{}\mbox{\boldmath$x$}\bigg{|}\frac{2}{\bar{y}+\rho(-\bm{f})}\mbox{\boldmath$x$}^{T}\bm{f}\leq\frac{2\bar{y}}{\bar{y}+\rho(-\bm{f})}\bigg{\\}}$
$\displaystyle={\mathcal{G}}(\gamma^{-1}\bm{f},{y}+1)$ (52)
$\displaystyle\text{and }{\mathcal{G}}(-\bm{f},\rho(-\bm{f}))$
$\displaystyle=\bigg{\\{}\mbox{\boldmath$x$}\bigg{|}-\frac{2}{\bar{y}+\rho(-\bm{f})}\mbox{\boldmath$x$}^{T}\bm{f}\leq\frac{2\rho(-\bm{f})}{\bar{y}+\rho(-\bm{f})}\bigg{\\}}$
$\displaystyle={\mathcal{G}}(-\gamma^{-1}\bm{f},-{y}+1),$ (53) on the other
hand. Finally, the proof (19e)–(19f) is achieved thusly:
$\displaystyle{\mathcal{E}}\cap{\mathcal{G}}(\bm{f},\bar{y})$
$\displaystyle={\mathcal{E}}\cap\Big{(}{\mathcal{G}}\big{(}\gamma^{-1}\bm{f},{y}+1\big{)}\cap{\mathcal{G}}\big{(}-\gamma^{-1}\bm{f},-{y}+1\big{)}\Big{)}$
(54)
$\displaystyle={\mathcal{E}}\cap{\mathcal{D}}\big{(}\gamma^{-1}\bm{f},{y}\big{)}.$
❑
### B.2 Proof of Theorem 4.2
Let ${\mathcal{E}}{\,:=\,}{\mathcal{E}}(\bm{c},\varsigma P)$. The signed
distance from the ellipsoid ${\mathcal{E}}$ to each of the two hyperplanes
${\mathcal{H}}(\bm{f},y\mp 1)$, bounding the strip ${\mathcal{D}}(\bm{f},y)$,
is
$\displaystyle\zeta{\,:=\,}\left\|\bm{f}\right\|^{-1}\big{(}\big{|}y\mp
1-\bm{f}^{T}\bm{c}\big{|}-\sqrt{\varsigma\bm{f}^{T}P\bm{f}}\big{)}.$ (55)
When $\zeta>0$, the ellipsoid doesn’t intersect any of both hyperplanes
meaning either that the ellipsoid is located outside the strip, in which case
(case 1), the intersection is empty or that it is situated between them i.e.,
contained in the strip (case 2). In the case 3, the interior of the ellipsoid
is outside the strip touching it in only one point and the case 3 of Thm 4.1
is then applicable:
${\mathcal{D}}(\bm{f},y)\cap{\mathcal{E}}={\mathcal{G}}(\bm{f},y+1)\cap{\mathcal{E}}$
(case 3.a) and
${\mathcal{D}}(\bm{f},y)\cap{\mathcal{E}}={\mathcal{G}}(-\bm{f},-y+1)\cap{\mathcal{E}}$
(case 3.b). In the case 4, where $\zeta\leq~{}0$, the intersection is not
empty. It is then possible to introduce the following lemma, based on the
results of [17] and [30]:
###### Lemma B.2
$\forall y\in{\mathop{\rm I\mkern-3.5muR}}$, $\bm{c}\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}$, $\bm{f}\in{\mathop{\rm I\mkern-3.5muR}}^{n}$,
$\sigma\in~{}{\mathop{\rm I\mkern-3.5muR}}_{+}^{*}$ and SPD
$P\in{{{\mathop{\rm I\mkern-3.5muR}}}^{n\times n}}$, if
${\mathcal{D}}(\bm{f},y)\cap{\mathcal{E}}(\bm{c},\sigma P)\neq\emptyset$, then
$\displaystyle\forall\omega\in{\mathop{\rm I\mkern-3.5muR}}_{+}^{*},$
$\displaystyle\quad{\mathcal{E}}(\tilde{\bm{c}}(\omega),\tilde{\sigma}(\omega)\tilde{P}\big{(}\omega)\big{)}\supset{\mathcal{D}}(\bm{f},y)\cap{\mathcal{E}}(\bm{c},\sigma
P),$ (56a) where $\displaystyle\tilde{P}(\omega)$
$\displaystyle{\,:=\,}P-\tfrac{\omega}{\omega\alpha+{1}}P{\bm{f}}{\bm{f}}^{T}P,\quad$
(56b) $\displaystyle\tilde{\bm{c}}(\omega)$
$\displaystyle{\,:=\,}\bm{c}+\tfrac{\omega}{\omega\alpha+{1}}\delta
P{\bm{f}}=\bm{c}+\omega\tilde{P}(\omega)^{-1}{\bm{f}}\delta,\quad$ (56c)
$\displaystyle\tilde{\sigma}(\omega)$
$\displaystyle{\,:=\,}\sigma+\omega(1-\tfrac{\alpha}{\omega+\alpha}\delta^{2}),$
(56d) $\displaystyle\delta$ $\displaystyle{\,:=\,}y-{\bm{f}}^{T}\bm{c}\text{
and $\alpha$ is given in \eqref{alpha_def}}.$ (56e)
This lemma is precisely the mono-output case of the “observation update” part
of Thm 1, [14]. (21d), (21e) and (21f) are obtained by setting
$\omega{\,:=\,}{\alpha\beta}({1-\beta})^{-1}$, thus
$\beta={\omega}({\alpha+\omega})^{-1}$. But before applying the lemma above,
it is suitable to reduce the strip ${\mathcal{D}}(\bm{f},y)$ in case where one
of the two hyperplanes does not intersect the ellipsoid ${\mathcal{E}}$, i.e.,
when either $y+1>\bar{\rho}$ or $y-1<-\underaccent{\bar}{\rho}$, by
translating the aforementioned hyperplane so that it becomes tangent to the
ellipsoid, as proposed in [35]. The new strip so obtained is
${\mathcal{D}}(\gamma^{-1}\bm{f},\breve{y})$, where $\gamma$ and
$\breve{y}{\,:=\,}y$ are given in (19f) and result from applying (case 4) of
Thm 4.1 to ${\mathcal{E}}\cap{\mathcal{G}}(-\bm{f},-y+1)$ and to
${\mathcal{E}}\cap{\mathcal{G}}(\bm{f},y+1)$.
### B.3 Proof of Theorem 4.3.1
Applying the generalization of the Sherman-Morrison formula to the pseudo-
inverse of the matrix (21d) (cf. Corollary 3.5 [38]), we can write
$\displaystyle\breve{P}(\beta)^{\dagger}=P^{\dagger}+\tfrac{\alpha\beta}{1-\beta}{P^{\dagger}P\bm{f}\bm{f}^{T}PP^{\dagger}}.$
(57)
Now, noticing that $(PP^{\dagger})^{T}=P^{\dagger}P$ and recalling that,
$\forall\bm{u}\in\mathpzc{R}(P)$, $\exists\mbox{\boldmath$v$}\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}$, $\bm{u}=P\mbox{\boldmath$v$}$, so
$PP^{\dagger}\bm{u}=PP^{\dagger}P\mbox{\boldmath$v$}=P\mbox{\boldmath$v$}=~{}\\!\bm{u}$,
in particular for $\bm{u}{\,:=\,}\mbox{\boldmath$x$}-\breve{\bm{c}}(\beta)$,
$\forall\mbox{\boldmath$x$}\in~{}\\!{\mathcal{E}}(\bm{c},\varsigma P)$, since
$\breve{\bm{c}}(\beta)\in{\mathcal{E}}(\bm{c},\varsigma P)$; replacing (57) in
(22a) leads then to
0pt
$\displaystyle{\mathscr{V}}_{\breve{P}(\beta)}\big{(}\bm{x}-\breve{\bm{c}}(\beta)\big{)}$
$\displaystyle{\,:=\,}\big{(}\bm{x}-\breve{\bm{c}}(\beta)\big{)}^{T}\Big{(}P^{\dagger}+\tfrac{\alpha\beta}{1-\beta}{P^{\dagger}P\bm{f}\bm{f}^{T}PP^{\dagger}}\Big{)}\big{(}\mbox{\boldmath$x$}-\breve{\bm{c}}(\beta)\big{)}$
$\displaystyle=\big{(}\bm{x}-\breve{\bm{c}}(\beta)\big{)}^{T}\Big{(}P^{\dagger}+\tfrac{\alpha\beta}{1-\beta}{\bm{f}\bm{f}^{T}}\Big{)}\big{(}\mbox{\boldmath$x$}-\breve{\bm{c}}(\beta)\big{)}.$
(58a) Inserting (21e) in (58a), we can show, by the mean of some standard
algebraic manipulations, that101010${\mathscr{V}}_{\breve{P}(\beta)}$ is
optimized on
${\mathcal{D}}(\breve{\bm{f}},\breve{y})\cap{\mathcal{E}}(\bm{c},\varsigma
P)$, it is then obvious that $\mbox{\boldmath$x$}\in\mathpzc{R}(P)$, since
$\bm{x}\in{\mathcal{E}}(\bm{c},\varsigma P)$., $\forall\bm{x}\in{{{\mathop{\rm
I\mkern-3.5muR}}}^{n}}$, _s.t._ $(\bm{x}-\bm{c})\in\mathpzc{R}(P)$ (since
$\bm{x}\in{\mathcal{E}}(\bm{c},\varsigma P)$),
$\displaystyle{\mathscr{V}}_{\breve{P}(\beta)}\big{(}\mbox{\boldmath$x$}-\breve{\bm{c}}(\beta)\big{)}$
|
# Independent Sets in Direct Products of Vertex-transitive Graphs
Huajun Zhang<EMAIL_ADDRESS>Department of Mathematics, Zhejiang Normal
University, Jinhua 321004, P.R. China Department of Mathematics, Shanghai
Normal University, Shanghai 200234, P.R. China
###### Abstract
The direct product $G\times H$ of graphs $G$ and $H$ is defined by:
$V(G\times H)=V(G)\times V(H)$
and
$E(G\times H)=\left\\{[(u_{1},v_{1}),(u_{2},v_{2})]:(u_{1},u_{2})\in
E(G)\mbox{\ and\ }(v_{1},v_{2})\in E(H)\right\\}.$
In this paper, we will prove that the equality
$\alpha(G\times H)=\max\\{\alpha(G)|H|,\alpha(H)|G|\\}$
holds for all vertex-transitive graphs $G$ and $H$, which provides an
affirmative answer to a problem posed by Tardif (Discrete Math. 185 (1998)
193-200). Furthermore, the structure of all maximum independent sets of
$G\times H$ are determined.
###### keywords:
direct product; primitivity; independence number; vertex-transitive
MSC: 05D05, 06A07
## 1 Introduction
Let $G$ and $H$ be two graphs. The direct product $G\times H$ of $G$ and $H$
is defined by
$\displaystyle V(G\times H)=V(G)\times V(H)$
and
$\displaystyle E(G\times
H)=\left\\{[(u_{1},v_{1}),(u_{2},v_{2})]:(u_{1},u_{2})\in E(G)\mbox{\ and\
}(v_{1},v_{2})\in E(H)\right\\}.$
It is easy to see this product is commutative and associative, and the product
of more than two graphs is well-defined. For a graph $G$, the products
$G^{n}=G\times G\times\cdots\times G$ is called the $n$-th powers of $G$.
An interesting problem is the independence number of $G\times H$. It is clear
that if $I$ is an independent set of $G$ or $H$, then the preimage of $I$
under projections is an independent set of $G\times H$, and so $\alpha(G\times
H)\geq\max\\{\alpha(G)|H|,\alpha(H)|G|\\}.$ It is natural to ask whether the
equality holds or not. In general, the equality does not hold for non-vertex-
transitive graphs (see [13]). So Tardif [17] posed the following problem.
###### Problem 1.1
(Tardif [17]) Does the equality
$\alpha(G\times H)=\max\\{\alpha(G)|H|,\alpha(H)|G|\\}$
hold for all vertex-transitive graphs $G$ and $H$?
Furthermore, it immediately raises another interesting problem:
###### Problem 1.2
When $\alpha(G\times H)=\max\\{\alpha(G)|H|,\alpha(H)|G|\\}$, is every maximum
independent set of $G\times H$ the preimage of an independent set of one
factor under projections?
If the answer is yes, we then say the direct product $G\times H$ is MIS-normal
(maximum-independent-set-normal). Furthermore, the direct products
$G_{1}\times G_{2}\times\cdots\times G_{n}$ is said to be MIS-normal if every
maximum independent set of it is the preimage of an independent set of one
factor under projections.
About these two problems, there are some progresses have been made for some
very special vertex-transitive graphs.
Let $n,r$ and $t$ be three integers with $n\geq r\geq t\geq 1$. The graph
$K(t,r,n)$ is defined by: whose vertices set is the set of all $r$-element
subsets of $[n]=\\{1,2,\ldots,n\\}$, and $A$ and $B$ of which are adjacent if
and only if $|A\cap B|<t$. If $n\geq 2r$, then $K(1,r,n)$ is the well-known
Kneser graph. The classical Erdős-Ko-Rado Theorem [8] states that
$\alpha(K(1,r,n))=\binom{n-1}{r-1}$ (where $n\geq 2r$), and Frankl [9] first
investigated the independence number of the direct products of Kneser graphs.
Subsequently, Ahlswede, Aydinian and Khachatrian investigated the general case
[2].
###### Theorem 1.3
Let $n_{i}\geq r_{i}\geq t_{i}$ for $i=1,2,\ldots,k$.
(i) (Frankl [9]) if $t_{1}=\cdots=t_{k}=1$ and
$\frac{r_{i}}{n_{i}}\geq\frac{1}{2}$ for $i=1,2,\ldots,k$, then
$\alpha\left(\prod_{1\leq i\leq
k}K(1,r_{i},n_{i})\right)=\max\left\\{\frac{r_{1}}{n_{1}},\frac{r_{2}}{n_{2}},\ldots,\frac{r_{k}}{n_{k}}\right\\}\prod_{1\leq
i\leq k}|K(1,r_{i},n_{i})|.$
(ii) (Ahlswede, Aydinian and Khachatrian [2])
$\alpha\left(\prod_{1\leq i\leq
k}K(t_{i},r_{i},n_{i})\right)=\max\left\\{\frac{\alpha(K(t_{i},r_{i},n_{i}))}{|K(t_{i},r_{i},n_{i})|}:1\leq
i\leq k\right\\}\prod_{1\leq i\leq k}|K(t_{i},r_{i},n_{i})|.$
The circular graph $Circ(r,n)$ ($n\geq 2r$) is defined by:
$V(Circ(r,n))=\mathbb{Z}_{n}=\\{0,1,\ldots,n-1\\}$
and
$E(Circ(r,n))=\left\\{(i,j):|i-j|\in\\{r,r+1,\ldots,n-r\\}\right\\}.$
It is well known that $\alpha(Circ(r,n))=r$. Mario and Juan [16] determined
the independence number of the direct products of circular graphs.
###### Theorem 1.4
(Mario and Juan [16]) Let $n_{i}\geq 2r_{i}$ for $i=1,2\ldots,k$. Then
$\alpha\left(\prod_{1\leq i\leq
k}Circ(r_{i},n_{i})\right)=\max\left\\{\frac{r_{1}}{n_{1}},\frac{r_{2}}{n_{2}},\ldots,\frac{r_{k}}{n_{k}}\right\\}\prod_{1\leq
i\leq k}n_{i}.$
For positive integers $n$, let $S_{n}$ denote the permutation group on $[n]$.
Two permutations $f$ and $g$ are said to be intersecting if there exists an
$i\in[n]$ such that $f(i)=g(i)$. We define a graph on $S_{n}$ as that two
permutations are adjacent if and only if they are not intersecting. For
brevity, this graph is also denoted by $S_{n}$. Deza and Frankl [7] first
obtained that $\alpha(S_{n})=(n-1)!$. Cameron and Ku [6] proved that each
maximum independent set of $S_{n}$ is a coset of the stabilizer of a point, to
which Larose and Malvenuto [14], Wang and Zhang [18] and Godsil and Meagher
[10] gave alternative proofs, respectively. Recently, Cheng and Wong [11]
further investigated the independence number and the MIS-normality of the
direct products of $S_{n}$.
###### Theorem 1.5
(Cheng and Wong[11]) Let $2\leq n_{1}=\cdots=n_{p}<n_{p+1}\leq\ldots,n_{q}$,
$1\leq p\leq q$. Then
$\alpha\big{(}S_{n_{1}}\times S_{n_{2}}\times\cdots\times
S_{n_{q}}\big{)}=(n_{1}-1)!\prod_{2\leq i\leq q}n_{i}!,$
and the direct products $S_{n_{1}}\times S_{n_{2}}\times\cdots\times
S_{n_{q}}$ is MIS-normal except for the following cases:
1. (i)
$n_{1}=\cdots=n_{p}<n_{p+1}=3\leq n_{p+2}\leq\cdots\leq n_{q}$;
2. (ii)
$n_{1}=n_{2}=3\leq n_{3}\leq\cdots\leq n_{q}$;
3. (iii)
$n_{1}=n_{2}=n_{3}\leq n_{4}\leq\cdots\leq n_{q}$.
In [15], Larose and Tardif investigated the relationship between projectivity
and the structure of maximum independent sets in powers of some vertex-
transitive graphs, and obtained the MIS-normality of the powers of Kneser
graphs and circular graphs.
###### Theorem 1.6
(Larose and Tardif [15]) Let $n$ and $r$ be two positive integers. If $n>2r$,
then both $K^{k}(1,r,n)$ and $Circ^{k}(r,n)$ are MIS-normal for all positive
integer $k$.
Besides the above results, Larose and Tardif [15] prove that if $G$ is vertex-
transitive, then $\alpha(G^{n})=\alpha(G)|V(G)|^{n-1}$ for all $n>1$. They
also ask whether or not $G^{n}$ is MIS-normal if $G^{2}$ is MIS-normal.
Recently, Ku and Mcmillan [12] gave an affirmative answer to this problem, and
we solved this problem in a more general setting [20].
In this paper we shall solve both Problem 1.1 and Problem 1.2. To state our
results we need to introduce some notations and notions.
For a graph $G$, let $I(G)$ denote the set of all maximum independent sets of
$G$. Given a subset $A$ of $V(G)$, we define
$N_{G}(A)=\\{b\in V(G):\mbox{$(a,b)\in E(G)$ for some $a\in A$}\\}$
$N_{G}[A]=N_{G}(A)\cup A\mbox{ and }\bar{N}_{G}[A]=V(G)-N_{G}[A].$
If $G$ is clear from the context, for simplicity, we will omit the index $G$.
In [20], by the so-called “No-Homomorphism” lemma of Albertson and Collins [1]
we proved the following result.
###### Proposition 1.7
([20]) Let $G$ be a vertex-transitive graph. Then, for every independent set
$A$ of $G$, $\frac{|A|}{|N_{G}[A]|}\leq\frac{\alpha(G)}{|V(G)|}$. Equality
implies that $|S\cap N_{G}[A]|=|A|$ for every $S\in I(G)$, and in particularly
$A\subseteq S$ for some $S\in I(G)$.
An independent set $A$ in $G$ is said to be imprimitive if $|A|<\alpha(G)$ and
$\frac{|A|}{|N[A]|}=\frac{\alpha(G)}{|V(G)|}$. And $G$ is called IS-
imprimitive if $G$ has an imprimitive independent set. In any other cases, $G$
is called _IS-primitive_. From definition we see that a disconnected vertex-
transitive graph $G$ is IS-imprimitive and hence an IS-primitive vertex-
transitive graph $G$ is connected.
The following Theorem is the main result of this paper.
###### Theorem 1.8
Let $G$ and $H$ be two vertex-transitive graphs with
$\frac{\alpha(G)}{|G|}\geq\frac{\alpha(H)}{|H|}$. Then
$\alpha(G\times H)=\alpha(G)|H|,$
and either:
1. (i)
$G\times H$ is MIS-normal, or
2. (ii)
$\frac{\alpha(G)}{|G|}=\frac{\alpha(H)}{|H|}$ and one of them is IS-
imprimitive, or
3. (iii)
$\frac{\alpha(G)}{|G|}>\frac{\alpha(H)}{|H|}$ and $H$ is disconnected.
We leave the proof of Theorem 1.8 to the next section, while in Section 3, we
discuss the MIS-normality of the direct products of more than two vertex-
transitive graphs.
## 2 Proof of Theorem 1.8
Let $S$ be a maximum independent set of $G\times H$. Then
$|S|\geq\alpha(G)|H|\geq|G|\alpha(H)$. We now prove $\alpha(G\times
H)\leq\alpha(G)|H|$.
For every $a\in G$, define
$X_{a}=\\{x\in H:(a,x)\in S\\}.$
Since $S$ is an independent set of $G\times H$, for each $x\in X_{a}$ and
$y\in X_{b}$, $(x,y)\not\in E(H)$ whenever $(a,b)\in E(G)$. In this case, we
say that $X_{a}$ and $X_{b}$ are cross-independent. This concept is equivalent
to cross-intersecting families in extremal set theory. We refer [19] for
details.
In the language of cross-intersecting families, Borg [3, 4, 5] introduce a
decomposition of $X_{a}$ as follows.
$X^{*}_{a}=\\{x\in X_{a}:N_{H}(x)\cap X_{a}=\emptyset\\},$
$X^{\prime}_{a}=\\{x\in X_{a}:N_{H}(x)\cap X_{a}\neq\emptyset\\}$
and
$X^{\prime}=\bigcup_{a\in V(G)}X^{\prime}_{a}.$
Clearly, $X_{a}^{*}$ is an independent set of $H$ for every $a\in V(G)$, and
$|S|=\sum_{a\in V(G)}|X_{a}|$. Here, the empty set is regarded as an
independent set.
We list all distinct $X^{*}_{a}$’s as $Y_{1},Y_{2},\ldots,Y_{k}$, and define
$B_{i}=\\{a\in V(G):X^{*}_{a}=Y_{i}\\},\ i=1,2,\ldots,k.$
We then obtain a partition of $V(G)$ as $V(G)=B_{1}\cup B_{2}\cup\cdots\cup
B_{k}$. Then
$\displaystyle|S|$ $\displaystyle=$ $\displaystyle\sum_{a\in
V(G)}|X_{a}|=\sum_{a\in V(G)}(|X_{a}^{*}|+|X^{\prime}_{a}|)=\sum_{i=1}^{k}\
\sum_{a\in B_{i}}|X_{a}^{*}|+\sum_{a\in V(G)}|X_{a}^{\prime}|$ (1)
$\displaystyle=$ $\displaystyle\sum_{i=1}^{k}|Y_{i}||B_{i}|+\sum_{x\in
X^{\prime}}|A_{x}|,$
where
$A_{x}=\\{a\in V(G):x\in X^{\prime}_{a}\\}.$
For every pair $a,b\in V(G)$, it is easy to verify that $(a,b)\not\in E(G)$ if
$X^{\prime}_{a}\cap X^{\prime}_{b}\neq\emptyset$. Therefore, $A_{x}$ is an
independent set of $G$. By Proposition 1.7 we have that
$|A_{x}|\leq\frac{\alpha(G)}{|V(G)|}|N_{G}[A_{x}]|,$ (2)
and equality holds if and only if $|A_{x}|=0$, or $|A_{x}|=\alpha(G)$, or
$A_{x}$ is an imprimitive independent set of $G$.
Suppose $x\in N_{H}[Y_{i}]=N_{H}(Y_{i})\cup Y_{i}$. If $x\in N_{H}(Y_{i})$,
then there exists $y\in Y_{i}$ such that $(x,y)\in E(H)$ and
$\\{(a,x),(b,y)\\}\subset S$ for any $b\in B_{i}$ and $a\in A_{x}$, hence
$(a,b)\not\in E(G)$ since $S$ is an independent set; if $x\in Y_{i}$, then for
each $a\in A_{x}$, there is a $z\in X_{a}$ with $(x,z)\in E(H)$ and
$\\{(a,z),(b,x)\\}\subset S$, yielding $(a,b)\not\in E(G)$. Thus proving that
$B_{i}\subseteq\bar{N}_{G}[A_{x}]$ if $x\in N_{H}[Y_{i}]$. From this it
follows that
$\sum_{i:x\in
N_{H}[Y_{i}]}|B_{i}|\leq|\bar{N}_{G}[A_{x}]|=|V(G)|-|N_{G}[A_{x}]|,$
i.e.,
$|N_{G}[A_{x}]|\leq|V(G)|-\sum_{i:x\in
N_{H}[Y_{i}]}|B_{i}|=\sum_{i:x\in\bar{N}_{H}[Y_{i}]}|B_{i}|.$ (3)
Note that
$X^{\prime}\subseteq\bigcup_{i=1}^{k}\bar{N}_{H}[Y_{i}].$ (4)
Together with (2), (3)and (4), we then obtain that
$\displaystyle\sum_{x\in
X^{\prime}}|A_{x}|\leq\frac{\alpha(G)}{|V(G)|}\sum_{x\in
X^{\prime}}\sum_{i:x\in\bar{N}_{H}[Y_{i}]}|B_{i}|$ (5) $\displaystyle\leq$
$\displaystyle\frac{\alpha(G)}{|V(G)|}\sum_{i=1}^{k}\sum_{x\in\bar{N}_{H}[Y_{i}]}|B_{i}|=\frac{\alpha(G)}{|V(G)|}\sum_{i=1}^{k}|B_{i}||\bar{N}_{H}[Y_{i}]|.$
Combining (1) and (5) gives that
$\displaystyle|S|$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{k}|Y_{i}||B_{i}|+\sum_{x\in X^{\prime}}|A_{x}|$
$\displaystyle\leq$
$\displaystyle\sum_{i=1}^{k}|Y_{i}||B_{i}|+\frac{\alpha(G)}{|V(G)|}\sum_{i=1}^{k}|B_{i}||\bar{N}_{H}[Y_{i}]|$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{k}|B_{i}|\left(\frac{\alpha(G)}{|V(G)|}|H|+|Y_{i}|-\frac{\alpha(G)}{|V(G)|}|N_{H}[Y_{i}]|\right)$
$\displaystyle=$
$\displaystyle\alpha(G)|H|+\sum_{i=1}^{k}|B_{i}|\left(|Y_{i}|-\frac{\alpha(G)}{|V(G)|}|N_{H}[Y_{i}]|\right)$
$\displaystyle\leq$ $\displaystyle\alpha(G)|H|.$
The last inequality follows from that
$\displaystyle|Y_{i}|-\frac{\alpha(G)}{|G|}|N_{H}[Y_{i}]|\leq|Y_{i}|-\frac{\alpha(H)}{|V(H)|}|N_{H}[Y_{i}]|\leq
0,$ (6)
by Proposition 1.7.
The maximum of $|S|$ implies that $|S|=\alpha(G)|H|$, from which it follows
that equalities (2), (3), (4) and (6) hold. Also, from Proposition 1.7,
equality (6) means that either $Y_{i}=\emptyset$, or
$\frac{\alpha(G)}{|G|}=\frac{\alpha(H)}{|V(H)|}$ and $Y_{i}$ is either
imprimitive or a maximum independent set of $H$ for $i=1,2,\ldots,k$.
We now prove that either $S$ is the preimages of projections of a maximum
independent set of $G$ or $H$, or (ii) or (iii) holds. There are two cases to
be considered.
Case 1: $\frac{\alpha(G)}{|G|}>\frac{\alpha(H)}{|H|}$. Then, equality (6)
means that $Y_{i}=\emptyset$ for all $i$, and so $X^{\prime}=V(H)$ by equality
(4). Hence, from equality (2) it follows that $A_{x}$ is a maximum independent
set of $G$ for all $x\in V(H)$. With this assumption we have that for any
$x,y\in V(H)$ with $(x,y)\in E(H)$, if $A_{x}\neq A_{y}$, there must exist
$a\in A_{x}$ and $b\in A_{y}$ with $(a,b)\in E(G)$ since both $A_{x}$ and
$A_{y}$ are maximum independent set, so $[(a,x),(b,y)]\in E(G\times H)$,
contradicting $\\{(a,x),(b,y)\\}\subset S$. Therefore, $A_{x}=A_{y}$ whenever
$(x,y)\in E(H)$, which implies that $S$ is the preimage of a maximum
independent set of $G$ under projections if $H$ is connected.
Case 2: $\frac{\alpha(G)}{|G|}=\frac{\alpha(H)}{|H|}$. Then, equality (6)
means that either $|Y_{i}|=0$ or $\alpha(H)$, or $Y_{i}$ is an imprimitive
independent set of $H$ for each index $i$. If $Y_{i}$ is an imprimitive
independent set of $H$ for some $i$, then $H$ is IS-imprimitive. If
$|Y_{i}|=\alpha(H)$ for all $i$, then $X_{a}=X_{a}^{*}$ is a maximum
independent set of $H$ for all $a\in V(G)$, and we can prove in the similar
way as in Case 1 that $S$ is the preimage of a maximum independent set of $H$
under projections if $G$ is connected. We now suppose that $|Y_{i}|=0$ for
some $i$. With this assumption, then equality (4) implies $X^{\prime}=V(H)$,
and then equality (3) means that either $A_{x}$ is either imprimitive or a
maximum independent set of $G$ for all $x\in V(H)$. If the former holds for
some $x\in V(H)$, we have that $H$ is IS-imprimitive; otherwise, the latter
holds for all $x\in V(H)$, and then we can prove in the similar way as in Case
1 that $S$ is the preimage of a maximum independent set of $G$ under
projections if $H$ is connected.
## 3 Concluding Remark.
Let $G_{1},G_{2},\ldots,G_{n}$ be $n$ non-empty vertex-transitive graphs, and
set $G=G_{1}\times G_{2}\times\cdots\times G_{n}$. From Theorem 1.8 it
immediately follows that
$\alpha(G)=\alpha(G_{1})\prod_{2\leq i\leq n}|G_{i}|.$
We now discuss the MIS-normality of $G$. For convenience, we say $G$ is MIS-
normal if $n=1$.
A graph $H$ is said to be non-empty if $E(H)\neq\emptyset$. It is well known
that if $H$ is a non-empty vertex-transitive graph, then
$\frac{\alpha(H)}{|H|}\leq\frac{1}{2}$, and equality holds if and only if $H$
is a bipartite graph.
Without loss of generality we may assume that
$\frac{1}{2}\geq\frac{\alpha(G_{1})}{|G_{1}|}=\cdots=\frac{\alpha(G_{\ell})}{|G_{\ell}|}>\frac{\alpha(G_{\ell+1})}{|G_{\ell+1}|}\geq\cdots\geq\frac{\alpha(G_{n})}{|G_{n}|}$,
and write $H_{0}=G_{1}\times\cdots\times G_{\ell}$ and $H_{i}=H_{i-1}\times
G_{\ell+i}$ for $i=1,\ldots,n-\ell$ subject to $n>\ell$. Then $G=H_{n-\ell}$
and with
$\frac{\alpha(H_{i-1})}{|H_{i-1}|}>\frac{\alpha(G_{\ell+i})}{|G_{\ell+i}|}$
for $i\geq 1$.
###### Proposition 3.1
Suppose $n>\ell$. Then $G$ is MIS-normal if and only if $H_{0}$ is MIS-normal
and $G_{\ell+1},\ldots,G_{n}$ are all connected.
Proof. Since $\alpha(G)=\alpha(H_{0})\prod_{i=\ell+1}^{n}|G_{i}|$, we have
that if $H_{0}$ is not MIS-normal, then $G$ is not MIS-normal. Furthermore, if
$G_{i}$ is not connected for for some $i\geq 1$, writing
$G_{i}=G_{i}^{\prime}\cup G_{i}^{\prime\prime}$, a union of disjoint
subgraphs, then, for all $I_{1},I_{2}\in I(H_{i-1})$ with $I_{1}\neq I_{2}$,
it is clear that $S=(I_{1}\times G_{i}^{\prime})\cup(I_{2}\times
G_{i}^{\prime\prime})\in I(H_{i})$, which is not a preimage of any independent
set of one factor under projections, i.e., $H_{i}$ is not MIS-normal, hence
$G$ is not MIS-normal.
Conversely, suppose $H_{0}$ is MIS-normal, and $G_{\ell+i}$ is connected for
$i\geq 1$. Since
$\frac{\alpha(H_{i-1})}{|H_{i-1}|}>\frac{\alpha(G_{\ell+i})}{|G_{\ell+i}|}$,
Theorem 1.8 implies that each maximal-sized independent set is of the form
$S\times G_{\ell+i}$, where $S\in I(H_{i-1})$, which means that $H_{i}$ is
MIS-normal for $i\geq 1$. We thus prove that $G$ is MIS-normal. ∎
We now discuss the case $n=\ell$, that is, each $G_{i}$ has the identical
independence ratio. To deal with this case we need a lemma as follows.
###### Lemma 3.2
Suppose that G is a vertex-transitive bipartite graph. Then $G$ is imprimitive
if and only if $G$ is disconnected.
Proof. It is clear that $G$ is imprimitive if $G$ is disconnected. On the
converse, if $G$ is imprimitive, then there is an imprimitive independent set
$A$ such that $\frac{|A|}{|N_{G}[A]|}=\frac{\alpha(G)}{|G|}=\frac{1}{2}.$ Set
$B=N_{G}(A)$. $|B|=|A|$ and $A\subseteq N_{G}(B)$ is clearly. If $N_{G}(B)\neq
A$, then we obtain that $\sum_{u\in A}d(u)\leq\sum_{v\in B}d(v)$, which
induces a contradiction. Hence $N_{G}(B)=A$, that is to say $G$ is
disconnected.
###### Proposition 3.3
Suppose that
$\frac{\alpha(G_{1})}{|G_{1}|}=\cdots=\frac{\alpha(G_{n})}{|G_{n}|}=\frac{\alpha(G)}{|G|}$.
Then $G$ is MIS-normal if and only if one of the following holds.
(i) $\frac{\alpha(G)}{|G|}<\frac{1}{2}$ and every $G_{i}$ is IS-primitive.
(ii) $\frac{\alpha(G_{1})}{|G_{1}|}=\frac{1}{2}$, $n=2$ and both $G_{1}$ and
$G_{2}$ are connected.
Proof. For $1\leq i\leq n$, set $\hat{G}_{i}=G_{1}\times\cdots\times
G_{i-1}\times G_{i+1}\times\cdots\times G_{n}$. Then $G=\hat{G}_{i}\times
G_{i}$ for $i=1,2,\ldots,n$. If $G_{i}$ is imprimitive, letting $A_{i}$ be an
imprimitive independent set of $G_{i}$, for every $I\in I(\hat{G}_{i})$, it is
easy to see that $S=(\hat{G}_{i}\times
A_{i})\cup(I\times\bar{N}_{G_{i}}[A_{i}])\in I(G)$, which is not a preimage of
any independent set of $\hat{G}_{i}$ or $G_{i}$ under projections, therefore,
$G$ is not MIS-normal. Conversely, if both $\hat{G}_{i}$ and $G_{i}$ are IS-
primitive, Theorem 1.8 implies that $G$ is MIS-normal. It remains to check
when $\hat{G}_{i}$ is IS-primitive. Summing up the above, $G$ is MIS-normal if
and only if both $\hat{G}_{i}$ and $G_{i}$ are IS-primitive. To complete the
proof, it remains to check when $\hat{G}_{i}$ is IS-primitive. We distinguish
two cases.
Case (i):$\frac{\alpha(G)}{|G|}<\frac{1}{2}$. In this case, Theorem 2.6 in
[20] says that if $G$ is MIS-normal, then both $\hat{G}_{i}$ and $G_{i}$ are
IS-primitive. The induction implies (i).
Case (ii): $\frac{\alpha(G_{1})}{|G_{1}|}=\frac{1}{2}$, i.e., every $G_{i}$ is
bipartite. From Lemma 3.2 it follows that $\hat{G}_{i}$ and $G_{i}$ is IS-
primitive if and only if both $\hat{G}_{i}$ and $G_{i}$ are connected.
However, it is well known that $\hat{G}_{i}$ is disconnected if $n>2$, thus
proving (ii). ∎
Combining Proposition 3.1 and Proposition 3.3 gives the following theorem.
###### Theorem 3.4
Let $G_{1},G_{2},\ldots,G_{n}$ be connected vertex-transitive graphs with
$\frac{1}{2}\geq\frac{\alpha(G_{1})}{|G_{1}|}=\cdots=\frac{\alpha(G_{\ell})}{|G_{\ell}|}>\frac{\alpha(G_{\ell+1})}{|G_{\ell+1}|}\geq\cdots\geq\frac{\alpha(G_{n})}{|G_{n}|}$,
where $n\geq 2$ and $1\leq\ell\leq n$. Then $G_{1}\times
G_{2}\times\cdots\times G_{n}$ is MIS-normal if and only if one of the
following holds:
(i) $\frac{\alpha(G_{1})}{|G_{1}|}<\frac{1}{2}$ and
$G_{1},G_{2},\ldots,G_{\ell}$ are all IS-primitive whenever $\ell>1$.
(ii) $\frac{\alpha(G_{1})}{|G_{1}|}=\frac{1}{2}$ and $\ell\leq 2$.
Acknowledgement The author is greatly indebted to Professor J. Wang for giving
useful comments, suggestions and helps that have considerably improved the
manuscript.
## References
* [1] M.O. Albertson and K.L. Collins, Homomorphisms of $3$-chromatic graphs, Discrete Math., 54 (1985) 127-132.
* [2] R. Ahlswede, H. Aydinian and L.H. Khachatrian, The Intersection Theorem for Direct Products, European J. Combin., 19 (1998) 649-661.
* [3] P. Borg, A short proof of a cross-intersection theorem of Hilton, Discrete Math., 309 (2009) 4750-4753.
* [4] P. Borg, Cross-intersecting families of permutations, J. Combin. Theory Ser. A, 117 (2010) 483-487.
* [5] P. Borg and I. Leader, Multiple cross-intersecting families of signed sets, J. Combin. Theory Ser. A, 117 (2010) 583-588.
* [6] P.J. Cameron and C.Y. Ku, Intersecting families of permutations, European J. Combin., 24 (2003) 881-890.
* [7] M. Deza and P. Frankl, On the maximum number of permutations with given maximal or minimal distance, J. Combin. Theory Ser. A, 22 (1977) 352-362.
* [8] P. Erdős, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford Ser., 2 (12) (1961) 313-318.
* [9] P. Frankl, An Erdős-Ko-Rado Theorem for direct products, European J. Combin., 17 (1996) 727-730.
* [10] C. Godsil and K. Meagher, A new proof of the Erdős-Ko-Rado theorem for intersecting families of permutations, Eurpean J. Combin., 30 (2008) 404-414.
* [11] C.Y. Ku and T.W.H. Wong, Intersecting families in the alternating group and direct product of symmetric groups, Electron. J. Combin., 14 (2007).
* [12] C.Y. Ku and B.B. Mcmillan, Independent sets of maximal size in tensor powers of vertex-transitive graphs, J. Graph Theory, 60 (2009) 295-301.
* [13] P.K. Jha and S. Klavz̆ar, Independence in direct-product graphs, Ars Combin., 50 (1998) 53-60.
* [14] B. Larose and C. Malvenuto, Stable sets of maximal size in Kneser-type graphs, European J. Combin., 25 (2004) 657-673.
* [15] B. Larose and C. Tardif, Projectivity and independent sets in powers of graph, J. Graph Theory, 40 (2002) 162-171.
* [16] V.P. Mario and V. Juan, Independence and coloring properties of direct products of some vertex-transitive graphs, Discrete Math., 306 (2006) 2275-2281.
* [17] C. Tardif, Graph products and the chromatic difference sequence of vertex-transitive graphs, Discrete Math., 185 (1998) 193-200.
* [18] J. Wang and S.J. Zhang, An Erdős-Ko-Rado-Type Theorem in Coxeter Groups, Eurpean J. Combin., 29 (2008) 1112-1115.
* [19] J. Wang and H.J. Zhang, Cross-intersecting families and primitivity of symmetric systems, submitted.
* [20] H.J. Zhang, Primitivity and independent sets in direct products of vertex-transitive graphs, J. Graph Theory, to appear.
|
# How will electric vehicles affect traffic congestion and energy consumption:
an integrated modelling approach
Artur Grigorev1, Tuo Mao1, Adam Berry1, Joachim Tan2, Loki Purushothaman1,
Adriana-Simona Mihaita1 1 Most authors are with the University of Technology
in Sydney, Faculty of Engineering and IT, School of Computer Science: 61
Broadway Ultimo, NSW, Australia. Corresponding authors contact:
<EMAIL_ADDRESS><EMAIL_ADDRESS>This
work was sponsored by AEMO the Australian Energy Market Operator. The authors
are highly grateful for the guidance and support of Joachim Tan, Adrian
Grantham, Greg Staib and Andrew Turley. Joachim Tan is a Principal Manager in
the Australian Energy Market Operators and main lead of the UTS AEMO project
collaboration.3 The authors are highly grateful for the guidance and data
support of Transport for NSW, Australia, more specifically Virginie Vernin,
Bill Chen, Blake Xu and Marwan Daizli.
###### Abstract
This paper explores the impact of electric vehicles (EVs) on traffic
congestion and energy consumption by proposing an integrated bi-level
framework comprising of: a) a dynamic micro-scale traffic simulation suitable
for modelling current and hypothetical traffic and charging demand scenarios
and b) a queue model for capturing the impact of fast charging station use,
informed by traffic flows, travel distances, availability of charging
infrastructure and estimated vehicle battery state of charge. To the best of
our knowledge, this paper represents the first integrated analysis of
potential traffic congestion and energy infrastructure impacts linked to EV
uptake, based on real traffic flows and the placement and design of existing
fast-charging infrastructure. Results showcase that the integrated queue-
energy-transport modelling framework can predict correctly the limitations of
the EV infrastructure as well as the traffic congestion evolution. The
modelling approach identifies concrete pain points to be addressed in both
traffic and energy management and planning. The code for this project can be
found at : https://github.com/Future-Mobility-Lab/EV-charging-impact
###### Index Terms:
electric vehicles, traffic simulation modelling, queue modelling, recharging
scenario evaluation, fast charge impact modelling
## I Motivation
The adoption of electric vehicles (EVs) at large scale around the globe is
largely dependent on two key factors: charging accessibility for its consumers
and availability of sufficient electrical grid capacity to support vehicle
charging, particularly where fast charging infrastructure is necessary. Under
ideal conditions for the consumer, access to an EV charging station should be
just as convenient as current access to a petrol station. Several studies have
been conducted in recent years to model and identify suitable locations for
such charging stations [2], [3]. The majority of these studies are based on
the behavioural pattern of combustion engine (CE) drivers. However, the
behaviour of electric vehicle (EV) drivers is significantly different in terms
of location selection and their waiting time to charge [2], [4]. Multiple
factors can influence the decision of EV drivers to choose a specific location
to charge, among which we cite: the travel time towards the closest EV
station, the state of charge of the battery (SoC) while driving, the waiting
time in queue at the EV station, the total charging time that a driver needs
to plan different activities around while the car is charging. The
stochasticity of this charging behaviour makes the modelling of EVs a hard
problem to solve, which may delay evidence-based infrastructure planning at a
government level and reduce investment confidence for commercial
infrastructure and electric vehicle businesses. Therefore, the inability to
charge conveniently in the traffic network coupled with limited charging stops
makes the problem challenging from both operational and planning perspectives.
Even more challenging, evaluating the grid impact of EV adoption comes along
not only with a consumer random behaviour affected as well by traffic
congestion variation, but with the physical constraints of both infrastructure
availability and maximal energy capacity of each serviced area.
These challenges represent the motivation of this study which to the best of
our knowledge is the first to propose an integrated bi-level framework
comprised of a dynamic traffic simulation modelling coupled with queueing
behaviour modelling for estimating the impact of traffic demand and congestion
on EV fast charging behaviour and vice-versa.
### I-A Related works
Figure 1: Bi-level framework of transport and energy modelling containing: A)
Upper Level focused on a hybrid transport and queue modelling approach for
quantifying EV charging on traffic congestion and vice-versa and B) Lower
level focus on building a home-and-destination charging model using substation
capacity data and the daily and dynamic energy load data in the study area.
Several studies [5]-[6], have been conducted to forecast the electricity
demand that will be created by the penetration of EVs into the market in the
future. There are also works to determine and forecast the impact this
electricity demand will have on the existing power grids [7]. However, the
majority of these studies either orient themselves towards analysing the
impact of the electricity consumption on the grid using hypothetical
assumptions of the charging locations or very broad estimations of the major
attraction points in specific areas (p-median or location-allocation problem)
such as shopping malls, schools, hospitals and workplaces [8]. Alternatively,
some studies that have considered the user equilibrium (UE) approach to
optimise the route choice so that the travel time is minimized when recharging
[9], [10]. However, these studies have several assumptions such as that
congestion will not affect the electricity consumption or that the path
lengths are within battery driving range. However, congestion represents an
important factor to be taken into consideration in the EV impact modelling,
especially when the fast charging stations reach their plug-in capacity. Some
studies have developed queuing models that take into consideration the station
capacity, the charging time and EV waiting time for example [11]; others have
considered as well the maximal charging demand which satisfies the user’s
maximal tolerable queuing time [12] or have proposed a global queue system
that would dispatch EVs to available charging stations [13]. While attention
was given to parameters describing the charging situations inside the EV
stations, studies did not take into consideration for example the battery
state of charge (SoC) when routing decisions are being made, nor the travel
time affected by increased demand. Other queuing approaches can be found in
[14] where they took consideration of the fact that charging stations can have
a different number of plugs with various charging capabilities.
In [15] there is a tentative approach to couple together simulation and
optimisation techniques for optimising a taxi fleet and similar approaches
have been proposed which consider for example the planning of routes according
to an activity-based model [16] or agent-based simulation [17]. We believe,
however, that there remains a key gap in the use of traffic simulation models
to explore the interrelationship between charging activity, traffic behaviour
and road congestion outside of strictly fleet-based settings. Similarly,
embedding EV queue modelling at each station in the simulation model remains
an under-explored research topic.
From the above discussions, we identify several gaps in the current
literature. Firstly, several studies do not take into consideration future
vehicle demand increases and how this may impact EV fast charging
accessibility, queuing times, traffic volumes and emergent congestion related
to the use of fast charging infrastructure. The inter-dependency between
congestion, demand and queue spill-back from EV charging stations should be
modelled together for both current and future traffic scenarios. Secondly, the
number of available plug-in slots and their charging rates should be embedded
in the queue model and constraints should be applied around realistic charging
queue lengths; this is to avoid assumptions of infinite queue models which can
accept all incoming EV requests. Thirdly, several studies propose an
optimisation of charging stations based on assumption that people charge
either at home or at their destination, whereas in reality, the charging
behaviour is dynamic and can include mid-route charging stops as well based on
battery SoC, charging availability and convenience (i.e. charging at a
shopping centre); this is a factor that would need to be embedded in any queue
modelling approach. Finally, analysis of electric vehicle behaviour should
include the electrical grid impact, as the shape and extent of charging demand
will inevitably impact electrical network infrastructure planning and
operation, particularly where charging demand is aligned with peak electrical
demand.
### I-B Paper Contributions
In our work we aim to address previous gaps by proposing an integrated
framework of traffic modelling with stochastic queue behaviour (the upper-
level which can be used for en-route charging behavioural modelling showcased
in this paper) and a data-driven energy consumption estimation (the lower-
level for home and work charging estimation which we will present in a future
paper extension). The approach has been applied on the Northern Sydney urban
area and is built using realistic data sets in terms of: traffic demand,
vehicle counting, EV charging stations (locations, number of plugs, type of
plugs, etc.), electricity zone substation capacity data and daily electricity
load profiles for those substations (sufficient for determining peak
electricity demand times).
Firstly, we propose an integrated traffic simulation and queue behavioural
model for capturing not only the dynamic nature of traffic congestion but also
the queue impact around existing EV charging stops. The microscopic traffic
simulation is calibrated against real-world traffic flow counts and is used to
evaluate traffic congestion under different EV and charging station settings.
We also explore the potential impact of hypothetical future demand scenarios.
The queue model which works on a first-in-first-out approach is tailored for
each station and ingests several simulation outputs (travel time to reach the
station, calibrated flow, travel distance, calibrated OD demand), as well as
the SoC of the batteries, the number of available charging plugs and their
charging rates. We denote this model as
$EV-Q(M/M/n_{ep}^{S_{i}}/N_{c}^{S_{i}})$ where $N_{c}^{S_{i}}$ is the maximum
number of car slots available at each station (for both waiting and charging)
and $n_{ep}^{S_{i}}$ is the number of plugs (chargers) for each station. The
EV-Q model generates queue length estimations which are then fed to the micro
traffic simulation model for congestion impact analysis. Secondly, we explore
the behaviour based on both observed road demand from 2016 and two
hypothetical scenarios: one with a $15\%$ increase in road use and another
with a $30\%$ increase. Across all scenarios, we explore impacts of different
levels of assumed EV penetration. Lastly, for all simulations, we estimate
total electrical demand associated with fast charging, enabling a review of
potential electrical infrastructure impacts.
The paper is organised as follows: Section II introduces the bi-level
framework for our integrated queue, energy and transport simulation modelling,
Section III presents a case study evaluation on the Northern Sydney area
including traffic simulation modelling (construction, calibration and
validation), queue modelling (construction and validation) and EV impact
evaluation across different penetration rates and demand scenarios. The last
section is reserved for conclusions, limitations of this work and further
directions.
## II Methodology
Figure 2: Network layout of the North Sydney traffic simulation model (NSTM)
including all existent electric vehicles charging stations (marked as red).
The bi-level framework we propose in this paper is illustrated in Fig. 1 and
is comprised of two major parts: A) the Upper level construction with the
purpose of building a hybrid model for analysing the quantified impact of EV
charging station availability on traffic congestion and vice-versa, and B) the
Lower level construction which focuses on building a home-and-destination
charging model using substation capacity data and the daily and dynamic energy
load data in the study area. Due to lack of space the current paper will only
include the upper level modelling and selected results. Each model comprises
of several modules coloured differently, with dedicated entry data streams and
output measures, summarized here below.
Figure 3: EV charging representation using a queue modelling approach of
vehicles arriving at each charging stations.
### II-A Transport simulation modelling
In order to evaluate the impact of current and future traffic demand on the
electric vehicle fast charging, one needs to be able to evaluate realistic
travel distances, travel times and traffic flows for the road network under
various dynamic traffic conditions. The traffic simulation we are building is
a microscopic dynamic model which is comprised of all vehicle movements, from
any internal origins to their destinations and intermediary charging stops
along the route.
We select North Sydney as an application area (see Fig. 2) and refer to our
simulation model as NSTM. All notations are provided in Table I. The area has
been selected due to its large business hubs, major train stations, new metro
line and major commuting trips to/from Sydney CBD on a daily basis. The NSTM
model consumes in its input: a) the network layout with real road structure
definition, b) OD demand matrices ($OD(T_{r})$), c) traffic signal control
definitions and d) the placement and characteristics of real-world EV charging
stations deployed in Northern Sydney ($S_{i},i\in\\{1,..25\\}$). Overall the
simulation model contains the daily trips of almost 100,000 people (according
to 2016 Australian Census data), 25 EV charging stations, 41 permanent traffic
count stations, 240 SCATS controlled intersections, 1,919 traffic flow
detectors and 2,750 road sections.
The micro-simulation model goes through several procedures of initialisation,
calibration and validation including: i) an initial static traffic assignment
(STA) procedure of the original traffic demand to available routes, ii) a
static OD adjustment (SOD) using permanent traffic flow counts, iii) a daily
demand profiling for AM and PM peaks, iv) a Dynamic OD adjustment (DOD) using
detector flow counts on an hourly basis for 24-hour time interval which helps
us to calibrate the initial demand matrix, v) a dynamic user equilibrium (DUE)
assignment which is validated against both permanent and detector flow counts,
and finally vi) a microscopic simulation run consuming on input as well the
estimated queue lengths on each road section in the vicinity of the EV
charging stops (this is an output of the EV-Q model described in the next sub-
section).
The major outputs of the micro NSTM model are the following: a) a calibrated
and more realistic OD demand matrix $OD_{c}(T_{r})$, b) the total travel times
for AM/PM peaks, c) the calibrated traffic flows and finally d) the average
travel distance (the last two used as inputs to the EV-Q charging model for
each station).
The NSTM is used for testing the impact of various scenarios: a baseline
scenario based on observed traffic demand for 2016, and two hypothetical
growth scenarios, with a $15\%$ and $30\%$ increase in total demand,
respectively. The scenarios allow us to explore how changes in traffic demand
interact with electric vehicle penetration rates and EV infrastructure
availability to effect road use, queuing times, congestion and electricity
demand.
### II-B Queue modelling of EV charging
While the traffic simulation model has the objective of providing dynamic
insights on the traffic congestion in the neighbourhood, it does not have the
capability of replicating the queueing behaviour of drivers that need EV
charging. Therefore, by using simulation outputs such as car average travel
distance and traffic flow on road sections, we develop a queue model to
quantify the impact of the waiting behaviour around existing EV stations and
feed this behaviour back to the simulation model. The traffic simulation and
the queue models are connected to each other and run in parallel consuming
each-other outputs. Fig. 3 presents the overall queue modelling framework
(EV-Q) which we further detail.
Table I: Notations in use for EV modelling.
Variable Definition $NSTM$ North Sydney Traffic Model, $NSTM_{c}$ North Sydney
Traffic Model calibrated, $T_{r}$ the 1h time interval during a 24-hour period
where $r\in\\{1...24\\}$; $OD(T_{r})$ the Origin-Destination matrix containing
the number of assigned car trips in the network in time period $T_{r}$
$OD_{c}(T_{r})$ the calibrated Origin-Destination matrix $Tf_{k}(T_{r})$ the
traffic flow for each road section $k$ in the network during time interval
$T_{r}$ $Avg_{d}(T_{r})$ the average travel distance of cars during $T_{r}$
$TT_{tot}^{e}(T_{r})$ the car total time during $T_{r}$ and specific $EV_{p}$
$N$ the total number of EV stations in the network, $S_{i},i\in\\{1,..,N\\}$
the EV station ID, $N_{c}^{S_{i}}$ the maximal number of cars inside a station
$EV_{p}$ the EV penetration rate in the network $\lambda^{S_{i}}$ mean-arrival
time of cars at an EV station $n_{ep}^{S_{i}}$ number of electric plugs per
station $B_{c}(T_{r})[kWh]$ battery size of arriving cars during $T_{r}$
$SoC^{S_{i}}$ the initial state of charge for vehicles arriving at station
$S_{i}$ $\rho^{S_{i}}$ the service density of a station $S_{i}$
$\overline{Q^{S_{i}}}(T_{r})$ (O1) Mean queue length of an EV station $S_{i}$
per hour $\overline{WT_{q}^{S_{i}}}(T_{r})$ (O2) Mean waiting time in queue at
an EV station $S_{i}$ $\mu_{t}^{S_{i}}(T_{r})$ (O3) Mean service time to
charge at an EV station $S_{i}$ $WT^{S_{i}}(T_{r})$ (O4) Total time spent
overall at an EV station $S_{i}$ $E^{S_{i}}(T_{r})$ (O5) Total energy
consumption of an EV station $S_{i}$ $MaxQ^{S_{i}}(T_{r})$ (O6) Maximum
recorded queue length of an EV station $S_{i}$ $MaxW_{q}^{S_{i}}(T_{r})$ (O7)
Maximum waiting time in queue at an EV station $S_{i}$ $MaxWT^{S_{i}}(T_{r})$
(O8) Maximum time spent overall at an EV station $S_{i}$ $MaxE^{S_{i}}(T_{r})$
(O9) Maximal energy consumption of an EV station $S_{i}$
Let $N_{c}(T_{r})$ be the total number of cars in the traffic network during
time interval $(T_{r})\in\\{0,1,2,...23h\\}$, $EV_{p}$ the percentage of
vehicles that will need to charge and $S$ the total number of EV charging
stations (25 in our study according to the real world set-up [18]). Each EV
station has a specific number of plugs (chargers) which we denote
$n_{ep}^{S_{i}}$ and which follows a real set-up arrangement in our network (7
stations have only 1 plug available, 3 have 4 plugs, 2 have 3 plugs, 10 have 2
plugs, 1 has 7 plugs and only 1 has 10 plugs available). We denote the queue
model as $EV-Q(M/M/n_{ep}^{S_{i}}/N_{c}^{S_{i}})$ where the first M stands for
EV arrivals following a Gaussian distribution with mean arrival time at each
station $\lambda^{S_{i}}$, the second M represents the distribution of the
charging times with $\mu^{S_{i}}$ mean-service time and $N_{c}^{S_{i}}$ is the
maximum number of car slots available at each station (for both waiting and
charging). For this paper, we follow the queue modelling procedures from [14,
19] which are using the Markov Chain theory for probability calculation of
charging at specific plugs of a station, with the assumption that the number
of waiting cars inside the stations are limited to a maximum threshold.
The cars arrive at each EV charging station following a normal distribution
with mean-inter-arrival (mIA) times denoted as $\lambda^{S_{i}}$ (this is
calculated from real data-streams of vehicles captured by the traffic count
detectors in the vicinity of each EV station in our network; mIA is calculated
hourly from traffic flow with a variation of $\pm 20\%$ following normal
distribution; therefore this is estimated based on simulation of traffic flow
at vicinity of the charging stations). For fairness of charging, the arriving
cars are being charged on a First-Come-First-Served (FCFS) approach to the
next available plug in the network (statistics on emergent queues are stored
in O1-Fig. 3). The service time (O3 in Fig. 3) is then calculated individually
for each arriving car by taking into consideration the car State of Charge
($SoC^{S_{i}}$) as follows:
$(O3):\mu^{S_{i}}(T_{r})=\frac{SoC^{S_{i}}(T_{r})*B_{c}(T_{r})}{Pp^{S_{i}}}$
(1)
where $SoC^{S_{i}}(T_{r})$ represents the state of charge for arriving cars,
$B_{c}(T_{r})$ is the battery size and $Pp^{S_{i}}$ is the charging rate of
the fast charger of a station $S_{i}$ (which follows the real EV station
information with the specification that majority have a 22kW capability [18]).
We assume that the initial State-of-Charge will follow a normal distribution
between $20\%-80\%$ according to efficient Li-Ion battery use [20]. Meanwhile
the battery size can take several values which have been used for sensitivity
and robustness analysis: $B_{c}(T_{r})[kWh]\in\\{50,58,66,74,82,90,98,106\\}$,
according to a 10% difference in battery size compared to Tesla Model 3
(Standard Range) [21]. Based on arrival and service times, one can then
express the EV charging density $\rho^{S_{i}}$ as:
$\rho^{S_{i}}(T_{r})=\lambda^{S_{i}}(T_{r})/\mu^{S_{i}}(T_{r})$ (2)
Based on the above formulations and the Markov Chain stationary conditions
regarding the matrix of transition probabilities $P$ and the stationary
distribution ($\vec{\pi}\cdot P=0$), we can first calculate the stationary
distribution (the probability that a vehicle will charge at a specific plug in
the EV station) in a recursive form as follows:
$\pi_{k}^{S_{i}}=\left\\{\begin{array}[]{ll}\frac{1}{k!}\left(\rho^{S_{i}}(T_{r})\right)^{k}\pi_{0}^{S_{i}}&\mbox{if
}\ k\leq{n_{ep}^{S_{i}}}\\\
\left(\frac{1}{n_{ep}^{S_{i}}}\right)^{(k-n_{ep}^{S_{i}})}\cdot\frac{1}{{n_{ep}^{S_{i}}}!}{\left(\rho^{S_{i}}(T_{r})\right)^{{n_{ep}^{S_{i}}}k}}\pi_{0}^{S_{i}}&\mbox{if
}\ k\geq{n_{ep}^{S_{i}}}\end{array}\right.$ (3)
By further replacing in the total probability condition ($\sum_{j}\pi_{j}=1$),
one can easily calculate the $\pi_{0}$ probability that no EV will charge at a
specific station $S_{i}$; Implicitly the average queue length (O1 in Fig. 3)
and the average waiting time (O2 in Fig. 3) can then be estimated based on the
surplus of vehicles waiting to charge, the capacity of a charging station and
the probability of these vehicles to arrive at a station $S_{i}$.
$\displaystyle(O1):Q^{S_{i}}(T_{r})$
$\displaystyle=\sum_{k=n_{ep}^{S_{i}}}^{N_{c}^{S_{i}}}(k-n_{ep}^{S_{i}})\cdot\pi_{k}^{S_{i}}$
(4) $\displaystyle(O2):\overline{WT_{q}^{S_{i}}}(T_{r})$
$\displaystyle=\frac{Q^{S_{i}}(T_{r})}{\lambda\left(1-\pi_{k}^{S_{i}}\right)}$
(5)
Other outputs of the EV-Q model can then be calculated on an hourly basis as
follows:
$\displaystyle(O4):WT^{S_{i}}(T_{r})$
$\displaystyle=W_{q}^{S_{i}}(T_{r})+\mu^{S_{i}}(T_{r})$ (6)
$\displaystyle(O5):E^{S_{i}}(T_{r})$
$\displaystyle=PP^{S_{i}}\cdot\mu^{S_{i}}(T_{r})$ (7)
where $WT^{S_{i}}(T_{r})$ represents the total waiting time to charge at a
station $S_{i}$ (O4 in Fig. 3), and $E^{S_{i}}(T_{r})$ is the total energy
consumption to charge during a specific time interval at a given EV station
(O5 in Fig. 3). Other model outputs are extracted such as minimum and maximal
queue length, waiting time and energy consumption (O6-O9). The EV-Q model
outputs are further used in the global traffic-energy simulation modelling
framework: a) the average queue length as an input in the NSTM calibration
process to realistically simulate the impact of queue waiting to charge even
outside of the EV stations and b) the energy consumption estimation for the
impact of fast charging on electricity distribution networks.
## III North Sydney Case Study
### III-A Simulation calibration and validation
Figure 4: Validation of NSTM model for the AM peak via a) Trip Length
Distribution and b) $R^{2}=0.92$ metric.
After the initial construction of the NSTM model, the 200-pair OD matrix
$OD(T_{r})$ is calibrated by using permanent traffic flow counts from 2016
records (marked in green in Fig. 10 from the supplementary material [1]). The
$OD(T_{r})$ matrices are split to contain $93\%$ of regular car trips and
$7\%$ of truck trips in the network according to real-life records. After the
calibration, the dynamic and hourly demand matrices are then validated against
traffic counts received from lane detectors placed at signalised intersections
(showcased in Fig. 11 and Fig. 12 from the supplementary material [1]). The
results for the AM-peak NTSM calibration and validation are provided in Fig.
4, where the Original versus Adjusted Trip Length distributions follow very
similar trends with less than $2\%$ difference between them (Fig. 4a));
similarly, the $R^{2}$ metric calculated between simulated and the real
traffic demand scored 0.92 in accuracy which validates once again the good
calibration of the model (Fig. 4b)). The NSTM calibration and validation for
the PM peak indicates similar performance of $R^{2}=0.92$ and are provided in
the complementary material [1]-Section -B. We make the observation that while
we validated the NSTM model against real-world traffic data from 2016, our two
hypothetical growth scenarios (OD15 and OD30) cannot be similarly validated
and therefore are best thought of as “what-if” scenarios. The outputs of the
simulation model are further used as inputs in both the queue modelling and
energy impact evaluation, as detailed in following section.
### III-B Queue Model results
The traffic flow $Tf_{k}(T_{r})$ and the average travel distance
$Avg_{d}(T_{r})$ have been used as inputs in the EV-Q model with the aim of
calculating the percentages of EVs needing recharging at each station, and the
travel distance they had to travel before recharging. The daily fluctuations
of traffic flow in the network can greatly influence the charging needs (see
Fig. 20 in [1]), and the number of EVs that will seek recharging based on the
nearest station.
Therefore, we apply the EV-Q model under several scenarios which allows an
extensive simulation of current and potential future traffic conditions, by
varying several input parameters such as the battery size of the vehicles
$B_{c}(T_{r})[kWh]$, the penetration rates of EVs that need to charge
$EV_{p}\in\\{0.01,0.02,..0.05,0.1,0.12,...0.2,0.21...0.3,0.5,1,2,5\\}[\%]$ as
well as our three traffic demand scenarios. Overall, the EV-Q modelling allows
us to analyse more than 90,000 different scenarios with modeling precision
down to a second-by-second.
Figure 5: The impact of EV penetration rates on queue lengths, waiting times,
service times and energy consumption.
The nine outputs of the EV-Q summarised in Table I are aggregated hourly and
used for analysing the energy impact when drivers decide to apply an in-route
charging approach (samples of O1-O5 are represented already in Fig. 3 and
several aggregations by station/area are extracted). A detailed selection of
averaged outputs from the EV-Q model is provided in Fig. 5, where we choose
$B_{c}=82kWh$ and vary $EV_{p}$ between $\\{0.01,0.02,..0.3\\}$ (higher
variations are presented in [1]-Fig. 21); we analyse the mean queue length
(O1-Fig. 5a)), the mean waiting time (O2-Fig. 5b)), the mean service time
(O3-Fig. 5c)) and the mean energy consumption (O5-Fig. 5d)). On each figure we
also plot the impact of the number of plugs on the EV-Q model; for example, in
Fig. 5a), the mean queue length recorded across EV stations with one plug
(blue dots and their confidence interval) indicates that cars start to queue
from an $EV_{p}=0.03\%$ and can easily reach 20 cars if $EV_{p}=0.3\%$;
stations with 7 charging plugs only start to accumulate cars in the queue from
$EV_{p}=0.21\%$ while reaching a mean of 10 cars in the queue for
$EV_{p}=0.3\%$; the best performing stations are the ones with 10 charging
plugs which present zero queue accumulation until $EV_{p}=1\%$ (see [1]-Fig.
22). The mean waiting time at stations (Fig. 5b)) is highly influenced by the
available number of fast charging plugs and for even a low $EV_{p}=0.04\%$ it
could reach 1 hour easily. The results highlighted in Fig. 5 indicate that
current fast charging infrastructure deployed in our case study area is likely
insufficient for even low levels of EV uptake and utilisation. Even assuming
that only $0.15\%$ of passing vehicles choose to recharge at a station,
average waiting times balloon to unrealistic and unsustainable levels
(exceeding 2 hours) in all but ten-plug stations. Separately, note that total
electricity consumption is highly sensitive to the placement of charging
infrastructure. The average total energy consumption of 7 plug stations, which
are located in high-volume traffic locations, far outstrip the consumption of
the ten plug stations located in lower volume areas (if $0.15\%$ of passing
vehicles charge, the 7 plug stations more than triple the energy consumption
levels of the 10 plug stations).
The average service time (O3) is under 1 hour for the majority of stations
regardless of the number of EVs waiting in the queue (see trend of Fig. 5c))
while the energy consumption is highly dependent on the traffic flow recorded
at specific stations and the utilisation of that station - for example in Fig.
5d) the 7-plug stations are consuming almost 5 times more than even stations
with ten plugs (at $EV_{p}=0.3\%$). The findings reveal not only the
limitation of the current EV infrastructure in the area, but also allow to
make predictions and planning decisions based on the popularity of specific
areas in the network, their capacity, traffic demand, etc.
### III-C EV impact on congestion
The mean queue results $\overline{Q^{S_{i}}}(T_{r})[veh]$ of the EV-Q
modelling is further utilised in the NSTM model to evaluate the impact that
queue spill-back outside of EV stations would have on the overall traffic
congestion in the network. Fig. 6 presents the daily total travel time spent
by all cars in the network, across all EV penetration rates (Ox axis), for
OD2016 (blue), OD15 (orange) and OD30 (grey). Assuming the traffic demand
remains constant in 2016, the total travel time would increase by almost
$4.9\%$ if the EV penetration rate would reach $5\%$ (see increase from point
A to B). More specifically, this represents an extra of almost $1074$ days
spent in traffic by all vehicles (equivalent to an extra 6.18 hours spent in
traffic for each driver - a highly penalising scenario). OD15 seems to add an
even higher burden: an $8.7\%$ increase from point C to D translates in $11.6$
extra hours of time spent in traffic per year. The OD30 demand scenario
translates in a $12.1\%$ increase assuming $EV_{p}=0.01$ remains the same, an
unrealistic assumption given the global EV uptake trajectories (point E versus
A).
Figure 6: Travel time evolution across several traffic demand and EV
penetration rates for OD2016, OD15, OD30 scenarios.
### III-D EV impact on energy consumption
While increasing the number of plugs can significantly reduce the waiting time
for drivers, the energy impact will also increase considerably; Fig. 7
showcases the regression trends of daily total energy $E^{S_{i}}$(O5) versus
$n_{ep}^{S_{i}}$; while stations with small $n_{ep}^{S_{i}}$ will not affect
the energy consumption (for $n_{ep}^{S_{i}}<2$ and $EV_{p}=5\%$ the
$E^{S_{i}}<1200kWh$), the stations with high number of plugs (7,9,10) will
induce almost a 4-times higher energy consumption in the network
($E^{S_{i}}=4,500kWH$ for $EV_{p}=5\%$); this can have a significant load on
the electric network and the current EV station capacity.
Figure 7: Energy consumption vs. the number of plugs for OD2016. Figure 8:
Energy consumption estimations for OD2016, OD15, OD30.
Furthermore, in Fig. 8 we show that our demand increase assumptions for OD15
and OD30 translates in a saturation of the average daily energy consumption
after an $EV_{p}=2\%$ (reaching 1400kWH) but follows similar trends across all
traffic demands that we tested; this is mostly related to: a) the saturation
of the traffic network when the demand increases considerably and vehicles
would queue everywhere in the network and b) the EV infrastructure limitation
which we assumed remains the same for our experiments despite the traffic
demand would increase even by $30\%$ in the next years. There is however a
significant impact on the energy consumption when switching from
$EV_{p}=0.01\%$ to $EV_{p}=0.1\%$ which translates in an extra 1000kWh
consumed daily on average across all stations. The impact of future EV
increases and how this affects differently all EV stations, with an additional
comparison is provided in [1]-Section -E and Fig. 23. An important aspect is
that EV stations with higher number of plugs (7,9,10) will be the ones that
will increase almost three times more their energy consumption when compared
to smaller stations (up to 4 plugs).
### III-E EV impact on state grid
Figure 9: 2016 state-level weekday energy consumption versus total estimated
EV consumption across all stations in the EV-Q model.
For analysing the EV impact on the electricity grid infrastructure, we compare
the shape of aggregated EV charging demand with the shape of state-wide
electricity demand in Fig. 9. The results suggest that EV fast charging may
exacerbate existing electricity peak demand issues. Indeed, during the
electricity peak demand period, modelled aggregate EV charging demand is
between $91\%$ and $95\%$ of its daily peak. In many ways, the coincidence of
electrical and charging demand is unsurprising: peak electrical demand in New
South Wales is driven by residential energy use, particularly in the period
following the end of the working day; traffic volume (and thus probable fast
charger use) is similarly linked to working patterns and transit from work to
home.
## IV Conclusion
This paper presents an integrated bi-level framework of dynamic traffic
modelling, data-driven queue and energy modelling with the purpose of
evaluating the EV impact on both the traffic network and the energy
consumption of the studies area. Results reveal existing limitations of the EV
charging infrastructure which would not be able to handle even slight
increases of EV rates, as well as significant travel time and waiting time
degradation for larger EV penetration rates rates at each station level. The
proposed bi-level framework was validated against real traffic and energy data
sources and is currently being used for an extended analysis of this case
study.
## References
* [1] online supplement, “Appendix: How will electric vehicles affect traffic congestion and energy consumption: an integrated modelling approach,” 2020, https://bit.ly/32gLSsY.
* [2] B. Csonka and C. Csiszár, “Determination of charging infrastructure location for electric vehicles,” _Transportation Research Procedia_ , vol. 27, pp. 768–775, 2017, 20th EURO Working Group on Transportation Meeting, EWGT 2017, 4-6 September 2017, Budapest, Hungary. [Online]. Available: https://bit.ly/3mNMNuc
* [3] Y. Motoaki, “Location-allocation of electric vehicle fast chargers—research and practice,” _World Electric Vehicle Journal_ , vol. 10, no. 1, 2019. [Online]. Available: https://www.mdpi.com/2032-6653/10/1/12
* [4] D. Chakraborty, D. S. Bunch, J. H. Lee, and G. Tal, “Demand drivers for charging infrastructure-charging behavior of plug-in electric vehicle commuters,” _Transportation Research Part D: Transport and Environment_ , vol. 76, pp. 255–272, 2019. [Online]. Available: https://bit.ly/3abRFo2
* [5] H. Moon, S. Y. Park, C. Jeong, and J. Lee, “Forecasting electricity demand of electric vehicles by analyzing consumers’ charging patterns,” _Transportation Research Part D: Transport and Environment_ , vol. 62, pp. 64–79, 2018. [Online]. Available: https://bit.ly/3uQcWvd
* [6] K. Qian, C. Zhou, M. Allan, and Y. Yuan, “Modeling of load demand due to ev battery charging in distribution systems,” _IEEE Transactions on Power Systems_ , vol. 26, no. 2, pp. 802–810, 2011.
* [7] L. I. Dulău and D. Bică, “Effects of electric vehicles on power networks,” _Procedia Manufacturing_ , vol. 46, pp. 370–377, 2020, 13th International Conference Interdisciplinarity in Engineering, INTER-ENG 2019, 3–4 October 2019, Targu Mures, Romania. [Online]. Available: https://bit.ly/2RFHjX7
* [8] M. Ghamami, Y. M. Nie, and A. Zockaie, “Planning charging infrastructure for plug-in electric vehicles in city centers,” _International Journal of Sustainable Transportation_ , vol. 10, no. 4, pp. 343–353, 2016. [Online]. Available: https://doi.org/10.1080/15568318.2014.937840
* [9] F. He, Y. Yin, and S. Lawphongpanich, “Network equilibrium models with battery electric vehicles,” _Transportation Research Part B: Methodological_ , vol. 67, pp. 306–319, 2014. [Online]. Available: https://bit.ly/3mLdH60
* [10] M. Xu, Q. Meng, and K. Liu, “Network user equilibrium problems for the mixed battery electric vehicles and gasoline vehicles subject to battery swapping stations and road grade constraints,” _Transportation Research Part B: Methodological_ , vol. 99, pp. 138–166, 2017. [Online]. Available: https://bit.ly/3g8pvxT
* [11] J. Zhu, Y. Li, J. Yang, X. Li, S. Zeng, and Y. Chen, “Planning of electric vehicle charging station based on queuing theory,” _The Journal of Engineering_ , vol. 2017, pp. 1867–1871, 2017.
* [12] Fang Lu and Guowei Hua, “A location-sizing model for electric vehicle charging station deployment based on queuing theory,” in _2015 International Conference on Logistics, Informatics and Service Sciences (LISS)_ , 2015, pp. 1–5.
* [13] S. Su, H. Li, and D. W. Gao, “Optimal planning of charging for plug-in electric vehicles focusing on users’ benefits,” _Energies_ , vol. 10, no. 7, 2017. [Online]. Available: https://www.mdpi.com/1996-1073/10/7/952
* [14] D. Xiao, S. An, H. Cai, J. Wang, and H. Cai, “An optimization model for electric vehicle charging infrastructure planning considering queuing behavior with finite queue length,” _Journal of Energy Storage_ , vol. 29, p. 101317, 2020. [Online]. Available: https://bit.ly/2PXAVdg
* [15] J. Jung, J. Y. Chow, R. Jayakrishnan, and J. Y. Park, “Stochastic dynamic itinerary interception refueling location problem with queue delay for electric taxi charging stations,” _Transportation Research Part C: Emerging Technologies_ , vol. 40, pp. 123–142, 2014. [Online]. Available: https://bit.ly/3e3eXNU
* [16] M. Usman, L. Knapen, A.-U.-H. Yasar, T. Bellemans, D. Janssens, and G. Wets, “Optimal recharging framework and simulation for electric vehicle fleet,” _Future Generation Computer Systems_ , vol. 107, pp. 745–757, 2020. [Online]. Available: https://bit.ly/32ihjCU
* [17] M. Klein, L. Lüpke, and M. Günther, “Home charging and electric vehicle diffusion: Agent-based simulation using choice-based conjoint data,” _Transportation Research Part D: Transport and Environment_ , vol. 88, p. 102475, 2020. [Online]. Available: https://bit.ly/3slF3kn
* [18] Electric vehicle Council, “EV Charger Map,” https://electricvehiclecouncil.com.au/about-ev/charger-map/, 2021, online; accessed 22 February18/03/2021.
* [19] Z. Tian, W. Hou, X. Gu, F. Gu, and B. Yao, “The location optimization of electric vehicle charging stations considering charging behavior,” _SIMULATION_ , vol. 94, no. 7, pp. 625–636, 2018. [Online]. Available: https://doi.org/10.1177/0037549717743807
* [20] S. Castano, L. Gauchia, E. Voncila, and J. Sanz, “Dynamical modeling procedure of a li-ion battery pack suitable for real-time applications,” _Energy Conversion and Management_ , vol. 92, pp. 396–405, 2015.
* [21] F. Lambert, “Tesla confirms new 82 kWh battery pack in Model 3, thanks to new cells,” https://bit.ly/3tk2jR7, 2020, [Online; accessed 20-March-2021].
This document is accompanying the submission How will electric vehicles affect
traffic congestion and energy consumption: an integrated modelling approach.
The information in this document complements the submission, and it is
presented here for completeness reasons. It is not required for understanding
the main paper, nor for reproducing the results.
### -A NSTM model representation
Figure 10: North Sydney Transport network layout including the connection of
centroids (main entry points of vehicles in the network; marked as circles
with blue interior) and permanent traffic count locations (marked in green
dots). Figure 11: Layout of detector traffic flow. Figure 12: Sample
representation of a signalised intersection including signal controls, phases,
turnings and detectors for vehicle counting per each turning.
### -B NSTM - PM peak validation
The model validation and calibration has been done as well for the PM peak
hours as represented in figures below.
Figure 13: Validation of PM-NSTM model via Trip Length Distribution Figure
14: Validation of the PM-NSTM model with $R^{2}=0.91$ metric.
While the PM peak distribution after calibration indicates a lower number of
adjusted vehicles travelling in the network, the trend profile still follows
the original profiling of the data set. In addition Fig. 14 indicates a high
$R^{2}=0.91$ which is a proof of model well calibrated, despite few cases of
over and under calibrated traffic counts.
### -C NSTM - car arrivals and mean distance travelled
One of the outputs of the NSTM model is the total number of arrivals and
departures from each area during AM and PM peaks. For modelling a home to
destination charging, the number of arriving cars in each urban area is
extremely important in order to be able to estimate the total number of EVs
that will require charging at specific times. Fig. 15 and Fig. 17 indicates
the AM and PM car arrivals for each of the urban Local Governmental Area in
our case study which allowed us to identify the top most attractive locations.
For example, Dalley is a small catchment, but this area connects two parts of
the city through Harbour Bridge.
Figure 15: NTSM car arrival by area - AM peak. Figure 16: NTSM mean distance
travelled - AM peak. Figure 17: NTSM car arrival by area - PM peak. Figure 18:
NTSM mean distance travelled - PM peak.
### -D Capacity and loading of 10 most popular EV stations
Figure 19: Zone substation area catchment.
The Zone substation catchment area helped to identify the mapping of EV
stations to the regional energy consumption and the analysis of the energy
load for the top 10 most popular stations. Findings have revealed that 7 out
of these 10 stations are operating at more than $75\%$ of their capacity,
which already reveals current EV infrastructure limitation for further EV
uptake.
Figure 20: Load and capacity of 10 most popular EV stations.
### -E Energy impact of various scenarios
Figure 21: Raw results for EV-Q model outputs for OD2016, OD15, OD30
aggregated across all stations. Figure 22: Mean queue variation for 10 plug EV
stations. Figure 23: Energy consumption of all stations vs $EV_{p}$,
|
# Subfrequency search at a laser as a signal of dark matter sector particles
Hye-Sung Lee Department of Physics, KAIST, Daejeon 34141, Korea Jiheon Lee
Department of Physics, KAIST, Daejeon 34141, Korea Jaeok Yi Department of
Physics, KAIST, Daejeon 34141, Korea
(January 2022)
###### Abstract
We introduce a new method to search for the dark matter sector particles using
laser light. Some dark matter particles may have a small mixing or interaction
with a photon. High-power lasers provide substantial test grounds for these
hypothetical light particles of exploding interests in particle physics. We
show that any light source can also emit a subfrequency light as a new physics
signal. Searching for this subfrequency light at a laser can be a simple and
effective way to investigate the new light particles, even in tabletop optics
experiments.
## I Introduction
One of the characteristic trends in particle physics research in the past
decade or two is investigating the very light dark matter sector particles.
While the standard model (SM) is the utmost successful model of the unveiled
particles and their interactions, cosmological observations indicate that 95%
of the Universe consists of nonluminous matter and energy, which cannot be
explained by the SM Aghanim _et al._ (2020). The mysterious dark sector may
have various kinds of particles as the SM sector, and the two sectors may have
a feeble interaction channel referred to portals Essig _et al._ (2013).
This tiny size of interactions allows extremely light mass for the dark matter
sector particles that often require an ingenious strategy to probe. It brings
the other fields such as optics and condensed matter physics into a new
physics search originating from the dark matter sector, permitting genuinely
interdisciplinary studies Geraci _et al._ (2018); Bartram _et al._ (2021);
Melcón _et al._ (2020); Trickle _et al._ (2020); Ehret _et al._ (2010);
Spector (2017); Robilliard _et al._ (2007); Chou _et al._ (2008); Afanasev
_et al._ (2009); Ballou _et al._ (2015); Cameron _et al._ (1993).
Among the popular dark sector particles are an axion ($a$) and a dark photon
($\gamma^{\prime}$). They are exceptional in the sense that they mix or couple
to a photon ($\gamma$), which chiefly distinguishes the dark sector from the
visible sector made of the SM particles. These particles have been searched
for using this connection to the light in the lab experiments as well as in
the astrophysical observations. (See Refs. Zyla _et al._ (2020); Kim and
Carosi (2010); Fabbrichesi _et al._ (2021) for the reviews on the related
subjects.)
Figure 1: Diagrams showing how the subfrequency photon can be emitted through
the combined dark sector portals ($\varepsilon$ and
$G_{a\gamma\gamma^{\prime}}$) when there is a source of a photon.
In this paper, we make one key observation regarding a photon and the dark
matter sector and study its implication in one experimental setup, which is a
laser experiment. Our observation is following. Whenever there is a source
that emits light, the same source may also emit subfrequency light due to the
connections of a photon to some dark sector particles. A dark photon may
couple to the electromagnetic current because of the kinetic mixing with a
photon. It decays to an axion and a lower energy (subfrequency) photon, which
can be the new physics signal. This is a very general property, which is
depicted in Fig. 1. As there is a great number of sources of light, the
potential applications of this possibility are far reaching. We find the
optical laser system is a simple and effective way to study this new physics.
Although there have been a few laser experiments to search for new physics,
this one is qualitatively different from any of them in generating and
detecting the new physics signal.
The rest of this paper is organized as followings. In Sec. II, we describe the
relevant Lagrangian for the new physics scenario. In Sec. III, we describe how
the subfrequency light can be produced from the dark photon decay. In Sec. IV,
we describe the kinematics of the relevant process. In Sec. V, we propose a
new laser experiment and the sensitivity of the experiment. In Sec. VI, we
discuss the general properties of the new idea and experiment. In Sec. VII, we
summarize our findings and depict outlooks.
## II New physics
The SM particles do not carry a charge of the dark sector gauge symmetry
$U(1)_{\text{dark}}$. The vector portal ($\varepsilon$) Holdom (1986) is a
dimensionless parameter, which is a small kinetic mixing between the
$U(1)_{\text{QED}}$ and the $U(1)_{\text{dark}}$. Through this
$\gamma$-$\gamma^{\prime}$ mixing, the dark photon can couple to the
electromagnetic current with a suppressed coupling constant $\varepsilon e$ in
contrast to the electromagnetic coupling constant $e$.
The dark axion portal ($G_{a\gamma\gamma^{\prime}}$) Kaneta _et al._ (2017)
is a nonrenormalizable $a$-$\gamma$-$\gamma^{\prime}$ vertex, whose actual
size depends on the details of the underlying theory. (By the axion, we mean
the axionlike particle in general, not necessarily related to the strong $CP$
problem Peccei and Quinn (1977); Weinberg (1978); Wilczek (1978).) Many
implications of this portal have been found in various contexts Choi _et al._
(2017); Kitajima _et al._ (2018); deNiverville _et al._ (2018); Arias _et
al._ (2021); Deniverville _et al._ (2021); Hook _et al._ (2021); Ge _et
al._ (2021).
The relevant part of the Lagrangian in the mass eigenstate is
$\mathcal{L}\sim-\big{(}A^{\text{phys}}_{\mu}+\varepsilon{A^{\prime}}^{\text{phys}}_{\mu}\big{)}J_{\text{em}}^{\mu}+\frac{G_{a\gamma\gamma^{\prime}}}{2}aF^{\text{phys}}_{\mu\nu}\tilde{F}_{\text{phys}}^{\prime\mu\nu}$
(1)
where $J_{\text{em}}^{\mu}$ is the electromagnetic current,
$F^{\text{phys}}_{\mu\nu}=\partial_{\mu}A^{\text{phys}}_{\nu}-\partial_{\nu}A^{\text{phys}}_{\mu}$
is the field strength tensor for the photon field $(A_{\mu}^{\text{phys}})$,
and ${F^{\prime}}^{\text{phys}}_{\mu\nu}$ is for the dark photon field
$({A^{\prime}}_{\mu}^{\text{phys}})$. For the dark photon heavier than the
axion, $\gamma^{\prime}\to a\gamma$ decay can occur with the
$G_{a\gamma\gamma^{\prime}}$. Although it is possible to consider the axion
decay into a pair of photons in the presence of the axion portal
($G_{a\gamma\gamma}$) allowing $\gamma^{\prime}\to
a\gamma\to\gamma\gamma\gamma$, we treat things as model independent as
possible and assume the axion portal is zero in this paper. In this sense, the
number of photon signals in our analysis will be conservative. The full
Lagrangian and the extended Maxwell’s equations in the presence of various
dark sector portals can be found in Ref. Huang and Lee (2019).
## III Subfrequency light from the dark photon decay
In the following, we use the number flux of laser photon $N_{\gamma}$, defined
as $P/\omega$, where $P$ is the laser power and $\omega$ is the frequency. At
the laser, a dark photon with mass $m_{\gamma^{\prime}}$ can be produced
through the suppressed coupling to the electromagnetic current. Alternatively,
this dark photon can be understood in terms of the photon-dark photon
oscillations in a similar fashion to the neutrino flavor oscillations, and it
has been searched for in the light shining through the wall (LSW) experiments
Ehret _et al._ (2010); Spector (2017); Robilliard _et al._ (2007); Chou _et
al._ (2008); Afanasev _et al._ (2009); Ballou _et al._ (2015); Cameron _et
al._ (1993). The largest mass of the dark photon that we can produce from the
optical laser is the laser frequency of the eV scale
($m_{\gamma^{\prime}}<\omega$).
In the presence of a lighter axion, the dark photon may decay into a photon
and an axion with a decay rate
$\Gamma\equiv\Gamma_{\gamma^{\prime}\rightarrow
a\gamma}=\frac{G_{a\gamma\gamma^{\prime}}^{2}}{96\pi}m_{\gamma^{\prime}}^{3}\bigg{(}1-\frac{m_{a}^{2}}{m_{\gamma^{\prime}}^{2}}\bigg{)}^{3}.$
(2)
The photon produced from this dark photon decay, which we call the signal
photon, has a smaller frequency than the original laser frequency.
In order to demonstrate the production of the signal photon, we investigate
the photon-dark photon oscillation together with the decay of massive dark
photons. We define the massless eigenstate as a photon mass eigenstate
$\ket{\gamma}$ and massive eigenstate as a dark photon mass eigenstate
$\ket{\gamma^{\prime}}$. The mass eigenstates propagate as plane waves of the
form
$\displaystyle\ket{\gamma(t,L)}$ $\displaystyle=e^{-i(\omega t-\omega
L)}\ket{\gamma}\,,$ (3) $\displaystyle\ket{\gamma^{\prime}(t,L)}$
$\displaystyle=e^{-i(\omega t-pL)}e^{-m_{\gamma^{\prime}}\Gamma
t/2\omega}\ket{\gamma^{\prime}}\,,$ (4)
where $p=\sqrt{\omega^{2}-m_{\gamma^{\prime}}^{2}}$ is the momentum of the
dark photon. The Lorentz factor $\gamma=\omega/m_{\gamma^{\prime}}$ reflects
the time dilation of the decay. Here, $t$ and $L$ are measured in the lab
frame.
When the photon is produced in the laser, it is produced in a flavor
eigenstate $\ket{A}$ and its orthogonal pair is $\ket{X}$. They are related to
the mass eigenstates $\ket{\gamma}$ and $\ket{\gamma^{\prime}}$ by $2\times 2$
unitary matrix.
$\begin{bmatrix}A\\\
X\end{bmatrix}=\begin{bmatrix}\sqrt{1-\varepsilon^{2}}&\varepsilon\\\
-\varepsilon&\sqrt{1-\varepsilon^{2}}\end{bmatrix}\begin{bmatrix}\gamma\\\
\gamma^{\prime}\end{bmatrix}$ (5)
Let our state emitted from the laser be $\ket{\psi(t,L)}$. Since the initial
state $\ket{\psi(0,0)}=\ket{A}$, the state after time $t$ and length $L$ of
propagation is
$\displaystyle\ket{\psi(t,L)}$
$\displaystyle=\sqrt{1-\varepsilon^{2}}e^{-i(\omega t-\omega L)}\ket{\gamma}$
$\displaystyle+\varepsilon e^{-i(\omega t-pL)-m_{\gamma^{\prime}}\Gamma
t/2\omega}\ket{\gamma^{\prime}}\,.$ (6)
We can obtain a probability that the initial $\ket{A}$ turns into $\ket{X}$
after propagation as
$\displaystyle\mathcal{P}_{A\rightarrow X}(L)$
$\displaystyle=|\braket{X}{\psi(t,L)}|^{2}$
$\displaystyle=\varepsilon^{2}\bigg{[}1+e^{-\frac{m_{\gamma^{\prime}}\Gamma
t}{\omega}}-2e^{-\frac{m_{\gamma^{\prime}}\Gamma t}{2\omega}}\cos(L\Delta
p)\bigg{]}$
$\displaystyle=\varepsilon^{2}\bigg{[}1+e^{-\frac{m_{\gamma^{\prime}}\Gamma
L}{p}}-2e^{-\frac{m_{\gamma^{\prime}}\Gamma L}{2p}}\cos(L\Delta p)\bigg{]}$
(7)
where $\Delta p=\omega-p=\omega-\sqrt{\omega^{2}-m_{\gamma^{\prime}}^{2}}$ and
the terms with $\mathcal{O}(\varepsilon^{4})$ are neglected. (We assume small
kinetic mixing.) We used $L=vt$ and $p=\omega v$ in the last line. Likewise, a
simple calculation gives the probability that $X$ results in decay products.
$\displaystyle\mathcal{P}_{X\rightarrow\text{decay}}(L)$
$\displaystyle=1-\mathcal{P}_{X\rightarrow A}(L)-\mathcal{P}_{X\rightarrow
X}(L)$
$\displaystyle=(1-\varepsilon^{2})\left[1-e^{-\frac{m_{\gamma^{\prime}}\Gamma
L}{p}}\right]$ (8)
Here, the oscillation behavior is suppressed with
$\mathcal{O}(\varepsilon^{4})$.
As we will describe in Sec. V, although it is not the only possible option,
our proposed setup is basically the LSW-type experiment in which there are two
baselines of length $L_{1}$ and $L_{2}$ that are separated by a “wall” in
between. So the laser light initially in the $\ket{A}$ travels the $L_{1}$,
then only the $\ket{X}$ component of the light travels the $L_{2}$ (as a
photon cannot go through the wall) decaying into an axion and a subfrequency
photon, which we measure at the end of $L_{2}$.
Therefore, the flux of the subfrequency signal photon is given as
$\displaystyle N_{\text{sub}}$ $\displaystyle=\mathcal{P}_{A\rightarrow
X}(L_{1})\mathcal{P}_{X\rightarrow\text{decay}}(L_{2})N_{\gamma}$
$\displaystyle=\varepsilon^{2}N_{\gamma}\bigg{[}1-e^{-\frac{m_{\gamma^{\prime}}\Gamma
L_{2}}{p}}\bigg{]}$
$\displaystyle\times\bigg{[}1+e^{-\frac{m_{\gamma^{\prime}}\Gamma
L_{1}}{p}}-2e^{-\frac{m_{\gamma^{\prime}}\Gamma L_{1}}{2p}}\cos(L_{1}\Delta
p)\bigg{]}.$ (9)
In this paper, we work on a “simple setup” described by the following
conditions.
1. (i)
$m_{a}\ll m_{\gamma^{\prime}}<\omega$.
2. (ii)
$\dfrac{m_{\gamma^{\prime}}\Gamma
L_{i}}{\sqrt{\omega^{2}-m_{\gamma^{\prime}}^{2}}}\ll 1$ (for $i=1,\ 2$).
3. (iii)
$L_{1}\Delta p\gg 2\pi$.
These are reasonable assumptions for the optical experiment in the laboratory
scale for the following reasons.
Condition (i) reflects that the dark photon should be produced from the laser
($m_{\gamma^{\prime}}<\omega$) and decay into a much lighter axion ($m_{a}\ll
m_{\gamma^{\prime}}$) and a subfrequency photon. The latter ensures the
subfrequency photon has a sizable number of events and a visible light
frequency range that can be detected with a typical photon detector, although
we will briefly discuss how the non-negligible axion mass ($m_{a}\sim
m_{\gamma^{\prime}}$) can change the result in Sec. VI.
Condition (ii) is the requirement to put the relevant exponential expressions
in linear form of $L$ as in Eq. (11), which is convenient for analysis. Under
this condition, the probability of the dark photon decay during the
propagation is much less than 1. Using Eq. (2), this can be rewritten as
$G_{a\gamma\gamma^{\prime}}^{2}\ll
96\pi\frac{\sqrt{\omega^{2}-m_{\gamma^{\prime}}^{2}}}{m_{\gamma^{\prime}}^{4}}\frac{1}{L_{2}}\,.$
(10)
This can be easily satisfied for the typical lab-size experiment scales if
$m_{\gamma^{\prime}}\ll\omega$. The left-hand side of condition (ii) diverges
when $m_{\gamma^{\prime}}\sim\omega$, but it hardly occurs because of the
finite line width of a laser. Thus only a tiny fraction of the dark photon can
have an enhancement effect, even if the $m_{\gamma^{\prime}}$ happens to
coincide the laser frequency.
Condition (iii) allows us to neglect the oscillation in
$\mathcal{P}_{\gamma\rightarrow\gamma^{\prime}}$. If $L_{1}\Delta p$ is much
larger than $2\pi$, the cosine function in
$\mathcal{P}_{\gamma\rightarrow\gamma^{\prime}}$ is highly oscillatory. The
linewidth of laser in the setup can alter $\Delta p$, and its small change can
alter the signal size uncontrollably. To treat this oscillation, we take the
average value of the cosine function in Eq. (9). Even in the low mass region,
this condition can be satisfied by our interested parameter region for
$L_{1}>1\text{m}$.
In the simple setup described above, the signal photon ratio can be obtained
by
$\frac{N_{\text{sub}}}{N_{\gamma}}=\frac{K^{2}}{48\pi}\frac{m_{\gamma^{\prime}}^{4}}{\sqrt{\omega^{2}-m^{2}_{\gamma^{\prime}}}}L$
(11)
where $L=L_{2}$ and the effective portal coupling $K$ is given as a product of
the two portal couplings
$K\equiv\varepsilon\,G_{a\gamma\gamma^{\prime}}.$ (12)
The fraction of the subfrequency light is given by two model parameters $K$,
$m_{\gamma^{\prime}}$, the laser frequency $\omega$, and a geometry factor
$L$. The factor of 2 difference between Eqs. (2) and (11) comes from the fact
that both mass eigenstates generate the $X$ state.
## IV Kinematics
Consider the rest frame of a dark photon, which decays into a subfrequency
signal photon and an axion. In our simple setup where an axion is considered
nearly massless, both the photon and axion have the same energy
$m_{\gamma^{\prime}}/2$. Since there is no preference of direction in the rest
frame, the probability density ($dp/d\Omega$) that the signal photon heads to
a specific solid angle $d\Omega$ is given by
$\frac{dp}{d\Omega}=\frac{1}{4\pi},\quad\int\frac{dp}{d\Omega}d\Omega=1.$ (13)
By denoting the solid angle with the polar angle $\theta_{\text{rest}}$ and
azimuthal angle $\phi_{\text{rest}}$, we have
$\int\frac{dp}{d\Omega}\sin\theta_{\text{rest}}d\theta_{\text{rest}}d\phi_{\text{rest}}=\int\frac{\sin\theta_{\text{rest}}}{2}d\theta_{\text{rest}}=1,$
(14)
where we used azimuthal symmetry in the first equality. Using this relation,
we can define the probability density function with respect to
$\theta_{\text{rest}}$:
$\frac{dp}{d\theta_{\text{rest}}}=\frac{\sin\theta_{\text{rest}}}{2}$ (15)
Let the $z$ axis be the direction a dark photon travels in. In the lab frame,
the dark photon has the energy $\omega=\gamma m_{\gamma^{\prime}}$, where
$\gamma=1/\sqrt{1-\beta^{2}}$ is the Lorentz factor. Thus, the lab frame
boosts with velocity $\beta=\sqrt{1-(m_{\gamma^{\prime}}/\omega)^{2}}$. Then
the energy $E$ and polar angle $\theta_{\text{lab}}$ of a signal photon in the
lab frame are
$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\omega}{2}(1+\beta\cos\theta_{\text{rest}}),$ (16)
$\displaystyle\tan\theta_{\text{lab}}$ $\displaystyle=$
$\displaystyle\frac{\sin\theta_{\text{rest}}}{\gamma(\cos\theta_{\text{rest}}+\beta)}.$
(17)
If $\beta\ll 1$ ($m_{\gamma^{\prime}}\approx\omega)$, then $E$ is not
dispersive. But $\theta_{\text{lab}}$ gets close to $\theta_{\text{rest}}$ and
the signal photons spread almost isotropically. On the other hand, $E$ is
highly dispersive if $\beta\approx 1$ ($m_{\gamma^{\prime}}\ll\omega)$. For a
fixed $\omega$, as $m_{\gamma^{\prime}}$ increases, the dispersion of $E$
decreases, but that of $\theta_{\text{lab}}$ increases. We can check these
dispersions by computing the probability density with respect to $E$ and
$\theta_{\text{rest}}$ for each. They are given by
$\frac{dp}{dE}=\frac{dp}{d\theta_{\text{rest}}}\left|\frac{d\theta_{\text{rest}}}{dE}\right|,\quad\frac{dp}{d\theta_{\text{lab}}}=\frac{dp}{d\theta_{\text{rest}}}\left|\frac{d\theta_{\text{rest}}}{d\theta_{\text{lab}}}\right|.$
(18)
These probability distributions and dispersive behaviors are illustrated in
Fig. 2.
Figure 2: Probability density function with respect to the energy $E$ of the
subfrequency signal photon and the angle in the lab frame
$\theta_{\text{lab}}$. As the dark photon mass $m_{\gamma^{\prime}}$
increases, the energy of the signal photon is focused around $\omega/2$ but
angular distribution gets more disperse.
One of the interesting features of the energy dispersion, as we can find in
Fig. 2, is that the distribution is uniform. This probability distribution
function is given by
$\frac{dp}{dE}=\frac{1}{\sqrt{\omega^{2}-m_{\gamma^{\prime}}^{2}}}~{}\text{
for }\frac{\omega}{2}(1-\beta)<E<\frac{\omega}{2}(1+\beta).$ (19)
The energy dispersion also shows that the frequency of resulting signal is
smaller than the laser frequency $\omega$, which justifies our naming
“subfrequency.” However, one should note that despite the dispersion of $E$, a
signal photon whose energy is much smaller than $\omega/2$ is produced very
little and in the backward direction as one can see from Eqs. (16)–(17).
Because of the angular dispersion of the subfrequency photon, the location and
the size of the detector could matter in detecting the signal. We want to
estimate how much signal can be collected in a circular detector of radius $a$
located at $L$ from the dark photon production. Often, a lens is installed in
front of a detector to focus the beam into the small detector, then the “$a$”
can be replaced by the radius of the lens as an effective detector size in the
conservative sense.
Figure 3: $\eta_{\text{kin}}$ of the subfrequency new physics signal in
various experimental setups. The angular dispersion of the signal photon is
large for high dark photon mass, and it weakens $\eta_{\text{kin}}$ unless a
suitable measure is taken (as in waveguide). Dotted curves denote the
waveguide with 98.5% reflectivity coating, while solid ones denote 100%. The
solid blue and solid purple curves (two curves with 100% reflectivity)
coincide. A small kink in the dotted curves are due to the limited wavelength
range of the mirror. (See texts in Sec. V for details.)
Let $\ell$ be the distance from the detector to the point a decay occurs. In
our simple setup, we can have
$\frac{d^{2}p}{d\ell
d\theta_{\text{lab}}}=\frac{1}{L}\frac{dp}{d\theta_{\text{lab}}}=\frac{1}{L}\frac{dp}{d\theta_{\text{rest}}}\left|\frac{d\theta_{\text{rest}}}{d\theta_{\text{lab}}}\right|.$
(20)
The probability that the photon reaches the detector is
$\eta_{\text{kin}}=\int_{0}^{L}d\ell\int_{0}^{\tan^{-1}(a/\ell)}d\theta_{\text{lab}}\
\frac{d^{2}p}{d\ell d\theta_{\text{lab}}},$ (21)
which is straightforward to calculate.
For the later use, we will briefly discuss substitution between
$\theta_{\text{lab}}$ and $E$. In many cases, calculations become much easier
by changing the variable from $\theta_{\text{lab}}$ to $E$ since $dp/dE$ is
uniform. The relations between $E$ and $\theta_{\text{lab}}$ are given by
$\displaystyle\theta_{\text{lab}}(E)$ $\displaystyle=\cos^{-1}\frac{2E\omega-
m_{\gamma^{\prime}}^{2}}{2E\sqrt{\omega^{2}-m_{\gamma^{\prime}}^{2}}},$ (22)
$\displaystyle E(\theta_{\text{lab}})$
$\displaystyle=\frac{m_{\gamma^{\prime}}^{2}\left(\omega+\sqrt{\omega^{2}-m_{\gamma^{\prime}}^{2}}\cos\theta_{\text{lab}}\right)}{2\left(\omega^{2}\sin^{2}\theta_{\text{lab}}+m_{\gamma^{\prime}}^{2}\cos^{2}\theta_{\text{lab}}\right)}.$
(23)
Using this substitution, Eq. (21) can be rewritten as
$\displaystyle\eta_{\text{kin}}$
$\displaystyle=\int_{0}^{L}d\ell\int^{E(\theta_{\text{lab}}=0)}_{E(\theta_{\text{lab}}=\tan^{-1}(a/\ell))}dE\
\frac{d^{2}p}{d\ell dE}$
$\displaystyle=\frac{1}{\sqrt{\omega^{2}-m_{\gamma^{\prime}}^{2}}}\int_{0}^{L}d\ell\int^{E(\theta_{\text{lab}}=0)}_{E(\theta_{\text{lab}}=\tan^{-1}(a/\ell))}dE$
$\displaystyle=\int_{0}^{L}\frac{m_{\gamma^{\prime}}^{2}\ell(\ell-\sqrt{\ell^{2}+a^{2}})+a^{2}\omega(\omega+\sqrt{\omega^{2}-m_{\gamma^{\prime}}^{2}})}{2L(\ell^{2}m_{\gamma^{\prime}}^{2}+a^{2}\omega^{2})}d\ell.$
(24)
In Fig. 3, we show $\eta_{\text{kin}}$ for various experimental setups. Some
of these setups are the existing LSW experiments that are partly sensitive to
our subfrequency new physics scenario. We used the parameter values specified
in Table 1. For $m_{\gamma^{\prime}}\approx\omega$, only a tiny fraction of
the signal photon can be detected. ALPS II has the lowest $\eta_{\text{kin}}$
due to its smallest $a/L$ ratio. It is clear how important to collect
dispersive signal photons. As we will detail later in this paper, our new
design adopts a waveguide that can fix this, and at least half of the signal
photons can reach the detector in ideal case. (See waveguide in Fig. 3.)
| a (mm) | $L$ (m) | $\omega$ (eV) | $N_{\gamma}$ (Hz) | $N_{\text{pass}}$ | $\eta_{\text{eff}}$ | $N_{d}$ (Hz) | $t_{s}$ (h)
---|---|---|---|---|---|---|---|---
Waveguide | $-$ | $1,\ 100$ | $1.17$ | $1.6\times 10^{20}$ | $5000$ | $0.54$ | $10^{-6}$ | $480$
ALPS II | $8.75$111The ALPS II uses an optic suitable to collect a 17.5 mm diameter beam Bähre _et al._ (2013). | $100$ | $1.17$ | $1.6\times 10^{20}$ | $5000$ | $0.95$ | $10^{-6}$ | $480$
ALPS | $7$222Because we are unaware of the radius of the lens, we take the maximum of the vacuum tube size that can be inserted in the HERA dipole magnet Ehret _et al._ (2009). | $7.6$ | $2.33$ | $2.6\times 10^{21}$ | $1$ | $0.9$ | $0.0018$ | $27$
GammeV | $25.5$333We take the size of the lens in front of the PMT Chou _et al._ (2008). | $7.2$ | $2.33$ | $6.6\times 10^{23}$ | $1$ | $0.25$ | $130$ | $24$
SPring-8 | $6$444We take the radius of crystal in Ge detector Inada _et al._ (2013). | $0.654$ | $7270\sim 26000$ | $4.3\times 10^{12}\sim 8.9\times 10^{13}$ | $1$ | $0.23\sim 0.83$ | $0.0019\sim 0.0142$ | $5.28\sim 8.88$
Table 1: Specifications of the experimental setups we use in our analysis. The
LSW experiments (GammeV, ALPS, ALPS II) are partly sensitive to the sub-
frequency new physics scenario. Our proposed experimental setup (Waveguide)
adopts a waveguide. The parameters for the LSW experiments including
$\eta_{\text{eff}}$ were taken from Refs. Ehret _et al._ (2010); Flambaum
_et al._ (2019); Bastidon _et al._ (2015); Inada _et al._ (2013). Some of
the parameter values may not properly reflect the actual experiments. For
instance, the $\eta_{\text{eff}}$ of the LSW experiments would change if the
frequency-dependence is properly applied.
## V New laser experiment
Figure 4: Proposed layout (waveguide). The red is the photon beam, and the
black is the dark photon beam. The mirror (M) reflects the laser beam
(photon), and only the dark photon can pass the mirror and decay to the
subfrequency signal photon. (We do not show the axion in the figure.) The
dashed line is the waveguide, which protects the signal from dispersion. A
lens is placed in front of the photon detector to focus the signal light into
the detector.
Figure 4 shows a schematic diagram of the laser experiment setup that can
explore the subfrequency new physics scenario. A small portion of laser light
converts to the dark photon, which subsequently decays into the photon and
axion. Because of the different frequencies, we can distinguish the
subfrequency signals from the original laser beam in several ways (for
instance, we can adopt a prism to distinguish the subfrequency signal light
from the original laser light), but here we adopt a rather simple method. We
use a mirror (M) to reflect the laser beam. Then only the dark photon beam
passes the mirror. The decay can occur anywhere in the propagation of the dark
photon, but the decay signal made before the M cannot pass the mirror. Thus,
the decay length ($L$) should be taken as the distance between the M and the
detector.
The Fabry-Perot cavity before the M is used to amplify the dark photon
transition rate through the mirrors, and the cavity amplification ratio
($\eta_{\text{cav}}$) is given by Ahlers _et al._ (2007)
$\eta_{\text{cav}}=(N_{\text{pass}}+1)/2$ (25)
where $N_{\text{pass}}$ is the number of the reflections of the laser beam
inside the cavity. This depends on the cavity factors and is limited by the
damage threshold of cavity mirrors at which the surface coating on the mirror
can be damaged. The authors of the Ref. Bähre _et al._ (2013) limit the
maximum intensity in the cavity mirror to 500 kW/cm2 (150 kW for 5 mm minimum
beam radius in ALPS II). One has a wide choice for the laser intensity and the
cavity amplification as long as the circulating power inside the cavity does
not exceed the damage threshold. We also take the 150 kW for our maximum
circulating power ($\omega N_{\gamma}N_{\text{pass}}<150$ kW).
To avoid losing signal photons from the angular dispersion described earlier,
we use a waveguide that can collect and guide all the signal photons in the
forward direction toward the detector, which will save at least half of the
signals. A long hollow pipe/rectanguloid with the inside coated by a highly
reflective metal would be suitable. The silver-coated mirror has high
reflectivity in the optical to near IR range, commercial mirror with more than
98.5$\%$ reflectivity in the 1100 to 20000 nm range is available. (In fact,
there is a reduced but still nonzero reflectivity below the 1100 nm wavelength
range, but we take it zero for the simplicity.) We choose the radius of
waveguide as the same as that of lens in ALPS II since we also want to use a
lens.
In the presence of the waveguide, one should consider the loss of the signal
due to the reflections at the mirror. Thus, for the waveguide setup,
$\eta_{\text{kin}}$ should be modified to
$\eta_{\text{kin}}=\frac{1}{\sqrt{\omega^{2}-m_{\gamma^{\prime}}^{2}}}\int_{0}^{L}d\ell\int^{E(\theta_{\text{lab}}=0)}_{E(\theta_{\text{lab}}=\pi/2)}dE\;\mathcal{R}(\theta_{\text{lab}}(E),\ell)\,.$
(26)
The $\mathcal{R}(\theta_{\text{lab}}(E),\ell)$ is a surviving fraction of the
photon, and we obtain it as
$\mathcal{R}(\theta_{\text{lab}}(E),\ell)=r(E)^{\frac{\ell}{2R}\tan\theta_{\text{lab}}+\frac{1}{2}}\,,$
(27)
where ${R}$ is the radius of the waveguide and $r$ is the reflectivity of the
mirror. The power of $r$ is the number of the reflections each signal photon
experiences. This expression slightly overcounts the reflection by a decimal,
but it is negligible when $r$ is close to $1$. The
$\mathcal{R}(\theta_{\text{lab}},\ell)$ is always less than 1 and
$\mathcal{R}(\theta_{\text{lab}},\ell)\rightarrow 1$ as $r\rightarrow 1$.
Figure 5: The averaged detector efficiency ($\bar{\eta}_{\text{eff}}$) with
the dark photon mass ($m_{\gamma^{\prime}}$) for various TES devices when
$\omega$=1.17 eV. We take tungsten (W) TES Lita _et al._ (2008),
titanium/silver (Ti/Au) TES Niwa _et al._ (2017), hafnium (Hf) TES Lita _et
al._ (2010) for the waveguide with $98.5\%$ reflectivity with $L=1$ m (dashed
curves), $L=100$ m (solid curves). Various TES are quite sensitive in the
given mass range.
In general, the SNR of the single photon detection is given by
$\text{SNR}=\frac{N_{s}\sqrt{t_{s}}}{\sqrt{N_{s}+N_{d}}}$ (28)
where $N_{s}$ ($N_{d}$) is the number of signal photon (noise) per second, and
$t_{s}$ is the total measurement time. To be sensitive to small coupling $K$,
it is required for the detector to have an extremely low noise. Also, one
needs to consider a detector efficiency ($\eta_{\text{eff}}$), which is a
ratio of the detected number of the photon to the incident photon to the
detector.
In our subfrequency search scenario, as we discussed in Sec. IV, the detector
needs to be sensitive to the frequency around $\omega/2$. Thus, the transition
edge sensor (TES) which has high sensitivity and low noise in the near IR to
the optical range, can be a suitable detector. Figure 5 shows the efficiency
of TES devices for the waveguide experiment with $L=1$ m and $L=100$ m. Since
the detector efficiency mainly depends on the energy of the incident signal,
one needs to consider the energy and angular dispersion of the subfrequency
signal.
We take the averaged detector efficiency as
$\bar{\eta}_{\text{eff}}=\frac{1}{\eta_{\text{kin}}}\frac{1}{\sqrt{\omega^{2}-m_{\gamma^{\prime}}^{2}}}\\\
\int_{0}^{L}d\ell\int^{E(\theta_{\text{lab}}=0)}_{E(\theta_{\text{lab}}=\pi/2)}dE\;\eta_{\text{eff}}(E)\mathcal{R}(\theta_{\text{lab}}(E),\ell)\,.$
(29)
When $m_{\gamma^{\prime}}\ll\omega$, the subfrequency signal is widely
dispersed in the energy domain (see Fig. 2). In that limit, the frequency-wide
average makes the sensitivity of each TES flat as shown in Fig. 5. In
contrast, when $m_{\gamma^{\prime}}\sim\omega$, the subfrequency signals focus
at $\omega/2$ and has a large angular dispersion. Therefore, the local shape
of the $\eta_{\text{eff}}(E)$ and the signal loss by the waveguide is fed into
the curve of $\bar{\eta}_{\text{eff}}$ near the $\omega/2$.
Finally, the ratio of the detectable signal photon number flux to the laser
flux is
$\frac{N_{s}}{N_{\gamma}}=\eta\frac{N_{\text{sub}}}{N_{\gamma}}=\eta\frac{K^{2}}{48\pi}\frac{m_{\gamma^{\prime}}^{4}}{\sqrt{\omega^{2}-m^{2}_{\gamma^{\prime}}}}L\,,$
(30)
with
$\eta\equiv\eta_{\text{kin}}\eta_{\text{cav}}\bar{\eta}_{\text{eff}}\,.$ (31)
## VI Discussions
So far, we considered only the case when $m_{a}\ll m_{\gamma^{\prime}}$. Non-
negligible axion mass has the following effects. The decay rate [see Eq. (2)]
and the energy of the each subfrequency photon decrease as
$\Gamma\rightarrow(1-m_{a}^{2}/m_{\gamma^{\prime}}^{2})^{3}\Gamma$,
$E\rightarrow(1-m_{a}^{2}/m_{\gamma^{\prime}}^{2})E$. These imply that the
subfrequency signal becomes weaker and no longer lies in the near IR to the
optical range as $m_{a}\rightarrow m_{\gamma^{\prime}}$. This would require
microwave or radio frequency detectors, which we do not consider in this
paper.
Figure 6: 1$\sigma$ C.L. exclusion/sensitivity limits in the mass
($m_{\gamma^{\prime}}$) and coupling ($K$) parameter space for various
experimental setups. For waveguides, we chose the same optical parameters and
the measurement time ($t_{s}=20$ days) as the ALPS II, which is the state-of-
the-art experiment for the high-power, low-signal measurement (see the
parameters and caveats in Table 1). Figure 7: A comparison of 1$\sigma$ C.L.
sensitivity limits for the waveguides of the $98.5\%$ (dashed curves) and
$100\%$ (solid curves) reflectivity coating. The signal loss from the
reflection is more critical when the length of waveguide is larger, since the
number of the reflection is proportional to the length.
Figure 6 shows the exclusion limits and the future sensitivities of various
experimental setups. The waveguide sensitivity curves are for the ideal,
100$\%$ reflectivity case. The comparison to the realistic case of the
$98.5\%$ reflectivity is shown in Fig. 7. We made a cutoff for each curve at
$m_{\gamma^{\prime}}=0.99\omega$. This is far below the spectrum of the laser
due to its linewidth. Also, the limit (10) is satisfied by this cutoff. Since
the decay rate of the dark photon (thus the signal) increases with its mass,
the experiment is particularly sensitive when the $m_{X}$ is close to the
laser frequency $\omega$, which can be seen from the negative slopes of
curves. In contrast, the subfrequency signal is more dispersive in the angular
domain as $m_{\gamma^{\prime}}\rightarrow\omega$. This reduces the signal
reaching the detector (see Fig. 3).
The new setup with a waveguide can excel the early LSW experiments (GammeV,
ALPS) by orders of magnitude. Even tabletop experiments ($L=1$ m) can
outperform them. ALPS II is an ongoing LSW experiment, which has the most
extended baseline ($L=100$ m). This long baseline clearly improves its
sensitivity, although it is not using a waveguide. However, the ALPS II gets
the most stringent degradation from the angular dispersion due to its large
baseline, as one can see from its gentle slope. As the curves show, using a
waveguide for a given $L$ significantly improves the sensitivity, especially
in the high mass region.
A rough estimate for the QCD axion can be done, if one assumes
$G_{a\gamma\gamma}\sim G_{a\gamma\gamma^{\prime}}$. The
experimental/observational bound is $G_{a\gamma\gamma}<10^{-10}$ GeV-1 for the
QCD axion of lighter than the eV scale Zyla _et al._ (2020). Our sensitivity
study in $K$ shown in Fig. 6 indicates the QCD axion cannot be detected with
the current technology even if we take the largest possible kinetic mixing
$\varepsilon\sim 1$.
The sensitivity would further increase if we can (i) increase the circulating
laser power, which is currently limited by the damage threshold of the mirrors
in the cavity (i.e., improve the mirror coating), (ii) reduce the detector
background noise, (iii) extend the baseline for the dark photon decay
(accompanied by a waveguide). It can also reach a higher mass using the higher
frequency light source. (For instance, see the mass coverage of GammeV and
ALPS, which use relatively high-frequency lasers.)
Especially, an x-ray source can cover high mass region where the decay rate is
large. In Fig. 8, we show the constraints from the null result of the light
shining through wall experiment SPring-8 using the x-ray Inada _et al._
(2013). Although a waveguide cannot be not used for the x-ray due to the high
transmittance, it already gives a sizable constraints.
Figure 8: 1$\sigma$ C.L. exclusion limits in the mass ($m_{\gamma^{\prime}}$)
and coupling ($K$) parameter space for SPring-8 experiments Inada _et al._
(2013). SPring-8 experiments consist of a series of LSW experiments with
various frequency x-ray sources. The sawtooth shape comes from the combination
of a few experiments.
One of the critical differences of the proposed experiment from the other
laser-based new physics searches is that it exploits the decay of a new
particle. Because of this property, it results in several differences such as
the angular deviation of the signal, need of sizeable decay length, no need of
wall in principle (we could use a prism instead of the wall), and different
signal frequency from the original laser frequency. These distinctions allow a
different way of optimization from the LSW, such as the waveguide we take.
## VII Summary and outlook
Our study shows a simple setup of an eV-scale optical laser and a photon
detector can do the entire process from the production of a new particle, its
decay into the SM particles, and the new physics signal detection. It was
considered possible only in high-energy accelerator experiments not long ago.
It is striking to note the whole setup can be as small as a tabletop
experiment of a 1-meter scale.
One of the crucial points is that the test of the combined portal
$K=\varepsilon G_{a\gamma\gamma^{\prime}}$ cannot be obtained just by
multiplying the bounds on $\varepsilon$ and $G_{a\gamma\gamma^{\prime}}$
obtained independently. For instance, if we use only $\varepsilon$, the dark
photon lighter than 1 MeV would not practically decay Pospelov _et al._
(2008). Also, the $\varepsilon$ bound from the stellar objects can be weakened
due to the reconversion of the dark photon with the combined portal. If we use
only $G_{a\gamma\gamma^{\prime}}$, on the other hand, then the dark photon
production will be greatly suppressed because it relies on an off-shell
photon. Therefore, we need a dedicated study for the implications and
constraints of the combined portal $K$. In this sense, it can be taken as a
practically new portal, which we name the “subfrequency portal.”
The essential part of this portal is that any source of light can produce
subfrequency light as a signal of new physics. While we focused on the new
physics search in the laser experiments, which cover the visible light
frequency, plus some x-ray examples, there are plenty of possible extensions
such as microwave cavities and isotope spectrum spectroscopy. Also, the
astrophysical/cosmological data from the stellar objects and cosmic microwave
backgrounds are wonderful places to study. Rich physics implications are
warranted.
###### Acknowledgements.
J. L. thanks J. Jung and H. Kim for helpful conversations about the optical
devices. This work was supported in part by the National Research Foundation
of Korea (Grant No. NRF-2021R1A2C2009718).
## References
* Aghanim _et al._ (2020) N. Aghanim _et al._ (Planck), Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
* Essig _et al._ (2013) R. Essig _et al._ , in _Community Summer Study 2013: Snowmass on the Mississippi_ (2013) arXiv:1311.0029 [hep-ph] .
* Geraci _et al._ (2018) A. A. Geraci _et al._ (ARIADNE), Springer Proc. Phys. 211, 151 (2018), arXiv:1710.05413 [astro-ph.IM] .
* Bartram _et al._ (2021) C. Bartram _et al._ (ADMX), Phys. Rev. Lett. 127, 261803 (2021), arXiv:2110.06096 [hep-ex] .
* Melcón _et al._ (2020) A. A. Melcón _et al._ (CAST), JHEP 21, 075 (2020), arXiv:2104.13798 [hep-ex] .
* Trickle _et al._ (2020) T. Trickle, Z. Zhang, and K. M. Zurek, Phys. Rev. Lett. 124, 201801 (2020), arXiv:1905.13744 [hep-ph] .
* Ehret _et al._ (2010) K. Ehret _et al._ , Phys. Lett. B 689, 149 (2010), arXiv:1004.1313 [hep-ex] .
* Spector (2017) A. Spector (ALPS), in _12th Patras Workshop on Axions, WIMPs and WISPs_ (2017) pp. 133–136, arXiv:1611.05863 [physics.ins-det] .
* Robilliard _et al._ (2007) C. Robilliard, R. Battesti, M. Fouche, J. Mauchain, A.-M. Sautivet, F. Amiranoff, and C. Rizzo, Phys. Rev. Lett. 99, 190403 (2007), arXiv:0707.1296 [hep-ex] .
* Chou _et al._ (2008) A. S. Chou, W. C. Wester, III, A. Baumbaugh, H. R. Gustafson, Y. Irizarry-Valle, P. O. Mazur, J. H. Steffen, R. Tomlin, X. Yang, and J. Yoo (GammeV (T-969)), Phys. Rev. Lett. 100, 080402 (2008), arXiv:0710.3783 [hep-ex] .
* Afanasev _et al._ (2009) A. Afanasev, O. K. Baker, K. B. Beard, G. Biallas, J. Boyce, M. Minarni, R. Ramdon, M. Shinn, and P. Slocum, Phys. Lett. B 679, 317 (2009), arXiv:0810.4189 [hep-ex] .
* Ballou _et al._ (2015) R. Ballou _et al._ (OSQAR), Phys. Rev. D 92, 092002 (2015), arXiv:1506.08082 [hep-ex] .
* Cameron _et al._ (1993) R. Cameron _et al._ , Phys. Rev. D 47, 3707 (1993).
* Zyla _et al._ (2020) P. A. Zyla _et al._ (Particle Data Group), PTEP 2020, 083C01 (2020).
* Kim and Carosi (2010) J. E. Kim and G. Carosi, Rev. Mod. Phys. 82, 557 (2010), [Erratum: Rev.Mod.Phys. 91, 049902 (2019)], arXiv:0807.3125 [hep-ph] .
* Fabbrichesi _et al._ (2021) M. Fabbrichesi, E. Gabrielli, and G. Lanfranchi, _The Physics of the Dark Photon_ (Springer Cham, 2021).
* Holdom (1986) B. Holdom, Phys. Lett. B 166, 196 (1986).
* Kaneta _et al._ (2017) K. Kaneta, H.-S. Lee, and S. Yun, Phys. Rev. Lett. 118, 101802 (2017), arXiv:1611.01466 [hep-ph] .
* Peccei and Quinn (1977) R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977).
* Weinberg (1978) S. Weinberg, Phys. Rev. Lett. 40, 223 (1978).
* Wilczek (1978) F. Wilczek, Phys. Rev. Lett. 40, 279 (1978).
* Choi _et al._ (2017) K. Choi, H. Kim, and T. Sekiguchi, Phys. Rev. D 95, 075008 (2017), arXiv:1611.08569 [hep-ph] .
* Kitajima _et al._ (2018) N. Kitajima, T. Sekiguchi, and F. Takahashi, Phys. Lett. B 781, 684 (2018), arXiv:1711.06590 [hep-ph] .
* deNiverville _et al._ (2018) P. deNiverville, H.-S. Lee, and M.-S. Seo, Phys. Rev. D 98, 115011 (2018), arXiv:1806.00757 [hep-ph] .
* Arias _et al._ (2021) P. Arias, A. Arza, J. Jaeckel, and D. Vargas-Arancibia, JCAP 05, 070 (2021), arXiv:2007.12585 [hep-ph] .
* Deniverville _et al._ (2021) P. Deniverville, H.-S. Lee, and Y.-M. Lee, Phys. Rev. D 103, 075006 (2021), arXiv:2011.03276 [hep-ph] .
* Hook _et al._ (2021) A. Hook, G. Marques-Tavares, and C. Ristow, JHEP 06, 167 (2021), arXiv:2105.06476 [hep-ph] .
* Ge _et al._ (2021) S.-F. Ge, X.-D. Ma, and P. Pasquini, Eur. Phys. J. C 81, 787 (2021), arXiv:2104.03276 [hep-ph] .
* Huang and Lee (2019) F. P. Huang and H.-S. Lee, Int. J. Mod. Phys. A 34, 1950012 (2019), arXiv:1806.09972 [hep-ph] .
* Bähre _et al._ (2013) R. Bähre _et al._ , JINST 8, T09001 (2013), arXiv:1302.5647 [physics.ins-det] .
* Ehret _et al._ (2009) K. Ehret _et al._ (ALPS), Nucl. Instrum. Meth. A 612, 83 (2009), arXiv:0905.4159 [physics.ins-det] .
* Inada _et al._ (2013) T. Inada, T. Namba, S. Asai, T. Kobayashi, Y. Tanaka, K. Tamasaku, K. Sawada, and T. Ishikawa, Physics Letters B 722, 301 (2013).
* Flambaum _et al._ (2019) V. V. Flambaum, I. B. Samsonov, and H. B. Tran Tan, Phys. Rev. D 99, 115019 (2019), arXiv:1904.02271 [hep-ph] .
* Bastidon _et al._ (2015) N. Bastidon, D. Horns, and A. Lindner (2015) arXiv:1509.02064 [physics.ins-det] .
* Ahlers _et al._ (2007) M. Ahlers, H. Gies, J. Jaeckel, J. Redondo, and A. Ringwald, Phys. Rev. D 76, 115005 (2007), arXiv:0706.2836 [hep-ph] .
* Lita _et al._ (2008) A. E. Lita, A. J. Miller, and S. W. Nam, Opt. Express 16, 3032 (2008).
* Niwa _et al._ (2017) K. Niwa, T. Numata, K. Hattori, and D. Fukuda, Scientific Reports 7 (2017), 10.1038/srep45660.
* Lita _et al._ (2010) A. Lita, B. Calkins, L. Pellouchoud, A. Miller, and S. Nam, Proceedings of SPIE - The International Society for Optical Engineering 7681 (2010), 10.1117/12.852221.
* Pospelov _et al._ (2008) M. Pospelov, A. Ritz, and M. B. Voloshin, Phys. Rev. D 78, 115012 (2008), arXiv:0807.3279 [hep-ph] .
|
# Modeling of dendritic solidification and numerical analysis of the phase-
field approach to model complex morphologies in alloys
Kunal Bhagat Department of Mechanical Engineering, University of Wisconsin-
Madison Shiva Rudraraju
Department of Mechanical Engineering, University of Wisconsin-Madison;
corresponding author
(Preprint)
## Abstract
Dendrites are one of the most widely observed patterns in nature and occur
across a wide spectrum of physical phenomena. In solidification and growth
patterns in metals and crystals, the multi-level branching structures of
dendrites pose a modeling challenge, and a full resolution of these structures
is computationally demanding. In the literature, theoretical models of
dendritic formation and evolution, essentially as extensions of the classical
moving boundary Stefan problem exist. Much of this understanding is from the
analysis of dendrites occurring during the solidification of metallic alloys,
as this is critical for understanding microstructure evolution during metal
manufacturing processes (MAM) that involve solidification of a liquid melt.
Motivated by the problem of modeling microstructure evolution from liquid
melts of pure metals and alloys during MAM, we developed a comprehensive
numerical framework for modeling a large variety of dendritic structures that
are relevant to metal solidification. In this work, we present a numerical
framework encompassing the modeling of Stefan problem formulations relevant to
dendritic evolution using a phase-field approach and a finite element method
implementation. Using this framework, we model numerous complex dendritic
morphologies that are physically relevant to the solidification of pure melts
and binary alloys. The distinguishing aspects of this work are - a unified
treatment of both pure-metals and alloys; novel numerical error estimates of
dendritic tip velocity; and the convergence of error for the primal fields of
temperature and the order parameter with respect to numerical discretization.
To the best of our knowledge, this is a first of its kind study of numerical
convergence of the phase-field equations of dendritic growth in a finite
element method setting. Further, we modeled various types of physically
relevant dendritic solidification patterns in 2D and 3D computational domains.
## 1 Introduction
Dendrites are tree-like patterns with complex multi-level branch structures
that are observed across a wide spectrum of physical phenomena - from snow
flakes to river basins; from bacterial colonies to lungs and vascular systems;
and ubiquitously in solidification and growth patterns in metals and crystals.
In the context of solidification problems involving a pure metal or metallic
alloys, dendritic “trees” with primary and secondary branches grow from a
nucleation point or a field perturbation. These solidification dendrites often
initiate at random nucleation points or at domain boundaries at the initial
stage, followed by anisotropic interface growth during the intermediate stage,
eventually leading to grain formation and coarsening. The theoretical
foundations of dendritic solidification lie in the classical Stefan problem, a
moving boundary problem, that describes the evolution of a solid-liquid phase
front [1]. The two-phase interface motion is obtained by solving the heat
equation in each phase, coupled with an evolving interface boundary condition
(Stefan condition) that explicitly sets the velocity of the moving interface.
While the evolution of dendrites in pure melts with no dissolved solutes is
well represented by the Stefan problem, for melts with dissolved solutes (like
alloys), mass diffusion in the constituent phases should also be accounted for
in the governing equations.
Many analytical and numerical approaches exist for treating the Stefan
problem, albeit under various simplifying assumptions on the problem geometry,
interface geometry and boundary conditions. In the numerical domain, front
tracking methods [2, 3] are useful in solving straight and curved interface
evolution in one-dimensional problems. Other popular methods that explicitly
track moving interface are the Landau transformation and the finite element
mesh-based moving node techniques. These methods are more suited when the
movement of the interface is not far from the initial position [4], but
bookkeeping of the interface movement can be an arduous task, especially for
anisotropic interphase growth conditions. The level set method is another
popular numerical technique that is extensively used to solve moving interface
problems, including dendrite solidification using the Stefan problem [5].
The phase-field method has gained popularity in the last two decades as an
alternate diffuse interface numerical technique to model solidification
processes. The phase-field model modifies the equations represented by the
Stefan problem by introducing an order parameter to distinguish between the
solid and liquid phases. Further, the previously sharp interface is
represented as a diffuse interface using this order parameter without
sacrificing much of the accuracy. The earliest of the simplified isotropic
phase-field model has been proposed by Fix [6], and Collins and Levine [7].
Caginalp [8, 9] and Kobayashi [10] introduced basic anisotropy into the phase-
field model and numerically predicted dendritic patterns seen in the
solidification of the pure melt. They also modeled the formation of the side
branching by introducing noise into the phase-field models.
The potential of the so far surveyed phase-field models was limited due to
constraints on the lattice size, undercooling, interface width, capillary
length, and interface kinetics. A modified thin interface phase-field model
proposed by Karma and Rappel [11] overcame these limitations. The dendrite
shape and tip velocities of the pure melt obtained from the thin interface
phase-field model were in excellent agreement with the steady-state results
reported in theoretical studies. Karma and Rappel [12] studied dendrite side-
branching by introducing a microscopic thermal noise, and the side branch
characteristics were in line with the linear WKB theory based stability
analysis. Plapp and Karma [13] proposed a hybrid technique where the phase-
field model and diffusion-based Monte-Carlo algorithm were used in conjunction
to simulate dendritic solidification at the low undercooling in two and three-
dimensional geometry.
The phase-field model for the solidification of a binary alloy was presented
by Warren and Boettinger [14]. Realistic patterns showing the primary and the
secondary arms of the dendrite were simulated using the interface thickness
far different from the asymptotic limit of the sharp interface models.
Loginova et al. [15] was the first to include the non-isothermal effects in
binary alloy solidification. However, the computations were time-consuming. An
artificial solute trapping was observed in the simulations with higher
undercooling. This artificial solute trapping was the result of diffuse
interface thickness. Karma [16] developed a modified thin interface phase-
field model for alloy solidification. Compared to the previous models, this
model was applicable for varied interface thicknesses, zero non-equilibrium
effects at the interface, and nearly zero diffusivity in the solid.
The thin interface phase-field model by Ramirez et al. [17] considered both
heat and solute diffusion along with zero kinetics at the interface. The
interface thickness used was an order of magnitude less than the radius of
curvature of the interface. Echebarria et al. [18] modeled numerous numerical
test cases and studied the convergence of the thin interface phase-field
model. They discussed the final form of anti-trapping current that is often
used in many alloy solidification models [16, 17, 19]. When the solid
diffusivity of the alloys is non-negligible, the form of anti-trapping current
previously used needs modification and this was rigorously derived in the work
of Ohno and Matsuura [20]
With the substantial review tracing Stefan problem and important theoretical
development of the phase-field model, we review literature focusing on the
numerical implementations of these models and error analysis. Several
important studies leveraged the finite-element and finite-difference method
based numerical schemes and their application to the phase-field model of
solidification. These studies focused on the accuracy and stability of
proposed numerical schemes. In this regard, the work of Feng and Prohl [21] is
important. They made use of fully discrete finite element methods and obtained
optimal error bounds in relation to the interface thickness for a phase-field
method of solidification. Using the error estimates they showed convergence of
finite element schemes to the solution to the phase field models in the limit
of the sharp interface. Gonzalez-Ferreiro et al. [22] presented a finite
element discretization in space and mid-point discretization in time of a
phase-field model that is consistent with both the thermodynamics laws. The
thermodynamically consistent numerical model was developed to have better
dendrite resolution and higher order accuracy in time. A first-order accurate
in time and energy stable numerical method was presented by Chen and Yang
[23]. The proposed method was applied to coupled Allen-Cahn, heat diffusion,
and modified Navier-Stokes equations. They resort to techniques that decoupled
these three equations with the use of implicit-explicit schemes in their
numerical implementation.
Better spatial and temporal accuracy and smaller error estimates were reported
in the work of Kessler and Scheid [24]. They applied a finite element method
to the phase-field model of binary alloy solidification where the error
convergence numerical tests were done on a physical example of Ni-Cu alloy
solidification with a simpler and non-branched solidification structure. The
adaptive meshing technique was used in many numerical studies of
solidification using a phase-field model. Hu et al. [25] presented a multi-
mesh adaptive finite-element based numerical scheme to solve the phase-field
method for the pure-melt solidification problem. The accuracy of their method
was tested with its ability to predict dendrite tip velocities with a range of
undercooling conditions. Work of Rosam et al. [26] using the mesh and time
adaptive fully implicit numerical scheme utilizing finite-difference method
applied to binary alloy solidification is very relevant. Such highly space-
time adaptive techniques saved computational time. Their focus was on the
implicit and explicit schemes error estimates using only derived variables
such as dendrite tip position, radius, and velocity. However, error analysis
reported in the literature did not explore the effect of basis continuity
(i.e, $C^{n}$-continuous basis) on capturing dendrite kinetics and
morphologies, and the convergence studied reported did not explicitly study
the error in the primal fields, i.e., temperature and phase-field order
parameter. As part of this manuscript, these two aspects of error analysis
will also be addressed.
A review by Tourret et al. [27] highlights several studies that applied the
phase-field methods to model solidification and obtained good comparisons with
the experimental results. Present challenges and extensions of phase-field
models were also discussed in their review. Wang et al. [28] discovered the
relation between lower and upper limit primary dendrite arm spacing to inter-
dendritic solute distribution and inter-dendritic undercooling by solving
phase-field models using finite-element method. Fallah et al. [29] work showed
the suitability of phase-field models coupled with heat transfer models to
reproduce experimentally known complex dendrite morphology and accurate
dendrite size of Ti-Nb alloys thereby demonstrating the suitability of the
numerical methods to laser deposition and industrial scale casting process.
More recently, microstructure evolution processes involving competitive grain
growth of columnar dendrites and the grain growth along the converging grain
boundary were modeled using the phase-field model by Tourret and Karma [30],
and Takaki et al. [31]. In manufacturing processes like welding and molding,
phase-field models were used to study solidification cracking susceptibility
in Al-Mg alloys by Geng et al. [32], and dendrite morphology in the melt pool
of Al-Cu alloys by Farzadi et al. [33]. Integrated phase-field model and
finite element methods are also used to study microstructure evolution
especially formation of laves phase in additive manufacturing of IN718 [34].
Phase-field models finds extensive applications in modeling rapid
solidification conditions prevalent in the metal additive manufacturing [35,
36, 37, 38, 39].
A lot of existing studies on solidification and related phase-field models
have focused only on either pure metals or alloys. However, in this work, we
present a unified treatment of both pure metals and alloys. We discuss
classical Stefan problems relevant to both these types of solidification, and
present their phase-field formulations and numerical implementations. Further,
we present novel numerical error estimates of dendritic tip velocity, and the
convergence of error for the primal fields of temperature and order parameter
with respect to the numerical discretization. Lastly, using this numerical
framework, various types of physically relevant dendritic solidification
patterns like single equiaxed, multi-equiaxed, single columnar and multi-
columnar dendrites are modeled in two-dimensional and three-dimensional
computational domains.
Following this literature review and summary of the advancements in numerical
modeling of solidification and dendritic growth, we now present an overview of
this manuscript. In Section 2 we present a detailed discussion of the
numerical models used in this work to model dendritic growth. Essentially, we
look at the appropriate Stefan problem formulations and their phase-field
representations for modeling the solidification problem of pure melts and
binary alloys. Then, in Section 3, we discuss the numerical framework and its
computational implementation. Further, we present a numerical error analysis
and the simulation results of various 2D and 3D dendritic solidification
problems. Finally, we present the concluding remarks in Section 4.
## 2 Numerical models of dendritic growth
In this section, we discuss the Stefan problem for modeling solidification of
a pure metal and a binary alloy. We elaborate on the complexities of the
Stefan problem and the physics associated with solidification. The Stefan
problem is then re-written as a phase-field formulation. The primary goal of
using a phase-field approach is to model the interface growth kinetics,
particularly the velocity of interface motion and the complex morphologies of
dendrites that occur during solidification, without the need to explicitly
track the dendritic interfaces. The interface growth can be influenced by the
surface anisotropy, heat diffusion, mass diffusion, interface curvature, and
the interface attachment kinetics. In Sections 2.1-2.2, the classical Stefan
problem and its phase-field representation relevant to solidification of pure
melts is presented. Then, in Sections 2.3-2.4, the extensions needed in the
Stefan problem and its phase-field representation for modeling solidification
of binary alloys are discussed.
In general, a pure melt is a single-component material (pure solvent) without
any solutes. Addition of one or more solute components results in an alloy. In
this work, we specifically consider a binary alloy (two-component alloy with a
solvent and one solute), as a representative of multi-component alloys. And as
will be shown, during the solidification of a binary alloy, we model the
diffusion of this solute concentration in the solid and liquid phases.
### 2.1 Stefan problem for modeling solidification of a pure melt
The Stefan problem describing the solidification of an undercooled pure metal
is presented. The set of Equations 1a-1c mathematically model the
solidification of an undercooled pure melt. Equations 2a-2c represent the same
governing equations in a non-dimensional form. Equation 1a models heat
conduction in the bulk liquid and the bulk solid regions of the pure metal. At
the solid-liquid interface, the balance of the heat flux from either of the
bulk regions is balanced by the freezing of the melt, and thus additional
solid bulk phase is formed. In other words, the solid-liquid interface moves.
The energy balance at the interface is given by the Equation 1b. The
temperature of the interface is not fixed and can be affected by multiple
factors such as the undercooling effect on the interface due to its curvature
(Gibbs-Thomson effect) and the interface attachment kinetics. These are
captured in the equation 1c.
$\frac{\partial T(\bm{x},t)}{\partial
t}=\frac{k}{c_{p}}\nabla^{2}T,\quad\bm{x}\in\Omega^{s},\Omega^{l}$ (1a)
$\bm{\nu}_{n}L=k\big{(}\partial_{n}T(\bm{x})\lvert^{+}-\;\partial_{n}T(\bm{x})\lvert^{-}\big{)},\quad\bm{x}\in\Gamma$
(1b) $T_{m}-T_{i}=\Delta T_{\Gamma_{g}}+\Delta
T_{\mu_{k}}=\Gamma_{g}\bm{\kappa}+\frac{\bm{\nu}_{n}}{\mu_{k}},\quad\bm{x}\in\Gamma$
(1c)
Here, $k$ and $c_{p}$ are thermal conductivity and specific heat capacity for
a pure solid or liquid metal. $T_{m}$ $\bm{\nu}_{n}$, $L$ is the melting
temperature, interface velocity, and latent heat of the metal.
$\partial_{n}T\lvert^{+}$ and $\partial_{n}T\lvert^{-}$ is the temperature
gradient normal to the interface in the liquid and the solid phase
respectively. $\Gamma_{g}$, $\bm{\kappa}$, and $\mu_{k}$ are the Gibbs-Thomson
coefficient, the curvature of the interface, and interface-attachment
coefficient, respectively. We re-write these equations in their non-
dimensional form, using the scaled temperature $u=\frac{c_{p}(T-T_{m})}{L}$
[4].
$\frac{\partial u(\bm{x},t)}{\partial
t}=D\nabla^{2}u,\quad\bm{x}\in\Omega^{s},\Omega^{l}$ (2a)
$\bm{\nu}_{n}=D\big{(}\partial_{n}u(\bm{x})\lvert^{+}-\;\partial_{n}u(\bm{x})\lvert^{-}\big{)},\quad\bm{x}\in\Gamma$
(2b)
$u^{*}=-d(\mathbf{n})\bm{\kappa}-\beta(\mathbf{n})\bm{\nu}_{n},\quad\bm{x}\in\Gamma$
(2c)
$D=\frac{k}{c_{p}}$ is the non-dimensional thermal diffusivity in the solid
and liquid phases. $d(\mathbf{n})=\gamma(\mathbf{n})T_{m}c_{p}/L^{2}$ is the
capillary length, $\gamma(\mathbf{n})$ is the surface tension, $\mathbf{n}$ is
the unit vector denoting normal to the interface, $\beta(\mathbf{n})$ is the
kinetic coefficient, and $\partial_{n}u\lvert^{+}$ and
$\partial_{n}u\lvert^{-}$ are the non-dimensional derivatives normal to the
interface in the solid and liquid region, respectively [40]. The driving force
for this solidification is the initial undercooling, i.e., the initial
temperature of the liquid melt below its melting temperature. The non-
dimensional undercooling is given by $\Delta=-c_{p}(\frac{Tm-T_{\infty}}{L})$.
The Stefan problem described using Equation 1 does not have a known analytical
solution, and its numerical implementation, as is the case with free boundary
problems, is challenging. Numerical methods like time-dependent boundary
integral formulations [11], a variational algorithm with zero or non-zero
interface kinetics, and methods that require the book-keeping of the
solidifying front and low grid anisotropy conditions have been used in the
past. Modeling correct dendritic growth involves getting the correct operating
dendritic tip conditions, mainly the tip radius and the tip velocity.
Variationally derived phase-field models, like the one described in Section
2.2, are now the method of choice to model complex dendritic solidification
problems.
### 2.2 Phase-field model describing the solidification of a pure melt
In this section, we describe the phase-field method developed by Plapp and
Karma [13] to model solidification in a pure metal. The governing equations
for the model are derived using variational principles. A phenomenological
free energy expression, $\Pi\left[\phi,u\right]$, in terms of a phase-field
order parameter ($\phi$) and temperature ($u$), is defined in Equation 3. This
functional form defines the thermodynamic state of a system, and has two terms
- first is $f(\phi,u)$, the bulk energy term, and second is an interface term
given by $\frac{1}{2}\lambda^{2}(\bm{n})|\bm{\nabla}\phi|^{2}$. The magnitude
of the interface term is controlled by the interface parameter, $\lambda$, and
the interface term is positive in the diffuse solid-liquid interface due to a
non-zero gradient of the order parameter; elsewhere, the interface term is
zero.
$\Pi\left[\phi,u\right]=\int_{\Omega}\left[f(\phi,u)+\frac{1}{2}\lambda^{2}(\mathbf{n})|\bm{\nabla}\phi|^{2}\right]\leavevmode\nobreak\
dV$ (3a) $f(\phi,u)=-\frac{1}{2}\phi^{2}+\frac{1}{4}\phi^{4}+\xi
u\phi\left(1-\frac{2}{3}\phi^{2}+\frac{1}{5}\phi^{4}\right)$ (3b)
The bulk energy term $f(\phi,u)$ in Equation 3b is given by the double well
potential that has a local minima at $\phi=1$ and $\phi=-1$.
$\Omega=\Omega_{s}\cup\Omega_{l}$ represents total volume encompassing the
bulk solid, the bulk liquid and the interface region. The value of $u$ then
tilts the equilibrium value of $f(\phi,u)$. The form of $f(\phi,u)=g(\phi)+\xi
uh(\phi)$ is given below. $\xi$ is the coupling parameter in the double well
function. The governing equations for the phase-field model of solidification
are obtained by minimizing the above functional with respect to the primal
fields, $\phi$ and $u$. This results in the partial differential equations
presented below:
$\frac{\partial u(\bm{x},t)}{\partial
t}=\bm{\nabla}.\Big{(}D\bm{\nabla}\frac{\delta\Pi}{\delta
u}\Big{)},\quad\bm{x}\in\Omega$ (4a) $\frac{\partial\phi(\bm{x},t)}{\partial
t}=-K_{\phi}\frac{\delta\Pi}{\delta\phi},\quad\bm{x}\in\Omega$ (4b)
Taking the variational derivatives results in,
$\frac{\partial u}{\partial
t}=D\nabla^{2}u+\frac{1}{2}\frac{\partial\phi}{\partial
t},\quad\bm{x}\in\Omega$ (5a)
$\displaystyle\tau(\bm{n})\frac{\partial\phi}{\partial t}=-\frac{\partial
f}{\partial\phi}+\bm{\nabla}\cdot\big{(}\lambda^{2}(\bm{n})\bm{\nabla}\phi\big{)}+\frac{\partial}{\partial\bm{x}}\left[|\bm{\nabla}\phi|^{2}\lambda(\bm{n})\frac{\partial\lambda(\bm{n})}{\partial\left(\frac{\partial\phi}{\partial\bm{x}}\right)}\right]+\frac{\partial}{\partial\bm{y}}\left[|\bm{\nabla}\phi|^{2}\lambda(\bm{n})\frac{\partial\lambda(\bm{n})}{\partial\left(\frac{\partial\phi}{\partial\bm{y}}\right)}\right],\bm{x}\in\Omega$
(5b)
where $\Omega$ is the problem domain. Anisotropy in the interface energy
representation is considered. Thus, $\lambda(\bm{n})=\lambda_{0}a_{s}(\bm{n})$
and $\tau(\bm{n})=\tau_{0}a_{s}^{2}(\bm{n})$, where anisotropy parameter is
defined as
$a_{s}=(1-3\epsilon_{4})\Big{[}1+\frac{4\epsilon_{4}}{(1-3\epsilon_{4})}\frac{(\partial_{x}\phi)^{4}+(\partial_{y}\phi)^{4}}{|\phi|^{4}}\Big{]}=1+\epsilon_{4}cos(m\theta)$.
The equivalence is easily established by taking
$\tan(\theta)=\frac{\partial\phi}{\partial y}/\frac{\partial\phi}{\partial
x}$. $\epsilon_{4}$ is the strength of anisotropy and m=4 corresponds to an
anisotropy with four-fold symmetry. The interface between the solid and liquid
phases is diffuse in the phase-field representation of a Stefan problem of
solidification, but in the asymptotic limit of $\lambda(\bm{n})\to 0$, the
Stefan problem is recovered [8, 9].
#### 2.2.1 Weak formulation
We now pose the above governing equations in their weak (integral) form. This
formulation is used to solve these equations within a standard finite element
method framework.
Find the primal fields $\\{u,\phi\\}$, where,
$\displaystyle u$
$\displaystyle\in\mathscr{S}_{u},\quad\mathscr{S}_{u}=\\{u\in\text{H}^{1}(\Omega)\leavevmode\nobreak\
|\leavevmode\nobreak\ u=\leavevmode\nobreak\ u^{\prime}\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \textbf{X}\in\Gamma^{u}\\},$ $\displaystyle\phi$
$\displaystyle\in\mathscr{S}_{\phi},\quad\mathscr{S}_{\phi}=\\{\phi\in\text{H}^{1}(\Omega)\leavevmode\nobreak\
|\leavevmode\nobreak\ \phi=\leavevmode\nobreak\
\phi^{\prime}\leavevmode\nobreak\ \forall\leavevmode\nobreak\
\textbf{X}\in\Gamma^{\phi}\\}$
such that,
$\displaystyle\forall\leavevmode\nobreak\ w_{u}$
$\displaystyle\in\mathscr{V}_{u},\quad\mathscr{V}_{u}=\\{w_{u}\in\text{H}^{1}(\Omega)\leavevmode\nobreak\
|\leavevmode\nobreak\ w_{u}=0\leavevmode\nobreak\ \forall\leavevmode\nobreak\
\textbf{X}\in\Gamma^{u}\\},$ $\displaystyle\forall\leavevmode\nobreak\
w_{\phi}$
$\displaystyle\in\mathscr{V}_{\phi},\quad\mathscr{V}_{\phi}=\\{w_{\phi}\in\text{H}^{1}(\Omega)\leavevmode\nobreak\
|\leavevmode\nobreak\ w_{\phi}=0\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \textbf{X}\in\Gamma^{\phi}\\}$
we have,
$\int_{\Omega}w_{u}\Big{(}\frac{\partial u}{\partial
t}-\frac{1}{2}\frac{\partial\phi}{\partial
t}\Big{)}\,dV+\int_{\Omega}D\big{(}\bm{\nabla}w_{u}.\bm{\nabla}u\big{)}\,dV=0$
(6a)
$\displaystyle\int_{\Omega}w_{\phi}\Big{(}\tau\frac{\partial\phi}{\partial
t}+\frac{\partial
f}{\partial\phi}\Big{)}\,dV+\int_{\Omega}\lambda^{2}\bm{\nabla}w_{\phi}.\bm{\nabla}\phi\,dV+\int_{\Omega}|\bm{\nabla}\phi|^{2}\lambda\frac{\partial
w_{\phi}}{\partial\bm{x}}\frac{\partial\lambda}{\partial\left(\frac{\partial\phi}{\partial\bm{x}}\right)}\,dV$
(6b) $\displaystyle+\int_{\Omega}|\bm{\nabla}\phi|^{2}\lambda\frac{\partial
w_{\phi}}{\partial\bm{y}}\frac{\partial\lambda}{\partial\left(\frac{\partial\phi}{\partial\bm{y}}\right)}\,dV=0$
here, $\Omega=\Omega_{s}\cup\Omega_{l}$, where $\Omega_{s}$ is the solid phase
and $\Omega_{l}$ is the liquid phase, respectively. The variations,
$\omega_{u}$ and $\omega_{\phi}$, belong to $\text{H}^{1}(\Omega)$ \- the
Sobolev space of functions that are square-integrable and have a square-
integrable derivatives.
### 2.3 Stefan problem for modeling solidification of a binary alloy
The Stefan problem describing the solidification of a binary alloy is
described in this section. As mentioned earlier, an alloy is a multi-component
system. A binary alloy is a specific case of a multi-component alloy with only
one solute component. A binary alloy consists of a solute that has a
composition of $c$, and a solvent that has a composition of $(1-c)$. During
the solidification of a binary alloy, we model the diffusion of the solute
concentration across the solid and the liquid phase. Mass diffusion is
governed by Equation 7a. Near the vicinity of the solid-liquid interface, the
composition of the solute in the solid and the liquid phase is dictated by the
binary alloy phase diagram given in the Figure 1a. The difference in the
concentration of the solute across the solid and liquid phase is balanced by
the mass diffusion flux. This is captured in Equation 7b. The temperature of
the interface is not fixed and can be affected by multiple factors. The
undercooling on the interface can be due to its curvature (Gibbs-Thomson
effect), the interface attachment kinetics, cooling rate ($\dot{T}$), and the
applied temperature gradient. This is mathematically represented in Equation
7c. Using the phase-diagram, we can write this condition in-terms of
composition, $c_{l}$ at the interface. We neglect the contribution from the
attachment kinetics.
$c_{l}=c^{0}_{l}-\frac{\Gamma_{g}\bm{\kappa}+Gy+\dot{T}t}{|m|}$ where $c$ is
the solute composition in either the solid or the liquid phase. $D$ is the
mass diffusivity of alloy in the liquid phase, $c_{l}$ and $c^{0}_{l}$ are the
solute concentration at the interface for the temperature $T_{i}$ and $T_{0}$
respectively. $\dot{T}=G\bm{\nu}_{p}$ is the cooling rate. We scale the solute
concentration into a variable $u_{c}=\frac{c-c^{0}_{l}}{c^{0}_{l}(1-k)}$ which
represents supersaturation. k is the partition coefficient which represents
the ratio of solid composition to the liquid composition.
$\frac{\partial c(\bm{x},t)}{\partial
t}=D\nabla^{2}c(\bm{x},t),\quad\bm{x}\in\Omega^{s},\Omega^{l}$ (7a)
$c_{l}(1-k)\bm{\nu}_{n}=-D\partial_{n}c(\bm{x},t)\lvert^{+},\quad\bm{x}\in\Gamma$
(7b)
$T_{m}-T_{i}=\Gamma\bm{\kappa}+\frac{\bm{\nu}_{n}}{\mu_{k}}+Gy+\dot{T}t,\quad\bm{x}\in\Gamma$
(7c)
here, $\Gamma$ is the solid-liquid interface that separates the solid phase,
$\Omega^{s}$, and the liquid phase, $\Omega^{l}$.
(a) (b)
Figure 1: Directional solidification in a binary alloy. Shown are (1a) a
simplified phase diagram of a binary alloy, and (1b) a schematic of a
directional solidification process. In the case of directional solidification,
typically, the melt is forced to solidify at a constant pulling velocity under
a fixed temperature gradient. The dendritic structures in this case evolve at
the solid-liquid interface, via a Mullins-Sekerka instability, when the
pulling velocity is above a critical value.
Rewriting the Equations 7a, 7b and 7c in-terms of the supersaturation,
$u_{c}$, we get,
$\frac{\partial u_{c}(\bm{x},t)}{\partial
t}=D\nabla^{2}u_{c}(\bm{x},t),\quad\bm{x}\in\Omega^{s},\Omega^{l}$ (8a)
$(1+(1-k)u^{*}_{c})\bm{\nu}_{n}=-D\partial_{n}u_{c}(\bm{x},t)\lvert^{+},\quad\bm{x}\in\Gamma$
(8b) $u^{*}_{c}=-d_{0}\bm{\kappa}+\dot{\theta}t-\gamma z,\quad\bm{x}\in\Gamma$
(8c)
Here $d_{0}=\frac{\Gamma_{g}}{\Delta T_{0}}$ is the chemical capillary length
, $\dot{\theta}=\frac{-\dot{T}}{\Delta T_{0}}$ is the non-dimensional cooling
rate and $\gamma=\frac{G}{\Delta T_{0}}$ is non-dimensional thermal gradient
and $\Delta T_{0}=|m|(1-k)c^{0}_{l}$ [41].
As can be expected, an analytical solution for the moving boundary problem in
Equations 8 is not known. In the next subsection, we discuss a phase-field
representation of this problem that is extensively used to model dendritic
solidification of binary alloys.
### 2.4 Phase-field model describing the solidification of a binary alloy
In this section, we describe the phase-field model for the solidification of a
binary alloy. Following the variational procedure adopted in the case of a
pure melt, we write the free energy functional form, $\Pi[\phi,c,T]$, for a
binary alloy. The free energy in this case has the additional dependence on
the solute composition, $c$, along with the order paramter, $\phi$, and the
temperature, $T$. This functional form, given by Equation 9, takes into
account bulk free energy of the system $f(\phi,T_{m})$, internal energy and
entropy $f_{AB}(\phi,c,T)$, and the interface term
$\frac{1}{2}\sigma|\bm{\nabla}\phi|^{2}$, where $\sigma\propto\lambda^{2}$.
$\Pi[\phi,c,T]=\int_{\Omega}\left[f(\phi,T_{m})+f_{AB}(\phi,c,T)+\frac{1}{2}\sigma|\bm{\nabla}\phi|^{2}\right]\leavevmode\nobreak\
dV$ (9)
here $\Omega=\Omega_{s}\cup\Gamma\cup\Omega_{l}$ represents the total volume
encompassing the bulk solid, the bulk liquid and the interface region. The
governing equations for the model are obtained by writing the diffusion
equations relevant to a conserved quantity - the solute composition, and a
non-conserved quantity - the order-parameter.
$\frac{\partial c(\bm{x},t)}{\partial
t}=\bm{\nabla}.\Big{(}M(\phi,c)\bm{\nabla}\frac{\delta\Pi}{\delta
c}\Big{)},\quad\bm{x}\in\Omega$ (10a) $\frac{\partial\phi(\bm{x},t)}{\partial
t}=-K_{\phi}\frac{\delta\Pi}{\delta\phi},\quad\bm{x}\in\Omega$ (10b) where
$M(\phi,c)$ is the solute mobility, and $K_{\phi}$ is the order parameter
mobility. It is convenient to scale the composition, c and write it as
$u_{c}=\frac{1}{1-k}\big{(}\frac{2c}{c^{0}_{l}(1-\phi+k(1+\phi))}-1\big{)}$.
The parameter $u_{c}$ can be understood as a concentration undercooling
parameter analogous to $u$ in Section 2.2. This change of variable enables us
to draw similarities with the earlier form of the phase-field model described
in Equations 5. $u_{c}$ is a continuous variable across the solid and liquid
region unlike the solute concentration, $c$ which is discontinuous. We present
the final form of the phase-field model in Equations 10c-10d, without the
accompanying mathematical derivation. For interested readers, a detailed
derivation can be found in [18].
$\displaystyle\Big{(}\frac{1+k}{2}-\frac{1-k}{2}\phi\Big{)}\frac{\partial
u_{c}}{\partial
t}=\bm{\nabla}\cdot\Big{(}\tilde{D}\big{(}\frac{1-\phi}{2}\big{)}\bm{\nabla}u_{c}-\bm{J}_{at}\Big{)}+\Big{(}1+(1-k)u_{c}\Big{)}\frac{1}{2}\frac{\partial\phi}{\partial
t},\quad\bm{x}\in\Omega$ (10c)
$\displaystyle\begin{split}\Big{(}1-(1-k)\frac{y-\tilde{\bm{\nu}}t}{\tilde{l}_{T}}\Big{)}a^{2}_{s}(\mathbf{n})\frac{\partial\phi}{\partial
t}=\bm{\nabla}\cdot\left(a^{2}_{s}(\mathbf{n})\bm{\nabla}\phi\right)+\frac{\partial}{\partial\bm{x}}\Big{(}|\bm{\nabla}\phi|^{2}a_{s}(\mathbf{n})\frac{\partial
a_{s}(\mathbf{n})}{\partial\left(\frac{\partial\phi}{\partial\bm{x}}\right)}\Big{)}\\\
+\frac{\partial}{\partial\bm{y}}\Big{(}|\bm{\nabla}\phi|^{2}a_{s}(\mathbf{n})\frac{\partial
a_{s}(\mathbf{n})}{\partial\left(\frac{\partial\phi}{\partial\bm{y}}\right)}\Big{)}+\phi-\phi^{3}-\xi(1-\phi^{2})^{2}\big{(}u_{c}+\frac{y-\tilde{\bm{\nu}}t}{\tilde{l}_{T}}\big{)}\end{split},\quad\bm{x}\in\Omega$
(10d)
The solute diffusivity in the solid region is small as compared to the solute
diffusivity in the liquid region. Thus mass diffusion is neglected in the
solid region. $\tilde{D}$ is the non-dimensional diffusivity in the liquid
phase. In the directional solidification process, dendrite growth is
influenced by the presence of a temperature gradient $G$ and pulling velocity
$\tilde{\bm{\nu}}_{p}$. This can be realized by the the term
$\frac{y-\tilde{\bm{\nu}}t}{\tilde{l}_{T}}$ in the Equation 10d, where
$\tilde{l}_{T}=\frac{|m|(1-k)c^{0}_{l}}{G}$ is called the thermal length. A
modified form of Equation 10c was suggested by Ohno and Matsuura[20] where the
mass diffusion in the solid phase is not neglected, and $D_{s}$ is comparable
to $D_{l}$. This results in the modification to the diffusivity part and anti-
trapping flux current as seen in the Equation 10e. If we substitute k=1 and
$D_{s}=D_{l}$ in this equation, Equation 10c is recovered.
$\begin{split}\Big{(}\frac{1+k}{2}-\frac{1-k}{2}\phi\Big{)}\frac{\partial
u_{c}}{\partial
t}=\bm{\nabla}\cdot\Bigg{(}\tilde{D}\big{(}\frac{1-\phi}{2}+k\frac{1+\phi}{2}\frac{D_{s}}{D_{l}}\big{)}\bm{\nabla
u_{c}}-(1-k\frac{D_{s}}{D_{l}})\bm{J}_{at}\Bigg{)}\\\
+\Big{(}1+(1-k)u_{c}\Big{)}\frac{1}{2}\frac{\partial\phi}{\partial
t}\end{split}$ (10e)
#### 2.4.1 Weak formulation
We now pose the above governing equations in their weak (integral) form. This
formulation is used to solve these equations within a standard finite element
framework.
Find the primal fields $\\{u_{c},\phi\\}$, where,
$\displaystyle u_{c}$
$\displaystyle\in\mathscr{S}_{u_{c}},\quad\mathscr{S}_{u_{c}}=\\{u_{c}\in\text{H}^{1}(\Omega)\leavevmode\nobreak\
|\leavevmode\nobreak\ u_{c}=\leavevmode\nobreak\
u_{c}^{\prime}\leavevmode\nobreak\ \forall\leavevmode\nobreak\
\textbf{X}\in\Gamma^{u_{c}}\\},$ $\displaystyle\phi$
$\displaystyle\in\mathscr{S}_{\phi},\quad\mathscr{S}_{\phi}=\\{\phi\in\text{H}^{1}(\Omega)\leavevmode\nobreak\
|\leavevmode\nobreak\ \phi=\leavevmode\nobreak\
\phi^{\prime}\leavevmode\nobreak\ \forall\leavevmode\nobreak\
\textbf{X}\in\Gamma^{\phi}\\}$
such that,
$\displaystyle\forall\leavevmode\nobreak\ w_{u_{c}}$
$\displaystyle\in\mathscr{V}_{u_{c}},\quad\mathscr{V}_{u_{c}}=\\{w_{u_{c}}\in\text{H}^{1}(\Omega)\leavevmode\nobreak\
|\leavevmode\nobreak\ w_{u_{c}}=0\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \textbf{X}\in\Gamma^{u_{c}}\\},$
$\displaystyle\forall\leavevmode\nobreak\ w_{\phi}$
$\displaystyle\in\mathscr{V}_{\phi},\quad\mathscr{V}_{\phi}=\\{w_{\phi}\in\text{H}^{1}(\Omega)\leavevmode\nobreak\
|\leavevmode\nobreak\ w_{\phi}=0\leavevmode\nobreak\
\forall\leavevmode\nobreak\ \textbf{X}\in\Gamma^{\phi}\\}$
we have,
$\displaystyle\int_{\Omega}w_{u_{c}}\Bigg{(}\Big{(}\frac{1+k}{2}-\frac{1-k}{2}\phi\Big{)}\frac{\partial
u_{c}}{\partial
t}-\Big{(}1+(1-k)u_{c}\Big{)}\frac{1}{2}\frac{\partial\phi}{\partial
t}\Bigg{)}\,dV$ (11a)
$\displaystyle+\int_{\Omega}\tilde{D}\big{(}\frac{1-\phi}{2}\big{)}\bm{\nabla}w_{u_{c}}.\big{(}\bm{\nabla}u_{c}-\bm{J}_{at}\big{)}\,dV=0$
$\displaystyle\int_{\Omega}w_{\phi}\Bigg{(}\Big{(}1-(1-k)\frac{y-\tilde{\bm{\nu}}t}{\tilde{l}_{T}}\Big{)}a^{2}_{s}(\mathbf{n})\frac{\partial\phi}{\partial
t}-\Big{(}\phi-\phi^{3}-\xi(1-\phi^{2})^{2}\big{(}u_{c}+\frac{y-\tilde{\bm{\nu}}t}{\tilde{l}_{T}}\big{)}\Big{)}\Bigg{)}\,dV$
(11b)
$\displaystyle+\int_{\Omega}a_{s}^{2}\bm{\nabla}w_{\phi}.\bm{\nabla}\phi\,dV+\int_{\Omega}|\bm{\nabla}\phi|^{2}a_{s}\frac{\partial
w_{\phi}}{\partial\bm{x}}\frac{\partial
a_{s}}{\partial\left(\frac{\partial\phi}{\partial\bm{x}}\right)}\,dV+\int_{\Omega}|\bm{\nabla}\phi|^{2}a_{s}\frac{\partial
w_{\phi}}{\partial\bm{y}}\frac{\partial
a_{s}}{\partial\left(\frac{\partial\phi}{\partial\bm{y}}\right)}\,dV=0$ here,
$\Omega=\Omega_{s}\cup\Omega_{l}$, where $\Omega_{s}$ is the solid phase and
$\Omega_{l}$ is the liquid phase, respectively. The variations,
$\omega_{u_{c}}$ and $\omega_{\phi}$, belong to $\text{H}^{1}(\Omega)$ \- the
Sobolev space of functions that are square-integrable and have a square-
integrable derivatives.
## 3 Numerical implementation and results
This section outlines the numerical implementation of the phase-field models
described in Section 2.2 and 2.4. In Section 3.1, we cover the computational
implementation. In Section 3.2, we summarize the input model parameters used
in the simulation of dendritic growth in a pure metal and a binary alloy, and
list the non-dimensional parameters used in these models. In Sections 3.3,
using various dendritic morphologies modeled in this work, we identify
important geometric features of dendrites that are tracked in the dendritic
shape studies presented later. In Sections 3.4.1 we simulate the classical
four fold symmetric dendrite shape occurring in undercooled pure melt and use
it as a basis for convergence studies of dendritic morphology and dendritic
tip velocity. Later, in Sections 3.5-3.8 , we model the evolution of complex
2D and 3D dendrite morphologies. These simulations cover various cases of
solidification ranging from a single equiaxed dendrite in a pure liquid melt
to a multi-columnar dendrites growing in a binary alloy. Finally, evolution of
equiaxed dendrite in 3D is demonstrated.
### 3.1 Computational implementation
The phase-field formulations presented in this work are solved using
computational implementations of two numerical techniques: (1) A
$C^{0}$-continuous basis code based on the standard Finite Element Method
(FEM); the code base is an in-house, C++ programing language based, parallel
code framework with adaptive meshing and adaptive time-stepping - build on top
of the deal.II open source Finite Element library [42], (2) A
$C^{1}$-continuous basis code based on the Isogeometric Analysis (IGA) method;
the code base is an in-house, C++ programing language based, parallel code
framework - build on top of the PetIGA open source IGA library [43]. As can be
expected, the IGA code base can model $C^{0}$-continuous basis also, but does
not currently support some capabilities like adaptive meshing that are needed
for 3D dendritic simulations. Both codes support a variety of implicit and
explicit time stepping schemes. The complete code base of the current work is
made available to the wider research community as an open source library [44].
### 3.2 Material properties and non-dimensional quantities in solidification
We now discuss the specific material properties and non-dimensional parameters
used in the modeling of pure metal and binary alloys. First, we consider the
case of dendrite growth in an undercooled melt of pure metal, whose governing
equations are given by Equations 5a and 5b. Undercooling of the liquid melt is
responsible for driving dendrite growth in this case and there is no external
imposition of a temperature gradient or a dendritic growth velocity. This is
commonly referred to as free dendritic growth. The set of non-dimensional
input parameters used in this model and their numerical values (adopted from
Karma and Rappel [45]) are summarized in Table 1. Here, $\tilde{D}$ is the
thermal diffusivity of heat conduction, $\epsilon_{4}$ is the strength of
interface surface energy anisotropy where the subscript indicates the assumed
symmetry of the dendritic structure, and in this simulation, we consider a
four-fold symmetry. The choice of the length-scale parameter $\lambda_{0}$,
identified as the interface thickness, and the time-scale parameter $\tau_{0}$
set the spatial and temporal resolution of the dendritic growth process. $\xi$
is a coupling parameter representing the ratio of interface thickness and
capillary length.
Non-dimensional parameter | Numerical value
---|---
Thermal diffusivity, $\tilde{D}$ | 1
Anisotropy strength, $\epsilon_{4}$ | 0.05
Characteristic length-scale, $\lambda_{0}$ | 1
Characteristic time-scale, $\tau_{0}$ | 1
Coupling parameter, $\xi$ | 1.60
Table 1: Input parameters for the phase-field model of solidification of a
pure metal. The length-scale parameter $\lambda_{0}$ and the time-scale
parameter $\tau_{0}$ set the spatial and temporal resolution of the dendritic
growth process.
Parameter | Numerical value (units)
---|---
Liquid phase diffusivity, $D_{l}$ | $2.4\times 10^{-9}$ $m^{2}s^{-1}$
Solid phase diffusivity, $D_{s}$ | $1.15\times 10^{-12}$ $m^{2}s^{-1}$
Partition coefficient, $k$ | 0.14
Anisotropy strength, $\epsilon_{4}$ | 0.05
Gibbs-Thomson coefficient, $\Gamma_{g}$ | $2.36\times 10^{-7}$ m-K
Liquidus slope, $m_{l}$ | -3.5 $K(wt\%)^{-1}$
Initial concentration, $c_{0}$ | 4wt%
Thermal gradient, $G$ | 3700 $Km^{-1}$
Pulling velocity, $\bm{\nu}_{p}$ | $1.35\times 10^{-5}$ $ms^{-1}$
Interface thickness, $\lambda_{0}$ | $1.058\times 10^{-6}$ m
Model constant, $a_{1}$ | 0.8839
Model constant, $a_{2}$ | 0.6267
Table 2: Material properties and phase-field model parameters for a binary
alloy Al-4wt.% Cu
Input parameters | Expression
---|---
Chemical capillary length, $d_{0}$ | $\frac{k\Gamma}{mC_{0}(k-1)}$
Ratio of interface thickness to capillary length, $\Xi$ | $\frac{\lambda_{0}}{d_{0}}$
Non-dimensional thermal diffusivity, $\tilde{D}$ | $\frac{D_{l}\tau_{0}}{\lambda_{0}^{2}}$
Non-dimensional pulling velocity, $\tilde{\bm{\nu}}$ | $\frac{\bm{\nu}\tau_{0}}{\lambda_{0}}$
Characteristic thermal length-scale, $\tilde{L_{T}}$ | $\frac{|m_{l}|(1-k)c_{0}}{kG{\lambda_{0}}}$
Characteristic time-scale, $\tau_{0}$ | $\frac{a_{1}a_{2}\xi\lambda_{0}^{2}}{D_{l}}$
Coupling parameter, $\xi$ | $a_{1}\Xi$
Table 3: Input parameters for the phase-field model of solidification of a
binary metallic alloy. The numerical values of the input parameters are
computed using the expressions shown here that are in terms of the properties
listed in Table 3.
Next we discuss the case of dendritic growth in binary alloys. As mentioned
earlier, chemically, alloys are distinct from pure melts due to the presence
of solute atoms of one or more alloying materials. During the solidification
of alloys, solute diffusion plays a critical role. The coupled governing
equations of solute diffusion and phase-field are given by the Equations
10c-10e. The solid-liquid interface is constrained with the imposition of the
temperature gradient, $G$, and the pulling velocity, $\bm{\nu}_{p}$. We solve
for two primal fields - composition undercooling, $u_{c}$, and the phase-field
order parameter, $\phi$. The input parameters used in the numerical model are
given in Tables 3-3. The parameters listed in the Table 3 are the required
material properties for modeling dendritic growth in a binary alloy (adopted
from Zhu et al.[46]): mass diffusivities $D_{l}$ and $D_{s}$ in the solid and
liquid phase, Gibbs-Thomson coefficient $\Gamma_{g}$, thermal gradient, $G$
and pulling velocity $\nu_{p}$. Since here we only model a generic dilute
binary alloy, the temperature and composition phase diagram of a physical
alloy is approximated by a linear relationship. This approximation is possible
if we assume the alloy to be a dilute binary alloy. This linearization of a
temperature-composition relationship results in two more parameters: the
partition coefficient $k$ and the slope of the liquidus line $m_{l}$. An
optimal choice of the phase-field interface thickness is made keeping in mind
the computational tractability of the final numerical model. Table 3 lists the
expressions used to calculate the input parameters relevant to the model.
### 3.3 Primary geometric features and shape variations in dendrite
morphologies
Dendrites occur in a great variety of morphologies and orientations. One of
most common, yet fascinating, occurrences of dendritic growth is in the
formation of snowflakes that are dendritic structures with a distinct six-fold
geometric symmetry. In more practical applications, as in the solidification
of metallic alloys, dendritic shapes can be broadly classified as “columnar”
or “equiaxed”. Columnar dendrites grow into a positive temperature gradient,
and the rate of growth is controlled by the constitutional undercooling of the
liquid melt ahead of the dendritic tip. On the other hand, growth of an
equiaxed dendrites in pure melt is controlled by the temperature undercooling
of the surrounding liquid melt.
The schematic shown in Figure 2 depicts these possible variations in dendritic
shapes, and also identifies important geometric features that are
traditionally used by the Materials Science and Metallurgy community to
quantify dendritic morphologies. At the top end of the schematic, shown is a
equiaxed dendrite with a six-fold symmetry (one-half of the symmetric dendrite
shown) that is typical of snowflakes and in certain alloy microstructures. At
the bottom, shown are columnar dendrites with their primary arms oriented
parallel and inclined at a $30^{\circ}$ angle to the vertical axis. These
primary arms develop secondary arms as they evolve, and the separation of the
primary arms (referred to as the primary dendrite arm spacing or PDAS) and
secondary arms (referred to as the secondary dendrite arm spacing or SDAS) are
important geometric characteristics of a given dendrite shape. Further,
towards the right side of the schematic is a columnar dendrite colony growing
homogeneously oriented along the horizontal axis, and shown towards the left
side of the schematic are two time instances (separated by an interval
$\Delta$t) of an equiaxed dendrite’s interface evolution and the movement of
the dendritic tip over this time interval is $\Delta\bm{x}$. The dendritic tip
velocity, a very important measure of dendritic kinetics, is then given by
$\bm{{\nu}}_{\text{tip}}=\frac{d\bm{x}}{dt}$. It is to be noted that all these
morphologies were produced by the authors of this manuscript in simulations of
various solidification conditions using the numerical framework presented in
this work, and are but one demonstration of the capabilities of this framework
in modeling complex dendritic morphologies.
Figure 2: Schematic showing important geometric features used to quantify
dendritic morphologies, and some typical variations of shapes and orientations
in columnar and equiaxed dendrites. Shown are the primary arms and primary
dendrite arm spacing (PDAS), secondary arms and secondary dendrite arm spacing
(SDAS), and the movement of the dendritic tip by a distance $\Delta$x over a
time interval $\Delta$t. Dendritic tip velocity is given by
$\bm{\nu}_{\text{tip}}$. All these shapes were generated using the
solidification numerical models presented in this work.
### 3.4 Single equiaxed dendrite evolution
Growth of a single equiaxed dendrite in an undercooled melt of pure metal is a
classical solidification problem, and much of traditional analysis of
dendritic growth is based on reduced order models for this problem. The phase-
field governing equations are given by Equations 6a-6b, and the primal fields
solved for are the temperature undercooling $u$ and the order parameter
$\phi$. These equations are solved using both a $C^{0}$ basis and a $C^{1}$
basis as part of the study of convergence with respect to variations in the
spatial discretization. Input parameters used in this model have been listed
in Section 3.2. Zero flux Boundary Conditions are considered on all the
boundaries for the fields $u$ and $\phi$, and the Initial Condition is a
uniformly undercooled liquid melt with $\Delta=-0.75$ everywhere. The phase-
field order parameter is set to $\phi=-1.0$ everywhere, except a very small
circular region at the center representing a solid seed formed due to
nucleation with $\phi=1.0$ in the seed. Recollect that, in all the models,
$\phi=1.0$ represents the bulk solid and $\phi=-1.0$ represents the bulk
liquid region and any value in between represents the diffuse interface
between these two regions. The non-dimensional size of the domain is 500x500
and it is uniformly discretized into elements with a length measure of $\Delta
x=2.0$. The non-dimensional time step for the implicit time stepping scheme
considered in this case is $\Delta t=0.04$.
The growth of dendrites out of the nucleation point at the center is dictated
by the degree of undercooling of the liquid melt. As solidification begins,
the surrounding undercooled liquid melt is at lower temperature than the solid
seed. As a result, the solid seed diffuses heat into the liquid region.
Theoretically, the solid-liquid interface of an undercooled pure melt is
always unstable, so small local numerical perturbations on circular solid seed
interface begin to grow. Due to the assumption of strong anisotropy (four-fold
symmetry) of the interface, perturbations to the interface grow rapidly along
the favorable orientations as compared to the non-favorable orientations.
Figure 3a represents $u$ at a particular time, t = 840. Diffusion takes place
over the entire stretch of the solid and liquid regions. From t = 0 to 840,
$u$ changes from the initial value of -0.75 as per the solution of the heat
diffusion equation. It should be noted that the bulk of the liquid region and
the solid dendrite are at a uniform temperature, and the temperature
variations are mostly localized to the vicinity of the interface. The bulk
liquid melt region shown in blue is at a low temperature and the bulk solid
dendrite in red is at a high temperature, as seen in Figure 3a. The level set
of $\phi=0$ is identified as the distinct solid-liquid interface and the
evolution of this interface is plotted in Figure 3b as a series of $\phi=0$
level sets at different time instances. As can be seen, at t=0, the circular
seed at the center of the domain, over time, evolves into a four-fold
symmetric interface. At time, t = 840, we see a fully developed equiaxed
dendrite shape. We also simulated this Initial Boundary Value Problem on
$C^{1}$-continuous mesh and found identical spatiotemporal growth of the
equiaxed dendrite.
(a) Temperature contour at t=840 (b) Time evolution of the interface
Figure 3: Growth of a single equiaxed dendrite in an undercooled melt of pure
metal. Shown are the (3a) Temperature contour $(u)$ at t = 840, and (3b)
Temporal evolution of the equiaxed dendrite interface, delineated by the level
set $\phi=0$, at different time instances.
#### 3.4.1 Numerical convergence studies
In this section, we look at the accuracy of the interface propagation
numerical results for the single equiaxed dendrite case presented above. We do
this by comparing the velocity of propagation of the dendritic tip with known
analytical solutions of tip velocity for this problem, and also study the
convergence of the tip velocity with mesh size and increasing continuity of
the numerical basis.
##### Velocity of the dendritic tip:
As can be seen in Figure 3, the phase-field model of solidification of a pure
metal successfully produces the freely growing dendrite shape and four-fold
symmetry as expected. However, predicting quantifiable features of the
dendritic evolution would provide a better validation of the proposed
numerical framework. In this context, the dendritic tip propagation velocity
(see Figure 2) for single equiaxed dendrites has been widely used in
analytical studies of dendritic propagation. In the literature, implicit
expressions for radius and velocity of the dendritic tip have been obtained
using semi-analytical approaches, and they provide estimates of
$\bm{\nu}_{tip}$. One of the more popular semi-analytical approaches to obtain
the tip velocity employs a Green’s function (GF) approach. The Green’s
function is an integral kernel that is widely used to solve of certain classes
of differential equations. We will not discuss this approach here, but only
use the dendritic tip velocity values predicted by this approach. These values
are denoted by $\bm{\nu}_{tip}(GF)$. Interested readers are referred to Karma
and Rappel[45] for details of the GF approach.
The model parameters used in this study are $\Delta=-0.65$,
$\epsilon_{4}=0.05$, $\tilde{D}=1$, $d_{0}/\lambda_{0}=0.544$,
$\lambda_{0}=1$, $\tau_{0}=1$, $\xi=1.596$. The numerical domain size and time
step size, in dimensionless units, are 500x500 and $\Delta t=0.04$,
respectively. For these values, the Green’s function approach predicts an
equilibrium tip velocity, $\bm{\nu}_{tip}(GF)=0.0469$. Freely growing equiaxed
dendrite in an infinite domain, under solidification conditions held constant
over time, attains a fixed equilibrium tip velocity. In this study, we
consider the effect of two discretization parameters, namely: (1) Comparison
of the numerical dendritic tip velocity ($\bm{\nu}_{tip}$) to the equilibrium
tip velocity obtained from Green’s function approach $\bm{\nu}_{tip}(GF)$
(Figure 4), and (2) Convergence of the tip velocity ($\bm{\nu}_{tip}$) with
mesh refinement, and its dependence on the order of continuity of the basis
(Figure 5).
The method of estimating the dendrite tip velocity from the numerical
simulations using the computational frameworks described above is described
briefly. Simulations were performed using both a $C^{0}$-continuous basis and
a $C^{1}$-continuous basis, to study the effect of basis continuity when
resolving fine dendritic morphologies. The dendritic tip velocity measurement
is done manually as a post-processing of the phase-field order parameter
contours of the numerical simulations. At each time instance, the tip of the
zero level set ($\phi=0$) of the order parameter is manually marked, and the
velocity of tip propagation at each time instance is computed using a simple
finite difference estimate of the derivative
$\bm{\nu}_{tip}=\frac{d\bm{x}}{dt}$. Two sources of error can potentially
cause minor variations to the computed velocity values. The first source of
error arises from not picking the exact tip location in subsequent time
snapshots as the dendrite interface shape is evolving with time. The second
source of error arises due to the use of only $C^{0}$ interpolation of the
nodal solutions inside the elements by the visualization tool. This error is
significant for $C^{1}$-basis, as the results are $C^{1}$-continuous but the
visualization of the results inside the elements is only $C^{0}$-continuous.
As can be seen in Figure 4, $\bm{\nu}_{tip}$ starts to approach the Green’s
function value, $\bm{\nu}_{tip}(GF)$, across all the simulations, but shows
better convergence to the Green’s function value for the $C^{1}$-continuous
basis. Further, we observe the percentage error of $\bm{\nu}_{tip}$, given by
Error$=\frac{|\bm{\nu}_{tip}-\bm{\nu}_{tip}(GF)|}{\bm{\nu}_{tip}(GF)}$,
consistently reduces with a decrease in the mesh size across both
$C^{0}$-continuous and $C^{1}$-continuous basis, as shown in Figures 5(a) and
5(b). This is an important numerical convergence result, as to the best of our
knowledge, this is the first time optimal convergence rates have been
demonstrated for non-trivial dendritic shape evolution with respect to mesh
refinement. The convergence rates are demonstrated in the plot in Figures
5(a), where the rate of convergence (slope of the percentage Error vs mesh
size plot) is close to two for a $C^{0}$-continuous basis, and close to three
for a $C^{1}$-continuous basis. While obtaining theoretical optimal converge
rates with respect to h-refinement of the discretization for this coupled
system of parabolic partial differential equations is very challenging, one
can observe that the convergence rates shown are comparable to the optimal
convergence rates of classical heat conduction and mass diffusion partial
differential equations [47]. This completes the estimation of error in
$\bm{\nu}_{tip}$, which is a measure of localized error in the order parameter
field, as the dendritic interface is given by the zero level set of the order
parameter field. Now we look at estimation of errors over the entire domain
for both the temperature field and the order parameter field.
$500$$1{,}000$$1{,}500$$2{,}000$$4.5$$5$$5.5$$6$$6.5$$7$$\cdot 10^{-2}$Time
$\bm{\nu}_{tip}$ (a) $\Delta x$=2.0
$500$$1{,}000$$1{,}500$$2{,}000$$4.5$$5$$5.5$$6$$6.5$$7$$7.5$$\cdot
10^{-2}$Time $\bm{\nu}_{tip}$ $C^{0}$ basis$C^{1}$ basisGreen’s function value
(b) $\Delta x$=1.0
Figure 4: Comparison of the numerical and analytical (Green’s function
approach) dendritic tip velocities over time during growth of a single
equiaxed dendrite for two different mesh sizes ($\Delta x$), and for both
$C^{0}$-continuous and $C^{1}$-continuous basis.
$10^{-0.1}$$10^{0}$$10^{0.1}$$10^{0.2}$$10^{0.3}$$10^{0}$$10^{1}$2131$y_{0}=2.04x_{0}+0.35$$y_{1}=3.72x_{1}-0.48$$y_{2}=4.11x_{2}-0.51$${\Delta
x}$ Error $C^{0}$ basis$C^{1}$ basis (a) Convergence of error in dendritic tip
velocity with mesh refinement
$C^{0}$ basis$C^{1}$ basis$0$$2$$4$$6$$8$$10$Error $\Delta x$ = 1.0 (b)
Comparison of error in dendritic tip velocity for a mesh size of $\Delta x$ =
1.0
Figure 5: Convergence of the percentage error in dendritic tip velocity, given
by Error=$\frac{|\bm{\nu}_{tip}-\bm{\nu}_{tip}(GF)|}{\bm{\nu}_{tip}(GF)}$,
with mesh refinement and its dependence on the order of continuity of the
basis. Subplot (a) shows optimal convergence of the error with respect to mesh
refinement in a log-log plot, and Subplot (b) shows the difference in
percentage errors between a $C^{0}$-continuous basis and a $C^{1}$-continuous
basis for one mesh size.
##### Numerical error estimates for the temperature and order parameter
fields:
For this study, we consider three different mesh refinements of a
$C^{0}$-continuous basis. For each of these refinements, the computational
domain size is 160x160, chosen time step is $\Delta t=0.12$, and the other
input parameters are as given in Table 1. As the Initial Condition, a small
nucleation seed was considered, and this seed grows into the surrounding
undercooled melt. For the problem of interest, there are no known analytical
solutions for the fields $u$ or $\phi$. As mentioned in the earlier
subsection, the only analytical result available is for the equilibrium
dendritic tip velocity. Thus, we follow a computational approach to obtain the
reference solution of both the primal fields. A separate problem is solved on
a very fine mesh ($\Delta x=$ 0.2) using the popular open source phase field
library PRISMS-PF [48], and the primal field solutions for this very fine mesh
are considered the reference solutions. It is to be noted that one of the
authors (S.R.) in this work was the lead developer of the PRISMS-PF library,
and this library is being successfully used by the wider Materials Science
community to solve numerous phase-field problems.
Using this numerical reference solution, error in the temperature and order
parameter fields of the solutions obtained for the numerical formulations
presented in this work, is calculated by comparing the fields and their
gradients using two types of norms relevant to the problem, and for three
different mesh refinements at one instance of time (time = 450) when the
dendritic shape is well established. The two error norms used in this analysis
are the $L_{2}$-norm and the $H_{1}$-norm, and these are defined as follows:
$|u-\hat{u}|_{L_{2}}=\sqrt{\int_{\Omega}|u_{h}-\hat{u}|^{2}\leavevmode\nobreak\
dV}$ (12a)
$|u-\hat{u}|_{H_{1}}=\sqrt{\int_{\Omega}\Big{(}|u_{h}-\hat{u}|^{2}+|\nabla
u_{h}-\nabla\hat{u}|^{2}\Big{)}\leavevmode\nobreak\ dV}$ (12b)
Here, $u_{h}$ is the finite element solution under consideration and $\hat{u}$
is the reference numerical solution. The integral is defined over the entire
numerical domain, so the estimated error, unlike the previous error in tip
velocity, is not localized to any particular region in the domain. The
convergence of this error in these two norms with mesh refinement is shown in
Figure 6 on a log-log scale. As can be seen from these plots, we observe good
convergence in both the norms for both the primal fields, with the rates of
convergence being in the vicinity of 1.5 to 2.0. We are not aware of any
theoretical optimal error estimates in any of the relevant error norms for
this problem, so this numerical strategy to obtain error estimates is seen as
filing this vital gap by providing numerical convergence estimates, thus
permitting the numerical analyses of these governing equations.
$10^{-0.2}$$10^{0}$$10^{0.2}$$10^{0.4}$$10^{0.5}$$10^{1}$$10^{1.5}$11.80y=1.80x+0.85$\mathbf{\Delta
x}$ $|\bm{u}-\hat{\bm{u}}|_{L_{2}}$ (a) $L^{2}$-norm of the temperature
$10^{-0.2}$$10^{0}$$10^{0.2}$$10^{0.4}$$10^{0.5}$$10^{1}$$10^{1.5}$11.79y=1.79x+0.86$\mathbf{\Delta
x}$ $|\bm{u}-\hat{\bm{u}}|_{H_{1}}$ Numerical dataLinear Fit (b) $H^{1}$-norm
of the temperature
$10^{-0.2}$$10^{0}$$10^{0.2}$$10^{0.4}$$10^{1}$$10^{2}$11.59y=1.59x+1.45$\mathbf{\Delta
x}$ $|\bm{\phi}-\hat{\bm{\phi}}|_{L^{2}}$ (c) $L^{2}$-norm of the order
parameter
$10^{-0.2}$$10^{0}$$10^{0.2}$$10^{0.4}$$10^{1.5}$$10^{2}$11.50y=1.50x+1.50$\mathbf{\Delta
x}$ $|\bm{\phi}-\hat{\bm{\phi}}|_{H^{1}}$ Numerical dataLinear Fit (d)
$H^{1}$-norm of the order parameter
Figure 6: Error estimates and order of convergence for the primal fields
$(u,\phi)$ shown on log-log plots. The error norms used in this study are the
$L_{2}$-norm and the $H_{1}$-norm of $(u,\phi)$. The linear fit expression is
also given for each plot.
### 3.5 Growth of multiple equiaxed dendrites
In Section 3.4, we discussed one particular type of dendritic growth, namely
the evolution of a single equiaxed dendrite during the solidification of pure
metal. Now we look at the evolution of multiple equiaxed dendrites during the
solidification of a more complex binary alloy solution. The input parameters
for the model are listed in Table 3. Growth of multi-equiaxed dendrites during
solidification of a binary alloy is governed by the Equations 11a-11b. The
primal fields in this problem are the non-dimensional composition, $u_{c}$,
and the phase-field order parameter, $\phi$. As an Initial Condition for this
problem, we consider multiple small nucleation seeds randomly distributed in
the numerical domain. The seed locations are completely random, and generated
using a random number generator. To ensure free equiaxed dendritic growth,
thermal gradient $G$ and pulling velocity $\bm{\nu}_{p}$ is taken to be zero.
The numerical domain of size 600x600 is subdivided into uniform elements with
a length measure of $\Delta x=1.2$. The time step size is $\Delta t=0.005$.
The average alloy composition in the entire domain is taken to be $c_{0}=4\%$,
which corresponds to $u_{c}=0$. Nucleation seeds of the solid phase are
randomly placed such that $\phi=1$ inside of the seed and $\phi=-1$ outside of
the seed. Zero flux Boundary Conditions are considered for both the fields.
During the metal casting process, many solid nuclei are randomly formed near
cavities or defects on the surface of a mold. These nuclei then start to grow
in an equiaxial manner with neighboring nuclei competing with each other. This
competition can shunt the growth of dendrites approaching each other. Solution
contours of the fields $u_{c}$ and $\phi$ are shown in the Figure 7. Here,
Subfigure 7a shows the solute composition in the numerical domain at t = 40.
Initially, the entire domain is at an alloy concentration of 4%. Due to
negligible mass diffusivity in the solid phase, solute diffusion primarily
takes place in the liquid region. A solute composition of $c_{0}=4\%$ is
highest near the interface, and as one moves away from the interface, the
solute concentration decreases. This is apparent in all the dendritic
interfaces shown in the Subfigure 7a. Solute composition inside the solid is
given by $c_{0}=0.56$ and it stays constant due to low negligible diffusivity
in the solid region. Subfigure 7b demonstrates the competition between the
dendrites growing out of the various seeds, the interaction of the dendritic
tips, and the growth of secondary dendrite arms. Material properties of an
alloy are often correlated with fine-scale microstructure length scales (at
the grain scale and sub-grain scale), and the secondary dendrite arm spacing
(SDAS) is one of the important microstructural properties of alloys.
(a) Composition (b) Zero level sets of the order parameter
Figure 7: Growth of multiple equiaxed dendrites. Shown are the (7a) contours
of solute composition, c, at time t = 40, and (7b) Time evolution of multiple
equiaxed dendrite interfaces at time t = 0, 25 and 40. The dendrite interfaces
are the zero level sets of the order parameter, $\phi=0.0$.
### 3.6 Growth of single columnar dendrites
In this section, we demonstrate the growth of columnar dendrites, growing from
a single seed during the solidification of a binary alloy. The numerical model
and input parameters used in this case are identical to the previous Section
3.5. The notable difference from the model in Section 3.5 to the current model
is the imposition of constraints like non-zero thermal gradient $G$ and
pulling velocity $\bm{\nu}_{p}$. Initially, we place a single solid seed at
the bottom of the numerical domain. The entire domain is at a solute
composition of $c_{0}=4\%$ which corresponds to the composition undercooling
of $u_{c}=0$. Zero flux Boundary Conditions are enforced on both the primal
fields, $u_{c}$ and $\phi$. Figure 8 depicts the evolution of the composition
field, and the dendritic interface as a function of time. The temperature
gradient (G) applied is in the vertical direction. As opposed to the
solidification in an undercooled pure metal, the liquid phase in this type of
solidification is at a higher temperature than the dendrite in a solid phase.
Any numerically localized perturbation to the solid-liquid interface will grow
as per the degree of constitutional undercooling. For the given temperature
gradient $G$ and pulling velocity $\bm{\nu}_{p}$, the solid-liquid interface
is unstable. This is evident from the interface $\phi=0$ evolution captured in
the Subfigure 8b. At t = 0, the solid seed starts to grow. By t = 35, the
solid seed has developed four primary branches due to the four-fold symmetry
in the anisotropy considered. The growth of only three primary arms is
captured in the simulation and the fourth arm pointing downward is not a part
of the computation. These three primary arms have developed secondary arms of
their own as seen in contour. By t = 40, the initial seed has grown into a
larger size dendrite and several of its secondary arms are growing in the
columnar fashion aligning with the direction in which constraints are applied,
i.e, vertically. The phenomenon of secondary arms growing in the vertical
direction becomes more obvious by t = 45 and t = 50 as seen in the Subfigure
8b. The diffusion of the solute is governed by the same mechanism discussed in
Section 3.5. Like Subfigure 7a, in this case as well, the solute composition
is maximum near the dendrite interface and it decreases as one moves away from
the interface. This is clearly shown in the Subfigure 8a.
(a) Composition (b) Zero level sets of the order parameter
Figure 8: Growth of a single columnar dendrite. Shown are the (8a) contours of
solute composition, c, at different time instances, and (8b) Time evolution of
single columnar dendrite interfaces at times t = 30 to 50.
### 3.7 Growth of multiple columnar dendrites
Columnar dendrites growing out of a flat boundary are the next type of
dendrites modeled. This type of dendrites occur in directional solidification
processes which are of significance in manufacturing of alloy components by
use of molds, where often the dendrite growth is primarily columnar. The
characteristic of the columnar dendrites are the growth of a colony of
dendrites along the direction of the imposed temperature gradient and the
pulling velocity. These dendrites then move at a steady-state tip velocity
which is determined by the pulling velocity. The other geometric features of
columnar dendrites which are typically observed in the experimental studies
include primary arms, secondary arms, spacing between arms, growth competition
between the neighboring dendrites, and the interaction of the diffusion field
between neighboring dendrites. The growth of multi-columnar dendrites is
governed by the same governing equations that were used to model the
solidification of a binary alloy in the previous Section 3.4-3.5, and so are
the input parameters. However for the Initial Condition, we model an initial
thin layer of solid solute on one of the boundaries, and set $\phi=1$ inside
the initial solute layer thickness and $\phi=-1$ in the rest of the domain.
Non-dimensional composition undercooling is set to $u_{c}=-1$ everywhere in
the domain. For the Boundary Conditions, zero flux conditions are enforced on
the boundaries for both the primal fields $u_{c}$ and $\phi$. Figure 9 depicts
the results of multiple columnar dendritic growth. As can be seen, the phase-
field contour interface ($\phi=0$) at time, t= 240, fills the entire domain
with a colony of columnar dendrites. The initial planar interface with a
solute composition seeds the dendrites that then compete to evolve into this
columnar dendritic structure. The emergence of seven primary arms of the
dendrites, as seen in Subfigure 9b, is the result of the competition for
growth among several more initial primary arms of dendrites. Some of these
underdeveloped arms can be seen in the lower portion of the figure. The
primary arm spacing between these seven primary arms appears to converge to a
constant value with the progress of the dendrites. Another feature of the
columnar dendrites that we were able to capture is the growth of the secondary
dendrite arms and their orientation. The concentration of the solute in the
solid region is nearly uniform at a value of 0.56. The region surrounding
these solid dendrites consists of the liquid melt where the highest
concentration is about 4.0. Due to mass diffusion, there exists a gradient in
the solute concentration in the liquid melt as can be seen in the Subfigure
9a.
(a) Composition (b) Zero level sets of the order parameter
Figure 9: Growth of multiple columnar dendrites. Shown are the (9a) Contours
of the solute composition, $c$, at time t = 240, and (9b) Growth profile of
multiple columnar dendrite interfaces at time t = 240.
### 3.8 Three dimensional growth of single equiaxed dendrite
We now simulate the phase-field model described by the Equations 5a-5b in
three dimensions. The initial solidification conditions consist of a small
spherical seed placed in a $500\times 500\times 500$ computational domain with
an adaptive mesh and the smallest element length measure of $\Delta x=2.0$ and
uniform time step $\Delta t=0.12$. The seed is placed in an undercooled melt
of pure metal. The input phase-field parameters used in the simulation are
listed in Table 1. The evolution of a spherical seed into a four-fold
symmetric dendrite structure is shown in the Figure 10. The fully grown
dendrite in blue at the end of the evolution has six primary arms as dictated
by the surface anisotropy. The distribution of an undercooling temperature $u$
on a fully grown dendrite, shown in red, can also be seen in the figure. Three
dimensional implementation of phase-field models of dendrites is
computationally very demanding, due to the need to resolve the fine scale
primary-arm and secondary-arm structures. To make this problem computationally
tractable, we use local mesh adaptivity (h-adaptivity), implicit time-stepping
schemes with adaptive time-step control, and domain decomposition using MPI.
Figure 10: Evolution of a 3D single equiaxed dendrite. The dendritic
structures shown in blue are the time evolution of the contours of the phase-
field parameter, $\phi$, and the dendritic structure shown in red is the
contour of the undercooling temperature, $u$, at one time instance.
## 4 Conclusion
In this work, we present a detailed development of the requisite numerical
aspects of dendritic solidification theory, and the computational
implementation of the corresponding phase-field formulations to model
dendritic growth in pure metals and binary alloys. A wide variety of
physically relevant dendritic solidification patterns are modeled by solving
the governing equations under various initial conditions and boundary
conditions. To validate the numerical framework, we simulate the classical
four-fold symmetric dendrite shape occurring in undercooled pure melt and
compare the numerically computed dendritic tip velocities with the
corresponding analytical values obtained using a Green’s function approach.
Further, this problem is used as a basis for performing error convergence
studies of dendritic tip velocity and dendritic morphology (primal fields of
temperature and order parameter). Further, using this numerical framework,
various types of dendritic solidification patterns like multi-equiaxed, single
columnar and multi-columnar dendrites are modeled in two-dimensional and
three-dimensional computational domains.
The distinguishing aspects of this work are - a unified treatment of both
pure-metals and alloys; novel numerical error estimates of dendritic tip
velocity; and the convergence of error for the primal fields of temperature
and the order parameter with respect to the numerical discretization. To the
best of our knowledge, this is a first of its kind study of numerical
convergence of the phase-field equations of dendritic growth in a finite
element method setting, and a unified computational framework for modeling a
variety of dendritic structures relevant to solidification in metallic alloys.
## Acknowledgement
The authors would like to thank Prof. Dan Thoma (University of Wisconsin-
Madison) and Dr. Kaila Bertsch (University of Wisconsin-Madison; now at
Lawrence Livermore National Laboratory) for very useful discussions on
dendritic growth and microstructure evolution in the context of additive
manufacturing of metallic alloys.
## Declarations
Conflict of interest: The authors declare that they have no conflict of
interest.
## References
* [1] L.I. Rubinstein. The stefan problem, transl. math. Monographs, 27:327–3, 1971.
* [2] Gunter H Meyer. The numerical solution of stefan problems with front-tracking and smoothing methods. Applied Mathematics and Computation, 4(4):283–306, 1978.
* [3] Guillermo Marshall. A front tracking method for one-dimensional moving boundary problems. SIAM journal on scientific and Statistical Computing, 7(1):252–263, 1986.
* [4] Jonathan A Dantzig and Michel Rappaz. Solidification: -Revised & Expanded. EPFL press, 2016.
* [5] S Chen, B Merriman, Smereka Osher, and P Smereka. A simple level set method for solving stefan problems. Journal of Computational Physics, 135(1):8–29, 1997.
* [6] G Fix. Phase field method for free boundary problems, in; free boundary problems, a. Fasanao and M. Primicerio, eds., Pit-mann, London, 1983.
* [7] Joseph B Collins and Herbert Levine. Diffuse interface model of diffusion-limited crystal growth. Physical Review B, 31(9):6119, 1985.
* [8] Gunduz Caginalp. An analysis of a phase field model of a free boundary. Archive for Rational Mechanics and Analysis, 92(3):205–245, 1986\.
* [9] G Caginalp. Stefan and hele-shaw type models as asymptotic limits of the phase-field equations. Physical Review A, 39(11):5887, 1989.
* [10] Ryo Kobayashi. Modeling and numerical simulations of dendritic crystal growth. Physica D: Nonlinear Phenomena, 63(3-4):410–423, 1993.
* [11] Alain Karma and Wouter-Jan Rappel. Phase-field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Physical review E, 53(4):R3017, 1996.
* [12] Alain Karma and Wouter-Jan Rappel. Phase-field model of dendritic sidebranching with thermal noise. Physical review E, 60(4):3614, 1999.
* [13] Mathis Plapp and Alain Karma. Multiscale finite-difference-diffusion-monte-carlo method for simulating dendritic solidification. Journal of Computational Physics, 165(2):592–619, 2000.
* [14] James A Warren and William J Boettinger. Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method. Acta Metallurgica et Materialia, 43(2):689–703, 1995.
* [15] Irina Loginova, Gustav Amberg, and John Ågren. Phase-field simulations of non-isothermal binary alloy solidification. Acta materialia, 49(4):573–581, 2001.
* [16] Alain Karma. Phase-field formulation for quantitative modeling of alloy solidification. Physical Review Letters, 87(11):115701, 2001.
* [17] JC Ramirez, C Beckermann, As Karma, and H-J Diepers. Phase-field modeling of binary alloy solidification with coupled heat and solute diffusion. Physical Review E, 69(5):051607, 2004.
* [18] Blas Echebarria, Roger Folch, Alain Karma, and Mathis Plapp. Quantitative phase-field model of alloy solidification. Physical Review E, 70(6):061604, 2004.
* [19] Robert F Almgren. Second-order phase field asymptotics for unequal conductivities. SIAM Journal on Applied Mathematics, 59(6):2086–2107, 1999.
* [20] Munekazu Ohno and Kiyotaka Matsuura. Quantitative phase-field modeling for dilute alloy solidification involving diffusion in the solid. Physical Review E, 79(3):031603, 2009.
* [21] Xiaobing Feng and Andreas Prohl. Analysis of a fully discrete finite element method for the phase field model and approximation of its sharp interface limits. Mathematics of computation, 73(246):541–567, 2004.
* [22] B Gonzalez-Ferreiro, Héctor Gómez, and Ignacio Romero. A thermodynamically consistent numerical method for a phase field model of solidification. Communications in Nonlinear Science and Numerical Simulation, 19(7):2309–2323, 2014.
* [23] Chuanjun Chen and Xiaofeng Yang. Efficient numerical scheme for a dendritic solidification phase field model with melt convection. Journal of Computational Physics, 388:41–62, 2019.
* [24] Daniel Kessler and J-F Scheid. A priori error estimates of a finite-element method for an isothermal phase-field model related to the solidification process of a binary alloy. IMA journal of numerical analysis, 22(2):281–305, 2002.
* [25] Xianliang Hu, Ruo Li, and Tao Tang. A multi-mesh adaptive finite element approximation to phase field models. Communications in Computational Physics, 5(5):1012–1029, 2009.
* [26] Jan Rosam, Peter K Jimack, and Andy Mullis. A fully implicit, fully adaptive time and space discretisation method for phase-field simulation of binary alloy solidification. Journal of Computational Physics, 225(2):1271–1287, 2007.
* [27] Damien Tourret, Hong Liu, and Javier LLorca. Phase-field modeling of microstructure evolution: Recent applications, perspectives and challenges. Progress in Materials Science, 123:100810, 2022.
* [28] Zhijun Wang, Junjie Li, Jincheng Wang, and Yaohe Zhou. Phase field modeling the selection mechanism of primary dendritic spacing in directional solidification. Acta Materialia, 60(5):1957–1964, 2012.
* [29] V Fallah, M Amoorezaei, N Provatas, SF Corbin, and A Khajepour. Phase-field simulation of solidification morphology in laser powder deposition of ti–nb alloys. Acta Materialia, 60(4):1633–1646, 2012.
* [30] Damien Tourret and Alain Karma. Growth competition of columnar dendritic grains: A phase-field study. Acta Materialia, 82:64–83, 2015.
* [31] Tomohiro Takaki, Munekazu Ohno, Takashi Shimokawabe, and Takayuki Aoki. Two-dimensional phase-field simulations of dendrite competitive growth during the directional solidification of a binary alloy bicrystal. Acta Materialia, 81:272–283, 2014.
* [32] Shaoning Geng, Ping Jiang, Xinyu Shao, Gaoyang Mi, Han Wu, Yuewei Ai, Chunming Wang, Chu Han, Rong Chen, Wei Liu, et al. Effects of back-diffusion on solidification cracking susceptibility of al-mg alloys during welding: A phase-field study. Acta Materialia, 160:85–96, 2018.
* [33] A Farzadi, Minh Do-Quang, S Serajzadeh, AH Kokabi, and Gustav Amberg. Phase-field simulation of weld solidification microstructure in an al–cu alloy. Modelling and Simulation in Materials Science and Engineering, 16(6):065005, 2008.
* [34] X Wang, PW Liu, Y Ji, Y Liu, MH Horstemeyer, and L Chen. Investigation on microsegregation of in718 alloy during additive manufacturing via integrated phase-field and finite-element modeling. Journal of Materials Engineering and Performance, 28(2):657–665, 2019.
* [35] Matthew R Rolchigo, Michael Y Mendoza, Peyman Samimi, David A Brice, Brian Martin, Peter C Collins, and Richard LeSar. Modeling of ti-w solidification microstructures under additive manufacturing conditions. Metallurgical and Materials Transactions A, 48(7):3606–3622, 2017\.
* [36] Supriyo Ghosh, Li Ma, Nana Ofori-Opoku, and Jonathan E Guyer. On the primary spacing and microsegregation of cellular dendrites in laser deposited ni–nb alloys. Modelling and simulation in materials science and engineering, 25(6):065002, 2017.
* [37] Xibing Gong and Kevin Chou. Phase-field modeling of microstructure evolution in electron beam additive manufacturing. Jom, 67(5):1176–1182, 2015.
* [38] Seshadev Sahoo and Kevin Chou. Phase-field simulation of microstructure evolution of ti–6al–4v in electron beam additive manufacturing process. Additive manufacturing, 9:14–24, 2016.
* [39] Trevor Keller, Greta Lindwall, Supriyo Ghosh, Li Ma, Brandon M Lane, Fan Zhang, Ursula R Kattner, Eric A Lass, Jarred C Heigel, Yaakov Idell, et al. Application of finite element, phase-field, and calphad-based methods to additive manufacturing of ni-based superalloys. Acta materialia, 139:244–253, 2017.
* [40] Vladimir Slavov, Stefka Dimova, and Oleg Iliev. Phase-field method for 2d dendritic growth. In International Conference on Large-Scale Scientific Computing, pages 404–411. Springer, 2003.
* [41] Hieram Neumann-Heyme. Phase-field modeling of solidification and coarsening effects in dendrite morphology evolution and fragmentation. PhD thesis, Technical University of Dresden, Dresden, Technical University of Dresden, Dresden, 8 2017. An optional note.
* [42] Daniel Arndt, Wolfgang Bangerth, Bruno Blais, Marc Fehling, Rene Gassmöller, Timo Heister, Luca Heltai, Uwe Köcher, Martin Kronbichler, Matthias Maier, Peter Munch, Jean-Paul Pelteret, Sebastian Proell, Konrad Simon, Bruno Turcksin, David Wells, and Jiaqi Zhang. The deal.II library, version 9.3. Journal of Numerical Mathematics, 29(3):171–186, 2021.
* [43] V.M. Calo N. Collier, L. Dalcin. PetIGA: High-performance isogeometric analysis. arxiv, (1305.4452), 2013. http://arxiv.org/abs/1305.4452.
* [44] K. Bhagat. Phase-field based dendritic modeling. https://github.com/cmmg/dendriticGrowth, 2022.
* [45] Alain Karma and Wouter-Jan Rappel. Quantitative phase-field modeling of dendritic growth in two and three dimensions. Physical review E, 57(4):4323, 1998.
* [46] Chang-sheng Zhu, Sheng Xu, Li Feng, Dan Han, and Kai-ming Wang. Phase-field model simulations of alloy directional solidification and seaweed-like microstructure evolution based on adaptive finite element method. Computational Materials Science, 160:53–61, 2019.
* [47] M Bieterman and I Babuška. The finite element method for parabolic equations. Numerische Mathematik, 40(3):373–406, 1982.
* [48] Stephen DeWitt, Shiva Rudraraju, David Montiel, W Beck Andrews, and Katsuyo Thornton. Prisms-pf: A general framework for phase-field modeling with a matrix-free finite element method. npj Computational Materials, 6(1):1–12, 2020.
|
# A note on an effective bound for the gonality conjecture
Alexander Duncan, Wenbo Niu, Jinhyung Park Department of Mathematical
Sciences, University of Arkansas, Fayetteville, AR 72701, USA
<EMAIL_ADDRESS>Department of Mathematical Sciences, University of Arkansas,
Fayetteville, AR 72701, USA<EMAIL_ADDRESS>Department of Mathematical
Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract.
The gonality conjecture, proved by Ein–Lazarsfeld, asserts that the gonality
of a nonsingular projective curve of genus $g$ can be detected from its
syzygies in the embedding given by a line bundle of sufficiently large degree.
An effective result obtained by Rathmann says that any line bundle of degree
at least $4g-3$ would work in the gonality theorem. In this note, we improve
the degree bound to $4g-4$ with two exceptional cases.
###### Key words and phrases:
gonality conjecture, syzygies of an algebraic curve, symmetric product of an
algebraic curve
###### 2020 Mathematics Subject Classification:
14Q20, 13A10
W. Niu was supported by the Simons Collaboration Grants for Mathematicians
J. Park was partially supported by the National Research Foundation (NRF)
funded by the Korea government (MSIT) (NRF-2021R1C1C1005479).
## 1\. Introduction
We work over the field $\mathbf{C}$ of complex numbers. Let $C$ be a
nonsingular projective curve of genus $g$, and $B$ and $L$ be line bundles on
$C$. Suppose that $L$ is globally generated, and write
$S=\operatorname{Sym}H^{0}(L)$. The associated section module
$R(C,B;L):=\bigoplus_{q\geq 0}H^{0}(B\otimes L^{q})$
is a finitely generated graded $S$-module. It admits a minimal graded free
resolution over $S$:
$\cdots\longrightarrow E_{p}\longrightarrow\cdots\longrightarrow
E\longrightarrow E_{1}\longrightarrow E_{0}\longrightarrow
R(C,B;L)\longrightarrow 0.$
Each graded free $S$-module $E_{p}$ in the resolution has the form
$E_{p}=\bigoplus_{q\in\mathbb{Z}}K_{p,q}(C,B;L)\otimes S(-p-q),$
where $K_{p,q}(C,B;L)$ is the Koszul cohomology group defined as the
cohomology at the middle of the Koszul-type complex
$\wedge^{p+1}H^{0}(L)\otimes H^{0}(B\otimes
L^{q+1})\longrightarrow\wedge^{p}H^{0}(L)\otimes H^{0}(B\otimes
L^{q})\longrightarrow\wedge^{p-1}H^{0}(L)\otimes H^{0}(B\otimes L^{q-1}).$
A particularly interesting case is when $B=\mathscr{O}_{C}$ and $L$ has large
degree. The Koszul groups $K_{p,q}(C;L):=K_{p,q}(C,\mathscr{O}_{C};L)$ give a
minimal free resolution of the section ring $R(C;L):=R(C,\mathscr{O}_{C};L)$.
Note that if $C\subseteq\mathbf{P}(H^{0}(L))$ is projectively normal, then
$R(C;L)$ is the homogeneous coordinate ring of $C$ in $\mathbf{P}(H^{0}(L))$.
If $L$ is nonspecial, then $K_{p,q}(C;L)$ vanishes for $q\geq 3$ and thus the
minimal free resolution essentially consists of two strands $K_{p,1}(C;L)$ and
$K_{p,2}(C;L)$. If $L$ has degree larger than $2g$, Green’s $(2g+1+p)$ theorem
([Gr84, Theorem 4.a.1], see also [GL88]) determines the strand $K_{p,2}(C;L)$
using the notion of $N_{p}$ property. The Green–Lazarsfeld gonality conjecture
[GL86] together with their nonvanishing theorem [Gr84, Appendix] predicts the
shape of the strand $K_{p,1}(C;L)$. This conjecture was proved by Ein-
Lazarsfeld [EL15] when $L$ has sufficiently large degree. Ranthmann then
showed an effective result (see [Ra16, Theorem 1.1]): if $\deg L\geq 4g-3$,
then
$K_{p,1}(C;L)\neq 0\Longleftrightarrow 1\leq p\leq\deg
L-g-\operatorname{gon}(C),$
where $\operatorname{gon}(C)$ is the gonality of $C$ which by definition is
the minimal degree of pencils on $C$. As pointed out in [Ra16], although the
degree bound above is not expected to be optimal, there is an example of a
plane quartic curve showing the degree bound $4g-4$ does not work. In this
short note, we investigate the failure of the gonality conjecture when $\deg
L=4g-4$. The main result is the following.
###### Theorem 1.1.
Let $C$ be a nonsingular projective curve of genus $g\geq 2$, and $L$ be a
line bundle on $C$ with $\deg L\geq 4g-4$. Then
$K_{p,1}(C;L)\neq 0\Longleftrightarrow 1\leq p\leq\deg
L-g-\operatorname{gon}(C)$
unless $L=\omega^{2}_{C}$ and either $g=2$ or $C$ is a plane quartic curve. In
the exceptional cases, $K_{\deg L-g-\operatorname{gon}(C)+1,1}(C;L)\neq 0$ but
$K_{\deg L-g-\operatorname{gon}(C)+2,1}(C;L)=0$.
An easy application of the theorem gives a uniform picture of syzygies of
pluricanonical embedding of curves, especially the second power of the
canonical divisor. It has been a long standing interest to understand the
syzygies of canonical curves. The shape of the minimal free resolution of
$R(C;\omega_{C})$ was predicted in Green’s conjecture [Gr84, Conjecture 5.1].
It was verified by Voisin [Voi1, Voi2] for general curves, but it is still
widely open in general. For pluricanonical embedding
$C\subseteq\mathbf{P}(H^{0}(\omega_{C}^{k}))$, the picture of syzygies turns
out to be complete, and we give a summary here. Let $C$ be a curve of genus
$g\geq 2$ and gonality $\operatorname{gon}(C)$. Put $L:=\omega^{k}_{C}$ and
write $r:=h^{0}(L)-1$. For $k\geq 3$, Greens’s $(2g+1+p)$-theorem and
Rathmann’s effective gonality theorem give the result that
$K_{p,1}(C;\omega^{k}_{C})\neq 0\Longleftrightarrow 1\leq p\leq
r-\operatorname{gon}(C).$
For $L=\omega^{2}_{C}$ and $r=3g-3$, Green’s $(2g+1+p)$-theorem and Theorem
1.2 gives us the following two cases
1. (1)
If either $g=2$ ($r=\operatorname{gon}(C)=2$), or $C$ is a plane quartic curve
($r=5$ and $\operatorname{gon}(C)=3$), then
$K_{p,1}(C;\omega^{2}_{C})\neq 0\Longleftrightarrow 1\leq p\leq
r-\operatorname{gon}(C)+1.$
2. (2)
Otherwise,
$K_{p,1}(C;\omega^{2}_{C})\neq 0\Longleftrightarrow 1\leq p\leq
r-\operatorname{gon}(C).$
In the setting of Theorem 1.1, Green–Lazarsfeld’s nonvanishing theorem [Gr84,
Appendix] shows that $K_{p,1}(C;L)\neq 0$ for $1\leq p\leq\deg
L-g-\operatorname{gon}(C)$. To prove the theorem, it is sufficient to prove
that $K_{\deg L-g-\operatorname{gon}(C)+1,1}(C;L)=0$. By the duality theorem
[Gr84, Theorem 2.c.6],
$K_{\deg
L-g-\operatorname{gon}(C)+1,1}(C;L)=K_{\operatorname{gon}(C)-2,1}(C,\omega_{C};L)^{\vee}.$
Notice that $\omega_{C}$ is $(\operatorname{gon}(C)-2)$-very ample. Recall
that $B$ is $p$-very ample if the restriction map on global sections
$H^{0}(B)\rightarrow H^{0}(B|_{\xi})$ is surjective for every effective
divisor $\xi$ of degree $p+1$, (in other words, $\xi$ imposes independent
conditions on the global sections of $B$). As in [EL15] and [Ra16], it is
natural to study more generally vanishing of $K_{p,1}(C,B;L)$ when $B$ is a
$p$-very ample line bundle and $\deg L\geq\deg B+2g-2$. The main result of
[Ra16] says that if $H^{1}(C,L\otimes B^{-1})=0$, then $K_{p,1}(C,B;L)=0$. For
our purpose, we only need to consider the case that $L=B\otimes\omega_{C}$.
Theorem 1.1 can be deduced from the following:
###### Theorem 1.2.
Let $C$ be a nonsingular projective curve of genus $g\geq 0$, $B$ be a
$p$-very ample line bundle on $C$, and $L:=B\otimes\omega_{C}$.
1. (1)
If $h^{0}(B)\geq p+3$, then $K_{p,1}(C,B;L)=0$.
2. (2)
If $h^{0}(B)=p+2$, then $K_{p,1}(C,B;L)=S^{p}H^{0}(\omega_{C})$.
3. (3)
If $h^{0}(B)=p+1$, then $K_{p,1}(C,B;L)=0$.
The idea to prove the theorem is to use the kernel bundles on the symmetric
products of the curve. We follow the approach introduced by Voisin [Voi1,
Voi2] and then used by Ein–Lazarsfeld [EL15], Rathmann [Ra16], and many others
to conduct a computation of Koszul cohomology groups on the symmetric products
of the curve. To be concrete, in our case,
$K_{p,1}(C,B;L)=H^{1}(C_{p+1},M_{p+1,B}\otimes N_{p+1,L}),$
where $M_{p+1,B}$ is the kernel bundle of the evaluation map
$H^{0}(C,B)\otimes\mathscr{O}_{C_{p+1}}\to E_{p+1,B}$ of the tautological
bundle $E_{p+1,B}$ and $N_{p+1,L}$ is a line bundle on $C_{p+1}$. More
generally, we establish the following vanishing:
$H^{i}(C_{p+1},\wedge^{k}M_{p+1,B}\otimes N_{p+1,L})=0~{}~{}\text{ for $i>0$}$
when $h^{0}(B)\geq p+k+2$. We hope that our results and methods may shed
lights on the similar problems for higher dimensional varieties.
Acknowledgments. The authors would like to thank Lawrence Ein for suggestions
and comments.
## 2\. Preliminaries
Let us start with setting up notations used throughout the paper. Let $C$ be a
nonsingular projective curve of genus $g$. For any $p\geq 0$, denote by
$C_{p+1}$ the $(p+1)$-th symmetric product. Write $U_{p+1}=C_{p}\times C$ to
be the universal family over $C_{p+1}$. One has a commutative diagram
$\textstyle{U_{p+1}=C_{p}\times C\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sigma_{p+1}}$$\scriptstyle{j}$$\textstyle{C_{p+1}\times
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{p+1}}$$\textstyle{C_{p+1}}$
in which $\pi_{p+1}$ is the projection map, $j$ is an embedding defined by
$j(\xi,x)=(\xi+x,x)$, and $\sigma_{p+1}=\pi_{p+1}|_{U_{p+1}}$ so that
$\sigma_{p+1}(\xi,x)=\xi+x$. Write $pr\colon U_{p+1}\rightarrow C$ to be the
projection map to $C$.
###### Definition 2.1.
Let $B$ be a line bundle on $C$. For $p\geq 0$, define
$E_{p+1,B}=\sigma_{p+1,*}(pr^{*}B)\text{ and }N_{p+1,B}=\det E_{p+1,B}.$
###### Remark 2.2.
For basic properties of the vector bundles $E_{p+1,B}$ and the line bundle
$N_{p+1,B}$, we refer the reader to the paper [ENP20]. Here we mention that
$N_{p+1,B}=S_{p+1,B}(-\delta_{p+1})$, where $S_{p+1,B}$ is the invariant
descend of
$B^{\boxtimes p+1}=\underbrace{B\boxtimes\cdots\boxtimes B}_{\text{$p+1$
times}}$
on $C^{p+1}$ to $C_{p+1}$ under the action of the permutations group
$\mathfrak{S}_{p+1}$ on $C^{p+1}$ and
$\mathscr{O}_{C_{p+1}}(-\delta_{p+1})=N_{p+1,\mathscr{O}_{C}}$.
Let $B$ be a $p$-very ample line bundle on $C$. As the fiber of $E_{p+1,B}$
over $\xi\in C_{p+1}$ is $H^{0}(B|_{\xi})$, the evaluation map
$H^{0}(B)\otimes\mathscr{O}_{C_{p+1}}\rightarrow E_{p+1,B}$ on global sections
is surjective. Define $M_{p+1,B}$ to be the kernel bundle of the evaluation
map. We obtain a short exact sequence
$0\longrightarrow M_{p+1,B}\longrightarrow
H^{0}(B)\otimes\mathscr{O}_{C_{p+1}}\longrightarrow E_{p+1,B}\longrightarrow
0.$
The following vanishing theorem about the kernel bundle $M_{p+1,B}$ is an
immediate consequence of Ranthmann’s vanishing theorem on Cartesian products
of the curve.
###### Proposition 2.3.
Let $B$ be a $p$-very ample line bundle on $C$, and $L$ be a globally
generated line bundle on $C$ such that $h^{1}(L)=h^{1}(L\otimes B^{-1})=0$.
Then one has
$H^{k}(C_{p+1},\wedge^{m}M_{p+1,B}\otimes N_{p+1,L})=0,\text{ for all
}k>0,m>0.$
###### Proof.
By [Ra16, Theorem 3.1], one has the vanishing
$H^{k}(C^{p+1},q^{*}(\wedge^{m}M_{p+1,B}\otimes N_{p+1,L}))=0\text{ for all
}k>0,m>0,$
where $q\colon C^{p+1}\rightarrow C_{p+1}$ is the quotient map. Since $q$ is
finite, $\mathscr{O}_{C_{p+1}}$ is a direct summand of
$q_{*}\mathscr{O}_{C^{p+1}}$. Thus by projection formula,
$\wedge^{m}M_{p+1,B}\otimes N_{p+1,L}$ is a direct summand of
$q_{*}(q^{*}(\wedge^{m}M_{p+1,B}\otimes N_{p+1,L}))$, from which the result
follows. ∎
Next we prove a crucial property of kernel bundle $M_{p+1,B}$, which is
important for us to use the inductive argument.
###### Proposition 2.4.
Let $B$ be a $p$-very ample line bundle. There is a short exact sequence
$0\longrightarrow\sigma^{*}_{p+1}M_{p+1,B}\longrightarrow
M_{p,B}\boxtimes\mathscr{O}_{C}\longrightarrow(\mathscr{O}_{C_{p}}\boxtimes
B)(-U_{p})\longrightarrow 0.$
###### Proof.
Denote by $\alpha\colon
M_{p,B}\boxtimes\mathscr{O}_{C}\rightarrow(\mathscr{O}_{C_{p}}\boxtimes
B)(-U_{p})$ the morphism appeared on the right hand side of the sequence. We
first show that it is surjective. Indeed, choose any $\xi\in C_{p}$, and
consider the fiber $C=\\{\xi\\}\times C\subseteq C_{p}\times C$ over $\xi$.
Restricting $\alpha$ to this fiber yields the evaluation map
$\alpha_{\xi}:H^{0}(B(-\xi))\otimes\mathscr{O}_{C}\longrightarrow B(-\xi).$
Since $B$ is $p$-very ample and $\xi$ has degree $p$, it follows that
$B(-\xi)$ is $0$-very ample and thus globally generated. Hence $\alpha_{\xi}$
is surjective. This means that $\alpha$ is surjective.
Next we consider the following fiber product diagram
$\begin{CD}C_{p}\times C\times C@>{\bar{\sigma}}>{}>C_{p+1}\times C\ \supseteq
U_{p+1}\\\ @V{\bar{\pi}}V{}V@V{}V{\pi_{p+1}}V\\\ U_{p+1}=C_{p}\times
<EMAIL_ADDRESS>
On $C_{p}\times C\times C$, we have two divisors $D_{0}$ and $D_{1}$ defined
in the way that $D_{0}$ is the image of
$C_{p}\times C\longrightarrow C_{p}\times C\times
C,\quad(\xi,x)\longmapsto(\xi,x,x),$
and $D_{1}$ is the image of
$C_{p-1}\times C\times C\longrightarrow C_{p}\times C\times
C,\quad(\xi,y,x)\longmapsto(\xi+x,y,x).$
Observe that
$\bar{\sigma}^{*}U_{p+1}=D_{0}+D_{1}\text{ and }D_{0}\cap D_{1}=C_{p-1}\times
C.$
It is easy to check that
$\sigma^{*}_{p+1}M_{p+1,B}=\bar{\pi}_{*}(pr^{*}B(-D_{0}-D_{1})\text{ and
}M_{p,B}\boxtimes\mathscr{O}=\bar{\pi}_{*}pr^{*}B(-D_{1}),$
where $pr:C_{p}\times C\times C\rightarrow C$ is the projection to the right
hand side component $C$. Now we can form a short exact sequence on
$C_{p}\times C\times C$,
$0\longrightarrow\mathscr{O}(-D_{0}-D_{1})\longrightarrow\mathscr{O}(-D_{1})\longrightarrow\mathscr{O}_{D_{0}}(-D_{1})\longrightarrow
0.$
Note that $\mathscr{O}_{D_{0}}(-D_{1})=\mathscr{O}_{C_{p}\times C}(-U_{p})$.
Tensoring the short exact sequence with $pr^{*}B$ and then pushing it down to
$C_{p}\times C$, we obtain the desired short exact sequence. ∎
###### Remark 2.5.
The proof above shows that for any line bundle $B$ (not necessarily $p$-very
ample), one has a short exact sequence
$0\longrightarrow
pr^{*}B(-U_{p})\longrightarrow\sigma^{*}_{p+1}E_{p+1,B}\longrightarrow
E_{p,B}\boxtimes\mathscr{O}_{C}\longrightarrow 0$
on the universal family $U_{p+1}$.
## 3\. Proofs of Main Results
In this section, we prove the main results of the paper – Theorems 1.1 and
1.2. We keep using the notations introduced in Section 2. On the universal
family $U_{p+1}$, consider the short exact sequence
(3.0.1)
$0\longrightarrow\mathscr{O}_{U_{p+1}}\longrightarrow\mathscr{O}_{U_{p+1}}(U_{p})\longrightarrow\mathscr{O}_{U_{p}}(U_{p})\longrightarrow
0$
associated to the divisor $U_{p}$. The normal sheaf
$\mathscr{O}_{U_{p}}(U_{p})$ of $U_{p}$ in $U_{p+1}$ can be expressed as
$\mathscr{O}_{U_{p}}(U_{p})\cong(\mathscr{O}_{C_{p-1}}\boxtimes\omega^{-1}_{C})(U_{p-1}).$
Let $L$ be a line bundle on $C$. Tensoring $pr^{*}L$ with the short exact
sequence (3.0.1), we obtain a short exact sequence
$0\longrightarrow\mathscr{O}_{C_{p}}\boxtimes
L\longrightarrow(\mathscr{O}_{C_{p}}\boxtimes
L)(U_{p})\longrightarrow(\mathscr{O}_{C_{p-1}}\boxtimes
L\otimes\omega^{-1}_{C})(U_{p-1})\longrightarrow 0$
on $U_{p+1}$. Pushing it down to $C_{p}$ by the projection map
$\pi_{p}:U_{p+1}\rightarrow C_{p}$ yields a connecting map $\delta$ in the
associated long exact sequence
$0\longrightarrow
H^{0}(L)\otimes\mathscr{O}_{C_{p}}\longrightarrow\pi_{p,*}((\mathscr{O}_{C_{p}}\boxtimes
L)(U_{p}))\longrightarrow\sigma_{p,*}((\mathscr{O}_{C_{p-1}}\boxtimes
L\otimes\omega^{-1}_{C})(U_{p-1}))\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}\cdots\hskip
56.9055pt$ $\hskip
199.16928pt\cdots\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}H^{1}(L)\otimes\mathscr{O}_{C_{p}}\longrightarrow
R^{1}\pi_{p,*}((\mathscr{O}_{C_{p}}\boxtimes L)(U_{p}))\longrightarrow 0,$
where $\sigma_{p}$ is the restriction of $\pi_{p}$ onto the divisor $U_{p}$.
To understand the connecting map $\delta$, we consider its dual map
$\delta^{\vee}$ by applying
$\operatorname{\mathscr{H}om}(-,\mathscr{O}_{C_{p+1}})$. It is easy to
calculate that
$\big{(}\sigma_{p,*}((\mathscr{O}_{C_{p-1}}\boxtimes
L\otimes\omega^{-1}_{C})(U_{p-1}))\big{)}^{\vee}=\sigma_{p,*}(\mathscr{O}_{C_{p-1}}\boxtimes
L^{-1}\otimes\omega_{C})=E_{p,L^{-1}\otimes\omega_{C}}.$
Then the dual map $\delta^{\vee}$ turns out to be the evaluation map
$H^{0}(L^{-1}\otimes\omega_{C})\otimes\mathscr{O}_{C_{p}}\stackrel{{\scriptstyle\delta^{\vee}}}{{\longrightarrow}}E_{p,L^{-1}\otimes\omega_{C}}.$
We shall only need the special case that $L=\omega_{C}$. In this case, the map
$\delta^{\vee}$ splits $E_{p,\mathscr{O}_{C}}$ by the trace map. As a
consequence of the splitting, we have
$(\sigma_{p,*}\mathscr{O}_{U_{p}})^{\vee}\cong\sigma_{p,*}(\mathscr{O}_{U_{p}}(U_{p-1}))\cong\mathscr{O}_{C_{p}}\oplus\mathscr{K}_{p},$
where the direct summand $\mathscr{K}_{p}$ is the kernel sheaf of the
connecting map $\delta$ fitting into a short exact sequence
$0\longrightarrow
H^{0}(\omega_{C})\otimes\mathscr{O}_{C_{p}}\longrightarrow\pi_{p,*}((\mathscr{O}_{C_{p}}\boxtimes\omega_{C})(U_{p}))\longrightarrow\mathscr{K}_{p}\longrightarrow
0.$
###### Theorem 3.1.
Let $B$ be a $p$-very ample line bundle on $C$. Consider a line bundle
$L:=B\otimes\omega_{C}$. Suppose that $h^{0}(B)\geq p+k+2$ for $k\geq 1$. Then
(3.1.1)
$H^{i}(U_{p+1},\sigma^{*}_{p+1}(\wedge^{k}M_{p+1,B})\otimes(N_{p,L}\boxtimes
L))=0\quad\text{ for }i>0.$
As a consequence, one has
$H^{i}(C_{p+1},\wedge^{k}M_{p+1,B}\otimes N_{p+1,L})=0\quad\text{ for }i>0.$
###### Proof.
First observe that by [ENP20, Lemma 3.5],
$\mathscr{O}_{C_{p+1}}(-\delta_{p+1})$ is a direct summand of the vector
bundle
$\sigma_{p+1,*}(\mathscr{O}_{C_{p}}(-\delta_{p})\boxtimes\mathscr{O}_{C})$.
Thus the bundle
$\sigma_{p+1,*}(\sigma_{p+1}^{*}(\wedge^{k}M_{p+1,B})\otimes(N_{p,L}\boxtimes
L))\cong\wedge^{k}M_{p+1,B}\otimes
S_{p+1,L}\otimes\sigma_{p+1,*}(\mathscr{O}_{C_{p}}(-\delta_{p})\boxtimes\mathscr{O}_{C})$
contains $\wedge^{k}M_{p+1,B}\otimes N_{p+1,L}$ as a direct summand. Since
$\sigma_{p+1}$ is a finite map, the second vanishing statement in the theorem
would follow from the first one. Thus in the sequel, it suffices to show the
first vanishing statement (3.1.1).
To this end, we use the short exact sequence in Lemma 2.4 to yield a locally
free resolution of $\sigma^{*}_{p+1}(\wedge^{k}M_{p+1,B})$ as follows:
$\cdots\longrightarrow(\wedge^{k+2}M_{p,B}\boxtimes
B^{-2})(2U_{p})\longrightarrow(\wedge^{k+1}M_{p,B}\boxtimes
B^{-1})(U_{p})\longrightarrow\sigma^{*}_{p+1}(\wedge^{k}M_{p+1,B})\longrightarrow
0.$
Tensoring it with $N_{p,L}\boxtimes L$ gives rise to a resolution
$\cdots\longrightarrow(\wedge^{k+2}M_{p,B}\otimes N_{p,L})\boxtimes(L\otimes
B^{-2})(2U_{p})\longrightarrow(\wedge^{k+1}M_{p,B}\otimes
N_{p,L})\boxtimes(L\otimes B^{-1})(U_{p})\longrightarrow\cdots\hskip
56.9055pt$ $\hskip
227.62204pt\cdots\longrightarrow\sigma^{*}_{p+1}(\wedge^{k}M_{p+1,B})\otimes(N_{p,L}\boxtimes
L)\longrightarrow 0.$
We make the following claim:
###### Claim 3.1.2.
One has
$R^{t}pr_{*}\Big{(}(\wedge^{k+j}M_{p,B}\otimes N_{p,L})\boxtimes(L\otimes
B^{-j})(jU_{p})\Big{)}=0~{}~{}\text{ for }t\geq 1,j\geq 2,$
where $pr\colon U_{p+1}\rightarrow C$ is the projection map.
Proof of Claim. For a point $x\in C$, the restriction of the sheaf
$(\wedge^{k+j}M_{p,B}\otimes N_{p,L})\boxtimes(L\otimes B^{-j})(jU_{p})$ onto
the fiber $pr^{-1}(x)\cong C_{p}$ equals $\wedge^{k+j}M_{p,B}\otimes
N_{p,L(jx)}$, and $H^{t}(\wedge^{k+j}M_{p,B}\otimes N_{p,L(jx)})=0$ for $t>0$
by Proposition 2.3. Thus the claimed vanishing holds by base change.
By the claim above and using Larry spectral sequence
$H^{s}(R^{t}pr_{*}((\wedge^{k+j}M_{p,B}\otimes N_{p,L})\boxtimes(L\otimes
B^{-j})(jU_{p})))\Rightarrow H^{s+t}(U_{p+1},(\wedge^{k+j}M_{p,B}\otimes
N_{p,L})\boxtimes(L\otimes B^{-j})(jU_{p})),$
we see that
$H^{i}(U_{p+1},(\wedge^{k+j}M_{p,B}\otimes N_{p,L})\boxtimes(L\otimes
B^{-j})(jU_{p}))=0,\text{ for }i\geq 2,j\geq 2.$
Thus chasing through the resolution of
$\sigma^{*}_{p+1}(\wedge^{k}M_{p+1,B})\otimes(N_{p,L}\boxtimes L)$, in order
to prove the vanishing (3.1.1), the only left thing is to show the case when
$j=1$, i.e., to show
(3.1.3) $H^{i}(U_{p+1},(\wedge^{k+1}M_{p,B}\otimes
N_{p,L})\boxtimes\omega_{C}(U_{p}))=0,$
where we use the fact $L\otimes B^{-1}\cong\omega_{C}$. To do this, we tensor
$(\wedge^{k+1}M_{p,B}\otimes N_{p,L})\boxtimes\omega_{C}$ with the short exact
sequence (3.0.1). Pushing down the resulting sequence to $C_{p}$ by the
projection map $\pi_{p}:U_{p+1}\rightarrow C_{p}$, we obtain a long exact
sequence
$0\longrightarrow\wedge^{k+1}M_{p,B}\otimes N_{p,L}\otimes
H^{0}(\omega_{C})\longrightarrow\wedge^{k+1}M_{p,B}\otimes
N_{p,L}\otimes\pi_{p,*}(\mathscr{O}_{C_{p}}\boxtimes\omega_{C}(U_{p}))\longrightarrow\ldots\hskip
56.9055pt$ $\hskip 71.13188pt\ldots\longrightarrow\wedge^{k+1}M_{p,B}\otimes
N_{p,L}\otimes\sigma_{p,*}(\mathscr{O}_{U_{p}}(U_{p-1}))\stackrel{{\scriptstyle\delta}}{{\longrightarrow}}\wedge^{k+1}M_{p,B}\otimes
N_{p,L}\longrightarrow\cdots$ $\hskip
227.62204pt\cdots\stackrel{{\scriptstyle\phantom{b}}}{{\longrightarrow}}R^{1}\pi_{p,*}(\wedge^{k+1}M_{p,B}\otimes
N_{p,L}\boxtimes\omega_{C}(U_{p}))\longrightarrow 0.$
As in the discussion located before the theorem, the connecting map $\delta$
splits. This means that $R^{1}\pi_{p,*}(\wedge^{k+1}M_{p,B}\otimes
N_{p,L}\boxtimes\omega_{C}(U_{p}))=0$ and $\wedge^{k+1}M_{p,B}\otimes N_{p,L}$
is a direct summand of $\wedge^{k+1}M_{p,B}\otimes
N_{p,L}\otimes\sigma_{p,*}(\mathscr{O}_{U_{p}}(U_{p-1}))$. Thus we reduce the
vanishing (3.1.3) to showing the vanishing
(3.1.4) $H^{i}(C_{p},\wedge^{k+1}M_{p,B}\otimes
N_{p,L}\otimes\sigma_{p,*}(\mathscr{O}_{U_{p}}(U_{p-1})))=0.$
Observe that
$N_{p,L}\otimes\sigma_{p,*}(\mathscr{O}_{U_{p}}(U_{p-1}))=\sigma_{p,*}(N_{p-1,L}\boxtimes
L).$
By projection formula, the vanishing (3.1.4) would follow from the following
vanishing
(3.1.5)
$H^{i}(U_{p},\sigma^{*}_{p}(\wedge^{k+1}M_{p,B})\otimes(N_{p-1,L}\boxtimes
L))=0.$
Repeating this argument and noticing that $B$ is $(p-1)$-very ample with
$h^{0}(B)\geq(p-1)+(k+1)+2$, we finally reduce the problem to showing the
vanishing
$H^{i}(C,\wedge^{k+p}M_{B}\otimes L)=0,$
Here we write $M_{B}=M_{1,B}$ for simplicity. The only nontrivial case is when
$i=1$. Write $b=\operatorname{rank}M_{B}$ and notice that $\det
M^{\vee}_{B}=B$. By Serre duality,
$H^{1}(C,\wedge^{k+p}M_{B}\otimes L)\cong H^{0}(C,\omega_{C}\otimes\det
M^{\vee}_{B}\otimes\wedge^{b-1-k-p}M_{B}\otimes
L^{-1})^{\vee}=H^{0}(C,\wedge^{b-1-k-p}M_{B})^{\vee}.$
Now as $\wedge^{b-1-k-p}M_{B}$ is a direct summand of $\otimes^{b-1-k-p}M_{B}$
and the latter has no global sections, we conclude
$H^{1}(C,\wedge^{k+p}M_{B}\otimes L)=0$ as desired. This completes the proof.
∎
###### Proposition 3.2.
Let $B$ be a $p$-very ample line bundle on a curve $C$. Consider a line bundle
$L=B\otimes\omega_{C}$.
1. (1)
If $h^{0}(B)=p+k+1$ for $k\geq 1$. Then
$H^{i}(C_{p+1},\wedge^{k}M_{p+1,B}\otimes
N_{p+1,L})=H^{i}(C_{p+1},S_{p+1,\omega_{C}})=S^{p+1-i}H^{0}(\omega_{C}).$
2. (2)
If $h^{0}(B)=p+k$ for $k\geq 1$, then $\wedge^{k}M_{p+1,B}=0$, and therefore
$H^{i}(C_{p+1},\wedge^{k}M_{p+1,B}\otimes N_{p+1,L})=0.$
###### Proof.
For (1), since $M_{p+1,B}$ has rank $k$ and $\wedge^{k}M_{p+1,B}\cong
N^{-1}_{p+1,B}\cong S_{p+1,B^{-1}}(\delta_{p+1})$, we compute
$\wedge^{k}M_{p+1,B}\otimes N_{p+1,L}\cong S_{p+1,L\otimes B^{-1}}\cong
S_{p+1,\omega_{C}}.$
The result then follows from [ENP20, Lemma 3.7]. For (2), since $M_{p+1,B}$
has rank $k-1$, the result follows immediately. ∎
We will only need Theorem 3.1 and Proposition 3.2 for the case $k=1$. In the
following proposition, we classify when a $p$-very ample line bundle $B$ can
have $h^{0}(B)\leq p+2$.
###### Proposition 3.3.
Let $B$ be a $p$-very ample line bundle on $C$, and $p\geq 0$.
1. (1)
$h^{0}(B)=p+1$ if and only if either $p=0$ and $B=\mathscr{O}_{C}$ or $p\geq
1$, $C=\mathbf{P}^{1}$ and $B=\mathscr{O}_{\mathbf{P}^{1}}(p)$.
2. (2)
$h^{0}(B)=p+2$ if and only if one of the following cases holds.
1. (i)
$g=0$, $p\geq 0$ and $B=\mathscr{O}_{\mathbf{P}^{1}}(p+1)$.
2. (ii)
$g=1$, $p\geq 0$ and $\deg B=p+2$.
3. (iii)
$g\geq 2$, either $p=0$ and $B$ is a base point free pencil, or $p=1$ and
$C\subseteq\mathbf{P}(H^{0}(B))$ is a plane curve of degree $\geq 4$.
###### Proof.
(1) If $p=0$, then $B$ is a globally generated line bundle with
$H^{0}(C,B)=1$. Then $B=\mathscr{O}_{C}$ since the only section of $B$ is
nowhere vanishing. Assume $p\geq 1$, so $B$ is very ample and gives an
embedding of $C$ into the space $\mathbf{P}^{p}=\mathbf{P}(H^{0}(B))$. As $B$
is $p$-very ample, any $p+1$ points of $C$ will span the whole space
$\mathbf{P}^{p}$, which means that the degree of $C$ would be smaller than
$p$. But $C$ is also nondegenerate in $\mathbf{P}^{p}$ and thus has degree
$\geq p$. Hence $C$ has degree exactly $p$, and therefore, it is a rational
normal curve.
(2) Since (i) and (ii) are obvious, we only need to prove (iii). If $p=0$,
then $B$ is a base point free pencil. Assume that $p\geq 2$. Take $p-1$ points
$x_{1},\ldots,x_{p-1}$ of $C$, and put $D:=x_{1}+\cdots+x_{p-1}$. Since $B$ is
a $p$-very ample, we see that $B(-D)$ is very ample with $h^{0}(B(-D))=3$ and
$h^{1}(B(-D+x_{1}))=h^{1}(B(-D))=h^{1}(B)$. This means $C$ is a plane curve of
some degree $d\geq 4$ embedded by $B(-D)$ into $\mathbf{P}^{2}$, and thus, the
canonical line bundle $\omega_{C}$ has the form $\omega_{C}=(B(-D))^{d-3}$ by
the adjunction formula. By duality, the equality
$h^{1}(B(-D+x_{1}))=h^{1}(B(-D))$ is the same as the equality
$h^{0}((B(-D))^{d-4}(-x_{1}))=h^{0}((B(-D))^{d-4})$, which is impossible
because $B(-D)$ is very ample. Thus we conclude $p=1$ and
$C\subseteq\mathbf{P}(H^{0}(B))$ is a plane curve of degree $\geq 4$. ∎
Recall that the gonality of $C$ captures the positivity of the canonical line
bundle $\omega_{C}$. More precisely, $\operatorname{gon}(C)\geq p+2$ if and
only if $\omega_{C}$ is $p$-very ample. In particular,
$\operatorname{gon}(C)=\max\\{p+2\mid\text{$\omega_{C}$ is $p$-very
ample}\\}.$
We can compare the gonality with the genus. The following proposition may be
well-known.
###### Corollary 3.4.
Assume that $g\geq 2$. Then $g\geq\operatorname{gon}(C)$, and the equality
holds if and only if either $g=2$ or $C$ is a plane quartic curve.
###### Proof.
Since $g\geq 2$, it follows that $\operatorname{gon}(C)\geq 2$. Write
$\operatorname{gon}(C)=p+2$. Then $\omega_{C}$ is $p$-very ample. Applying
Proposition 3.3 to the case $B=\omega_{C}$, we see that $g\geq p+2$ and the
equality holds if either $g=2$ (i.e., $g=\operatorname{gon}(C)=2$), or $C$ is
a plane curve of $g=3$ which is a plane quartic curve (i.e.,
$g=\operatorname{gon}(C)=3$). ∎
###### Proof of Theorem 1.2.
In (1) and (2), $B$ is ample and thus $h^{1}(L)=0$. This implies
$h^{1}(C_{p+1},N_{p+1,L})=0$ and thus [EL15, Lemma 1.1] yields
$K_{p,1}(C,B;L)=H^{1}(C_{p+1},M_{p+1,B}\otimes N_{p+1,L}).$
So the assertion (1) follows from Theorem 3.1 by taking $k=1$, and the
assertion (2) follows from Proposition 3.2 by taking $k=1$. For the assertion
(3), if $p=0$, then $B=\mathscr{O}_{C}$ and then $K_{0,1}(C;\omega_{C})=0$ by
definition of Koszul cohomology group. If $p\geq 1$, then by Proposition 3.3,
$C=\mathbf{P}^{1}$ and $B=\mathscr{O}_{\mathbf{P}^{1}}(p)$. Then
$K_{p,1}(\mathbf{P}^{1},\mathscr{O}_{\mathbf{P}^{1}}(p);\mathscr{O}_{\mathbf{P}^{1}}(p-2))=0$
by a direct computation. ∎
###### Corollary 3.5.
Assume that $g\geq 2$. Let $B$ be a $p$-very ample line bundle on $C$, and $L$
be a line bundle on $C$. Suppose that $\deg(L\otimes B^{-1})\geq 2g-2$. Then
one has
$K_{p,1}(C,B;L)=0$
unless $L=B\otimes\omega_{C}$ and either $(1)$ $p=0$ and $B$ is a base point
free pencil, or $(2)$ $p=1$ and $C\subseteq\mathbf{P}(H^{0}(B))$ is a plane
curve. In the exceptional cases, $K_{p,1}(C,B;L)\neq 0$ but
$K_{p-1,1}(C,B;L)=0$.
###### Proof.
If $L\otimes B^{-1}\neq\omega_{C}$, then $h^{1}(L\otimes B^{-1})=0$ so that
one can use Rathmann’s theorem [Ra16, Theorem 1.1] to get the desired result.
Assume that $L\otimes B^{-1}=\omega_{C}$. By Theorem 1.2, $K_{p,1}(C,B;L)=0$
if $h^{0}(B)\neq p+2$, and $K_{p,1}(C,B;L)\neq 0$ if $h^{0}(B)=p+2$. In the
latter case, $K_{p-1,1}(C,B;L)=0$ by Theorem 1.2 since $B$ is $(p-1)$-very
ample and $h^{0}(B)=(p-1)+3$. However, if $h^{0}(B)=p+2$, then Proposition 3.3
shows that either $(1)$ $p=0$ and $B$ is a base point free pencil, or $(2)$
$p=1$ and $C\subseteq\mathbf{P}(H^{0}(B))$ is a plane curve. ∎
###### Proof of Theorem 1.1.
By Green–Lazarsfeld’s nonvanishing theorem [Gr84, Appendix] and the duality
theorem [Gr84, Theorem 2.c.6], we only need to know when
$K_{\operatorname{gon}(C)-2,1}(C,\omega_{C};L)=0$ vanishes. As $\omega_{C}$ is
$(\operatorname{gon}(C)-2)$-very ample, the theorem follows from Corollary
3.5. ∎
## References
* [AN10] Marian Aprodu and Jan Nagel. Koszul Cohomology and Algebraic Geometry. University Lecture Series, 52, Amer. Math. Soc., Providence, RI, 2010.
* [EL15] Lawrence Ein and Robert Lazarsfeld. The gonality conjecture on syzygies of algebraic curves of large degree. Publ. Math. Inst. Hautes Études Sci., 122:301–313, 2015.
* [ENP20] Lawrence Ein, Wenbo Niu, and Jinhyung Park. Singularities and syzygies of secant varieties of nonsingular projective curves. Invent. Math., 222(2):615–665, 2020.
* [Gr84] Mark Green. Koszul cohomology and the geometry of projective varieties. J. Differential Geom., 19(1):125–171, 1984.
* [GL86] Mark Green and Robert Lazarsfeld. On the projective normality of complete linear series on an algebraic curve. Invent. Math., 83(1):73–90, 1986.
* [GL88] Mark Green and Robert Lazarsfeld. Some results on the syzygies of finite sets and algebraic curves. Compositio Math., 67(3):301–314, 1988.
* [La04] Robert Lazarsfeld. Positivity in algebraic geometry. I. Classical setting: line bundles and linear series. II. Positivity for vector bundles, and multiplier ideals. Springer-Verlag, 2004
* [Ra16] Juergen Rathmann. An effective bound for the gonality conjecture.
* [Voi1] Claire Voisin. Green’s generic syzygy conjecture for curves of even genus lying on a K3 surface. J. Eur. Math. Soc. (JEMS) 4 (2002), 363–404.
* [Voi2] Claire Voisin. Green’s canonical syzygy conjecture for generic curves of odd genus. Compositio Math. 141 (2005), 1163–1190.
|
# Gender Biases in Error Mitigation by Voice Assistants
Amama Mahmood<EMAIL_ADDRESS>Johns Hopkins University3400 N. Charles
StBaltimoreMarylandUSA21218 and Chien-Ming Huang<EMAIL_ADDRESS>Johns Hopkins University3400 N. Charles StBaltimoreMarylandUSA21218
(2018)
###### Abstract.
Commercial voice assistants are largely feminized and associated with
stereotypically feminine traits such as warmth and submissiveness. As these
assistants continue to be adopted for everyday uses, it is imperative to
understand how the portrayed gender shapes the voice assistant’s ability to
mitigate errors, which are still common in voice interactions. We report a
study (N=40) that examined the effects of voice gender (feminine, ambiguous,
masculine), error mitigation strategies (apology, compensation) and
participant’s gender on people’s interaction behavior and perceptions of the
assistant. Our results show that AI assistants that apologized appeared warmer
than those offered compensation. Moreover, male participants preferred
apologetic feminine assistants over apologetic masculine ones. Furthermore,
male participants interrupted AI assistants regardless of perceived gender
more frequently than female participants when errors occurred. Our results
suggest that the perceived gender of a voice assistant biases user behavior,
especially for male users, and that an ambiguous voice has the potential to
reduce biases associated with gender-specific traits.
error mitigation, smart speaker, gender stereotypes, ambiguous voices, voice
assistance
††copyright: acmcopyright††journalyear: 2018††doi: XXXXXXX.XXXXXXX††ccs:
Human-centered computing Empirical studies in HCI††ccs: Computing
Methodologies Artificial intelligence
Figure 1. In this project, we investigate how the portrayed gender of a voice
assistant, error mitigation strategy, and participant gender impact people’s
perceptions of and behavior towards the AI assistant in a collaborative task.
This figure illustrates that users react to errors made by voice assistants,
to apology or compensation offered by voice assistants to mitigate negative
effects of those errors, and to other aspects of interaction in general.
[Voice assistants elicit non-verbal reactions from user in different
conversational scenarios]The figure consists of images of users reacting to
three different conversational scenarios. There are total of 9 pictures. Each
picture has a person sitting on a chair conversing with the smart speaker
placed on a table in front of them. The The first three pictures show
reactions of people to errors made by AI agent. The reactions are laughter by
a male participant in the first picture, laughter by a female participant in
the second picture, and a male participant frowning in the third picture. Next
two pictures show reactions to mitigation statements that are; a frown by a
male participant and laughter by a female participant. Next 4 pictures show
reactions to other aspects of interactions which are; a female participant
laughing, a female participant nodding, a female participant smiling, and a
male participant raising eyebrows.
## 1\. Introduction
Intelligent voice assistants are increasingly being used for a variety of
assistive tasks in various personal and professional settings. For example,
voice assistants, such as Alexa and Siri, are capable of supporting people
while performing functional tasks, such as retrieving information, setting
alarms and reminders, sending texts, and making calls; and engaging in social
tasks, such as telling jokes, and playing music and games (e.g., “Simon
says”). Moreover, voice assistants are positioned to play an essential role in
future collaborative learning and work (Baker et al., 2021; Paraiso and
Barthès, 2006; Sáiz-Manzanares et al., 2020; Schmid et al., 2022).
These modern voice assistants are powered by data-driven deep learning models
to recognize speech commands and user intent to ensure smooth collaboration.
While becoming powerful, these data-driven models are susceptible to errors
(Pearl, 2016), and when not mitigated and managed, these errors are likely to
result in users distrusting and halting the collaboration. As a result,
growing research has studied various strategies for mitigating AI errors. For
instance, apology as an error mitigation strategy from an agent makes it
likeable (Mahmood et al., 2022) and perceived as polite (Lee et al., 2010),
whereas compensation increases users’ satisfaction after error recovery (Lee
et al., 2010).
On top of experiencing imperfect interactions with voice assistants, users
tend to personify (Habler et al., 2019) and characterize (Abercrombie et al.,
2021) these assistants as feminine (UNESCO, 2019). Such gender
characterization is detrimental (for instance, it welcomes sexualised comments
and insults (Curry and Rieser, 2019; Curry et al., 2020)) because it fortifies
women’s amenable and unassertive role in the society (Loideain and Adams,
2020; Guzman, 2016). Existing gender biases due to association of stereotypes
in human-agent interactions (Ahn et al., 2022; Lee et al., 2019) raise the
following research questions that we aim to investigate: “how gender biases
may potentially manifest when users interact with voice assistants,
particularly in the context of erroneous interactions” , and “how the
effectiveness of error mitigation strategies may be modulated by the portrayed
gender of voice assistants”. The growing availability of (imperfect) voice
assistants in digital services and the rooted gender biases in our society
propel our inquiry into the relationships among the portrayed gender of a
voice assistant, its mitigation strategy for errors, and the gender of the
user when the user is interacting with a voice assistant.
In this work, we experimentally manipulated the gender of voice assistant
(ambiguous, feminine, and masculine) and error mitigation strategy (apology
and compensation), and studied how these factors along with participant gender
(female (women) and male (men)) may influence people’s perceptions of and
behavior toward erroneous voice assistants. We developed a mock smart speaker
and conducted an in-person experiment where participants interacted with voice
assistants to complete a series of shopping tasks (Figure 1). In this study,
our emphasis is on the “assistance” scenarios, where commercial voice
assistants excel in tasks such as online shopping, setting reminders,
retrieving information, and creating lists. Therefore, we select the online
shopping task as it exemplifies a typical assistance-type task performed by
these voice assistants and has the potential for a longer, and multi-turn
interaction between user and the voice assistant.
Our experiment reveals several findings identifying nuanced dynamics among the
portrayed gender of a voice assistant, mitigation strategy, and participant
gender:
* •
Participants reacted more socially, with a higher number of non-verbal
reactions, to voice assistants that offer apology rather than compensation
when mitigating errors.
* •
Male participants interrupted voice assistants, regardless of their portrayed
gender and use of mitigation strategy, more often than female participants.
* •
Male participants interacted more socially with “apologetic” voice assistants
that were portrayed as feminine compared to their masculine counterparts.
Overall, we also observed that male participants reacted more socially to the
feminine assistants than masculine ones and that apology promotes the
perception of warmth for voice assistants while compensation increases
participants’ satisfaction of service recovery, and perception of assistants’
competence.
## 2\. Background and Related Work
Given that the primary focus of this paper is to explore gender biases, error
mitigation strategies, and their intersection in user interactions with voice
assistants, we first present related work on various error mitigation
strategies for repairing human-agent relationships damaged by inevitable
disruptions. We then provide an overview of previous studies that examine the
stereotypes associated with gendered AI agents, with a specific emphasis on
voice assistants.
### 2.1. Error Mitigation Strategies in Human-Agent Interaction
It is crucial to repair relationships between agents111In our discussion of
prior work, we have defined agents in a broad sense with respect to their
embodiment. These agents encompass a wide range of forms, such as physical
robots, AI aids, virtual avatars, voice assistants, and other similar
entities. and humans after service breakdowns for continued use of technology.
Mitigation strategies can help increase people’s acceptance of faulty agents
(Lee et al., 2010). Moreover, these mitigation strategies can have varying
effects on different dimensions of people’s perceptions of the agents. For
instance, a robot that apologizes for its mistake is considered more likeable
and increases people’s willingness to use it (Cameron et al., 2021); however,
it is not perceived equally capable to a robot that demonstrates the knowledge
of why the error occurred and how the error can be fixed (Cameron et al.,
2021). Within apologies, a neutral apology, as opposed to humorous one, is
found to be more sincere and effective in repairing human-agent relationship
after errors made by smart speakers (Ge et al., 2019). For voice-based
interactions, combining acceptance of blame for the error and seriousness of
tone while apologizing results in better user satisfaction after recovery from
errors (Mahmood et al., 2022); moreover, such agents are perceived to be more
intelligent and likeable compared to the agents that lacked either seriousness
or acceptance of blame (Mahmood et al., 2022).
While apologies create an “emotional shift toward forgiveness” to promote
reconciliation and regain trust during relationship repair in both human-human
interactions (Govier and Verwoerd, 2002) and human-agent interactions (Lee et
al., 2010), other mitigation strategies such as compensation may offer better
user satisfaction while recovering from service failures. For instance,
compensating with a free drink on a robot’s failure to bring correct drink to
the user resulted in higher service recovery satisfaction than acknowledging
and apologizing for the error (Lee et al., 2010). For deeper insights into the
comparative aspects of apology and compensation strategies, we can look into
the decades of research on mitigation strategies in repairing broken human-
human relationships, which has a stronger foundation compared to research on
human-agent interactions. For instance, human interactions in e-commerce show
that the type of trust violation (competence- or integrity-based) also affects
whether apologizing with internal or external attribution of blame, and
compensating equal to or more than the loss would be more effective in
regaining trust (Cui et al., 2018). Even in purely financial settings,
apologies can facilitate preservation of relationships especially in the case
of undercompensation that fails to redress the loss of the affected party;
moreover, not surprisingly, overcompensation also repairs the relationship
better than under or equal compensation (Haesevoets et al., 2013). Since in
human-human interactions, apology is identified as a relational while
compensation is termed as a financial (utilitarian) strategy (Haesevoets et
al., 2013), we may find similar preferences towards apologies and compensation
offered by intelligent agents. In the domain of human-robot interaction, we
see that users’ orientation towards service can affect how the error
mitigation strategies are perceived. People who want to maintain amicable
relationship with service providers (stronger relational orientation) prefer
the apology strategy and have as much of negative impression for compensation
strategy as for no strategy (Lee et al., 2010). However, people who care more
about quality of service than relationship with providers (stronger
utilitarian orientation) find the compensation strategy to be more appealing
(Lee et al., 2010). However, it is unclear how the preference towards apology
and compensation might hold for voice assistants that do not have
anthropomorphic physical embodiment. In this work, we seek to understand how
these two mitigation strategies compare in a voice-based interactions
contextualized in an online shopping task using a smart speaker.
### 2.2. Gender Stereotypes in Human-Agent Interaction
This section aims to provide an overview of the relevant literature on gender
stereotypes and biases during human-agent interactions; in particular, we
review the stereotypes and biases in 1) the broader literature on human-agent
interactions, and 2) specific research focused on voice assistants.
Additionally, we discuss the research regarding the utilization of different
voice variations for assistants as a means of combating gender stereotypes and
biases.
#### 2.2.1. Gender Stereotypes in Human-Robot and Human-AI Interactions
Design of “masculine” and “feminine” AI agents and robots may introduce gender
based stereotypes. The terms “masculine” or “feminine” when referring to
agents and robots encompass various methods that may contribute to their
gendering. These methods may include manipulating their appearance, voice, and
the use of specific names and pronouns in experiment instructions and surveys.
Such gendering can be influenced by one or a combination of these factors.
Thus, when we refer to an agent as “masculine” or “feminine”, it means an
agent that may be perceived to look, sound or act as a stereotypical male or
female.
Prior work in HRI supports that masculine robots are considered better at
stereotypical male tasks while feminine ones are perceived to be friendlier
and warmer (Nomura, 2017). Even with minimal gender markers, such as their
voices or names, people tend to react in alignment with gender stereotypes
seen in human interactions. For instance, robots with feminine names or voices
are expected to be emotionally intelligent (Chita-Tegmark et al., 2019).
Additionally, personality of the agent, perceived as feminine or masculine,
influences perception of trustworthiness, reliability and competence. Context
and task at hand further shape how the gendered agents are perceived in these
dimensions. For instance, a robot with a feminine personality reflected
through agreeable and warm statements may be considered less trustworthy,
reliable, and competent as opposed to one with masculine personality that is
dominant, confident, and assertive, in a male-oriented task (e.g., dispatching
taxis (Kraus et al., 2018)).
The perception of individuals’ competence is often associated with factors
such as intelligence, efficiency, and skill, while warmth is more related to
friendliness, sincerity, and helpfulness (Fiske et al., 2007). Thus,
considering masculine agents to be more competent while assigning higher
warmth to feminine agents (Ahn et al., 2022), depicts association of gender
stereotypes based on agent characteristics. Such biased perceptions also have
implications for consumer preferences when interacting with AI agents in
relation to different types of products. For example, in a chatbot-based
interaction, masculine AI is preferred while recommending utilitarian products
(Ahn et al., 2022) (such as headphones for work from home), which are
practical and require competence from the seller to successfully convince
buyers of their value (Bennett and Hill, 2012). On the contrary, feminine AI
fairs better in promoting hedonic products (related to happiness: drawing
supplies) (Ahn et al., 2022), where warmth in the salesperson’s personality
influences purchasing decisions (Bennett and Hill, 2012).
In summary, various factors such as agent appearance, voice, personality,
task, and context may significantly influence how the perceived gender of an
agent may cause stereotyping. For this study, our focus is primarily on
examining the interaction between gendered voices of AI agents and error
mitigation strategies in assistance-type task scenarios. We aim to understand
how apology (relational/hedonic strategy) and compensation
(financial/utilitarian strategy) interact with gendered voices of AI agents to
influence user’s perceptions.
#### 2.2.2. Stereotypes Associated with Gendering of Voice Assistants
Most voice assistants in our present digital world are gendered as female; for
example, Apple’s Siri, Amazon’s Alexa, and Google voice assistant all have
feminine voices set as default. These agents are intentionally feminized and
marketed as “fembots” to increase their familiarity among users (Phan, 2019;
Strengers and Kennedy, 2021; Woods, 2018). Gendering of commercial voice
assistants poses societal harm because association of female gender by default
to these personal voice assistants reinforces women’s submissive role of
taking and fulfilling orders in the society and welcomes harassment, abuse,
and discriminatory behavior (Loideain and Adams, 2020; Guzman, 2016; Curry et
al., 2020; Curry and Rieser, 2019). In recent work, concerns about
stereotypical assignment of female gender to agent voices have been raised,
and power dynamic between the user and the woman-like voice assistants have
been discussed (Hwang et al., 2019). A report by United Nations (UN) (UNESCO,
2019) states that despite what the creators claim, “nearly all of these
assistants have been feminized—in name, in voice, in patterns of speech and in
personality. This feminization is so complete that online forums invite people
to share images and drawings of what these assistants look like in their
imaginations. Nearly all of the depictions are of young, attractive women.”
The influence of context and task on gender biases. Prior research has
yielded mixed results on the influence of task on gender biases in voice
assistants. On one hand, gender biases exist in the preferences for voice
assistants and are dependent on contextual settings and task at hand. For
example, a masculine voice is often favored for informative tasks, while a
feminine voice is preferred for social interactions (Lee et al., 2019).
Similarly, feminine voice assistant gained more trust in automation system
targeted for home while masculine voice assistant gained more trust in office
setting (Damen and Toh, 2019). Moreover, in health-critical situations, users
exhibit greater trust in feminine voice assistants compared to their masculine
counterparts (Goodman and Mayhorn, 2023). However, on the other hand, type of
task may not always affect the perception of voice assistants. For instance,
no gender biases were seen while performing functional (information retrieval
and mathematics calculations) and social (humours questions) tasks with Siri
(Lee et al., 2021). Such contradictory evidence on how task, especially
assistive tasks such as controlling automated devices (Damen and Toh, 2019)
and retrieving information (Lee et al., 2021), may affect the perception of
voice assistants motivated our choice of an assistance-type task—online
shopping—in our investigation of gender biases in error mitigation by voice
assistants.
The influence of participant gender on gender biases. In addition to context
and task, user’s own gender may shape their perceptions of voice assistants.
For instance, prior work has demonstrated effect of participant’s gender on
their perception of trust (Lee et al., 2021). Participants tend to trust
voices of the same gender more, even when uncertain about the accuracy of the
advice or suggestion being given by voice assistants such as Siri (Lee et al.,
2021). Additionally, female participants have shown a higher level of
skepticism compared to male participants, regardless of whether the voice they
were interacting with belonged to a masculine or feminine assistant (Lee et
al., 2021). On the contrary, when interacting with animated pedagogical
agents, female students showed a preference for a matched gender, whereas male
students rated the female agent more favorably (Ozogul et al., 2013).
Similarly, other studies indicate varied gender preferences (Krämer et al.,
2016; Kim et al., 2007), suggesting existence of potential gender stereotypes
and biases. Therefore, in this work, we also look at the role of gender of
users, in addition to gender of assistants, in association of stereotypes with
voice assistants.
#### 2.2.3. Intervention for Mitigating Gender Stereotypes and Biases
To address the issue of stereotypes and biases associated with gendered
voices, researchers have explored various methods to mitigate their influence.
One approach discussed to mitigate or eliminate the association of stereotypes
with gendered voices is the creation of ambiguous voices through the
manipulation of voice characteristics, such as pitch. Research has shown that
there were no discernible differences in user trust formation between voice
assistants with gendered voices, and those with claimed ambiguous voices
(Tolmeijer et al., 2021); hence, ambiguous voices can potentially be used
commercially in place of gendered assistants as a tool to reduce biases.
However, it is worth mentioning that the gender ambiguous voice employed in
the aforementioned study is unable to completely eradicate implicit biases,
which are still evident similar to the biases associated with the feminine
voice(Tolmeijer et al., 2021), drawing our attention to lack of empirical
evidence for effectiveness of ambiguous voices to culminate gender biases,
implicit or explicit. Hence, in this work, in addition to masculine and
feminine sounding assistants, we also study effects of ambiguous sounding
assistants on people’s perceptions of voice assistants in an online shopping
task using a smart speaker.
#### 2.2.4. Gaps in Methodological Approach for Empirical Studies on Gender
Biases
Within the context of empirical studies on gender biases, a significant gap
exists in the methodological approach employed. Most of prior studies on
exploration of gender stereotypes and biases in voice assistants are based on
online reviews (Phan, 2019), crowd sourced evaluations (Curry and Rieser,
2019), case studies (Woods, 2018), surveys (Obinali, 2019; Curry et al.,
2020), chat bots, and online interfaces (Damen and Toh, 2019; Tolmeijer et
al., 2021; Goodman and Mayhorn, 2023). While there are a few exceptions, such
as studies conducted with phone-based voice assistant Siri (Lee et al., 2021),
or using off-the-shelf speakers controlled by experimenters (Habler et al.,
2019) , these studies primarily depend on subjective evaluations through
questionnaires to gain insights into stereotypes. In our work, we strive to
address the methodological gap by introducing a simulated physical smart
speaker platform through the implementation of the Wizard of Oz (WoZ) paradigm
for the voice assistant. Moreover, we augment the subjective measures with an
analysis of users’ behavioral data to investigate implicit expression of
biases associated with gender stereotypes.
### 2.3. Hypotheses
Based on the prior work on mitigation strategies to repair human-agent
relationship, and understanding various factors that impact gender stereotypes
in human-agent interaction outlined above, we propose the following
hypotheses:
* •
Hypothesis 1. Compensation, as a mitigation strategy, improves perceptions of
service recovery satisfaction and competence more than apology while apology
increases perceptions of warmth of the assistants. Hypothesis 1 is informed by
previous research showing that 1) compensation is a financial strategy while
apology is a relational strategy (Haesevoets et al., 2013), and 2) apologies
make an agent or robot more likeable (Cameron et al., 2021) and appear more
polite (Lee et al., 2010), while compensation results in better service
recovery satisfaction (Lee et al., 2010).
* •
Hypothesis 2. Feminine sounding assistants are perceived warmer than masculine
sounding assistants whereas masculine assistants are considered more
competent. Hypothesis 2 is informed by previous work showing that masculine AI
agent is rated more competent while feminine AI agent is more likeable and
warm (Ahn et al., 2022; Nomura, 2017; Chita-Tegmark et al., 2019). In addition
to testing this hypothesis, we will explore if the gender ambiguous voice can
culminate these biases.
* •
Hypothesis 3. Assistants with feminine voice gain better user perceptions when
they apologize for their mistake while assistants with masculine voice are
perceived more positively when they compensate for their mistake. Hypothesis 3
is formulated based on evidence from prior work in human-human interactions
showing that 1) typically women are more apologetic than men (Turiman et al.,
2013), 2) men are characterized to have authoritative and practical roles
while women are considered better at submissive roles (Powell et al., 2002;
Koenig et al., 2011); and 3) apology is a relational strategy while offering
compensation is a financial (utilitarian) strategy (Haesevoets et al., 2013)
and may be considered as a feminine and a masculine trait, respectively.
Moreover, we explore where gender ambiguous voices would fall on this spectrum
and if ambiguous assistants can reduce gender biases faced by AI agents while
still being perceived as effective.
Lastly, as discussed above, prior work on how participant gender may influence
perception of masculine or feminine agents and robots has shown conflicting
findings (Lee et al., 2021; Mitchell et al., 2011; Siegel et al., 2009). Thus,
in this study we also consider possible influence of participants’ gender on
their perceptions of gendered voice assistants while mitigating AI errors.
Moreover, we conduct this study in the context of an assistance-type task and
contextualize our results in that space.
## 3\. Methods
### 3.1. Study Design and Experimental Task
To test our hypotheses, we conducted an in-person experiment considering three
factors: error mitigation strategy—(apology vs. compensation), gendered voice
of AI assistants—(ambiguous vs. feminine vs. masculine), and participant
gender—(female vs. male). Error mitigation strategy and gendered voice of AI
assistants were within-subjects factors while participant gender was a
between-subjects factor.
Our experiment consisted of six online shopping tasks; for each task,
participants were asked to collaborate with one of the six AI voice assistants
to complete the task. We presented the six tasks as beta testing of a voice
assistant developed for our prototype smart speaker. Each task involved
working with a different voice assistant on a different shopping list. The six
AI assistants were named Galaxy, Dot, Cloud, Mist, Phoenix, Gamma. We picked
non-human and gender neutral names to reduce any biases in participant’s
preconception of the assistant’s gender. The order of the assistant names and
shopping lists was same for all the participants. For instance, the assistant
in the first session is always named Galaxy and the participant ordered items
for a party. Balanced Latin square is used to counterbalance the six
experimental conditions.
In each task, the user was given a list of five items to order through a
speech-based interaction with the AI assistant (Fig. 2). The user would prompt
the assistant to order a specific item and confirm the addition of item to the
cart once the assistant presents the user with the item it found. The
assistant would let the user know after adding the item to the cart. The user
could also ask the assistant to show more options. The AI assistant would make
a mistake on either the second, third, or fourth item on the list. In this
study, we considered intent recognition errors, which are described as
“recognized but incorrectly” (Pearl, 2016). The predetermined errors in this
study were the use of homonyms—words that sound the same but may mean
different things. For example, “flour” could be heard as flour or flower. As
there are only a few possible items, mostly two, associated with the chosen
homonyms, the assistant recovered from the error in the second attempt by
suggesting correct item.
The user could report a mistake by verbally indicating that the item suggested
by the assistant is not the correct one when the assistant asked if it should
add the item to the cart. On indication of error by the user, the assistant
prompted them to try again. On the second try, the assistant fixed the mistake
by indicating the correct item. After fixing the error, the assistant
performed the recovery strategy by initiating either apology or offering
compensation based on the experimental condition. After ordering the five
items on the list, the user was directed to solve a riddle while they “waited
for the items to arrive”. The experimenter handed over the ordered items
printed on a piece of paper with title and images to the user while they are
solving the riddle. The user was prompted to see if the ordered items were
correct. The user could start a return process with the assistant in case any
of the arrived items was incorrect. The assistant would initiate the return
and notify that the item was returned and refunded. It was up to the user if
they wanted to order the required item after returning the incorrect one or
not. The assistant either apologized or offered compensation based on
experimental condition once the return process was complete. Next, we describe
the implementation details of our mock smart speaker used in the experiment.
Figure 2. Experimental setup. The participant interacts with the smart speaker
using voice commands. The wired speaker is connected to the laptop. The LED
lights ring on the speaker is activated by the Arduino once either of the two
applications running on the laptop signals that assistant or participant
speech is detected. The experimenter accesses the web interface on the laptop
to control the speech of the mock smart speaker using TeamViewer on the
Desktop (WoZ). The participant and the experimenter are in the same room
separated by a physical partition.
[Experimental set up explaining WoZ interface and interaction with user.]The
figure shows our experimental set up. To the left of the figure, the
experimenter is shown with the desktop computer. To the right of a physical
partition which separates the participant from the experimenter, the
participant is shown with a wired speaker connected to Arduino and the laptop.
To the very right, a picture of actual participant interacting with the smart
speaker is shown where the person is sitting in front of the speaker which is
placed on a table. The WoZ remote control is indicated between the desktop and
the laptop. The experimenter simulates the assistant speech by controlling the
laptop next to the wired speaker remotely through the desktop computer.
### 3.2. Mock Smart Speaker Setup
We developed a mock smart speaker (Fig. 2) that can be controlled by an
experimenter through Wizard of Oz (WoZ) paradigm. The speaker consisted of
following components:
* •
Arduino controlled LED lights ring for signaling when the assistant was
speaking or listening. LED lights were activated for signaling assistant’s
speech as soon as the experimenter played any audio on behalf of the
assistant. For listening, the the LED light ring was activated on wake words
which were either the name of the assistant, “Hello”, “Hey”, or “Hi” for the
first time the assistant is addressed. Once the interaction started, if any
speech from participant was detected right after the wake work or the
assistant’s turn of speaking (assistant waits for the participant’s response
after talking every time), the LED ring lighted up indicating that the speaker
was listening to them. Each assistant had a different colored ring (blue,
purple, green, pink, orange, and yellow) which was decided by the row of Latin
square the participant was assigned to. We coded this behavior using two
applications in $C\\#$; 1) to detect audio output from the assistant, and 2)
to detect and recognize the participant’s speech. On detection of either
participant or assistant’s speech (as described above), the respective
application will send a signal to Arduino to activate the lights ring. Both
applications are running on the laptop (see Fig. 2).
* •
Wired speakers for playing audio were connected to the laptop (see Fig. 2. The
laptop powered and controlled the wired speaker, and the Arduino. The laptop
was controlled remotely by the experimenter using
TeamViewer222https://www.teamviewer.com/en-us/ on a desktop computer. The
laptop screen was closed so that it is not visible to the participant. An iPad
is also placed next to the wired speaker for the participants to complete
puzzles and surveys throughout the study.
* •
Wizard of Oz interface is used to control the smart speaker’s audio. The
interface website, hosted on GitHub, dictated the flow of experiment and was
running on the laptop connected to the wired speakers. The experimenter
operated the website remotely using TeamViewer on the desktop computer to
control the assistant on the mock smart speaker using WoZ paradigm. The
experimenter could click on various buttons to activate assistant responses to
the participant. Apart from standard responses for each item in each shopping
list such as “Okay, I found Duracell Double A batteries. Should I add it
cart”, “Let’s try that again” for fixing the error, and “The item has been
added to your cart”, the web interface also had generic responses to handle a
participant’s divergent requests. For instance, statements such as “I am doing
alright. How I can help you today?” to respond to the participant’s questions
about how the assistant is doing. The simple responses such as “Yes”, “No”,
“Hi”, and “How can I help you today?” were also included. For requests that
could not be fulfilled by the speaker such as the participant asking to read
out all the items in their cart, the assistant responded with “Unfortunately,
that is currently beyond my capabilities.” and presented a possible solution
to continue the conversation, “but you can ask me if a specific item is in the
cart or not?”. To handle such specific queries, we also added “Yes, that item
is already in the cart” and “No, that item is not in the cart yet, would you
like me to add it?”. If the participant requests to remove any particular item
from the cart, the assistant obliged and responded, “The item has been removed
from the cart.” If the participant forgot to ask the voice assistant to check
out at the end of the task, the assistant prompted the participant by asking
“Would you like to check out?” On the participant’s request or confirmation to
check out, the assistant said “Okay, checking out. Your items should arrive
shortly”. To handle return process, there were a few statements at the
experimenter’s disposal such as “Okay, I can start a return process for the
item”, “Which item would you like to return?”, and “Your item has been
returned and a refund has been issued” in addition to the error mitigation
statement.
Next, we discuss design of our manipulations for error mitigation strategies
and gendered voices.
### 3.3. Manipulation of Error Mitigation Strategies
In this study, we focused on two error mitigation strategies, namely apology
and compensation; the strategy of apology has been employed in prior research
with a similar online shopping context (Mahmood et al., 2022), whereas the
strategy of compensation is relevant given the nature of the shopping task.
* •
Apology. The design of our apology strategy followed a similar design of error
responses in a prior study on online shopping with a smart speaker in a
simulated online experiment (Mahmood et al., 2022); in particular, the study
showed that an apology offered in a serious tone with the acceptance of blame
for the AI error is preferred over the apologies that lacked seriousness,
acceptance of blame, or both. In our study, the agent apologized for the AI
error by saying “I am sorry for the inconvenience. I confused the items
because there are multiple items for this keyword. From time to time, I have
difficulty distinguishing among such items.”
* •
Compensation. For the compensation strategy, the agent offered a 10 dollar
credit to the user by saying “I will give you a 10 dollar credit that can be
applied to your future purchases with us for the inconvenience caused.”
### 3.4. Manipulation of Gendered Voices
We manipulated the gender of AI voice to sound feminine, masculine and
ambiguous. We chose the feminine and masculine voices from two free text-to-
speech websites “Sound of Text”333https://soundoftext.com/ and
“wideo”444https://wideo.co/text-to-speech/ (US Jack Bailey), respectively. The
voices had American accent, and default pitch and speed. Inspired from prior
work (Tolmeijer et al., 2021), we generated gender ambiguous voices by
shifting the pitch of the above feminine and masculine voices by 2 or 3
semitones where shifting by one semitone is equivalent to shifting by 5.946%.
We used an online pitch shifting tool555https://onlinetonegenerator.com/pitch-
shifter.html. The masculine voice was increased by 2 and 3 semitones while the
feminine voice was lowered by 2 and 3 semitones to obtain 4 candidates for
gender ambiguous voices. To select a gender ambiguous voice, we conducted two
pilot studies:
#### 3.4.1. Pilot Study 1: Picking 2 gender ambiguous voices from 4
candidates
To pick one ambiguous voice between the shifted masculine voices and one
between the shifted feminine voices, we conducted an online poll with nine
participants who listened to the 4 voices and were asked to choose two
ambiguous voices, one from each gender category. Moreover, they were asked to
rank the voices from most ambiguous to least ambiguous. Six of the 9
participants picked feminine-3 semitones to be more ambiguous among shifted
feminine voices which is consistent with the results from prior study
(Tolmeijer et al., 2021). Whereas, seven of the 9 participants picked the
masculine+2 semitones as more ambiguous from the shifted masculine voices.
Overall, masculine+2 semitones was picked by eight participants to be in top 2
when they were asked to rank the voices for most ambiguous to least ambiguous
voices. While feminine-3 semitones was picked by four participants to be in
top 2. Based on the results from this poll, we kept masculine+2 semitones
(masculine-ambiguous) and feminine-3 semitones (feminine-ambiguous) as two
candidates for gender ambiguous voice for the next pilot study.
#### 3.4.2. Pilot Study 2: Picking 1 gender ambiguous voice from 2 candidates
and validating the manipulation of all gendered AI voices.
To verify that the gendered voices, especially ambiguous voices, are perceived
as intended regardless of the nature of a task, we ran a within-subjects study
with two factors: voice gender—(feminine, masculine feminine-ambiguous,
masculine-ambiguous) and nature of task—(stereotypically masculine, neutral,
stereotypically feminine). For the manipulation of task nature, we picked
following three statements narrated by the voice agents:
* •
Stereotypically masculine task (fixing a car). “If the car’s charging port is
not working, check to see if the fuse for the power outlet has blown”.
* •
Neutral task (defining a term). “A recession is a period of temporary economic
decline during which trade and industrial activity are reduced”.
* •
Stereotypically feminine task (nursing). “The doctor will be with you shortly.
Meanwhile, please fill out the consent forms”.
We recruited 16 participants (gender: 9 female, 7 male; age:
$M=23.33,SD=2.61$) through convenience sampling from the local community. The
participants were presented with 12 conditions in a randomized order. In each
condition, the participants listened to the voice of one of the AI agents
saying one of the above statements and then rated the agent in terms of
perceived gender (1 being feminine, 3 being ambiguous and 5 being masculine)
and humanlikeness (1 being robotic and 5 being human) on 5-point scales.
Furthermore, participants were asked to give a name to the agent. Two coders
independently assigned gender (masculine, feminine, or ambiguous) to the names
provided by the participants. In case of disagreement between two coders, they
discussed and reached a consensus on the perceived gender based on the name
given to the agent.
The data collected from the second pilot study was analyzed to select the
voices for the voice assistants. To validate the manipulation of perceived
gender of the AI voice, we compared the rating of perceived gender to the
baseline, the value 3 (ambiguous) on a 5-point scale. One-way analysis of
variance (ANOVA) revealed that the AI voice had a significant main effect on
perceived gender, $F(4,235)=146.04,p<.001$. Pairwise comparisons using
Dunnett’s test with control (baseline: perceived gender = 3, ambiguous)
revealed that all the voices except masculine-ambiguous ($M=3.25,SD=0.84$) is
significantly different from the baseline. The masculine voice
($M=4.52,SD=0.74$) is significantly higher on perceived gender (5-point scale,
1 being feminine, 3 being ambiguous, and 5 being masculine) than baseline
($M=3,SD=0$), $p<.001$ while the feminine voice ($M=1.14,SD=.36$), $p<.001$,
and feminine-ambiguous ($M=2.65,1.02$), $p=.046$, voices are significantly
lower on perceived gender than baseline. These results verified that our
manipulation for masculine and feminine voices was adequate. A chi-square test
of independence was performed to examine the effect of gender of AI voice and
the perceived gender of the names given by participants. The relation between
these variables was significant ($\chi^{2}(6,N=192)=169.21,p<.001$). Our
masculine-ambiguous AI voice was given gender ambiguous names ($66.7\%$) more
often than the feminine-ambiguous AI voice ($62.5\%$). Based on these results,
we concluded that masculine-ambiguous voice was perceived to be more gender
ambiguous than the feminine-ambiguous voice. As a result, we used the
masculine-ambiguous voice as a gender ambiguous voice in our main study.
Furthermore, on running two-way repeated measures ANOVA with voice gender and
nature of task as factors, there is no significant main effect of nature of
task on perceived gender. A two-way repeated measures ANOVA revealed no
significant main effect of gender and nature of task on humanlikeness.
However, there is a significant interaction effect between voice gender and
nature of task, $F(6,90)=2.46,p=.030,\eta^{2}=.141$; Tukey’s HSD post hoc test
shows no significant pairwise comparisons.
### 3.5. Measures
For the main study, we used a range of metrics to measure user perceptions of
agent characteristics, subjective evaluations of the voice assistants, and
user behavior towards the assistants.
#### 3.5.1. Perceptions of Agent Characteristics
* •
Perceived gender. To check if our manipulation of the gender of AI voices was
perceived as intended, we asked the participants to rate the voice gender on a
5-point scale (1 being feminine, 3 being ambiguous, and 5 being masculine).
* •
Humanlikeness. We asked the participants to rate humanlikness of the agent on
a 5-point scale (1 being robotic and 5 being human).
#### 3.5.2. Subjective Measures of User Experience and Perceptions of AI
* •
Service recovery satisfaction (Three items; Cronbach’s $\alpha=.91$). We used
three questions (“The AI assistant was responsive to any errors.”, “I am happy
with how the error was handled” and “In my opinion, the AI assistant provided
a satisfactory response to the error”) as informed by prior work on apology
(Roschk and Kaiser, 2013) in the domain of consumer services (Tax et al.,
1998; Maxham III and Netemeyer, 2003) to measure service recovery
satisfaction.
* •
Warmth (Four items; Cronbach’s $\alpha=87$). We asked the participants to rate
their impression of the agent on these dimensions: 1) Cold – Warm, 2)
Unfriendly – Friendly, 3) Insincere – Sincere, and 4) Boastful – Modest.
* •
Competence (Three items; Cronbach’s $\alpha=.86$). We asked the participants
to rate their impression of the agent on these dimensions: 1) Unskillful –
Skillful, 2) Impractical – Practical, and 3) Foolish – Intelligent as informed
by prior work on gender stereotypes on AI recommendations (Ahn et al., 2022)
and social cognition dimensions (Fiske et al., 2007) similar to ‘warmth’.
#### 3.5.3. Behavioral Measures
To understand participants’ behaviors toward voice assistants during their
interactions, we asked two human coders to annotate the interaction videos and
extracted the following behavioral measures based on the annotation. The two
coders annotated same 10% of the participant videos for inter-coder
reliability; the rest of the videos were 50:50 split between the two coders.
For dichotomous variables, percentage agreement was reported for assessing
inter-coder reliability. For continuous variables, Intraclass Correlation
Coefficient (ICC) estimates and their 95% confident intervals were calculated
and reported using SPSS assuming single measurement, consistency, two-way
mixed-effects model. ICC coefficients between 0.75 and 0.90, and greater than
0.90 are indicative of good and excellent reliability, respectively (Koo and
Li, 2016).
* •
Participant interruption during error (Dichotomous). This measure encodes
whether the participant interrupted the assistant after realizing that an
error had occurred while the assistant was still talking. Essentially, this
measure indicates if the participant started talking before the assistant
stopped. For instance, during the statement with error, “Okay, I found red
archery bow. Should I add it to the cart?”, the interruption would be recorded
if the participant starts talking before the assistant finishes saying “Should
I add it to the cart?”. There was a 100% agreement between the two coders.
* •
Non-verbal reaction to error (Dichotomous). This measure encodes whether the
participant reacted to the error non-verbally (e.g., changes in face and/or
upper body; see Fig. 1 for example reactions). There was a 100% agreement
between the two coders.
* •
Verbal response to mitigation statement (Dichotomous). We look at whether the
participant responds verbally to the error mitigation statement (compensation
or apology). For instance, participants saying “Thank you” and “It’s okay”
would would count as verbal responses to error mitigation strategy. There was
a 100% agreement between the two coders.
* •
Non-verbal response to mitigation statement (Dichotomous). We look at whether
the participant reacts non-verbally (changes in face and/or upper body; see
Fig. 1 for example responses) to the error mitigation statement (compensation
or apology). There was a 100% agreement between the two coders.
* •
Number of other interruptions (interruptions that were not concerned with the
error). Interruptions are defined as when the participant and the assistant
interrupts or talks over each other. This measure counts the number of
interruptions, apart form those to error as mentioned above, during the
interaction, trying to capture the overall interaction fluency. A good degree
of reliability was found between 2 raters, $ICC(3,1)=.835[.644,.928]$.
* •
Number of reactions (verbal and non-verbal). Verbal reactions are recorded
when the participant says phrases other than direct response to the speaker
such as “Thank you”, “You’re good” etc. Non-verbal reactions, such as smile,
laugh, and raising eye brows, are recorded when the participant’s face and/or
upper body (visible in video recordings) changes as reaction to the
assistant’s speech (see Fig. 1 for example non-verbal reactions). Involuntary
movements were not recorded as non-verbal reactions. We do not include verbal
and non-verbal responses to error in this measure because we are interested in
understanding, in general, when nothing unexpected occurs how social the
interaction of the participant with the smart speaker is. Hence, we focus only
on the conversational aspect of the interaction, which includes responses to
mitigation statements. A good degree of reliability was found between 2 raters
for number of verbal reactions, $ICC(3,1)=.837[.648,.929]$. Similarly, a good
degree of reliability was also found between 2 raters for number of non-verbal
reactions, $ICC(3,1)=.896[.766,.955]$.
### 3.6. Procedure
This study consisted of five phases:
1. (1)
Introduction and consent. At the start of the study, participants were
provided with a brief description of the study. The description stated that
participants would be ordering items from shopping lists while collaborating
with six AI assistants to test our smart speaker prototype. The participation
was voluntary; participants agreed to continue the study by signing the
consent form approved by our Institutional Review Board (IRB).
2. (2)
Pre-study questionnaire. The participants filled a pre-study questionnaire
inspired from prior work (Bradley and LaFleur, 2016) to evaluate users’
utilitarian or relational orientation regarding shopping experience.
3. (3)
Experimental task: simulated shopping. Participants were assigned one of the
rows in a Latin square of order 6, dictating the order of experimental tasks.
They interacted with the AI assistant on the mock smart speaker. Participants
were prompted to do a puzzle while waiting for their order to arrive. After
the experimenter handed them the items (printed on a piece of paper) that they
ordered, participants verified if the items were correct, and started a return
with the speaker if needed.
4. (4)
Perception survey. After interacting with the AI assistant, participants
completed a questionnaire about their perceptions of the AI assistant. They
continued onto the next condition and repeated phases 3 and 4.
5. (5)
Post-study questionnaire. After completing all the conditions, participants
filled out a post-study demographics questionnaire.
The study was approved by our IRB. The study took approximately 45 minutes to
complete. The participants were compensated with a $15 USD gift card for their
participation in the study.
### 3.7. Participants
We recruited 43 participants using university mailing lists and flyers around
the campus. Two of the participants were excluded from data analysis because
they failed to identify the targeted error correctly in 5 out of 6 trials. A
total of 41 participants (21 females (women), 19 males (men), 1 non-binary)
were considered for data analysis. The participants were aged between 18 to 58
($M=26.20,SD=7.92$) and had a variety of education backgrounds, including
computer science, engineering and technology, healthcare, life sciences, media
sciences, music, and education. Twenty seven participants indicated their
ethnicity to be Asian, 2 African American, 6 Caucasian, 5 Spanish origin of
any race, and 1 Caucasian and Spanish origin of any race.
Table 1. Results of perceived gender of agent voice(1: feminine, 3: ambiguous, and 5: masculine) and humanlikeness. Mixed-model repeated measures ANOVA for voice gender, mitigation strategy, and participant gender. ${}^{\ast}p\leq 0.05$, ${}^{\ast\ast}p\leq 0.01$, and ${}^{\ast\ast\ast}p\leq 0.001$ Perceived Voice Gender (1: feminine 3: ambiguous 5: masculine) | | |
---|---|---|---
Variables | $F$ | $p$ | $\eta_{p}^{2}$
Voice Gender | $F(2,76)=323.55$ | $<.001^{\ast\ast\ast}$ | $.895$
Feminine ($M=1.14,SD=0.44$) ¡ Masculine ($M=4.89,SD=0.34$) | | $<.001^{\ast\ast\ast}$ |
Feminine ($M=1.14,SD=0.44$) ¡ Ambiguous ($M=3.93,SD=1.18$) | | $<.001^{\ast\ast\ast}$ |
Ambiguous ($M=3.93,SD=1.18$) ¡ Masculine ($M=4.89,SD=0.34$) | | $<.001^{\ast\ast\ast}$ |
Mitigation Strategy | $F(1,38)=0.15$ | $.700$ | $.004$
Participant Gender | $F(1,38)=5.96$ | $.019^{\ast}$ | $.136$
Male ($M=3.17,SD=1.82$) ¡ Female ($M=3.45,SD=1.70$) | | |
Participant Gender $\times$ Voice Gender | $F(2,76)=2.86$ | $.063$ | $.070$
Participant Gender $\times$ Mitigation Strategy | $F(1,38)=3.77$ | $.072$ | $.083$
Voice Gender $\times$ Mitigation Strategy | $F(2,76)=1.69$ | $.192$ | $.043$
Participant Gender $\times$ Voice Gender & Mitigation Strategy | $F(2,76)=0.52$ | $.598$ | $.013$
Humanlikeness | | |
Variables | $F$ | $p$ | $\eta_{p}^{2}$
Voice Gender | $F(2,76)=21.18$ | $<.001^{\ast\ast\ast}$ | $.358$
Masculine ($M=2.96,SD=1.06$) ¡ Feminine ($M=3.36,SD=1.02$) | | $.003^{\ast\ast}$ |
Ambiguous ($M=2.59,SD=1.13$) ¡ Feminine ($M=3.36,SD=1.02$) | | $<.001^{\ast\ast\ast}$ |
Ambiguous ($M=3.93,SD=1.18$) ¡ Masculine ($M=2.96,SD=1.06$) | | $.007^{\ast\ast}$ |
Mitigation Strategy | $F(1,38)=0.21$ | $.646$ | $.006$
Participant Gender | $F(1,38)=0.90$ | $.348$ | $.023$
Participant Gender $\times$ Voice Gender | $F(2,76)=1.26$ | $.291$ | $.032$
Participant Gender $\times$ Mitigation Strategy | $F(1,38)=3.56$ | $.067$ | $.086$
Voice Gender $\times$ Mitigation Strategy | $F(2,76)=0.48$ | $.620$ | $.013$
Participant Gender $\times$ Voice Gender & Mitigation Strategy | $F(2,76)=0.54$ | $.583$ | $.014$
## 4\. Results
Our data analysis included a total of 246 trials from 41 participants (six
trials for six conditions per participant). In 9 of the 246 trials, the
intended errors were not correctly identified. To handle these missing values
for the questionnaire measures, first, we analyzed whether these values are
missing completely at random (MCAR) or not. Little’s MCAR test (Little, 1988)
was not significant suggesting that it is safe to assume that data is MCAR,
$\chi^{2}(197,N=43)=158.28$, $p=.98$. Multiple Imputation (MI) (Rubin, 2004)
is one of the optimal techniques to handle the missing data and gives unbiased
results under MCAR (van Ginkel et al., 2020). Thus, before proceeding with our
analysis, we replaced the missing data using MI in SPSS by computing five
imputations and pooling the results, taking into account variation across
these imputations.
For the analyses reported in this paper, we included participant gender as a
between-subjects factor to explore the nuanced relationships among voice
gender, mitigation strategy, and participant gender. Due to the significant
imbalance in participant gender, with only one participant who self reported
having a non-binary gender, we excluded this participant in the following
analyses. Thus, a total of 40 participants (21 females (women), 19 males
(men)) aged between 18 to 58 ($M=26.23,SD=8.08$) were included for analysis.
For the results reported below, we used mixed-model repeated measures analysis
of variance (ANOVA) unless stated otherwise. The gender of AI voice, error
mitigation strategy, and participant’s gender were set as the fixed effect and
participants as a random effect. Voice gender and error mitigation strategy
are within-subjects factor while participant gender is between-subjects
factor. For all the statistical tests reported below, $p<.05$ is considered as
a significant effect. We only describe significant results below. Complete
results are presented in Tables 1, 2, 3, and 4. We follow Cohen’s guidelines
on effect size and considered $\eta_{p}^{2}=0.01$ a small effect,
$\eta_{p}^{2}=0.06$ a medium effect, and $\eta_{p}^{2}=0.14$ a large effect
(Cohen, 1988).
Figure 3. Results of perceived gender and humalikeness of the assistant.
Mixed-model repeated measures ANOVAs were conducted to discover effects of
voice gender—ambiguous (Amb.), feminine (Fem.), masculine (Masc.)—, mitigation
strategy—apology (Apol.), compensation (Comp.)—, and participant (PCP)
gender—female, male— on participants’ perceptions of agent characteristics.
All pairwise comparisons were conducted using Fisher’s LSD method with
Bonferroni correction. Error bars represent standard error (SE) and only the
significant comparisons ($p<.05$) are highlighted.
[Results of perceived gender and humalikeness of the assistant.]The figure
shows results of perceived gender and humalikeness of the assistant. The
figure further consists of 2 sub graphs. All are bar plots and y-axis is
ranged from 1 to 5. The first of the 2 sub graphs from left to right further
consists of 3 bar plots and shows results on perceived gender which is the
y-axis for all three bar plots. One means most feminine and 5 means most
masculine while 3 means ambiguous. First of the 3 bar plots has voice gender
on x-axis and shows significant main effect of voice gender on perceived
gender of assistants. The p-value of all the significant comparisons discussed
is less than .001 unless stated otherwise. Feminine assistant with a mean of
1.1 is rated more feminine than masculine assistant with a mean of 4.9. While,
ambiguous with mean of about 3.9 is also rated more masculine than feminine
and more feminine than masculine assistant. Second of the 3 bar plots has
mitigation strategy on x-axis and shows no significant main effect of
mitigation strategy on assistant’s perceived gender. Assistants employing
compensation and apology strategy have mean of about 3.5. Third of the 3 bar
plots has participant gender on x-axis and shows significant effect of
participant gender on assistants’ perceived gender with a p-value of.019. Male
participants with a mean of 3.2 perceived the assistants’ gender to be lower
on the scale which is more feminine than female participants with a mean of
3.4. The figure shows results of perceived gender and humalikeness of the
assistant. The figure further consists of 2 sub graphs. All are bar plots and
y-axis is ranged from 1 to 5. The second of the 2 sub graphs from left to
right further consists of 3 bar plots and shows results on perceived
humanlikeness which is the y-axis for all three bar plots. One means least
humanlike and 5 means most humanlike. First of the 3 bar plots has voice
gender on x-axis and shows significant main effect of voice gender on
perceived gender of assistants. Feminine assistant with a mean of 3.4 is rated
more humanlike than masculine assistant with a mean of 3.0 and a p-value of
.003. While, ambiguous with mean of about 2.6 is rated less humanlike as
compared to masculine with a p-value of .007 and feminine assistants with a
p-value of less than .001. Second of the 3 bar plots has mitigation strategy
on x-axis and shows no significant main effect of mitigation strategy on
assistant’s perceived humanlikeness. Assistants employing compensation and
apology strategy have mean of about 2.9 and 3.0 respectively. Third of the 3
bar plots has participant gender on x-axis and shows no significant main
effect of participant gender on assistants’ perceived humalikeness. Male
participants have a mean of 3 and female participants have a mean of 3.2.
### 4.1. Perceptions of Agent Characteristics
Fig. 3 and Table 1 summarize our results for perceptions of agent
characteristics.
#### 4.1.1. Perceived Voice Gender
We saw a significant main effect of the voice gender on participants’
perception of agent’s gender confirming that the manipulation of gendered
voices for assistants in the study was successful (Fig. 3 a and Table 1). The
agent with feminine voice was rated more feminine than the agents with
masculine and ambiguous voice. Masculine voice was rated more masculine than
ambiguous voice. Moreover, we observed a significant main effect of
participant gender on their perceptions of agent’s gender. Male participants
perceived assistants’ gender to be lower on the scale (more feminine) than
female participants.
#### 4.1.2. Humanlikeness
We found a significant main effect of voice gender on participants’ perception
of humanlikeness of the agent (Fig. 3 b and Table 1). The agent with feminine
voice was perceived more humanlike than the agent with masculine voice.
Ambiguous voice was perceived more robotic than both feminine and masculine
voice.
Table 2. Results of subjective measures: service recovery satisfaction, warmth, and competence. Mixed-model repeated measures ANOVA for voice gender, mitigation strategy, and participant gender. ${}^{\ast}p\leq 0.05$, ${}^{\ast\ast}p\leq 0.01$, and ${}^{\ast\ast\ast}p\leq 0.001$ Service Recovery Satisfaction | | |
---|---|---|---
Variables | $F$ | $p$ | $\eta_{p}^{2}$
Voice Gender | $F(2,76)=0.99$ | $.377$ | $.025$
Mitigation Strategy | $F(1,38)=4.81$ | $.034^{\ast}$ | $.112$
Apology ($M=4.22,SD=0.84$) ¡ Compensation ($M=4.47,SD=0.76$) | | |
Participant Gender | $F(1,38)=1.93$ | $.173$ | $.048$
Participant Gender $\times$ Voice Gender | $F(2,76)=.310$ | $.720$ | $.039$
Participant Gender $\times$ Mitigation Strategy | $F(1,38)=3.77$ | $.060$ | $.090$
Voice Gender $\times$ Mitigation Strategy | $F(2,76)=0.19$ | $.831$ | $.005$
Participant Gender $\times$ Voice Gender & Mitigation Strategy | $F(2,76)=0.29$ | $.712$ | $.008$
Warmth | | |
Variables | $F$ | $p$ | $\eta_{p}^{2}$
Voice Gender | $F(2,76)=3.93$ | $.024^{\ast}$ | $.094$
Ambiguous ($M=3.79,SD=0.82$) ¡ Feminine ($M=4.03,SD=0.70$) | | $.027^{\ast}$ |
Mitigation Strategy | $F(1,38)=6.39$ | $.016^{\ast}$ | $.144$
Compensation ($M=3.80,SD=0.77$) ¡ Apology ($M=3.98,SD=0.74$) | | |
Participant Gender | $F(1,38)=0.96$ | $.334$ | $.025$
Participant Gender $\times$ Voice Gender | $F(2,76)=1.52$ | $.225$ | $.039$
Participant Gender $\times$ Mitigation Strategy | $F(1,38)=0.05$ | $.826$ | $.001$
Voice Gender $\times$ Mitigation Strategy | $F(2,76)=0.42$ | $.656$ | $.011$
Participant Gender $\times$ Voice Gender & Mitigation Strategy | $F(2,76)=1.953$ | $.149$ | $.023$
Competence | | |
Variables | $F$ | $p$ | $\eta_{p}^{2}$
Voice Gender | $F(2,76)=3.56$ | $.033^{\ast}$ | $.086$
Masculine ($M=3.83,SD=0.85$) ¡ Feminine (($M=4.00,SD=0.84$)) | | $.021^{\ast}$ |
Mitigation Strategy | $F(1,38)=6.39$ | $.016^{\ast}$ | $.144$
Compensation ($M=3.80,SD=0.77$) ¡ Apology ($M=3.98,SD=0.74$) | | |
Participant Gender | $F(1,38)=1.93$ | $.173$ | $.048$
Participant Gender $\times$ Voice Gender | $F(2,76)=1.20$ | $.308$ | $.030$
Participant Gender $\times$ Mitigation Strategy | $F(1,38)=0.00$ | $.997$ | $.000$
Voice Gender $\times$ Mitigation Strategy | $F(2,76)=0.18$ | $.794$ | $.005$
Participant Gender $\times$ Voice Gender & Mitigation Strategy | $F(2,76)=0.29$ | $.712$ | $.008$
### 4.2. Subjective Measures
Fig. 4 and Table 2 summarize our results for subjective measures.
#### 4.2.1. Service Recovery Satisfaction
We observed a significant main effect of mitigation strategy of participants’
satisfaction of service recovery; in particular, compensation resulted in
better service recovery satisfaction than apology (Fig. 4 a and Table 2).
Figure 4. Results of participants’ service recovery satisfaction, and
perceived warmth and competence of the assistant. Mixed-model repeated
measures ANOVAs were conducted to discover effects of voice gender—ambiguous
(Amb.), feminine (Fem.), masculine (Masc.)—-, mitigation strategy—apology
(Apol.), compensation (Comp.)—, and participant (PCP) gender—female, male— on
subjective measures. All pairwise comparisons were conducted using Fisher’s
LSD method with Bonferroni correction. Error bars represent standard error
(SE) and only the significant comparisons ($p<.05$) are highlighted.
[Results of participants’ service recovery satisfaction, and perceived warmth
and competence of the assistant.]The figure shows results of participants’
service recovery satisfaction, and perceived warmth and competence of the
assistant. The figure further consists of 3 sub graphs. All are bar plots and
y-axis is ranged from 1 to 5. The first of the 3 sub graphs from left to right
further consists of 3 bar plots and shows results on service recovery
satisfaction which is the y-axis for all three bar plots. First of the 3 bar
plots has voice gender on x-axis and shows no significant main effect of voice
gender on service recovery satisfaction. the mean for ambiguous is 4.2, and
for feminine and masculine sounding assistants is closer to 4.3. Second of the
3 bar plots has mitigation strategy on x-axis and shows a significant main
effect of mitigation strategy on service recover satisfaction with a p-value
of .034. Assistant employing compensation strategy with a mean of about 4.5 is
rated better than the assistant that apologizes with a mean of 4.2. Third of
the 3 bar plots has participant gender on x-axis and shows no significant main
effect of participant gender on service recovery satisfaction. Male
participants have a mean of 4.2 and female participants have a mean of 4.3.
The second of the 3 sub graphs from left to right further consists of 3 bar
plots and shows results on warmth which is the y-axis for all three bar plots.
First of the 3 bar plots has voice gender on x-axis and shows a significant
difference between ambiguous sounding and Feminine sounding assistants with a
p-value of 0.027. Feminine sounding assistant with a mean of about 4.0 is
perceived to be warmer than ambiguous sounding one with a mean of 3.8.
Masculine assistant has a mean warmth of 3.9. Second of the 3 bar plots has
mitigation strategy on x-axis and shows a significant main effect of
mitigation strategy on warmth with a p-value of .016. Assistant employing
apology strategy with a mean of about 4.0 is rated better than the assistant
that compensates with a mean of 3.8. Third of the 3 bar plots has participant
gender on x-axis and shows no significant main effect of participant gender on
assistants’ perceived warmth. Male participants have a mean of about 3.8 and
female participants have a mean of about 4.0. The third of the 3 sub graphs
from left to right further consists of 3 bar plots and shows results on
competence which is the y-axis for all three bar plots. First of the 3 bar
plots has voice gender on x-axis and shows a significant difference between
feminine sounding and masculine sounding assistants with a p-value of 0.021.
Feminine sounding assistant with a mean of about 4.0 is perceived to be more
competent than masculine sounding one with a mean of 3.8. Ambiguous assistant
has a mean competence of 3.9. Second of the 3 bar plots has mitigation
strategy on x-axis and shows no significant main effect of mitigation strategy
on warmth. Assistant employing compensation strategy has a mean of about 4.0
and the assistant that apologizes with a mean of about 3.8. Third of the 3 bar
plots has participant gender on x-axis and shows no significant main effect of
participant gender on assistants’ perceived competence. Male participants have
a mean of about 3.7 and female participants have a mean of about 4.2.
#### 4.2.2. Warmth
We observed a a significant main effect of voice gender on participants’
perception of warmth of agent. The agent with a feminine voice was perceived
warmer than the agent with an ambiguous voice (Fig. 4 b and Table 2).
#### 4.2.3. Competence
We observed a significant main effect of voice gender on participants’
perception of competence of voice assistant (Fig. 4 c and Table 2). The
assistant with a feminine voice was perceived more competent than the
assistant with a masculine voice. Moreover, there was a significant main
effect of the mitigation strategy on perceived competence; assistants offered
compensation were perceived to be warmer than assistants that apologized.
Table 3. Results of behavioral analysis: Related to error and error mitigation strategy. ${}^{\ast}p\leq 0.05$, ${}^{\ast\ast}p\leq 0.01$, and ${}^{\ast\ast\ast}p\leq 0.001$ Participant interruption during error | |
---|---|---
Variables | $\chi^{2}$ | $p$
Voice Gender | $\chi^{2}(2)=1.66$ | $.435$
Participant Gender | $\chi^{2}(1)=4.93$ | $.026^{\ast}$
Female($9.6\%$) ¡ Male($19.8\%$) $95\%$ CI: $[1.10,4.89]$ | |
Non-verbal reaction to error | |
Variables | $\chi^{2}$ | $p$
Voice Gender | $\chi^{2}(2)=3.08$ | $.857$
Participant Gender | $\chi^{2}(1)=0.03$ | $.857$
Verbal response to error mitigation strategy | |
Variables | $\chi^{2}$ | $p$
Voice Gender | $\chi^{2}(2)=0.44$ | $.802$
Mitigation Strategy | $\chi^{2}(1)=0.09$ | $.766$
Participant Gender | $\chi^{2}(1)=0.06$ | $.812$
Non-verbal response to error mitigation strategy | |
Variables | $\chi^{2}$ | $p$
Voice Gender | $\chi^{2}(2)=1.50$ | $.471$
Mitigation Strategy | $\chi^{2}(1)=0.01$ | $.937$
Participant Gender | $\chi^{2}(1)=3.32$ | $.069$
Figure 5. Results of interruption (Intrp.) related behavioral metrics.
Generalized Estimating Equations (GEE) was used to fit a logistic regression
to discover effects of voice gender, and participant gender on participants’
interruption to error. Moreover, a mixed model ANOVA was conducted to study
effects of voice gender—ambiguous, feminine, masculine—-, mitigation
strategy—apology (Apol.), compensation (Comp.)—, and participant (PCP)
gender—female, male— on number of interruptions. All pairwise comparisons were
conducted using Tukey’s HSD method. Error bars represent standard error (SE)
and only the significant comparisons ($p<.05$) are highlighted.
[Results of interruption related behavioral metrics]The figures has results on
interruption related user behavior metrics. There are 3 sub graphs. First sub
graph from left to right further consists of two mosaic plots. Generalized
Estimating Equations (GEE) to fit a repeated measures logistic regression was
run for this analysis of interrupted error. First of the two mosaic plots has
participant gender on x-axis and interrupted error (dichotomous variable)
which shows a significant difference with p-value of .026 between female and
male participants. Female participants interrupted 12 times while did not
interrupt at all 113 times where as male participants interrupted 21 times and
did not interrupt 84 times. The second mosaic plot has portrayed AI gender on
x-axis while y-axis is the same which is interrupted error. It shows no
significant effect of AI gender on whether participants interrupted during
mitigation statement delivery. For ambiguous assistant there was an
interruption in 11 of total 76 trials, for feminine assistant in 14 of 80
trials and for masculine assistant in 8 of 75 trials. The second and third
graphs from left to right show results from a mixed model ANOVA to study
effects of our factors on number of interruptions. The y-axis for both plots
is average count of interruptions with a range of 0 to 0.8. The second from
left to right has two bar plots. First of the two bar plots is for female
participants only and the x-axis is mitigation strategies showing no
significant interaction effect between the two and mean for both apology and
compensation is around 0.1. For the second of 2 bar graphs, mitigation
strategies only for male participants are on x-axis. It shows that apologetic
assistants had lesser interruptions with mean about 0.2 than compensating
assistants with mean about 0.4 when looking at male participants only. The
p-value for this interaction effect is .047. The third and the last graph from
left to right has mitigation strategies on x-axis only for ambiguous sounding
assistants. Apologetic assistants with mean of about 0.1 were interrupted less
than compensating ones with mean of about 0.3 when looking at ambiguous
sounding assistants only. The p-value for this interaction effect is .018.
Table 4. Results of behavioral analysis: Relevant to overall interaction. ${}^{\ast}p\leq 0.05$, ${}^{\ast\ast}p\leq 0.01$, and ${}^{\ast\ast\ast}p\leq 0.001$ Number of interruptions | |
---|---|---
Variables | $F$ | $p$
Voice Gender | $F(2,75.89)=0.16$ | $.849$
Mitigation Strategy | $F(1,38.06)=3.45$ | $.071$
Compensation ($M=3.80,SD=0.77$) ¡ Apology ($M=3.98,SD=0.74$) |
Participant Gender | $F(1,38.30)=0.49$ | $.489$
Participant Gender $\times$ Voice Gender | $F(2,75.89)=0.78$ | $.464$
Participant Gender $\times$ Mitigation Strategy | $F(1,38.06)=4.67$ | $.037^{\ast}$
Male, Apology ($M=0.17,SD=0.64$) ¡ | |
Male, compensation ($M=0.34,SD=0.90$) | | $.047^{\ast}$
Voice Gender $\times$ Mitigation Strategy | $F(2,75.29)=3.80$ | $.027^{\ast}$
Ambiguous, Apology ($M=0.08,SD=0.36$) | |
¡ Ambiguous, Compensation ($M=0.33,SD=0.77$) | | $.018^{\ast}$
Participant Gender $\times$ Voice Gender & Mitigation Strategy | $F(2,75.29)=0.58$ | $.562$
Number of other verbal reactions | |
Variables | $F$ | $p$
Voice Gender | $F(2,63.99=0.04$ | $.960$
Mitigation Strategy | $F(1,38.35)=2.59$ | $.116$
Participant Gender | $F(1,37.34)=0.73$ | $.397$
Participant Gender $\times$ Voice Gender | $F(2,63.99)=0.95$ | $.391$
Participant Gender $\times$ Mitigation Strategy | $F(1,38.35)=0.27$ | $.606$
Voice Gender $\times$ Mitigation Strategy | $F(2,66.31)=2.10$ | $.130$
Participant Gender $\times$ Voice Gender & Mitigation Strategy | $F(2,66.31)=0.34$ | $.710$
Number of other non-verbal reactions | |
Variables | $F$ | $p$
Voice Gender | $F(2,73.65)=2.80$ | $.067$
Mitigation Strategy | $F(1,35.1)=7.14$ | $.011^{\ast}$
Compensation ($M=0.40,SD=0.77$) ¡ Apology ($M=0.74,SD=1.27$) | |
Participant Gender | $F(1,38.30)=1.88$ | $.178$
Participant Gender $\times$ Voice Gender | $F(2,73.65)=4.54$ | $.014^{\ast}$
Male, Masculine ($M=0.53,SD=0.90$) ¡ Male, Feminine ($M=1.08,SD=1.62$) | | $.029^{\ast}$
Participant Gender $\times$ Mitigation Strategy | $F(1,35.10)=.69$ | $.411$
Voice Gender $\times$ Mitigation Strategy | $F(2,76.79)=0.19$ | $.828$
Participant Gender $\times$ Voice Gender & Mitigation Strategy | $F(2,76.79)=4.83$ | $.011^{\ast}$
Male, Masculine, Apology ($M=0.61,SD=0.98$) | |
¡ Male, Feminine, Apology ($M=1.47,SD=1.98$) | | $.030^{\ast}$
### 4.3. Behavioral Analysis
We did not include the nine trials in which participants did not recognize the
error (one trial from a female participant and eight trials from male
participants). As a result, the following analyses included 231 trials. Tables
3 and 4 summarize the results of behavioral analysis.
#### 4.3.1. Participant interruption during error (Dichotomous variable:
0/Yes, 1/No)
For this analysis, we did not consider error mitigation as a factor because
the participants did not experience the mitigation strategy until after the
error was fixed. In other words, whether the participants interrupted the
agent during the error was not affected by the mitigation strategy. We set up
Generalized Estimating Equations (GEE) to fit a logistic regression to examine
whether gender of voice and participant gender had an effect on whether the
participant interrupted the error or not. The model estimated that the odds
for male participants ($19.8\%$) interrupting the assistant after realizing
the error has occurred while assistant was still talking are approximately
2.32 times higher than female participants ($9.6\%$) interrupting the
assistant (Fig. 5 a and Table 3).
Figure 6. Results of number of other non-verbal reactions (RXN). Mixed-model
repeated measures ANOVA was conducted to discover effects of voice
gender—ambiguous (Amb.), feminine (Fem.), masculine (Masc.)—, mitigation
strategy—apology (Apol.), compensation (Comp.)—, and participant (PCP)
gender—female, male— on number of other non-verbal reactions. All pairwise
comparisons were conducted using Tukey’s HSD method. Error bars represent
standard error (SE) and only the significant comparisons ($p<.05$) are
highlighted.
[Results of number of other non-verbal reactions]There are four bar charts for
mixed model repeated measures ANOVAs to show effects of our factors on number
of other non-verbal reactions. The range of y-axis is 0 to 1.6 for all graphs.
First bar chart from left to right further consists of 3 bar charts. The first
among the three bar charts shows voice gender on the x-axis to present that
there is no significant main effect of gender of AI voice on number of other
non-verbal reactions. The mean for ambiguous sounding assistant is around 0.4,
female sounding assistant is around 0.6 and male one is around 0.5. The second
of the 3 bar charts shows mitigation strategy on x-axis to present that there
is significant main effect of gender on the number of non-verbal reactions,
p-value is .025 where mean for apology is about 0.6 and is higher than mean
for compensation which is about 0.4. the third of the 3 bar charts shows
participant gender on the x-axis and shows no significant effect of
participant gender on count of other non-verbal reactions. The mean for female
participants is around 0.4 while mean for male participants is around 0.6. The
second bar chart from left to right further consists of one bar chart for
showing interaction of voice gender and participant gender on the number of
other non-verbal reactions. It only shows plot for error mitigation strategies
for male participants. Thus, the x-axis is ambiguous with mean of about 0.6,
female with mean of about 0.9 and male assistants with mean of about 0.4 for
male participants only. There is only a significant difference between female
and male assistants with female having higher mean than male assistants while
looking at male participants only. The p-value for this interaction is .032.
The third bar chart from left to right further consists of 3 bar charts. It
only shows all the graphs for male participants only. The first of the three
bar charts shows mitigation strategies for male voice assistants only, on
x-axis. There is no significant difference between apology with a mean of
about 0.8 and compensation strategy with a mean of about 0.6 when it comes to
number of non-verbal reactions. The second of the three bar charts shows
mitigation strategies for only female voice assistants on x-axis while the
third of the three bar charts only shows mitigation strategies for male voice
assistants. There is only one significant interaction effect between these two
bar charts which is that male participants react non-verbally more to
apologetic female assistants with mean of about 1.2 than apologetic male
assistants with mean of about 0.4. The p-value for this interaction is .042.
Mean for female voice assistants that compensate is about 0.6 and for male
voice assistant that compensated is about 0.4.
#### 4.3.2. Non-verbal reaction to error (Dichotomous variable: 0/Yes, 1/No)
For this measure, we also did not considering error mitigation as factor as
well because the participants did not interact with this manipulation till
after error was fixed. Hence, whether the participants reacted non-verbally to
the agent during the error was not affected by the mitigation strategy. We set
up GEE to fit a logistic regression to examine whether voice gender and
participant gender influenced non-verbal reaction to error. We saw no
significant results.
#### 4.3.3. Verbal response to error mitigation strategy (Dichotomous
variable: 0/Yes, 1/No)
We set up GEE to fit a logistic regression to examine whether voice gender,
mitigation strategy, and participant gender influenced verbal response to
error mitigation strategy. We did not see any significant effects.
#### 4.3.4. Non-verbal response to error mitigation strategy (Dichotomous
variable: 0/Yes, 1/No)
We set up GEE to fit a logistic regression to examine whether voice gender,
mitigation strategy, and participant gender influenced non-verbal response to
error mitigation strategy. No significant effects were observed.
#### 4.3.5. Number of interruptions
A mixed repeated measures ANOVA with voice gender and mitigation as within-
subjects factors, and participant gender as a between-subjects factor revealed
a significant interaction effect of participant gender and mitigation on count
of interruptions. Post-hoc comparisons using Tukey’s HSD revealed that there
were more interruptions between male participants and agents compensating than
the agents apologizing, regardless of assistant’s gender (see Fig. 5 b and
Table 4). Moreover, a significant interaction effect of voice gender and
mitigation on count of interruptions was also observed. For ambiguous sounding
agents, there were more interruptions when they compensated rather than
apologized for the errors (see Fig. 5 c and Table 4).
#### 4.3.6. Number of other verbal reactions
Using mixed model repeated measures ANOVA with voice gender and mitigation
strategy as within-subjects factors, and participant gender (male, female) as
a between-subjects factor, we observed no significant effects.
#### 4.3.7. Number of other non-verbal reactions
Fig. 6 and Table 4 summarize our results for number of other non-verbal
reactions. Using mixed model repeated measures ANOVA with voice gender and
mitigation strategy as within-subjects factors, and participant gender (male,
female) as a between-subjects factor, we observed a significant main effect of
mitigation strategy on the number of non-verbal responses. Participants
reacted non-verbally more to agents in the apology condition than in the
compensation condition. There was a significant interaction effect between
participant gender and the voice gender of AI. Tukey’s HSD shows that male
participants reacted non-verbally more to agents with feminine voice than to
agents with masculine voices. Similarly, a significant interaction effect was
found between participant gender, mitigation strategy, and the voice gender of
AI. Tukey’s HSD shows that male participants reacted non-verbally more to
apology coming from the agent with feminine voice than from the agent with
masculine voice.
## 5\. Discussion
Mitigating AI errors during recovery from service failures is essential to
restoring and maintaining human-AI collaboration. AI agents in tasks of
assistance nature are mostly defaulted to manifesting feminine voices and
characteristics, which may reinforce biases because of association of gender
stereotypes. Users’ gender may also influence perceptions of gendered AI
agents. Gender ambiguous voices have been explored as a way to mitigate these
biases. This work explored how error mitigation strategy (apology vs.
compensation), gender of AI assistant voice (feminine vs. ambiguous vs.
masculine), and participant’s gender (female vs. male) may affect
participants’ satisfaction with the service recovery, perception of, and their
behavior around voice assistants during an assistive task exemplified as an
online shopping task. Below, we discuss our results, their implications,
limitations of this study, and future work.
### 5.1. Apology Communicates Warmth and Compensation Leads to User
Satisfaction
Our hypothesis 1 predicts that apology during recovery from errors by voice
assistants portrays the assistant as warmer while offered compensation creates
greater service recovery satisfaction and perceptions of competence. Our
results support this hypothesis (Fig. 4 and Table 2). Prior work has
demonstrated that humans exhibit social reactions in response to
conversational agents when these agents exhibit social personality such as
engaging in small talk, displaying gender, and age characteristics, using
gestures, or expressing facial expressions (Feine et al., 2019). In our study,
we see that voice assistants that were apologetic —exhibiting a social
personality— elicited more non-verbal reactions than the ones that offered
compensation —exhibiting a utilitarian personality— (Fig. 6 a and Table 4).
For example, one participant laughed 13 times when interacting with assistants
that apologized while only 5 times for assistants that compensated. These non-
verbal reactions such as smile, laugh, and nod are reflective of social
reactions. Since, apologizing or offering compensation can be seen as
indications of distinct personalities or traits communicated through speech,
the level of non-verbal user reactions can serve as an indicator of users’
perceptions of voice assistant’s personality.
However, further looking into the reactions, their valence may vary for
different participants. For instance, one participant reacted a total of six
times to apologetic agents in which two of the reactions were “nodding”, two
were “shaking head”, and two were “hands in the air” gestures indicating
confusion. While both of his reactions to compensating agents were “hands in
the air” gestures indicating confusion. The apologetic agents were able to
elicit a greater variety of expressions with both positive and negative
connotations. Though may be participant dependent, social reactions can be
informative of users’ perceptions of agent’s personality traits such as warmth
and competence. This observation has design implications for using social
reactions to understand users’ perceptions of assistants and is similar to the
prior findings illustrating that social reactions from people are indicative
of their surprise and experiences of unexpected events (e.g., errors) during
their interactions with robots (Stiber et al., 2022, 2023). However, further
research is needed to study how the valence of non-verbal reactions may be
used to model and estimate users’ perceptions of AI agents, which could then
be used to adapt the agents’ behaviors towards achieving desirable interaction
outcomes.
### 5.2. Portrayed Voice Gender Influences User Perceptions of the Voice
Assistant
Our hypothesis 2 predicts that masculine voice assistants are viewed as more
competent while feminine voice assistants are considered warmer. Our results
do not support this hypothesis since we observe that the feminine assistants
are perceived as more competent than masculine assistants (Fig. 4 and Table
2). We speculate that this preference of feminine voice assistants in the
context of online shopping assistance may be influenced by the broader
societal stereotypes associating women with competence in serving supportive
roles in assistance-type tasks (Powell et al., 2002; Koenig et al., 2011). It
is noteworthy that commonly available voice assistants such as Siri, Alexa,
and Cortana are marketed as “personal assistants” and typically employ
feminine voices (UNESCO, 2019); such de facto voice setting has shown to
attract abusive behavior (Curry and Rieser, 2019; Curry et al., 2020). Such
stereotypes have been translated to the preference of feminine voice
assistants in tasks that are “perceived” as domestic chores such as shopping
for groceries, schools supplies, and more (Lee et al., 2019). While digital
personal assistance should not be inherently associated with a specific
gender, when a gendered voice (e.g., feminine voice) is assigned to a voice
assistant, user perceptions of the online shopping task may align with the
gender stereotypes and biases seen in prior work (Lee et al., 2019; Ahn et
al., 2022; Damen and Toh, 2019; Tay et al., 2014). Hence, the association of
competence with feminine agents for assistance-type tasks found in our study
possibly hint at implicit preferences and biases driven by gender stereotypes.
The text-to-speech systems are evolving to generate increasingly realistic and
humanlike voices for AI agents (Seaborn et al., 2021). In this context, our
study revealed that the feminine voice is perceived to be more humanlike
($M=3.36$, 1 being robotic and 5 being humanlike) as compared to masculine
($M=2.96$) and ambiguous one ($M=2.59$). We speculate that the higher
perception of humanlikeness for feminine voices may again be attributed to
users’ familiarity with feminine assistants in real life, such as Alexa, Siri,
and Bixby. However, as voice assistants become more humanlike due to addition
of social cues to their behavior (Bucher et al., 2022), biases encountered by
these assistants could potentially reinforce gender preferences and biases in
actual human interactions. Therefore, designers and developers must carefully
consider the portrayed gender of voices while designing AI agents for specific
tasks to support people effectively while minimizing potential negative
consequences in both human-agent and human-human interactions.
### 5.3. Men Show More Biased Behavior
Our results highlight the impact of user gender on their behavior towards
gendered voice assistants while repairing human-agent relationship using
different mitigation strategies. Male participants generally reacted non-
verbally (socially) more to feminine assistants than masculine ones (Fig. 6 b
and Table 4). For example, a male participant smiled at feminine assistants
more than at masculine assistants. Similarly, another male participant
presented various types of reactions (two nodes, one smile, and one frown) to
feminine assistants whereas he only frowned once at the masculine assistants.
Moreover, male participants registered more non-verbal reactions to
“apologetic” agents when they sounded feminine as opposed to masculine (Fig. 6
c and Table 4). For instance, a male participant laughed six times when
interacting with feminine sounding apologetic assistant while only one time to
the masculine counterpart. Similarly, another male participant nodded twice,
smiled once, and raised an eyebrow (frowned) once in reaction to the feminine
sounding apologetic assistant, whereas no reaction was given to the masculine
sounding apologetic assistant. As previously discussed in Section 5.1, we see
that the valence and variety of reactions may vary depending on the individual
participant. Future research could explore how such diverse reactions
contribute to the reflection of biased behavior in assistance scenarios.
Prior work has also shown similar discrepancy in behavior between men and
women. Men found robots with feminine voices more trustworthy and engaging,
leading them to donate more money. On the other hand, women show no preference
between robots with feminine or masculine voices (Siegel et al., 2009).
Consistent with these prior findings, our results imply that male participants
engage more with feminine assistants, especially during apologetic
interactions which highlights the importance of considering user gender and
its potential impact on interactions with gendered voice assistants,
especially considering that these assistants will inevitably make mistakes in
their interactions with people in complex, real-world settings.
We further see biased behavior towards voice assistants by men—regardless of
the gender of voice assistants and error mitigation strategy, male
participants were found to interrupt the assistants on realization of error
before the assistant stopped talking almost twice as many times as female
participants would (Fig. 5 a and Table 3). These interruptions can be seen as
impolite behavior (Hutchby, 2008), possibly stemming from implicit power
dynamics where men are typically in authoritative roles (Powell et al., 2002).
Previous research in human-agent interactions, has also indicated that men
show less inclination to assist the agent, such as in the case of an AI study
companion (Lubold et al., 2016). Such biased behavior towards voice assistants
by men highlights the need for further examination of implicit power dynamics
and their impact on human-agent interactions.
We also observe that that male participants ($M=3.17$), as opposed to their
female counterparts ($M=3.45)$, overall, perceived the voice assistants to be
more feminine regardless of the portrayed gender of the voice and mitigation
strategy (Fig. 3 a and Table 1). Although this perception may arise from a
greater familiarity with “feminine commercial voice assistants” as discussed
in section 5.2, it can still have a negative impact on interactions with AI
assistants. Hence, it is of utmost importance to further understand impacts of
association of any gender to voice assistants, and interaction of agents’ and
users’ gender before continuing to create gendered voice assistants aimed to
assist and support people.
Despite male participants interrupting the voice assistants more during error,
we see that the interruptions between the male participants and the voice
assistants were fewer in number when the assistants were apologizing for
errors instead of compensating the participants (Fig. 5 b and Table 4). A
possible explanation for this finding is that it may be considered rude to
interrupt while someone is apologizing. Thus, voice assistants can potentially
utilize apologies to deter people from interrupting, thereby fostering more
polite, smooth, and respectful interactions. This finding implies that we can
tailor agent behavior such as using apology as a mitigation strategy to elicit
positive attitude from people in face of imperfect interactions.
### 5.4. Gender Ambiguous Voices as a Plausible Means to Reduce Biases
Gender ambiguous voices (e.g., “Q, the
genderless”666https://www.genderlessvoice.com/) is presented as a tool to
culminate gender stereotypes and biases in prior work on voice assistants
(Tolmeijer et al., 2021). Therefore, in addition to our hypotheses 2 and 3, we
explore if an ambiguous voice can eliminate or reduce the effects of
stereotypes associated with gendered voices for AI agents. Our results show
that ambiguous sounding assistants are perceived less warm compared to
feminine ones (Fig. 4 b and Table 2), suggesting that ambiguous voices can
potentially be employed to isolate gender stereotypes around warm personality
(feminine agents or robots are considered warmer than masculine ones (Ahn et
al., 2022; Nomura, 2017)) for voice assistants. This finding contradicts a
previous study where feminine and ambiguous voices are not disassociated with
stereotypical feminine traits such as family-orientation, delicacy, and
sensitivity (Tolmeijer et al., 2021). The distinct perception of warmth of our
gender-ambiguous voice, as compared to feminine one, indicates the potential
for using ambiguity of gender in voice to create less biased voice
interactions with AI agents.
Looking into behavioral measures, the interruptions between users and
ambiguous voice assistants were more frequent when the assistants compensated
instead of apologizing for the error (Fig. 5 c and Table 4). Fewer
interruptions generally reflect increased politeness (Hutchby, 2008).
Therefore, by employing a relational strategy like apologizing, ambiguous
assistants were able to elicit politeness and courtesy from users without
significant interruptions, despite being perceived as more robotic than
gendered assistants. Therefore, a design implication of our work is that
ambiguous voices for AI agents have the potential to maintain certain
benefits, such as promoting politeness and attentive listening, while reducing
the association with stereotypical feminine traits (such as warmth and
humanlikness) when the agent’s role is assistive in nature.
Our study additionally shows that ambiguous sounding assistants are perceived
to be the most robotic when compared to feminine and masculine ones. This
result, in combination with prior work showing that majority of people
preferred a robotic voice ($32.4\%$), followed by feminine ($20.6\%$) and
masculine ($11.8\%$) voice for conversational assistants (Curry et al., 2020),
highlights potential of gender ambiguous voices being welcomed by users.
However, further research is needed to establish the effectiveness of
ambiguous voices in a variety of assistance-type tasks; this work only serves
as an initial exploration in this direction.
### 5.5. Limitations and Future work
This work has a few limitations that can inform future work. First, online
shopping with voice assistants is novel to most people because of limited
capabilities and hindrance in adoption of the smart speakers for functional
tasks with higher risks than putting alarms and reminders, controlling smart
appliances, information retrieval, and playing music. Second, the experiment
was conducted in a lab setting where participants might not be fully
comfortable to interact freely with the speaker. Future work should
investigate the integration of voice assistants into people’s personal
physical spaces to better understand the organic interaction between humans
and agents in commonplace and everyday tasks.
Third, limited voices were explored for masculine, feminine and ambiguous
agents in this study. Future work can focus on understanding various
characteristics of voice generation towards creation and testing of genderless
and ambiguous voices. Moreover, there are other aspects, such as personality,
names assigned, and representations in form of color and appearance, to design
of voice assistants than just voice that may influence perception of their
gender (Sutton, 2020).
Fourth, our participants do not represent the full spectrum of gender, age,
ethnicity, and education. Thus, our findings and implications should be
interpreted with this context in mind. Specifically, due to large imbalance in
gender of participants, we could not include non-binary participant gender in
our analyses. Therefore, effects of gendered and ambiguous voices on non-
binary population is unexplored. Future research should aim to recruit
participants representative of all genders to understand interaction of users’
and assistants’ gender. Moreover, age of users may affect how they interact
with voice assistants as these assistants are likely to make more errors when
interacting with older adults due to the diminished ability in their speech
such as difficulty in constructing a structured sentence for a command (Kim et
al., 2021).
Overall, future research should explore voice-based interactions with virtual
assistants to assist people in realistic settings and how various aspects of
agent design such as voice, personality, and style of speech may be
manipulated to reduce biases caused due to the association of stereotypes
based on humanlike characteristics, such as race and gender, projected onto AI
agents.
## 6\. Conclusion
The growing use of intelligent voice assistants, their susceptibility to
errors, and the potential reinforcement of gender biases in user interactions
motivated us to study how gender portrayal through gendered voice influences
error mitigation strategies employed by these voice assistants. Our results
reveal that people exhibit higher social engagement, as evidenced by non-
verbal reactions, when interacting with voice assistants that apologize rather
than compensate for errors, possibly because apologetic voice assistants are
perceived as warmer while compensating ones are viewed as more competent.
Moreover, we observe that men prefer interacting with feminine voice
assistants that apologize for mistakes during assistive tasks. Furthermore,
our study reveals a notable gender difference in interruption patterns—men
interrupted voice assistants more frequently than women when errors occurred,
indicating potential sociocultural and power dynamics at play. Apart from
illustrating biases in users’ perceptions and behaviors towards gendered voice
assistants, our study highlights the potential of using ambiguous voices to
mitigate the association of stereotypical feminine traits, such as warmth,
with AI agents designed to assist people. This work takes a step towards
designing characteristics and behavior of voice assistants to foster
interactions that are engaging, while simultaneously addressing potential
gender biases, with the ultimate goal of enhancing the overall user
experience.
## References
* (1)
* Abercrombie et al. (2021) Gavin Abercrombie, Amanda Cercas Curry, Mugdha Pandya, and Verena Rieser. 2021. Alexa, Google, Siri: What are Your Pronouns? Gender and Anthropomorphism in the Design and Perception of Conversational Assistants. _arXiv preprint arXiv:2106.02578_ (2021).
* Ahn et al. (2022) Jungyong Ahn, Jungwon Kim, and Yongjun Sung. 2022. The effect of gender stereotypes on artificial intelligence recommendations. _Journal of Business Research_ 141 (2022), 50–59.
* Baker et al. (2021) Matthew Baker, Xiaohui Hu, Gennaro De Luca, and Yinong Chen. 2021. Intelligent Voice Instructor-Assistant System for Collaborative and Interactive Classes. _Journal of Artificial Intelligence and Technology_ 1, 2 (2021), 121–130.
* Bennett and Hill (2012) Aronté Marie Bennett and Ronald Paul Hill. 2012. The universality of warmth and competence: A response to brands as intentional agents. _Journal of Consumer Psychology_ 22, 2 (2012), 199–204.
* Bradley and LaFleur (2016) Gregory T Bradley and Elizabeth K LaFleur. 2016. Toward the development of hedonic-utilitarian measures of retail service. _Journal of Retailing and Consumer Services_ 32 (2016), 60–66.
* Bucher et al. (2022) Andreas Bucher, Dario Staehelin, Mateusz Dolata, and Gerhard Schwabe. 2022. Form Follows Function: Designing For Tensions Of Conversational Agents In Service Encounters. (2022).
* Cameron et al. (2021) David Cameron, Stevienna de Saille, Emily C Collins, Jonathan M Aitken, Hugo Cheung, Adriel Chua, Ee Jing Loh, and James Law. 2021. The effect of social-cognitive recovery strategies on likability, capability and trust in social robots. _Computers in Human Behavior_ 114 (2021), 106561.
* Chita-Tegmark et al. (2019) Meia Chita-Tegmark, Monika Lohani, and Matthias Scheutz. 2019. Gender effects in perceptions of robots and humans with varying emotional intelligence. In _2019 14th ACM/IEEE International Conference on Human-Robot Interaction (HRI)_. IEEE, 230–238.
* Cohen (1988) Jacob Cohen. 1988. Statistical power analysis for the behavioral sciences. _England: Routledge_ (1988).
* Cui et al. (2018) Yubao Cui, Xuehe Zhang, Xiaobao Peng, and Jianxun Chu. 2018. How to use apology and compensation to repair competence-versus integrity-based trust violations in e-commerce. _Electronic Commerce Research and Applications_ 32 (2018), 37–48.
* Curry and Rieser (2019) Amanda Cercas Curry and Verena Rieser. 2019. A crowd-based evaluation of abuse response strategies in conversational agents. _arXiv preprint arXiv:1909.04387_ (2019).
* Curry et al. (2020) Amanda Cercas Curry, Judy Robertson, and Verena Rieser. 2020. Conversational assistants and gender stereotypes: Public perceptions and desiderata for voice personas. In _Proceedings of the Second Workshop on Gender Bias in Natural Language Processing_. 72–78.
* Damen and Toh (2019) Nicole Damen and Christine Toh. 2019. Designing for trust: Understanding the role of agent gender and location on user perceptions of trust in home automation. _Journal of Mechanical Design_ 141, 6 (2019).
* Feine et al. (2019) Jasper Feine, Ulrich Gnewuch, Stefan Morana, and Alexander Maedche. 2019. A taxonomy of social cues for conversational agents. _International Journal of Human-Computer Studies_ 132 (2019), 138–161.
* Fiske et al. (2007) Susan T Fiske, Amy JC Cuddy, and Peter Glick. 2007. Universal dimensions of social cognition: Warmth and competence. _Trends in cognitive sciences_ 11, 2 (2007), 77–83.
* Ge et al. (2019) Xiang Ge, Dan Li, Daisong Guan, Shihui Xu, Yanyan Sun, and Moli Zhou. 2019. Do smart speakers respond to their errors properly? A study on human-computer dialogue strategy. In _International Conference on Human-Computer Interaction_. Springer, 440–455.
* Goodman and Mayhorn (2023) Kylie L Goodman and Christopher B Mayhorn. 2023. It’s not what you say but how you say it: Examining the influence of perceived voice assistant gender and pitch on trust and reliance. _Applied Ergonomics_ 106 (2023), 103864.
* Govier and Verwoerd (2002) Trudy Govier and Wilhelm Verwoerd. 2002. The promise and pitfalls of apology. _Journal of social philosophy_ 33, 1 (2002), 67–82.
* Guzman (2016) Andrea L Guzman. 2016. Making AI safe for humans: A conversation with Siri. In _Socialbots and their friends_. Routledge, 85–101.
* Habler et al. (2019) Florian Habler, Valentin Schwind, and Niels Henze. 2019. Effects of Smart Virtual Assistants’ Gender and Language. In _Proceedings of Mensch und Computer 2019_. 469–473.
* Haesevoets et al. (2013) Tessa Haesevoets, Chris Reinders Folmer, David De Cremer, and Alain Van Hiel. 2013. Money isn’t all that matters: The use of financial compensation and apologies to preserve relationships in the aftermath of distributive harm. _Journal of Economic Psychology_ 35 (2013), 95–107.
* Hutchby (2008) Ian Hutchby. 2008. Participants’ orientations to interruptions, rudeness and other impolite acts in talk-in-interaction. _Journal of Politeness Research-language Behaviour Culture_ 4 (08 2008), 221–241. https://doi.org/10.1515/JPLR.2008.011
* Hwang et al. (2019) Gilhwan Hwang, Jeewon Lee, Cindy Yoonjung Oh, and Joonhwan Lee. 2019. It sounds like a woman: Exploring gender stereotypes in South Korean voice assistants. In _Extended Abstracts of the 2019 CHI Conference on Human Factors in Computing Systems_. 1–6.
* Kim et al. (2021) Sunyoung Kim et al. 2021\. Exploring How Older Adults Use a Smart Speaker–Based Voice Assistant in Their First Interactions: Qualitative Study. _JMIR mHealth and uHealth_ 9, 1 (2021), e20427.
* Kim et al. (2007) Yanghee Kim, Amy L Baylor, and Entong Shen. 2007. Pedagogical agents as learning companions: the impact of agent emotion and gender. _Journal of Computer Assisted Learning_ 23, 3 (2007), 220–234.
* Koenig et al. (2011) Anne M Koenig, Alice H Eagly, Abigail A Mitchell, and Tiina Ristikari. 2011. Are leader stereotypes masculine? A meta-analysis of three research paradigms. _Psychological bulletin_ 137, 4 (2011), 616.
* Koo and Li (2016) Terry K Koo and Mae Y Li. 2016. A guideline of selecting and reporting intraclass correlation coefficients for reliability research. _Journal of chiropractic medicine_ 15, 2 (2016), 155–163.
* Krämer et al. (2016) Nicole C Krämer, Bilge Karacora, Gale Lucas, Morteza Dehghani, Gina Rüther, and Jonathan Gratch. 2016. Closing the gender gap in STEM with friendly male instructors? On the effects of rapport behavior and gender of a virtual agent in an instructional interaction. _Computers & Education_ 99 (2016), 1–13.
* Kraus et al. (2018) Matthias Kraus, Johannes Kraus, Martin Baumann, and Wolfgang Minker. 2018. Effects of gender stereotypes on trust and likability in spoken human-robot interaction. In _Proceedings of the eleventh international conference on language resources and evaluation (LREC 2018)_.
* Lee et al. (2010) Min Kyung Lee, Sara Kiesler, Jodi Forlizzi, Siddhartha Srinivasa, and Paul Rybski. 2010. Gracefully mitigating breakdowns in robotic services. In _2010 5th ACM/IEEE International Conference on Human-Robot Interaction (HRI)_. IEEE, 203–210.
* Lee et al. (2019) Sanguk Lee, Rabindra Ratan, and Taiwoo Park. 2019. The voice makes the car: Enhancing autonomous vehicle perceptions and adoption intention through voice agent gender and style. _Multimodal Technologies and Interaction_ 3, 1 (2019), 20.
* Lee et al. (2021) Sun Kyong Lee, Pavitra Kavya, and Sarah C Lasser. 2021. Social interactions and relationships with an intelligent virtual agent. _International Journal of Human-Computer Studies_ 150 (2021), 102608.
* Little (1988) Roderick JA Little. 1988. A test of missing completely at random for multivariate data with missing values. _Journal of the American statistical Association_ 83, 404 (1988), 1198–1202.
* Loideain and Adams (2020) Nora Ni Loideain and Rachel Adams. 2020. From Alexa to Siri and the GDPR: the gendering of virtual personal assistants and the role of data protection impact assessments. _Computer Law & Security Review_ 36 (2020), 105366.
* Lubold et al. (2016) Nichola Lubold, Erin Walker, and Heather Pon-Barry. 2016. Effects of voice-adaptation and social dialogue on perceptions of a robotic learning companion. In _2016 11th ACM/IEEE International Conference on Human-Robot Interaction (HRI)_. IEEE, 255–262.
* Mahmood et al. (2022) Amama Mahmood, Jeanie W Fung, Isabel Won, and Chien-Ming Huang. 2022. Owning Mistakes Sincerely: Strategies for Mitigating AI Errors. In _CHI Conference on Human Factors in Computing Systems_. 1–11.
* Maxham III and Netemeyer (2003) James G Maxham III and Richard G Netemeyer. 2003. Firms reap what they sow: the effects of shared values and perceived organizational justice on customers’ evaluations of complaint handling. _Journal of Marketing_ 67, 1 (2003), 46–62.
* Mitchell et al. (2011) Wade J Mitchell, Chin-Chang Ho, Himalaya Patel, and Karl F MacDorman. 2011. Does social desirability bias favor humans? Explicit–implicit evaluations of synthesized speech support a new HCI model of impression management. _Computers in Human Behavior_ 27, 1 (2011), 402–412.
* Nomura (2017) Tatsuya Nomura. 2017. Robots and gender. _Gender and the Genome_ 1, 1 (2017), 18–25.
* Obinali (2019) Chidera Obinali. 2019. The perception of gender in voice assistants. In _SAIS 2019 Proceedings of the Southern Association for Information Systems Conference_. 22–23.
* Ozogul et al. (2013) Gamze Ozogul, Amy M Johnson, Robert K Atkinson, and Martin Reisslein. 2013. Investigating the impact of pedagogical agent gender matching and learner choice on learning outcomes and perceptions. _Computers & Education_ 67 (2013), 36–50.
* Paraiso and Barthès (2006) Emerson Cabrera Paraiso and Jean-Paul A Barthès. 2006. An intelligent speech interface for personal assistants in R&D projects. _Expert Systems with Applications_ 31, 4 (2006), 673–683.
* Pearl (2016) Cathy Pearl. 2016. _Designing voice user interfaces: Principles of conversational experiences_. ” O’Reilly Media, Inc.”.
* Phan (2019) Thao Phan. 2019. Amazon Echo and the aesthetics of whiteness. _Catalyst: Feminism, Theory, Technoscience_ 5, 1 (2019), 1–38.
* Powell et al. (2002) Gary N Powell, D Anthony Butterfield, and Jane D Parent. 2002. Gender and managerial stereotypes: have the times changed? _Journal of management_ 28, 2 (2002), 177–193.
* Roschk and Kaiser (2013) Holger Roschk and Susanne Kaiser. 2013. The nature of an apology: An experimental study on how to apologize after a service failure. _Marketing Letters_ 24, 3 (2013), 293–309.
* Rubin (2004) Donald B Rubin. 2004. _Multiple imputation for nonresponse in surveys_. Vol. 81. John Wiley & Sons.
* Sáiz-Manzanares et al. (2020) María Consuelo Sáiz-Manzanares, Raúl Marticorena-Sánchez, and Javier Ochoa-Orihuel. 2020. Effectiveness of using voice assistants in learning: A study at the time of COVID-19. _International journal of environmental research and public health_ 17, 15 (2020), 5618.
* Schmid et al. (2022) Damaris Schmid, Dario Staehelin, Andreas Bucher, Mateusz Dolata, and Gerhard Schwabe. 2022. Does Social Presence Increase Perceived Competence? Evaluating Conversational Agents in Advice Giving Through a Video-Based Survey. _Proceedings of the ACM on Human-Computer Interaction_ 6, GROUP (2022), 1–22.
* Seaborn et al. (2021) Katie Seaborn, Norihisa P Miyake, Peter Pennefather, and Mihoko Otake-Matsuura. 2021. Voice in human–agent interaction: a survey. _ACM Computing Surveys (CSUR)_ 54, 4 (2021), 1–43.
* Siegel et al. (2009) Mikey Siegel, Cynthia Breazeal, and Michael I Norton. 2009. Persuasive robotics: The influence of robot gender on human behavior. In _2009 IEEE/RSJ International Conference on Intelligent Robots and Systems_. IEEE, 2563–2568.
* Stiber et al. (2022) Maia Stiber, Russell Taylor, and Chien-Ming Huang. 2022. Modeling human response to robot errors for timely error detection. In _2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)_. IEEE, 676–683.
* Stiber et al. (2023) Maia Stiber, Russell H Taylor, and Chien-Ming Huang. 2023. On using social signals to enable flexible error-aware HRI. In _Proceedings of the 2023 ACM/IEEE International Conference on Human-Robot Interaction_. 222–230.
* Strengers and Kennedy (2021) Yolande Strengers and Jenny Kennedy. 2021. _The smart wife: Why Siri, Alexa, and other smart home devices need a feminist reboot_. MIT Press.
* Sutton (2020) Selina Jeanne Sutton. 2020. Gender ambiguous, not genderless: Designing gender in voice user interfaces (VUIs) with sensitivity. In _Proceedings of the 2nd conference on conversational user interfaces_. 1–8.
* Tax et al. (1998) Stephen S Tax, Stephen W Brown, and Murali Chandrashekaran. 1998. Customer evaluations of service complaint experiences: implications for relationship marketing. _Journal of marketing_ 62, 2 (1998), 60–76.
* Tay et al. (2014) Benedict Tay, Younbo Jung, and Taezoon Park. 2014. When stereotypes meet robots: the double-edge sword of robot gender and personality in human–robot interaction. _Computers in Human Behavior_ 38 (2014), 75–84.
* Tolmeijer et al. (2021) Suzanne Tolmeijer, Naim Zierau, Andreas Janson, Jalil Sebastian Wahdatehagh, Jan Marco Marco Leimeister, and Abraham Bernstein. 2021. Female by Default?–Exploring the Effect of Voice Assistant Gender and Pitch on Trait and Trust Attribution. In _Extended Abstracts of the 2021 CHI Conference on Human Factors in Computing Systems_. 1–7.
* Turiman et al. (2013) Syamimi Turiman, Amelia Leong, and Fauziah Hassan. 2013. Are Men More Apologetic Than Women? _Pertanika Journal of Social Sciences & Humanities_ 21, 3 (2013).
* UNESCO (2019) EQUALS Skills Coalition UNESCO. 2019. I’d blush if I could: closing gender divides in digital skills through education. https://unesdoc.unesco.org/ark:/48223/pf0000367416.page=1. Accessed: 2021-09-02.
* van Ginkel et al. (2020) Joost R van Ginkel, Marielle Linting, Ralph CA Rippe, and Anja van der Voort. 2020. Rebutting existing misconceptions about multiple imputation as a method for handling missing data. _Journal of personality assessment_ 102, 3 (2020), 297–308.
* Woods (2018) Heather Suzanne Woods. 2018. Asking more of Siri and Alexa: Feminine persona in service of surveillance capitalism. _Critical Studies in Media Communication_ 35, 4 (2018), 334–349.
|
# Ground state preparation with shallow variational warm-start
Youle Wang<EMAIL_ADDRESS>Chenghong Zhu<EMAIL_ADDRESS>Mingrui Jing<EMAIL_ADDRESS>Xin Wang<EMAIL_ADDRESS>Institute
for Quantum Computing, Baidu Research, Beijing 100193, China
###### Abstract
Preparing the ground states of a many-body system is essential for evaluating
physical quantities and determining the properties of materials. This work
provides a quantum ground state preparation scheme with shallow variational
warm-start to tackle the bottlenecks of current algorithms, i.e., demand for
prior ground state energy information and lack of demonstration of efficient
initial state preparation. Particularly, our methods would not experience the
instability for small spectral gap $\Delta$ during pre-encoding the phase
factors since our methods involve only $\widetilde{O}(1)$ factors while
$\widetilde{O}(\Delta^{-1})$ is requested by the near-optimal methods. We
demonstrate the effectiveness of our methods via extensive numerical
simulations on spin-$1/2$ Heisenberg models. We also show that the shallow
warm-start procedure can process chemical molecules by conducting numerical
simulations on the hydrogen chain model. Moreover, we extend research on the
Hubbard model, demonstrating superior performance compared to the prevalent
variational quantum algorithms.
## I Introduction
The ground state is one crucial quantum state appearing in condensed matter
physics, quantum chemistry, and combinatorial optimisation. Physical reality
tends to force a quantum system to its ground state at low temperature, which
can exist with stability Feynman and Cohen (1956). Therefore, preparing the
exact form of the ground state becomes significant for evaluating the
expectations of physical quantities and determining the properties of
materials Sakurai and Commins (1995). Classical methods have raised success in
predicting matter behaviours. Based on quantum mechanics, traditional methods,
including, Hartree-Fock (HF) method Sherrill (2009), full configuration
interaction Knowles and Handy (1989), density functional method Parr (1980),
and Monte Carlo simulations Blankenbecler _et al._ (1981) have boosted
developments of quantum chemistry by providing accurate eigenstate
evaluations. However, the exponential growth of Hilbert space and multi-
determinant interactions Cade _et al._ (2020); Dagotto (1994); Nelson _et
al._ (2019) demand an explosive amount of memories leading to a restricted
efficiency regime and forcing the problem into an NP-complete complexity
Barahona (1982). The fermionic statistics also raise the sign problem, which
pulls down the convergence speed of approximation methods Gilks _et al._
(1995); Del Moral _et al._ (2006); Scalapino (2006); LeBlanc _et al._
(2015).
Quantum computer, originally proposed by Feynman Feynman (1982), is naturally
embedded in the quantum Hilbert space, indicating promising and competitive
near-term applications, including solving ground states of different physical
models and molecules Peruzzo _et al._ (2014); Wecker _et al._ (2015). The
typical results from variational quantum eigensolver (VQE) Peruzzo _et al._
(2014) demonstrated an excellent perspective for solving the ground state
energies of small molecules Kandala _et al._ (2017). Advanced variational
ansatzes have been designed and achieved better performance predicting the
physical properties and electronic behaviours Wecker _et al._ (2015); Dagotto
(1994), such as charge and spin densities, metal-insulator transitions, and
dynamics of distinct models, which could further energise the industrial
development Sapova and Fedorov (2022) covering new material design Ostaszewski
_et al._ (2021) and studies on high-temperature superconductivity Dallaire-
Demers _et al._ (2019); Cade _et al._ (2020). Another category, namely,
adiabatic quantum algorithms has practically solved the ground state of the
Ising model, demonstrating the potential of handling the combinatorial
optimisation problems Johnson _et al._ (2011).
One of the most promising technical routes to prepare the ground state is to
utilise phase estimation. Under a reasonable assumption that the initial state
$\rho$ has a non-zero overlap $\gamma$ regarding the exact ground state and
the ground state energy (within the required precision) is known, one can
directly apply phase estimation Kitaev (1995) together with amplitude
amplification Brassard _et al._ (2002) can project out the ground state from
$\rho$ with a probability proportional to $\gamma^{2}$. However, phase
estimation usually demands an extensive ancilla system, making the realisation
of preparing high-fidelity states challenging. Recently, Ge _et al._ (2019)
proposed a method to exponentially and polynomially improve the run-time on
the allowed error $\varepsilon$ and the spectral gap $\Delta$ and the initial
state overlap $\gamma$ dependencies, respectively, where $\Delta$ is the
difference between the ground and the first excited state energies. The same
authors also discussed that polylogarithmic scaling in $\varepsilon^{-1}$
could be attained using the method proposed by Poulin and Wocjan (2009)
requiring more ancilla qubits.
Quantum singular value transformation (QSVT) Gilyén _et al._ (2019) is
another promising route for realising the ground state preparation. Notably,
Lin and Tong (2020) have shown a method using constant number of ancilla
qubits to achieve a quantum complexity
$\mathcal{O}(\Delta^{-1}\gamma^{-1}\log(\varepsilon^{-1}))$111The
normalisation factor $\alpha$ in the block encoding is omitted.. Their
algorithms use the block encoding input model of the Hamiltonian Berry _et
al._ (2015) rather than the evolution operator used in phase estimation. They
also provide lower bounds of quantum complexity to show the algorithms’
dependence optimality on the spectral gap and initial overlap. Subsequently,
Dong _et al._ (2022) proposed a technique called “quantum eigenvalue
transformation of unitary matrices” (QET-U), which is closely related to
quantum signal processing Low and Chuang (2017a) and QSVT Gilyén _et al._
(2019). With QET-U and controlled Hamiltonian evolution, they further reduce
the ancilla qubits to two while reaching the same asymptotic scaling in
$\varepsilon$, $\Delta$, $\gamma$ as Lin and Tong (2020). Remarkably, both Lin
and Tong (2020) and Dong _et al._ (2022) assume an upper bound $\mu$ of
ground state energy at priority. If no such $\mu$ is known, it can be obtained
by first employing ground state energy estimation Abrams and Lloyd (1999);
Higgins _et al._ (2007); Berry _et al._ (2009); Poulin and Wocjan (2009);
Knill _et al._ (2007); Lin and Tong (2020); Dong _et al._ (2022); Lin and
Tong (2022); Ding and Lin (2022); O’Brien _et al._ (2019); Wang _et al._
(2019); Somma (2019); Ge _et al._ (2019); Wan _et al._ (2022); Wang _et
al._ (2022a) with precision $\mathcal{O}(\Delta)$, which would increase the
cost of state preparation.
Fig 1: The outline of our ground state preparation with shallow variational
warm-start involving (left) the three applicable physical models of Heisenberg
spin-$1/2$ chain model, chemical molecular model and Fermi-Hubbard model could
be efficiently solved via our scheme; (middle) the two main components of our
scheme of (a) shallow-depth VQE as the warm-start and (b) the ground state
preparation algorithms provided the warm-start initial state; (right) our
demonstrations from testing (top) warm-start with a massive number of models
and (bottom) the principle of the ground state preparation. The sign function
$F(x)$ is simulated via the ground state preparation circuit.
To solve ground-state problems in physics and chemistry, we notice the demand
for overcoming limitations set by the current algorithms, such as the prior
information of the ground-state energy and an efficient initial state
preparation. This paper addresses these issues using shallow-depth VQE and the
recent advances in quantum phase estimation Wang _et al._ (2022b).
Intuitively, one would expect to prepare the ground state with very high
fidelity when estimating the ground state energy. However, it cannot be
realised using the conventional phase estimation due to the imperfect residual
states, as discussed in Poulin and Wocjan (2009); Ge _et al._ (2019).
Fortunately, this expectation can be fulfilled by the algorithm of Wang _et
al._ (2022b), which uses only one ancilla qubit and can apply to block
encoding and Hamiltonian evolution input models. We show that measuring the
ancilla qubit can estimate the ground state energy up to the desired precision
with a probability $\approx\gamma^{2}$. Setting the precision as
$\mathcal{O}(\Delta)$, the ground state can be prepared up to exponentially
high precision at a cost
$\widetilde{\mathcal{O}}(\gamma^{-2}\Delta^{-1})$222Notation
$\widetilde{\mathcal{O}}$ hides the logarithmic factors.. As a notable
property, the prior ground state energy information is no longer necessary.
Methods | Input | #Phase | Query | #Ancilla Qubits | Need prior | Need amp.
---|---|---|---|---|---|---
Model | Factors | Complexity | GSE info.? | amplif.?
This work | HE | $\widetilde{\mathcal{O}}(1)$ | $\widetilde{\mathcal{O}}(\Delta^{-1}\gamma^{-2})$ | $1$ | No | No
BE | $\widetilde{\mathcal{O}}(\alpha\Delta^{-1}\gamma^{-2})$ | 2
Dong _et al._ (2022) | HE | $\widetilde{\mathcal{O}}(\Delta^{-1})$ | $\widetilde{\mathcal{O}}(\Delta^{-1}\gamma^{-2})$ | 1 | Yes | No
$\widetilde{\mathcal{O}}(\Delta^{-1}\gamma^{-1})$ | 2 | Yes
Lin and Tong (2020) | BE | $\widetilde{\mathcal{O}}(\Delta^{-1})$ | $\widetilde{\mathcal{O}}(\alpha\Delta^{-1}\gamma^{-1})$ | $\mathcal{O}(1)$ | Yes | Yes
Ge _et al._ (2019) | HE | — | $\widetilde{\mathcal{O}}(\Delta^{-1}\gamma^{-1})$ | $\mathcal{O}\left(\log(\Delta^{-1})+\log\log(\varepsilon^{-1})\right)$ | Yes | Yes
Table 1: Comparison to the current quantum ground-state preparation
algorithms. $\Delta$ is a lower bound of the spectral gap of the Hamiltonian.
Here, we use $\gamma=\sqrt{\operatorname{tr}(\rho\Pi)}$ to represent the
overlap between the initial and ground states, where $\Pi$ denotes a
projection onto the subspace generated by ground states. “GSE” represents the
ground state energy, and “amp. amplif.” is the abbreviation of amplitude
amplification. “HE” means Hamiltonian evolution $U=e^{iH}$, and “BE” stands
for block encoding $U_{H}=\left[\begin{smallmatrix}H/\alpha&\cdot\\\
\cdot&\cdot\end{smallmatrix}\right]$, where $\alpha$ is the normalisation
factor. If $U_{H}$ is constructed by the linear-combination-of-unitaries
technique Childs _et al._ (2018), the ancillas of our method can be further
reduced to 1. The number of ancillas of Lin and Tong (2020) is at least three,
which are used to block-encode the projector by QSVT. The ancillas for
implementing amplitude amplification need to be counted as well.
In our algorithms, we apply a sequence of single-qubit rotations on the
ancilla qubit and interleave the power of the input models. The structures of
QSVT and QET-U are similar to ours but use block encoding and Hamiltonian
evolution, respectively. Rotation angles, usually referred to as the phase
factors, are chosen so that the circuit can simulate a specific function, such
as sign function Lin and Tong (2020); Wang _et al._ (2022b) and shifted sign
function Dong _et al._ (2022). The number of phase factors is crucial for
preparing a high-fidelity ground state. In Lin and Tong (2020); Dong _et al._
(2022), the number scales as $\mathcal{O}(\Delta^{-1}\log(\varepsilon^{-1}))$,
given the spectral gap $\Delta$ and an approximation error $\varepsilon$.
However, classically computing the factors would suffer from numerical
instability, especially when $\Delta$ is small. A convex-optimisation-based
method is proposed in Dong _et al._ (2022) to generate a near-optimal
approximation. In contrast, our algorithms advantageously require
$\widetilde{\mathcal{O}}(1)$ number of factors, endowing better applicability
in practice than Lin and Tong (2020); Dong _et al._ (2022). In particular,
our numerical experiments have used circuits with fixed phase factors to
prepare ground states of more than 200 random Hamiltonian instances. To better
understand our algorithms, we provide a schematic diagram in Fig. 1 and
summarize the comparisons to Ge _et al._ (2019); Lin and Tong (2020); Dong
_et al._ (2022) in Table 1.
Although algorithms of Lin and Tong (2020); Dong _et al._ (2022) have better
scaling in $\gamma$, the scheme relies on amplitude amplification incurring
more ancilla qubits and deeper circuits. The quantum complexity can be
compensated by preparing an initial state with an overlap larger than 0.1
since the quadratic scaling in overlap would make no big difference. To this
end, we have conducted extensive zero-noise numerical simulations to apply a
depth-three VQE to warm-start ground-state preparation of many-body systems
and find that a shallow-depth VQE could play a good overlap initialiser.
Firstly, we randomly test 10000 two-body interactions, spin-$1/2$ Heisenberg-
type models for every case $\leq$ 10 qubits, 200 and 100 models for the cases
of 12 and 14 qubits, respectively. We observe that the optimised overlaps
highly cluster between 0.4 and 1, and ground states with the fidelity of
almost one are prepared via our algorithms. Secondly, we investigate the
shallow-depth warm-start on molecular ground state preparation offering
satisfactory performances for small bond distances. Thirdly, we apply our
methods to solve the Hubbard model which the 2D case is recognised as a
classical challenge LeBlanc _et al._ (2015); Scalapino (2006); hub (2013);
Cade _et al._ (2020) and quantumly demonstrated only by VQE-based methods
Cade _et al._ (2020); Wecker _et al._ (2015). Our scheme derives extremely
accurate charge-spin densities and the chemical potentials of a specific 1D
model, reaching an absolute error around $10^{-5}$ beyond the chemical
accuracy with respect to the full configuration interaction (FCI) method.
Organisation. In Sec. II, we give technical details of algorithms for ground
state preparation and provide a method to analyse the success probability. In
Sec. III, we show extensive numerical results that demonstrate the efficacy of
a constant-depth VQE in processing Heisenberg models up to 14 qubits. We also
show correctness by applying the algorithms to the optimised states of VQE. In
Sec. IV, we numerically examine the constant-depth warm-start on a hydrogen
chain model. In Sec. V, we use our algorithms to study the Fermi-Hubbard
models and accurately evaluate the model’s physical properties. Finally, we
summarize our results and discuss potential applications in Sec. VI.
## II Ground state preparation
In this section, we present ground-state preparation algorithms. The core of
our approaches has two steps: firstly, prepare an initial state with a
considerable overlap, such as states generated by shallow-depth VQE, Gibbs-VQE
Wang _et al._ (2021), or Hartree-Fock states (which will be discussed in
numerical experiments); secondly, use a quantum algorithm to project out the
ground state from the initial state with high probability. The projection can
be realised using the quantum algorithmic tool, namely the quantum phase
search algorithm (QPS) Wang _et al._ (2022b), which can classify the
eigenvectors of a unitary operator. Using QPS, the probability of sampling the
ground state is highly approximate to the initial overlap. Therefore, by
repeating these two steps sufficient times, one can obtain the desired state
with arbitrarily high precision.
Motivated by the previous works Dong _et al._ (2022); Ge _et al._ (2019);
Lin and Tong (2020), we establish the algorithms for the ground state
preparation with two different Hamiltonian input modes: one is the real-time
Hamiltonian evolution $U=e^{iH}$, and the other uses the block encoding
operator Low and Chuang (2019), which is a unitary matrix
$U_{H}=\left[\begin{smallmatrix}H/\alpha&\cdot\\\
\cdot&\cdot\end{smallmatrix}\right]$ with a scaled Hamiltonian in the upper-
left corner. Since the size of $U_{H}$ is larger than that of $H$, it would
require an ancilla system to implement $U_{H}$ on quantum computers. A common
strategy to block-encode the Hamiltonian of a many-body system is the linear-
combination-of-unitaries (LCU) technique Childs _et al._ (2018). We,
therefore, could regard the LCU-type block encoding as the input of our
algorithms. To better describe our algorithms, we first make some assumptions:
the ancilla system for implementing $U_{H}$ is composed of $m$ qubits, the
spectrum of the matrix $H$ is restricted in the region $(-\pi,\pi)$, the
spectral gap of $H$ is larger than $\Delta>0$, and $\|H\|\leq\alpha$ regarding
the spectral norm. With these, the results of the algorithms are summarized in
Theorem 1.
###### Theorem 1 (Ground state preparation)
Suppose a Hamiltonian of the form
$H=\sum_{j=0}^{2^{n}-1}\lambda_{j}\op{\psi_{j}}{\psi_{j}}$, where
$-\pi\leq\lambda_{0}\leq\lambda_{1}\leq\ldots\leq\lambda_{2^{n}-1}\leq\pi$.
Let $\Delta>0$ be a lower bound of the spectral gap. Assume access to a
quantum state $\rho$ that has a non-zero overlap
$\gamma=\sqrt{\operatorname{tr}(\rho\Pi)}$, where $\Pi$ denotes a projection
operator onto the ground state subspace. Then there exists a quantum algorithm
that prepares the ground state with a precision of at least $1-\varepsilon$.
* •
If the Hamiltonian is accessed through a Hamiltonian evolution operator
$U=e^{iH}$, then the total queries to controlled $U$ and $U^{\dagger}$ is
$\mathcal{O}(\gamma^{-2}\Delta^{-1}\log(\varepsilon^{-1}\log(\Delta^{-1})))$.
The number of used ancilla qubits is $1$ and used copies of $\rho$ is
$\mathcal{O}(\gamma^{-2})$.
* •
If the Hamiltonian is accessed through a block encoding operator
$U_{H}=\left[\begin{smallmatrix}H/\alpha&\cdot\\\
\cdot&\cdot\end{smallmatrix}\right]$ that satisfies $\left\langle
0^{m}\middle|U_{H}\middle|0^{m}\right\rangle=H/\alpha$, then the total queries
to controlled $U_{H}$ and $U_{H}^{\dagger}$ is
$\mathcal{O}(\alpha\gamma^{-2}\Delta^{-1}\log(\varepsilon^{-1}\log(\alpha\Delta^{-1})))$.
The number of used ancilla qubits is $2$ and used copies of $\rho$ is
$\mathcal{O}(\gamma^{-2})$.
The trace distance of quantum states characterises the precision, and the
notation $|0^{m}\rangle$ really indicates $|0^{m}\rangle\otimes I$, where we
have omitted the identity operator on the main system.
Next, we concretely discuss the algorithm using Hamiltonian evolution in Sec.
II.1. We provide the detailed procedure of finding the ground state via QPS
and the analysis of the output probability. The algorithm using block encoding
is further discussed in Sec. II.2. Since the analysis of the algorithm’s
correctness is similar from the previous literature, we mainly discuss the
differences from using the Hamiltonian evolution.
### II.1 Algorithm using real-time Hamiltonian evolution
Given access to the controlled versions of the unitary $U=e^{iH}$ and
$U^{\dagger}=e^{-iH}$, we construct the quantum circuit of QPS as shown in
Fig. 2.
$\Qcircuit@C=0.2em@R=1em{\lstick{|0\rangle}&\gate{R_{z}(\phi_{L})}\gate{R_{y}(\theta_{L})}\ctrl{1}\gate{R_{z}(\phi_{L-1})}\gate{R_{y}(\theta_{L-1})}\ctrlo{1}\gate{R_{z}(\phi_{L-2})}\gate{R_{y}(\theta_{L-2})}\qw\cdots\ctrlo{1}\gate{R_{z}(\phi_{0})}\gate{R_{y}(\theta_{0})}\gate{R_{z}(\omega)}\qw\meter\\\
\lstick{\rho}/^{n}\qw\qw\gate{U}\qw\qw\gate{U^{\dagger}}\qw\qw\qw\cdots\gate{U^{\dagger}}\qw\qw\qw\qw}$
Fig 2: Quantum circuit for quantum phase search. The unitary $U$ would be
adaptively changed with its power. Parameters $\bm{\phi}$, $\bm{\theta}$, and
$\omega$ are chosen such that the circuit can implement a trigonometric
polynomial transformation to each phase of the unitary operator. The integer
$L$ is the order of the polynomial.
There is an ancilla system consisting of only one qubit, which is applied with
a sequence of single-qubit rotations. Meanwhile, several controlled unitaries
are interleaved between rotations. The phase factors $\bm{\phi}$,
$\bm{\theta}$, and $\omega$ are chosen such that the circuit implements a
trigonometric polynomial transformation to each phase of the unitary, i.e.,
$f\mathrel{\mathop{\mathchar
58\relax}}\lambda\mapsto\sum_{j=-L}^{L}c_{j}e^{ij\lambda}$, where each
$c_{j}\in\mathbb{C}$ and satisfies $\sum_{j}|c_{j}|\leq 1$. For a certain
trigonometric polynomial, the corresponding factors can be classically
computed beforehand Dong _et al._ (2021); Wang _et al._ (2022b). The action
of running the circuit to an eigenvector of the unitary $U$ is presented in
the lemma below.
###### Lemma 2 (Quantum phase evolution Wang _et al._ (2022b))
Given a unitary operator
$U=\sum_{j=0}^{2^{n}-1}e^{i\lambda_{j}}|\psi_{j}\rangle\\!\langle\psi_{j}|$,
for any trigonometric polynomial $f(x)=\sum_{j=-L}^{L}c_{j}e^{ijx}$ with
$\sum_{j}|c_{j}|\leq 1$, there exist parameters $\omega\in{{\mathbb{R}}}$ and
$\bm{\theta},\bm{\phi}\in{{\mathbb{R}}}^{L+1}$ such that the circuit in Fig. 2
transforms the initial state $|0,\psi\rangle$ into
$\left(\sqrt{(1+f(\lambda))/2}|0\rangle+\sqrt{(1-f(\lambda))/2}|1\rangle\right)|\psi\rangle$,
where $|\psi\rangle$ denotes an eigenvector of $U$ with a phase
$\lambda\in(-\pi,\pi)$.
For QPS, the trigonometric polynomial $f(x)=\sum_{j=-L}^{L}c_{j}e^{ijx}$ is
chosen to approximate the sign function. Technically, supposing a constant
$\kappa\in(0,1/2)$, we could find a $f(x)$ with order
$L\in\mathcal{O}(\kappa^{-1}\log(\varepsilon^{-1}))$ such that
$|f(x)-1|\leq\varepsilon$ for all $x\in(\kappa,\pi-\kappa)$ and
$|f(x)+1|\leq\varepsilon$ for all $x\in(-\pi+\kappa,-\kappa)$ Low and Chuang
(2017b); Wang _et al._ (2022b). Particularly, given the relation between the
order $L$ and precision $\varepsilon$, we could choose $\varepsilon$ to be
exponentially small at the expense of a modest increase in the circuit depth.
Therefore, eigenvectors fall in the region
$(-\pi+\kappa,-\kappa)\cup(\kappa,\pi-\kappa)$ are labelled by states 0 or 1,
and the collapsed state from the measured ancilla qubit could indicate the
label. For instance, for any state
$|\psi\rangle=\sum_{j}\alpha_{j}|\psi_{j}\rangle$, the circuit of QPS
transforms the initial state $|0,\psi\rangle$ into the following form.
$\displaystyle|0\rangle\sum_{j}\alpha_{j}\sqrt{\frac{1+f(\lambda_{j})}{2}}|\psi_{j}\rangle+|1\rangle\sum_{j}\alpha_{j}\sqrt{\frac{1-f(\lambda_{j})}{2}}|\psi_{j}\rangle.$
(1)
We readily know that $|0\rangle$ is associated with eigenvectors in the region
$(-\kappa,\pi+\kappa)$, and $|1\rangle$ is associated with eigenvectors in the
region $(-\pi-\kappa,\kappa)$. The only possible source of error comes from
the intrinsic sign function approximating error from the searching procedures
discussed previously, which raises negligible effect. This property enables
one to efficiently locate the region where the ground state energy falls
through a binary search.
#### Rough ground state energy search
By executing the binary search, the region containing the ground state energy
would shrink gradually, and the probability of outputting the ground state is
analyzed later. After each iteration, the underlying region is cut into two
halves, and one of them is chosen, where the post-measurement state falls in,
for the next iteration. Since the region’s size decreases by nearly half, the
search could halt in a logarithmic time, producing an exponentially-small
region. When the final region of the ground state is attained, any point in
the final region could be taken as an estimate of ground state energy. In
particular, if the estimation accuracy is smaller than half of the spectral
gap, the ground state would eventually fall into a region whose size is
smaller than the spectral gap, and excited states would be filtered out. At
that time, the post-measurement state on the primary system is the desired
ground state.
To better understand the algorithm, we sketch the search procedure below. For
remark, the final post-measurement state is not the desired ground state. We
defer the discussion of improving the estimation accuracy.
1. 1.
Input: unitary $U=e^{iH}$, constant $\kappa$ (irrelevant to $H$), error
tolerance $\varepsilon$, and state $\rho$ that non-trivially overlaps with the
ground state.
2. 2.
Compute parameters $\bm{\phi}$, $\bm{\theta}$, and $\omega$ according to
parameters $\kappa$ and $\varepsilon$.
3. 3.
Choose the initial region $(x_{0},x_{1})$, where $x_{0}=-\pi$ and $x_{1}=\pi$.
4. 4.
Calculate the middle point of the region $(x_{0},x_{1})$, which is $x_{\rm
mid}=\frac{x_{0}+x_{1}}{2}$.
5. 5.
Set the circuit of QPS using controlled $\tilde{U}=e^{-ix_{\rm mid}}U$ and
$\tilde{U}^{\dagger}$ and the calculated parameters from the step 2.
6. 6.
Input the initial state $\op{0}{0}\otimes\rho$ and run the circuit.
7. 7.
Measure the ancilla qubit of the circuit and update the region according to
the measurement outcome: when $x_{1}-x_{0}>2\pi-2\kappa$, execute the update
$\left\\{\begin{matrix}(x_{0},x_{1})\leftarrow(x_{\rm
mid}-\kappa,x_{1}+\kappa),&\textrm{if the outcome is 0;}\\\\[5.0pt]
(x_{0},x_{1})\leftarrow(x_{0}-\kappa,x_{\rm mid}+\kappa),&\textrm{if the
outcome is 1.}\end{matrix}\right.$ (2)
When $x_{1}-x_{0}\leq 2\pi-2\kappa$, execute the update
$\left\\{\begin{matrix}(x_{0},x_{1})\leftarrow(x_{\rm
mid}-\kappa,x_{1}),&\textrm{if the outcome is 0;}\\\\[5.0pt]
(x_{0},x_{1})\leftarrow(x_{0},x_{\rm mid}+\kappa),&\textrm{if the outcome is
1.}\end{matrix}\right.$ (3)
8. 8.
Go to step 4 and repeat till the region converges.
9. 9.
Output: final region $(x_{0},x_{1})$.
After executing the rough search procedure, the size of the region would
converge to $2\kappa$. Specifically, after the first iteration, the size of
the region is $\kappa+\pi/2$. Since then, the region is updated as
$(\theta-\kappa,x_{1})$ or $(x_{0},\theta+\kappa)$, and the size of the new
region is $\kappa+(x_{1}-x_{0})/2$. Then we can inductively find the size of
the final region. Let $(x_{0}^{(t)},x_{1}^{(t)})$ denote the region in the
$t$-iteration and $Q$ denote the total number of iterations, then we have
$\displaystyle|x_{1}^{(Q)}-x_{0}^{(Q)}|$
$\displaystyle=\kappa+\frac{|x_{1}^{(Q-1)}-x_{0}^{(Q-1)}|}{2}=\kappa+\frac{1}{2}\left[\kappa+\frac{|x_{1}^{(Q-2)}-x_{0}^{(Q-2)}|}{2}\right]=\cdots=2\kappa+\frac{\pi}{2^{Q}}.$
(4)
#### Improving search accuracy
Suppose the ground state energy falls in the region
$(x_{0}^{(Q)},x_{1}^{(Q)})$, then the above procedure would return a rough
estimate of the ground state energy up to precision $\kappa+\pi/2^{Q+1}$.
However, the estimation precision is not guaranteed to be smaller than
$\Delta/2$. Thus more processes are needed to improve the precision. It is
realised by applying the power of the unitary operator that amplifies the
phases of the post-measurement state. By repeatedly running the circuit with
the unitary power, the ground state energy is efficiently constrained to a
smaller region. To be specific, firstly, run the rough search procedure once
with unitary $U^{(0)}=U$, attaining a region of size $2\kappa+\pi/2^{Q}$. For
clarity, denote this region by $(\zeta_{0}^{(1)},\zeta_{1}^{(1)})$ and
$\bar{\kappa}=\kappa+\pi/2^{Q+1}$, where
$|\zeta_{1}^{(1)}-\zeta_{0}^{(1)}|=2\bar{\kappa}$. Let
$\lambda^{(1)}=(\zeta_{0}^{(1)}+\zeta_{1}^{(1)})/2$ denote the middle point of
the region. Secondly, consider a modified unitary
$U^{(1)}=(e^{-i\lambda^{(1)}}U^{(0)})^{\lfloor 1/\bar{\kappa}\rfloor}$333The
notation $\lfloor x\rfloor$ denotes the floor function that gives as output
the greatest integer less than or equal to $x$.. For $U^{(1)}$, phases of
eigenvectors in the post-measurement state are rescaled to $(-1,1)$. Now,
running the rough search procedure with the new unitary operator would give a
new region $(\zeta_{0}^{(2)},\zeta_{1}^{(2)})$.
Executing the above two steps would exponentially fast improve the estimation
accuracy compared with the rough searching scheme. For state
$|\psi_{0}\rangle$, let $\lambda_{0}$ denote its original phase in $U$. Its
phase in unitary $U^{(1)}$ is $(\lambda_{0}-\lambda^{(1)})\lfloor
1/\bar{\kappa}\rfloor$ and falls in the region
$(\zeta_{0}^{(2)},\zeta_{1}^{(2)})$. Let
$\lambda^{(2)}=(\zeta_{0}^{(2)}+\zeta_{1}^{(2)})/2$ denote the middle point of
the second region $(\zeta_{0}^{(2)},\zeta_{1}^{(2)})$. Then we readily give an
inequality that characterises the estimation error
$\left|\lambda^{(2)}-(\lambda_{0}-\lambda^{(1)})\lfloor
1/\bar{\kappa}\rfloor\right|\leq\bar{\kappa}$. Rewrite this inequality as
$\left|\lambda_{0}-\left(\lambda^{(1)}+\lambda^{(2)}/\lfloor
1/\bar{\kappa}\rfloor\right)\right|\leq\bar{\kappa}/\lfloor
1/\bar{\kappa}\rfloor\leq\lfloor 1/\bar{\kappa}\rfloor^{-2}$. From this
equation, we can see that the first two steps of the scheme give an estimate
of the phase with an error of $\lfloor 1/\bar{\kappa}\rfloor^{-2}$. So,
inductively repeating the above procedure leads to a sequence
$(\lambda^{(1)},\lambda^{(2)},\ldots)$ that could be taken as an estimate of
the ground state energy. Assuming repeats $j$ times, the estimate is given by
$\displaystyle\left|\lambda_{0}-\left(\lambda^{(1)}+\frac{\lambda^{(2)}}{\lfloor
1/\bar{\kappa}\rfloor}+\frac{\lambda^{(3)}}{\lfloor
1/\bar{\kappa}\rfloor^{2}}+\cdots+\frac{\lambda^{(j)}}{\lfloor
1/\bar{\kappa}\rfloor^{j-1}}\right)\right|\leq\frac{\bar{\kappa}}{\lfloor
1/\bar{\kappa}\rfloor^{j-1}}\leq\lfloor 1/\bar{\kappa}\rfloor^{-j}.$ (5)
Clearly, reaching the halt condition only requires repeating at most
$\mathcal{O}(\log(1/\Delta)/\log(\lfloor 1/\bar{\kappa}\rfloor))$ times.
#### Output probability analysis
Due to many intermediate measurements in QPS, the search procedure is
conducted randomly. The event that the ground state is output would occur with
a probability. However, how the analysis of this probability is missing in
Wang _et al._ (2022b). We use an example to explain how to analyse the output
probability, which can be used in more general cases.
Example. If we collect all measurement outcomes together to represent a
trajectory that the ground state moves along, the search process of QPS can be
depicted using a binary tree as shown in Fig. 3. The root node represents the
initial state $\rho$, and other nodes represent the post-measurement state
with a non-zero overlap concerning the ground state. Each leaf node in the
tree represents a possible searched ground state from the program. Each
trajectory consists of four nodes, meaning that QPS finds the ground state
after three iterations. In addition, the line goes left, meaning that the
measurement outcome is 0, and goes right, meaning that the measurement outcome
is 1.
$\rho$$\rho_{1,1}$$\rho_{2,1}$$\rho_{3,1}$$\rho_{2,2}$$\rho_{3,2}$$\rho_{1,2}$$\rho_{2,3}$$\rho_{3,3}$$\rho_{3,4}$
Fig 3: An example of a binary tree diagram for sketching phase search. The
node with notation $\rho_{l,k}$ represents the $k$th post-measurement state at
$l$ iteration.
By the law of total probability, the probability of finding the ground state
is the sum of the probability of following each trajectory. We take the most
left trajectory that reaches the leaf node $\rho_{3,1}$ as an example to
explain how to compute the probability of a trajectory. Let
$\Pr[\rho_{i,j}|\rho_{p,l}]$ denote the conditional probability of reaching
node $\rho_{i,j}$ from node $\rho_{p,l}$. Clearly, the probability that
$\rho_{p,l}$ transits to $\rho_{i,j}$ only depends on the state $\rho_{p,l}$.
Then the probability of reaching the leaf node $\rho_{3,1}$ from the root node
can be given by
$\displaystyle\Pr[\rho_{3,1}]=\Pr[\rho_{3,1}|\rho_{2,1}]\times\Pr[\rho_{2,1}|\rho_{1,1}]\times\Pr[\rho_{1,1}|\rho].$
(6)
To represent probability $\Pr[\rho_{1,1}|\rho]$, we define linear maps
$\mathcal{M}_{(1,1)}^{(0)}$ that transform the input density matrix $\rho$ to
an un-normalised post-measurement state, where $k\in\\{0,1\\}$ indicates the
measurement outcome. Recalling Theorem 6 of Wang _et al._ (2022b), the
measurement probabilities are
$\Pr[0]=\frac{1}{2}(1+\sum_{j}\left\langle\psi_{j}\middle|\rho\middle|\psi_{j}\right\rangle
f(\lambda_{j}))$ and
$\Pr[1]=\frac{1}{2}(1-\sum_{j}\left\langle\psi_{j}\middle|\rho\middle|\psi_{j}\right\rangle
f(\lambda_{j}))$. Then the action of $\mathcal{M}_{(1,1)}^{(k)}$ is described
as follows.
$\displaystyle\mathcal{M}_{(1,1)}^{(0)}\mathrel{\mathop{\mathchar
58\relax}}\rho=\sum_{l}p_{l}\op{\phi_{l}}{\phi_{l}}\mapsto\sum_{l}p_{l}\op{\widetilde{\phi}_{l}^{(0)}}{\widetilde{\phi}_{l}^{(0)}},\quad\mathrm{where}\quad|\widetilde{\phi}_{l}^{(0)}\rangle=\sum_{j}\sqrt{\frac{1+f(\lambda_{j})}{2}}\langle\psi_{j}|\phi_{l}\rangle|\psi_{j}\rangle.$
(7) $\displaystyle\mathcal{M}_{(1,1)}^{(1)}\mathrel{\mathop{\mathchar
58\relax}}\rho=\sum_{l}p_{l}\op{\phi_{l}}{\phi_{l}}\mapsto\sum_{l}p_{l}\op{\widetilde{\phi}_{l}^{(1)}}{\widetilde{\phi}_{l}^{(1)}},\quad\mathrm{where}\quad|\widetilde{\phi}_{l}^{(1)}\rangle=\sum_{j}\sqrt{\frac{1-f(\lambda_{j})}{2}}\langle\psi_{j}|\phi_{l}\rangle|\psi_{j}\rangle.$
(8)
Note that $|\widetilde{\phi}_{l}^{(0)}\rangle$ and
$|\widetilde{\phi}_{l}^{(1)}\rangle$ are not quantum states but represent
column vectors. For other probabilities, the maps are defined similarly. That
is, the map that transforms $\rho_{i,j}$ to $\rho_{p,l}$ is denoted by
$\mathcal{M}_{(i,j)\to(p,l)}^{(0)}$. With these notations, the probabilities
are re-expressed below.
$\displaystyle\Pr[\rho_{1,1}|\rho]$
$\displaystyle=\operatorname{tr}\left(\mathcal{M}_{(1,1)}^{(0)}(\rho)\right)$
(9) $\displaystyle\Pr[\rho_{2,1}|\rho_{1,1}]$
$\displaystyle=\operatorname{tr}\left(\mathcal{M}_{(1,1)\to(2,1)}^{(0)}(\rho_{1,1})\right)=\frac{\operatorname{tr}\left(\mathcal{M}_{(1,1)\to(2,1)}^{(0)}\circ\mathcal{M}_{(1,1)}^{(0)}(\rho)\right)}{\operatorname{tr}\left(\mathcal{M}_{(1,1)}^{(0)}(\rho)\right)}$
(10) $\displaystyle\Pr[\rho_{3,1}|\rho_{2,1}]$
$\displaystyle=\operatorname{tr}\left(\mathcal{M}_{(2,1)\to(3,1)}^{(0)}(\rho_{2,1})\right)=\frac{\operatorname{tr}\left(\mathcal{M}_{(2,1)\to(3,1)}^{(0)}\circ\mathcal{M}_{(1,1)\to(2,1)}^{(0)}\circ\mathcal{M}_{(1,1)}^{(0)}(\rho)\right)}{\operatorname{tr}\left(\mathcal{M}_{(1,1)\to(2,1)}^{(0)}\circ\mathcal{M}_{(1,1)}^{(0)}(\rho)\right)}$
(11)
Eventually, the probability $\Pr[\rho_{3,1}]$ can be written as
$\displaystyle\Pr[\rho_{3,1}]=\operatorname{tr}\left(\mathcal{M}_{(2,1)\to(3,1)}^{(0)}\circ\mathcal{M}_{(1,1)\to(2,1)}^{(0)}\circ\mathcal{M}_{(1,1)}^{(0)}(\rho)\right).$
(12)
As shown in Eq. (7), the circuit of QPS allocates weight to each eigenvector
conditional on the state of the ancilla qubit. Specifically, each eigenvector
$|\psi_{j}\rangle$ is allocated with a weight $\sqrt{(1+f(\lambda_{j}))/2}$ if
the ancilla qubit is $|0\rangle$; otherwise, the weight is
$\sqrt{(1-f(\lambda_{j}))/2}$. Since $f(x)$ is an approximation of the sign
function in the sense that $|f(x)-\mathrm{sign}(x)|\leq\varepsilon$ for all
$x\in(-\pi+\kappa,-\kappa)\cup(\kappa,\pi-\kappa)$, and the approximation
error $\varepsilon$ could be exponentially small, we can see that some
eigenvectors are filtered from the un-normalised post-measurement state.
$\begin{array}[]{cc}|\widetilde{\phi}_{l}^{(0)}\rangle=\sum_{j\mathrel{\mathop{\mathchar
58\relax}}\lambda_{j}\not\in(-\pi+\kappa,-\kappa)}\sqrt{\frac{1+f(\lambda_{j})}{2}}\langle\psi_{j}|\phi_{l}\rangle|\psi_{j}\rangle,&|\widetilde{\phi}_{l}^{(1)}\rangle=\sum_{j\mathrel{\mathop{\mathchar
58\relax}}\lambda_{j}\not\in(\kappa,\pi-\kappa)}\sqrt{\frac{1-f(\lambda_{j})}{2}}\langle\psi_{j}|\phi_{l}\rangle|\psi_{j}\rangle.\end{array}$
(13)
Hence, after the operation of
$\mathcal{M}_{(2,1)\to(3,1)}^{(0)}\circ\mathcal{M}_{(1,1)\to(2,1)}^{(0)}\circ\mathcal{M}_{(1,1)}^{(0)}$,
the initial state $\rho$ is filtered, and only the ground state remains. Using
the linearity of the map $\mathcal{M}$ and trace, we can see that
$\displaystyle\Pr[\rho_{3,1}]$
$\displaystyle=\sum_{ljk}p_{l}\langle\psi_{j}|\phi_{l}\rangle\langle\phi_{l}|\psi_{k}\rangle\operatorname{tr}\left(\mathcal{M}_{(2,1)\to(3,1)}^{(0)}\circ\mathcal{M}_{(1,1)\to(2,1)}^{(0)}\circ\mathcal{M}_{(1,1)}^{(0)}(\op{\psi_{j}}{\psi_{k}})\right)$
(14) $\displaystyle=\sum_{l,j\mathrel{\mathop{\mathchar
58\relax}}\Pi|\psi_{j}\rangle\neq 0,\atop k\mathrel{\mathop{\mathchar
58\relax}}\Pi|\psi_{k}\rangle\neq
0}p_{l}\langle\psi_{j}|\phi_{l}\rangle\langle\phi_{l}|\psi_{k}\rangle\operatorname{tr}\left(\mathcal{M}_{(2,1)\to(3,1)}^{(0)}\circ\mathcal{M}_{(1,1)\to(2,1)}^{(0)}\circ\mathcal{M}_{(1,1)}^{(0)}(\op{\psi_{j}}{\psi_{k}})\right)$
(15)
$\displaystyle=\operatorname{tr}\left(\mathcal{M}_{(2,1)\to(3,1)}^{(0)}\circ\mathcal{M}_{(1,1)\to(2,1)}^{(0)}\circ\mathcal{M}_{(1,1)}^{(0)}(\Pi\rho\Pi)\right).$
(16)
where $\Pi$ denotes the projection operator onto the subspace of the ground
state. The second equality holds since excited eigenvectors would be assigned
zero weight. Conducting the same process would give expressions for the rest
probabilities $\Pr[\rho_{3,2}]$, $\Pr[\rho_{3,3}]$, and $\Pr[\rho_{3,4}]$.
Finally, simple calculations would immediately give the probability of finding
the ground state.
$\displaystyle\Pr[\rho_{3,1}]+\Pr[\rho_{3,2}]+\Pr[\rho_{3,3}]+\Pr[\rho_{3,4}]=\operatorname{tr}(\rho\Pi).$
(17)
To consider the error of approximating the sign function, the above discussion
can be reduced to the case where the initial state is an eigenvector. Then the
output probability is highly approximate to $\operatorname{tr}(\rho\Pi)$. We
refer the interested readers to Wang _et al._ (2022b) for more complexity
analysis.
For a general case, we can sketch the search process as a binary tree and
calculate the probability of finding the ground state in the same manner.
Hence, we conclude that the probability is highly approximate to the initial
state overlap under the assumption that the Hamiltonian evolution is simulated
very accurately.
### II.2 Algorithm using block encoding
When the input mode is a block encoding operator
$U_{H}=\left[\begin{smallmatrix}H/\alpha&\cdot\\\
\cdot&\cdot\end{smallmatrix}\right]$ of the Hamiltonian, QPS is similarly
employed to prepare the target ground state. The discussion on probability and
correctness is similar to the previous section. In contrast, there are several
differences in using the Hamiltonian evolution. Firstly, implementing $U_{H}$
requires an ancilla system; thus, making the circuit wider. Secondly, QPS
outputs the eigenvalue and eigenvector of the block encoding rather than the
Hamiltonian; hence further processing is demanded. Specifically, when the QPS
finishes, computing the cosine function of the output gives an estimation of
the eigenvalue of the Hamiltonian. Measure the ancilla system of the final
output state to achieve the ground state once the post-measurement state has
the outcomes all zeros; otherwise, one has to restart the QPS procedure with
the initial state or the post-measurement state and halt until all-zero
outcomes are observed. Thirdly, the estimation accuracy is set smaller than
$\Delta/2\alpha$. The corresponding circuit is shown in Fig. 4. More
discussions on implementing block encoding in QPS can be found in Wang _et
al._ (2022b).
$\Qcircuit@C=0.2em@R=1em{\lstick{|0\rangle}&\gate{R_{z}(\phi_{L})}\gate{R_{y}(\theta_{L})}\ctrl{1}\gate{R_{z}(\phi_{L-1})}\gate{R_{y}(\theta_{L-1})}\ctrlo{1}\gate{R_{z}(\phi_{L-2})}\gate{R_{y}(\theta_{L-2})}\qw\cdots\ctrlo{1}\gate{R_{z}(\phi_{0})}\gate{R_{y}(\theta_{0})}\gate{R_{z}(\omega)}\qw\meter\\\
\lstick{|0^{m}\rangle}/^{m}\qw\qw\multigate{1}{U_{H}}\qw\qw\multigate{1}{U_{H}^{\dagger}}\qw\qw\qw\cdots\multigate{1}{U_{H}^{\dagger}}\qw\qw\qw\qw\\\
\lstick{\rho}/^{n}\qw\qw\ghost{U_{H}}\qw\qw\ghost{U_{H}^{\dagger}}\qw\qw\qw\cdots\ghost{U_{H}^{\dagger}}\qw\qw\qw\qw}$
Fig 4: Block-encoding based QPS circuit. $U_{H}$ is a block encoding of the
Hamiltonian. The second line represents the block-encoding’s ancilla system of
size $m$. The third line stands for the main system with $n$ qubits.
A general block encoding cannot be directly employed to prepare the ground
state unless it satisfies an invariant property. When the invariant property
is satisfied, eigenvalues of $U_{H}$ and $H$ can be connected via a cosine
function. Specifically, the action of the block encoding can be described as
$U_{H}|0^{m},\psi_{j}\rangle=\lambda_{j}/\alpha|0^{m},\psi_{j}\rangle+\sqrt{1-(\lambda_{j}/\alpha)^{2}}|\bot_{j}\rangle$,
where $|\bot_{j}\rangle$ denotes a normalised column vector satisfying
$(\op{0^{m}}{0^{m}}\otimes I)|\bot_{j}\rangle=0$. Let $\mathcal{H}_{j}$ denote
a two-dimensional plane spanned by vectors
$\\{|0^{m},\psi_{j}\rangle,|\bot_{j}\rangle\\}$. If $U_{H}$ is invariant in
$\mathcal{H}_{j}$, i.e., $U_{H}\mathrel{\mathop{\mathchar
58\relax}}\mathcal{H}_{j}\to\mathcal{H}_{j}$ is a rotation, we could easily
find the relation between $\lambda_{j}$ and the corresponding phase
$\pm\tau_{j}$ of $U_{H}$, which is $\lambda_{j}/\alpha=\cos(\pm\tau_{j})$. If
$U_{H}$ does not satisfy the invariant property, this issue can be addressed
by using the “qubitization” technique Low and Chuang (2019). The core of
qubitization is constructing a new block encoding operator using one more
ancilla qubit and querying the controlled-$U_{H}$ and its inverse operator
once. Furthermore, the LCU-type block encoding satisfies the aforementioned
invariant property; hence, qubitization is no longer needed in this case.
To output an eigenvector of $U_{H}$ corresponding to the ground state, the
estimation accuracy is set smaller than $\Delta/2\alpha$. Note that the cosine
function is monotonically increasing and decreasing in the region $(-\pi,0)$
and $(0,\pi)$, respectively. We can know that the phase gap of the unitary is
larger than the scaled spectral gap $\Delta/\alpha$. Thus setting accuracy as
$\Delta/2\alpha$ suffices to isolate the ground state. Suppose we have
attained an eigenvector, e.g., $(|0^{m},\psi_{0}\rangle\pm
i|\bot_{0}\rangle)/\sqrt{2}$. Directly measuring on the ancilla system of the
block encoding, the post-measurement state is the ground state if the outcomes
are all zeros. The probability of this happening is exactly $1/2$. If all-zero
outcomes are not observed, one can restart the whole procedure. Or one can
send the post-measurement state into QPS again. Afterwards, the state will
collapse to either $(|0^{m},\psi_{0}\rangle+i|\bot_{0}\rangle)/\sqrt{2}$ or
$(|0^{m},\psi_{0}\rangle-i|\bot_{0}\rangle)/\sqrt{2}$. This process is
repeated until the outcomes are all zeros. Particularly, the failure
probability will fast decay as $(1/2)^{r}$, where $r$ means the number of
repeats.
## III Application to Heisenberg Hamiltonians
The general Heisenberg-typed model is arguably one of the most commonly used
models in the research of quantum magnetism and quantum many-body physics that
researching in the model’s ground state indicates the characteristics of spin
liquid and inspires crystal topology design Yan _et al._ (2011). The
Hamiltonian can be expressed as
$H=\sum_{\langle
i,j\rangle}(J_{x}S_{i}^{x}S_{j}^{x}+J_{y}S_{i}^{y}S_{j}^{y}+J_{z}S_{i}^{z}S_{j}^{z})+\sum_{i}h_{z}S_{i}^{z},$
(18)
with $\langle i,j\rangle$ depends on the specific lattice structure,
$J_{x},J_{y},J_{z}$ describe the spin coupling strength respectively in the
$x$,$y$,$z$ directions and $h_{z}$ is the magnetic field applied along the $z$
direction. Classical approaches involving Bethe ansatz and thermal limitations
Bonechi _et al._ (1992); Faddeev (1996); Rojas _et al._ (2001) have been
developed and the model can be exactly solved. Besides, variational methods,
in particular, VQE provides an alternative practical approach.
For no-noise numerical experiments, we evaluate the performance of constant-
depth VQE on randomly generated Hamiltonians with shuffled spin coupling
strengths. Specifically, we consider the 1D model with adjacent interactions
and randomly assign $J_{xyz}$ coefficients in the interval $[-1,1]$ while
ignoring magnetic fields involving periodic boundary conditions. Furthermore,
we utilise two commonly used parameterised quantum circuits or ansatzes namely
hardware efficient ansatz (HEA) Kandala _et al._ (2017) and alternating
layered ansatz (ALT) Cerezo _et al._ (2021), as illustrated in Fig. 5. And
for all simulations, we randomly initialise circuit parameters and perform 200
optimisation iterations using Adam optimiser with 0.1 learning rate.
$\Qcircuit<EMAIL_ADDRESS>\qw\gate{R_{y}}\gate{R_{z}}\targ\ctrl{1}\qw\gate{R_{y}}\gate{R_{z}}\qw\\\
\qw\gate{R_{y}}\gate{R_{z}}\qw\targ\ctrl{1}\gate{R_{y}}\gate{R_{z}}\qw\\\
\qw\gate{R_{y}}\gate{R_{z}}\qw\qw\targ\gate{R_{y}}\gate{R_{z}}\qw\inputgroupv{1}{4}{0.4em}{3em}{n=4\quad}\gategroup{1}{5}{4}{9}{1.2em}{--}\\\
\\\ \quad\times p}$
$\Qcircuit<EMAIL_ADDRESS>\qw\gate{R_{y}}\qw\ctrl{0}\gate{R_{y}}\ctrl{1}\gate{R_{y}}\qw\qw\\\
\qw\gate{R_{y}}\qw\ctrl{1}\gate{R_{y}}\ctrl{0}\gate{R_{y}}\qw\qw\\\
\qw\gate{R_{y}}\qw\ctrl{0}\gate{R_{y}}\qw\qw\qw\qw\inputgroupv{1}{4}{0.4em}{3em}{n=4\quad}\gategroup{1}{5}{4}{8}{2.4em}{--}\\\
\\\ \times p}$
Fig 5: Four-qubit templates for the experimental ansatz (with $p$ repeated
circuit block) diagrams of the left: Hardward Efficient ansatz (HEA) and the
right: Alternating Layered ansatz (ALT).
#### Algorithm effectiveness via 1D Heisenberg model
First, we investigate the performance of initial state preparation as the
circuit depth increases. In our simulation, we randomly sample 200 $8$-qubit
Heisenberg Hamiltonians to demonstrate the initial state preparation by using
ALT ansatz with circuit depth $d\in\\{1,3,5,7,9\\}$. As shown in Fig. 6, we
categorise the trained overlaps into ranges $(0-0.4)$, $(0.4-1.0)$,
$(0.6-1.0)$, and $(0.8-1.0)$ and designate them with purple, blue, brown and
orange lines, respectively. We observe that as the circuit depth increases,
the probability of $(0.6-1.0)$ interval rises from $26\%$ to $98\%$ and
$(0.8-1.0)$ overlap interval rises from $0$ to $58.5\%$, while $(0-0.4)$
interval drops from $12.5\%$ to $1\%$. We also find that VQE with circuit
depth can generate the initial state within $(0.4-1.0)$ overlap range with at
least $97\%$ confidence. This finding indicates that depth-3 ansatz can
reliably prepare the initial state for the ground state preparation of the 1D
Heisenberg model and a deeper-depth circuit may further boost the overlap. As
depth-3 ansatz is already capable of achieving satisfactory performance, we
choose it for additional investigations.
Fig 6: The left figure shows the overlap range prepared with ALT ansatz depth
$[1,3,4,5,7]$ for 200 8-qubit Heisenberg Hamiltonians. The purple, blue, brown
and orange lines indicate the intervals $(0.0-0.4)$, $(0.4-1.0)$, $(0.6-1.0)$
and $(0.8-1.0)$ in which the sampled probability of overlaps that prepared
shallow-depth VQE may fall. The right figure shows the overlap prepared by
depth-3 VQE with HEA ansatz and our algorithm. The blue zone represents
overlap generated by shallow-depth VQE, while the red portion represents
overlap produced by our method.
We then demonstrate the validity of our ground state preparation algorithms by
applying the depth-3 VQE for preparing the initial state. The experimental
results are shown in Fig. 6. There are 200 8-qubit Heisenberg Hamiltonians
being sampled, and HEA ansatz with circuit depth 3 is used. The blue region
depicts the overlap generated by a randomly initialised depth-3 VQE, while the
red line represents the overlap boosted by our algorithm given the initial
states prepared before. We can see that the shallow-depth VQE provides an
initial state with a minimum overlap of 0.5 with the ground state. Then, by
inputting the optimised state into our program, we can effectively prepare the
ground state with a fidelity error of less than $10^{-4}$ regarding the true
ground state. For remark, we use the Hamiltonian evolution operator $U=e^{iH}$
and set the estimation accuracy as $1/3$ of the spectral gap.
Previous two experiments demonstrate the depth-3 VQE has satisfactory
performance in the 8-qubit Heisenberg model. To further understand the
scalability of shallow-depth VQE in other system sizes, we conduct extensive
noiseless numerical simulations for system sizes up to 14. Specifically, we
sample 10000 randomly generated Heisenberg Hamiltonians for each qubit size
between 4 to 10, and sample 200 and 100 random Hamiltonians for qubit sizes 12
and 14. As a comparison, ALT and HEA ansatz are both used. The experimental
results are depicted in Fig. 7.
Fig 7: The frequency of overlap prepared by using depth-3 VQE ansatz of ALT
(up) and HEA (down) for varied system sizes. The x-axis labels the overlap
regarding the randomly generated Heisenberg Hamiltonians’ ground states. For
every system with a size less than or equal to 10 qubits, $10000$ random
Hamiltonians are sampled, and for qubit sizes 12 and 14, $200$ and $100$
random Hamiltonians are sampled, respectively. The result shows the efficacy
of depth-3 VQE of two ansatz templates in preparing initial states with an
overlap greater than 0.4 for the intermediate-scaled Heisenberg models.
According to the numerical results, the shallow-depth VQE is shown to be
effective for Heisenberg Hamiltonians. The majority of the results indicate
that the prepared state can have a greater overlap with the ground state than
0.4, making it an acceptable initial state for our algorithm. Also, as the
system size of Heisenberg Hamiltonian increases, the prepared states have a
trend that approximately converged to an overlap between 0.5 and 0.75 and the
full width at half maximum of the sampled probability distribution expands,
implying that, with a randomly initialised shallow-depth parameterised quantum
circuit, the optimised state can be prepared with substantial overlap to the
ground state with high probability. Particularly, our results show that the
technical concern from Ge _et al._ (2019); Dong _et al._ (2022); Lin and
Tong (2020) can be fulfilled by a shallow-depth VQE, indicating the
foreseeable possibility of realising the warm-start applications on large
systems.
Exceptionally, the experiments show that the overlap is less than 0.1 for a
small number of Hamiltonians. We alternatively utilise the Gibbs-VQE Wang _et
al._ (2021) method. In comparison to the shallow-depth VQE, Gibbs-VQE is less
prone to prepare state with small overlap and has a more steady performance.
More discussions can be found in Appendix. A. Furthermore, due to the
computational limitations, we are limited to observe the phenomenon up to the
14-qubit Heisenberg model. To further study the effectiveness of shallow-depth
VQE, the Ising model is considered with system sizes greater than 20. A
detailed discussion can be found in Appendix. B. Finally, barren plateaus
McClean _et al._ (2018) is a significant obstacle to the efficient use of
VQE. It refers to the gradient of the cost function vanishing exponentially
with the number of qubits for a randomly initialised PQC with sufficient
depth. We show that shallow-depth VQE will not be affected by barren plateaus
and is scalable to large-scale models, which can be referred to Appendix. C.
## IV Application to chemical molecules
Fig 8: The molecular geometry of the hydrogen chain $H_{4}$ with $4$ atoms
with equal spacing $R$. The right shows the qubit representation of the spin
orbitals, and the subscripts label the qubit indices in the quantum circuit.
The four spin-up orbitals are in the $|1\rangle$ state while no spin-down
orbitals are occupied, representing the HF initial state of $H_{4}$.
Previous results have demonstrated the effectiveness of our algorithms for
solving the Heisenberg model under intermediate-size situations and have the
potential for large-size applications. Nevertheless, the systematic analysis
of the ground state preparation performances with shallow-depth warm-start
stays ambiguous for other physical models. In this section, we aim to extend
the usage of our algorithm into the quantum chemical regime by numerically
demonstrating the warm-start performance on handling the small molecular model
while taking the well-known Hartree-Fock (HF) state as a standard.
To our best knowledge, both VQE and HF state methods own abilities to prepare
the high-accurate ground states Dong _et al._ (2021); Lin and Tong (2020);
Wecker _et al._ (2015) that can be used in quantum chemistry calculations for
determining molecular geometries, and ground energies Bartlett and Stanton
(1994); Peruzzo _et al._ (2014); Kandala _et al._ (2017). In the HF
approximation, electrons are treated independently and indistinguishably Omar
(2005) that interact with nuclei potential and an electronic mean field Seeger
and Pople (1977). Quantum chemistry assumes discrete molecular orbitals
determining the wave function of each electron appearing in the molecule.
General molecular Hamiltonian fermionic configuration can be represented in a
second quantisation form,
$H=\sum_{pq}h_{pq}c_{p}^{\dagger}c_{q}+\frac{1}{2}\sum_{pqrs}h_{pqrs}c_{p}^{\dagger}c_{q}^{\dagger}c_{r}c_{s},$
(19)
where $c_{p}^{\dagger},c_{p}$ are the creation and annihilation operators,
respectively. The coefficients $h_{pq}$ and $h_{pqrs}$ denote the one- and
two-electron Coulomb integrals Fermann and Valeev (2020). In the quantum
computing regime, the molecular electronic wave function can be represented in
the occupation number basis that each qubit defines a spin-orbital. By adding
one fermion (electron) into the system, each orbital can be in an occupied
state $|1\rangle$ or a non-occupied state $|0\rangle$ represented in the
computational basis. The single Slater-determinant state of $N$ orbitals can
be defined as,
$|\psi_{\theta}\rangle=U_{\theta}|\eta\rangle,\quad
U_{\theta}=\exp{\left(\sum_{j,k=1}^{N}\theta_{jk}c_{j}^{\dagger}c_{k}\right)}$
(20)
where $|\eta\rangle=c_{\eta}^{\dagger}\cdots c_{1}^{\dagger}|0\rangle$ in the
core orbital basis. The Hartree-Fock state is the Slater-determinant having
the lowest energy. Preparing the HF state classically requires a large amount
of computational resource Quantum _et al._ (2020). Even with quantum
technologies Tubman _et al._ (2019); Kivlichan _et al._ (2018), one have to
resolve the ansatz design for finding a basis to achieve relatively low
circuit complexity Babbush _et al._ (2018). Besides, the extreme number of
terms in the molecular Hamiltonian, writing in molecular orbital basis, could
raise scalable challenging to simulate and measure which requires
factorisation strategies Quantum _et al._ (2020); Berry _et al._ (2019).
(a)
(b)
Fig 9: Figures showing the ground state energy estimation and ground state
preparation via shallow-depth (depth-3) VQE (blue), deep-depth (depth-50) VQE
(green) and Gibbs-VQE (depth-3) (orange) on $H_{4}$ chain with bond distance
$R$ equals to $0.5$ (a) and $3.0$ (b). The above diagrams illustrate the
energy variations through the optimisation iterations, while the red lines
represent the exact values. The bottom shows the corresponding prepared state
overlaps, while the red shows the HF states’ overlaps.
We specifically consider the 1D hydrogen chain of $4$ atoms at fixed equal
spacing $R$ containing $4$ electrons. Each hydrogen atom has one molecular
orbital involving two spin orbitals that, in total, demands $8$ qubits for
VQE. We choose hydrogen chain $H_{4}$ as the objective since its simplicity
matches our current computational resource and its historical meaning as a
benchmark for quantum chemistry Motta _et al._ (2017). We use openfermion
built-in Psi4 kernel to compute the fermionic Hamiltonian components (19) of
$h_{pq}$ and $h_{pqrs}$ via self-consistent field method using the STO-3G
basis. We perform the simulations with $R=0.5$ and $3.0$ a.u. corresponding to
the strong and weak interaction regimes, respectively. Before running the VQE,
we need to map the fermionic Hamiltonian into the qubit representation via the
Jordan-Wigner transformation Seeley _et al._ (2012). During all simulations,
the VQE ansatz is chosen to be the HEA optimised in $200$ iterations of $0.1$
fixed learning rate by the Adam optimiser.
#### Comparing VQE and HF initial state on hydrogen chain
Results have been illustrated in Fig. 9. The top two diagrams illustrate the
energy curves while the bottom shows the corresponding output states’ overlap
regarding the exact ground states. The decreasing energy curves’ observation
in the left plot matches the growth in the optimised overlap for all the cases
of shallow-depth VQE (blue) and HF state (red) in (a) bottom. As the energy
converges to around $-1.65$ Ha in (a) top, both the two schemes reach an
overlap of almost one ($0.995$) in (a) bottom. Intuitively, bond distance
$R=0.5$ a.u. creates strong electron-electron interaction making the mean
field a reasonable approximation so that the HF state obtained a nearly
perfect fidelity. This also explains the reduced overlap from the HF results
reaching about $0.54$ when $R=3.0$ a.u. in (b) bottom since the approximation
deviates. Counterintuitively, the VQE results, in blue, experience a distinct
decline in the overlap though the corresponding energy values almost reach the
ground of $-1.868$ Ha of error $10^{-3}$ compared to the exact diagonalisation
results. However, increasing the depth to $50$ for VQE ansatz (green) could
gradually improve the overlap to $0.46$ within $200$ optimisation steps. We
also include the Gibbs-VQE (yellow) stated in the previous sections as a
reference scheme that showed similar behaviours from the shallow-depth VQE.
We conclude that the shallow-depth VQE may be a scalable and preferable warm-
starting strategy for molecular ground state preparation, showing great
performance of generating high-overlap initial states for short bond
distances. Interestingly, for larger bond distances, both methods imply
reduced initial overlap values. As the mean field decoupled, the accuracy of
the HF state decreases. Besides, the energy gaps between eigenstates shrunk
traps the VQE into the local minimum Wierichs _et al._ (2020) since the close
energy values carrying by the excited states could disturb the minimisation
processes leading to a zero overlap by the orthogonality. Deeper VQE circuit
could mitigate the effect but with a slow growth of the overlap. Therefore,
the ground energy estimation still performs well for governed by the
variational principle, however, might be insufficient for the efficient
initial state preparation. The fundamental pictures of the effect and the
coping strategies remain open for extending the usage of our method on
molecular models.
## V Application to Fermi-Hubbard models
Researching the Fermi-Hubbard model is one of the fundamental priorities in
condensed matter physics covering metal-insulator transitions and high-
temperature superconductivity Cade _et al._ (2020); Dagotto (1994). However,
the model involves a wide range of correlated electrons requiring multi-
determinant calculations and numerous computational resources, hence impeding
the classical exploration in the area. Therefore, designing the quantum method
for the model stays as a compelling application for quantum computing. The
potential of solving the 1D Hubbard model has been proposed by Dong _et al._
(2022) while only the VQE-typed algorithms have been examined via both
numerical simulations and real devices Cheuk _et al._ (2016); Wecker _et
al._ (2015); Lin and Tong (2020), as far as we know. In this section, we aim
to provide a first numerical demonstration for the quantum ground state
preparation scheme on the performance of predicting and estimating the
physical properties of a classically challenging Fermi-Hubbard model other
than commonly utilised VQE Stanisic _et al._ (2022); Quantum _et al._
(2020).
Fig 10: Qubit layout for implementing our $1\times 5$ Hubbard instances. The
lower illustrate the model’s physical site configuration where each site is
associated with an index $1\leq i\leq 5$ labelling from left to right. Every
site can hold a pair of spin-up and -down electrons which requires two qubits
to encode all possible spin orbitals. Each qubit has perfect connections with
the other qubits.
For an $n_{x}\times n_{y}$ square lattice physical system, for example, a
metallic crystal, each lattice point, called a site, is assigned with an
index. The Hubbard model Hamiltonian has a fermionic, second quantisation
form,
$H=-J\sum_{\langle
i,j\rangle,\sigma}(a^{\dagger}_{i\sigma}a_{j\sigma}+a^{\dagger}_{j\sigma}a_{i\sigma})+U\sum_{i}n_{i\uparrow}n_{i\downarrow}+H_{\text{local}},$
(21)
where $a^{\dagger}_{i\sigma},a_{i\sigma}$ are fermionic creation and
annihilation operators; $n_{i\sigma}=a^{\dagger}_{i\sigma},a_{i\sigma}$, are
the number operators; the notation $\langle i,j\rangle$ associates adjacent
sites in the $n_{x}\times n_{y}$ rectangular lattice;
$\sigma\in\\{\uparrow,\downarrow\\}$ labels the spin orbital. The first term
in Eq. (21) is the hopping term with $J$ being the tunnelling amplitude, and
$U$ in the second term is the on-site Coulomb repulsion. The last term defines
the local potential from nuclear-electron interaction. The local interaction
term is determined by the Gaussian form Wecker _et al._ (2015),
$H_{\text{local}}=\sum_{j=1}\sum_{\nu=\uparrow,\downarrow}\epsilon_{j,\nu}n_{j,\nu};\quad\epsilon_{j,\nu}=-\lambda_{\nu}e^{-\frac{1}{2}(j-m_{\nu})^{2}/\sigma_{\nu}^{2}}.$
(22)
We consider an $1\times 5$ lattice model with $J=2,U=3$ and
$\lambda_{\uparrow,\downarrow}=3,0.1$, $m_{\uparrow,\downarrow}=3,3$. The
standard deviation for both spin-up and -down potential is set to $1$. Such a
setup guarantees a charge-spin symmetry around the centre site ($i=3$) of the
entire system.
As before, representing the Hubbard Hamiltonian on a quantum computer in the
qubit representation requires fermionic encoding via efficient JW
transformation. For a 1D Hubbard model with $N$ sites, based on the Pauli
exclusion principle Deutsch (1951), each site can contain at most a pair of
spin-up and -down, two electrons. In total, there are $2N$ possible electronic
orbitals represented by the same number of qubits which covers all possible
orbitals for the system. The qubit layout of our five-qubit model has been
shown in Fig. 10 where all pairs of qubits are connected perfectly during
simulations. We study the medium-scaled model having non-degenerate ground
space. The fermionic Hamiltonian and qubitic transformations could be derived
using openfermion and paddle quantum python libraries. The VQE uses an HEA
circuit of depth $3$ and $50$ approaching realistic applications, and we
choose the Adam optimiser (fixed learning rate $0.1$) as before training the
circuit within $200$ iterations.
#### Evaluating physical quantities of Hubbard model
Fig 11: The (a) charge $\rho^{+}$ and (b) spin $\rho^{-}$ densities evaluated
from different produced ground states of our stated Hubbard model ($5$ sites)
by direct diagonalisation (green dashed), shallow-depth VQE (blue solid),
deep-depth VQE (orange solid) and our algorithm (red star) methods. The
shallow- and deep-depth VQE access HEA circuit with depth $3$ and $50$,
respectively. In both cases, the learning rate is set to $0.1$, and the total
number of iterations is set to $200$. The Adam optimiser powers all the
training procedures.
The first discussion is about predicting the charge-spin behaviours of the
given solid system. We study the global ground state of the Hubbard model
allowing the accessibility of every orbital with a sufficient number of
electrons in the qubit representation. After deriving the ground state of the
Hubbard Hamiltonian, we could compute the expectation value of different
physical observables, for example, charge and spin densities defined as,
$\rho^{\pm}_{j}=\langle n_{j\uparrow}\rangle\pm\langle
n_{j\downarrow}\rangle,$ (23)
where $j$ is the target site index. The densities could indicate the model’s
electronic and magnetic structures which further relate to the metal-metal
bonds of given materials Epsztein _et al._ (2018) and can be experimentally
demonstrated via electron paramagnetic resonance experiments Jacob and Reiher
(2012), respectively. The density diagrams illustrate charge-spin
distributions which could be used in designing solid structures of materials.
We compare the evaluated ground state densities from different preparation
schemes shown in Fig. 11. As we could see both the densities show a
centralized feature that matches our model configuration. The initial state of
our algorithm is prepared via shallow-depth VQE deriving the corresponding
density values (red stars) in the figure. A dramatic observation here shows
our scheme outperforms the traditional VQE methods, in which the derived
densities match the exact results while deep-depth VQE produces a visible gap
regarding the true values (green dashed). Notice that a deeper circuit
increases the VQE accuracy of both evaluated densities due to the better
expressibility. However, such a circuit approximates 2-design which
significantly drops the convergence speed of VQE. As a result, with the deep-
depth circuit, VQE could only prepare a defective ground state and predict
approximate densities (orange) within 200 iterations.
Besides, our algorithm only requires a non-zero-overlap initial state coming
from the shallow-depth VQE experiencing no trainability predicament. The
searched state should theoretically be the exact ground state and predict
exact density results. We show our method could derive high-precision charge-
spin densities which can be used to verify the theoretical developments on
Luttinger liquid and predict separation dynamics Arute _et al._ (2020).
Fig 12: Chemical potential $\mu$ calculations of Hubbard ground states from
different occupation numbers $N_{\text{occ}}$. The green dashed and blue solid
lines illustrate the exact and shallow-depth VQE-prepared ground states,
respectively. The stars label our algorithm’s results by initialising the
shallow-depth states. The subplot at the left corner indicates the
corresponding potential values’ absolute error $\varepsilon(\mu)$ (log-scale),
where the green shaded area marks the points satisfying chemical accuracy
regarding the FCI results.
Apart from that, chemical potential is another physical quantity determining
the energy variations during chemical and electronic interactions Stanisic
_et al._ (2022). In the second quantisation, the potential fields are
considered with discrete orbital states. Chemical potential describes the
energy requirement by adding or subtracting an electron from the system. Such
an electron would occupy an orbital state or leave a state empty,
respectively. The value can be computed via subtracting the ground state
energies of different occupation numbers (number of orbitals that are filled
with particles), having the following expression,
$\mu(N_{\text{occ}})=E(N_{\text{occ}})-E(N_{\text{occ}}-1).$ (24)
In the simulations, we derive the Hubbard Hamiltonian with different
occupation numbers by classically applying fermionic projection maps to the
previous global model by $\tilde{H}=P_{N_{\text{occ}}}HP_{N_{\text{occ}}}$.
The resulting $\tilde{H}$ could have image space spanning by the standard
basis elements with a fixed number of $|1\rangle$’s, and hence restrict the
number of electrons in the system. Such a projection can also be realised by
so-called the number-preserving ansatz introduced in Anselmetti _et al._
(2021). Previous literature works on the deep HVA-based ansatzes ($30$ layers)
and directly prepare the Hubbard ground states using VQE Stanisic _et al._
(2022). We instead, run the shallow-depth VQE on each $\tilde{H}$ to prepare
an initial state followed by our searching algorithm to locate the ground
states of each occupation number with constant scaled phase factor. The
corresponding energies can be simultaneously estimated from our algorithm.
As from Fig. 12, the algorithm could successfully extract the ground state
from initial shallow-depth VQE’s states. By zooming into the subplot
illustrating the absolute error of the chemical potentials at each
$N_{\text{occ}}$. Our estimated values obtain an absolute error of almost
$10^{-5}$ which have all reached the chemical precision with respect to the
FCI calculations via noiseless simulations in the green-shaded area. The
observation significantly proves the effectiveness of our method on other
high-accurate condensed matter calculations which inspires that our scheme
could promote a variety of physical research and provide a new paradigm for
studying and predicting hard-to-solve model behaviours from both classical and
quantum horizons.
## VI Conclusion and outlook
In this paper, we provide quantum algorithms to prepare ground states and
demonstrate the correctness and effectiveness through extensive numerical
simulations on Heisenberg models and chemical molecules. We also apply our
algorithms to estimate physical quantities of interest, such as charge and
spin densities and chemical potential of Fermi-Hubbard models having no
generally efficient classical methods, to our best knowledge. Only VQE and its
variants have been applied and demonstrated with numerical and practical
experiments from recent literature. A notable property of our algorithms is
that the ground state can be prepared simultaneously with the ground state
energy estimation, thus requiring no prior information on the ground state
energy. In contrast, existing ground state preparation algorithms involving
quantum phase estimation have to estimate a proper ground state energy
beforehand Ge _et al._ (2019), leading to indefinite practicability. For
other works Lin and Tong (2020); Dong _et al._ (2022) based on QSVT and QET-U
algorithms, a reasonable upper bound estimate of the ground state energy is
pre-requested. Since such information is usually not known as priorities, one
has to expensively estimate the ground state energy before preparing the
state. Another worth-noting point is that our algorithms only require to
compute phase factors with a scaling $\widetilde{\mathcal{O}}(1)$ while QSVT
and QET-U based algorithms require $\widetilde{\mathcal{O}}(\Delta^{-1})$,
which means our algorithms have advantages in avoiding challenging phase
factor evaluations when $\Delta$ is small.
Besides algorithmic improvements, we also contribute to showing that shallow-
depth VQE can be a favourable method for warm-starting the ground state
preparation of the intermediate-scaled many-body systems. In contrast, the
previous works Ge _et al._ (2019); Dong _et al._ (2022); Lin and Tong
(2020); Dong _et al._ (2022) all assume the state with a considerable overlap
is available and can be reused. Our scheme provides a potential paradigm for
using shallow-depth VQE to solve challenging physical models. By conducting
simulations to a 10-qubit Hubbard model, our method outperforms the VQE-based
evaluations of the charge and spin densities and the chemical potentials.
Meanwhile, we efficiently and qualitatively predict the ground state density
behaviours of the potential centralised model and derive extremely accurate
chemical potentials of distinct occupations up to an absolute error of
$10^{-5}$. Such evaluations have reached the chemical accuracy of the FCI
results, which is foreseeable to own advantages to exploring high-temperature
superconductivity and quantum chemistry problems. Our simulations have
demonstrated the potential of our algorithm on solving the 1D Hubbard model
and prepared for further research on the classical challenging 2D models Cheuk
_et al._ (2016).
Ground state preparation is significant to the fundamental research in physics
and chemistry. To establish a synthetic quantum solution for this task, there
are still some remaining problems to resolve in future. Since our works can
not offer suggestions on the depth of the warm-start ansatz, a theoretical
guarantee of reaching the large initial state overlap using sufficient-depth
ansatzes stays required. Recall that the algorithms are based on the prior
information on the spectral gap, especially, when the spectral gap is small,
it would be better to amplify it economically. Hence, an efficient and
effective method to compute the spectral gap and even amplify the gap is
necessary for applying quantum computers in solving ground-state problems. In
addition to addressing many-body systems, we believe our techniques can also
be used for combinatorial optimization problems e.g. the travelling
salesperson problem. Based on the results in this paper, we believe that
shallow variational warm-start would find more applications in physics,
chemistry, and quantum machine learning, which is worth further studying in
theory and experiments.
## Acknowledgement
Y. W., C. Z., and M. J. contributed equally to this work. Part of this work
was done when Y. W., C. Z., and M. J. were research interns at Baidu Research.
Y. W. acknowledges support from the Baidu-UTS AI Meets Quantum project. The
authors would like to thank Lloyd C.L. Hollenberg and Haokai Zhang for the
valuable comments and Lei Zhang for the helpful discussion.
## References
* Feynman and Cohen (1956) R. P. Feynman and Michael Cohen, “Energy spectrum of the excitations in liquid helium,” Phys. Rev. 102, 1189–1204 (1956).
* Sakurai and Commins (1995) J. J. Sakurai and Eugene D. Commins, “Modern quantum mechanics, revised edition,” American Journal of Physics 63, 93–95 (1995), https://doi.org/10.1119/1.17781 .
* Sherrill (2009) C. David Sherrill, “An introduction to hartree-fock molecular orbital theory,” (2009).
* Knowles and Handy (1989) Peter J. Knowles and Nicholas C. Handy, “A determinant based full configuration interaction program,” Computer Physics Communications 54, 75–83 (1989).
* Parr (1980) Robert G Parr, “Density functional theory of atoms and molecules,” in _Horizons of Quantum Chemistry: Proceedings of the Third International Congress of Quantum Chemistry Held at Kyoto, Japan, October 29-November 3, 1979_ (Springer, 1980) pp. 5–15.
* Blankenbecler _et al._ (1981) Richard Blankenbecler, DJ Scalapino, and RL Sugar, “Monte carlo calculations of coupled boson-fermion systems. i,” Physical Review D 24, 2278 (1981).
* Cade _et al._ (2020) Chris Cade, Lana Mineh, Ashley Montanaro, and Stasja Stanisic, “Strategies for solving the fermi-hubbard model on near-term quantum computers,” Physical Review B 102 (2020), 10.1103/physrevb.102.235122.
* Dagotto (1994) Elbio Dagotto, “Correlated electrons in high-temperature superconductors,” Rev. Mod. Phys. 66, 763–840 (1994).
* Nelson _et al._ (2019) James Nelson, Rajarshi Tiwari, and Stefano Sanvito, “Machine learning density functional theory for the hubbard model,” Physical Review B 99 (2019), 10.1103/physrevb.99.075132.
* Barahona (1982) Francisco Barahona, “On the computational complexity of ising spin glass models,” Journal of Physics A: Mathematical and General 15, 3241 (1982).
* Gilks _et al._ (1995) W. R. Gilks, N. G. Best, and K. K. C. Tan, “Adaptive rejection metropolis sampling within gibbs sampling,” Journal of the Royal Statistical Society: Series C (Applied Statistics) 44, 455–472 (1995), https://rss.onlinelibrary.wiley.com/doi/pdf/10.2307/2986138 .
* Del Moral _et al._ (2006) Pierre Del Moral, Arnaud Doucet, and Ajay Jasra, “Sequential monte carlo samplers,” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 68, 411–436 (2006).
* Scalapino (2006) Douglas J. Scalapino, “Numerical studies of the 2d hubbard model,” arXiv: Strongly Correlated Electrons , 495–526 (2006).
* LeBlanc _et al._ (2015) J. P. F. LeBlanc, Andrey E. Antipov, Federico Becca, Ireneusz W. Bulik, Garnet Kin-Lic Chan, Chia-Min Chung, Youjin Deng, Michel Ferrero, Thomas M. Henderson, Carlos A. Jiménez-Hoyos, E. Kozik, Xuan-Wen Liu, Andrew J. Millis, N. V. Prokof’ev, Mingpu Qin, Gustavo E. Scuseria, Hao Shi, B. V. Svistunov, Luca F. Tocchio, I. S. Tupitsyn, Steven R. White, Shiwei Zhang, Bo-Xiao Zheng, Zhenyue Zhu, and Emanuel Gull (Simons Collaboration on the Many-Electron Problem), “Solutions of the two-dimensional hubbard model: Benchmarks and results from a wide range of numerical algorithms,” Phys. Rev. X 5, 041041 (2015).
* Feynman (1982) Richard P. Feynman, “Simulating physics with computers,” International Journal of Theoretical Physics 21, 467–488 (1982).
* Peruzzo _et al._ (2014) Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, and Jeremy L. O’Brien, “A variational eigenvalue solver on a photonic quantum processor,” Nature Communications 5 (2014), 10.1038/ncomms5213.
* Wecker _et al._ (2015) Dave Wecker, Matthew B. Hastings, and Matthias Troyer, “Progress towards practical quantum variational algorithms,” Physical Review A 92 (2015), 10.1103/physreva.92.042303.
* Kandala _et al._ (2017) Abhinav Kandala, Antonio Mezzacapo, Kristan Temme, Maika Takita, Markus Brink, Jerry M. Chow, and Jay M. Gambetta, “Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets,” Nature 549, 242–246 (2017).
* Sapova and Fedorov (2022) Mariia D. Sapova and Aleksey K. Fedorov, “Variational quantum eigensolver techniques for simulating carbon monoxide oxidation,” Communications Physics 5, 199 (2022).
* Ostaszewski _et al._ (2021) Mateusz Ostaszewski, Lea M. Trenkwalder, Wojciech Masarczyk, Eleanor Scerri, and Vedran Dunjko, “Reinforcement learning for optimization of variational quantum circuit architectures,” in _Advances in Neural Information Processing Systems_, Vol. 34, edited by M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan (Curran Associates, Inc., 2021) pp. 18182–18194.
* Dallaire-Demers _et al._ (2019) Pierre-Luc Dallaire-Demers, Jonathan Romero, Libor Veis, Sukin Sim, and Alán Aspuru-Guzik, “Low-depth circuit ansatz for preparing correlated fermionic states on a quantum computer,” Quantum Science and Technology 4, 045005 (2019).
* Johnson _et al._ (2011) M. W. Johnson, M. H. S. Amin, S. Gildert, T. Lanting, F. Hamze, N. Dickson, R. Harris, A. J. Berkley, J. Johansson, P. Bunyk, E. M. Chapple, C. Enderud, J. P. Hilton, K. Karimi, E. Ladizinsky, N. Ladizinsky, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, C. J. S. Truncik, S. Uchaikin, J. Wang, B. Wilson, and G. Rose, “Quantum annealing with manufactured spins,” Nature 473, 194–198 (2011).
* Kitaev (1995) A. Yu. Kitaev, “Quantum measurements and the abelian stabilizer problem,” (1995).
* Brassard _et al._ (2002) Gilles Brassard, Peter Hoyer, Michele Mosca, and Alain Tapp, “Quantum amplitude amplification and estimation,” Contemporary Mathematics 305, 53–74 (2002).
* Ge _et al._ (2019) Yimin Ge, Jordi Tura, and J Ignacio Cirac, “Faster ground state preparation and high-precision ground energy estimation with fewer qubits,” Journal of Mathematical Physics 60, 022202 (2019).
* Poulin and Wocjan (2009) David Poulin and Pawel Wocjan, “Preparing ground states of quantum many-body systems on a quantum computer,” Physical Review Letters 102 (2009), 10.1103/physrevlett.102.130503.
* Gilyén _et al._ (2019) András Gilyén, Yuan Su, Guang Hao Low, and Nathan Wiebe, “Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics,” in _Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing_ (ACM, 2019).
* Lin and Tong (2020) Lin Lin and Yu Tong, “Near-optimal ground state preparation,” Quantum 4, 372 (2020).
* Berry _et al._ (2015) Dominic W. Berry, Andrew M. Childs, and Robin Kothari, “Hamiltonian simulation with nearly optimal dependence on all parameters,” in _2015 IEEE 56th Annual Symposium on Foundations of Computer Science_ (IEEE, 2015).
* Dong _et al._ (2022) Yulong Dong, Lin Lin, and Yu Tong, “Ground-state preparation and energy estimation on early fault-tolerant quantum computers via quantum eigenvalue transformation of unitary matrices,” PRX Quantum 3 (2022), 10.1103/prxquantum.3.040305.
* Low and Chuang (2017a) Guang Hao Low and Isaac L. Chuang, “Optimal hamiltonian simulation by quantum signal processing,” Physical Review Letters 118 (2017a), 10.1103/physrevlett.118.010501.
* Abrams and Lloyd (1999) Daniel S. Abrams and Seth Lloyd, “Quantum algorithm providing exponential speed increase for finding eigenvalues and eigenvectors,” Physical Review Letters 83, 5162–5165 (1999).
* Higgins _et al._ (2007) B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wiseman, and G. J. Pryde, “Entanglement-free heisenberg-limited phase estimation,” Nature 450, 393–396 (2007).
* Berry _et al._ (2009) D. W. Berry, B. L. Higgins, S. D. Bartlett, M. W. Mitchell, G. J. Pryde, and H. M. Wiseman, “How to perform the most accurate possible phase measurements,” Physical Review A 80 (2009), 10.1103/physreva.80.052114.
* Knill _et al._ (2007) Emanuel Knill, Gerardo Ortiz, and Rolando D. Somma, “Optimal quantum measurements of expectation values of observables,” Phys. Rev. A 75, 012328 (2007).
* Lin and Tong (2022) Lin Lin and Yu Tong, “Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers,” PRX Quantum 3 (2022), 10.1103/prxquantum.3.010318.
* Ding and Lin (2022) Zhiyan Ding and Lin Lin, “Even shorter quantum circuit for phase estimation on early fault-tolerant quantum computers with applications to ground-state energy estimation,” (2022).
* O’Brien _et al._ (2019) Thomas E O’Brien, Brian Tarasinski, and Barbara M Terhal, “Quantum phase estimation of multiple eigenvalues for small-scale (noisy) experiments,” New Journal of Physics 21, 023022 (2019).
* Wang _et al._ (2019) Daochen Wang, Oscar Higgott, and Stephen Brierley, “Accelerated variational quantum eigensolver,” Phys. Rev. Lett. 122, 140504 (2019).
* Somma (2019) Rolando D Somma, “Quantum eigenvalue estimation via time series analysis,” New Journal of Physics 21, 123025 (2019).
* Wan _et al._ (2022) Kianna Wan, Mario Berta, and Earl T. Campbell, “Randomized quantum algorithm for statistical phase estimation,” Physical Review Letters 129 (2022), 10.1103/physrevlett.129.030503.
* Wang _et al._ (2022a) Guoming Wang, Daniel Stilck-Franca, Ruizhe Zhang, Shuchen Zhu, and Peter D Johnson, “Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision,” arXiv preprint arXiv:2209.06811 (2022a).
* Wang _et al._ (2022b) Xin Wang, Youle Wang, Zhan Yu, and Lei Zhang, “Quantum phase processing: Transform and extract eigen-information of quantum systems,” arXiv preprint arXiv:2209.14278 (2022b).
* Childs _et al._ (2018) Andrew M Childs, Dmitri Maslov, Yunseong Nam, Neil J Ross, and Yuan Su, “Toward the first quantum simulation with quantum speedup,” Proceedings of the National Academy of Sciences 115, 9456–9461 (2018).
* hub (2013) “The hubbard model at half a century,” Nature Physics , 523 (2013).
* Wang _et al._ (2021) Youle Wang, Guangxi Li, and Xin Wang, “Variational quantum gibbs state preparation with a truncated taylor series,” Physical Review Applied 16 (2021), 10.1103/physrevapplied.16.054035.
* Low and Chuang (2019) Guang Hao Low and Isaac L. Chuang, “Hamiltonian simulation by qubitization,” Quantum 3, 163 (2019).
* Dong _et al._ (2021) Yulong Dong, Xiang Meng, K. Birgitta Whaley, and Lin Lin, “Efficient phase-factor evaluation in quantum signal processing,” Physical Review A 103 (2021), 10.1103/physreva.103.042419.
* Low and Chuang (2017b) Guang Hao Low and Isaac L Chuang, “Hamiltonian simulation by uniform spectral amplification,” arXiv preprint arXiv:1707.05391 (2017b).
* Yan _et al._ (2011) Simeng Yan, David A. Huse, and Steven R. White, “Spin-liquid ground state of the $\langle s\rangle$ = 1/2 kagome heisenberg antiferromagnet,” Science 332, 1173–1176 (2011).
* Bonechi _et al._ (1992) F Bonechi, E Celeghini, R Giachetti, E Sorace, and M Tarlini, “Heisenberg XXZ model and quantum galilei group,” Journal of Physics A: Mathematical and General 25, L939–L943 (1992).
* Faddeev (1996) LD Faddeev, “How algebraic bethe ansatz works for integrable model,” arXiv preprint hep-th/9605187 (1996).
* Rojas _et al._ (2001) Onofre Rojas, S. M. de Souza, E. V. Corrêa Silva, and M. T. Thomaz, “Thermodynamics of the Limiting Cases of the XXZ Model Without Bethe Ansatz,” Brazilian Journal of Physics 31, 577–582 (2001).
* Cerezo _et al._ (2021) M. Cerezo, Akira Sone, Tyler Volkoff, Lukasz Cincio, and Patrick J. Coles, “Cost function dependent barren plateaus in shallow parametrized quantum circuits,” Nature Communications 12 (2021), 10.1038/s41467-021-21728-w.
* McClean _et al._ (2018) Jarrod R. McClean, Sergio Boixo, Vadim N. Smelyanskiy, Ryan Babbush, and Hartmut Neven, “Barren plateaus in quantum neural network training landscapes,” Nature Communications 9 (2018), 10.1038/s41467-018-07090-4.
* Bartlett and Stanton (1994) Rodney J Bartlett and John F Stanton, “Applications of post-hartree—fock methods: A tutorial,” Reviews in computational chemistry , 65–169 (1994).
* Omar (2005) Yasser Omar, “Indistinguishable particles in quantum mechanics: an introduction,” Contemporary Physics 46, 437–448 (2005).
* Seeger and Pople (1977) Rolf Seeger and John A. Pople, “Self‐consistent molecular orbital methods. xviii. constraints and stability in hartree–fock theory,” The Journal of Chemical Physics 66, 3045–3050 (1977), https://doi.org/10.1063/1.434318 .
* Fermann and Valeev (2020) Justin T. Fermann and Edward F. Valeev, “Fundamentals of molecular integrals evaluation,” (2020).
* Quantum _et al._ (2020) Google AI Quantum, Collaborators*†, Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Sergio Boixo, Michael Broughton, Bob B Buckley, _et al._ , “Hartree-fock on a superconducting qubit quantum computer,” Science 369, 1084–1089 (2020).
* Tubman _et al._ (2019) Norm Tubman, Carlos Mejuto Zaera, Jeffrey Epstein, Diptarka Hait, Daniel Levine, William Huggins, Zhang Jiang, Jarrod McClean, Ryan Babbush, Martin Head-Gordon, and Birgitta Whaley, “Postponing the orthogonality catastrophe: efficient state preparation for electronic structure simulations on quantum devices,” in _APS March Meeting Abstracts_ , APS Meeting Abstracts, Vol. 2019 (2019) p. E42.004.
* Kivlichan _et al._ (2018) Ian D. Kivlichan, Jarrod McClean, Nathan Wiebe, Craig Gidney, Alán Aspuru-Guzik, Garnet Kin-Lic Chan, and Ryan Babbush, “Quantum simulation of electronic structure with linear depth and connectivity,” Phys. Rev. Lett. 120, 110501 (2018).
* Babbush _et al._ (2018) Ryan Babbush, Nathan Wiebe, Jarrod McClean, James McClain, Hartmut Neven, and Garnet Kin-Lic Chan, “Low-depth quantum simulation of materials,” Phys. Rev. X 8, 011044 (2018).
* Berry _et al._ (2019) Dominic W. Berry, Craig Gidney, Mario Motta, Jarrod R. McClean, and Ryan Babbush, “Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization,” Quantum 3, 208 (2019).
* Motta _et al._ (2017) Mario Motta, David M. Ceperley, Garnet Kin-Lic Chan, John A. Gomez, Emanuel Gull, Sheng Guo, Carlos A. Jiménez-Hoyos, Tran Nguyen Lan, Jia Li, Fengjie Ma, Andrew J. Millis, Nikolay V. Prokof’ev, Ushnish Ray, Gustavo E. Scuseria, Sandro Sorella, Edwin M. Stoudenmire, Qiming Sun, Igor S. Tupitsyn, Steven R. White, Dominika Zgid, and Shiwei Zhang (Simons Collaboration on the Many-Electron Problem), “Towards the solution of the many-electron problem in real materials: Equation of state of the hydrogen chain with state-of-the-art many-body methods,” Phys. Rev. X 7, 031059 (2017).
* Seeley _et al._ (2012) Jacob T. Seeley, Martin J. Richard, and Peter J. Love, “The bravyi-kitaev transformation for quantum computation of electronic structure,” The Journal of Chemical Physics 137, 224109 (2012), https://doi.org/10.1063/1.4768229 .
* Wierichs _et al._ (2020) David Wierichs, Christian Gogolin, and Michael Kastoryano, “Avoiding local minima in variational quantum eigensolvers with the natural gradient optimizer,” Physical Review Research 2 (2020), 10.1103/physrevresearch.2.043246.
* Cheuk _et al._ (2016) Lawrence W Cheuk, Matthew A Nichols, Katherine R Lawrence, Melih Okan, Hao Zhang, Ehsan Khatami, Nandini Trivedi, Thereza Paiva, Marcos Rigol, and Martin W Zwierlein, “Observation of spatial charge and spin correlations in the 2d fermi-hubbard model,” Science 353, 1260–1264 (2016).
* Stanisic _et al._ (2022) Stasja Stanisic, Jan Lukas Bosse, Filippo Maria Gambetta, Raul A. Santos, Wojciech Mruczkiewicz, Thomas E. O’Brien, Eric Ostby, and Ashley Montanaro, “Observing ground-state properties of the Fermi-Hubbard model using a scalable algorithm on a quantum computer,” Nature Commun. 13, 5743 (2022), arXiv:2112.02025 [quant-ph] .
* Deutsch (1951) Martin Deutsch, “Introductory nuclear physics.” Science 113, 443–443 (1951).
* Epsztein _et al._ (2018) Razi Epsztein, Evyatar Shaulsky, Nadir Dizge, David M. Warsinger, and Menachem Elimelech, “Role of ionic charge density in donnan exclusion of monovalent anions by nanofiltration,” Environmental Science & Technology 52, 4108–4116 (2018), pMID: 29510032, https://doi.org/10.1021/acs.est.7b06400 .
* Jacob and Reiher (2012) Christoph R. Jacob and Markus Reiher, “Spin in density-functional theory,” International Journal of Quantum Chemistry 112, 3661–3684 (2012).
* Arute _et al._ (2020) Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph C Bardin, Rami Barends, Andreas Bengtsson, Sergio Boixo, Michael Broughton, Bob B Buckley, _et al._ , “Observation of separated dynamics of charge and spin in the fermi-hubbard model,” arXiv preprint arXiv:2010.07965 (2020).
* Anselmetti _et al._ (2021) Gian-Luca R Anselmetti, David Wierichs, Christian Gogolin, and Robert M Parrish, “Local, expressive, quantum-number-preserving vqe ansätze for fermionic systems,” New Journal of Physics 23, 113010 (2021).
* Egger _et al._ (2021) Daniel J. Egger, Jakub Mareč ek, and Stefan Woerner, “Warm-starting quantum optimization,” Quantum 5, 479 (2021).
* Nakaji and Yamamoto (2021) Kouhei Nakaji and Naoki Yamamoto, “Expressibility of the alternating layered ansatz for quantum computation,” Quantum 5, 434 (2021).
Supplementary Material for
Ground state preparation with shallow variational warm-start
## Appendix A Initial State Preparation with Gibbs-VQE
The experiments in Fig. 7 show that it is unsuccessful for a small number of
Hamiltonians. This may be caused by the local minima problem, which we re-
initialise all the circuit parameters and re-run the circuit again may
significantly amplify the overlap. To mitigate this issue, we consider
applying variational Gibbs-state preparationWang _et al._ (2021) (Gibbs-VQE)
alternatively.
We begin by briefly reviewing the background of the Gibbs state and the
protocol Gibbs-VQE. For a Hamiltonian with an $n$-qubit system, the Gibbs
state $\rho_{G}$ at temperature $T$ is defined as follows:
$\rho_{G}=\frac{e^{-\beta H}}{\operatorname{tr}(e^{-\beta H})},$ (A.1)
where $-\beta H$ is the matrix exponential of matrix $-\beta H$.
$\beta=\frac{1}{kT}$ is the inverse temperature of the system and $k$ is the
Boltzmann’s constant. To prepare a Gibbs state we use the protocol as
introduced in Wang _et al._ (2021). The main idea is to consider the
variational principle of the system’s free energy, where parameterised quantum
circuits are used to minimize the free energy. Specifically, we set the loss
function as $L=L_{1}+L_{2}+L_{3}$, where
$L_{1}=\operatorname{tr}{H_{\rho_{B}}},\quad
L_{2}=2\beta^{-1}\operatorname{tr}{\rho_{B}^{2}},\quad
L_{3}=\frac{-\beta^{-1}(\operatorname{tr}{\rho_{B}^{2}}+3)}{2}.$ (A.2)
Fig S1: The optimised overlap using shallow-depth VQE and shallow-depth Gibbs-
VQE. For each sampled Hamiltonian, 10 optimisations are performed to determine
the overlap between the optimised and ground states. The blue circle indicates
the overlap prepared with shallow-depth VQE and the red diamond represents the
case with the Gibbs-VQE.
To compare the performance of Gibbs-VQE and shallow-depth VQE, 200 random
Hamiltonians are sampled and 10 random initialisations of circuit parameters
are performed for each Hamiltonian. For both simulations of two methods, we
employ depth-3 HEA ansatz and 200 Adam optimiser optimisation iterations. The
experiment results are shown in Fig. S1. The blue circle indicates the overlap
prepared by shallow-depth VQE, and the red diamond indicates the overlap
prepared by shallow-depth Gibbs-VQE. Simulation results indicate that Gibbs-
VQE has a more steady performance than shallow-depth VQE and is less prone to
become stuck in local minima. It can be seen that $\sim 11\%$ number of
shallow-VQE prepare the overlap less than 0.4, whereas, for Gibbs-VQE, the
ratio reduces to $\sim 2\%$. Thus, Gibbs-VQE is superior for solving physical
and chemical problems because it generates initial states with reliable
overlap.
## Appendix B Initial State Preparation with Tensor Product Ansatz
The process that initialises an approximating state for a quantum algorithm
via classical methods is usually called a warm-start quantum method. By
inheriting the initial state, quantum algorithm could require fewer resources
for solving the problem. Famous result, for example, warm-start QAOA Egger
_et al._ (2021) combining Goemans-Williamson (GW) randomized rounding
algorithm, guarantees the performance of solving the max-cut problem with
${\cal O}(n)$ qubits and low-depth quantum circuits.
From the previous section III, we observe the efficiency of initial state
preparation on Heisenberg Hamiltonians through shallow-depth VQE. Due to the
computational limitations, we are limited to observe the phenomenon on
Hamiltonians up to $14$-qubit. To further investigate the efficacy of shallow-
depth VQE, we use the tensor product ansatz to examine the max-cut problem.
Specifically, the max-cut problem can be naturally mapped to the QUBO
Hamiltonian. A general QUBO Hamiltonian has an expression,
$H_{\text{QUBO}}=\frac{1}{2}\sum_{p\neq
q}Z_{p}Z_{q}-\frac{n}{2}\mathbb{I}^{\otimes n}.$ (B.1)
This is closely related to the continuous relaxation of the max-cut problem
which can be approximately solved using semidefinite programming and GW
rounding. Such a technique generates the initial state for the problem
Hamiltonian, which can be efficiently prepared by applying a single layer of
$R_{Y}$ rotations. This indicates the possibility of the approximating ground
state of specific Hamiltonian models via shallow-depth quantum circuits.
Taking the max-cut problem as a starting point, we could have the following
observation.
###### Lemma S1
For a product state ansatz composed with only one layer of single-qubit
$R_{y}$ rotation, s.t., $|\psi(\theta)\rangle=\bigotimes
R_{y}(\theta_{p})|0\rangle$, we have
$\langle\psi(\theta)|H|\psi(\theta)\rangle=\frac{1}{2}\sum_{p\neq
q}\langle\psi(\theta)|Z_{p}Z_{q}|\psi(\theta)\rangle-\frac{n}{2}\langle\psi(\theta)|\mathbb{I}^{\otimes
n}|\psi(\theta)\rangle=\frac{1}{2}\sum_{p\neq
q}\cos{(\theta_{p})}\cos{(\theta_{q})}-\frac{n}{2}.$ (B.2)
As from above, we could work out the analytic gradient for any k-local
Hamiltonian with the product state ansatz $|\psi(\theta)\rangle$. For QUBO
case, setting cost
$C(\bm{\theta})=\langle\psi(\theta)|H_{\text{QUBO}}|\psi(\theta)\rangle$, we
could easily derive,
$\partial_{\alpha}C=-\sin{(\theta_{\alpha})}\sum_{p\neq\alpha}\cos{(\theta_{p})}$
We then use QUBO Hamiltonian and the result in Lemma S1 to further investigate
the performance of shallow-depth VQE. With powerful classical machine learning
techniques, we could classically derive the optimal rotation angles via
minimizing the cost (B.2). The preparation of the approximating initial state
is then implemented by applying the corresponding rotations on $|0\rangle$. In
this experiment, we investigate QUBO Hamiltonians with a size of 15 to 22
qubits and calculate the overlap using a brute-force technique. The experiment
results are shown in Fig. S2.
Fig S2: The highest overlap between the optimised state and one of the ground
states of different given QUBO Hamiltonian models (each model probably has
degenerate ground state space) vs. the system sizes. For the number of qubits
between 15 to 19 and 20 to 22, we sample $500$ and $100$ random max-cut
graphs’ qubo Hamiltonians respectively. The true ground states of each model
were first determined using brute force algorithm and stored in classical
memory. Each model will then be processed in VQE with the tensor product
ansatz and the optimisation was done w.r.t the analytic form Eq. (B.2) to
produce an optimised state.
As we seen from the diagram, the x-axis represents the size of QUBO
Hamiltonian and y-axis is the overlap between the initial state prepared by
analytic form according to E.q. B.2 and the ground states. The probability of
getting high overlap states regarding one of the ground states is slightly
higher than trapping into the excited states (local minima). The experiment
trend suggests that shallow-depth VQE may continue to perform well on large-
scale Hamiltonians.
## Appendix C Numerical Simulations on the Gradient of Shallow-Depth Circuit
The scalability is a significant obstacle to the efficient use of variational
quantum eigensolver. Numerous studies have demonstrated that the gradient of
the cost function vanishes exponentially with the number of qubits for a
randomly initialised PQC with sufficient depth McClean _et al._ (2018). This
phenomenon is termed the barren plateaus (BP). It has been shown that hardware
efficient ansatz with deep depth leads to exponentially vanishing gradients
McClean _et al._ (2018). While Alternating Layered Ansatz, a specific
structure of the HEA ansatz Cerezo _et al._ (2021), has been demonstrated
that when circuit depth is shallow, it does not exhibit the gradient vanishing
problem Nakaji and Yamamoto (2021). As we explore using the HEA and ALT ansatz
as an overlap initialiser, it is crucial to determine if these two ansatz with
shallow depth suffer from the gradient vanishing issue.
As shallow-depth ALT ansatz is BP-free, we choose to use the depth-3 HEA
ansatz as an illustration and conduct numerical experiments to sample the
variance of the gradient. The experiment results are shown in Fig. S3. Similar
to the experimental setting described in McClean _et al._ (2018), we analyse
the Pauli $ZZ$ operator acting on the first and second qubits such that
$H=Z_{1}Z_{2}$. The gradient is then sampled on the first parameter
$\theta_{1,1}$ by taking 1000 samples and the system size is evaluated up to
18 qubits.
Fig S3: Semi-log plot displaying the sample variance of the gradient of a two-
local Pauli term as a function of the number of qubits. The blue line
represents exponential decay as a function of the number of qubits, the dark
blue line represents the logarithmic decay and the orange line indicates that
the shallow-depth VQE do not exhibit exponential decay.
As shown in the figure, the orange line indicates the sample variance of the
gradient of the shallow-depth VQE while the blue and dark blue lines depict
exponential and logarithmic decay, respectively. Additionally, we identify the
blue area below the exponential decay line as the BP area and the purple area
above the logarithmic decay as BP-free. It is evident that the depth-3 HEA
ansatz does not suffer from the gradient vanishing problem. Therefore, it is
scalable to use shallow-depth VQE for initial state preparation.
|
# ERASING THE VARIABLE: EMPIRICAL FOREGROUND DISCOVERY FOR GLOBAL $21$ cm
SPECTRUM EXPERIMENTS
Eric R. Switzer11affiliation: NASA Goddard Space Flight Center, Greenbelt, MD
20771, USA 22affiliation: Canadian Institute for Theoretical Astrophysics,
University of Toronto, 60 St. George St., Toronto, ON, M5S 3H8, Canada and
Adrian Liu33affiliation: Department of Astronomy, UC Berkeley, Berkeley, CA
94720, USA 44affiliation: Berkeley Center for Cosmological Physics, UC
Berkeley, Berkeley, CA 94720, USA<EMAIL_ADDRESS>
###### Abstract
Spectral measurements of the $21$ cm monopole background have the promise of
revealing the bulk energetic properties and ionization state of our universe
from $z\sim 6-30$. Synchrotron foregrounds are orders of magnitude larger than
the cosmological signal, and are the principal challenge faced by these
experiments. While synchrotron radiation is thought to be spectrally smooth
and described by relatively few degrees of freedom, the instrumental response
to bright foregrounds may be much more complex. To deal with such
complexities, we develop an approach that discovers contaminated spectral
modes using spatial fluctuations of the measured data. This approach exploits
the fact that foregrounds vary across the sky while the signal does not. The
discovered modes are projected out of each line of sight of a data cube. An
angular weighting then optimizes the cosmological signal amplitude estimate by
giving preference to lower-noise regions. Using this method, we show that it
is essential for the passband to be stable to at least $\sim 10^{-4}$. In
contrast, the constraints on the spectral smoothness of the absolute
calibration are mainly aesthetic if one is able to take advantage of spatial
information. To the extent it is understood, controlling polarization to
intensity leakage at the $\sim 10^{-2}$ level will also be essential to
rejecting Faraday rotation of the polarized synchrotron emission.
###### Subject headings:
dark ages, reionization, first stars – methods: data analysis – methods:
statistical
## 1\. Introduction
One of the richest yet least understood narratives in cosmology is the
formation of the complex structure that we see today out of the simple initial
conditions implied by the cosmic microwave background (CMB). The first
luminous objects are thought to have formed at $z\sim 20-30$ through collapse
in $10^{6}-10^{8}\,{\rm M}_{\odot}$ halos (Barkana & Loeb, 2001; Bromm, 2013).
The radiation from these objects heated and then reionized the intergalactic
medium (IGM). There are several sources of complementary information about the
evolution of ionization in this epoch.
The CMB temperature anisotropy is damped by the total Thomson depth to free
electrons. The Planck collaboration has used this effect, combined with a
constraint on the scalar amplitude from gravitational lensing, to measure the
optical depth through reionization (Planck Collaboration et al., 2013). In
addition, Thomson scattering through the reionization epoch generates a unique
polarization signature on large angular scales. WMAP has measured the total
optical depth using this polarization signature (Bennett et al., 2013). These
are integral constraints on the free electron abundance and can be translated
into a central reionization redshift of $10.6\pm 1.1$ (Bennett et al., 2013).
Once the IGM is highly ionized, it is transparent to Ly-$\alpha$ photons.
Absorption measurements along sight lines to high-redshift quasars indicate
that reionization must have ended by $z<6$ (Fan et al., 2006). Absorption
saturates at low abundance, so these should be taken as bounds on the end of
reionization, which could still have been largely complete at redshifts higher
than $6$.
Recently, two methods have been developed to place much more direct bounds on
the duration of reionization. The ionization process is thought to be
spatially patchy, as local sources of radiation blow ionized bubbles that
coalesce into the fully reionized IGM. CMB photons scattering from this patchy
screen produce an additional kinetic Sunyaev-Zel’dovich anisotropy appearing
most clearly at $\ell>3000$, where the primary CMB is negligible (Knox et al.,
1998). Upper limits on this effect translate into a model-dependent upper
bound on the duration of reionization Zahn et al. (2012) and hold the promise
of direct detection of patchy structure in the near future.
The patchy structure of reionization can also be observed directly in three
dimensions using emission of neutral hydrogen through its $21$ cm line
(Furlanetto et al., 2006). Recent bounds from GMRT (Paciga et al., 2013), MWA
(Dillon et al., 2014), and PAPER (Parsons et al., 2014; Pober et al., 2013)
are marching down to the expected level of fluctuations, in parallel to
efforts at LOFAR (van Haarlem et al., 2013). An alternative to measuring the
$21$ cm anisotropy is to measure the signal of its global emission (or
absorption at earlier times) (Pritchard & Loeb, 2008, 2010), which reveals the
bulk energetic properties and ionization state of the universe during
reionization and preceding epochs when the first luminous structures were
forming. Global $21$ cm experiments include EDGES (Bowman & Rogers, 2010) and
SCI-HI Voytek et al. (2014) (which have both reported bounds),
LEDA111http://www.cfa.harvard.edu/LEDA/science.html and the proposed DARE
mission (Burns et al., 2012).
The frequencies of interest in these global studies are $\sim 50-200$ MHz, and
fiducial theoretical models suggest a maximum contrast of $\sim 100$ mK
relative to the synchrotron emission of the galaxy, which can vary $\sim
10^{2}-10^{5}$ K across the sky and frequency range. Astrophysical synchrotron
emission is thought to be fully described by a handful of spectrally smooth
functions that can be distinguished from the variation of the global
reionization signal.
Extremely bright foregrounds make the measurement susceptible to instrumental
systematics. For example, if an instrument with a $1\%$ perturbation to the
spectral calibration observed a $500$ K power law, subtraction of a pure power
law would leave a $5$ K residual, significantly larger than the signal.
Through careful instrumental design, the level of instrumental systematics may
be controlled but generally cannot be nulled entirely.
Here, we develop methods that can be used to constrain the cosmological signal
in these heavily contaminated data. Following the monopole nature of the
signal, experiments to date have mapped the sky with very large beams (Bowman
& Rogers, 2010; Voytek et al., 2014). However, a unique trait of the
foregrounds is that they vary across the sky, while the signal is constant.
This regime is the opposite of the situation normally found in analysis of
small signals, where a modulated signal is pulled out of foregrounds. Liu et
al. (2013) (henceforth L13) proposed that experiments seeking to measure the
global signal should also resolve the sky with an instrumental beam. This
allows selective weighting against regions of high contamination and allows
for the use of angular correlation information to reject foregrounds. The
additional spatial resolution yields higher fidelity recovery of the
cosmological $21$ cm spectrum.
Here, we extend the ideas in L13 to a method that uses the spatial
fluctuations of foregrounds in the data to discover contaminated spectral
modes. A similar idea has been employed successfully in $21$ cm large-scale
structure measurements Switzer et al. (2013); Masui et al. (2013); Chang et
al. (2010) at $z\sim 1$ and has been suggested for cleaning ionospheric
contamination (Vedantham et al., 2014).
Discovery of foreground spectral modes in the measured data makes the method
more robust to assumptions about the foreground contamination. For example,
now if the instrumental passband has a $1\%$ ripple, the largest foreground
mode discovered in our foreground cleaning method will also self-consistently
exhibit this ripple. Generally, instrumental systematics take relatively clean
and smooth functions of frequency from synchrotron emission and convert them
into a more complex structure that requires additional spectral functions to
describe. We argue that the primary goal in instrumental design should be to
prevent proliferation of bright, new foreground modes in the data. Each new
foreground degree of freedom produced by instrumental response to foregrounds
results in more signal loss and makes discovery of the signal more ambiguous.
The methods described here of (1) using spatial variation to discover spectral
foreground modes, which can then be projected out and (2) down-weighting known
spatial areas of high contamination (the galaxy) provide the strongest methods
for recovering the global $21$ cm signal in the absence of additional prior
information about the foregrounds or instrumental response. While the
algorithm of mode subtraction and angular weighting is intuitive, we develop
it from the ground up to expose several implicit choices and possible
pitfalls.
Recently, Bernardi et al. (2014) argued that a dipole gain pattern can be
calibrated in an interferometric array. However, additional variations in
spectral response due to factors such as the analog-to-digital converter,
reflection, and signal loss after the antenna were not included. Our goal here
is to understand how data analysis can be made more robust to this class of
instrumental response (or any other source of foreground covariance), or
alternately how tightly certain instrumental tolerances must be constrained.
In Section 2 we review the basic properties of the global signal and describe
our foreground model. Section 3 builds up the estimator for joint foreground
and signal estimation using a simplified model of spectra along independent
sight lines. Section 4 considers implications of this model for passband
calibration. Section 5 develops spatial weights, and Section 6 combines the
estimators with spatial and spectral weights. Section 7 describes a number of
considerations for using the methods developed here and for global $21$ cm
signal estimation in general. We discuss telescope beam width, the foreground
monopole, how aggressively foreground modes should be removed, and
susceptibility to Faraday rotation. We also consider mode removal of the pre-
reionization absorption feature and extensions of the simple template
amplitude constraint considered throughout. We summarize our conclusions in
Section 8.
## 2\. Global Signal and Foreground Models
From radiative transport arguments, the brightness temperature of $21$ cm
radiation is
$T_{b}\approx 27(1-\bar{x}_{i})\left(\frac{T_{S}-T_{\rm
CMB}}{T_{S}}\right)\left(\frac{1+z}{10}\right)^{1/2}\,{\rm mK},$ (1)
where $\bar{x}_{i}$ is the mean ionization fraction, $T_{S}$ is the spin
temperature of the hyperfine transition, and $T_{\rm CMB}$ is the CMB
temperature. The basic physics of the spin-temperature coupling is well
understood Pritchard & Loeb (2010), but the detailed astrophysical processes
that determine the coupling strength and the gas temperature are still
conjectural.
For $z>200$, the universe is dense enough that electron collisional
interactions drive the spin temperature to the gas temperature, which is
cooling faster than the CMB. This produces absorption. By $z\sim 30$, the
universe is sufficiently rarefied that the spin temperature is better coupled
to the CMB bath, and absorption is expected to subside. Once the first
luminous objects form, these produce radiation that drives the Wouthuysen-
Field (Wouthuysen, 1952; Field, 1958) coupling of $T_{S}$ again to the gas
temperature, leading to a second absorption feature. Then, X-ray heating of
the gas can drive the spin temperature above the CMB temperature, leading to
emission. As these luminous processes proceed and increase, they also ionize
the IGM, which causes the signal to disappear as $1-\bar{x}_{i}\rightarrow 0$
(Pritchard & Loeb, 2010).
We will find it convenient to have a reionization model with a small number of
parameters, rather than a full spectrum. For concreteness, we will spend most
of the paper focusing on the evolution of the brightness temperature as the
universe is reionized, rather than the absorption dip as heating begins.
However, our methods are applicable at all redshifts, and in Section 7.6 we
will briefly discuss the pre-reionization dip.
At the beginning of the reionization epoch, it is thought that the spin
temperature is strongly coupled to the gas and that the gas is heated, driving
$21$ cm emission. As the ionization proceeds, this emission dies away. A
simple way of parameterizing this is (Pritchard & Loeb, 2010)
$\bar{x}_{i}(z)=\frac{1}{2}\left[1+\tanh\left(\frac{z_{r}-z}{\Delta
z}\right)\right],$ (2)
so that the brightness temperature scales as (Figure 1)
$T_{b}(z)=\frac{27}{2}\left(\frac{1+z}{10}\right)^{1/2}\left[1+\tanh\left(\frac{z_{r}-z}{\Delta
z}\right)\right]\,{\rm mK}$ (3)
where we have assumed that $T_{S}\gg T_{\rm CMB}$.
Figure 1.— Brightness temperature of the global $21$ cm emission in a
simplified two-parameter model of reionization. The central redshift is taken
from the WMAP9 constraint at $z=10.6$, and several representative values for
the duration of reionization are shown.
Our model for the diffuse intensity (Stokes-$I$) of the foregrounds is based
on an extended version of the Global Sky Model (GSM) of de Oliveira-Costa et
al. (2008) that is developed in L13. The original GSM used a principal
component analysis to extrapolate and interpolate between previous galactic
surveys at a wide range of frequencies. It is based on three spectral
components and is accurate down to $\approx 10^{\circ}$. L13 extends this
model by adding additional (less smooth) spectral eigenmodes as well as a
population of power-law sources with randomly distributed spectral indices,
drawn from the $dN/dS$ brightness distribution of Di Matteo et al. (2002). The
goal of adding these components is to boost the rank of the foreground
covariance to reflect spectral fluctuations that can be expected on the real
sky. These cannot be probed in the original rank-3 GSM. An example map of our
foreground model at $80\,\textrm{MHz}$ is shown in Figure 2.
Figure 2.— Foreground model at 80 MHz, displayed as $\log(T)$.
In Section 7.7, we will additionally consider the impact of Faraday rotation
and polarized foregrounds. Because this is somewhat peripheral to the
development of our data analysis methods in the next few sections, we will
defer our discussion of our foreground polarization model until then.
## 3\. Global $21$ cm Observations Along Independent Lines of Sight
A global $21$ cm experiment that maps the sky will have a finite number of
resolution elements across its field, defined by the beam. In each one of
these resolution elements, it observes the spectrum of incoming radiation. A
reasonable approximation is to split the survey into $N_{\theta}$ spectra of
length $N_{\nu}$ along all of the angular resolution elements. Let
$\boldsymbol{y}_{i}$ be that vector, where $i$ indexes the sight line ($1$ to
$N_{\theta}$) and the length of the vector is $N_{\nu}$. In this section, we
will build up infrastructure and intuition using this simplified case of
$N_{\theta}$ independent and identically distributed sight lines. Section 6
extends this to a method that treats realistic foregrounds with proper angular
correlations. Because angular correlations become largely irrelevant when
considering wide-beam experiments with essentially no angular sensitivity,
much of the intuition developed here is directly transferable to the analysis
of such experiments.
In writing an estimator, there are many choices for aspects of the global
signal that could be estimated. The goal could be to constrain (1) an
arbitrary spectrum of the cosmological $21$ cm evolution, (2) some modes of
its variation that relate to physical parameters, (3) some amplitudes that are
based on external information about the expected redshift of reionization, or
(4) the amplitude of a template of a provisional global $21$ cm signal. For
our purposes here, it is simplest to develop estimators for the amplitude of
an assumed $21$ cm signal template. Initially, experiments will simply be
stepping down in bounds and seeking some evidence of $z\sim 20$ absorption and
heating or a $z\sim 11$ reionization signal—an amplitude constraint could
provide the simplest, clear indication of a cosmological signal. Section 7.5
considers other estimation regimes in light of the methods developed here.
Let the assumed template of the global signal be a vector $\boldsymbol{x}$ of
length $N_{\nu}$ and normalized to have a maximum of $1$. Multiply by some
amplitude $\alpha$ to get the $21$ cm signal. Then the observed spectrum is
the sum of signal, thermal noise $\boldsymbol{n}_{{\rm inst},i}$, and
foregrounds $\boldsymbol{n}_{{\rm fg},i}$:
$\boldsymbol{y}_{i}=\alpha\boldsymbol{x}+\boldsymbol{n}_{{\rm
fg},i}+\boldsymbol{n}_{{\rm inst},i}.$ (4)
Initially we will assume that the foregrounds and noise are identically and
independently normally distributed along each line of sight $i$,
$\boldsymbol{n}_{{\rm fg},i}\sim N(0,\mathbf{\Sigma}_{\rm fg})$ and
$\boldsymbol{n}_{{\rm inst},i}\sim N(0,\mathbf{\Sigma}_{\rm inst})$. In
reality, the foreground field is strongly correlated in angle and non-
Gaussian. These issues will be examined in subsequent sections. In contrast,
thermal noise is uncorrelated between sight lines and normally distributed:
$\boldsymbol{n}_{{\rm inst},i}\sim N(0,\mathbf{\Sigma}_{\rm inst})$ will
remain an excellent approximation. Throughout, the $N_{\nu}\times N_{\nu}$
matrix $\mathbf{\Sigma}\equiv\mathbf{\Sigma}_{\rm inst}+\mathbf{\Sigma}_{\rm
fg}$ will refer to the total $(\nu,\nu^{\prime})$ covariance.
### 3.1. Known Covariance
If the foregrounds and thermal noise are drawn from a total, known covariance
$\mathbf{\Sigma}$, then the maximum likelihood estimate for the template
amplitude is
$\hat{\alpha}_{\rm
ML}=(\boldsymbol{x}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{\bar{y}}$
(5)
where
$\boldsymbol{\bar{y}}=N_{\theta}^{-1}\sum_{i=1}^{N_{\theta}}\boldsymbol{y}_{i}$
(6)
is the mean spectrum along all lines of sight.
This can be understood as “deweighting” the foregrounds
($\mathbf{\Sigma}^{-1}\boldsymbol{\bar{y}}$), projecting onto the signal
template ($\boldsymbol{x}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{\bar{y}}$), and
then applying a normalization to account for the weights. The estimated
amplitude is normally distributed, and its error is
${\rm Var}(\hat{\alpha}_{\rm
ML})|_{\mathbf{\Sigma}}=(\boldsymbol{x}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{x})^{-1}.$
(7)
To convert this into a more intuitive cleaning process, take the
eigendecomposition of the covariance
$\mathbf{\Sigma}=\boldsymbol{V}\mathbf{\Lambda}\boldsymbol{V}^{T}$. The
cleaning operation $\mathbf{\Sigma}^{-1}\boldsymbol{\bar{y}}$ is then
$\boldsymbol{V}\mathbf{\Lambda}^{-1}\boldsymbol{V}^{T}\boldsymbol{\bar{y}}$.
Here $\boldsymbol{V}^{T}$ projects the data onto a basis where the
contaminated modes $\boldsymbol{v}_{j}$ (the $j$’th row of $\boldsymbol{V}$)
are orthonormal. Those modes are weighted against
$\lambda_{j}^{-1}=[\mathbf{\Lambda}^{-1}]_{jj}$, where the highest variance,
most contaminated modes are down-weighted. Then, $\boldsymbol{V}$ projects
this down-weighted basis back onto the spectral basis.
In the usual assumptions, the foregrounds are not full rank, even if the
spectral covariance $\boldsymbol{\Sigma}$—which contains instrumental noise—is
full rank. Often, the foregrounds are instead described as some set of
contaminated modes $\boldsymbol{v}_{j}$ where $j$ ranges from one to the
number of contaminated modes $N_{\rm fg}$, or
$\boldsymbol{n}_{{\rm fg},i}=\sum_{j}^{N_{\rm fg}}a_{i,j}\boldsymbol{v}_{j}.$
(8)
If the foreground covariance is described by only $N_{\rm fg}$ highly
contaminated modes, a robust cleaning method is to null those modes entirely,
setting $\lambda\rightarrow\infty$ artificially (Liu & Tegmark, 2012). This is
equivalent to fitting and subtracting those spectral modes from each line of
sight, as
$\boldsymbol{\bar{y}}_{\rm clean}=\sum_{j}^{N_{\rm
fg}}(1-\boldsymbol{v}_{j}\boldsymbol{v}_{j}^{T})\boldsymbol{\bar{y}},$ (9)
where the $\boldsymbol{v}_{i}$ are normalized so that
$\boldsymbol{v}^{T}_{i}\boldsymbol{v}_{i}=1$.
Here we have assumed that the foreground spectral functions are known in
advance and can be projected out. This is essentially the same as arguments
for subtracting polynomials or power laws along the line of sight. A crucial
difference is that there is no assumption of smoothness—if the foregrounds
were known to have a particular spectral shape, that could be represented in
the vectors $\boldsymbol{v}_{i}$. The essential property of the foregrounds
that allows for their removal is not that they are smooth, but that they are
described by few functions. In the limit that the number of orthogonal
functions removed approaches $N_{\nu}$, all of the signal is removed because
$\\{\boldsymbol{v}_{i},i=1\ldots N_{\nu}\\}$ spans the space, assuming
orthogonality of the $\boldsymbol{v}_{i}$. The rank of bright foregrounds is
the primary determinant of cosmological signal that can be extracted.
Subtraction of bright foregrounds has proven to be significantly more
challenging in practice due to instrumental effects (Switzer et al., 2013).
Figure 3 shows an example of $0.1\%$ calibration for a power-law foreground
spectrum similar in amplitude to the one reported by Voytek et al. (2014) and
scaled to the slightly different frequencies of interest here. This represents
the case where the $\boldsymbol{v}_{i}$ mode removed assumes a pure power law
but the actual measurement reflects the instrument’s response to
$\boldsymbol{v}_{i}$. Residuals are considerably larger than the signal.
Figure 3.— Impact of calibration uncertainty. Top: a power-law foreground
spectrum similar to the one observed in Voytek et al. (2014), multiplied by a
$0.1\%$ calibration error. Bottom: residuals when a smooth power-law spectrum
is subtracted from data with a calibration error. For contrast, the global
signal at these redshifts is expected to be $\sim 30$ mK. Also note that the
calibration error will not integrate down thermally.
One approach to treating the passband is to control it through instrumental
design and to measure it very precisely prior to the experiment. In principle,
this could be achieved by long integrations on a calibration reference that is
known to be smooth. This carries the challenge of calibrating on a source
other than the sky. An alternative is to recognize that the foregrounds are a
bright spectral reference, viewed through the same survey program as the
primary science. One calibration scheme would then consist of integrating down
on bright synchrotron emission and renormalizing the spectrum under the
assumption that the emission is spectrally smooth (Voytek et al., 2014).
An alternative to calibrating the instrument to meet the smooth mode functions
$\\{\boldsymbol{v}_{i}\\}$ is to adjust the mode functions to reflect
imperfections in the instrument. In the next section we develop a method for
discovering foreground spectral mode functions in the data based on their
spatial variations. The central idea is that the cosmological signal is a
monopole spectrum, so the spectrum of anything that fluctuates across the sky
must reflect some foreground modes (hence the title “erasing the variable”).
If there are monopole foreground components, they are formally
indistinguishable from the cosmological $21$ cm monopole in this picture (in
the absence of prior information).
Measurement of $\\{\boldsymbol{v}_{i}\\}$ modes within the data may also
discover unanticipated or unconstrained instrumental response systematics.
Examples include (1) a passband that varies with time or pointing, which
causes a response to synchrotron radiation that varies across the sky, (2)
polarization leakage, which can cause Faraday (spectral) rotation that varies
across the sky, (3) terrestrial interference that varies with instrument
pointing, and (4) unmodeled frequency dependence of the beam. Even if the
foregrounds were intrinsically a simple rank-1 spectral function, the
instrument response will tend to proliferate the rank of the foreground
covariance. This limits the ultimate sensitivity to the global signal. While
many of these sources of systematics may be tightly controlled by the
instrument’s construction, residual imperfections will not be known in advance
and could be discovered by this method.
We will assume throughout that the true synchrotron foregrounds are described
by a small number of modes $N_{\rm fg,true}$. The instrument observes these
bright foregrounds and records spectral data that may need some $N_{\rm
fg}\geq N_{\rm fg,true}$ functions to describe the contamination.
### 3.2. Covariance from the Measurements
This section considers the joint determination of the global signal amplitude
$\alpha$ and the contaminated foreground modes. The statistical methods
described and extended here were developed by Rao (1967). The central idea in
these methods is to begin with a poor, contaminated estimator for the $21$ cm
signal amplitude. This initial estimator can then be cleaned up by projecting
out correlations with the foregrounds.
To reflect the reality that often one does not know the covariance a priori,
first consider a simple estimator that does not use any information from the
spectral covariance. This will serve as our starting point for constructing an
estimator that simultaneously estimates the spectral covariance from the data.
Again, let $\boldsymbol{x}$ be the spectral template of the cosmological
signal that we would like to constrain. The simplest (and arguably worst)
estimator we could develop for the amplitude is
$T(\parallel\boldsymbol{x})=(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}\boldsymbol{\bar{y}},$
(10)
which takes the dot product of the $21$ cm template and observed data
($\boldsymbol{x}^{T}\boldsymbol{\bar{y}}$) and normalizes through
$(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}$. We refer to this as
$T(\parallel\boldsymbol{x})$ because it estimates the amplitude of the
component of $\boldsymbol{\bar{y}}$ parallel to the signal $\boldsymbol{x}$.
This is exactly the estimator that would be used if there were no frequency
correlations in the noise. What makes this a very poor estimator is that it is
strongly correlated with the foreground modes and will therefore contain
strong foreground residuals. In contrast, Equation (5) in the previous section
used the known frequency covariance to deweight the foreground contamination.
We can begin to improve upon our simple estimator by constructing spectral
modes that contain no cosmological signal by design. Using a Gram-Schmidt
process, we can make a set of vectors $\boldsymbol{z}_{i}$ that span the space
orthogonal to the theoretical template of the cosmological signal
$\boldsymbol{x}$, so that $\boldsymbol{z}_{i}^{T}\boldsymbol{x}=0$. There are
$N_{\nu}-1$ such vectors, and the choice of these vectors is completely
arbitrary at this point. Any spectrum can be represented as a sum
$\boldsymbol{y}=\alpha\boldsymbol{x}+\sum_{i}\beta_{i}\boldsymbol{z}_{i}$ (11)
because $\\{\boldsymbol{x},\boldsymbol{z}_{i}\\}$ span the
$N_{\nu}$-dimensional space. We can pack these vectors $\boldsymbol{z}_{i}$
into a matrix $\boldsymbol{Z}$ and write an estimator for all of the spectral
information orthogonal to $\boldsymbol{x}$,
$T(\perp\boldsymbol{x})=\boldsymbol{Z}^{T}\boldsymbol{\bar{y}}$. This
represents all of the components of $\boldsymbol{\bar{y}}$ orthonormal to the
signal and is dominated by foregrounds. (Note that we could choose to
normalize this for general vectors $\boldsymbol{z}$ and find
$(\boldsymbol{Z}^{T}\boldsymbol{Z})^{-1}\boldsymbol{Z}^{T}\boldsymbol{\bar{y}}$.
This would make the equations more complex but would not modify the estimator
for the cosmological quantity.)
The covariance between $T(\parallel\boldsymbol{x})$ and
$T(\perp\boldsymbol{x})$ is
$\displaystyle{\rm Cov}_{\parallel,\perp}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\boldsymbol{C}_{\parallel,\parallel}&\boldsymbol{C}_{\parallel,\perp}\\\
\boldsymbol{C}_{\perp,\parallel}&\boldsymbol{C}_{\perp,\perp}\end{array}\right)$
(14) $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}\mathbf{\Sigma}\boldsymbol{x}(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}&(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}\mathbf{\Sigma}\boldsymbol{Z}\\\
\boldsymbol{Z}^{T}\mathbf{\Sigma}\boldsymbol{x}(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}&\boldsymbol{Z}^{T}\mathbf{\Sigma}\boldsymbol{Z}\end{array}\right).$
(17)
The off-diagonal terms
$(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}\mathbf{\Sigma}\boldsymbol{Z}$
represent correlations between $T(\parallel\boldsymbol{x})$ and
$T(\perp\boldsymbol{x})$ caused by the $(\nu,\nu^{\prime})$ covariance of
foregrounds in $\mathbf{\Sigma}$. If the noise terms were only thermal
fluctuations, $\mathbf{\Sigma}=\sigma_{\rm inst}^{2}\mathbf{1}$, and by the
construction $\boldsymbol{x}^{T}\boldsymbol{Z}=0$, the off-diagonal terms
vanish. In other words, an estimator $T(\perp\boldsymbol{x})$ that is known to
contain only contaminants (and no cosmological signal) is correlated with our
estimate of the cosmological signal, so the latter must be contaminated.
With knowledge of spectral correlations, we can form an improved, adjusted
estimator
$\hat{\alpha}=T(\parallel\boldsymbol{x})-\boldsymbol{C}_{\parallel,\perp}\boldsymbol{C}_{\perp,\perp}^{-1}T(\perp\boldsymbol{x})$
(18)
that projects foreground frequency correlations out of the
$T(\parallel\,\boldsymbol{x})$ estimator. Intuitively, this estimator
instructs us to make an estimate $T(\perp\boldsymbol{x})$ of the portion of
the measurement that is known to contain only foregrounds. Off-diagonal
elements of the covariance matrix then allow the level of foreground leakage
(into our estimate $T(\parallel\boldsymbol{x})$ of the cosmological signal) to
be predicted and subtracted off. A similar technique was used recently in a
power spectrum analysis of PAPER data (Parsons et al., 2014).
So far, we have assumed that $\mathbf{\Sigma}$ is known, but the same
covariance adjustment can be performed with respect to an estimated
$\mathbf{\hat{\Sigma}}$. Without perfect knowledge of $\mathbf{\Sigma}$, the
error bars grow but the estimator remains unbiased. The $(\nu,\nu^{\prime})$
covariance can be estimated empirically using
$\mathbf{\hat{\Sigma}}=(N_{\theta}-1)^{-1}\sum_{i=1}^{N_{\theta}}(\boldsymbol{y}_{i}-\boldsymbol{\bar{y}})(\boldsymbol{y}_{i}-\boldsymbol{\bar{y}})^{T}$
(19)
Writing out Equation (18),
$\hat{\alpha}=(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}(1-\mathbf{\Pi})\boldsymbol{\bar{y}}$
(20)
where
$\mathbf{\Pi}=\mathbf{\hat{\Sigma}}\boldsymbol{Z}(\boldsymbol{Z}^{T}\mathbf{\hat{\Sigma}}\boldsymbol{Z})^{-1}\boldsymbol{Z}^{T}$.
Remarkably, under our current assumptions where $\boldsymbol{Z}$ spans the
rest of the $N_{\nu}$ dimensions, this estimator is equivalent to (see
Appendix A)
$\hat{\alpha}=(\boldsymbol{x}^{T}\mathbf{\hat{\Sigma}}^{-1}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}\mathbf{\hat{\Sigma}}^{-1}\boldsymbol{\bar{y}},$
(21)
which is precisely the same as Equation (5), except with the estimated
spectral covariance $\mathbf{\hat{\Sigma}}$ replacing the known covariance
$\mathbf{\Sigma}$.
Gleser & Olkin (1972) develop the same result through a completely independent
method. They write down the joint likelihood of the covariance and mean
(cosmological signal), where the covariance is Wishart-distributed and the
mean is normally distributed. They then maximize the likelihood and find the
same result, showing that the choice
$\mathbf{\Sigma}\rightarrow\mathbf{\hat{\Sigma}}$ coincides with the maximum
likelihood.
In summary, the proposed procedure is to
* •
Find the sample mean and $(\nu,\nu^{\prime})$ covariance across observed lines
of sight.
* •
Invert the measured covariance and use it to down-weight contaminated modes in
the data.
* •
Find the inner product of the cleaned data and the signal template, then
normalize.
While the maximum of the likelihood has the same form as the case where the
covariance is known, the distribution of the estimated $21$ cm global signal
template amplitude is no longer Gaussian, and is generally broader. These
changes are due to the fact that the data are also used for the covariance
estimation, which uses up degrees of freedom in determining the foreground
modes. Rao (1967) shows that
${\rm
Var}(\hat{\alpha})=\frac{N_{\theta}-1}{N_{\theta}-1-r}[\boldsymbol{C}_{\parallel,\parallel}-\boldsymbol{C}_{\parallel,\perp}\boldsymbol{C}_{\perp,\perp}^{-1}\boldsymbol{C}_{\perp,\parallel}],$
(22)
where $r$ is the rank of the additional degrees of freedom that are estimated.
In our case here, $r=N_{\nu}-1$ ($\boldsymbol{Z}$ spans the rest of the
spectral space) so that
${\rm
Var}(\hat{\alpha})=\left(1+\frac{N_{\nu}-1}{N_{\theta}-N_{\nu}}\right)(\boldsymbol{x}^{T}\mathbf{\hat{\Sigma}}^{-1}\boldsymbol{x})^{-1},$
(23)
where we have also plugged in the explicit forms for
$\boldsymbol{C}_{\parallel,\parallel}$,
$\boldsymbol{C}_{\parallel,\perp}$,$\boldsymbol{C}_{\perp,\perp}$, and
$\boldsymbol{C}_{\perp,\parallel}$ from Equation (14). We see that the
covariance is enhanced relative to the case of perfect foreground covariance
knowledge (Equation (7)) by a factor related to the number of frequencies and
independent lines of sight. The error diverges when $N_{\theta}$ approaches
$N_{\nu}$ from above. The rank $r(\mathbf{\hat{\Sigma}})\leq{\rm
min}(N_{\theta},N_{\nu})$, and for $N_{\theta}\gg N_{\nu}$, the covariance is
well measured. This suggests that the optimal limit is to have many more
resolution elements than spectral bins.
There is little instrumental limitation on the number of spectral bins
$N_{\nu}$ over the bands observed. In contrast, there is a hard limit on the
number of “independent” lines of sight that can be observed, $N_{\theta}$. At
the frequencies of interest for the global $21$ cm signal, the beam size tends
to be large by diffraction. Angular correlations and noise generally limit the
number of independent samples of the spectrum (see Section 7.1).
The requirement $N_{\theta}>N_{\nu}$ stems from the fact that
$\mathbf{\hat{\Sigma}}$ is trying to estimate a full $N_{\nu}$ rank
covariance. In general, the sensitivity to $\alpha$ should not depend on the
number of spectral bins in the survey, once they become fine enough to resolve
the signal. The solution is to instead estimate some number $N_{\rm fg}$ of
contaminated spectral functions, less than the number of spectral bins. Then
the estimation becomes independent of the $N_{\nu}$, so long as
$N_{\nu}>N_{\rm fg}$.
### 3.3. Empirically Determined Foregrounds of Limited Rank
The key step in the covariance adjustment scheme of the previous section was
the formation and application of
$\mathbf{\Pi}=\mathbf{\hat{\Sigma}}\boldsymbol{Z}(\boldsymbol{Z}^{T}\mathbf{\hat{\Sigma}}\boldsymbol{Z})^{-1}\boldsymbol{Z}^{T}$,
which modeled contaminated modes in the original “poor” estimator for later
subtraction. The modes formed a basis $\boldsymbol{Z}$ orthogonal to the
signal $\boldsymbol{x}$. The specific choice of vectors in $\boldsymbol{Z}$
was arbitrary so long as they were orthonormal (for simplicity) and spanned
the spectral subspace orthogonal to the signal. For concreteness, we suggested
forming this basis blindly using a Gram-Schmidt process. While formally a
solution to our problem, this is not a particularly efficient way to implement
our recipe in practice, for each of the resulting basis vectors will be a
linear combination of noise and foreground modes. On the other hand, the
foregrounds should be describable by a small number of modes (Liu & Tegmark,
2012; Bernardi et al., 2014). A sensible alternative would therefore be to
intelligently partition the basis $\boldsymbol{Z}$ into two sub-bases, one
that is of a relatively low rank $N_{\rm fg}$ containing the foregrounds and
another of rank $N_{\nu}-N_{\rm fg}-1$ consisting of the remaining noise-
dominated modes. Computationally, this means that rather than estimating the
full $N_{\nu}$ rank covariance from the data, we only need to estimate a
subset of contaminated modes and their amplitudes. This limits the number of
degrees of freedom $r$ that need to be estimated from the data, which as we
saw from Equation (22) is crucial for keeping the final error bars small.
In the rank-restricted foreground approximation, the data along a line of
sight are the cosmological signal plus some amplitudes $\boldsymbol{\beta}$
times foreground modes $\boldsymbol{F}$ plus thermal noise, as
$\boldsymbol{y}=\alpha\boldsymbol{x}+\boldsymbol{F}\boldsymbol{\beta}+\boldsymbol{n}_{t}\Rightarrow\mathbf{\Sigma}=\boldsymbol{F}\mathbf{\Gamma}\boldsymbol{F}^{T}+\sigma_{\rm
inst}^{2}\boldsymbol{1},$ (24)
where we have taken the thermal noise to be stationary for simplicity and let
$\mathbf{\Gamma}$ be the eigenvalue matrix of the foreground spectral
covariance
$\boldsymbol{F}\langle\boldsymbol{\beta}\boldsymbol{\beta}^{T}\rangle\boldsymbol{F}^{T}$,
quantifying the power of the foreground modes $\boldsymbol{F}$. The amplitude-
ordered eigenvalue spectrum of the total covariance $\mathbf{\Sigma}$ in this
case would show some large contaminant amplitudes followed by a noise floor
set by $\sigma$.
Unlike $\boldsymbol{Z}$ of the previous section, the foreground mode vectors
in $\boldsymbol{F}$ alone do not span the entire spectral subspace orthogonal
to the cosmological signal. The remaining portion of the space is spanned by
basis vectors that describe instrumental noise, under the assumptions of
Equation (24). These vectors can be formed, like before, using a Gram-Schmidt
process (this time relative to the signal $\boldsymbol{x}$ and foregrounds
$\boldsymbol{F}$) and packed into a matrix $\boldsymbol{G}$. The observed
spectrum $\boldsymbol{y}$ can then be written as
$\boldsymbol{y}=\alpha\boldsymbol{x}+\sum_{j=1}^{N_{\rm
fg}}\beta_{j}\boldsymbol{f}_{j}+\sum_{j=N_{\rm
fg}+1}^{N_{\nu}}\gamma_{j}\boldsymbol{g}_{j},$ (25)
which is to be contrasted with Equation (11).
Again, we can form poor estimators for the amplitudes of the signal
($\hat{\alpha}=(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}\boldsymbol{\bar{y}}$),
foreground modes
($\boldsymbol{\hat{\beta}}=\boldsymbol{F}^{T}\boldsymbol{\bar{y}}$), and
thermal noise modes
($\boldsymbol{\hat{\gamma}}=\boldsymbol{G}^{T}\boldsymbol{\bar{y}}$). To do
so, however, we must first specify how the foreground modes $\boldsymbol{F}$
are identified and defined, and in the following section we outline two
different methods for this.
### 3.4. Two Outlooks on Separation of Signal and Foregrounds
The first method is closely related to the treatment that we have presented so
far. In particular, we considered modes in the matrices $\boldsymbol{Z}$ (full
rank) or $\boldsymbol{F}$ (finite rank) that represent components of the data
that are orthogonal to the cosmological signal. There is, however, nothing
preventing real foregrounds from being parallel to the signal, and generally
$\boldsymbol{n}_{\rm fg}$ will be the sum of $\boldsymbol{n}_{{\rm
fg},\parallel\boldsymbol{x}}$ (foregrounds parallel to the cosmological
signal) and $\boldsymbol{n}_{{\rm fg},\perp\boldsymbol{x}}$ (foregrounds
perpendicular to the signal). For example, if the signal and foregrounds share
a slowly varying spectral component, it will fall in $\boldsymbol{n}_{{\rm
fg},\parallel\boldsymbol{x}}$.
An alternative is to form foreground spectral modes that best describe the
covariance of the fluctuating terms on the sky without regard for the
cosmological signal. Then, the signal has a piece that is parallel to those
foreground modes and a piece that is perpendicular, or
$\boldsymbol{x}=\boldsymbol{x}_{\parallel\boldsymbol{F}}+\boldsymbol{x}_{\perp\boldsymbol{F}}$.
In equations, these two methods are
$\displaystyle\textrm{ Method 1:}~{}~{}~{}~{}\boldsymbol{y}$ $\displaystyle=$
$\displaystyle\boldsymbol{x}+\boldsymbol{n}_{{\rm
fg},\parallel\boldsymbol{x}}+\boldsymbol{n}_{{\rm
fg},\perp\boldsymbol{x}}~{}~{}~{}~{}\textrm{delete:}~{}~{}\boldsymbol{n}_{{\rm
fg},\perp\boldsymbol{x}}$ $\displaystyle\textrm{ Method
2:}~{}~{}~{}~{}\boldsymbol{y}$ $\displaystyle=$
$\displaystyle\boldsymbol{x}_{\perp\boldsymbol{F}}+\boldsymbol{x}_{\parallel\boldsymbol{F}}+\boldsymbol{n}_{\rm
fg}~{}~{}~{}~{}\textrm{delete:}~{}~{}\boldsymbol{x}_{\parallel\boldsymbol{F}}+\boldsymbol{n}_{\rm
fg}.$
In the first method, we can develop a cleaning operation that
removes/deweights $\boldsymbol{n}_{{\rm fg},\perp\boldsymbol{x}}$, that is,
components of the foreground variance orthogonal to the signal. This estimator
does not touch the signal by design, so it is guaranteed to have no
cosmological signal loss. However, substantial foregrounds will remain in the
estimated signal with this rather conservative approach. In the second method,
we can develop cleaning that removes any foreground spectral modes that vary
spatially on the sky along with any component of the signal parallel to that.
If implemented carelessly, this aggressive method will entail cosmological
signal loss, although the problem is rectifiable.
Method 1 is described in Rao (1967) and is a simply a slight modification of
the prescription in Section 3.2. Method 2 is the one we develop here and
advocate for global $21$ cm signal recovery.
We demonstrate the differences in approach using simple simulations. These
have $N_{\theta}$ lines of sight and a number of power laws in the synchrotron
emission (to limit the rank). The indices of these power laws are
$\\{-2.1,-2.5,-2.9\\}$ and the amplitude is uniformly distributed and
positive. Here, $-2.5$ is a characteristic synchrotron index at these
frequencies Rogers & Bowman (2012); Voytek et al. (2014), and the two spanning
it give additional modes in our initial toy model. Again, as amplitudes we
take representative values from the recent observations of Voytek et al.
(2014) and scale to the slightly higher frequencies of interest here. For the
purposes here we will neglect thermal noise and assume the integration is deep
relative to the cosmological signal. Figure 4 shows one realization of the
foreground model. This model is only meant to be pedagogical, showing the main
point of the algorithms here for independent sight lines in a simplified
setting with three spectral modes. Beginning in Section 5 we will use the full
extended GSM model data cubes described in Section 2.
Figure 4.— Simple simulation of 10 lines of sight with a random combination of
three power laws.
Figure 5.— Modes discovered from the $(\nu,\nu^{\prime})$ covariance of simple
simulations with three synchrotron spectral indices (Figure 4). These include
only the foregrounds and simple signal model of Equation (3) with $\Delta
z=0.5$ and $z_{r}=10.6$. Top left: the largest three eigenvectors of the
covariance $\mathbf{\hat{\Sigma}}$ that fully describe the foregrounds. Top
right: the filtered data
$(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{\bar{y}}$. The input signal
in red is largely recovered after cleaning the three foreground eigenmodes in
the top left. In this toy model, the foregrounds are removed entirely, and
only the signal (minus components parallel to the subtracted foregrounds) is
left. Bottom left: the largest three eigenvectors of the restricted covariance
$\mathbf{\hat{\Sigma}}|_{\boldsymbol{x}}=[1-\boldsymbol{x}(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}]\mathbf{\hat{\Sigma}}$.
These are all orthogonal to the cosmological signal. Bottom right: the data
after subtracting the restricted eigenvectors [or
$(1-\boldsymbol{F}_{\perp}\boldsymbol{F}_{\perp}^{T})\boldsymbol{\bar{y}}$]
are exactly the signal template shape by the construction of the mode
functions. The amplitude, however, is clearly dominated by residual
foregrounds (note that the signal is too small to be seen on this vertical
scale). These are the foreground components parallel to the cosmological
signal. While this procedure is immune from loss of cosmological signal, it is
insufficiently aggressive when dealing with foregrounds, and the final result
is strongly influenced by the foreground covariance. We note that the apparent
recovery of the reionization step’s location in the bottom-right subplot is
spurious, for it is almost entirely driven by our choice of which theoretical
template we use. The location of the “step” in the top-right subplot, on the
other hand, is truly constrained by the data, although signal loss means that
it no longer appears as a clean step.
#### 3.4.1 Method 1: Foreground Modes Orthogonal to Signal
To find foreground spectral modes that are orthogonal to the cosmological
signal $\boldsymbol{x}$, we can find the largest eigenmodes of the restricted
covariance
$\mathbf{\hat{\Sigma}}\biggl{|}_{\perp\boldsymbol{x}}=[1-\boldsymbol{x}(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}]\mathbf{\hat{\Sigma}},$
(26)
which in practice will be eigenmodes that represent foregrounds. By
construction, any eigenmode $\boldsymbol{f}_{i}$ of
$\mathbf{\hat{\Sigma}}\bigl{|}_{\perp\boldsymbol{x}}$ will be orthogonal to
$\boldsymbol{x}$, i.e. $\boldsymbol{f}_{i}^{T}\boldsymbol{x}=0$. Let
$\boldsymbol{F}_{\perp}$ hold the restricted eigenvectors
$\boldsymbol{f}_{i}$. The top-left subplot of Figure 5 shows the unrestricted
eigenvectors of an input foreground simulation, while the bottom-left subplot
shows the restricted eigenvectors relative to a signal template.
Having identified our foreground eigenvectors, we can construct our estimator
for the signal amplitude $\alpha$ using the methods introduced in Section 3.2.
Once again, we first form a poor initial estimator of the signal, which we
then adjust using the covariance between the poor estimator and the
foregrounds (which is nonzero because a substantial foreground contamination
remains in the estimate of the signal). The covariance of this estimator takes
the form
${\rm
Cov}_{\hat{\alpha},\boldsymbol{\hat{\beta}}}=\left(\begin{array}[]{cc}(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}\mathbf{\hat{\Sigma}}\boldsymbol{x}(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}&(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}\mathbf{\hat{\Sigma}}\boldsymbol{F}_{\perp}\\\
\boldsymbol{F}^{T}_{\perp}\mathbf{\hat{\Sigma}}\boldsymbol{x}(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}&\boldsymbol{F}^{T}_{\perp}\mathbf{\hat{\Sigma}}\boldsymbol{F}_{\perp}\end{array}\right),$
(27)
which looks exactly like Equation (14) but with all occurrences of
$\boldsymbol{Z}$ replaced by $\boldsymbol{F}_{\perp}$. This is unsurprising
because the modes in $\boldsymbol{G}$ represent a diagonal piece of the
covariance, $\sigma_{\rm inst}^{2}\boldsymbol{1}$, and thus are not correlated
in frequency with either the foreground modes or the signal.
Projecting out the piece of the signal estimator that is correlated with the
foregrounds,
$\displaystyle\hat{\alpha}$ $\displaystyle=$
$\displaystyle(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}(1-\mathbf{\Pi})\boldsymbol{\bar{y}}$
(28) $\displaystyle\mathbf{\Pi}$ $\displaystyle=$
$\displaystyle\mathbf{\hat{\Sigma}}\boldsymbol{F}_{\perp}(\boldsymbol{F}_{\perp}^{T}\mathbf{\hat{\Sigma}}\boldsymbol{F}_{\perp})^{-1}\boldsymbol{F}_{\perp}^{T}.$
(29)
That is, project foreground-contaminated modes in the data and then dot
against the signal template. The operation $\mathbf{\Pi}$ puts the signal in
the basis of foreground modes, weights those by their covariance, and moves
back to the spectral space. Note that $\boldsymbol{F}_{\perp}$ does not span
the rest of the spectral space like $\boldsymbol{Z}$ in the previous section.
Here, Equation (28) is not equivalent to
$\hat{\alpha}=(\boldsymbol{x}^{T}\mathbf{\hat{\Sigma}}^{-1}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}\mathbf{\hat{\Sigma}}^{-1}\boldsymbol{\bar{y}}$.
By construction, our estimator suffers from no loss of cosmological signal
(despite the projecting-out of foregrounds) because
$(1-\mathbf{\Pi})\boldsymbol{x}=\boldsymbol{x}-\mathbf{\hat{\Sigma}}\boldsymbol{F}_{\perp}(\boldsymbol{F}_{\perp}^{T}\mathbf{\hat{\Sigma}}\boldsymbol{F}_{\perp})^{-1}\boldsymbol{F}_{\perp}^{T}\boldsymbol{x}=\boldsymbol{x}$.
In the last step, we used the fact that
$\boldsymbol{F}_{\perp}^{T}\boldsymbol{x}=0$ through the choice of forming
$\boldsymbol{F}_{\perp}$ with eigenvectors restricted to be orthogonal to
$\boldsymbol{x}$, Equation (26).
The error on $\hat{\alpha}$ is
$\displaystyle{\rm Var}(\hat{\alpha})$ $\displaystyle\propto$
$\displaystyle[\boldsymbol{C}_{\parallel,\parallel}-\boldsymbol{C}_{\parallel,\perp}\boldsymbol{C}_{\perp,\perp}^{-1}\boldsymbol{C}_{\perp,\parallel}]$
(30) $\displaystyle\propto$
$\displaystyle(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}(1-\mathbf{\Pi})\mathbf{\hat{\Sigma}}\boldsymbol{x}(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}.$
(31)
Here $\boldsymbol{x}^{T}(1-\mathbf{\Pi})\mathbf{\hat{\Sigma}}\boldsymbol{x}$
can be interpreted as projecting all of the foreground modes orthogonal to the
signal out of the $(\nu,\nu^{\prime})$ covariance and finding the noise with
respect to that residual covariance. The residual covariance originates from
foreground components parallel to the signal and from thermal noise—any
$(\nu,\nu^{\prime})$ covariance component with a nonzero dot product into the
signal $\boldsymbol{x}$ will contribute to the variance of $\hat{\alpha}$.
This property can be seen in Fig 5. The recovered signal has exactly the shape
of the signal template $\boldsymbol{x}$, but the amplitude is dominated by
foregrounds. This feature is clearly undesirable in the regime where
foregrounds vastly exceed the signal.
Another undesirable property of this estimator is that the quoted error ${\rm
Var}(\hat{\alpha})$ relies on the foregrounds being normally distributed. Real
foregrounds are strongly non-Gaussian, and we would like to avoid describing
those components in the errors of $\hat{\alpha}$. We can make a simple
modification to the estimator in Rao (1967) to treat both of these
shortcomings.
#### 3.4.2 Method 2: Signal Component Orthogonal to Foregrounds
In some sense, the method described in Section 3.4.1 was too conservative. By
limiting our labeling of foregrounds to modes that are orthogonal to the
signal, we arrived at an estimator with no formal signal loss, but one in
which substantial foreground residuals remained in the final answer. Here we
consider an estimator that more aggressively projects out foregrounds. The
result will be a lossy treatment, but we will also show how this can be
rectified.
Instead of isolating the foregrounds orthogonal to the signal, consider the
component of the signal that is orthogonal to the foregrounds. The signal is
then partitioned as
$\boldsymbol{x}=\boldsymbol{x}_{\parallel\boldsymbol{F}}+\boldsymbol{x}_{\perp\boldsymbol{F}}$
(32)
For this decomposition to be meaningful, we must once again decide on a
definition for our foreground modes. We let $\boldsymbol{F}$ be the largest
foreground eigenvectors of $\mathbf{\hat{\Sigma}}$ (without the orthogonality
restrictions of the previous section). In the calculation for
$\mathbf{\hat{\Sigma}}$, the mean $\boldsymbol{\bar{y}}$ contains the
cosmological signal and is subtracted out. For this reason, the cosmological
signal does not perturb the $(\nu,\nu^{\prime})$ covariance or its eigenmodes.
This is in contrast to inhomogeneous $21$ cm mapping, where the cosmological
signal is stochastic, cannot be subtracted from the covariance, and does
perturb the eigenvectors (Switzer et al., 2013; Masui et al., 2013). (The
foreground modes can optionally be isolated from thermal noise by forming the
$(\nu,\nu^{\prime})$ cross-variance of maps acquired at two different times,
as was done in Switzer et al. 2013.)
To project the identified foreground modes out of the estimator, we apply
$1-\mathbf{\Pi}=1-\boldsymbol{F}\boldsymbol{F}^{T}$. This operation splits the
spectrum into a foreground-contaminated and a (theoretically) foreground-clean
subspace as
$\left(\begin{array}[]{c}\mathbf{\Pi}\\\
1-\mathbf{\Pi}\end{array}\right)\boldsymbol{y}=\left(\begin{array}[]{c}\mathbf{\Pi}\\\
1-\mathbf{\Pi}\end{array}\right)\alpha\boldsymbol{x}+\left(\begin{array}[]{c}\boldsymbol{n}_{\rm
fg}\\\ \mathbf{0}\end{array}\right)+\boldsymbol{n}_{\rm inst}.$ (33)
Unlike the previous estimator Equation (28), however, the foreground modes are
no longer constructed to be orthogonal to the signal. By projecting out the
foregrounds, then, it is likely that some cosmological signal will be lost.
Mathematically, $(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{x}$ equals
$\boldsymbol{x}_{\perp\boldsymbol{F}}$, not $\boldsymbol{x}$, and the
estimator
$\hat{\alpha}=(\boldsymbol{x}^{T}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}(1-\mathbf{\Pi})\boldsymbol{\bar{y}}$
is an incorrectly normalized estimator of the signal amplitude (i.e., it
suffers from a multiplicative bias). Correcting this, we propose the revised
estimator
$\hat{\alpha}=\frac{\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{\bar{y}}}{\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{x}}.$
(34)
In the error calculation from Equation (31), the key term
$(1-\mathbf{\Pi})\mathbf{\hat{\Sigma}}$ now reduces to $\mathbf{1}\sigma_{\rm
inst}^{2}$, under the assumption that all foregrounds are removed to a good
approximation and only Gaussian thermal noise remains. While the foregrounds
may be strongly non-Gaussian, we will assume that the residuals after the
modes $\boldsymbol{F}$ have been subtracted are Gaussian and due to the
thermal noise floor. This would need to be verified in an analysis of real
data.
The overall error is now
${\rm Var}(\hat{\alpha})\sim\left(1+\frac{N_{\rm fg}}{N_{\theta}-1-N_{\rm
fg}}\right)\frac{\sigma_{\rm
inst}^{2}}{[\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{x}]^{2}}$
(35)
To get some rough intuition, assume that the foreground modes all have about
the same overlap with the cosmological signal. Then
$\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{x}\propto
N_{\nu}-N_{\rm fg}$. As $N_{\rm fg}$ approaches $N_{\nu}$, the foreground
modes do a progressively better job of spanning the $N_{\nu}$ spectral space.
In the limit that they fully span the space (there is no signal
distinguishable from foregrounds), the signal is also nulled and the error
diverges. Alternately, as $N_{\rm fg}$ approaches $N_{\theta}$, the foreground
discovery process uses up all of the spatial degrees of freedom and the error
diverges. In this formulation, then, we are self-consistently including the
possibility of signal loss in our error analysis.
## 4\. Constraints on Passband Calibration
The abstract design guidance suggested by this method is that (1) the
experiment should not increase the rank of the foreground covariance (keep
$N_{\rm fg}$ small) and (2) if there are foregrounds with $N_{\rm fg}$
spectral degrees of freedom, at least $N_{\theta}=N_{\rm fg}$ samples of its
variations need to be observed if those foregrounds are to be subtracted.
A time-varying passband calibration is the worst instrumental systematic in
this sense. Here each line of sight sees a slightly different foreground
spectrum and so requires a different spectral function for cleaning. In the
limit that each line of sight can have a different foreground, $N_{\rm
fg}\rightarrow N_{\theta}$ and the errors diverge. In contrast, a constant
passband error only modulates the foreground covariance spectral functions,
but it does not increase the rank of its covariance. Its impact is more
aesthetic, can be corrected by recalibrating off the brightest mode, and does
not fundamentally limit the investigation. (This is proven in Section 6 using
the specific notation developed there.)
Figure 6 shows the 10 simulated lines of sight with a constant $10\%$
calibration error. Here we multiply the spectrum along each line of sight by a
constant $1+\boldsymbol{\delta}$, where $\boldsymbol{\delta}$ is $N_{\nu}$
long and normally distributed. Fitting a smooth power law to data with this
constant $1+\boldsymbol{\delta}$ calibration error would leave unacceptable
residuals, but the modes are able to discover the constant calibration error
and subtract the foreground with modest signal loss. This parameterization of
a calibration error is a matter of convenience. In simulations where the
errors $\boldsymbol{\delta}$ are a smoothly varying function of frequency or
even have the same shape as the signal template, the conclusions are the same.
(In the pathological case that the calibration error interacts with the
foreground power law to produce modes identical to the cosmological template,
indeed signal would be lost. This loss is self-consistently accounted for in
the method and could be discovered by visual inspection of the modes.)
In contrast, Figure 7 shows that even a $0.1\%$ _variable_ passband
calibration error spreads foreground variance into many modes that are all
significantly larger than the signal ($10^{-5}$ there). To implement a
variable passband, we multiply each line of sight by a different
$1+\boldsymbol{\delta}_{i}$, representing a change in the instrument response
with time.
The rule of thumb here is that if foregrounds are $R$ times larger than the
signal, then the per-pointing calibration needs to be stable to $\sim 1/R$.
Averaging over sight lines tends to relax this constraint. In the simulations
shown in Section 7.1, we find that the stability must be better than $\sim
10^{-4}$ to have no effect. The constraint degrades rapidly above that. In the
other limit of extremely high stability, we find that once calibration
stability is suppressed below the thermal noise, there is no further gain in
foreground discrimination.
An important point is that the method developed here can discover any response
to foregrounds that varies across the sky (non-monopole). This does not
necessarily need to originate from passband calibration or differential
polarization response—these are just plausible sources of error from
instrumental response.
Figure 6.— Toy model showing that an absolute passband calibration error that
does not vary with time does not impact primordial signal recovery. Top left:
observations of the spectrum across 10 lines of sight with a stable $10\%$
passband calibration error between bins (here neglecting thermal noise). The
input synchrotron is taken from a mixture of three power laws, so it is
described by three spectral modes in the data. Bottom left: foreground modes
discovered in the data. Because of passband calibration error, these modes are
not smooth. A constant calibration error does not increase the rank of the
three input power laws. Right: recovered cosmological signal compared to the
input signal, and the cosmological signal with the three modes on the left
removed. The fixed calibration error is cosmetic. Figure 7.— Residual
foregrounds as a function of modes removed in a toy model with three
foreground degrees of freedom. Amplitudes are normalized to one for zero modes
removed and in rms rather than variance. In the case with no variable
calibration error, the variance in the spectrum quickly reaches thermal noise
after three modes are removed. If the passband calibration varies between
lines of sight, the residuals become significantly worse. Generally, for
foregrounds $10^{5}$ times larger than the signal, the passband should be
stable to $\sim 10^{-5}$. Passband calibration instability spreads the
variance of the three intrinsic foreground modes across many more degrees of
freedom.
## 5\. Incorporating Spatial Information
The previous discussion assumed that the foregrounds are stationary and
independent between different lines of sight. It is for this reason that the
first step in the prescription outlined above was to form $\overline{y}$, the
unweighted average of all lines of sight—with stationary statistics, there is
no reason to prefer one direction of the sky over another, and an unweighted
average provides the best signal-to-noise ratio. This is the implicit
assumption that is being made by experiments that average over large regions
of sky at once (such as single-dipole experiments) and produce a single
spectrum as their measurement. The results in preceding sections therefore
also apply to experiments with no angular sensitivity, except without that
angular information, error bars and foreground cleaning methods cannot be
informed by the data itself and must be derived a priori.
In reality, foregrounds are neither stationary nor independent. The
synchrotron brightness varies across the sky (violating stationarity), and in
addition the foregrounds are known to be spatially correlated (violating
independence).
Angular information can be leveraged in several ways. First, nonstationarity
of the foregrounds allows the estimator to down-weight parts of the sky that
are known to be particularly noisy, such as the galactic center. Second,
correlation information allows foreground properties in one part of the sky to
be inferred from observations of another part of the sky. For example, in the
unrealistically extreme limit of perfectly correlated foregrounds, a
measurement of foregrounds in any part of the sky automatically allows a
perfect prediction of foreground brightness in any other part of the sky.
To take advantage of angular information, L13 considered an optimal estimator
given full-sky maps at all frequencies of interest as well as a known
$N_{\nu}N_{\rm pix}\times N_{\nu}N_{\rm pix}$ covariance $\mathbf{N}$ between
all pixels and all frequencies. Let $\boldsymbol{d}$ be a vector of length
$N_{\nu}N_{\theta}$ containing the measured sky maps. Here we use $N_{\rm
pix}$ to denote the number of pixel indices in a full-sky HEALPix map (Górski
et al., 2005). For reasons we will see in Section 7.1, this also prevents
confusion between the $N_{\theta}$ independent sight lines in the previous
section and the present discussion, which is made more complex by beam
convolution.
The observations $\boldsymbol{d}$ are related to the global $21$ cm spectrum
$\boldsymbol{s}$ via the measurement equation
$\boldsymbol{d}=\boldsymbol{A}\boldsymbol{s}+\boldsymbol{n},$ (36)
where
$\boldsymbol{n}\equiv\mathbf{n}_{\textrm{fg}}+\mathbf{n}_{\textrm{inst}}$ is
the generalized noise, containing $\mathbf{n}_{\textrm{fg}}$ and
$\mathbf{n}_{\textrm{inst}}$ as the foreground and instrumental noise
contribution to $\boldsymbol{d}$, respectively. The $N_{\rm pix}N_{\nu}\times
N_{\nu}$ matrix $\boldsymbol{A}$ is given by
$\boldsymbol{1}\otimes\boldsymbol{e}_{0}$, where $\otimes$ is the Kronecker
product, $\boldsymbol{1}$ is an $N_{\nu}\times N_{\nu}$ identity matrix, and
$\boldsymbol{e}_{0}$ is the spatial monopole, i.e., an $N_{\rm pix}$-long
vector of 1 s. Its function is to copy, at every frequency, the value of the
global spectrum $\boldsymbol{s}$ to all of the spatial pixels comprising the
sky map. The maximum-likelihood estimate $\hat{\mathbf{s}}_{\textrm{ML}}$ for
the full global spectrum is given by
$\hat{\boldsymbol{s}}_{\textrm{ML}}=(\boldsymbol{A}^{T}\boldsymbol{N}^{-1}\boldsymbol{A})^{-1}\boldsymbol{A}^{T}\boldsymbol{N}^{-1}\boldsymbol{d},$
(37)
where
$\boldsymbol{N}\equiv\langle\boldsymbol{n}\boldsymbol{n}^{T}\rangle-\langle\boldsymbol{n}\rangle\langle\boldsymbol{n}\rangle^{T}$
is the covariance matrix of the generalized noise, which is assumed to be
known. To facilitate comparisons with previous work, we will begin by
considering the estimator $\hat{\boldsymbol{s}}$ of the full global spectrum.
Section 6 returns to the template amplitude constraint.
### 5.1. Spatial Weighting to Recover the Spectrum
L13 used a series of analytical manipulations to show that incorporating
angular information via Equation (37) could in principle lead to large
reductions in foreground contamination. In practice, however, this
prescription is difficult to implement for a number of reasons. First, one may
not possess sufficiently accurate models of the instrument and the foregrounds
to write down the matrix $\boldsymbol{N}$, placing a priori expectations on
the instrument response. This matrix cannot be derived from the data (unlike
$\boldsymbol{\Sigma}$ from previous sections) because it has dimensions
$N_{\nu}N_{\theta}\times N_{\nu}N_{\theta}$ and therefore contains many more
degrees of freedom than the number of measurements $N_{\nu}N_{\theta}$.
Moreover, even if the matrix is somehow known, its large size makes its
inversion in Equation (37) computationally challenging.
To deal with these challenges, consider a modified recipe where the sky maps
are dealt with frequency by frequency (unlike in Equation (37) where all
frequencies and all lines of sight are mixed together) and an estimator for
the global signal at each frequency is formed by spatially averaging with
nontrivial weights (unlike in Equation (6), where different lines of sight
were equally weighted). Let $\boldsymbol{d}_{\beta}$ be a vector of length
$N_{\rm pix}$ that represents the map within $\boldsymbol{d}$ corresponding to
the $\beta$th frequency channel. The global signal at this frequency channel
can then be estimated by computing the weighted average
$\hat{\boldsymbol{s}}_{\beta}=\boldsymbol{w}_{\beta}^{T}\boldsymbol{d}_{\beta},$
(38)
where $\boldsymbol{w}_{\beta}$ is a vector of length $N_{\rm pix}$ with weight
appropriate for the $\beta$th frequency channel. Throughout this paper, we
adopt a convention where summations are written explicitly. Repeated indices
therefore do _not_ imply summations. The variance of this estimator—which we
will seek to minimize—is
$\textrm{Var}(\hat{\boldsymbol{s}}_{\beta})\equiv\langle\hat{\boldsymbol{s}}_{\beta}^{2}\rangle-\langle\hat{\boldsymbol{s}}_{\beta}\rangle^{2}=\boldsymbol{w}_{\beta}^{T}\mathbf{\Phi}_{\beta}\boldsymbol{w}_{\beta},$
(39)
where $\mathbf{\Phi}_{\beta}$ is the $N_{\rm pix}\times N_{\rm pix}$ _spatial_
covariance matrix of the map at the $\beta$th frequency. For our final
estimator to be correctly normalized, we require the weights at each frequency
to sum to unity, i.e., $\boldsymbol{w}_{\beta}^{T}\boldsymbol{e}_{0}=1$. To
obtain the minimum-variance solution for our weights, we use a Lagrange
multiplier $\mu$ to impose our normalization constraint and minimize the
quantity
$L=\boldsymbol{w}_{\beta}^{T}\mathbf{\Phi}_{\beta}\boldsymbol{w}_{\beta}-\mu\boldsymbol{w}_{\beta}^{T}\boldsymbol{e}_{0}.$
(40)
Doing so gives
$\boldsymbol{w}_{\beta}=\frac{\mathbf{\Phi}^{-1}_{\beta}\boldsymbol{e}_{0}}{\boldsymbol{e}_{0}^{T}\mathbf{\Phi}^{-1}_{\beta}\boldsymbol{e}_{0}},$
(41)
and therefore
$\hat{\boldsymbol{s}}_{\beta}=\frac{\boldsymbol{e}_{0}^{T}\mathbf{\Phi}^{-1}_{\beta}\boldsymbol{d}_{\beta}}{\boldsymbol{e}_{0}^{T}\mathbf{\Phi}^{-1}_{\beta}\boldsymbol{e}_{0}}.$
(42)
In words, these weights tell us to whiten the sky maps at each frequency using
the inverse spatial covariance before summing together different lines of
sight and normalizing. Because the whitening takes the form of a nondiagonal
matrix multiplication, this operation not only down-weights brighter parts of
the sky but also makes use of angular correlation information to better
estimate the monopole signal. Tegmark et al. (2003) tackled the transpose of
this problem for CMB foreground removal. Note that in modeling
$\boldsymbol{\Phi}_{\beta}$, it is essential to capture the nonstationary
nature of our galaxy’s foreground emission. It is insufficient, for example,
to describe the galaxy using an angular power spectrum alone, which assumes
statistical isotropy. This would not only be physically unrealistic but would
also make our weights constant across the sky (a fact that can be derived by
applying Parseval’s theorem to our expression for $\boldsymbol{w}_{\beta}$),
defeating the purpose of using spatial weights in the first place.
### 5.2. Spatial Weights Within a Separable Covariance Model
The spatial weight recipe that we have just specified can also be considered a
special limit of the maximum-likelihood estimator, albeit with one small
modification. Suppose that the full $N_{\nu}N_{\theta}\times
N_{\nu}N_{\theta}$ covariance $\boldsymbol{N}$ is separable so that we can
write it as $\boldsymbol{\Sigma}\otimes\boldsymbol{\Phi}$, where
$\boldsymbol{\Sigma}$ is the $N_{\nu}\times N_{\nu}$ spectral covariance from
previous sections, $\boldsymbol{\Phi}$ is an $N_{\rm pix}\times N_{\rm pix}$
spatial covariance, and $\otimes$ is the Kronecker product.
Using the identities
$(\boldsymbol{G}\otimes\boldsymbol{H})^{-1}\equiv\boldsymbol{G}^{-1}\otimes\boldsymbol{H}^{-1}$
and
$(\boldsymbol{G}\otimes\boldsymbol{H})(\boldsymbol{J}\otimes\boldsymbol{K})\equiv\boldsymbol{G}\boldsymbol{J}\otimes\boldsymbol{H}\boldsymbol{K}$,
we have
$\boldsymbol{A}^{T}\boldsymbol{N}^{-1}=\boldsymbol{\Sigma}^{-1}\otimes\boldsymbol{e}_{0}^{T}\boldsymbol{\Phi}^{-1}$
(43)
and
$(\boldsymbol{A}^{T}\boldsymbol{N}^{-1}\boldsymbol{A})^{-1}=\boldsymbol{\Sigma}\otimes(\boldsymbol{e}_{0}^{T}\boldsymbol{\Phi}^{-1}\boldsymbol{e}_{0})^{-1}.$
(44)
Inserting these into Equation (37), the final estimator in this separable
approximation is then
$\hat{\boldsymbol{s}}_{\beta}\Bigg{|}_{\textrm{sep}}=\frac{\boldsymbol{e}_{0}^{T}\boldsymbol{\Phi}^{-1}\boldsymbol{d}_{\beta}}{\boldsymbol{e}_{0}^{T}\boldsymbol{\Phi}^{-1}\boldsymbol{e}_{0}},$
(45)
which is almost identical to Equation (42). If we slightly relax the
assumption of perfect separability by allowing $\boldsymbol{\Phi}$ to acquire
a frequency dependence (so that
$\boldsymbol{\Phi}\rightarrow\boldsymbol{\Phi}_{\beta}$), the correspondence
becomes exact. Including this frequency-dependent nonseparability also allows
us to incorporate the fact that the instrumental beam will generally broaden
toward lower frequencies, in addition to intrinsic nonseparability of the
foregrounds. Section 6.1 discusses nonseparability caused by the differences
between foregrounds and thermal noise.
Interestingly—but unsurprisingly—the spectral covariance $\boldsymbol{\Sigma}$
drops out of the optimal estimator for the global spectrum once separability
is invoked. As discussed in L13, this is due to the fact that once we
spatially average over the maps, we are left with a spectrum consisting of
$N_{\nu}$ numbers. If our final goal is to measure a cosmological spectrum,
that is also of length $N_{\nu}$, the constraint that our estimator be
correctly normalized and unbiased means that there is nothing left to do.
Mathematically, this manifested itself in our derivation when the factors of
$\boldsymbol{\Sigma}$ in Equations (43) and (44) canceled each other out when
forming
$(\boldsymbol{A}^{T}\boldsymbol{N}^{-1}\boldsymbol{A})^{-1}\boldsymbol{A}^{T}\boldsymbol{N}^{-1}$.
If we return to our method in previous sections, however, and attempt to
constrain the amplitude $\alpha$ of a theoretical template, the spectral
covariance $\boldsymbol{\Sigma}$ re-enters the discussion. When constraining
$\alpha$, one is seeking to measure a single number from $N_{\nu}$
measurements, which in general ought to be combined nonuniformly if these
measurements have different error properties (as captured by
$\boldsymbol{\Sigma}$).
## 6\. Analyzing Global $21$ cm Data Cubes
In this section we combine the finite-rank, in situ estimate of spectral
foregrounds with a model of angular correlations. The end product will be an
estimator for the amplitude of a cosmological signal template that can be
applied to experimental global $21$ cm data cubes. The angular part of the
estimator uses assumed spatial distributions to improve the amplitude
estimate. In contrast, the central aspect of this estimator is that it does
not assume a set of _spectral_ modes or spectral covariance of the data.
It is notationally convenient to write the $N_{\rm pix}N_{\nu}$-long data
matrix $\boldsymbol{d}$ by reshaping it into an $N_{\nu}\times N_{\rm pix}$
matrix $\boldsymbol{Y}$. This “data cube” is a stack of maps at all of the
observed frequencies. The original data vector $\boldsymbol{d}$ can be
recovered through the serializing “vec()” operation ${\rm
vec}(\boldsymbol{Y})=\boldsymbol{d}$. This data cube form naturally
accommodates weighting operations by the separable covariance. A left-
multiplication acts on the spectral direction, and a right-multiplication acts
on the spatial direction.
The essential weighting operation on the full data set is
$\boldsymbol{N}^{-1}\boldsymbol{d}$, which under the separable covariance
assumption becomes
$\boldsymbol{N}^{-1}\boldsymbol{d}=(\mathbf{\Sigma}\otimes\mathbf{\Phi})^{-1}\boldsymbol{y}=\mathbf{\Sigma}^{-1}\boldsymbol{Y}\mathbf{\Phi}^{-1}$
(46)
using identities of the Kronecker and vec operations.
A model for the noise on $\boldsymbol{Y}$ is to draw some matrix
$\boldsymbol{E}$ with shape $N_{\nu}\times N_{\rm pix}$ of normal deviates
from $N(\mu=0,\sigma^{2}=1)$ and then correlate the spectral and spatial parts
as $\mathbf{\Sigma}^{1/2}\boldsymbol{E}\mathbf{\Phi}^{1/2}$ to give a noise
realization drawn from the covariance $\mathbf{\Sigma}\otimes\mathbf{\Phi}$.
The data model in this presentation is
$\boldsymbol{Y}=\alpha\boldsymbol{x}\boldsymbol{e}_{0}^{T}+\mathbf{\Sigma}^{1/2}\boldsymbol{E}\mathbf{\Phi}^{1/2}.$
(47)
This is a specialized case of the “growth curve” model (Kollo & von Rosen,
2005), reviewed in Appendix B. We can transfer the separable method developed
in Section 5.2 to the present case of constraining the template amplitude
$\alpha$ by letting $\boldsymbol{s}\rightarrow\alpha$ and
$\boldsymbol{A}=\boldsymbol{1}\otimes\boldsymbol{e}_{0}\rightarrow\boldsymbol{x}\otimes\boldsymbol{e}_{0}$
(because we are now assuming a form for $\boldsymbol{x}$ as a theoretical
template). Here, the maximum-likelihood estimate for the amplitude of some
spectral template $\boldsymbol{x}$ is
$\hat{\alpha}=(\boldsymbol{x}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{x})^{T}\boldsymbol{x}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{Y}\mathbf{\Phi}^{-1}\boldsymbol{e}_{0}(\boldsymbol{e}_{0}^{T}\mathbf{\Phi}^{-1}\boldsymbol{e}_{0})^{-1}.$
(48)
This applies the familiar maximum-likelihood estimators for the spatial
monopole (with respect to covariance $\mathbf{\Phi}$) and spectral template
(with respect to covariance $\mathbf{\Sigma}$) to the right and left side of
our data cube $\boldsymbol{Y}$.
Following Section 3.2, we can seek to replace $\mathbf{\Sigma}$ with a
$(\nu,\nu^{\prime})$ covariance measured in the data, $\mathbf{\hat{\Sigma}}$.
In Section 3.2, there was no information distinguishing different lines of
sight, so the mean subtraction in Equation (19) simply used the unweighted
average of $\boldsymbol{y}_{i}$. We now have information about the angular
covariance and can improve the mean spectrum estimation by including the
spatial weight, as $\boldsymbol{Y}_{\rm
sub}^{T}=(1-\boldsymbol{e}_{0}\boldsymbol{w}^{T})\boldsymbol{Y}^{T}$. Here
$\boldsymbol{w}=\boldsymbol{\Phi}^{-1}\boldsymbol{e}_{0}(\boldsymbol{e}_{0}^{T}\boldsymbol{\Phi}^{-1}\boldsymbol{e}_{0})^{-1}$
is an $N_{\rm pix}$-long weight map. Another way of looking at the operation
$(1-\boldsymbol{e}_{0}\boldsymbol{w}^{T})$ is that it subtracts the monopole
spectrum out of the map so that the eigenvectors of $\mathbf{\hat{\Sigma}}$
will only contain spatially variable spectral modes, not the monopole.
Procedurally, one (1) applies the angular weight $\boldsymbol{\Phi}^{-1}$, (2)
dots against the spatial monopole template $\boldsymbol{e}_{0}$, (3) accounts
for the impact of the spatial weight through
$(\boldsymbol{e}_{0}^{T}\boldsymbol{\Phi}^{-1}\boldsymbol{e}_{0})^{-1}$, and
4) projects this spectral template as a monopole across the map (under
$\boldsymbol{e}_{0}$) and subtracts this from the measured map. Using the
mean-subtracted map, we can find the weighted covariance as
$\mathbf{\hat{\Sigma}}=\boldsymbol{Y}_{\rm
sub}\boldsymbol{\Phi}^{-1}\boldsymbol{Y}_{\rm sub}^{T}~{}~{}{\rm
with}~{}~{}\boldsymbol{Y}_{\rm
sub}^{T}=(1-\boldsymbol{e}_{0}\boldsymbol{w}^{T})\boldsymbol{Y}^{T}.$ (49)
The estimator for the monopole spectrum template amplitude is then
$\hat{\alpha}=(\boldsymbol{x}^{T}\mathbf{\hat{\Sigma}}^{-1}\boldsymbol{x})^{-1}\boldsymbol{x}^{T}\mathbf{\hat{\Sigma}}^{-1}\boldsymbol{Y}\boldsymbol{w}$
(50)
Kollo & von Rosen (2005) show that this choice of weight yields the maximum-
likelihood $\hat{\alpha}$. The inverse $\mathbf{\Phi}^{-1}$ of the $N_{\rm
pix}\times N_{\rm pix}$ spatial covariance appears explicitly only in the
construction of the sample variance. Elsewhere, it is collapsed onto the
single simple weight map $\boldsymbol{w}$. Section 6.2 describes choices for
the form of $\boldsymbol{\Phi}$. There we advocate a diagonal form of
$\boldsymbol{\Phi}$ based on thermal noise, whose amplitude is determined by a
position-dependent sky temperature and integration time. Diagonal
$\boldsymbol{\Phi}$ makes estimation of $\mathbf{\hat{\Sigma}}^{-1}$
computationally simple.
We will argue in Section 6.2 that the angular weights only need to treat the
residual foregrounds and thermal noise variations across the sky. We choose to
let $\mathbf{\hat{\Sigma}}^{-1}$ do the legwork of determining the spectral
modes to remove from the data itself. In this role,
$\mathbf{\hat{\Sigma}}^{-1}$ accounts for instrumental systematics, whereas
$\mathbf{\Phi}^{-1}$ can improve the estimate of $\hat{\alpha}$ by using a
foreground model to down-weight noisier regions.
Similar to the discussion in Section 3.3, we can replace the
$\mathbf{\hat{\Sigma}}^{-1}$ operation by a projection that removes a finite
number of contaminated spectral modes. This is equivalent to (1) identifying a
data subspace parallel to foregrounds that is contaminated and fully removed
and (2) keeping a data subspace orthogonal to foregrounds.
Let $\boldsymbol{F}$ contain the largest eigenvectors of the sample covariance
$\mathbf{\hat{\Sigma}}$. Then
$\hat{\alpha}=\frac{\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})}{\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{x}}\boldsymbol{Y}\boldsymbol{w}.$
(51)
The denominator
$\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{x}$ is a
scalar normalization.
This is analogous to Equation (34), except that instead of a simple
$\boldsymbol{\bar{y}}$, each spatial slice of the map $\boldsymbol{Y}$ is
weighted to find the mean as $\boldsymbol{Y}\boldsymbol{w}$.
Using this notation, we can easily prove that constant passband calibration
errors do not increase the rank of foreground spectral modes. Let
$1+\delta\boldsymbol{c}$ be an $N_{\nu}$-long vector that represents constant
miscalibration of the passband. If a constant calibration error multiplies all
lines of sight, $\boldsymbol{C}={\rm diag}(1+\delta\boldsymbol{c})$ modifies
the data as $\boldsymbol{C}\boldsymbol{Y}$. So long as the response is not
zeroed out at some frequencies, $\boldsymbol{C}$ is invertible, and ${\rm
rank}(\boldsymbol{C}\mathbf{\hat{\Sigma}}\boldsymbol{C}^{T})={\rm
rank}(\mathbf{\hat{\Sigma}})$ for invertible $\boldsymbol{C}$. Intuitively, a
constant passband error can be thought of as just a per-frequency rescaling of
units. Because such a rescaling does not affect the rank of the foregrounds,
there is no increase in the number of foreground modes that need to be
constrained. Our self-discovery scheme for spectral modes will therefore be
just as effective when constant passband miscalibrations are present.
### 6.1. Deriving the Final Estimator: A Two-component Covariance
In deriving Equation (51), the key assumption that we made was that the total
covariance is separable into spatial and spectral components. This confuses
foregrounds and thermal noise, which each have very different angular
distributions. For example, we know that a synchrotron spectral mode is
associated with broad spatial distributions of galactic emission. In contrast,
thermal noise is always uncorrelated across angular pixels, but may change in
variance across the sky. In the limit that spectral foreground subtraction is
perfect, only Gaussian thermal noise will remain. In contrast, the separable
assumption would dictate that it has approximately the same angular
correlations as the galaxy. This issue ultimately translates into the angular
weight $\boldsymbol{w}$ that should be applied.
The solution to this problem is to let the total covariance be nonseparable
but comprised of independently separable foreground and noise components.
Here, $\boldsymbol{N}=\boldsymbol{N}_{f}+\boldsymbol{N}_{g}$ where
$\boldsymbol{N}_{f}=\mathbf{\Sigma}_{f}\otimes\mathbf{\Phi}_{f}$ and
$\boldsymbol{N}_{g}=\mathbf{\Sigma}_{g}\otimes\mathbf{\Phi}_{g}$. For
concreteness, imagine that the “$f$” component consists of modeled foreground
modes in $\boldsymbol{F}$, and the “$g$” component consists of any residuals
after the projection operation. This is exactly analogous to the
$\boldsymbol{F}/\boldsymbol{G}$ split in Rao (1967) described in Section 3.3.
If our discovered foreground modes happen to perfectly capture the true
foregrounds, then the “g” component would consist only of Gaussian thermal
noise. In general, however, we need not make this assumption.
The central problem with the model
$\boldsymbol{N}=\mathbf{\Sigma}\otimes\mathbf{\Phi}$ is that
$(1-\boldsymbol{F}\boldsymbol{F}^{T})$ only acts on the $\mathbf{\Sigma}$ part
but not on $\mathbf{\Phi}$. In contrast, in the two-component model, when
$(1-\boldsymbol{F}\boldsymbol{F}^{T})$ acts on the data, the remaining
covariance is only $\mathbf{\Sigma}_{g}\otimes\mathbf{\Phi}_{g}$, e.g., it
correctly ascribes $\mathbf{\Phi}_{f}$’s spatial correlations to the modes
that $\boldsymbol{F}$ removes.
We can now reassess the $\boldsymbol{w}$ in Equation (51) that yields the
minimum-variance estimate of $\alpha$. To clarify the discussion, let
$\boldsymbol{q}\equiv[\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{x}]^{-1}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{x}$.
The first step of Equation (51) in this notation is to form
$\boldsymbol{q}^{T}\boldsymbol{Y}$, which can be interpreted as a series of
estimators for $\alpha$, one for each line of sight.
With our two-component covariance, a little algebra reveals that the variance
to minimize is given by
$\textrm{Var}(\hat{\alpha})=(\boldsymbol{q}^{T}\boldsymbol{\Sigma}_{f}\boldsymbol{q})(\boldsymbol{w}^{T}\boldsymbol{\Phi}_{f}\boldsymbol{w})+(\boldsymbol{q}^{T}\boldsymbol{\Sigma}_{g}\boldsymbol{q})(\boldsymbol{w}^{T}\boldsymbol{\Phi}_{g}\boldsymbol{w}).$
(52)
The first term is zero by construction because
$\boldsymbol{q}^{T}\boldsymbol{\Sigma}_{f}\propto(\boldsymbol{1}-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{F}=0$.
Minimizing the remaining term subject to the constraint that the weights sum
to unity requires minimizing the quantity
$L=(\boldsymbol{q}^{T}\boldsymbol{\Sigma}_{g}\boldsymbol{q})(\boldsymbol{w}^{T}\boldsymbol{\Phi}_{g}\boldsymbol{w})-\mu\boldsymbol{w}^{T}\boldsymbol{e}_{0},$
(53)
where $\mu$ is again a Lagrange multiplier. Because
$\boldsymbol{q}^{T}\boldsymbol{\Sigma}_{g}\boldsymbol{q}$ is just a constant,
it can be absorbed into $\mu$ by a simple rescaling. One then sees that $L$ is
identical to Equation (40), except that $\boldsymbol{\Phi}_{g}$ (the spatial
covariances of the residuals) takes the place of $\boldsymbol{\Phi}_{\beta}$
(the full spatial covariance at the $\beta$th frequency). We may thus import
our previous solution for the spatial weights to find
$\boldsymbol{w}=\boldsymbol{\Phi}^{-1}_{g}\boldsymbol{e}_{0}(\boldsymbol{e}_{0}^{T}\boldsymbol{\Phi}^{-1}_{g}\boldsymbol{e}_{0})^{-1}$
(54)
In conclusion, we see that a simple reinterpretation of Equation (51) is all
that is required to accommodate a two-component covariance model. The new
weights $\boldsymbol{w}$ reflect the spatial covariance of residuals after the
frequency mode subtraction rather than the spatial covariance of the sky
itself. In spirit, this is reminiscent of the approach in L13, where a best-
guess model of the foregrounds was first subtracted from the sky maps before
the variance of the residuals was minimized. The difference here is that our
best-guess cleaning is based on the data rather than on a model. In addition,
the estimator operates with separable spectral and spatial steps that are
computationally trivial (a feature that we have managed to retain even with
the _non_ separable two-component covariance of this section).
### 6.2. Angular Weighting
Angular weighting is applied after the majority of the foreground covariance
has been removed by the spectral mode subtraction
$(1-\boldsymbol{F}\boldsymbol{F}^{T})$. In the language of the previous
section, it is not the full angular covariance $\boldsymbol{\Phi}$ of the
foregrounds that acts as a weight, but instead just $\boldsymbol{\Phi}_{g}$,
the angular covariance of the residuals.
If instrumental systematics are well controlled, the eigenvalue spectrum of
the $\nu-\nu^{\prime}$ covariance will fall rapidly and settle to a thermal
noise floor. Once the full complement of spectral modes down to the noise
floor are removed, the map would then be dominated by thermal noise. The
optimal angular weight in that case corresponds to dividing by the foreground
template squared. Rather than suppressing any foregrounds, this operation
simply weighs more strongly against regions of higher thermal noise due to
brighter foregrounds that have boosted the $T_{\rm sys}$ (assuming foregrounds
dominate the system temperature in these frequencies).
Our primary argument throughout this paper has been that foreground models
should not be trusted to define spectral modes for foreground cleaning because
they will invariably be too simple due to instrumental effects. In the case of
angular weighting of thermal noise, though, the operation of dividing by the
foreground template simply needs to be a best-guess thermal noise weight.
Slight misspecification reduces the optimality of $\hat{\alpha}$ but does not
introduce bias.
In experimental reality, the eigenvalue spectrum may fall more gradually and
merge into the noise floor. This could be caused by the calibration, beam and
polarization effects described in Section 7. As the eigenvalue falls to the
thermal noise floor, the thermal noise spectral modes are strongly mixed with
foreground modes, making the foregrounds difficult to cleanly separate.
Further, Section 7.4 describes how the number of foreground modes to remove
may be ambiguous because of thermal noise. An $\ell$-by-$\ell$ weighting of
the residuals then has the potential to improve the estimate of $\hat{\alpha}$
by down-weighting angular correlations that are consistent with residual
foregrounds. Detecting some angular power spectrum in the
$(1-\boldsymbol{F}\boldsymbol{F}^{T})$-cleaned map that is measurably
different from Gaussian thermal noise could indicate that a better weight
$w_{\ell}$ is possible.
Given that the integration depth of an experiment is at the level of the
cosmological signal (which is much smaller than the foregrounds), by
definition most of the foreground spectral modes will be well measured.
Deleting this set of spectral modes will remove the vast majority of
foregrounds. Even if there are residual foregrounds mixed with the thermal
noise, we expect that the optimality of the angular weight would not depend
strongly on the assumed foreground spatial template.
Following L13, we split the problem of nonstationarity and angular correlation
of the angular covariance by writing
$\boldsymbol{\Phi}_{g}={\rm diag}(\boldsymbol{m})\boldsymbol{Q}{\rm
diag}(\boldsymbol{m})^{T},$ (55)
where $\boldsymbol{m}$ is the foreground mean map. This is equivalent to
dividing the foreground by the expected mean and then encoding the angular
correlations of that normalized map in the matrix $\boldsymbol{Q}$. Let these
correlations be full sky, isotropic, and diagonal. If $\boldsymbol{T}$ is a
matrix that converts from real space to spherical harmonics, then the
correlations will be diagonal
$\mathbf{\Lambda}=\boldsymbol{T}\boldsymbol{Q}\boldsymbol{T}^{T}$, where
$\mathbf{\Lambda}$ is just some function of $\ell$. The operation
$\mathbf{\Phi}^{-1}$ can be interpreted algorithmically as
* •
Divide each spatial slice of the data by a foreground model map (a synchrotron
template).
* •
Transform to spherical harmonic space and deweight $\ell$-by-$\ell$ by
$w_{\ell}$ (defined below).
* •
Transform back to map space and divide by the foreground model map again.
A simple, two-parameter model for the correlations is to let the residuals
have some amplitude $\xi$ relative to thermal noise and correlation length
$\sigma$, as
$w_{\ell}=\left[\xi
e^{-\sigma^{2}\ell(\ell+1)/2}+e^{\theta_{b}\ell(\ell+1)}\right]^{-1}.$ (56)
Here we have assumed that the input data are deconvolved by the beam before
applying angular weights (dividing by $B_{\ell}$). The term
$e^{\theta_{b}\ell(\ell+1)}$ describes thermal noise in that case. By picking
the $\sigma$ and $\xi$ to give the lowest error, we may give preference to
parameters where the error spuriously scatters low. A better alternative is to
use the known instrumental thermal noise to inform $\xi$ and assume residuals
in the data are correlated across the beam scale $\theta_{b}$ (not determining
$w_{l}$ based on the error of $\hat{\alpha}$). Ideally, the residuals would be
dominated by nearly Gaussian thermal noise, and $w_{\ell}=1$ would be a
sufficiently optimal weight.
Figure 8.— Top: spatial weights at $80\,\textrm{MHz}$ for noise-dominated
residuals ($\xi=0.5$; $\sigma=20^{\circ}$). The main purpose of spatial
weighting in this regime is to average down noise, hence the weight is
positive across the sky. Bright regions of the sky (which cause more
instrumental noise for sky-noise-dominated instruments) are down-weighted.
Bottom: spatial weights at $80\,\textrm{MHz}$ for foreground-dominated
residuals ($\xi=50$; $\sigma=20^{\circ}$). In this regime, the spatial weights
play a role in foreground subtraction, and thus the weights are both positive
and negative, as different parts of the sky are differenced to mitigate
residual foregrounds. The beam size assumed in both cases is
$\theta_{b}=10^{\circ}$. The weights are defined to sum to one, but only the
shape, not normalization, is relevant.
In Figure 8 we show example spatial weights computed from Equations (54),
(55), and (56) in two different regimes. The top panel shows a noise-dominated
case ($\xi=0.5$), and the bottom panel shows a foreground residual-dominated
case ($\xi=50$). In the noise-dominated case, the spatial weights are designed
to simply average down the noise, and are therefore all positive. The galactic
plane and the bright point sources are seen to receive almost zero weight
because they contribute the most thermal noise in a sky-noise-dominated
instrument. In the foreground-dominated case, the spatial weights attempt to
further suppress residual foreground contamination. There are thus both
negative and positive weights because different parts of the sky can be
differenced to subtract off foregrounds. In both regimes, the right half of
the sky is given more weight because the galaxy is dimmer there (see Figure
2).
An important property of our spatial weighting scheme is that the
normalization is irrelevant and only the shape matters. This is a particularly
attractive property because most models of the sky at our frequencies of
interest are based on interpolations and extrapolations from other
frequencies. The amplitude may have large uncertainties, but the shape
information is likely to be much more reliable. If one is particularly
confident about the available shape information, it may be prudent to go one
step further and set $w_{\ell=0}=0$ by hand. Examining Equations. (55) and
(56) reveals that this projects out any component of the measured sky that has
precisely the same spatial shape as our foreground model. Note that even
though this is the $\ell=0$ weight, we do not destroy the global monopole
signal that we seek to measure because the $w_{\ell}$ act in a prewhitened
space _after_ dividing by $\boldsymbol{m}$. The global signal therefore takes
the form $1/\boldsymbol{m}$ and is no longer just the $\ell=0$ mode. It is
instead spread out over a wide range of $\ell$ values. Setting $w_{\ell=0}=0$
then sacrifices the sensitivity in one of the modes within this range,
resulting in slightly increased error bars as one deviates slightly from the
optimal prescription described above. However, this may be a cost that is
worth bearing for the sake of robustness in an aggressive campaign against
foreground systematics.
### 6.3. Summary of the Proposed Algorithm
The final algorithm suggested here for analyzing global $21$ cm data cubes is
* •
Use a spatially weighted average to project the monopole spectrum out of the
map and find the $(\nu,\nu^{\prime})$ sample covariance.
* •
Find the largest eigenvectors of the sample covariance and project them out of
the map.
* •
Combine lines of sight using a prescription for the angular correlation of the
residuals (which conservatively is to divide by the synchrotron template
squared).
* •
Dot this against the signal template to find the amplitude and perform error
analysis.
While this algorithm is intuitive, our goal here has been to describe some of
the implicit choices: (1) discovering frequency modes within the data (and
implications for errors), (2) choosing to remove the part of the signal
parallel to the foreground modes, and (3) assuming a two-component separable
form for the thermal noise and foregrounds, respectively. The next section
describes several considerations for using estimators of this type and
challenges in global $21$ cm signal measurement.
## 7\. Considerations for Analyzing Global $21$ Maps
In the next few subsections, we will consider (1) the notion of resolution
elements and the choice of beam size, (2) combined passband stability and
resolution considerations, (3) challenges in error estimation from residual
monopoles, (4) determination of the number of modes to remove, (5) extensions
to simple amplitude constraints, (6) applications of our methods to the pre-
reionization absorption feature, and (7) complications in foreground
mitigation that result from Faraday rotation.
### 7.1. Resolution Elements and Foreground Modes
In Section 3, the data were spectra of independent lines of sight. There, the
number of foreground modes removed ($N_{\rm fg}$) compared to the number of
sight lines ($N_{\theta}$) entered the errors as $\sim(N_{\rm
fg}-N_{\theta})^{-1}$. The sense of $N_{\theta}$ is more complex in our
present case of an $N_{\nu}\times N_{\rm pix}$ $\boldsymbol{Y}$ data cube
where the spatial part of the sky signal is convolved by an instrumental
beam,.
Intuition suggests that the number of independent sight lines is approximately
the number of beam spots on the sky, but this is incorrect. If each spatial
slice of a noiseless full-sky map is written in terms of its spherical
harmonics $T_{\ell m}$, convolution by the beam is multiplication by
$B_{\ell}$. Unless the beam falls to zero somewhere, this operation is
invertible and so does not modify the rank of $\mathbf{\hat{\Sigma}}$. The
rank of $\boldsymbol{Y}$ is $N_{\rm fg}$ so long as $N_{\rm pix}>N_{\rm fg}$
and $N_{\nu}>N_{\rm fg}$. Hence, all of the spectral modes can still be
recovered from the noiseless beam-convolved map.
With wider beams, higher eigenvalue spectral eigenmodes (typically the less-
smooth modes) are washed out by the beam. These modes do not disappear
entirely, but their amplitude diminishes. We can develop a rough rule for this
effect by finding the eigenvalue spectrum of full-rank white noise that is
convolved by a given beam size. A fitting form for the suppression of
eigenvalues (normalized to give no suppression for infinitely fine beams) is
$b_{n}\approx 10^{-(\theta_{\rm FWHM}/300^{\circ})^{2}f_{\rm sky}^{-1}n}$, as
a function of eigenvalue $n$ and full width at half max (FWHM) of the beam.
At first sight, it would appear that a more rapidly falling eigenvalue
spectrum is beneficial because it means that the foregrounds are better
described by fewer covariance modes. When the suppression occurs because of
beam convolution, however, it can be problematic in the presence of
instrumental noise. With instrumental noise, the information about a mode may
be suppressed below the noise floor. That foreground can therefore no longer
be cleanly detected and removed and may contaminate the signal and lead to
larger errors. Such detection and removal is potentially crucial because
higher eigenvalues can still greatly exceed the signal (even after beam
suppression) because of the hierarchy of scales.
We therefore come to the conclusion that even if angular resolution does not
impose a hard limit on the number of foreground modes that may be discovered
(as suggested in the case of independent sight lines), it does put a practical
limit on the efficacy of the cleaning. Note that the eigenvalues of
$\mathbf{\Sigma}=\mathbf{\Sigma}_{\rm fg}+\mathbf{\Sigma}_{\rm inst}$ will be
different from $\mathbf{\Sigma}_{\rm fg}$. Adding noise has the effect of
mixing the high-variance foreground modes with the thermal noise modes, making
it harder to cleanly separate foregrounds.
A rule of thumb for finding the number of modes discoverable before they are
confused by noise is to plot the eigenvalue spectrum of noiseless foregrounds
and the thermal noise, then find where they intersect. Modes some margin above
this point will still be well measured. Consider a normalized eigenvalue
spectrum of foregrounds that follows $\lambda_{n}\approx b_{n}\cdot 10^{-an}$
(Liu & Tegmark, 2012). The eigenvalues of thermal noise are relatively flat in
$n$ in comparison. Finding the intersection with the noise level, the number
of foreground modes that can be learned in the data is
$N_{\rm fg}\sim\log({\rm FNR})\left[\left(\frac{\theta_{\rm
FWHM}}{300^{\circ}}\right)^{2}f_{\rm sky}^{-1}+a\right]^{-1},$ (57)
where FNR is the foreground-to-noise ratio (in map variance). In our simple
models here, $a$ may be a factor of a few, for only a few modes in total. In
this idealized setting, the rather lenient scale of $300^{\circ}$ in our
fitting form for $b_{n}$ means that an instrument could cover the full sky at
poor resolution and still discover all of the foreground modes. In reality,
instrumental effects will drive a higher number of required modes. For
example, Switzer et al. (2013) had to remove tens of modes to clean
foregrounds even though only approximately four intrinsic synchrotron modes
were expected.
An advantageous factor here is that the sky signal is convolved by the beam
whereas instrumental thermal noise is not. Hence, thermal noise can be
estimated within the data using $N_{\rm pix}$ independent samples. In an ideal
experiment, all of the foreground modes can be discovered and subtracted in
$\boldsymbol{F}$ so that only thermal noise remains. Then errors could be
assessed from the map’s thermal noise, and signal loss could be corrected
against the subtracted modes as
$\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{x}$.
It is generally beneficial to cover larger areas with finer resolution to
better constrain and discover foreground modes. Higher resolution also helps
focus deweighting of particular contaminated sky regions. An important
exception to this occurs when the number of foreground modes scales with the
FWHM or sky fraction. Both Faraday rotation and passband instability will
generally increase the number of modes needed for studies on larger areas with
smaller beams. Each new beam that is observed could see a new rotation measure
or be observed with a different passband.
### 7.2. Summarizing the Effects of Passband Stability and Angular Resolution
Figure 9.— Error of the recovered cosmological template amplitude as a
function of calibration stability and instrumental resolution where the
fiducial amplitude is $\alpha=1$. The thermal noise is set so that in the
limit of no calibration or beam effects the amplitude can be constrained to
$5\sigma$. Here we consider foregrounds with four intrinsic spectral degrees
of freedom. The constraint degrades for FWHM above $\sim 50^{\circ}$ and
$10^{-4}$ stability in passband calibration. At low resolution, the modes
cannot be discovered. Calibration stability worse than the thermal noise
produces many more modes that must be constrained and removed. The Monte Carlo
estimates in the upper right become very noisy, and we saturate the color
scale. Figure 10.— Spectral eigenmodes of the foreground model described in
Section 2. The second, third, and fourth modes are $3\times 10^{-4}$, $8\times
10^{-7}$, and $6\times 10^{-8}$ down from the first mode (in variance),
respectively.
Figure 9 summarizes two of the main points of our paper. Here, we use the
galaxy model described in Section 2 to perform a Monte Carlo simulation for
the errors on $\alpha$ as a function of calibration stability and instrument
resolution. Figure 10 shows the four largest modes of this extended GSM model.
All further modes of the model itself (without noise) are negligible at the
level of machine precision. Note that in contrast to the simple pedagogical
models in Section 3 (with three modes), the full data cube simulation is more
representative of the synchrotron sky.
The sense of the passband calibration stability in our Monte Carlo is pixel to
pixel (e.g., not beam convolved), so it is not affected by the beam
convolution operation. We set the thermal noise level to produce a $5\sigma$
detection of $\alpha$ in the absence of beam and calibration effects. For
beams that are too large, not all foreground modes can be recovered and
subtracted. In the simulations here with four modes, the constraint on
$\alpha$ worsens gradually as the FWHM exceeds $\sim 50^{\circ}$. For
calibration stabilities worse than $\sim 10^{-4}$, each line of sight responds
differently to bright foregrounds, so each requires a new spectral function to
remove. As soon as these passband calibration perturbations exceed the noise
floor, they rapidly degrade the constraint.
Decreasing the thermal noise through longer integration times loosens the beam
resolution requirement (because foreground modes are better measured). It does
not modify the passband stability requirement. Passband stability is driven by
the separation in scales between the foregrounds and the cosmological signal
and the fact that each line of sight has a different response to the
foregrounds. Even if the modes are well measured relative to thermal noise,
they quickly overwhelm the signal.
Figure 11.— Spectral eigenmodes of foregrounds in a diffraction-limited
instrument where a Gaussian ${\rm FWHM}\propto\lambda$. Here, the beam’s
frequency dependence mixes spatial structure into spectral structure. This
dramatically proliferates the number of spectral modes that are discovered and
need to be removed, resulting in the loss of essentially all of the signal. A
treatment for this effect in a mapping experiment is to convolve to a common
resolution using the measured beams. Imperfections in the beam model will then
generically produce additional modes of the spectral covariance.
Throughout, we have assumed that the angular resolution at all frequencies is
constant. This is difficult to arrange experimentally. More typically, ${\rm
FWHM}\propto\lambda$ by diffraction. In the frequency range considered here, a
diffraction-limited FWHM would change by a factor of 2.5. Variation of the
beam will mix spatial structure into frequency structure because different
beams at each frequency see different angular contamination patterns (Morales
et al., 2012). Implicitly, we have assumed that the beam is known well enough
to convolve all of the frequency slices to a common resolution. In contrast,
if no correction is performed, beam effects will proliferate the number of
modes that need to be removed. Figure 11 shows the first $10$ bright spectral
modes from the simulations described in Section 2, which originally had four
modes. These new spectral modes describe the spatial structure that is mixed
into the spectral direction. If these spectral modes had to be removed, most
of the signal would also be projected out. The frequency-dependent beam must
therefore be corrected to achieve efficient cleaning. Errors in the beam model
executing this correction then translate into new spectral contamination
modes.
The tolerance of beam models is experiment-dependent and is beyond the scope
of the present work but must be carefully considered. We emphasize that
frequency-dependent beam effects impact any global $21$ cm experiment, whether
they choose to take advantage of angular information or not. The new spectral
mixing modes reflect a best effort of our algorithm to describe a new spectral
covariance that must be cleaned to reach the $21$ cm monopole. Careful
experimental design and measurement of the beam will control these covariance
modes (which remain inaccessible in a single-beam measurement).
Here we have limited the intrinsic foreground rank to four. In reality, it may
be discovered that foregrounds on the sky have higher rank, pushing the
resolution constraint. Also, we assume a full sky here—an experiment covering
a smaller fraction of the sky would need $f_{\rm sky}^{-1}$ more resolution in
our simple approximations here.
### 7.3. Challenges in Error Estimation
In the absence of prior information, monopole contamination is
indistinguishable from the signal. This contamination may arise from a
synchrotron monopole or from an additive $T_{\rm rx}(\nu)$ of the receiver
that is temporally constant but has some spectral structure. The foreground
modes are estimated from the $(\nu,\nu^{\prime})$ sample covariance
$\mathbf{\hat{\Sigma}}$ of the map with the mean spectrum removed. The modes
in $\boldsymbol{F}$ represent the frequency components of the spatially
fluctuating part of the foregrounds, and the operation
$1-\boldsymbol{F}\boldsymbol{F}^{T}$ is uninformed by a monopole signal.
The monopole foregrounds may have significant overlap with the spectral
functions of the spatially fluctuating foregrounds. In our simulations here,
the foregrounds are randomly drawn from a fixed set of spectral modes. For
example, if there is a spatially varying synchrotron foreground with a
spectral index of $-2$, that synchrotron emission will also be subtracted from
the monopole by $1-\boldsymbol{F}\boldsymbol{F}^{T}$ .
In reality, we cannot rule out the possibility of a foreground monopole that
remains even after cleaning the spatially-varying synchrotron modes. Further,
the spectral pattern of a constant instrumental $T_{\rm rx}(\nu)$ will never
be discoverable by mapping across the sky and could be confused with the
global signal. These present serious challenges for rigorous error estimation.
Performing the angular operations on the right of Equation (51) first, the
noise terms of $\hat{\alpha}$ can be isolated as
$\hat{\alpha}=\alpha|_{\rm
true}+\frac{\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})}{\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{x}}(\boldsymbol{n}_{\rm
fg}+\boldsymbol{n}_{\rm inst})$ (58)
where $\boldsymbol{n}_{\rm fg}$ and $\boldsymbol{n}_{\rm inst}$ are the
foregrounds and thermal noise of the monopole through
$\boldsymbol{Y}\boldsymbol{w}$. We lump the constant $T_{\rm rx}(\nu)$ in with
$\boldsymbol{n}_{\rm fg}$ rather than introducing a new term.
While the thermal noise contribution
$\propto\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{n}_{\rm
inst}$ has a well-defined distribution and can easily be included in the
errors,
$\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{n}_{\rm
fg}$ is much more challenging. If
$\boldsymbol{x}^{T}(1-\boldsymbol{F}\boldsymbol{F}^{T})\boldsymbol{n}_{\rm
fg}$ were always positive, then $\hat{\alpha}$ could be interpreted as an
upper bound. This practice is common in treating foregrounds to the power
spectrum because their contribution is always positive in the quadratic
quantity Switzer et al. (2013); Parsons et al. (2014); Dillon et al. (2014);
Paciga et al. (2013). Here, we have no guarantee that the residual foreground
monopole dotted into the signal is positive. Recall that even though
synchrotron radiation may be a strong power law, the residuals here will
largely represent unknown instrumental factors that could dot with an
arbitrary sign with the signal template. It has long been known that higher-
order polynomials lead to better subtractions. This suggests that residuals
will generally produce two-sided errors. Further, any monopole residual
foregrounds are likely to be non-Gaussian, inheriting from the non-Gaussianity
of the bright foregrounds.
These “monopole” foregrounds do not strictly need to be a monopole on the
sky—they only have to be constant over the area of the survey, if that only
covers part of the sky. This again bolsters the argument that global $21$ cm
experiments should observe as much of the sky as possible with modest
resolution to maximize their sensitivity to spatial variations that could lead
to significantly cleaner signal recovery.
Barring any prior information about the foregrounds or instrumental response,
the best ways to immunize the analysis against this response are (1) to
document that the spectral cube cleaned with
$(1-\boldsymbol{F}\boldsymbol{F}^{T})$ is consistent with Gaussian thermal
noise and (2) to cover a large area of the sky with modest resolution to
capture additional spatial variation of foregrounds. Additional information
about spectral smoothness and instrument response can be used (Bowman &
Rogers, 2010; Voytek et al., 2014) to further clean the spectral monopole.
Quoted errors would be based only on thermal noise, and boosted to account for
signal lost in the cleaning process, Equation (35). Another possibility would
be to conduct, for example, surveys of the north and south celestial poles
with different receiver architectures. If these were performed independently
and had the same result up to thermal errors, it could lend some confidence
that any residual monopoles in the respective surveys are small compared to
noise.
### 7.4. Determining the Number of Modes to Remove
Figure 12.— Fraction of the root mean squared of the reionization signal
remaining after the first four foreground modes are removed. The signal is
given by Equation (3) for a simple reionization scenario with central redshift
$z_{r}$ and width $\Delta z$. The first mode is a power law (see Figure 10),
and most of the signal remains for these reionization scenarios. The other
three modes carry more redshift information and tend to express signal
variations at high redshift better. In all cases, as the signal becomes
smoother ($\Delta z$ increasing), it becomes harder to differentiate from the
foreground modes.
Figure 12 shows the fractional rms of the cosmological reionization signal
remaining after removing the four foreground modes of the extended GSM
(Section 2), shown in Figure 10. We use the operation
$(1-\boldsymbol{F}\boldsymbol{F}^{T})$ described previously. The foreground
spectral functions overlap strongly with the signal, especially for slower,
spectrally smoother reionization histories (high $\Delta z$ in the plot).
Nulling four foregrounds causes significant signal loss over a range of
reionization models. Any instrumental response that increases the rank of
these foregrounds will result in additional signal loss from the new modes it
introduces.
In the case of these simulations, we know the precise number of spectral
degrees of freedom. In experimental practice, the number of contaminated modes
cannot be determined directly. This can be made especially ambiguous by an
eigenvalue spectrum that drops slowly rather than reaching a clear noise
floor.
A central challenge of any global $21\,\textrm{cm}$ signal estimator is
determining the number of modes to remove (Rao, 1967; Kollo & von Rosen,
2005). In similar methods applied to the $21$ cm autopower (Switzer et al.,
2013), an advantage is that the bias from foregrounds is purely additive. In
this case, an experiment can report the constraint with errors as a function
of modes removed. This will generically fall as more contaminated modes are
removed. This decline will level out if the foreground have been discovered.
As further modes are removed, the errors will increase due to cosmological
signal loss, which is self-consistently included in our formalism. A
reasonable prescription for a bound is to find the number of modes that gives
the most stringent upper bound.
In the global spectrum case, foreground residuals after $N$ spectral modes are
removed can have either sign in the dot product with the signal template. This
two-sided bias should stabilize as most of the foreground modes are
discovered. For more modes removed, the errors will grow, but the central
value should remain stable if most of the foreground covariance is described
and removed. The summary results of a global $21$ cm experiment should include
plots of the spectral modes removed and characterization of the constraint as
a function of modes removed.
Both this issue and the residual monopole in Section 7.3 cannot be addressed
rigorously. An irreducible challenge of global $21$ cm measurement is
determining whether any foreground bias remains. The methods here provide
excellent prospects for cleaning the data and guiding an experiment, but they
do not solve these central issues.
### 7.5. Extending the Spectral Template
We have discussed constraints on the amplitude of a cosmological $21$ cm
signal template that is known in advance. The primary goal of the current
generation of instruments is discovery of this amplitude. Subsequent
experiments will then constrain the spectrum itself. The methods developed
here can also be applied to that case.
There is a continuum of models between the template estimate here and a full
spectrum. One possibility would be to use prior information about the redshift
of reionization and estimate two amplitudes, $\alpha(z<z_{r})$ and
$\alpha(z>z_{r})$, and then examine the significance of a step feature.
Another option would be to constrain a fixed number of spectral spline
amplitudes. In the limit that each frequency bin has a different amplitude,
the methods developed here can find the component of that signal orthogonal to
the identified foreground modes.
The choice between models must juggle (1) use of prior information, (2)
discriminatory power, and (3) insight about the trajectory of the global $21$
cm signal. This model selection problem is deferred to future work.
### 7.6. Application to Pre-reionization Absorption
Figure 13.— Same as Figure 12, except that we now consider a pre-reionization
absorption feature parameterized by a Gaussian. The duration is given as the
$\sigma$ width of the Gaussian. Note that significantly more signal remains.
Throughout the paper, we have considered the global $21$ cm signal from
reionization. This choice was primarily pedagogical, to have a simple step
around which the foregrounds are subtracted. The maximum contrast of this
feature is $\sim 30$ mK, and for even modest $\Delta z$ it becomes smooth and
more easily subtracted by the foreground modes. In contrast, in the period
prior to reionization, the $21$ cm is thought to go into absorption at $\sim
100$ mK (Pritchard & Loeb, 2010). Here, the spin temperature is strongly
radiatively coupled to the cold gas. However, while synchrotron radiation
contaminating $z\sim 10$ $21$ cm radiation around the time of reionization is
$\sim 500$ K (averaged across the sky), the contamination of an absorption
signature at $z\sim 20$ is $\sim 3000$ K.
While the synchrotron emission from higher redshift $21$ cm emission may be
greater, the absorption signature has the additional feature that it is two-
sided, both falling and rising. This generally makes foreground modes less
parallel to the signal and leads to less signal loss. For our purposes here,
we can again develop a simple model where $T_{b}$ is a Gaussian function
parameterized by a central redshift and width $\sigma_{z}$. Figure 13 is
analogous to Figure 12 except that it considers this era. Generally much more
cosmological signal remains orthogonal to the foreground modes.
### 7.7. Susceptibility to Faraday Rotation
In addition to the Stokes-$I$ component of the foregrounds, synchrotron
emission is also polarized. The polarized emission is subject to Faraday
rotation as it traverses various column densities of free electrons in
magnetic fields. If a receiver were designed to be purely sensitive to
Stokes-$I$, then this spectral fluctuation could be ignored. Generally though,
a receiver will have some level of response to polarized signals, and Faraday
rotation produces a signal that oscillates in frequency (Moore et al., 2013).
The rotation angle is $({\rm RM})\cdot\lambda^{2}$, where RM is the rotation
measure. Even for modest RM, the rotation angle can vary rapidly over our
band. This is especially problematic for the global $21$ cm signal because the
rotation measure varies over the sky, increasing the number of contaminated
modes in the data.
Extragalactic sources are subject to Faraday rotation through the screen of
the entire Milky Way. This has been measured recently by Oppermann et al.
(2012, 2014), and is shown in Figure 14. Each point source will generally have
a different rotation measure and require a new degree of freedom to describe.
Bernardi et al. (2013) has conducted deep, wide-field observations of
polarization of galactic synchrotron emission at $189$ MHz. They find only one
source out of 70 with $S>4$ Jy with polarization at the level of $1.8\%$ in
$2400\,{\rm deg}^{2}$, with all others falling below the $2\%$ polarization
fraction.
Polarization and Faraday rotation of galactic synchrotron emission is more
complex because synchrotron emission is interspersed with the Faraday screen.
Bernardi et al. (2013) find peak polarized emission at $\sim 13$ K and
rotation measures mostly below $10~{}{\rm rad}\,{\rm m}^{-2}$. The low
rotation measures and polarization fraction are thought to be due to a
depolarization horizon (Landecker et al., 2010; Uyaniker et al., 2003) within
the galaxy, a distance beyond which most of the emission is depolarized along
the line of sight.
The global $21$ cm experiments to date have been single element, making it
difficult to make a precise comparison with interferometric galaxy
polarization surveys conducted at similar frequencies (Pen et al., 2009;
Bernardi et al., 2013). Vinyaikin et al. (1996) measured $3.5\pm 1.0$ K at
$88$ MHz ($24^{\circ}$ FWHM, phased array) and $2.15\pm 0.25$ K at $200$ MHz
($8^{\circ}$ FWHM, single dish) for the value of $\sqrt{Q^{2}+U^{2}}$ toward
the north celestial pole.
To assess the galactic polarization field, we use the model of Waelkens et al.
(2009), which self-consistently simulates the polarized emission and rotation
measure. Figure 14 shows the eigenvalue spectrum of the Stokes-$Q$ spectral
modes. Unlike Stokes-$I$, polarization is spectrally much more complex, and
its covariance is spread among many modes. Ideally, all of these modes would
be suppressed well below the noise level of the experiment so they do not need
to be estimated and subtracted. Based on the measured amplitudes of the
polarized signals above, suppose that polarization is $10^{-3}$ of intensity.
To achieve a $10^{5}$ suppression, the polarization leakage must be kept below
$1\%$.
Figure 14.— Top: galactic rotation measure map of Oppermann et al. (2014) in
units of ${\rm rad}\,{\rm m}^{-2}$, with color range truncated at $\pm
100~{}{\rm rad}\,{\rm m}^{-2}$ to show more structure out of the plane.
Bottom: the eigenvalue spectrum of Stokes-$Q$ emission in the galactic model
of Waelkens et al. (2009), normalized to one. This component falls off very
slowly because the rotation measure introduces many degrees of freedom that
vary across the sky.
## 8\. Conclusion
Measurements of the global $21$ cm signal are very challenging due to (1)
astrophysical foregrounds and their interaction with instrumental systematics,
(2) terrestrial radio interference, and (3) the ionosphere. Here we have
examined the first issue, especially in regard to instrumental response.
Following $z\sim 1$ $21$ cm literature (Switzer et al., 2013; Masui et al.,
2013; Chang et al., 2010), we develop a new method where foregrounds can be
jointly estimated with the monopole spectrum. This relies on the fact that
foregrounds vary across the sky while the cosmological signal is constant.
This idea also extends L13, who argue that surveys with moderate angular
resolution covering much of the sky are able to better discriminate between
the monopole $21$ cm signal and foregrounds.
The key observation arising from our cleaning method is that the instrument
should be designed to minimize the generation of new spectral degrees of
freedom from the foregrounds. For example, if each line of sight has a
slightly different passband calibration, it also requires a new spectral
degree of freedom to describe the foregrounds there. In this sense, the
instrument “spreads out” the variance over modes that ultimately require more
aggressive cleaning and signal loss. In contrast, a constant passband
calibration error does not increase the rank of the foreground spectral
covariance, and its effect is primarily aesthetic. This is fortuitous because
obtaining a smooth spectral calibration of an instrument at these frequencies
would require very large, expensive structures that are black in radio
wavelengths. Simply requiring that galactic foregrounds be smooth to a few
percent provides a sufficient passband calibration for signal recovery. In
contrast, significant effort must be put into maintaining passband stability
to at least $\sim 10^{-4}$. This may require using much of the instrument’s
sensitivity to integrate against a stable reference.
Polarization to intensity leakage is another example of an instrumental
systematic that increases the rank of the foreground covariance. It allows
spectral oscillations in the Faraday-rotating polarization signal to
contaminate the spectral intensity measurement. Because of depolarization, sky
polarization is only a fraction of the intensity, and our estimates of the
instrumental constraints are more lax, $<10^{-2}$.
In attempting to measure the global $21\,\textrm{cm}$ signal, one is faced
with bright contaminating foregrounds that can interact with instrumental
systematics in nontrivial ways, overwhelming the cosmological signal that one
seeks to measure. In this paper, we have developed methods that bear
similarities to various previously proposed intuitive data-analysis
techniques. However, our methods arise from a rigorous, self-consistent
framework that allows unknown and unanticipated foreground properties to be
derived from real data. Such a framework also provides guidance for instrument
design. If design requirements can be adequately met, high-significance
measurements of the global $21\,\textrm{cm}$ signal will be possible,
providing direct access to the rich and complex astrophysics of the first
luminous objects and reionization.
E.S. thanks Ue-Li Pen for conversations and feedback, and for stimulating the
approach in collaboration on $z\sim 1$ $21$ cm data analysis from the Green
Bank Telescope. A.L. thanks Matt McQuinn for conversations. We thank Harish
Vedantham for comments and the Ohio $21$ cm workshop for fostering
conversations that initiated the project. A.L. acknowledges support from NSF
grants AST-0804508, AST-1129258, and AST-1125558.
## Appendix A A. Relation Between Covariance Adjustment and the Maximum-
Likelihood Linear Estimator
In Section 3.2, we claim that the estimator
$\boldsymbol{m}=(\boldsymbol{X}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{X})^{-1}\boldsymbol{X}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{x}$
is equivalent to the projection operation,
$\boldsymbol{m}=(\boldsymbol{X}^{T}\boldsymbol{X})^{-1}\boldsymbol{X}^{T}(1-\mathbf{\Pi})\boldsymbol{y}$
where
$\mathbf{\Pi}=\mathbf{\Sigma}\boldsymbol{Z}(\boldsymbol{Z}^{T}\mathbf{\Sigma}\boldsymbol{Z})^{-1}\boldsymbol{Z}^{T}$.
The two essential conditions here are that
$\boldsymbol{X}^{T}\boldsymbol{Z}=0$ (that the signal and nonsignal vectors
are orthogonal) and that the vectors in $\boldsymbol{X}$ and $\boldsymbol{Z}$
together span the space. The operations $\boldsymbol{X}^{T}$ and
$\boldsymbol{Z}^{T}$ can be understood as projecting onto the signal basis and
the everything-but-signal basis, respectively. This proof follows Lemma 2b of
Rao (1967). Writing out the terms, we would like to prove
$(\boldsymbol{X}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{X})^{-1}\boldsymbol{X}^{T}\mathbf{\Sigma}^{-1}=(\boldsymbol{X}^{T}\boldsymbol{X})^{-1}\boldsymbol{X}^{T}-(\boldsymbol{X}^{T}\boldsymbol{X})^{-1}\boldsymbol{X}^{T}\mathbf{\Sigma}\boldsymbol{Z}(\boldsymbol{Z}^{T}\mathbf{\Sigma}\boldsymbol{Z})^{-1}\boldsymbol{Z}^{T}.$
(A1)
Multiplying on the right by $\boldsymbol{X}$ is trivially true because
$\boldsymbol{Z}^{T}\boldsymbol{X}=0$ and the other terms are the identity
matrix. This checks just the $\boldsymbol{X}$ subspace of the equality. To
prove general equality we need to check the $\boldsymbol{Z}$ subspace.
Multiply on the right by $\boldsymbol{Z}$ and from the left by
$\boldsymbol{X}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{X}$ (which will allow
simplification)
$\displaystyle\boldsymbol{X}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{Z}=-(\boldsymbol{X}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{X})(\boldsymbol{X}^{T}\boldsymbol{X})^{-1}\boldsymbol{X}^{T}\mathbf{\Sigma}\boldsymbol{Z}(\boldsymbol{Z}^{T}\mathbf{\Sigma}\boldsymbol{Z})^{-1}\boldsymbol{Z}^{T}\boldsymbol{Z}$
(A2)
$\displaystyle\boldsymbol{X}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{Z}=-(\boldsymbol{X}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{X})(\boldsymbol{X}^{T}\boldsymbol{X})^{-1}\boldsymbol{X}^{T}\mathbf{\Sigma}\boldsymbol{Z}[(\boldsymbol{Z}^{T}\boldsymbol{Z})^{-1}(\boldsymbol{Z}^{T}\mathbf{\Sigma}\boldsymbol{Z})]^{-1}$
(A3)
$\displaystyle\boldsymbol{X}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{Z}[(\boldsymbol{Z}^{T}\boldsymbol{Z})^{-1}(\boldsymbol{Z}^{T}\mathbf{\Sigma}\boldsymbol{Z})]=-(\boldsymbol{X}^{T}\mathbf{\Sigma}^{-1}\boldsymbol{X})(\boldsymbol{X}^{T}\boldsymbol{X})^{-1}\boldsymbol{X}^{T}\mathbf{\Sigma}\boldsymbol{Z}$
(A4)
$\displaystyle\boldsymbol{X}^{T}\mathbf{\Sigma}^{-1}[\boldsymbol{Z}(\boldsymbol{Z}^{T}\boldsymbol{Z})^{-1}\boldsymbol{Z}^{T}+\boldsymbol{X}(\boldsymbol{X}^{T}\boldsymbol{X})^{-1}\boldsymbol{X}^{T}]\mathbf{\Sigma}\boldsymbol{Z}=0.$
(A5)
Because $\boldsymbol{X}$ and $\boldsymbol{Z}$ span the space,
$\boldsymbol{Z}(\boldsymbol{Z}^{T}\boldsymbol{Z})^{-1}\boldsymbol{Z}^{T}+\boldsymbol{X}(\boldsymbol{X}^{T}\boldsymbol{X})^{-1}\boldsymbol{X}^{T}=\boldsymbol{1}$,
then noting that $\boldsymbol{X}^{T}\boldsymbol{Z}=0$ proves the identity. In
many of the methods in the paper, the projection $\mathbf{\Pi}$ is reduced to
$(1-\boldsymbol{F}\boldsymbol{F}^{T})$, completely projecting out a set of
modes. The modes in $\boldsymbol{F}$ clearly cannot span the space. Further,
we let the modes $\boldsymbol{F}$ overlap with the signal. These choices make
the estimator less formally optimal but more robust to very bright, non-
|
[a]Andrés E. Rentería-Olivo
# Grover’s Quantum Search Algorithm of Causal Multiloop Feynman Integrals.
###### Abstract
A proof-of-concept application of a quantum algorithm to multiloop Feynman
integrals in the Loop-Tree Duality (LTD) framework is applied to a
representative four-loop topology. Bootstrapping causality in the LTD
formalism, is a suitable problem to address with quantum computers given the
straightforward possibility to encode the two on-shell states of a propagator
on the two states of a qubit. A modification of Grover’s quantum search
algorithm is developed and the quantum algorithm is successfully implemented
on IBM Quantum and QUTE simulators.
## 1 Introduction
High-precision theoretical predictions for current and future high-energy
particle colliders require novel techniques to deal with the higher orders in
perturbation theory of scattering amplitudes. In this context, the loop-tree
duality (LTD) [1, 3, 2] formalism exhibits interesting mathematical properties
of high potential to overcoming current limitations. A remarkable property of
LTD is the possibility of representing the causal nature of Feynman diagrams
and scattering amplitudes, leading to an intuitive understanding of the
singular structure of loop integrals. This manifestly causal representation
[3, 4, 5] allows to replace the original multiloop Feynman diagrams into a
class of multiloop topologies defined by collapsing propagators into edges
where the two on-shell states of propagators are naturally encoded by the two
states of a qubit, leading to explore the application of quantum algorithms,
for instance, a Grover’s quantum search algorithm [6] and a Variational
Quantum Eigensolver [7] approach. In this work we focus on the former
approach.
## 2 Loop-Tree Duality and Causality
The causal representation of scattering amplitudes in the LTD formalism is
obtained through the calculation of nested residues. In Refs. [2, 3] it is
shown that scattering amplitudes can be written as
${\cal
A}_{D}^{(L)}=\int_{\vec{\ell}_{1}\ldots\vec{\ell}_{L}}\frac{1}{x_{n}}\sum_{\sigma\in\Sigma}{\cal
N}_{\sigma}\,\prod_{i=1}^{n-L}\,\frac{1}{\lambda_{\sigma(i)}^{h_{\sigma(i)}}}\
+(\lambda^{+}\leftrightarrow\lambda^{-})\leavevmode\nobreak\ ,$ (1)
with $x_{n}=\prod_{i}2q_{i,0}^{(+)}$, $h_{\sigma(i)}=\pm 1$,
$\cal{N}_{\sigma}$ a numerator determined by the interaction vertices of a
specific theory and $\int_{\vec{\ell_{s}}}=-\mu^{4-d}(2\pi)^{1-d}\int{\rm
d}^{d-1}\ell_{s}$, the integration measure in the loop three-momentum space.
The Eq. (1) only involves denominators with positive on-shell energies
$q_{i,0}^{(+)}=(\vec{q}_{i}^{\,2}+m_{i}^{2}-\imath 0)^{1/2}$, added together
in same-sign combinations in the so-called causal propagators,
$1/\lambda_{\sigma(i)}^{\pm}$, with
$\lambda_{\sigma(i)}^{\pm}\equiv\lambda_{p}^{\pm}=\sum_{i\in
p}q_{i,0}^{(+)}\pm k_{p,0}\leavevmode\nobreak\ ,$ (2)
where $\sigma(i)$ stands for a partition $p$ of the set of on-shell energies
and the orientation of the energy components of the external momenta,
$k_{p,0}$. The causal structure of $\lambda_{p}^{\pm}$ is defined by the sign
of $k_{p,0}$ when the propagators in the partition $p$ are set on-shell. Each
causal propagator is in a one-to-one correspondence with any possible
threshold singularity of the amplitude, which contains overlapped thresholds
that are known as causal entangled thresholds. The combinations of entangled
causal propagators represent causal thresholds that can occur simultaneously,
which are collected in the set $\Sigma$.
## 3 Causal query of multiloop Feynman integrals: the four-eloop case
We develop a modified version [6] of Grover’s algorithm [8] for querying the
causal configurations from the LTD framework. We take as example the four-
eloop topology, which consists in four loops made of edges (eloops) with a
central four-particle interaction vertex and we implement it on IBM
Quantum***https://quantum-computing.ibm.com/. The implementation of this
quantum algorithm requires four registers, namely, $|{e}\rangle$,
$|{c}\rangle$, $|{a}\rangle$ and $|{\it out}\rangle$. In the first register,
$|{e}\rangle$, we encode the state of the $N=2^{n}$ edges of the topology and
initialise it with a uniform superposition by applying Hadamard gates,
$|{e}\rangle=H^{\otimes n}|{0}\rangle$. The second register, $|{c}\rangle$,
encodes the comparison between two adjacent edges $e_{i}$ and $e_{j}$ in
binary Boolean clauses defined by
$\displaystyle
c_{ij}\equiv(e_{i}=e_{j})\qquad\text{and}\qquad\bar{c}_{ij}\equiv(e_{i}\neq
e_{j})\leavevmode\nobreak\ .$ (3)
Each clause $\bar{c}_{ij}$ is implemented through two CNOT gates, each taking
as control the corresponding qubits $e_{i}$ and $e_{j}$ respectively, and both
taking as target the same qubit in the register $|{c}\rangle$. The clause
$c_{ij}$ requires an extra NOT gate on the target qubit on the register
$|{c}\rangle$. The third register, $|{a}\rangle$, stores the eloop clauses
that probe the adjacent edges that compose a cyclic circuit within the
diagram. Specifically for the four-eloop topology, taking the conventional
orientation of the edges in Fig. (1), the clauses are given by
$\displaystyle a_{0}^{(4)}$ $\displaystyle=\neg(c_{01}\wedge c_{12}\wedge
c_{23})\leavevmode\nobreak\ ,$ $\displaystyle a_{1}^{(4)}$
$\displaystyle=\neg(\bar{c}_{05}\wedge\bar{c}_{45})\leavevmode\nobreak\ ,$
$\displaystyle a_{2}^{(4)}$
$\displaystyle=\neg(\bar{c}_{16}\wedge\bar{c}_{56})\leavevmode\nobreak\ ,$
$\displaystyle a_{3}^{(4)}$
$\displaystyle=\neg(\bar{c}_{27}\wedge\bar{c}_{67})\leavevmode\nobreak\ ,$
$\displaystyle a_{4}^{(4)}$
$\displaystyle=\neg(\bar{c}_{34}\wedge\bar{c}_{47})\leavevmode\nobreak\ .$ (4)
Each $a_{i}^{(4)}$ requires a multicontrolled Toffoli gate that takes as
control its specific qubits $|{c_{ij}}\rangle$ and as target a qubit in the
register $|{a}\rangle$ followed by a NOT gate. The last register, $|{\it
out}\rangle$, requires a single qubit initialised in the
$|{-}\rangle=HX|{0}\rangle$ and is used as the Grover’s marker. It stores the
output of the quantum algorithm and is implemented through a multicontrolled
Toffoli gate taking as control all the qubits from the register $|{a}\rangle$
and, if required, a qubit from the register $|{e}\rangle$. The oracle is
defined as
$\displaystyle U_{w}|{e}\rangle|{c}\rangle|{a}\rangle|{\it
out}\rangle=(-1)^{f(a,e)}|{e}\rangle|{c}\rangle|{a}\rangle|{\it
out}\rangle\leavevmode\nobreak\ .$ (5)
When the causal conditions are satisfied, $f(a,e)=1$, and marks the
corresponding states; otherwise, if $f(a,e)=0$, they are left unchanged. For
the four-eloop topology, the required Boolean marker is given by
$\displaystyle f^{(4)}(a,e)=\left(\leavevmode\nobreak\
\bigwedge_{i=0}^{4}a_{i}^{(4)}\leavevmode\nobreak\ \right)\wedge
e_{0}\leavevmode\nobreak\ .$ (6)
After storing the marked states in $|{\it out}\rangle$, the registers
$|{a}\rangle$ and $|{c}\rangle$ are rotated back to the state $|{0}\rangle$ by
applying the oracle operations in inverse order. The last step in the
algorithm is the amplification of the marked states by applying the diffuser
opertator on the register $|{e}\rangle$. We use the diffuser defined on IBM
Quantum†††https://qiskit.org.
missing
$e_{0}$
missing
$e_{1}$
missing
$e_{2}$
missing
$e_{3}$
missing
$e_{4}$
missing
$e_{5}$
missing
$e_{6}$
missing
$e_{7}$
Figure 1: Left: Four-eloop topology with a fixed conventional orientation.
Right: corresponding quantum circuit for the four-eloop topology.
## 4 Conclusions
The quantum algorithm developed in this work allowed us to successfully
identify the causal singular configurations of the four-eloop topology in the
LTD framework. The algorithm was successfully implemented in IBM Quantum and
QUTE simulators‡‡‡https://qute.ctic.es. The result of the quantum algorithm is
used to bootstrap the causal representation in LTD of representative multiloop
topologies, allowing us a better understanding of this approach and enabling
us to explore other techniques, such as a Variation Quantum approach, This
challenging problem is closely related to the identification of directed
acyclic graphs in graph theory, which can find application in other fields
beyond particle physics.
## 5 Acknowledgements
I would like to thank the team that collaborated in the development of this
work, G. Rodrigo, S. Ramírez-Uribe, G. F. R. Sborlini and L. Vale-Silva. To
IBM Quantum for the access to IBMQ and also to Fundación Centro Tecnológico de
la Información y Comunicación (CTIC) for the access to Quantum Testbed (QUTE).
This work is supported by the PRE2018-085925 grant from the Consejo Superior
de Investigaciones Científicas (CSIC) of the Spanish Government,
MCIN/AEI/10.13039/501100011033, and Grant No. PID2020-114473GB-I00.
## References
* [1] S. Catani, T. Gleisberg, F. Krauss, G. Rodrigo and J.-C. Winter, _JHEP_ 09 (2008) 065, [0804.3170].
* [2] J. de Jesús Aguilera-Verdugo, F. Driencourt-Mangin, R. J. Hernández-Pinto, J. Plenter, R. M. Prisco, N. S. Ramírez-Uribe, A. E. Rentería-Olivo, G. Rodrigo, G. Sborlini and W. J. Torres Bobadilla, et al. Symmetry 13 (2021) no.6, 1029 [2104.14621 [hep-ph].
* [3] J. J. Aguilera-Verdugo, F. Driencourt-Mangin, R. J. Hernández-Pinto, J. Plenter, S. Ramirez-Uribe, A. E. Renteria Olivo, G. Rodrigo, G. F. R. Sborlini, W. J. Torres Bobadilla and S. Tracz, Phys. Rev. Lett. 124 (2020) no.21, 211602 [2001.03564 [hep-ph]].
* [4] J. J. Aguilera-Verdugo, R. J. Hernandez-Pinto, G. Rodrigo, G. F. R. Sborlini and W. J. Torres Bobadilla, _JHEP_ 01 (2021) 069, [2006.11217].
* [5] S. Ramírez-Uribe, R. J. Hernández-Pinto, G. Rodrigo, G. F. R. Sborlini and W. J. Torres Bobadilla, _JHEP_ 04 (2021) 129, [2006.13818].
* [6] S. Ramírez-Uribe, A. E. Rentería-Olivo, G. Rodrigo, G. F. R. Sborlini and L. Vale Silva, JHEP 05 (2021) 100, [2105.08703].
* [7] G. Clemente, A. Crippa, K. Jansen, S. Ramírez-Uribe, A. E. Rentería-Olivo, G. Rodrigo, G. F. R. Sborlini and L. Vale Silva, [2210.13240].
* [8] L. K. Grover, _Phys. Rev. Lett._ 79 (1997) 325--328, [quant-ph/9706033].
|
# Task-Free Continual Learning via
Online Discrepancy Distance Learning
Fei Ye and Adrian G. Bors
Department of Computer Science
University of York
York, YO10 5GH, UK
<EMAIL_ADDRESS>
###### Abstract
Learning from non-stationary data streams, also called Task-Free Continual
Learning (TFCL) remains challenging due to the absence of explicit task
information. Although recently some methods have been proposed for TFCL, they
lack theoretical guarantees. Moreover, forgetting analysis during TFCL was not
studied theoretically before. This paper develops a new theoretical analysis
framework which provides generalization bounds based on the discrepancy
distance between the visited samples and the entire information made available
for training the model. This analysis gives new insights into the forgetting
behaviour in classification tasks. Inspired by this theoretical model, we
propose a new approach enabled by the dynamic component expansion mechanism
for a mixture model, namely the Online Discrepancy Distance Learning (ODDL).
ODDL estimates the discrepancy between the probabilistic representation of the
current memory buffer and the already accumulated knowledge and uses it as the
expansion signal to ensure a compact network architecture with optimal
performance. We then propose a new sample selection approach that selectively
stores the most relevant samples into the memory buffer through the
discrepancy-based measure, further improving the performance. We perform
several TFCL experiments with the proposed methodology, which demonstrate that
the proposed approach achieves the state of the art performance.
## 1 Introduction
Continual learning (CL) and its extension to lifelong learning, represents one
of the most desired functions in an artificial intelligence system,
representing the capability of learning new concepts while preserving the
knowledge of past experiences [32]. Such an ability can be used in many real-
time applications such as robotics, health investigative systems, autonomous
vehicles [20] or for guiding agents exploring artificial (meta) universes,
requiring adapting to a changing environment. Unfortunately, modern deep
learning models suffer from a degenerated performance on past data after
learning novel knowledge, a phenomenon called catastrophic forgetting [13].
A popular attempt to relieve forgetting in CL is by employing a small memory
buffer to preserve a few past samples and replay them when training on a new
task [1, 6]. However, when there are restrictions on the available memory
capacity, memory-based approaches would suffer from degenerated performance on
past tasks, especially when aiming to learn an infinite number of tasks.
Recently, the Dynamic Expansion Model (DEM) [51] has shown promising results
in CL, aiming to guarantee optimal performance by preserving the previously
learnt knowledge through the parameters of frozen components trained on past
data, while adding a new component when learning a novel task. However, such
approaches require knowing where and when the knowledge associated with a
given task is changed, which is not always applicable in a real environment.
In this paper, we address a more realistic scenario, called Task-Free
Continual Learning (TFCL) [3], where task identities are not available while
the model can only access a small batch of samples at a given time. Most
existing CL methods requiring the task label can be adapted to TFCL by
removing the task information dependency. For instance, memory-based
approaches can store a few past samples from the data stream at each training
time and replay them during later training steps [8, 12]. However, such an
approach requires to carefully design the sample selection criterion to avoid
memory overload. The key challenge for the memory-based approach is the
negative backward transfer caused by the stored samples that interfere with
the model’s updating with incoming samples [6]. This issue can be relieved by
DEM in which previously learnt samples are preserved into frozen components
and do not interfere with the learning of probabilistic representations of new
data [24, 38]. However, these approaches do not provide any theoretical
guarantees and there are no studies analysing the trade-off between the
model’s generalization and its complexity under TFCL.
Recent attempts have provided the theoretical analysis for CL from different
perspectives including the risk bound [46, 51], NP-hard problem [17], Teacher-
Student framework [23, 58] and game theory [37]. However all these approaches
require strong assumptions, such as defining the task identities, which is not
available in TFCL. This inspires us to bridge the gaps between the underlying
theory and the algorithm implementation for TFCL. We propose a theoretical
classification framework, which provides new insights in the forgetting
behaviour analysis and guidance for algorithm design addressing catastrophic
forgetting. The primary motivation behind the proposed theoretical framework
is that we can formulate forgetting as a generalization error in the domain
adaptation theory. Based on this analysis we extend the domain adaptation
theory [29] to derive time-dependent generalization risk bounds, explicitly
explaining the forgetting process at each training step.
Inspired by the theory, we devise the Online Discrepancy Distance Learning
(ODDL) method which introduces a new expansion mechanism based on the
discrepancy distance estimation for implementing TFCL. The proposed expansion
mechanism detects the data distribution shift by evaluating the variance of
the discrepancy distance during the training. This model enables a trade-off
mechanism between the model’s generalization and complexity. We also propose a
new sample selection approach based on the discrepancy-based criterion, which
guides storing diverse samples with respect to the already learnt knowledge,
further improving performance. Our contributions are :
* •
This paper is the first research study to propose a new theoretical framework
for TFCL, which provides new insights into the forgetting behaviour of the
model in classification tasks.
* •
Inspired by the theoretical analysis, we develop a novel dynamic expansion
approach, which ensures a compact model architecture enabled by optimal
performance.
* •
We propose a new sample selection approach that selects appropriate data
samples for the memory buffer, further improving performance.
* •
The proposed method achieves state of the art results on TFCL benchmarks,
## 2 Related works
Continual learning
defines a learning paradigm which aims to learn a sequence of tasks without
forgetting. Catastrophic forgetting is a major challenge in continual
learning. One of the most popular approaches to relieve forgetting is by
imposing a regularization loss within the optimization procedure [7, 11, 13,
16, 19, 25, 26, 31, 34, 35, 40, 41, 57], where the network’s parameters which
are important to the past tasks are penalized when updating. Another kind of
approaches for continual learning focuses on the memory system, which usually
employs a small memory buffer [1, 5, 6, 28, 36, 44, 59] to store a few past
data or trains a generator to provide the replay samples when learning new
tasks [38, 43, 46, 47, 52, 58, 53]. However, these approaches usually rely on
knowing the task information, which is not applicable in TFCL.
Task-free continual learning
is a special scenario in CL where a model can only see one or very few samples
in each training step/time without having any task labels. Using a small
memory buffer to store past samples has shown benefits for TFCL and was
firstly investigated in [3, 54, 56]. This memory replay approach was then
extended by employing Generative Replay Mechanisms (GRMs) for training both a
Variational Autoencoder (VAEs) [15] and a classifier, where a new retrieving
mechanism is used to select specific data samples, called the Maximal
Interfered Retrieval (MIR), [2]. The Gradient Sample Selection (GSS) [1] is
another sample selection approach that treats sample selection as a
constrained optimization reduction. More recently, a Learner-Evaluator
framework is proposed for TFCL, called the Continual Prototype Evolution
(CoPE) [8] which stores the same number of samples for each class in the
memory in order to ensure the balance replay. Another direction for the
memory-based approaches is to edit the stored samples which would increase the
loss in the upcoming model updates, called the Gradient based Memory EDiting
(GMED) [12], which can be employed in the existing CL models to further
enhance their performance.
Dynamic expansion models
aim to automatically increase the model’s capacity to adapt to new tasks by
adding new hidden layers and units. Such an approach, called the Continual
Unsupervised Representation Learning (CURL) [38], dynamically builds new
inference models when meeting the expansion criterion. However, since CURL
still requires a GRM to relieve forgetting, it would lead to a negative
knowledge transfer when updating the network’s parameters to adapt to a new
task. This issue can be addressed by using Dirichlet processes by adding new
components while freezing all previously learnt members, in a model called the
Continual Neural Dirichlet Process Mixture (CN-DPM), [24]. However, the
expansion criterion used by these approaches relies on the change of the loss
when training each time, which does not have any theoretical guarantees.
## 3 Theoretical analysis of TFCL
In this section, we firstly introduce learning settings and notations, and
then we analyze the forgetting behaviour for a single as well as for a dynamic
expansion model by deriving their Generalization Bounds (GBs).
### 3.1 Preliminary
Let $\mathcal{X}$ be the input space and $\mathcal{Y}$ represent the output
space which is $\\{-1,1\\}$ for binary classification and
$\\{1,2,\dots,n^{\prime}\\},n^{\prime}>2$ for multi-class classification. Let
$\mathcal{D}^{T}_{i}=\\{{\bf x}^{T}_{j},y^{T}_{j}\\}^{N^{T}_{i}}_{j=1}$ and
$\mathcal{D}^{S}_{i}=\\{{\bf x}^{S}_{j},y^{S}_{j}\\}^{N^{S}_{i}}_{j=1}$
represent the training and testing sets for the $i$-th dataset where ${\bf
x}^{T}_{j}\in\mathcal{X}$ and $y^{T}_{j}\in{\mathcal{Y}}$ are the image and
the associated ground truth label. $N^{T}_{i}$ and $N^{S}_{i}$ are the total
number of samples for ${\mathcal{D}}^{T}_{i}$ and ${\mathcal{D}}^{S}_{i}$,
respectively. In the paper, we mainly focus on the task-free class-incremental
learning, described as follows.
Definition 1.
(Data stream.)
For a given $t$-th training dataset $\mathcal{D}^{S}_{t}$ with $C^{S}_{t}$
data categories, let us consider a data stream $\mathcal{S}$ which consists of
samples $\mathcal{D}^{S}_{t,j}$ from each category, expressed by
$S=\bigcup_{j=1}^{C^{S}_{t}}\mathcal{D}^{S}_{t,j}$. Let
${\mathcal{D}}^{T}_{t,j}$ represent the set of samples drawn from the $j$-th
category of ${\mathcal{D}}^{T}_{t}$. Let ${\mathbb{P}}^{T}_{t,j}$ and
${\mathbb{P}}^{S}_{t,j}$ represent the distributions for
${\mathcal{D}}^{T}_{t,j}$ and ${\mathcal{D}}^{S}_{t,j}$, respectively. Let
${\mathbb{P}}^{T,{\mathcal{X}}}_{t,j}$ represent the marginal distribution
over $\mathcal{X}$.
Definition 2.
(Learning setting.)
Let $\mathcal{T}_{i}$ represent the $i$-th training step. For a given data
stream $\mathcal{S}$, we assume that there are a total of $n$ training steps
for $\mathcal{S}$, where each training step $\mathcal{T}_{i}$ is associated
with a small batch of paired samples $\\{{\bf X}^{b}_{i},{\bf Y}^{b}_{i}\\}$
drawn from $\mathcal{S}$, expressed by ${\mathcal{S}}=\bigcup_{i=1}^{n}\\{{\bf
X}_{i}^{b},{\bf Y}^{b}_{i}\\},\;\;\\{{\bf X}_{i}^{b},{\bf
Y}^{b}_{i}\\}\cap\\{{\bf X}_{j}^{b},{\bf Y}^{b}_{j}\\}=\varnothing$, where
$i\neq j$ and a model (classifier) can only access $\\{{\bf X}^{b}_{i},{\bf
Y}^{b}_{i}\\}$ at $\mathcal{T}_{i}$ while all previous batches are not
available. After finishing all training steps, we evaluate the classification
accuracy of the model on the testing set $\mathcal{D}^{T}_{t}$. In the
following, we define the model and the memory buffer.
Definition 3.
(Model and memory.)
Let us consider $h$ a model implemented by a classifier, and
$\mathcal{H}=\\{h\,|\,h\colon{\mathcal{X}}\to\mathcal{Y}\\}$ the space of
classifiers. Let $\mathcal{M}_{i}$ be a memory buffer at $\mathcal{T}_{i}$. We
assume that $\mathcal{M}_{i}$ randomly removes samples from the memory buffer
while continually adding new data to its memory at each training step. Let
$\mathbb{P}_{{\mathcal{M}}_{i}}$ represent the probabilistic representation of
${\mathcal{M}}_{i}$ and $|{\mathcal{M}}_{i}|$ its cardinality.
### 3.2 Measuring the distribution shift
In TFCL, the distance between the target domain (testing set) and the source
domain (memory) would be dynamically changed during each training step. We can
use the discrepancy distance [29] to measure this gap through the analysis of
the model’s risk.
Definition 4.
(The risk.)
For a given distribution ${\mathbb{P}}^{S}_{t,j}$, the risk of a model $h$ is
defined as
${\mathcal{R}}\big{(}h,{\mathbb{P}}^{S}_{t,j}\big{)}\buildrel\Delta\over{=}{\mathbb{E}}_{\\{{\bf
x},y\\}\sim{\mathbb{P}}^{S}_{t,j}}\big{[}{\mathcal{L}}\big{(}y,h({\bf
x})\big{)}\big{]}$ where
$\mathcal{L}\colon{\mathcal{Y}}\times{\mathcal{Y}}\to[0,1]$ is the loss
function.
Definition 5.
(Discrepancy distance.)
For two given distributions ${{\mathbb{P}}}^{T}_{t,j}$ and
${{\mathbb{P}}}^{S}_{t,j}$, the discrepancy distance $\mathcal{L}_{d}$ is
defined on two marginals as :
$\displaystyle{\mathcal{L}}_{d}\big{(}{\mathbb{P}}^{T,{\mathcal{X}}}_{t,j},{{\mathbb{P}}}^{S,{\mathcal{X}}}_{t,j}\big{)}$
$\displaystyle\buildrel\Delta\over{=}\mathop{\sup}\limits_{\left(h,h^{\prime}\right)\in\mathcal{H}^{2}}\Big{|}{\mathbb{E}}_{{\bf
x}\sim{\mathbb{P}}^{T,{\mathcal{X}}}_{t,j}}\big{[}{\mathcal{L}}\big{(}h({\bf
x}),h^{\prime}({\bf x})\big{)}\big{]}-{\mathbb{E}}_{{\bf
x}\sim{\mathbb{P}}^{S,{\mathcal{X}}}_{t,j}}\big{[}{\mathcal{L}}\big{(}h({\bf
x}),h^{\prime}({\bf x})\big{)}\big{]}\Big{|}\,,$ (1)
where $\\{h,h^{\prime}\\}\in\mathcal{H}$. In practice, the discrepancy
distance ${\mathcal{L}}_{d}(\cdot,\cdot)$ can be estimated as the upper bound
based on the Rademacher complexity which is used in the domain adaptation
theory as a measure of richness for a particular hypothesis space [30, 60].
Corollary 1.
[29] For two given domains ${{\mathbb{P}}}^{T,{\mathcal{X}}}_{t,j}$ and
${{\mathbb{P}}}^{S,{\mathcal{X}}}_{t,j}$, let $U_{\mathcal{P}}$ and
$U_{\mathbb{P}}$ represent sample sets of sizes $m_{\mathcal{P}}$ and
$m_{\mathbb{P}}$, drawn independently from
${{\mathbb{P}}}^{T,{\mathcal{X}}}_{t,j}$ and
${{\mathbb{P}}}^{S,{\mathcal{X}}}_{t,j}$. Let
${\widehat{\mathbb{P}}}^{T,{\mathcal{X}}}_{t,j}$ and
${\widehat{\mathbb{P}}}^{S,{\mathcal{X}}}_{t,j}$ represent the empirical
distributions for $U_{\mathcal{P}}$ and $U_{\mathbb{P}}$. Let
$\mathcal{L}({\bf x^{\prime}},{\bf x})=|{\bf x^{\prime}}-{\bf x}{|^{1}}$ be a
loss function (L1-Norm), satisfying
$\forall({\bf{x}},{\bf{x^{\prime}}})\in\mathcal{X},\mathcal{L}({\bf{x}},{\bf{x^{\prime}}})>M$,
where $M>0$. Then, with the probability $1-\delta$, we have :
$\displaystyle{\mathcal{L}_{d}}({{\mathbb{{P}}}}^{T,{\mathcal{X}}}_{t,j},{{\mathbb{{P}}}}^{S,{\mathcal{X}}}_{t,j})$
$\displaystyle\leq{\mathcal{L}_{d}}(\widehat{\mathbb{P}}^{T,{\mathcal{X}}}_{t,j},\widehat{\mathbb{P}}^{S,{\mathcal{X}}}_{t,j})+C^{\star}+3M\left({\sqrt{\frac{{\log\left({\frac{4}{\delta}}\right)}}{{2{m_{\mathcal{P}}}}}}+\sqrt{\frac{{\log\left({\frac{4}{\delta}}\right)}}{{2{m_{\mathbb{P}}}}}}}\right),$
(2)
where $C^{\star}=4q\left({{{\mathop{\rm
Re}\nolimits}}_{{U_{\mathcal{P}}}}}\left(\mathcal{H}\right)+{{{\mathop{\rm
Re}\nolimits}}_{{U_{\mathbb{P}}}}}\left(\mathcal{H}\right)\right)$ and ${\rm
Re}_{U_{\mathcal{P}}}({\mathcal{H}})$ is the Rademacher complexity (Appendix-B
from Supplemental Material (SM)). Let
${\mathcal{L}_{\widehat{d}}}({{\mathbb{{P}}}}^{T,{\mathcal{X}}}_{t,j},{{\mathbb{{P}}}}^{S,{\mathcal{X}}}_{t,j})$
represent the Right-Hand Side (RHS) of Eq. (2). We also assume that
${\mathcal{L}}\colon\mathcal{Y}\times\mathcal{Y}\to[0,1]$ is a symmetric and
bounded loss function
$\forall(y,y^{\prime})\in{\mathcal{Y}^{2}},{\mathcal{L}}(y,y^{\prime})\leq{U^{\prime}}$,
and ${\mathcal{L}}(\cdot,\cdot)$ obeys the triangle inequality, where
${U^{\prime}}$ is a positive number.
### 3.3 GB for a single model
Based on the definitions from Section 3.2, we firstly derive the GB that can
describe the learning process of a single model under TFCL.
Theorem 1.
Let $\mathcal{P}_{i}$ represent the distribution of all visited training
samples (including all previous batches) drawn from $\mathcal{S}$ at
$\mathcal{T}_{i}$. Let
$h_{\mathcal{P}_{i}}=\arg\min_{h\in\mathcal{H}}{\mathcal{R}}(h,{\mathcal{P}}_{i})$
and
$h_{{\mathcal{M}}_{i}}=\arg\min_{h\in\mathcal{H}}{\mathcal{R}}(h,{\mathbb{P}}_{{\mathcal{M}}_{i}})$
represent the ideal classifiers for ${\mathcal{P}}_{i}$ and
${\mathbb{P}}_{{\mathcal{M}}_{i}}$, respectively. We derive the GB between
$\mathcal{P}_{i}$ and ${\mathbb{P}}_{{\mathcal{M}}_{i}}$, based on the results
from Corollary 1 :
$\displaystyle{\mathcal{R}}\big{(}h,{\mathcal{P}}_{i}\big{)}$
$\displaystyle\leq{\mathcal{R}}\big{(}h,h_{{\mathcal{M}}_{i}},{\mathbb{P}}_{{\mathcal{M}_{i}}}\big{)}+{\mathcal{L}}_{\widehat{d}}\big{(}{\mathcal{P}}^{\mathcal{X}}_{i},{\mathbb{P}}^{\mathcal{X}}_{{\mathcal{M}}_{i}}\big{)}+\eta\big{(}{\mathcal{P}}_{i},{\mathbb{P}}_{{\mathcal{M}}_{i}}\big{)}\,,$
(3)
where $\eta\big{(}{\mathcal{P}}_{i},{\mathbb{P}}_{{\mathcal{M}}_{i}}\big{)}$
is the optimal combined error
${\mathcal{R}}(h_{{\mathcal{P}}_{i}},h_{{\mathcal{M}}_{i}},{{\mathcal{P}}_{i}})+{\mathcal{R}}(h_{{\mathcal{P}}_{i}},h_{{\mathcal{P}}_{i}}^{\star},{{\mathcal{P}}_{i}})$
where
${\mathcal{R}}(h_{{\mathcal{P}_{i}}},h_{{\mathcal{M}}_{i}},{\mathbb{P}}_{{\mathcal{M}}_{i}})$
is the risk, expressed by ${\mathbb{E}}_{{{\bf
x}}\sim{\mathbb{P}}_{{\mathcal{M}}_{i}}}[{\mathcal{L}}(h_{{\mathcal{P}}_{i}}({\bf
x}),h_{{\mathcal{M}}_{i}}({\bf x}))]$ and $h^{\star}_{{\mathcal{P}}_{i}}$ is
the true labeling function for ${\mathcal{P}}_{i}$.
The proof is provided in Appendix-A from the SM. Compared to the GB used in
the domain adaptation [29], Theorem 1 provides an explicit way to measure the
gap between the model’s prediction and the true labels in each training step
(${\mathcal{T}}_{i}$). During the initial training stages (when $i$ is very
small), the memory ${\mathcal{M}}_{i}$ can store all previous samples and GB
is tight. However, as the number of training steps increases, the discrepancy
distance
${\mathcal{L}}_{\widehat{d}}\big{(}{\mathcal{{P}}}^{\mathcal{X}}_{i},{\mathbb{{P}}}^{\mathcal{X}}_{{\mathcal{M}}_{i}}\big{)}$
would increase because ${\mathcal{M}}_{i}$ would lose the knowledge about
previously learnt samples. This can lead to a degenerated performance on
${\mathcal{P}}_{i}$, corresponding to the forgetting process. Next we extend
Theorem 1 to analyze the generalization performance on testing sets.
Theorem 2.
For a given target domain ${\mathbb{P}}^{T}_{t,j}$, we derive the GB for a
model at the training step ${\mathcal{T}}_{i}$ :
$\displaystyle{\mathcal{R}}\big{(}h,{\mathbb{P}}^{T}_{t,j}\big{)}$
$\displaystyle\leq{\mathcal{R}}\big{(}h,h_{{\mathcal{M}}_{i}},{\mathbb{P}}_{{\mathcal{M}_{i}}}\big{)}+{\mathcal{L}}_{\widehat{d}}\big{(}{\mathbb{P}}^{T,{\mathcal{X}}}_{t,j},{\mathbb{P}}^{\mathcal{X}}_{{\mathcal{M}}_{i}}\big{)}+\eta\big{(}{\mathbb{P}}^{T}_{t,j},{\mathbb{P}}_{{\mathcal{M}}_{i}}\big{)}\,,$
(4)
The proof is similar to that for Theorem 1. We can observe that the
generalization performance on a target domain ${\mathbb{P}}^{T}_{t,j}$, by a
model $h$ is relying mainly on the discrepancy distance between
${\mathbb{P}}^{T,{\mathcal{X}}}_{t,j}$ and ${\mathbb{P}}_{{\mathcal{M}}_{i}}$.
In practice, we usually measure the generalization performance of $h$ on
several data categories where each category is represented by a different
underlying distribution. In the following, we extend Theorem 2 for multiple
target distributions.
Lemma 1.
For a given data stream $S=\bigcup_{j=1}^{C^{S}_{t}}{\mathcal{D}^{S}_{t,j}}$
consisting of samples from ${\mathcal{D}}^{S}_{t}$, let
${\mathcal{D}}^{T}_{t}$ be the corresponding testing set and
${\mathbb{P}}^{T}_{t,j}$ represent the distribution of samples for the $j$-th
category from ${\mathcal{D}}^{T}_{t}$, we derive the GB for multiple target
domains as:
$\displaystyle\sum\nolimits_{j=1}^{C^{T}_{t}}\Big{\\{}{\mathcal{R}}\big{(}h,{\mathbb{P}}^{T}_{t,j}\big{)}\Big{\\}}$
$\displaystyle\leq\sum\nolimits_{j=1}^{C^{T}_{t}}\Big{\\{}{\mathcal{R}}\big{(}h,h_{{\mathcal{M}}_{i}},{\mathbb{P}}_{{\mathcal{M}_{i}}}\big{)}+{\mathcal{L}}_{\widehat{d}}\big{(}{\mathbb{P}}^{T,{\mathcal{X}}}_{t,j},{\mathbb{P}}^{\mathcal{X}}_{{\mathcal{M}}_{i}}\big{)}+\eta\big{(}{\mathbb{P}}^{T}_{t,j},{\mathbb{P}}_{{\mathcal{M}}_{i}}\big{)}\Big{\\}}\,,\vspace*{-0.4cm}$
(5)
where $C^{T}_{t}$ represents the number of testing data streams.
Remarks.
Lemma 1 had the following observations : 1) The optimal performance of the
model $h$ on the testing set can be achieved by minimizing the discrepancy
distance between each target domain ${\mathbb{P}}^{T,{\mathcal{X}}}_{t,j}$ and
the distribution ${\mathbb{P}}_{{\mathcal{M}}_{i}}$ at the training step
${\mathcal{T}}_{i}$. 2) [17]employs the set theory to theoretically
demonstrate that a perfect memory is crucial for CL. In contrast, we evaluate
the memory quality using the discrepancy distance in Eq. (5), which provides a
practical way to investigate the relationship between the memory and
forgetting behaviour of existing approaches [6, 8] at each training step
without requiring any task information (See more details in Appendix-D from
SM). 3) [51]introduces a similar risk bound for forgetting analysis, which
still requires the task information. In contrast, the proposed GB can be used
in a more realistic CL scenario. Moreover, we provide the theoretical analysis
for component diversity (Appendix-C from SM), which is missing in [51].
### 3.4 GB for the dynamic expansion mechanism
As discussed in Lemma 1, a memory of fixed capacity would lead to degenerated
performance on all target domains. The other problem for a single memory
system is the negative backward transfer [27] in which the performance of the
model is decreased due to samples being drawn from entirely different
distributions [14]. A Dynamic Expansion Model (DEM) can address these
limitations from two aspects : 1) DEM relieves the negative transfer by
preserving the previously learnt knowledge into a frozen component from a
mixture system; 2) DEM would achieve better generalization performance under
TFCL by allowing each component to model one or only a few similar underlying
data distributions. We derive GB for DEM and show the advantages of DEM for
TFCL.
Definition 7.
(Dynamic expansion mechanism.)
Let ${\mathcal{G}}$ represent a dynamic expansion model and ${G}_{j}$
represent the $j$-th component in ${\mathcal{G}}$, implemented by a single
classifier. ${\mathcal{G}}$ starts with training its first component during
the initial training phase and would add new components during the following
training steps. In order to overcome forgetting, only the newly created
component is updated each time, while all previously trained components have
their parameters frozen.
Theorem 3.
For a given data stream ${\mathcal{S}}=\\{{\bf X}^{b}_{1},\cdots,{\bf
X}^{b}_{n}\\}$, let ${\mathcal{P}}_{(i,j)}$ represent the distribution of the
$j$-th training batch ${\bf X}^{b}_{j}$ (visited) drawn from ${\mathcal{S}}$
at $\mathcal{T}_{i}$. We assume that ${\mathcal{G}}=\\{G_{1},\cdots,G_{c}\\}$
trained $c$ components at $\mathcal{T}_{i}$. Let
${\mathcal{T}}=\\{{\mathcal{T}}_{k_{1}},\cdots,{\mathcal{T}}_{k_{c}}\\}$ be a
set of training steps, where $G_{j}$ was frozen at $\mathcal{T}_{k_{j}}$. We
derive the GB for $\mathcal{G}$ at ${\mathcal{T}}_{i}$ as:
$\displaystyle\frac{1}{i}\sum\nolimits_{j=1}^{i}\big{\\{}{\mathcal{R}}\big{(}h,{\mathcal{P}}_{(i,j)}\big{)}\big{\\}}$
$\displaystyle\leq\frac{1}{i}\sum\nolimits_{j=1}^{i}\big{\\{}{\rm
F}_{S}\big{(}{\mathcal{P}}_{(i,j)},{\mathcal{G}}\big{)}\big{\\}}\,,$ (6)
where ${\rm F}_{S}(\cdot,\cdot)$ is the selection function, defined as :
$\displaystyle{\rm F}_{S}\big{(}{\mathcal{P}}_{(i,j)},{\mathcal{G}}\big{)}$
$\displaystyle\buildrel\Delta\over{=}\mathop{\min}\limits_{k_{1},\cdots,k_{c}}\Big{\\{}{\mathcal{R}}\big{(}h,h_{{\mathcal{M}}_{k_{i}}},{\mathbb{P}}_{{\mathcal{M}_{k_{i}}}}\big{)}+{\mathcal{L}}_{\widehat{d}}\big{(}{\mathcal{P}}^{\mathcal{X}}_{(i,j)},{\mathbb{P}}^{\mathcal{X}}_{{\mathcal{M}}_{k_{i}}}\big{)}+\eta\big{(}{\mathcal{P}}_{(i,j)},{\mathbb{P}}_{{\mathcal{M}}_{k_{i}}}\big{)}\Big{\\}}\,,$
(7)
where ${\mathbb{P}}_{{\mathcal{M}_{k_{i}}}}$ represents the memory
distribution at $\mathcal{T}_{k_{i}}$. The proof is provided in Appendix-B
from SM. It notes that ${\rm F}_{S}(\cdot,\cdot)$ can be used for arbitrary
distributions. We assume an ideal model selection in Eq. (7), where always the
component with the minimal risk is chosen. DEM can achieve the minimal upper
bound to the risk (Left Hand Side (LHS) of Eq. (6)) when comparing with a
single model (Theorem 3),. Then, we derive the GB for analyzing the
generalization performance of ${\mathcal{G}}$ on multiple target
distributions.
Lemma 2.
For a given data stream $S=\bigcup^{C^{S}_{t}}_{j}{\mathcal{D}^{S}_{t,j}}$
consisting of samples from ${\mathcal{D}}^{S}_{t}$, we have a set of target
sets $\\{{\mathcal{D}}^{T}_{t,1},\cdots,{\mathcal{D}}^{T}_{t,C^{T}_{t}}\\}$,
where each ${\mathcal{D}}^{T}_{t,j}$ contains $C^{b}_{t,j}$ batches of
samples. Let ${\mathbb{P}}^{T}_{t,j}(d)$ represent the distribution of the
$d$-th batch of samples in ${\mathcal{D}}^{T}_{t,j}$. We assume that
$\mathcal{G}$ consists of $c$ components trained on samples from
$\mathcal{S}$, at ${\mathcal{T}}_{i}$. We derive the GB for multiple target
domains as :
$\displaystyle\sum\nolimits_{j=1}^{C^{T}_{t}}\Big{\\{}\sum\nolimits_{d=1}^{C^{b}_{t,j}}{\mathcal{R}}\big{(}h,{\mathbb{P}}^{T}_{t,j}(d)\big{)}\Big{\\}}$
$\displaystyle\leq\sum\nolimits_{j=1}^{C^{T}_{t}}\Big{\\{}\sum\nolimits_{d=1}^{C^{b}_{t,j}}\big{\\{}{\rm
F}_{S}\big{(}{\mathbb{P}}^{T}_{t,j}(d),{\mathcal{G}}\big{)}\big{\\}}\Big{\\}}.$
(8)
Remark.
We have several observations from Lemma 2 : 1) The generalization performance
of $\mathcal{G}$ is relying on the discrepancy distance between each target
distribution ${\mathbb{P}}^{T}_{t,j}$ and the memory distribution
${\mathbb{P}}_{{\mathcal{M}}_{k_{i}}}$ of the selected component (Also see
details in Appendix-C from SM). 2) Eq. (8) provides the analysis of the
trade-off between the model’s complexity and generalization for DEM [24, 38].
By adding new components, $\mathcal{G}$ would capture additional information
of each target distribution and thus improve its performance. On the other
hand, the selection process ensures a probabilistic diversity of stored
information, aiming to capture more knowledge with a minimal number of
components.
In practice, we usually perform the model selection for ${\mathcal{G}}$ by
using a certain criterion that only accesses the testing samples without task
labels. Therefore, we introduce a selection criterion ${\widehat{\rm
F}}(\cdot,\cdot)$, implemented by comparing the sample log-likelihood :
$\displaystyle{\widehat{\rm
F}}\big{(}{\mathbb{P}}^{T}_{t,j}(d),{\mathcal{G}}\big{)}$
$\displaystyle\buildrel\Delta\over{=}\mathop{\arg\max}\limits_{k_{1},\cdots,k_{c}}\big{\\{}{\mathbb{E}}_{{\bf
x}\sim{\mathbb{P}}^{T}_{t,j}(d)}[\hat{f}({\bf x},G_{k_{j}})]\big{\\}},$ (9)
where $j=\\{1,\cdots,c\\}$ and ${\hat{f}}(\cdot,\cdot)$ is a pre-defined
sample-log likelihood function. Then Eq. (9) is used for model selection:
$\displaystyle{\widehat{\rm
F}}_{S}\big{(}{\mathbb{P}}_{t,j}^{T}(d),{\mathcal{G}}\big{)}$
$\displaystyle=\big{\\{}{\mathcal{R}}\big{(}h,h_{{\mathcal{M}}_{s}},{\mathbb{P}}_{{\mathcal{M}_{s}}}\big{)}+{\mathcal{L}}_{\widehat{d}}\big{(}{\mathbb{P}}^{T,{\mathcal{X}}}_{t,j}(d),{\mathbb{P}}^{\mathcal{X}}_{{\mathcal{M}}_{s}}\big{)}+\eta\big{(}{\mathbb{P}}^{T}_{t,j}(d),{\mathbb{P}}_{{\mathcal{M}}_{s}}\big{)}\,|\,$
$\displaystyle s={\widehat{\rm
F}}\big{(}{\mathbb{P}}^{T}_{t,j}(d),{\mathcal{G}}\big{)}\big{\\}}\,.$ (10)
We rewrite Eq. (8) using Eq. (3.4), resulting in :
$\displaystyle\sum\nolimits_{j=1}^{C^{T}_{t}}\Big{\\{}$
$\displaystyle\sum\nolimits_{d=1}^{C^{b}_{t,j}}{\mathcal{R}}\big{(}h,{\mathbb{P}}^{T}_{t,j}(d)\big{)}\Big{\\}}\leq\sum\nolimits_{j=1}^{C^{T}_{t}}\Big{\\{}\sum\nolimits_{d=1}^{C^{b}_{t,j}}\big{\\{}{\widehat{\rm
F}}_{S}\big{(}{\mathbb{P}}_{t,j}^{T}(d),{\mathcal{G}}\big{)}\big{\\}}\Big{\\}}\,.$
(11)
Compared with an ideal solution (Eq. (8)), Eq. (11) would involve the extra
error terms caused by the selection process (Eq. (3.4)), expressed as
$\sum\nolimits_{j=1}^{C^{T}_{t}}\Big{\\{}\sum\nolimits_{d=1}^{C^{b}_{t,j}}\big{\\{}{{\widehat{\rm
F}}}_{S}\big{(}{\mathbb{P}}_{t,j}^{T}(d),{\mathcal{G}}\big{)}-{{\rm
F}}_{S}\big{(}{\mathbb{P}}_{t,j}^{T}(d),{\mathcal{G}}\big{)}\big{\\}}\Big{\\}}$.
In Section 4, we introduce a new CL framework according to the theoretical
analysis.
### 3.5 Parallels with other studies defining lifelong learning bounds
In this section, we discuss the differences between the results in this paper
and those from other studies proposing lifelong learning bounds. The bound
(Theorem 1) in [30] assumes that the model is trained on all previous samples,
which is not practical under a TFCL setting. In contrast, in Theorem 1 from
this paper, the model is trained on a memory buffer, which can be used in the
context of TFCL. In addition, the bounds in [30] mainly provide the
theoretical guarantees for the performance when learning a new data
distribution (Theorem 1 from [30]), which does not provide an analysis of the
model’s forgetting. In contrast, this paper studies the forgetting analysis of
our model and its theoretical developments can be easily extended for
analyzing the forgetting behaviour of a variety of continual learning methods,
while this cannot be said about the study from [30] (See Appendix-D from
supplemental material).
When comparing with [33], our theoretical analysis does not rely on explicit
task boundaries, which is a more realistic assumption for TFCL. In addition,
[33] employs the KL divergence to measure the distance between two
distributions, which would require knowing the explicit distribution form and
is thus hard to evaluate in practice during the learning. However, our theory
employs the discrepancy distance, which can be reliably estimated. Moreover,
the study from [33] does not develop a practical algorithm to be used
according to the theoretical analysis. In contrast, our theory provides
guidelines for the algorithm design under the TFCL assumption. Finally,
inspired by the proposed theoretical analysis, this paper develops a
successful continual learning approach for TFCL.
Theorem 1 in [4], similarly to [30], assumes that the model is trained on
uncertain data sets over time, which would not be suitable for TFCL since the
model can not access previously learnt samples under TFCL. In contrast,
Theorem 1 in our theory represents a realistic TFCL scenario in which the
model is trained on a fixed-length memory buffer and is evaluated on all
previously seen samples. Therefore, the analysis for the forgetting behaviour
of the model under TFCL relies on Theorem 1 and its consequences. In addition,
the study from [4], similarly to [30, 33], only provides a theoretical
guarantee for a single model. In contrast, we extend our theoretical analysis
to the dynamic expansion model, which motivates us to develop a novel
continual learning approach for TFCL. Moreover, this paper is the first work
to provide the theoretical analysis for the diversity of knowledge recorded by
different components (See Appendix-C from supplemental material). This
analysis indicates that by maintaining the knowledge diversity among different
components we ensure a good trade-off between the model’s complexity and
generalization performance, thus providing invaluable insights into algorithm
design for TFCL.
## 4 Methodology
### 4.1 Network architecture
Each component $G_{j}\in{\mathcal{G}}$ consists of a classifier $h_{j}$
implemented by a neural network $f_{\varsigma_{j}}({\bf x})$ with trainable
parameters $\varsigma_{j}$, and a variational autoencoder (VAE) model
implemented by an encoding network ${\rm enc}_{\omega_{j}}$ as well as the
corresponding decoding network ${\rm dec}_{\theta_{j}}$, with trainable
parameters $\\{\omega_{j},\theta_{j}\\}$. Due to its robust generation and
inference mechanisms, VAE has been widely used in many applications [48, 55,
49, 50, 51, 61]. This paper employs a VAE for discrepancy estimation and
component selection. The loss function for the $j$-th component at
$\mathcal{T}_{i}$ is defined as :
$\displaystyle{\mathcal{L}}_{class}(G_{j},{\mathcal{M}}_{i})\buildrel\Delta\over{=}\frac{1}{{|{\mathcal{M}}_{i}|}}\sum\nolimits_{t=1}^{|{\mathcal{M}}_{i}|}\big{\\{}{\mathcal{L}}_{ce}\big{(}h_{j}({\bf
x}_{t}),{y}_{t}\big{)}\big{\\}}\,,$ (12)
$\displaystyle{\mathcal{L}}_{VAE}(G_{j},{\mathcal{M}}_{i})\buildrel\Delta\over{=}$
$\displaystyle{\mathbb{E}_{{q_{{\omega_{j}}}}({\bf{z}}\,|\,{\bf{x}})}}\left[{\log{p_{{\theta_{j}}}}({\bf{x}}_{t}\,|\,{\bf{z}})}\right]-{D_{KL}}\left[{{q_{{\omega_{j}}}}({\bf{z}}\,|\,{\bf{x}}_{t})\,||\,p({\bf{z}})}\right],$
(13)
where $\\{{\bf x}_{t},y_{t}\\}\sim\mathcal{M}_{i}$, where $|\mathcal{M}_{i}|$
is the memory buffer size, and ${\mathcal{L}}_{ce}(\cdot)$ is the cross-
entropy loss. $\mathcal{L}_{VAE}(\cdot,\cdot)$ is the VAE loss [15],
${p_{{\theta_{j}}}}({\bf{x}}_{t}\,|\,{\bf{z}})$ and
${q_{{\omega_{j}}}}({\bf{z}}\,|\,{\bf{x}}_{t})$ are the encoding and decoding
distributions, implemented by ${\rm enc}_{\omega_{j}}$ and ${\rm
dec}_{\theta_{j}}$, respectively. We also implement $\hat{f}(\cdot,\cdot)$ in
Eq. (9) by $-\mathcal{L}_{VAE}(\cdot,\cdot)$ for the component selection at
the testing phase. The training algorithm for the proposed ODDL consists of
three stages (See more information in Appendix-E from SM). In the initial
training stage, we aim to learn the initial knowledge of the data stream and
preserve it into a frozen component ${G}_{1}$, which can provide information
for the dynamic expansion and sample selection evaluation in the subsequent
learning. During the evaluator training stage we train the current component
as the evaluator that estimates the discrepancy distance between each
previously learnt component and the memory buffer, providing appropriate
signals for the model expansion. In the sample selection stage, we train the
current component with new data, while we aim to promote knowledge diversity
among components.
In the following, we firstly propose a new dynamic expansion mechanism based
on the discrepancy criterion. Then we introduce a new sample selection
approach that can further improve the performance of the model. Finally, we
provide the detailed algorithm implementation.
### 4.2 Discrepancy based mixture model expansion
From Lemma 2, we observe that the probabilistic diversity of trained
components in $\mathcal{G}$ can ensure a compact network architecture while
maintaining a good generalization performance (See Theorem 4 in Appendix-C
from SM). In order to achieve this, we maximize the discrepancy between each
trained component of ${\mathcal{G}}$ and the current memory buffer, during the
training, by using the discrepancy distance (notations are defined in Theorem
3 ), expressed as :
$\displaystyle{\mathcal{M}}^{\star}=\mathop{\arg\max}\limits_{{\mathcal{M}}_{i}}\sum\nolimits_{j}^{\rm{c}}\big{\\{}{\mathcal{L}}_{\widehat{d}}({\mathbb{P}}^{\mathcal{X}}_{{\mathcal{M}}_{k_{j}}},{\mathbb{P}}^{\mathcal{X}}_{{\mathcal{M}}_{i}})\big{\\}}\,,$
(14)
where $i=k_{c}+1,\cdots,n$ represents the index of the training steps and
$k_{c}$ is the $c$-th component trained at ${\mathcal{T}}_{k_{c}}$.
${\mathcal{M}}^{\star}$ is the optimal memory. Eq. (14) can be seen as a
recursive optimization problem when ${\mathcal{G}}$ dynamically adds new
components ($c$ is increased) during the training. In order to provide a
practical way to solve Eq. (14) while balancing the model’s complexity and
performance, we derive an expansion criterion based on the discrepancy
distance :
$\displaystyle\min\Big{\\{}{\mathcal{L}}_{\widehat{d}}\big{(}{\mathbb{P}}^{\mathcal{X}}_{{\mathcal{M}}_{k_{j}}},{\mathbb{P}}^{\mathcal{X}}_{{\mathcal{M}}_{i}}\big{)}\,|\,j=1,\cdots,c\Big{\\}}\geq\lambda\,,$
(15)
where $\lambda\in[0,4]$ is an architecture expansion threshold. If the current
memory distribution ${\mathbb{P}}_{{\mathcal{M}}_{i}}$ is sufficiently
different from each component (satisfies Eq. (15)), ${\mathcal{G}}$ will add a
new component to preserve the knowledge of the current memory
${\mathcal{M}}_{i}$, while also encouraging the probabilistic diversity among
the trained components.
In the following, we describe the implementation. We start by training the
first component $G_{1}$ which consists of a classifier $h_{1}$ (classification
task) and a VAE model $v_{1}$ (model selection at the testing phase). We also
train an additional component called the evaluator $G_{e}=\\{h_{e},v_{e}\\}$,
at the initial training stage, which aims to capture the future information
about the data stream. Once the memory $\mathcal{M}_{j}$ reaches its maximum
size at $\mathcal{T}_{j}$, we freeze the weights of the first component to
preserve the previously learnt knowledge while continually training the
evaluator during the following training steps. Since the evaluator continually
captures the knowledge from the current memory ${\mathcal{M}}_{i}$, we check
the model expansion using Eq. (15) at ${\mathcal{T}}_{i}$ :
$\displaystyle{\mathcal{L}}^{\star}_{d}\big{(}{\mathbb{P}}^{{\mathcal{X}}}_{{\mathcal{M}}_{k_{1}}},{\mathbb{P}}^{\mathcal{X}}_{{\mathcal{M}}_{i}}\big{)}\geq\lambda\,.$
(16)
It notes that we can not access the previously learnt memory distribution
${\mathbb{P}}^{\mathcal{X}}_{{\mathcal{M}}_{k_{1}}}$ and we approximate it by
using the auxiliary distribution ${\mathbb{P}}^{{\mathcal{X}}}_{v^{j}_{1}}$
formed by samples drawn from $v_{1}^{j}$ of $G_{1}^{j}$, where the superscript
$j$ represents the first component finishing the training at
${\mathcal{T}}_{j}$. ${\mathcal{L}}^{\star}_{d}(\cdot,\cdot)$ is the estimator
of the discrepancy distance, achieved by $h^{i}_{e}$ and $h^{j}_{1}$.
$\displaystyle{\mathcal{L}}^{\star}_{d}\big{(}{\mathbb{P}}^{{\mathcal{X}}}_{v^{j}_{1}},{\mathbb{P}}_{{\mathcal{M}}_{i}}\big{)}$
$\displaystyle\buildrel\Delta\over{=}\Big{|}{\mathbb{E}}_{{\bf
x}\sim{\mathbb{P}}^{{\mathcal{X}}}_{v^{j}_{1}}}\big{[}{\mathcal{L}}\big{(}h^{j}_{1}({\bf
x}),h^{i}_{e}({\bf x})\big{)}\big{]}-{\mathbb{E}}_{{\bf
x}\sim{\mathbb{P}}_{{\mathcal{M}}_{i}}}\big{[}{\mathcal{L}}\big{(}h^{j}_{1}({\bf
x}),h^{i}_{e}({\bf x})\big{)}\big{]}\Big{|}.$ (17)
If Eq. (16) is fulfilled, then we add $G^{i}_{e}$ into the model $\mathcal{G}$
while building a new evaluator $(G^{i+1}_{e}=G_{3}^{i+1})$ at
$\mathcal{T}_{i+1}$; otherwise, we train $G^{i}_{e}\to G^{i+1}_{e}$ at the
next training step, $\mathcal{T}_{i+1}$. Furthermore, for satisfying the
diversity of knowledge in the components from the mixture model (See Lemma 2),
we clear the memory $\mathcal{M}_{i}$ when performing the expansion in order
to ensure that we would learn non-overlapping distributions during the
following training steps. We can also extend Eq. (16) to the expansion
criterion for $\mathcal{G}=\\{G_{1},\cdots,G_{s-1}\\}$ that has already
preserved $(s-1)$ components, illustrated in Fig. 1:
$\displaystyle\min\Big{\\{}{\mathcal{L}}^{\star}_{d}\big{(}{\mathbb{P}}^{{\mathcal{X}}}_{v^{k_{j}}_{j}},{\mathbb{P}}^{\mathcal{X}}_{{\mathcal{M}}_{i}}\big{)}\,|\,j=1,\cdots,s-1\Big{\\}}\geq\lambda\,,\vspace{-20pt}$
(18)
where ${\mathbb{P}}^{{\mathcal{X}}}_{v^{k_{j}}_{j}}$ is the distribution of
the generated samples by $G^{k_{j}}_{j}$ and denote
$S_{j}={\mathcal{L}}^{\star}_{d}\big{(}{\mathbb{P}}^{{\mathcal{X}}}_{v^{k_{j}}_{j}},{\mathbb{P}}_{{\mathcal{M}}_{i}}\big{)}$.
### 4.3 Sample selection
Figure 1: We generate the knowledge by using the VAE ($V_{j}$) of each
previous component $G_{j},j=1,\cdots,s-1$, which is used to evaluate the
discrepancy distance
$S_{j}={\mathcal{L}}^{\star}_{d}\big{(}{\mathbb{P}}^{{\mathcal{X}}}_{v_{j}},{\mathbb{P}}_{{\mathcal{M}}_{i}}\big{)}$
at ${\mathcal{T}}_{i}$ (Eq. (17)) between $G_{j}$ and the memory buffer. Then
we use these discrepancy scores $\\{S_{1},\cdots,S_{s-1}\\}$ to check the
model expansion (Eq. (18))
According to Lemma 2, the probabilistic diversity of the knowledge accumulated
in the components is crucial for the performance. In the following, we also
introduce a novel sample selection approach from the memory buffer that
further encourages component diversity. The primary motivation behind the
proposed sample selection approach is that we desire to store in the memory
buffer those samples that are completely different from the data used for
training the other components. This mechanism enables the currently created
component to capture a different underlying data distribution. To implement
this goal, we estimate the discrepancy distance on a pair of samples as the
diversity score :
$\displaystyle{\cal L}_{d}^{s}({\cal G},{{\bf{x}}_{c}}){\rm{}}$
$\displaystyle\buildrel\Delta\over{=}\frac{1}{s}\sum\nolimits_{t=1}^{s}\big{|}{\cal
L}(h_{t}^{k_{t}}({\bf{x}}_{c}),h_{e}^{i}({\bf{x}}_{c}))-{\cal
L}(h_{t}^{k_{t}}({{\bf{x}}^{k_{t}}_{t}}),h_{e}^{i}({\bf{x}}^{k_{t}}_{t}))\big{|}\,,$
(19)
where ${\bf x}_{c}$ and ${\bf{x}}^{k_{t}}_{t}$ are the $c$-th sample from
$\mathcal{M}_{i}$ and the generated one drawn from $G^{k_{t}}_{t}$,
respectively. $h_{t}^{k_{t}}({\bf{\cdot}})$ is the classifier of
$G^{k_{t}}_{t}$ in $\mathcal{G}$. Eq. (19) evaluates the average discrepancy
distance between the knowledge generated by each trained component and the
stored samples, which guides for the sample selection at the training step
(${\mathcal{T}}_{i}$) as :
$\displaystyle\vspace{-10pt}{\mathcal{M}}_{i}=\bigcup\nolimits^{|{{\mathcal{M}}}^{\prime}_{i}|-b}_{j=1}{{\mathcal{M}}}^{\prime}_{i}[j]\,,$
(20)
where ${{\mathcal{M}}}^{\prime}_{i}$ is the sorted memory that satisfies the
condition
${\mathcal{L}}^{s}_{d}({\mathcal{G}},{{\mathcal{M}}}^{\prime}_{i}[a])>{\mathcal{L}}^{s}_{d}({\mathcal{G}},{{\mathcal{M}}}^{\prime}_{i}[q])$
for $a<q$. ${{\mathcal{M}}}^{\prime}_{i}[j]$ represents the $j$-th stored
sample and $b=10$ is the batch size. We name the approach with sample
selection as ODDL-S.
(a) Risk on the training set. (b) Risk on all target sets. (c) Results on
MNIST.
Figure 2: The forgetting analysis of a single model and DEM in (a) and (b)
where the batch size is 64 in the data stream. Data distribution shift and
increasing the number of components in ODDL-S during the training in (c).
### 4.4 Algorithm implementation
We provide the pseudocode ( Algorithm 1 ) in Appendix-E from SM. The
algorithm has three main stages :
* •
(Initial training).
We start by building two components $\\{G_{1},G_{2}\\}$ where we only add
$G_{1}$ in $\mathcal{G}$ and consider $G_{2}=G_{e}$ as the Evaluator. $G_{1}$
and $G_{2}$ are trained jointly using Eq. (13) and Eq. (12) during the initial
training stage until the memory $\mathcal{M}_{j}$ reaches its maximum size
$|{\mathcal{M}}|^{max}$ at a certain training step ${\mathcal{T}}_{j}$. Then
we freeze the first component $G_{1}$ and perform the second stage.
* •
(Evaluator training).
In this stage, we only update the Evaluator using Eq. (13) and Eq. (12) on
${\mathcal{M}}_{j+1}$ at ${\mathcal{T}}_{j+1}$. If
$|\mathcal{M}_{j+1}|\geq|\mathcal{M}|^{max}$, then we evaluate the discrepancy
distance using Eq. (17) to check the expansion (Eq. (18)). If the expansion
criterion is satisfied then we add $G_{e}$ to $\mathcal{G}$ and build a new
Evaluator while cleaning up the memory $\mathcal{M}_{j+1}$, otherwise, we
perform the sample selection.
* •
(Sample selection).
We evaluate the diversity score for each stored sample in $\mathcal{M}_{j+1}$
using Eq. (19). We then perform the sample selection for $\mathcal{M}_{j+1}$
using Eq. (20) and return back to the second stage.
## 5 Experiments
We perform the experiments to address the following research questions: 1)
What factors would cause the model’s forgetting, and how to explain such
behaviour? 2) How efficient is the proposed ODDL-S under TFCL benchmarks? 3)
How important is each module in OODL-S?
In this experiment, we adapt the TFCL setting from [8] which employs several
datasets including Split MNIST [22], Split CIFAR10 [18] and Split CIFAR100
[18]. The detailed information for datasets, hyperparameters and network
architectures is provided in Appendix-F from the supplementary material. The
code is available at https://dtuzi123.github.io/ODDL/.
Table 1: The accuracy of various continual learning models for five independent runs. Methods | Split MNIST | Split CIFAR10 | Split CIFAR100
---|---|---|---
finetune* | 19.75 $\pm$ 0.05 | 18.55 $\pm$ 0.34 | 3.53 $\pm$ 0.04
GEM* | 93.25 $\pm$ 0.36 | 24.13 $\pm$ 2.46 | 11.12 $\pm$ 2.48
iCARL* | 83.95 $\pm$ 0.21 | 37.32 $\pm$ 2.66 | 10.80 $\pm$ 0.37
reservoir* | 92.16 $\pm$ 0.75 | 42.48 $\pm$ 3.04 | 19.57 $\pm$ 1.79
MIR* | 93.20 $\pm$ 0.36 | 42.80 $\pm$ 2.22 | 20.00 $\pm$ 0.57
GSS* | 92.47 $\pm$ 0.92 | 38.45 $\pm$ 1.41 | 13.10 $\pm$ 0.94
CoPE-CE* | 91.77 $\pm$ 0.87 | 39.73 $\pm$ 2.26 | 18.33 $\pm$ 1.52
CoPE* | 93.94 $\pm$ 0.20 | 48.92 $\pm$ 1.32 | 21.62 $\pm$ 0.69
ER + GMED${\dagger}$ | 82.67 $\pm$ 1.90 | 34.84 $\pm$ 2.20 | 20.93 $\pm$ 1.60
ERa \+ GMED${\dagger}$ | 82.21 $\pm$ 2.90 | 47.47 $\pm$ 3.20 | 19.60 $\pm$ 1.50
CURL* | 92.59 $\pm$ 0.66 | - | -
CNDPM* | 93.23 $\pm$ 0.09 | 45.21 $\pm$ 0.18 | 20.10 $\pm$ 0.12
Dynamic-OCM | 94.02 $\pm$ 0.23 | 49.16 $\pm$ 1.52 | 21.79 $\pm$ 0.68
ODDL | 94.85 $\pm$ 0.02 | 51.48 $\pm$ 0.12 | 26.20 $\pm$ 0.72
ODDL-S | 95.75 $\pm$ 0.05 | 52.69 $\pm$ 0.11 | 27.21 $\pm$ 0.87
### 5.1 Empirical results for the forgetting analysis
In this section, we investigate the forgetting behaviour of the model
according to the proposed theoretical framework. Firstly, we train a single
classifier $h$ under Split MNIST database, as a baseline, with a memory buffer
of the maximum size of 2000, and we randomly remove a batch of stored samples
(batch size is 10) when the memory is full. Then we estimate
${\mathcal{R}}(h,{\mathcal{P}}_{i})$ (target risk on all visited training
samples),
${\mathcal{R}}(h,h_{{\mathcal{M}}_{i}},{\mathbb{P}}_{{\mathcal{M}}_{i}})$
(source risk on the memory). We plot the results in Fig. 2-a, where “Random
(Source risk)” represents the source risk of a single classifier. The results
show that the source risk always keeps stable, and the target risk is small
for a few initial training steps since the memory can capture all information
of visited samples. However, as the number of training steps grows, the target
risk is increased, which is caused by the memory that loses previous samples,
theoretically explained in Theorem 1. We also evaluate the risk of the
baseline on all testing sets (target risk) and plot the results in Fig. 2-b
where it is observed that the single model always leads to a large target risk
during the training.
We evaluate the source and target risks for the proposed ODDL under Split
MNIST and plot the results in Fig. 2. The performance of ODDL on the
distribution ${\mathcal{P}}_{i}$ (Target risk) does not degenerate during the
whole training phase. At the same time, the baseline tends to increase the
target risk on ${\mathcal{P}}_{i}$ as the training steps increase, as shown in
Fig. 2-a. Finally, the risk on all target distributions (all categories in the
testing set) from each training step is shown in Fig. 2-b, where the proposed
ODDL minimizes the target risk as gaining more knowledge during the training.
In contrast, the baseline invariably leads to a large target risk even when
the number of training steps increases. These results show that ODDL can
relieve forgetting and achieve better generalization than the random approach
on all target sets.
### 5.2 Results on TFCL benchmark
We provide the results in Tab. 1 where * and ${\dagger}$ denote the results
cited from [8] and [12], respectively. We compare with several baselines
including: finetune that directly trains a classifier on the data stream, GSS
[1], Dynamic-OCM [54], MIR [2], Gradient Episodic Memory (GEM) [27],
Incremental Classifier and Representation Learning (iCARL) [39], Reservoir
[45], CURL, CNDPM, CoPE, ER + GMED and ERa \+ GMED [12] where ER is the
Experience Replay [42] and ERa is ER with data augmentation. The number of
components in ODDL-S and ODDL for Split MNIST, Split CIFAR10 and Split
CIFAR100 is 7, 9, 7, respectively. The proposed approach outperforms CNDPM,
which uses more parameters, on all datasets and achieves state-of-the-art
performance.
Table 2: Classification accuracy for 20 runs when testing various models on Split MImageNet and Permuted MNIST. Methods | Split MImageNet | Permuted MNIST
---|---|---
ERa | 25.92 $\pm$ 1.2 | 78.11 $\pm$ 0.7
ER + GMED | 27.27 $\pm$ 1.8 | 78.86 $\pm$ 0.7
MIR+GMED | 26.50 $\pm$ 1.3 | 79.25 $\pm$ 0.8
MIR | 25.21 $\pm$ 2.2 | 79.13 $\pm$ 0.7
CNDPM | 27.12 $\pm$1.5 | 80.68 $\pm$ 0.7
ODDL | 27.45 $\pm$ 0.9 | 82.33 $\pm$ 0.6
ODDL-S | 28.68 $\pm$ 1.5 | 83.56 $\pm$ 0.5
In the following, we also evaluate the performance of the proposed approach on
the large-scale dataset (MINI-ImageNet [21]), and Permuted MNIST [9]. We split
MINI-ImageNet into 20 tasks (See details in Appendix-F1 from SM), namely Split
MImageNet. We follow the setting from [2] where the maximum memory size is
$10K$, and a smaller version of ResNet-18 [10] is used as the classifier. The
hyperparameter $\lambda$ used for learning Split MINI-ImageNet and Permuted
MNIST is equal to 1.2 and 1.5, respectively. We compare with several state-of-
the-art methods under Split MImageNet, reported in Tab. 2, where the results
of other baselines are cited from [12]. These results show that the proposed
ODDL-S outperforms different baselines under the challenging dataset.
### 5.3 Ablation study
We investigate whether the proposed discrepancy-based criterion can provide
better signals for the expansion of ODDL-S. We train ODDL-S on Split MNIST,
where we record the variance of tasks and the number of components in each
training step. We plot the results in Fig. 2-c where “task” represents the
number of tasks in each training step. We observe that the proposed
discrepancy-based criterion can detect the data distribution shift accurately,
allowing ODDL-S to expand the network architecture each time when detecting
the data distribution shift. This also encourages the proposed ODDL-S to use a
minimal number of components while achieving optimal performance, as discussed
in Lemma 2. We also provide the analysis of how to maximize the memory bound
in Appendix-F2.2, while more ablation study results are provided in
Appendix-F.2 from the supplementary material.
## 6 Conclusion
In this paper, we develop a novel theoretical framework for Task-Free
Continual Learning (TFCL), by defining a statistical discrepancy distance.
Inspired by the theoretical analysis, we propose the Online Discrepancy
Distance Learning enabled by a memory buffer sampling (ODDL-S) model, which
trades off between the model’s complexity and performance. The memory buffer
sampling mechanism ensures the information diversity learning. The proposed
theoretical analysis provides new insights into the model’s forgetting
behaviour during each training step of TFCL. Experimental results on several
TFCL benchmarks show that the proposed ODDL-S achieves state-of-the-art
performance.
## References
* [1] R. Aljundi, M. Lin, B. Goujaud, and Y. Bengio. Gradient based sample selection for online continual learning. In Advances in Neural Information Processing Systems (NeurIPS), arXiv preprint arXiv:1903.08671, 2019.
* [2] Rahaf Aljundi, Eugene Belilovsky, Tinne Tuytelaars, Laurent Charlin, Massimo Caccia, Min Lin, and Lucas Page-Caccia. Online continual learning with maximal interfered retrieval. In Advances in Neural Information Processing Systems (NeurIPS), arXiv preprint arXiv:1908.04742, 2019.
* [3] Rahaf Aljundi, Klaas Kelchtermans, and Tinne Tuytelaars. Task-free continual learning. In Proc. of IEEE/CVF Conf. on Computer Vision and Pattern Recognition, pages 11254–11263, 2019.
* [4] Verónica Álvarez, Santiago Mazuelas, and José Antonio Lozano. Minimax classification under concept drift with multidimensional adaptation and performance guarantees. In Proc. International Conference on Machine Learning (ICML), vol. PMLR 162, pages 486–499, 2022.
* [5] Jihwan Bang, Heesu Kim, YoungJoon Yoo, Jung-Woo Ha, and Jonghyun Choi. Rainbow memory: Continual learning with a memory of diverse samples. In Proc. of IEEE/CVF Conf. on Computer Vision and Pattern Recognition (CVPR), pages 8218–8227, 2021.
* [6] A. Chaudhry, M. Rohrbach, M. Elhoseiny, T. Ajanthan, P. Dokania, P. H. S. Torr, and M.’A. Ranzato. On tiny episodic memories in continual learning. arXiv preprint arXiv:1902.10486, 2019.
* [7] W. Dai, Q. Yang, G. R. Xue, and Y. Yu. Boosting for transfer learning. In Proc. Int Conf. on Machine Learning (ICML), pages 193–200, 2007\.
* [8] Matthias De Lange and Tinne Tuytelaars. Continual prototype evolution: Learning online from non-stationary data streams. In Proc. of the IEEE/CVF Int. Conf. on Computer Vision (ICCV), pages 8250–8259, 2021.
* [9] Ian J Goodfellow, Mehdi Mirza, Da Xiao, Aaron Courville, and Yoshua Bengio. An empirical investigation of catastrophic forgetting in gradient-based neural networks. In arXiv preprint arXiv:1312.6211, 2014.
* [10] K. He, X. Zhang, S. Ren, and J. Sun. Deep residual learning for image recognition. In Proc. of IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), pages 770–778, 2016.
* [11] G. Hinton, O. Vinyals, and J. Dean. Distilling the knowledge in a neural network. In Proc. NIPS Deep Learning Workshop, arXiv preprint arXiv:1503.02531, 2014.
* [12] Xisen Jin, Arka Sadhu, Junyi Du, and Xiang Ren. Gradient-based editing of memory examples for online task-free continual learning. In Advances in Neural Information Processing Systems (NeurIPS), arXiv preprint arXiv:2006.15294, 2021.
* [13] H. Jung, J. Ju, M. Jung, and J. Kim. Less-forgetting learning in deep neural networks. In Proc. AAAI Conf. on Artificial Intelligence, volume 32, pages 3358–3365, 2018.
* [14] Zixuan Ke, Bing Liu, Nianzu Ma, Hu Xu, and Lei Shu. Achieving forgetting prevention and knowledge transfer in continual learning. Advances in Neural Information Processing Systems, 34, 2021.
* [15] D. P. Kingma and M. Welling. Auto-encoding variational Bayes. arXiv preprint arXiv:1312.6114, 2013.
* [16] J. Kirkpatrick, R. Pascanu, N. Rabinowitz, J. Veness, G. Desjardins, A. A. Rusu, K. Milan, J. Quan, T. Ramalho, A. Grabska-Barwinska, D. Hassabis, C. Clopath, D. Kumaran, and R. Hadsell. Overcoming catastrophic forgetting in neural networks. Proc. of the National Academy of Sciences (PNAS), 114(13):3521–3526, 2017.
* [17] Jeremias Knoblauch, Hisham Husain, and Tom Diethe. Optimal continual learning has perfect memory and is NP-hard. In Proc. Int. Conf. on Machine Learning (ICML), vol PMLR 119, pages 5327–5337, 2020.
* [18] Alex Krizhevsky and Geoffrey Hinton. Learning multiple layers of features from tiny images. Technical report, Univ. of Toronto, 2009.
* [19] Richard Kurle, Botond Cseke, Alexej Klushyn, Patrick van der Smagt, and Stephan Günnemann. Continual learning with Bayesian neural networks for non-stationary data. In Int. Conf. on Learning Representations (ICLR), 2020.
* [20] Sampo Kuutti, Richard Bowden, Yaochu Jin, Phil Barber, and Saber Fallah. A survey of deep learning applications to autonomous vehicle control. IEEE Transactions on Intelligent Transportation Systems, 22(2):712–733, 2020.
* [21] Ya Le and Xuan Yang. Tiny imageNet visual recognition challenge. Technical report, Univ. of Stanford, 2015.
* [22] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document recognition. Proc. of the IEEE, 86(11):2278–2324, 1998.
* [23] Sebastian Lee, Sebastian Goldt, and Andrew Saxe. Continual learning in the teacher-student setup: Impact of task similarity. In Proc. Int. Conf. on Machine Learning (ICML), vol. PMLR 139, pages 6109–6119, 2021.
* [24] Soochan Lee, Junsoo Ha, Dongsu Zhang, and Gunhee Kim. A neural Dirichlet process mixture model for task-free continual learning. In Int. Conf. on Learning Representations (ICLR), arXiv preprint arXiv:2001.00689, 2020.
* [25] Z. Li and D. Hoiem. Learning without forgetting. IEEE Trans. on Pattern Analysis and Machine Intelligence, 40(12):2935–2947, 2017.
* [26] Sen Lin, Li Yang, Deliang Fan, and Junshan Zhang. TRGP: Trust region gradient projection for continual learning. In Int. Conf. on Learning Representations (ICLR), arXiv preprint arXiv:2202.02931, 2022.
* [27] David Lopez-Paz and Marc’Aurelio Ranzato. Gradient episodic memory for continual learning. In Advances in Neural Information Processing Systems, pages 6467–6476, 2017.
* [28] Divyam Madaan, Jaehong Yoon, Yuanchun Li, Yunxin Liu, and Sung Ju Hwang. Representational continuity for unsupervised continual learning. In Int. Conf. on Learning Representations (ICLR), arXiv preprint arXiv:2110.06976, 2022.
* [29] Yishay Mansour, Mehryar Mohri, and Afshin Rostamizadeh. Domain adaptation: Learning bounds and algorithms. In Proc. Conf. on Learning Theory (COLT), arXiv preprint arXiv:2002.06715, 2009.
* [30] Mehryar Mohri and Andres Munoz Medina. New analysis and algorithm for learning with drifting distributions. In International Conference on Algorithmic Learning Theory, pages 124–138. Springer, 2012.
* [31] Cuong V Nguyen, Yingzhen Li, Thang D Bui, and Richard E Turner. Variational continual learning. In Int. Conf. on Learning Representations (ICLR), arXiv preprint arXiv:1710.10628, 2018.
* [32] G. I. Parisi, R. Kemker, J. L. Part, C. Kanan, and S. Wermter. Continual lifelong learning with neural networks: A review. Neural Networks, 113:54–71, 2019.
* [33] Anastasia Pentina and Christoph H Lampert. Lifelong learning with non-iid tasks. In Proc. Advances in Neural Information Processing Systems (NIPS), pages 1540–1548, 2015.
* [34] R. Polikar, L. Upda, S. S. Upda, and Vasant Honavar. Learn++: An incremental learning algorithm for supervised neural networks. IEEE Trans. on Systems Man and Cybernetics, Part C, 31(4):497–508, 2001.
* [35] Mozhgan PourKeshavarzi, Guoying Zhao, and Mohammad Sabokrou. Looking back on learned experiences for class/task incremental learning. In Int. Conf. on Learning Representations (ICLR), 2022.
* [36] Ameya Prabhu, Philip HS Torr, and Puneet K Dokania. GDumb: A simple approach that questions our progress in continual learning. In Proc. European Conference on Computer Vision (ECCV), vol. LNCS 12347, pages 524–540, 2020.
* [37] Krishnan Raghavan and Prasanna Balaprakash. Formalizing the generalization-forgetting trade-off in continual learning. Advances in Neural Information Processing Systems, 34, 2021.
* [38] Dushyant Rao, Francesco Visin, Andrei A. Rusu, Yee Whye Teh, Razvan Pascanu, and Raia Hadsell. Continual unsupervised representation learning. In Proc. Neural Inf. Proc. Systems (NIPS), pages 7645–7655, 2019\.
* [39] Sylvestre-Alvise Rebuffi, Alexander Kolesnikov, Georg Sperl, and Christoph H Lampert. iCaRL: Incremental classifier and representation learning. In Proc. of the IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), pages 2001–2010, 2017.
* [40] B. Ren, H. Wang, J. Li, and H. Gao. Life-long learning based on dynamic combination model. Applied Soft Computing, 56:398–404, 2017.
* [41] Hippolyt Ritter, Aleksandar Botev, and David Barber. Online structured Laplace approximations for overcoming catastrophic forgetting. In Advances in Neural Information Processing Systems (NeurIPS), volume 31, pages 3742–3752, 2018.
* [42] David Rolnick, Arun Ahuja, Jonathan Schwarz, Timothy P. Lillicrap, and Gregory Wayne. Experience replay for continual learning. In Advances in Neural Information Processing Systems 34 (NeurIPS), pages 348–358, 2019.
* [43] H. Shin, J. K. Lee, J. Kim, and J. Kim. Continual learning with deep generative replay. In Advances in Neural Inf. Proc. Systems (NIPS), pages 2990–2999, 2017.
* [44] Shengyang Sun, Daniele Calandriello, Huiyi Hu, Ang Li, and Michalis Titsias. Information-theoretic online memory selection for continual learning. In Int. Conf. on Learning Representations (ICLR), arXiv preprint arXiv:2204.04763, 2022.
* [45] Jeffrey S Vitter. Random sampling with a reservoir. ACM Transactions on Mathematical Software (TOMS), 11(1):37–57, 1985\.
* [46] Fei Ye and Adrian G. Bors. Learning latent representations across multiple data domains using lifelong VAEGAN. In Proc. European Conf. on Computer Vision (ECCV), vol. LNCS 12365, pages 777–795, 2020.
* [47] Fei Ye and Adrian G. Bors. Lifelong learning of interpretable image representations. In Proc. Int. Conf. on Image Processing Theory, Tools and Applications (IPTA), pages 1–6, 2020.
* [48] Fei Ye and Adrian G Bors. Mixtures of variational autoencoders. In Proc. Int. Conf. on Image Processing Theory, Tools and Applications (IPTA), pages 1–6, 2020.
* [49] Fei Ye and Adrian G. Bors. InfoVAEGAN: Learning joint interpretable representations by information maximization and maximum likelihood. In Proc. IEEE Int. Conf. on Image Processing (ICIP), pages 749–753, 2021.
* [50] Fei Ye and Adrian G Bors. Learning joint latent representations based on information maximization. Information Sciences, 567:216–236, 2021.
* [51] Fei Ye and Adrian G. Bors. Lifelong infinite mixture model based on knowledge-driven Dirichlet process. In Proc. of the IEEE Int. Conf. on Computer Vision (ICCV), pages 10695–10704, 2021.
* [52] Fei Ye and Adrian G. Bors. Lifelong mixture of variational autoencoders. IEEE Transactions on Neural Networks and Learning Systems, pages 1–14, 2021.
* [53] Fei Ye and Adrian G. Bors. Lifelong twin generative adversarial networks. In Proc. IEEE Int. Conf. on Image Processing (ICIP), pages 1289–1293, 2021.
* [54] Fei Ye and Adrian G Bors. Continual variational autoencoder learning via online cooperative memorization. arXiv preprint arXiv:2207.10131, 2022.
* [55] Fei Ye and Adrian G. Bors. Deep mixture generative autoencoders. IEEE Transactions on Neural Networks and Learning Systems, 33(10):5789–5803, 2022.
* [56] Fei Ye and Adrian G Bors. Learning an evolved mixture model for task-free continual learning. arXiv preprint arXiv:2207.05080, 2022.
* [57] Fei Ye and Adrian G. Bors. Lifelong generative modelling using dynamic expansion graph model. Proceedings of the AAAI Conference on Artificial Intelligence, 36(8):8857–8865, Jun. 2022.
* [58] Fei Ye and Adrian G. Bors. Lifelong teacher-student network learning. IEEE Transactions on Pattern Analysis and Machine Intelligence, 44(10):6280–6296, 2022.
* [59] Jaehong Yoon, Divyam Madaan, Eunho Yang, and Sung Ju Hwang. Online coreset selection for rehearsal-based continual learning. In Int. Conf. on Learning Representations (ICLR), arXiv preprint arXiv:2106.01085, 2022.
* [60] Yuchen Zhang, Tianle Liu, Mingsheng Long, and Michael Jordan. Bridging theory and algorithm for domain adaptation. In Proc. International Conference on Machine Learning (ICML), vol. PMLR 97, pages 7404–7413, 2019.
* [61] Shengjia Zhao, Jiaming Song, and Stefano Ermon. InfoVAE: Balancing learning and inference in variational autoencoders. In Proc. AAAI Conf. on Artif. Intel., volume 33, pages 5885–5892, 2019.
|
111 Mathematics Subject Classification (2020). Primary: 34L40; Secondary:
34L20. Keywords: Dirac operator, degenerate boundary conditions, spectrum
On the spectrum of $2\times 2$ Dirac operator
with degenerate boundary conditions
Alexander Makin
Peoples Friendship University of Russia
117198, Miklukho-Maklaya str. 6, Moscow, Russia
> We study the spectral problem for the Dirac operator with degenerate
> boundary conditions and a complex-valued summable potential. Sufficient
> conditions are found under which the spectrum of the problem under
> consideration coincides with the spectrum of the corresponding unperturbed
> operator.
1\. Introduction
In the present paper, we study the Dirac system
$B\mathbf{y}^{\prime}+V\mathbf{y}=\lambda\mathbf{y},$ $None$
where
$B=\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\quad V=\begin{pmatrix}p(x)&q(x)\\\
q(x)&-p(x)\end{pmatrix},$
the functions $p,q\in L_{1}(0,\pi)$, with two-point boundary conditions
$\begin{array}[]{c}U_{1}(\mathbf{y})=a_{11}y_{1}(0)+a_{12}y_{2}(0)+a_{13}y_{1}(\pi)+a_{14}y_{2}(\pi)=0,\\\
U_{2}(\mathbf{y})=a_{21}y_{1}(0)+a_{22}y_{2}(0)+a_{23}y_{1}(\pi)+a_{24}y_{2}(\pi)=0,\end{array}$
$None$
where the coefficients $a_{jk}$ are arbitrary complex numbers, and rows of the
matrix
$A=\begin{pmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\\
a_{21}&a_{22}&a_{23}&a_{24}\end{pmatrix}$
are linearly independent.
We consider the operator
$\mathbb{L}\mathbf{y}=B\mathbf{y}^{\prime}+V\mathbf{y}$ as a linear operator
on the space $\mathbb{H}=L_{2}(0,\pi)\oplus L_{2}(0,\pi)$, with the domain
$D(\mathbb{L})=\\{\mathbf{y}\in
W_{1}^{1}[0,\pi]:\,\mathbb{L}\mathbf{y}\in\mathbb{H}$, $U_{j}(\mathbf{y})=0$
$(j=1,2)\\}$.
Denote by $J_{jk}$ the determinant composed of the $j$th and $k$th columns of
the matrix $A$. Denote $J_{0}=J_{12}+J_{34}$, $J_{1}=J_{14}-J_{23}$,
$J_{2}=J_{13}+J_{24}$.
Boundary conditions (2) are called degenerate if
$J_{1}=J_{2}=0;\quad J_{0}=0,\quad J_{1}+iJ_{2}\neq 0,\quad
J_{1}-iJ_{2}=0;\quad J_{0}=0,\quad J_{1}+iJ_{2}=0,\quad J_{1}-iJ_{2}\neq 0,$
otherwise they are nondegenerate.
There is an enormous literature related to the spectral theory for Dirac
operators with nondegenerate boundary conditions. The case of degenerate
conditions has been investigated much less although in the last decade
interest in the study of these spectral problems has increased significantly
[see 1-9 and the references therein]. The main goal of present paper is to
establish conditions on the potential $V$ under which the spectrum of the
problem under consideration with degenerate boundary conditions coincides with
the spectrum of the corresponding unperturbed operator (3)
$B\mathbf{y}^{\prime}=\lambda\mathbf{y},\quad U(\mathbf{y})=0.$ $None$
2\. Main results
Theorem 1. Suppose the following conditions are valid
$J_{14}=J_{23}=J_{13}+J_{24}=0,$ $None$
$p(\pi-x)=-p(x),\quad q(\pi-x)=q(x),$ $None$
where $0\leq x\leq\pi$. Then the spectrum of problem (1) (2) coincides with
the spectrum of the corresponding unperturbed operator (3).
Proof. First of all, we rewrite system (1) in scalar form
$\left\\{\begin{array}[]{rcl}y_{2}^{\prime}+p(x)y_{1}+q(x)y_{2}=\lambda
y_{1}\\\ -y_{1}^{\prime}+q(x)y_{1}-p(x)y_{2}=\lambda y_{2}.\\\
\end{array}\right.$ $None$
Denote by
$E(x,\lambda)=\begin{pmatrix}e_{11}(x,\lambda)&e_{12}(x,\lambda)\\\
e_{21}(x,\lambda)&e_{22}(x,\lambda)\end{pmatrix}$ $None$
the matrix of the fundamental solution system to system (1) with boundary
condition $E(\frac{\pi}{2},\lambda)=I$, where $I$ is the unit matrix. It is
well known that
$e_{11}(x,\lambda)e_{22}(x,\lambda)-e_{12}(x,\lambda)e_{21}(x,\lambda)=1$
$None$
for any $x,\lambda$.
Substituting the first column of matrix (7) in system (6), we obtain
$\left\\{\begin{array}[]{rcl}e_{21}^{\prime}(x,\lambda)+p(x)e_{11}(x,\lambda)+q(x)e_{21}(x,\lambda)=\lambda
e_{11}(x,\lambda)\\\
-e_{11}^{\prime}(x,\lambda)+q(x)e_{11}(x,\lambda)-p(x)e_{21}(x,\lambda)=\lambda
e_{21}(x,\lambda).\\\ \end{array}\right.$ $None$
Replacing $x=\pi-t$ in relations (9), we find
$\left\\{\begin{array}[]{rcl}-e_{21}^{\prime}(\pi-t,\lambda)+p(\pi-t)e_{11}(\pi-t,\lambda)+q(\pi-t)e_{21}(\pi-t,\lambda)=\lambda
e_{11}(\pi-t,\lambda)\\\
e_{11}^{\prime}(\pi-t,\lambda)+q(\pi-t)e_{11}(\pi-t,\lambda)-p(\pi-t)e_{21}(\pi-t,\lambda)=\lambda
e_{21}(\pi-t,\lambda).\\\ \end{array}\right.$ $None$
It follows from (5) and (10) that
$\left\\{\begin{array}[]{rcl}-e_{21}^{\prime}(\pi-t,\lambda)-p(t)e_{11}(\pi-t,\lambda)+q(t)e_{21}(\pi-t,\lambda)=\lambda
e_{11}(\pi-t,\lambda)\\\
e_{11}^{\prime}(\pi-t,\lambda)+q(t)e_{11}(\pi-t,\lambda)+p(t)e_{21}(\pi-t,\lambda)=\lambda
e_{21}(\pi-t,\lambda).\\\ \end{array}\right.$ $None$
Denote $z_{2}(t,\lambda)=e_{11}(\pi-t,\lambda)$,
$z_{1}(t,\lambda)=e_{21}(\pi-t,\lambda)$. It follows from (11) that
$\left\\{\begin{array}[]{rcl}z_{2}^{\prime}+p(t)z_{1}+q(t)z_{2}=\lambda
z_{1}\\\ -z_{1}^{\prime}+q(t)z_{1}-p(t)z_{2}=\lambda z_{2}.\\\
\end{array}\right.$ $None$
Obviously, $z_{2}(\frac{\pi}{2},\lambda)=1$, $z_{1}(\frac{\pi}{2},\lambda)=0$,
therefore, by virtue of the uniqueness of the solution to the Cauchy problems
for systems (9) and (12) $z_{2}(t,\lambda)=e_{22}(t,\lambda)$,
$z_{1}(t,\lambda)=e_{21}(t,\lambda)$, hence, we obtain
$e_{22}(t,\lambda)=e_{11}(\pi-t,\lambda),\quad
e_{21}(t,\lambda)=e_{21}(\pi-t,\lambda).$
The last relations imply
$e_{21}(\pi)=e_{12}(0),\quad e_{12}(\pi)=e_{21}(0),\quad
e_{11}(\pi)=e_{22}(0),\quad e_{22}(\pi)=e_{11}(0).$ $None$
The eigenvalues of problem (1), (2) are the roots of the characteristic
equation
$\Delta(\lambda)=0,$
where
$\Delta(\lambda)=\left|\begin{array}[]{cccc}U_{1}(E^{[1]}(\cdot,\lambda))&U_{1}(E^{[2]}(\cdot,\lambda))\\\
U_{2}(E^{[1]}(\cdot,\lambda))&U_{2}(E^{[2]}(\cdot,\lambda))\\\
\end{array}\right|,$
$E^{[k]}(x,\lambda)$ is the $k$th column of matrix (7). Simple computations
together with relations (4), (8), and (13) show that
$\Delta(\lambda)=\left|\begin{array}[]{cccc}a_{11}e_{11}(0)+a_{12}e_{21}(0)+a_{13}e_{11}(\pi)+a_{14}e_{21}(\pi)&a_{11}e_{12}(0)+a_{12}e_{22}(0)+a_{13}e_{12}(\pi)+a_{14}e_{22}(\pi)\\\
a_{21}e_{11}(0)+a_{22}e_{21}(0)+a_{23}e_{11}(\pi)+a_{24}e_{21}(\pi)&a_{21}e_{12}(0)+a_{22}e_{22}(0)+a_{23}e_{12}(\pi)+a_{24}e_{22}(\pi)\\\
\end{array}\right|$
$\begin{array}[]{c}=[a_{11}e_{11}(0)+a_{12}e_{21}(0)+a_{13}e_{11}(\pi)+a_{14}e_{21}(\pi)][a_{21}e_{12}(0)+a_{22}e_{22}(0)+a_{23}e_{12}(\pi)+a_{24}e_{22}(\pi)]\\\
-[a_{21}e_{11}(0)+a_{22}e_{21}(0)+a_{23}e_{11}(\pi)+a_{24}e_{21}(\pi)][a_{11}e_{12}(0)+a_{12}e_{22}(0)+a_{13}e_{12}(\pi)+a_{14}e_{22}(\pi)]\\\
\\\
=[a_{11}e_{11}(0)+a_{12}e_{21}(0)+a_{13}e_{22}(0)+a_{14}e_{12}(0)][a_{21}e_{12}(0)+a_{22}e_{22}(0)+a_{23}e_{21}(0)+a_{24}e_{11}(0)]\\\
-[a_{21}e_{11}(0)+a_{22}e_{21}(0)+a_{23}e_{22}(0)+a_{24}e_{12}(0)][a_{11}e_{12}(0)+a_{12}e_{22}(0)+a_{13}e_{21}(0)+a_{14}e_{11}(0)]\\\
\\\
=\par[a_{11}e_{11}(0)a_{21}e_{12}(0)+a_{11}e_{11}(0)a_{22}e_{22}(0)+a_{11}e_{11}(0)a_{23}e_{21}(0)+a_{11}e_{11}(0)a_{24}e_{11}(0)\\\
+a_{12}e_{21}(0)a_{21}e_{12}(0)+a_{12}e_{21}(0)a_{22}e_{22}(0)+a_{12}e_{21}(0)a_{23}e_{21}(0)+a_{12}e_{21}(0)a_{24}e_{11}(0)\\\
+a_{13}e_{22}(0)a_{21}e_{12}(0)+a_{13}e_{22}(0)a_{22}e_{22}(0)+a_{13}e_{22}(0)a_{23}e_{21}(0)+a_{13}e_{22}(0)a_{24}e_{11}(0)\\\
+a_{14}e_{12}(0)a_{21}e_{12}(0)+a_{14}e_{12}(0)a_{22}e_{22}(0)+a_{14}e_{12}(0)a_{23}e_{21}(0)+a_{14}e_{12}(0)a_{24}e_{11}(0)]\\\
-[a_{21}e_{11}(0)a_{11}e_{12}(0)+a_{21}e_{11}(0)a_{12}e_{22}(0)+a_{21}e_{11}(0)a_{13}e_{21}(0)+a_{21}e_{11}(0)a_{14}e_{11}(0)\\\
+a_{22}e_{21}(0)a_{11}e_{12}(0)+a_{22}e_{21}(0)a_{12}e_{22}(0)+a_{22}e_{21}(0)a_{13}e_{21}(0)+a_{22}e_{21}(0)a_{14}e_{11}(0)\\\
+a_{23}e_{22}(0)a_{11}e_{12}(0)+a_{23}e_{22}(0)a_{12}e_{22}(0)+a_{23}e_{22}(0)a_{13}e_{21}(0)+a_{23}e_{22}(0)a_{14}e_{11}(0)\\\
+a_{24}e_{12}(0)a_{11}e_{12}(0)+a_{24}e_{12}(0)a_{12}e_{22}(0)+a_{24}e_{12}(0)a_{13}e_{21}(0)+a_{24}e_{12}(0)a_{14}e_{11}(0)]\\\
\\\ =\par
e_{11}(0)e_{22}(0)(a_{11}a_{22}+a_{13}a_{24}-a_{21}a_{12}-a_{23}a_{14})+e^{2}_{11}(0)(a_{11}a_{24}-a_{21}a_{14})+e^{2}_{22}(0)(a_{13}a_{22}-a_{23}a_{12})\\\
+e_{12}(0)e_{21}(0)(a_{12}a_{21}+a_{14}a_{23}-a_{24}a_{13}-a_{22}a_{11})+e_{11}(0)e_{12}(0)(a_{11}a_{21}+a_{14}a_{24}-a_{21}a_{11}-a_{24}a_{14})\\\
+e_{11}(0)e_{21}(0)(a_{11}a_{23}+a_{12}a_{24}-a_{21}a_{13}-a_{22}a_{14})\\\
+e^{2}_{21}(0)(a_{12}a_{23}-a_{22}a_{13})+e^{2}_{12}(0)(a_{14}a_{21}-a_{24}a_{11})\\\
+e_{21}(0)e_{22}(0)(a_{12}a_{22}+a_{13}a_{23}-a_{22}a_{12}-a_{23}a_{13})\\\
+e_{22}(0)e_{12}(0)(a_{13}a_{21}+a_{14}a_{22}-a_{23}a_{11}-a_{24}a_{12})\\\
\\\ =\par
e_{11}(0)e_{22}(0)(J_{12}+J_{34})+e^{2}_{11}(0)J_{14}+e^{2}_{22}(0)J_{32}\\\
+e_{12}(0)e_{21}(0)(J_{21}+J_{43})+e_{11}(0)e_{21}(0)(J_{13}+J_{24})\\\
+e^{2}_{21}(0)J_{23}+e^{2}_{12}(0)J_{41}+e_{22}(0)e_{12}(0)(J_{31}+J_{42})\\\
\\\
=\par[e_{11}(0)e_{22}(0)-e_{12}(0)e_{21}(0)](J_{12}+J_{34})+e^{2}_{11}(0)J_{14}+e^{2}_{22}(0)J_{32}+e^{2}_{21}(0)J_{23}+e^{2}_{12}(0)J_{41}\\\
+[e_{11}(0)e_{21}(0)-e_{22}(0)e_{12}(0)](J_{13}+J_{24)})\\\ \\\ =\par
J_{12}+J_{34}+[e^{2}_{11}(0)-e^{2}_{12}(0)]J_{14}+[e^{2}_{21}(0)-e^{2}_{22}(0)]J_{23}+[e_{11}(0)e_{21}(0)-e_{22}(0)e_{12}(0)](J_{13}+J_{24})=J_{12}+J_{34}.\par\par\par\par\par\end{array}$
This completes the proof.
Remark 1. Condition (4) holds if, for example,
$A=\begin{pmatrix}1&0&0&b\\\ 0&1&b&0\end{pmatrix}\quad\mbox{or}\quad
A=\begin{pmatrix}0&b&1&0\\\ b&0&0&1\end{pmatrix}.$ $None$
In both cases $J_{12}+J_{34}=1-b^{2}$, hence, if $b^{2}\neq 1$ the spectrum is
empty, and if $b^{2}=1$ the spectrum fills all complex plane. Notice, that if
$b=0$ conditions (14) are the Cauchy boundary conditions. The Cauchy problem
has no spectrum for any potential $V$.
Remark 2. Let us consider system (1) with boundary conditions
$\tilde{A}=\begin{pmatrix}1&0&0&1\\\ 0&1&1&0\end{pmatrix}.$ $None$
Denote by
$\tilde{E}(x,\lambda)=\begin{pmatrix}c_{1}(x,\lambda)&-s_{2}(x,\lambda)\\\
s_{1}(x,\lambda)&c_{2}(x,\lambda)\end{pmatrix}$
the matrix of the fundamental solution system to system (1) with boundary
condition $\tilde{E}(0,\lambda)=I$, where $I$ is the unit matrix. It is easy
to see that $J_{14}=J_{23}=J_{12}+J_{34}=0,J_{13}=1,J_{24}=-1$. Trivial
computation shows that the characteristic determinant of problem (1), (15) can
be reduced to the form
$\tilde{\Delta}(\lambda)=s_{1}(\pi,\lambda)-s_{2}(\pi,\lambda).$
Let condition (5) hold. From Theorem 1, we get that any complex $\lambda$ is
an eigenvalue of problem (1), (15), hence, $\tilde{\Delta}(\lambda)\equiv 0$,
therefore, $s_{1}(\pi,\lambda)\equiv s_{2}(\pi,\lambda)$.
## References
* [1]
* [2] A. P. Kosarev, A. A. Shkalikov, _Spectral asymptotics of solutions of a $2\times 2$ system of first-order ordinary differential equations,_ Math. Notes. 110 (2021) 967-971.
* [3] A. A. Lunyov and M. M. Malamud, _On Spectral Synthesis for Dissipative Dirac Type Operators,_ Integr. Equ. Oper. Theory 90 (2014), 79-106.
* [4] A. A. Lunyov, M. M. Malamud, _On the completeness and Riesz basis property of root subspaces of boundary value problems for first order systems and applications,_ J. Spectr. Theory. 5 (2015), 17-70.
* [5] A. A. Lunyov and M. M. Malamud, _On the Riesz basis property of the root vector system for Dirac-type $2\times 2$ systems,_ Dokl. Math. 90 (2014), 556-561.
* [6] A. A. Lunyov, M. M. Malamud, _On the Riesz basis property of root vectors system for $2\times 2$ Dirac type operators,_ J. Math. Anal. Appl. 441 (2016), 57-103.
* [7] A. A. Lunyov, M. M. Malamud, _On transformation operators and Riesz basis property of root vectors system for $n\times n$ Dirac type operators. Application to the Timoshenko beam model,_ arXiv:2112.07248.
* [8] A. A. Lunyov, M. M. Malamud, _On the trace formulas and completeness property of root vectors systems for $2\times 2$ Dirac type operators,_ arXiv:2312.15933.
* [9] A. S. Makin, _On the completeness of root function system of the Dirac operator with two-point boundary conditions_ , Math. Nachr. (2024), accepted.
* [10] A. S. Makin, _On the completeness of root function system of the $2\times 2$ Dirac operators with non-regular boundary conditions,_ arXiv:2401.02232.
email<EMAIL_ADDRESS>
|
Subsets and Splits
Filtered Text Samples
Retrieves 100 samples of text containing the specific phrase "You are a helpful assistant", providing limited insight into the dataset.
Helpful Assistant Text Samples
Returns a limited set of rows containing the phrase 'helpful assistant' in the text, providing basic filtering of relevant entries.